```python
import matplotlib.pyplot as plt
import numpy as np
import sympy as sym

import models
import payoffs
import plotting
import selection_functions
import symbolics
```

```python
%matplotlib inline
plt.style.use("ggplot")
```

# One-locus model

Start from full model. Set $x1=x_2=0, x_3, x_4=1 - x3$. Monomorphic with $\gamma=g$ only.

```python
x3 = sym.symbols('x3', real=True, positive=True)
T, R, P, S = sym.symbols('T, R, P, S', real=True, positive=True)
M, epsilon = sym.symbols("M, epsilon", real=True, positive=True)

UGA = symbolics.UGA
UgA = symbolics.UgA
```

# Total offspring (fitness)

```python
x = np.array([[0], [0], [x3], [1 - x3]])
payoff_kernel = np.array([[R, S], [T, P]])
W = models.generalized_sexual_selection(x, UGA, UgA, payoff_kernel, M=0, m=0, mutation_rate=0)
N, = models.total_offspring(W, x)
```

```python
N = sym.simplify(N)
```

```python
sym.factor(N, UgA(x3))
```

$\displaystyle 4.0 \left(0.5 P + \left(- 1.0 P + 0.5 S + 0.5 T\right) \operatorname{UgA}{\left(x_{3} \right)} + \left(0.5 P + 0.5 R - 0.5 S - 0.5 T\right) \operatorname{UgA}^{2}{\left(x_{3} \right)}\right)$

After a bit of algebra we arrive at the following expression for total offspring.

\begin{align}
   % N(x_1) =& 2.0*P*UGA(x1)**2 - 4.0*P*UGA(x1) + 2.0*P + 2.0*R*UGA(x1)**2 - 2.0*S*UGA(x1)**2 + 2.0*S*UGA(x1) - 2.0*T*UGA(x1)**2 + 2.0*T*UGA(x1)\\
   % N(x_1) =& 2\bigg(PU_{GA}(x_1)^2 - 2PU_{GA}(x_1) + P + RU_{GA}(x_1)^2 - SU_{GA}(x_1)^2 + SU_{GA}(x_1) - TU_{GA}(x_1)^2 + TU_{GA}(x_1)\bigg) \\
   % N(x_1) =& 2\bigg(RU_{GA}(x_1)^2 + PU_{GA}(x_1)^2 - TU_{GA}(x_1)^2 - SU_{GA}(x_1)^2 + SU_{GA}(x_1) + TU_{GA}(x_1) - 2PU_{GA}(x_1) + P\bigg) \\
    N(x_1) =& 2\bigg(\big((R + P) - (T + S)\big)U_{GA}(x_1)^2 + \big((T + S) - 2P\big)U_{GA}(x_1) + P\bigg) \\
\end{align}

# The locus of potential equilibria (LPE)

```python
_, _, (f,), _ = models.offspring_genotypes_evolution(W, x)
```

```python
f
```

$\displaystyle - x_{3} + \frac{x_{3} \left(0.5 \left(1 - \operatorname{UgA}{\left(x_{3} \right)}\right) \left(2 P \left(1 - \operatorname{UgA}{\left(x_{3} \right)}\right) + 2 T \operatorname{UgA}{\left(x_{3} \right)}\right) + 1.0 \left(2 R \operatorname{UgA}{\left(x_{3} \right)} + 2 S \left(1 - \operatorname{UgA}{\left(x_{3} \right)}\right)\right) \operatorname{UgA}{\left(x_{3} \right)}\right) + 0.5 \left(1 - x_{3}\right) \left(2 R \operatorname{UgA}{\left(x_{3} \right)} + 2 S \left(1 - \operatorname{UgA}{\left(x_{3} \right)}\right)\right) \operatorname{UgA}{\left(x_{3} \right)}}{0.5 x_{3} \left(1 - \operatorname{UgA}{\left(x_{3} \right)}\right) \left(2 P \left(1 - \operatorname{UgA}{\left(x_{3} \right)}\right) + 2 T \operatorname{UgA}{\left(x_{3} \right)}\right) + x_{3} \left(0.5 \left(1 - \operatorname{UgA}{\left(x_{3} \right)}\right) \left(2 P \left(1 - \operatorname{UgA}{\left(x_{3} \right)}\right) + 2 T \operatorname{UgA}{\left(x_{3} \right)}\right) + 1.0 \left(2 R \operatorname{UgA}{\left(x_{3} \right)} + 2 S \left(1 - \operatorname{UgA}{\left(x_{3} \right)}\right)\right) \operatorname{UgA}{\left(x_{3} \right)}\right) + 0.5 \left(1 - x_{3}\right) \left(2 R \operatorname{UgA}{\left(x_{3} \right)} + 2 S \left(1 - \operatorname{UgA}{\left(x_{3} \right)}\right)\right) \operatorname{UgA}{\left(x_{3} \right)} + \left(1 - x_{3}\right) \left(1.0 \left(1 - \operatorname{UgA}{\left(x_{3} \right)}\right) \left(2 P \left(1 - \operatorname{UgA}{\left(x_{3} \right)}\right) + 2 T \operatorname{UgA}{\left(x_{3} \right)}\right) + 0.5 \left(2 R \operatorname{UgA}{\left(x_{3} \right)} + 2 S \left(1 - \operatorname{UgA}{\left(x_{3} \right)}\right)\right) \operatorname{UgA}{\left(x_{3} \right)}\right)}$

```python
f.subs({x3: 0})
```

$\displaystyle 0$

```python
Ug = sym.symbols("Ug", real=True, nonnegative=True)
```

```python
# solving for the LPE using Python
x3_star, = sym.solve(f, x3, implicit=True)
x3_star =  sym.cancel(x3_star.subs({UgA(x3): Ug}))
```

```python
sym.factor(x3_star, UgA_star)
```

$\displaystyle \frac{2.0 Ug \left(R Ug - S Ug + S\right)}{2.0 P Ug^{2} - 4.0 P Ug + 2.0 P + 2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug - 2.0 T Ug^{2} + 2.0 T Ug}$

```python
sym.solve(f.subs({UgA(x3): x3}), x3)
```

    [1.00000000000000, (P - S)/(P + R - S - T)]

```python
sym.simplify(N.subs({UgA(x3): x3}).subs({x3: (P - S) / (R + P - S - T)}))
```

$\displaystyle \frac{2.0 \left(P R - S T\right)}{P + R - S - T}$

Setting the equation of motion equal to zero, rearranging the resulting equation yields the locus of potential equilibria (LPE). The LPE represents, for each possible $x_1$, the value that $U_{GA}$ would have to take in order for that $x_1$ to be an equilbrium.

\begin{align}
  L(x^*) \equiv x_1^* - 2\left(\frac{\big(R - S\big)U_{GA}^{*2} + \big(S - M\big)U_{GA}^* - 2\epsilon\big((T - P)U_{GA}^{*2} - (T - 2P)U_{GA}^* - P\big)}{N(U_{GA}^*)(1 + 2\epsilon)}\right) = 0\\
\end{align}

We already know that $x_1^*=0, U_{GA}^*=0$ and $x_1^*=1, U_{GA}^*=1$ are equilibria, so it follows that $L(0)=0$ and $L(1)=0$.

Differentiate $x_1^*$ with respect to $U_{GA}^*$ yields an expression that will be useful for assessing stability and invadability of an equilibrium.

```python
f
```

    -x1 + (x1*((-epsilon + 1)*(2*R*UGA(x1) + 2*S*(-UGA(x1) + 1))*UGA(x1) + 0.5*(2*P*(-UGA(x1) + 1) + 2*T*UGA(x1))*(-UGA(x1) + 1)) + (-x1 + 1)*(epsilon*(2*P*(-UGA(x1) + 1) + 2*T*UGA(x1))*(-UGA(x1) + 1) + 0.5*(2*R*UGA(x1) + 2*S*(-UGA(x1) + 1))*UGA(x1)))/(x1*(epsilon*(2*R*UGA(x1) + 2*S*(-UGA(x1) + 1))*UGA(x1) + 0.5*(2*P*(-UGA(x1) + 1) + 2*T*UGA(x1))*(-UGA(x1) + 1)) + x1*((-epsilon + 1)*(2*R*UGA(x1) + 2*S*(-UGA(x1) + 1))*UGA(x1) + 0.5*(2*P*(-UGA(x1) + 1) + 2*T*UGA(x1))*(-UGA(x1) + 1)) + (-x1 + 1)*(epsilon*(2*P*(-UGA(x1) + 1) + 2*T*UGA(x1))*(-UGA(x1) + 1) + 0.5*(2*R*UGA(x1) + 2*S*(-UGA(x1) + 1))*UGA(x1)) + (-x1 + 1)*((-epsilon + 1)*(2*P*(-UGA(x1) + 1) + 2*T*UGA(x1))*(-UGA(x1) + 1) + 0.5*(2*R*UGA(x1) + 2*S*(-UGA(x1) + 1))*UGA(x1)))

```python
sym.factor(f.subs({UGA(x1): U}), x1)
```

    -1.5*(-0.333333333333333*P*epsilon + x1**3*(0.333333333333333*P*epsilon + 0.166666666666667*P + 0.333333333333333*R*d**2*epsilon + 0.166666666666667*R*d**2 - 0.333333333333333*S*d*epsilon - 0.166666666666667*S*d - 0.333333333333333*T*d*epsilon - 0.166666666666667*T*d) + x1**2*(-1.0*P*epsilon - 0.333333333333333*P - 0.166666666666667*R*d**2 + 0.333333333333333*S*d*epsilon + 0.333333333333333*S*d + 0.666666666666667*T*d*epsilon + 0.166666666666667*T*d) + x1*(1.0*P*epsilon + 0.166666666666667*P - 0.166666666666667*S*d - 0.333333333333333*T*d*epsilon))/(0.5*P + x1**2*(0.5*P + 0.5*R*d**2 - 0.5*S*d - 0.5*T*d) + x1*(-1.0*P + 0.5*S*d + 0.5*T*d))

```python
n, _ = sym.fraction(sym.factor(f.subs({UGA(x1): U}), x1))
```

```python
sym.factor(N.subs({UGA(x1): U}), x1)
```

    4.0*(0.5*P + x1**2*(0.5*P + 0.5*R*d**2 - 0.5*S*d - 0.5*T*d) + x1*(-1.0*P + 0.5*S*d + 0.5*T*d))/(x1*(d - 1) + 1)**2

```python
p = sym.Poly(n, x1)
```

```python
sym.solve(sym.factor(p.discriminant(), epsilon).subs({epsilon**2: 0}), epsilon)
```

    [(-0.25*P**2*R**2*d**2 + 0.5*P**2*R*T*d - 0.25*P**2*T**2 + 0.5*P*R**2*S*d**3 - P*R*S*T*d**2 + 0.5*P*S*T**2*d - 0.25*R**2*S**2*d**4 + 0.5*R*S**2*T*d**3 - 0.25*S**2*T**2*d**2)/(-2.0*P**3*R + 5.0*P**2*R*S*d + P**2*S*T - 2.0*P*R**3*d**4 + 5.0*P*R**2*T*d**3 - 4.0*P*R*S**2*d**2 - 4.0*P*R*T**2*d**2 - 2.0*P*S**2*T*d + P*T**3*d + R**2*S*T*d**4 + R*S**3*d**3 - 2.0*R*S*T**2*d**3 + S**3*T*d**2 + S*T**3*d**2)]

```python
sym.factor(sym.expand(1.6875 *(-1.0*P**2*R**2*d**2 + 0.666666666666667*P*R*S*T*d**2 + 0.148148148148148*P*S**3*d + 0.148148148148148*R*T**3*d**3 + 0.037037037037037*S**2*T**2*d**2)))
```

    1.6875*d*(-1.0*P**2*R**2*d + 0.666666666666667*P*R*S*T*d + 0.148148148148148*P*S**3 + 0.148148148148148*R*T**3*d**2 + 0.037037037037037*S**2*T**2*d)

```python
roots = sym.solve(n, x1)
```

```python
sym.factor(sym.together(roots[0]), epsilon)
```

```python
roots[0]
```

    -0.333333333333333*(-3.0*(6.0*P*epsilon + P - S*d - 2.0*T*d*epsilon)/(2.0*P*epsilon + P + 2.0*R*d**2*epsilon + R*d**2 - 2.0*S*d*epsilon - S*d - 2.0*T*d*epsilon - T*d) + (-6.0*P*epsilon - 2.0*P - R*d**2 + 2.0*S*d*epsilon + 2.0*S*d + 4.0*T*d*epsilon + T*d)**2/(2.0*P*epsilon + P + 2.0*R*d**2*epsilon + R*d**2 - 2.0*S*d*epsilon - S*d - 2.0*T*d*epsilon - T*d)**2)/(-27.0*P*epsilon/(2.0*P*epsilon + P + 2.0*R*d**2*epsilon + R*d**2 - 2.0*S*d*epsilon - S*d - 2.0*T*d*epsilon - T*d) + 0.5*(-4.0*(-3.0*(6.0*P*epsilon + P - S*d - 2.0*T*d*epsilon)/(2.0*P*epsilon + P + 2.0*R*d**2*epsilon + R*d**2 - 2.0*S*d*epsilon - S*d - 2.0*T*d*epsilon - T*d) + (-6.0*P*epsilon - 2.0*P - R*d**2 + 2.0*S*d*epsilon + 2.0*S*d + 4.0*T*d*epsilon + T*d)**2/(2.0*P*epsilon + P + 2.0*R*d**2*epsilon + R*d**2 - 2.0*S*d*epsilon - S*d - 2.0*T*d*epsilon - T*d)**2)**3 + (-54.0*P*epsilon/(2.0*P*epsilon + P + 2.0*R*d**2*epsilon + R*d**2 - 2.0*S*d*epsilon - S*d - 2.0*T*d*epsilon - T*d) - 9.0*(6.0*P*epsilon + P - S*d - 2.0*T*d*epsilon)*(-6.0*P*epsilon - 2.0*P - R*d**2 + 2.0*S*d*epsilon + 2.0*S*d + 4.0*T*d*epsilon + T*d)/(2.0*P*epsilon + P + 2.0*R*d**2*epsilon + R*d**2 - 2.0*S*d*epsilon - S*d - 2.0*T*d*epsilon - T*d)**2 + 2.0*(-6.0*P*epsilon - 2.0*P - R*d**2 + 2.0*S*d*epsilon + 2.0*S*d + 4.0*T*d*epsilon + T*d)**3/(2.0*P*epsilon + P + 2.0*R*d**2*epsilon + R*d**2 - 2.0*S*d*epsilon - S*d - 2.0*T*d*epsilon - T*d)**3)**2)**0.5 - 4.5*(6.0*P*epsilon + P - S*d - 2.0*T*d*epsilon)*(-6.0*P*epsilon - 2.0*P - R*d**2 + 2.0*S*d*epsilon + 2.0*S*d + 4.0*T*d*epsilon + T*d)/(2.0*P*epsilon + P + 2.0*R*d**2*epsilon + R*d**2 - 2.0*S*d*epsilon - S*d - 2.0*T*d*epsilon - T*d)**2 + (-6.0*P*epsilon - 2.0*P - R*d**2 + 2.0*S*d*epsilon + 2.0*S*d + 4.0*T*d*epsilon + T*d)**3/(2.0*P*epsilon + P + 2.0*R*d**2*epsilon + R*d**2 - 2.0*S*d*epsilon - S*d - 2.0*T*d*epsilon - T*d)**3)**(1/3) - 0.333333333333333*(-27.0*P*epsilon/(2.0*P*epsilon + P + 2.0*R*d**2*epsilon + R*d**2 - 2.0*S*d*epsilon - S*d - 2.0*T*d*epsilon - T*d) + 0.5*(-4.0*(-3.0*(6.0*P*epsilon + P - S*d - 2.0*T*d*epsilon)/(2.0*P*epsilon + P + 2.0*R*d**2*epsilon + R*d**2 - 2.0*S*d*epsilon - S*d - 2.0*T*d*epsilon - T*d) + (-6.0*P*epsilon - 2.0*P - R*d**2 + 2.0*S*d*epsilon + 2.0*S*d + 4.0*T*d*epsilon + T*d)**2/(2.0*P*epsilon + P + 2.0*R*d**2*epsilon + R*d**2 - 2.0*S*d*epsilon - S*d - 2.0*T*d*epsilon - T*d)**2)**3 + (-54.0*P*epsilon/(2.0*P*epsilon + P + 2.0*R*d**2*epsilon + R*d**2 - 2.0*S*d*epsilon - S*d - 2.0*T*d*epsilon - T*d) - 9.0*(6.0*P*epsilon + P - S*d - 2.0*T*d*epsilon)*(-6.0*P*epsilon - 2.0*P - R*d**2 + 2.0*S*d*epsilon + 2.0*S*d + 4.0*T*d*epsilon + T*d)/(2.0*P*epsilon + P + 2.0*R*d**2*epsilon + R*d**2 - 2.0*S*d*epsilon - S*d - 2.0*T*d*epsilon - T*d)**2 + 2.0*(-6.0*P*epsilon - 2.0*P - R*d**2 + 2.0*S*d*epsilon + 2.0*S*d + 4.0*T*d*epsilon + T*d)**3/(2.0*P*epsilon + P + 2.0*R*d**2*epsilon + R*d**2 - 2.0*S*d*epsilon - S*d - 2.0*T*d*epsilon - T*d)**3)**2)**0.5 - 4.5*(6.0*P*epsilon + P - S*d - 2.0*T*d*epsilon)*(-6.0*P*epsilon - 2.0*P - R*d**2 + 2.0*S*d*epsilon + 2.0*S*d + 4.0*T*d*epsilon + T*d)/(2.0*P*epsilon + P + 2.0*R*d**2*epsilon + R*d**2 - 2.0*S*d*epsilon - S*d - 2.0*T*d*epsilon - T*d)**2 + (-6.0*P*epsilon - 2.0*P - R*d**2 + 2.0*S*d*epsilon + 2.0*S*d + 4.0*T*d*epsilon + T*d)**3/(2.0*P*epsilon + P + 2.0*R*d**2*epsilon + R*d**2 - 2.0*S*d*epsilon - S*d - 2.0*T*d*epsilon - T*d)**3)**(1/3) - 0.333333333333333*(-6.0*P*epsilon - 2.0*P - R*d**2 + 2.0*S*d*epsilon + 2.0*S*d + 4.0*T*d*epsilon + T*d)/(2.0*P*epsilon + P + 2.0*R*d**2*epsilon + R*d**2 - 2.0*S*d*epsilon - S*d - 2.0*T*d*epsilon - T*d)

```python
sym.expand(roots[1])
```

    (-0.333333333333333*((-27.0*P*epsilon*(2.0*P*epsilon + P + 2.0*R*d**2*epsilon + R*d**2 - 2.0*S*d*epsilon - S*d - 2.0*T*d*epsilon - T*d)**2 + 0.5*(-4.0*(-3.0*(6.0*P*epsilon + P - S*d - 2.0*T*d*epsilon)*(2.0*P*epsilon + P + 2.0*R*d**2*epsilon + R*d**2 - 2.0*S*d*epsilon - S*d - 2.0*T*d*epsilon - T*d) + (-6.0*P*epsilon - 2.0*P - R*d**2 + 2.0*S*d*epsilon + 2.0*S*d + 4.0*T*d*epsilon + T*d)**2)**3 + (-54.0*P*epsilon*(2.0*P*epsilon + P + 2.0*R*d**2*epsilon + R*d**2 - 2.0*S*d*epsilon - S*d - 2.0*T*d*epsilon - T*d)**2 - 9.0*(6.0*P*epsilon + P - S*d - 2.0*T*d*epsilon)*(-6.0*P*epsilon - 2.0*P - R*d**2 + 2.0*S*d*epsilon + 2.0*S*d + 4.0*T*d*epsilon + T*d)*(2.0*P*epsilon + P + 2.0*R*d**2*epsilon + R*d**2 - 2.0*S*d*epsilon - S*d - 2.0*T*d*epsilon - T*d) + 2.0*(-6.0*P*epsilon - 2.0*P - R*d**2 + 2.0*S*d*epsilon + 2.0*S*d + 4.0*T*d*epsilon + T*d)**3)**2)**0.5*(2.0*P*epsilon + P + 2.0*R*d**2*epsilon + R*d**2 - 2.0*S*d*epsilon - S*d - 2.0*T*d*epsilon - T*d)**3*Abs(2.0*P*epsilon + P + 2.0*R*d**2*epsilon + R*d**2 - 2.0*S*d*epsilon - S*d - 2.0*T*d*epsilon - T*d)**(-3.0) - 4.5*(6.0*P*epsilon + P - S*d - 2.0*T*d*epsilon)*(-6.0*P*epsilon - 2.0*P - R*d**2 + 2.0*S*d*epsilon + 2.0*S*d + 4.0*T*d*epsilon + T*d)*(2.0*P*epsilon + P + 2.0*R*d**2*epsilon + R*d**2 - 2.0*S*d*epsilon - S*d - 2.0*T*d*epsilon - T*d) + (-6.0*P*epsilon - 2.0*P - R*d**2 + 2.0*S*d*epsilon + 2.0*S*d + 4.0*T*d*epsilon + T*d)**3)/(2.0*P*epsilon + P + 2.0*R*d**2*epsilon + R*d**2 - 2.0*S*d*epsilon - S*d - 2.0*T*d*epsilon - T*d)**3)**(2/3)*(-0.5 + 0.866025403784439*I)*(2.0*P*epsilon + P + 2.0*R*d**2*epsilon + R*d**2 - 2.0*S*d*epsilon - S*d - 2.0*T*d*epsilon - T*d)**2 - 0.333333333333333*((-27.0*P*epsilon*(2.0*P*epsilon + P + 2.0*R*d**2*epsilon + R*d**2 - 2.0*S*d*epsilon - S*d - 2.0*T*d*epsilon - T*d)**2 + 0.5*(-4.0*(-3.0*(6.0*P*epsilon + P - S*d - 2.0*T*d*epsilon)*(2.0*P*epsilon + P + 2.0*R*d**2*epsilon + R*d**2 - 2.0*S*d*epsilon - S*d - 2.0*T*d*epsilon - T*d) + (-6.0*P*epsilon - 2.0*P - R*d**2 + 2.0*S*d*epsilon + 2.0*S*d + 4.0*T*d*epsilon + T*d)**2)**3 + (-54.0*P*epsilon*(2.0*P*epsilon + P + 2.0*R*d**2*epsilon + R*d**2 - 2.0*S*d*epsilon - S*d - 2.0*T*d*epsilon - T*d)**2 - 9.0*(6.0*P*epsilon + P - S*d - 2.0*T*d*epsilon)*(-6.0*P*epsilon - 2.0*P - R*d**2 + 2.0*S*d*epsilon + 2.0*S*d + 4.0*T*d*epsilon + T*d)*(2.0*P*epsilon + P + 2.0*R*d**2*epsilon + R*d**2 - 2.0*S*d*epsilon - S*d - 2.0*T*d*epsilon - T*d) + 2.0*(-6.0*P*epsilon - 2.0*P - R*d**2 + 2.0*S*d*epsilon + 2.0*S*d + 4.0*T*d*epsilon + T*d)**3)**2)**0.5*(2.0*P*epsilon + P + 2.0*R*d**2*epsilon + R*d**2 - 2.0*S*d*epsilon - S*d - 2.0*T*d*epsilon - T*d)**3*Abs(2.0*P*epsilon + P + 2.0*R*d**2*epsilon + R*d**2 - 2.0*S*d*epsilon - S*d - 2.0*T*d*epsilon - T*d)**(-3.0) - 4.5*(6.0*P*epsilon + P - S*d - 2.0*T*d*epsilon)*(-6.0*P*epsilon - 2.0*P - R*d**2 + 2.0*S*d*epsilon + 2.0*S*d + 4.0*T*d*epsilon + T*d)*(2.0*P*epsilon + P + 2.0*R*d**2*epsilon + R*d**2 - 2.0*S*d*epsilon - S*d - 2.0*T*d*epsilon - T*d) + (-6.0*P*epsilon - 2.0*P - R*d**2 + 2.0*S*d*epsilon + 2.0*S*d + 4.0*T*d*epsilon + T*d)**3)/(2.0*P*epsilon + P + 2.0*R*d**2*epsilon + R*d**2 - 2.0*S*d*epsilon - S*d - 2.0*T*d*epsilon - T*d)**3)**(1/3)*(-6.0*P*epsilon - 2.0*P - R*d**2 + 2.0*S*d*epsilon + 2.0*S*d + 4.0*T*d*epsilon + T*d)*(2.0*P*epsilon + P + 2.0*R*d**2*epsilon + R*d**2 - 2.0*S*d*epsilon - S*d - 2.0*T*d*epsilon - T*d) - 0.333333333333333*(-0.5 - 0.866025403784439*I)*(-3.0*(6.0*P*epsilon + P - S*d - 2.0*T*d*epsilon)*(2.0*P*epsilon + P + 2.0*R*d**2*epsilon + R*d**2 - 2.0*S*d*epsilon - S*d - 2.0*T*d*epsilon - T*d) + (-6.0*P*epsilon - 2.0*P - R*d**2 + 2.0*S*d*epsilon + 2.0*S*d + 4.0*T*d*epsilon + T*d)**2))/(((-27.0*P*epsilon*(2.0*P*epsilon + P + 2.0*R*d**2*epsilon + R*d**2 - 2.0*S*d*epsilon - S*d - 2.0*T*d*epsilon - T*d)**2 + 0.5*(-4.0*(-3.0*(6.0*P*epsilon + P - S*d - 2.0*T*d*epsilon)*(2.0*P*epsilon + P + 2.0*R*d**2*epsilon + R*d**2 - 2.0*S*d*epsilon - S*d - 2.0*T*d*epsilon - T*d) + (-6.0*P*epsilon - 2.0*P - R*d**2 + 2.0*S*d*epsilon + 2.0*S*d + 4.0*T*d*epsilon + T*d)**2)**3 + (-54.0*P*epsilon*(2.0*P*epsilon + P + 2.0*R*d**2*epsilon + R*d**2 - 2.0*S*d*epsilon - S*d - 2.0*T*d*epsilon - T*d)**2 - 9.0*(6.0*P*epsilon + P - S*d - 2.0*T*d*epsilon)*(-6.0*P*epsilon - 2.0*P - R*d**2 + 2.0*S*d*epsilon + 2.0*S*d + 4.0*T*d*epsilon + T*d)*(2.0*P*epsilon + P + 2.0*R*d**2*epsilon + R*d**2 - 2.0*S*d*epsilon - S*d - 2.0*T*d*epsilon - T*d) + 2.0*(-6.0*P*epsilon - 2.0*P - R*d**2 + 2.0*S*d*epsilon + 2.0*S*d + 4.0*T*d*epsilon + T*d)**3)**2)**0.5*(2.0*P*epsilon + P + 2.0*R*d**2*epsilon + R*d**2 - 2.0*S*d*epsilon - S*d - 2.0*T*d*epsilon - T*d)**3*Abs(2.0*P*epsilon + P + 2.0*R*d**2*epsilon + R*d**2 - 2.0*S*d*epsilon - S*d - 2.0*T*d*epsilon - T*d)**(-3.0) - 4.5*(6.0*P*epsilon + P - S*d - 2.0*T*d*epsilon)*(-6.0*P*epsilon - 2.0*P - R*d**2 + 2.0*S*d*epsilon + 2.0*S*d + 4.0*T*d*epsilon + T*d)*(2.0*P*epsilon + P + 2.0*R*d**2*epsilon + R*d**2 - 2.0*S*d*epsilon - S*d - 2.0*T*d*epsilon - T*d) + (-6.0*P*epsilon - 2.0*P - R*d**2 + 2.0*S*d*epsilon + 2.0*S*d + 4.0*T*d*epsilon + T*d)**3)/(2.0*P*epsilon + P + 2.0*R*d**2*epsilon + R*d**2 - 2.0*S*d*epsilon - S*d - 2.0*T*d*epsilon - T*d)**3)**(1/3)*(2.0*P*epsilon + P + 2.0*R*d**2*epsilon + R*d**2 - 2.0*S*d*epsilon - S*d - 2.0*T*d*epsilon - T*d)**2)

```python
dx1star_dUGA_star = sym.factor(sym.cancel(sym.diff(x1_star, UGA_star, 1)))
```

```python
dx1star_dUGA_star
```

    1.0*(2.0*epsilon - 1.0)*(2.0*P*R*UGA_star**2 - 2.0*P*R*UGA_star - 1.0*P*S*UGA_star**2 + 2.0*P*S*UGA_star - 1.0*P*S - 1.0*R*T*UGA_star**2)/((2.0*epsilon + 1.0)*(1.0*P*UGA_star**2 - 2.0*P*UGA_star + 1.0*P + 1.0*R*UGA_star**2 - 1.0*S*UGA_star**2 + 1.0*S*UGA_star - 1.0*T*UGA_star**2 + 1.0*T*UGA_star)**2)

```python
optimal_x1_star = sym.factor(sym.cancel(x1_star.subs({UGA_star: optimal_UGA_star})))
```

```python
optimal_x1_star
```

    1.0*(2.0*P**2*R + 4.0*P*R**2*epsilon - 4.0*P*R*S*epsilon - 2.0*P*R*T + 1.0*P*S**2*epsilon - 1.0*P*S**2 - 1.0*P*S*T - 1.0*P*T**2*epsilon - 0.5*R*S**2 - 2.0*R*S*T*epsilon - 2.0*R*T**2*epsilon + 0.5*R*T**2 + 0.5*S**3 + 1.0*S**2*T*epsilon + 1.0*S**2*T + 2.0*S*T**2*epsilon + 0.5*S*T**2 + 1.0*T**3*epsilon)/((1.0*epsilon + 0.5)*(0.5*P + 0.5*R - 0.5*S - 0.5*T)*(2.0*P*R - 0.5*S**2 - 1.0*S*T - 0.5*T**2))

## Interactive plot of the locus of potential equilibria

```python
_locus_of_potential_equilibria = sym.lambdify((UGA_star, T, R, P, S, epsilon),
                                              x1_star,
                                              modules="numpy")

def plot_locus_of_potential_equilibria(selection_function, d1, T, R, P, S, epsilon):
    assert T > R > P > S
    us = np.linspace(0, 1, 100)
    xs = _locus_of_potential_equilibria(us, T, R, P, S, epsilon)

    fig, ax = plt.subplots(1, 1, figsize=(10, 8))
    ax.plot(xs, us, label="LPE")
    ax.set_xlabel(r"$x_1^*$", fontsize=25)
    ax.set_ylabel(r"$U_{GA}^*$", rotation="horizontal", fontsize=25)
    ax.plot(us, us, 'k--')

    # create the selection functions
    if selection_function == "kirkpatrick":
        UGA = lambda x_A: selection_functions.kirkpatrick_selection(x_A, d1)
    elif selection_function == "seger":
        UGA = lambda x_A: selection_functions.seger_selection(x_A, d1)
    elif selection_function == "wright":
        UGA = lambda x_A: selection_functions.wright_selection(x_A, d1)
    else:
        valid_funcs = ("kirkpatrick", "seger", "wright")
        msg = "Selection_function must be one of {}, {}, or {}.".format(*valid_funcs)
        raise ValueError(msg)

    # add selection functions for comparison
    ax.plot(xs, UGA(xs), label=selection_function)
    ax.legend()

    plt.show()
```

```python
#mpld3.enable_notebook()

# sliders used to control the Prisoner's Dilemma Payoffs
T_slider = widgets.FloatSlider(value=25, min=0, max=100, step=0.1, description=r"$T$")
R_slider = widgets.FloatSlider(value=3, min=0, max=100, step=0.1, description=r"$R$")
P_slider = widgets.FloatSlider(value=2, min=0, max=100, step=0.1, description=r"$P$")
S_slider = widgets.FloatSlider(value=1, min=0, max=100, step=0.1, description=r"$S$")

# slider used to control metabolic costs
e_slider = widgets.FloatSlider(value=0, min=0, max=1, step=1e-3, description=r"$\epsilon$", readout_format=".3f")

# slider used to control which selection function is being used
UGA_slider = widgets.Dropdown(options=["kirkpatrick", "seger", "wright"], index=0, description=r"$U_{\gamma(j)A}$")

# slider that controls the parameters of the selection function
d1_slider = widgets.FloatSlider(value=1.0, min=0.0, max=10, step=0.05, description=r"$d_1$")

w = widgets.interactive(plot_locus_of_potential_equilibria, selection_function=UGA_slider, d1=d1_slider,
                        T=T_slider, R=R_slider, P=P_slider, S=S_slider, epsilon=e_slider)
display(w)
```

    interactive(children=(Dropdown(description='$U_{\\gamma(j)A}$', options=('kirkpatrick', 'seger', 'wright'), va…

# Stability

Necessary and sufficient conditions for stability require that the derivative of the equation of motion with respect to $x_1$ be strictly negative when evaluated at the potential equilibrium.

```python
f_jac = f.diff(x1, 1)
```

```python
f_jac
```

    (x1*((-epsilon + 1)*(2*R*UGA(x1) + 2*S*(-UGA(x1) + 1))*UGA(x1) + 0.5*(2*P*(-UGA(x1) + 1) + 2*T*UGA(x1))*(-UGA(x1) + 1)) + (-x1 + 1)*(epsilon*(2*P*(-UGA(x1) + 1) + 2*T*UGA(x1))*(-UGA(x1) + 1) + 0.5*(2*R*UGA(x1) + 2*S*(-UGA(x1) + 1))*UGA(x1)))*(epsilon*(2*P*(-UGA(x1) + 1) + 2*T*UGA(x1))*(-UGA(x1) + 1) - epsilon*(2*R*UGA(x1) + 2*S*(-UGA(x1) + 1))*UGA(x1) - x1*(epsilon*(2*R*UGA(x1) + 2*S*(-UGA(x1) + 1))*U_prime(x1) + epsilon*(2*R*U_prime(x1) - 2*S*U_prime(x1))*UGA(x1) - 0.5*(2*P*(-UGA(x1) + 1) + 2*T*UGA(x1))*U_prime(x1) + (-2*P*U_prime(x1) + 2*T*U_prime(x1))*(-0.5*UGA(x1) + 0.5)) - x1*((-epsilon + 1)*(2*R*UGA(x1) + 2*S*(-UGA(x1) + 1))*U_prime(x1) + (-epsilon + 1)*(2*R*U_prime(x1) - 2*S*U_prime(x1))*UGA(x1) - 0.5*(2*P*(-UGA(x1) + 1) + 2*T*UGA(x1))*U_prime(x1) + (-2*P*U_prime(x1) + 2*T*U_prime(x1))*(-0.5*UGA(x1) + 0.5)) + (-epsilon + 1)*(2*P*(-UGA(x1) + 1) + 2*T*UGA(x1))*(-UGA(x1) + 1) - (-epsilon + 1)*(2*R*UGA(x1) + 2*S*(-UGA(x1) + 1))*UGA(x1) - (-x1 + 1)*(-epsilon*(2*P*(-UGA(x1) + 1) + 2*T*UGA(x1))*U_prime(x1) + epsilon*(-2*P*U_prime(x1) + 2*T*U_prime(x1))*(-UGA(x1) + 1) + (1.0*R*UGA(x1) + 1.0*S*(-UGA(x1) + 1))*U_prime(x1) + (1.0*R*U_prime(x1) - 1.0*S*U_prime(x1))*UGA(x1)) - (-x1 + 1)*(-(-epsilon + 1)*(2*P*(-UGA(x1) + 1) + 2*T*UGA(x1))*U_prime(x1) + (-epsilon + 1)*(-2*P*U_prime(x1) + 2*T*U_prime(x1))*(-UGA(x1) + 1) + (1.0*R*UGA(x1) + 1.0*S*(-UGA(x1) + 1))*U_prime(x1) + (1.0*R*U_prime(x1) - 1.0*S*U_prime(x1))*UGA(x1)) - 1.0*(2*P*(-UGA(x1) + 1) + 2*T*UGA(x1))*(-UGA(x1) + 1) + 1.0*(2*R*UGA(x1) + 2*S*(-UGA(x1) + 1))*UGA(x1))/(x1*(epsilon*(2*R*UGA(x1) + 2*S*(-UGA(x1) + 1))*UGA(x1) + 0.5*(2*P*(-UGA(x1) + 1) + 2*T*UGA(x1))*(-UGA(x1) + 1)) + x1*((-epsilon + 1)*(2*R*UGA(x1) + 2*S*(-UGA(x1) + 1))*UGA(x1) + 0.5*(2*P*(-UGA(x1) + 1) + 2*T*UGA(x1))*(-UGA(x1) + 1)) + (-x1 + 1)*(epsilon*(2*P*(-UGA(x1) + 1) + 2*T*UGA(x1))*(-UGA(x1) + 1) + 0.5*(2*R*UGA(x1) + 2*S*(-UGA(x1) + 1))*UGA(x1)) + (-x1 + 1)*((-epsilon + 1)*(2*P*(-UGA(x1) + 1) + 2*T*UGA(x1))*(-UGA(x1) + 1) + 0.5*(2*R*UGA(x1) + 2*S*(-UGA(x1) + 1))*UGA(x1)))**2 - 1 + (-epsilon*(2*P*(-UGA(x1) + 1) + 2*T*UGA(x1))*(-UGA(x1) + 1) + x1*((-epsilon + 1)*(2*R*UGA(x1) + 2*S*(-UGA(x1) + 1))*U_prime(x1) + (-epsilon + 1)*(2*R*U_prime(x1) - 2*S*U_prime(x1))*UGA(x1) - 0.5*(2*P*(-UGA(x1) + 1) + 2*T*UGA(x1))*U_prime(x1) + (-2*P*U_prime(x1) + 2*T*U_prime(x1))*(-0.5*UGA(x1) + 0.5)) + (-epsilon + 1)*(2*R*UGA(x1) + 2*S*(-UGA(x1) + 1))*UGA(x1) + (-x1 + 1)*(-epsilon*(2*P*(-UGA(x1) + 1) + 2*T*UGA(x1))*U_prime(x1) + epsilon*(-2*P*U_prime(x1) + 2*T*U_prime(x1))*(-UGA(x1) + 1) + (1.0*R*UGA(x1) + 1.0*S*(-UGA(x1) + 1))*U_prime(x1) + (1.0*R*U_prime(x1) - 1.0*S*U_prime(x1))*UGA(x1)) + 0.5*(2*P*(-UGA(x1) + 1) + 2*T*UGA(x1))*(-UGA(x1) + 1) - 0.5*(2*R*UGA(x1) + 2*S*(-UGA(x1) + 1))*UGA(x1))/(x1*(epsilon*(2*R*UGA(x1) + 2*S*(-UGA(x1) + 1))*UGA(x1) + 0.5*(2*P*(-UGA(x1) + 1) + 2*T*UGA(x1))*(-UGA(x1) + 1)) + x1*((-epsilon + 1)*(2*R*UGA(x1) + 2*S*(-UGA(x1) + 1))*UGA(x1) + 0.5*(2*P*(-UGA(x1) + 1) + 2*T*UGA(x1))*(-UGA(x1) + 1)) + (-x1 + 1)*(epsilon*(2*P*(-UGA(x1) + 1) + 2*T*UGA(x1))*(-UGA(x1) + 1) + 0.5*(2*R*UGA(x1) + 2*S*(-UGA(x1) + 1))*UGA(x1)) + (-x1 + 1)*((-epsilon + 1)*(2*P*(-UGA(x1) + 1) + 2*T*UGA(x1))*(-UGA(x1) + 1) + 0.5*(2*R*UGA(x1) + 2*S*(-UGA(x1) + 1))*UGA(x1)))

```python
UGA_prime_star = sym.symbols("UGA_prime_star", real=True, nonnegative=True)
evaluated_f_jac = f_jac.subs({sym.Derivative(UGA(x1), x1): UGA_prime_star, UGA(x1): UGA_star, x1: x1_star})
```

```python
sym.expand(evaluated_f_jac)
```

```python
numerator, denominator = sym.fraction(sym.factor(sym.together(evaluated_f_jac)))
```

```python
# denominator is positive
denominator
```

    (0.5*P*UGA_star**2 - 1.0*P*UGA_star + 0.5*P + 0.5*R*UGA_star**2 - 0.5*S*UGA_star**2 + 0.5*S*UGA_star - 0.5*T*UGA_star**2 + 0.5*T*UGA_star)**2

```python
numerator
```

    -0.25*P**2*UGA_star**4*epsilon - 0.125*P**2*UGA_star**4 + 1.0*P**2*UGA_star**3*epsilon + 0.5*P**2*UGA_star**3 - 1.5*P**2*UGA_star**2*epsilon - 0.75*P**2*UGA_star**2 + 1.0*P**2*UGA_star*epsilon + 0.5*P**2*UGA_star - 0.25*P**2*epsilon - 0.125*P**2 - 0.5*P*R*UGA_star**4*epsilon - 0.25*P*R*UGA_star**4 + 1.0*P*R*UGA_star**3*epsilon + 0.5*P*R*UGA_star**3 + 0.5*P*R*UGA_star**2*epsilon*U_prime(4.0*P*UGA_star**2*epsilon/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) - 8.0*P*UGA_star*epsilon/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) + 4.0*P*epsilon/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) + 2.0*R*UGA_star**2/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) - 2.0*S*UGA_star**2/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) + 2.0*S*UGA_star/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) - 4.0*T*UGA_star**2*epsilon/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) + 4.0*T*UGA_star*epsilon/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star)) - 0.5*P*R*UGA_star**2*epsilon - 0.25*P*R*UGA_star**2*U_prime(4.0*P*UGA_star**2*epsilon/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) - 8.0*P*UGA_star*epsilon/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) + 4.0*P*epsilon/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) + 2.0*R*UGA_star**2/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) - 2.0*S*UGA_star**2/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) + 2.0*S*UGA_star/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) - 4.0*T*UGA_star**2*epsilon/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) + 4.0*T*UGA_star*epsilon/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star)) - 0.25*P*R*UGA_star**2 - 0.5*P*R*UGA_star*epsilon*U_prime(4.0*P*UGA_star**2*epsilon/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) - 8.0*P*UGA_star*epsilon/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) + 4.0*P*epsilon/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) + 2.0*R*UGA_star**2/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) - 2.0*S*UGA_star**2/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) + 2.0*S*UGA_star/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) - 4.0*T*UGA_star**2*epsilon/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) + 4.0*T*UGA_star*epsilon/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star)) + 0.25*P*R*UGA_star*U_prime(4.0*P*UGA_star**2*epsilon/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) - 8.0*P*UGA_star*epsilon/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) + 4.0*P*epsilon/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) + 2.0*R*UGA_star**2/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) - 2.0*S*UGA_star**2/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) + 2.0*S*UGA_star/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) - 4.0*T*UGA_star**2*epsilon/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) + 4.0*T*UGA_star*epsilon/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star)) + 0.5*P*S*UGA_star**4*epsilon + 0.25*P*S*UGA_star**4 - 1.5*P*S*UGA_star**3*epsilon - 0.75*P*S*UGA_star**3 - 0.25*P*S*UGA_star**2*epsilon*U_prime(4.0*P*UGA_star**2*epsilon/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) - 8.0*P*UGA_star*epsilon/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) + 4.0*P*epsilon/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) + 2.0*R*UGA_star**2/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) - 2.0*S*UGA_star**2/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) + 2.0*S*UGA_star/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) - 4.0*T*UGA_star**2*epsilon/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) + 4.0*T*UGA_star*epsilon/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star)) + 1.5*P*S*UGA_star**2*epsilon + 0.125*P*S*UGA_star**2*U_prime(4.0*P*UGA_star**2*epsilon/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) - 8.0*P*UGA_star*epsilon/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) + 4.0*P*epsilon/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) + 2.0*R*UGA_star**2/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) - 2.0*S*UGA_star**2/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) + 2.0*S*UGA_star/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) - 4.0*T*UGA_star**2*epsilon/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) + 4.0*T*UGA_star*epsilon/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star)) + 0.75*P*S*UGA_star**2 + 0.5*P*S*UGA_star*epsilon*U_prime(4.0*P*UGA_star**2*epsilon/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) - 8.0*P*UGA_star*epsilon/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) + 4.0*P*epsilon/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) + 2.0*R*UGA_star**2/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) - 2.0*S*UGA_star**2/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) + 2.0*S*UGA_star/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) - 4.0*T*UGA_star**2*epsilon/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) + 4.0*T*UGA_star*epsilon/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star)) - 0.5*P*S*UGA_star*epsilon - 0.25*P*S*UGA_star*U_prime(4.0*P*UGA_star**2*epsilon/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) - 8.0*P*UGA_star*epsilon/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) + 4.0*P*epsilon/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) + 2.0*R*UGA_star**2/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) - 2.0*S*UGA_star**2/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) + 2.0*S*UGA_star/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) - 4.0*T*UGA_star**2*epsilon/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) + 4.0*T*UGA_star*epsilon/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star)) - 0.25*P*S*UGA_star - 0.25*P*S*epsilon*U_prime(4.0*P*UGA_star**2*epsilon/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) - 8.0*P*UGA_star*epsilon/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) + 4.0*P*epsilon/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) + 2.0*R*UGA_star**2/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) - 2.0*S*UGA_star**2/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) + 2.0*S*UGA_star/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) - 4.0*T*UGA_star**2*epsilon/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) + 4.0*T*UGA_star*epsilon/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star)) + 0.125*P*S*U_prime(4.0*P*UGA_star**2*epsilon/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) - 8.0*P*UGA_star*epsilon/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) + 4.0*P*epsilon/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) + 2.0*R*UGA_star**2/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) - 2.0*S*UGA_star**2/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) + 2.0*S*UGA_star/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) - 4.0*T*UGA_star**2*epsilon/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) + 4.0*T*UGA_star*epsilon/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star)) + 0.5*P*T*UGA_star**4*epsilon + 0.25*P*T*UGA_star**4 - 1.5*P*T*UGA_star**3*epsilon - 0.75*P*T*UGA_star**3 + 1.5*P*T*UGA_star**2*epsilon + 0.75*P*T*UGA_star**2 - 0.5*P*T*UGA_star*epsilon - 0.25*P*T*UGA_star - 0.25*R**2*UGA_star**4*epsilon - 0.125*R**2*UGA_star**4 + 0.5*R*S*UGA_star**4*epsilon + 0.25*R*S*UGA_star**4 - 0.5*R*S*UGA_star**3*epsilon - 0.25*R*S*UGA_star**3 + 0.5*R*T*UGA_star**4*epsilon + 0.25*R*T*UGA_star**4 - 0.5*R*T*UGA_star**3*epsilon - 0.25*R*T*UGA_star**3 - 0.25*R*T*UGA_star**2*epsilon*U_prime(4.0*P*UGA_star**2*epsilon/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) - 8.0*P*UGA_star*epsilon/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) + 4.0*P*epsilon/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) + 2.0*R*UGA_star**2/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) - 2.0*S*UGA_star**2/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) + 2.0*S*UGA_star/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) - 4.0*T*UGA_star**2*epsilon/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) + 4.0*T*UGA_star*epsilon/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star)) + 0.125*R*T*UGA_star**2*U_prime(4.0*P*UGA_star**2*epsilon/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) - 8.0*P*UGA_star*epsilon/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) + 4.0*P*epsilon/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) + 2.0*R*UGA_star**2/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) - 2.0*S*UGA_star**2/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) + 2.0*S*UGA_star/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) - 4.0*T*UGA_star**2*epsilon/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) + 4.0*T*UGA_star*epsilon/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star)) - 0.25*S**2*UGA_star**4*epsilon - 0.125*S**2*UGA_star**4 + 0.5*S**2*UGA_star**3*epsilon + 0.25*S**2*UGA_star**3 - 0.25*S**2*UGA_star**2*epsilon - 0.125*S**2*UGA_star**2 - 0.5*S*T*UGA_star**4*epsilon - 0.25*S*T*UGA_star**4 + 1.0*S*T*UGA_star**3*epsilon + 0.5*S*T*UGA_star**3 - 0.5*S*T*UGA_star**2*epsilon - 0.25*S*T*UGA_star**2 - 0.25*T**2*UGA_star**4*epsilon - 0.125*T**2*UGA_star**4 + 0.5*T**2*UGA_star**3*epsilon + 0.25*T**2*UGA_star**3 - 0.25*T**2*UGA_star**2*epsilon - 0.125*T**2*UGA_star**2

Now we can solve the numerator for $U'_{GA}(x^*)$.  This will allow us to write the condition for which the Jacobian will be negative as an upper bound on $U'_{GA}(x^*)$.

```python
upper_bound, = sym.solve(numerator, UGA_prime_star)
```

```python
sym.factor(upper_bound)
```

    1.0*(2.0*epsilon + 1.0)*(1.0*P*UGA_star**2 - 2.0*P*UGA_star + 1.0*P + 1.0*R*UGA_star**2 - 1.0*S*UGA_star**2 + 1.0*S*UGA_star - 1.0*T*UGA_star**2 + 1.0*T*UGA_star)**2/((2.0*epsilon - 1.0)*(2.0*P*R*UGA_star**2 - 2.0*P*R*UGA_star - 1.0*P*S*UGA_star**2 + 2.0*P*S*UGA_star - 1.0*P*S - 1.0*R*T*UGA_star**2))

After fiddling around with the above expression we arrive at the following condition for a negative Jacobian and a stable interior equilibrium at $0 < x^* < 1$.

\begin{align}
    U'_{GA}(x^*) <& \left(\frac{1 + 2\epsilon}{1 - 2\epsilon}\right)\frac{\bigg(\big((R + P) - (T + S)\big)U_{GA}^{*2} + \big((T + S) - 2P\big)U_{GA}^* + P\bigg)^2}{\big(TR + PS - 2RP\big)U_{GA}^{*2} + 2P(R - S)U_{GA}^* + PS}\\
\end{align}

Can also use a graphical approach to check the correctness of the above condition.  The above condition should be the zero contour in a contour plot of the value of the Jacobian as a function of $U_{GA}$ and $U'_{GA}$.

```python
# vectorized numerical function created from our symbolic expression
_numeric_f_jac = sym.lambdify((UGA_star, UGA_prime_star, T, R, P, S, epsilon), evaluated_f_jac, modules="numpy")
```

```python
def plot_jacobian(T, R, P, S, epsilon):

    fig, ax = plt.subplots(1, 1, figsize=(20, 10))
    ax.set_ylabel(r"$U_{GA}^*$", fontsize=20, rotation="horizontal")
    ax.set_xlabel(r"$U_{GA}^{'*}$", fontsize=20)
    ax.set_title(r"Negative values of $\frac{\partial \dot{x}_1}{\partial x_1}$ indicate stability!", fontsize=25)
    ax.grid("off")

    equilibrium_selection_probs = np.linspace(0, 1, 100).reshape(-1, 1)
    equilibrium_selection_derivs = np.linspace(0, 10, 1000).reshape(1, -1) # sensible UGA'(0) >= 1
    Z = _numeric_f_jac(equilibrium_selection_probs, equilibrium_selection_derivs, T, R, P, S, epsilon)
    cax = ax.imshow(Z, origin="lower", aspect="auto", vmin=-0.5, vmax=1.0)

    levels = np.arange(-0.25, 1.25, 0.25)
    contours = ax.contour(Z, levels=levels, colors='w', origin='lower')
    ax.clabel(contours, contours.levels, inline=True, fontsize=10)

    # adjust the tick labels
    locs, _ = plt.xticks()
    plt.xticks(locs[1:], np.linspace(0, 10, locs.size-1))
    locs, _ = plt.yticks()
    plt.yticks(locs[1:], np.linspace(0, 1, locs.size-1))

    plt.show()
```

```python
mpld3.disable_notebook()  # don't need interactive plotting for this!

# sliders used to control the Prisoner's Dilemma Payoffs
T_slider = widgets.FloatSlider(value=25, min=0, max=100, step=0.1, description=r"$T$")
R_slider = widgets.FloatSlider(value=3, min=0, max=100, step=0.1, description=r"$R$")
P_slider = widgets.FloatSlider(value=2, min=0, max=100, step=0.1, description=r"$P$")
S_slider = widgets.FloatSlider(value=1, min=0, max=100, step=0.1, description=r"$S$")

e_slider = widgets.FloatSlider(value=0, min=0, max=1, step=1e-3, description=r"$\epsilon$", readout_format=".3f")

w = widgets.interactive(plot_jacobian, T=T_slider, R=R_slider, P=P_slider, S=S_slider, epsilon=e_slider)
display(w)
```

    A Jupyter Widget

# Invadability

Restrict the full model to obtain a monomorphic $\gamma$ equilibrium, add a small amount of individuals carrying the $g$ allele of the $\gamma$ gene and simulate.

```python
UG = symbolics.UGA
Ug = symbolics.UgA

x1, x2, x3 = sym.symbols("x1, x2, x3", real=True, positive=True)
x4 = 1 - x1 - x2 - x3
x = np.array([[x1], [x2], [x3], [x4]])
W = models.generalized_sexual_selection(x, UG, Ug, payoff_kernel, M=M)
f1, f2, f3, _ = models.offspring_genotypes_evolution(W, x)
```

```python
sym.factor(models._offspring_by_genotype(W, x)[0, 0], M, UGA, UgA, deep=True)
```

$\displaystyle \frac{M \left(1.5 x_{1}^{3} \operatorname{UGA}{\left(x_{1} + x_{3} \right)} + 0.5 x_{1}^{3} + 1.5 x_{1}^{2} x_{2} \operatorname{UGA}{\left(x_{1} + x_{3} \right)} - 0.5 x_{1}^{2} x_{2} + 2.0 x_{1}^{2} x_{3} \operatorname{UGA}{\left(x_{1} + x_{3} \right)} + 1.0 x_{1}^{2} x_{3} - 1.5 x_{1}^{2} \operatorname{UGA}{\left(x_{1} + x_{3} \right)} - 0.5 x_{1}^{2} + 2.0 x_{1} x_{2} x_{3} \operatorname{UGA}{\left(x_{1} + x_{3} \right)} - 0.5 x_{1} x_{2} x_{3} - 1.0 x_{1} x_{2} \operatorname{UGA}{\left(x_{1} + x_{3} \right)} + 0.5 x_{1} x_{3}^{2} \operatorname{UGA}{\left(x_{1} + x_{3} \right)} + 0.5 x_{1} x_{3}^{2} - 0.5 x_{1} x_{3} \operatorname{UGA}{\left(x_{1} + x_{3} \right)} - 0.5 x_{1} x_{3} + 0.5 x_{2} x_{3}^{2} \operatorname{UGA}{\left(x_{1} + x_{3} \right)} - 0.5 x_{2} x_{3} \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) - 0.5 P x_{1}^{3} \operatorname{UGA}^{2}{\left(x_{1} + x_{3} \right)} + 1.0 P x_{1}^{3} \operatorname{UGA}{\left(x_{1} + x_{3} \right)} - 0.5 P x_{1}^{3} + 0.5 P x_{1}^{2} x_{2} \operatorname{UGA}^{2}{\left(x_{1} + x_{3} \right)} - 1.0 P x_{1}^{2} x_{2} \operatorname{UGA}{\left(x_{1} + x_{3} \right)} + 0.5 P x_{1}^{2} x_{2} - 1.0 P x_{1}^{2} x_{3} \operatorname{UGA}^{2}{\left(x_{1} + x_{3} \right)} + 2.0 P x_{1}^{2} x_{3} \operatorname{UGA}{\left(x_{1} + x_{3} \right)} - 1.0 P x_{1}^{2} x_{3} + 0.5 P x_{1}^{2} \operatorname{UGA}^{2}{\left(x_{1} + x_{3} \right)} - 1.0 P x_{1}^{2} \operatorname{UGA}{\left(x_{1} + x_{3} \right)} + 0.5 P x_{1}^{2} + 0.5 P x_{1} x_{2} x_{3} \operatorname{UGA}^{2}{\left(x_{1} + x_{3} \right)} - 1.0 P x_{1} x_{2} x_{3} \operatorname{UGA}{\left(x_{1} + x_{3} \right)} + 0.5 P x_{1} x_{2} x_{3} \operatorname{UgA}^{2}{\left(x_{1} + x_{3} \right)} - 1.0 P x_{1} x_{2} x_{3} \operatorname{UgA}{\left(x_{1} + x_{3} \right)} + 1.0 P x_{1} x_{2} x_{3} - 0.5 P x_{1} x_{3}^{2} \operatorname{UGA}^{2}{\left(x_{1} + x_{3} \right)} + 1.0 P x_{1} x_{3}^{2} \operatorname{UGA}{\left(x_{1} + x_{3} \right)} - 0.5 P x_{1} x_{3}^{2} + 0.5 P x_{1} x_{3} \operatorname{UGA}^{2}{\left(x_{1} + x_{3} \right)} - 1.0 P x_{1} x_{3} \operatorname{UGA}{\left(x_{1} + x_{3} \right)} + 0.5 P x_{1} x_{3} + 0.5 P x_{2} x_{3}^{2} \operatorname{UgA}^{2}{\left(x_{1} + x_{3} \right)} - 1.0 P x_{2} x_{3}^{2} \operatorname{UgA}{\left(x_{1} + x_{3} \right)} + 0.5 P x_{2} x_{3}^{2} - 2.0 R x_{1}^{3} \operatorname{UGA}^{2}{\left(x_{1} + x_{3} \right)} + 0.5 R x_{1}^{3} \operatorname{UgA}^{2}{\left(x_{1} + x_{3} \right)} - 1.0 R x_{1}^{2} x_{2} \operatorname{UGA}^{2}{\left(x_{1} + x_{3} \right)} + 0.5 R x_{1}^{2} x_{2} \operatorname{UgA}^{2}{\left(x_{1} + x_{3} \right)} - 3.0 R x_{1}^{2} x_{3} \operatorname{UGA}^{2}{\left(x_{1} + x_{3} \right)} + 2.0 R x_{1}^{2} \operatorname{UGA}^{2}{\left(x_{1} + x_{3} \right)} - 1.0 R x_{1}^{2} \operatorname{UgA}^{2}{\left(x_{1} + x_{3} \right)} - 1.5 R x_{1} x_{2} x_{3} \operatorname{UGA}^{2}{\left(x_{1} + x_{3} \right)} + 0.5 R x_{1} x_{2} x_{3} \operatorname{UgA}^{2}{\left(x_{1} + x_{3} \right)} + 1.0 R x_{1} x_{2} \operatorname{UGA}^{2}{\left(x_{1} + x_{3} \right)} - 0.5 R x_{1} x_{2} \operatorname{UgA}^{2}{\left(x_{1} + x_{3} \right)} - 1.0 R x_{1} x_{3}^{2} \operatorname{UGA}^{2}{\left(x_{1} + x_{3} \right)} - 0.5 R x_{1} x_{3}^{2} \operatorname{UgA}^{2}{\left(x_{1} + x_{3} \right)} + 1.0 R x_{1} x_{3} \operatorname{UGA}^{2}{\left(x_{1} + x_{3} \right)} + 0.5 R x_{1} \operatorname{UgA}^{2}{\left(x_{1} + x_{3} \right)} - 0.5 R x_{2} x_{3}^{2} \operatorname{UGA}^{2}{\left(x_{1} + x_{3} \right)} + 0.5 R x_{2} x_{3} \operatorname{UGA}^{2}{\left(x_{1} + x_{3} \right)} + 2.0 S x_{1}^{3} \operatorname{UGA}^{2}{\left(x_{1} + x_{3} \right)} - 2.0 S x_{1}^{3} \operatorname{UGA}{\left(x_{1} + x_{3} \right)} - 0.5 S x_{1}^{3} \operatorname{UgA}^{2}{\left(x_{1} + x_{3} \right)} + 0.5 S x_{1}^{3} \operatorname{UgA}{\left(x_{1} + x_{3} \right)} + 1.0 S x_{1}^{2} x_{2} \operatorname{UGA}^{2}{\left(x_{1} + x_{3} \right)} - 1.0 S x_{1}^{2} x_{2} \operatorname{UGA}{\left(x_{1} + x_{3} \right)} - 0.5 S x_{1}^{2} x_{2} \operatorname{UgA}^{2}{\left(x_{1} + x_{3} \right)} + 0.5 S x_{1}^{2} x_{2} \operatorname{UgA}{\left(x_{1} + x_{3} \right)} + 3.0 S x_{1}^{2} x_{3} \operatorname{UGA}^{2}{\left(x_{1} + x_{3} \right)} - 3.0 S x_{1}^{2} x_{3} \operatorname{UGA}{\left(x_{1} + x_{3} \right)} - 2.0 S x_{1}^{2} \operatorname{UGA}^{2}{\left(x_{1} + x_{3} \right)} + 2.0 S x_{1}^{2} \operatorname{UGA}{\left(x_{1} + x_{3} \right)} + 1.0 S x_{1}^{2} \operatorname{UgA}^{2}{\left(x_{1} + x_{3} \right)} - 1.0 S x_{1}^{2} \operatorname{UgA}{\left(x_{1} + x_{3} \right)} + 1.5 S x_{1} x_{2} x_{3} \operatorname{UGA}^{2}{\left(x_{1} + x_{3} \right)} - 1.5 S x_{1} x_{2} x_{3} \operatorname{UGA}{\left(x_{1} + x_{3} \right)} - 0.5 S x_{1} x_{2} x_{3} \operatorname{UgA}^{2}{\left(x_{1} + x_{3} \right)} + 0.5 S x_{1} x_{2} x_{3} \operatorname{UgA}{\left(x_{1} + x_{3} \right)} - 1.0 S x_{1} x_{2} \operatorname{UGA}^{2}{\left(x_{1} + x_{3} \right)} + 1.0 S x_{1} x_{2} \operatorname{UGA}{\left(x_{1} + x_{3} \right)} + 0.5 S x_{1} x_{2} \operatorname{UgA}^{2}{\left(x_{1} + x_{3} \right)} - 0.5 S x_{1} x_{2} \operatorname{UgA}{\left(x_{1} + x_{3} \right)} + 1.0 S x_{1} x_{3}^{2} \operatorname{UGA}^{2}{\left(x_{1} + x_{3} \right)} - 1.0 S x_{1} x_{3}^{2} \operatorname{UGA}{\left(x_{1} + x_{3} \right)} + 0.5 S x_{1} x_{3}^{2} \operatorname{UgA}^{2}{\left(x_{1} + x_{3} \right)} - 0.5 S x_{1} x_{3}^{2} \operatorname{UgA}{\left(x_{1} + x_{3} \right)} - 1.0 S x_{1} x_{3} \operatorname{UGA}^{2}{\left(x_{1} + x_{3} \right)} + 1.0 S x_{1} x_{3} \operatorname{UGA}{\left(x_{1} + x_{3} \right)} - 0.5 S x_{1} \operatorname{UgA}^{2}{\left(x_{1} + x_{3} \right)} + 0.5 S x_{1} \operatorname{UgA}{\left(x_{1} + x_{3} \right)} + 0.5 S x_{2} x_{3}^{2} \operatorname{UGA}^{2}{\left(x_{1} + x_{3} \right)} - 0.5 S x_{2} x_{3}^{2} \operatorname{UGA}{\left(x_{1} + x_{3} \right)} - 0.5 S x_{2} x_{3} \operatorname{UGA}^{2}{\left(x_{1} + x_{3} \right)} + 0.5 S x_{2} x_{3} \operatorname{UGA}{\left(x_{1} + x_{3} \right)} + 0.5 T x_{1}^{3} \operatorname{UGA}^{2}{\left(x_{1} + x_{3} \right)} - 0.5 T x_{1}^{3} \operatorname{UGA}{\left(x_{1} + x_{3} \right)} - 0.5 T x_{1}^{2} x_{2} \operatorname{UGA}^{2}{\left(x_{1} + x_{3} \right)} + 0.5 T x_{1}^{2} x_{2} \operatorname{UGA}{\left(x_{1} + x_{3} \right)} + 1.0 T x_{1}^{2} x_{3} \operatorname{UGA}^{2}{\left(x_{1} + x_{3} \right)} - 1.0 T x_{1}^{2} x_{3} \operatorname{UGA}{\left(x_{1} + x_{3} \right)} - 0.5 T x_{1}^{2} \operatorname{UGA}^{2}{\left(x_{1} + x_{3} \right)} + 0.5 T x_{1}^{2} \operatorname{UGA}{\left(x_{1} + x_{3} \right)} - 0.5 T x_{1} x_{2} x_{3} \operatorname{UGA}^{2}{\left(x_{1} + x_{3} \right)} + 0.5 T x_{1} x_{2} x_{3} \operatorname{UGA}{\left(x_{1} + x_{3} \right)} - 0.5 T x_{1} x_{2} x_{3} \operatorname{UgA}^{2}{\left(x_{1} + x_{3} \right)} + 0.5 T x_{1} x_{2} x_{3} \operatorname{UgA}{\left(x_{1} + x_{3} \right)} + 0.5 T x_{1} x_{3}^{2} \operatorname{UGA}^{2}{\left(x_{1} + x_{3} \right)} - 0.5 T x_{1} x_{3}^{2} \operatorname{UGA}{\left(x_{1} + x_{3} \right)} - 0.5 T x_{1} x_{3} \operatorname{UGA}^{2}{\left(x_{1} + x_{3} \right)} + 0.5 T x_{1} x_{3} \operatorname{UGA}{\left(x_{1} + x_{3} \right)} - 0.5 T x_{2} x_{3}^{2} \operatorname{UgA}^{2}{\left(x_{1} + x_{3} \right)} + 0.5 T x_{2} x_{3}^{2} \operatorname{UgA}{\left(x_{1} + x_{3} \right)}}{\left(x_{1} + x_{3}\right) \left(- x_{1} - x_{3} + 1\right)}$

```python
sym.simplify(models._offspring_by_genotype(W, x)[0, 0]).subs
```

$\displaystyle \frac{- x_{1} \left(0.5 \left(x_{1} + x_{3}\right) \left(\left(M - P\right) \left(\operatorname{UGA}{\left(x_{1} + x_{3} \right)} - 1\right) - \left(M - T\right) \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \left(\operatorname{UGA}{\left(x_{1} + x_{3} \right)} - 1\right) \left(x_{1} - x_{2} + x_{3} - 1\right) + \left(2.0 x_{1} + x_{3}\right) \left(\left(M - R\right) \operatorname{UGA}{\left(x_{1} + x_{3} \right)} - \left(M - S\right) \left(\operatorname{UGA}{\left(x_{1} + x_{3} \right)} - 1\right)\right) \left(x_{1} + x_{3} - 1\right) \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) + x_{3} \left(x_{1} \left(R \operatorname{UgA}{\left(x_{1} + x_{3} \right)} - S \left(\operatorname{UgA}{\left(x_{1} + x_{3} \right)} - 1\right)\right) \left(x_{1} + x_{3} - 1\right) \operatorname{UgA}{\left(x_{1} + x_{3} \right)} - 0.5 x_{2} \left(x_{1} + x_{3}\right) \left(P \left(\operatorname{UgA}{\left(x_{1} + x_{3} \right)} - 1\right) - T \operatorname{UgA}{\left(x_{1} + x_{3} \right)}\right) \left(\operatorname{UgA}{\left(x_{1} + x_{3} \right)} - 1\right)\right) - \left(0.5 x_{1} \left(R \operatorname{UgA}{\left(x_{1} + x_{3} \right)} - S \left(\operatorname{UgA}{\left(x_{1} + x_{3} \right)} - 1\right)\right) \left(x_{1} + x_{2} + x_{3} - 1\right) \operatorname{UgA}{\left(x_{1} + x_{3} \right)} + x_{2} \left(x_{1} + 0.5 x_{3}\right) \left(\left(M - R\right) \operatorname{UGA}{\left(x_{1} + x_{3} \right)} - \left(M - S\right) \left(\operatorname{UGA}{\left(x_{1} + x_{3} \right)} - 1\right)\right) \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \left(x_{1} + x_{3} - 1\right)}{\left(x_{1} + x_{3}\right) \left(x_{1} + x_{3} - 1\right)}$

```python
n, d = sym.fraction(sym.together(models._offspring_by_genotype(W, x)[0, 0]))
```

```python
sym.factor(n, M)
```

$\displaystyle M \left(1.5 x_{1}^{3} \operatorname{UGA}{\left(x_{1} + x_{3} \right)} + 0.5 x_{1}^{3} + 1.5 x_{1}^{2} x_{2} \operatorname{UGA}{\left(x_{1} + x_{3} \right)} - 0.5 x_{1}^{2} x_{2} + 2.0 x_{1}^{2} x_{3} \operatorname{UGA}{\left(x_{1} + x_{3} \right)} + 1.0 x_{1}^{2} x_{3} - 1.5 x_{1}^{2} \operatorname{UGA}{\left(x_{1} + x_{3} \right)} - 0.5 x_{1}^{2} + 2.0 x_{1} x_{2} x_{3} \operatorname{UGA}{\left(x_{1} + x_{3} \right)} - 0.5 x_{1} x_{2} x_{3} - 1.0 x_{1} x_{2} \operatorname{UGA}{\left(x_{1} + x_{3} \right)} + 0.5 x_{1} x_{3}^{2} \operatorname{UGA}{\left(x_{1} + x_{3} \right)} + 0.5 x_{1} x_{3}^{2} - 0.5 x_{1} x_{3} \operatorname{UGA}{\left(x_{1} + x_{3} \right)} - 0.5 x_{1} x_{3} + 0.5 x_{2} x_{3}^{2} \operatorname{UGA}{\left(x_{1} + x_{3} \right)} - 0.5 x_{2} x_{3} \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) - 0.5 P x_{1}^{3} \operatorname{UGA}^{2}{\left(x_{1} + x_{3} \right)} + 1.0 P x_{1}^{3} \operatorname{UGA}{\left(x_{1} + x_{3} \right)} - 0.5 P x_{1}^{3} + 0.5 P x_{1}^{2} x_{2} \operatorname{UGA}^{2}{\left(x_{1} + x_{3} \right)} - 1.0 P x_{1}^{2} x_{2} \operatorname{UGA}{\left(x_{1} + x_{3} \right)} + 0.5 P x_{1}^{2} x_{2} - 1.0 P x_{1}^{2} x_{3} \operatorname{UGA}^{2}{\left(x_{1} + x_{3} \right)} + 2.0 P x_{1}^{2} x_{3} \operatorname{UGA}{\left(x_{1} + x_{3} \right)} - 1.0 P x_{1}^{2} x_{3} + 0.5 P x_{1}^{2} \operatorname{UGA}^{2}{\left(x_{1} + x_{3} \right)} - 1.0 P x_{1}^{2} \operatorname{UGA}{\left(x_{1} + x_{3} \right)} + 0.5 P x_{1}^{2} + 0.5 P x_{1} x_{2} x_{3} \operatorname{UGA}^{2}{\left(x_{1} + x_{3} \right)} - 1.0 P x_{1} x_{2} x_{3} \operatorname{UGA}{\left(x_{1} + x_{3} \right)} + 0.5 P x_{1} x_{2} x_{3} \operatorname{UgA}^{2}{\left(x_{1} + x_{3} \right)} - 1.0 P x_{1} x_{2} x_{3} \operatorname{UgA}{\left(x_{1} + x_{3} \right)} + 1.0 P x_{1} x_{2} x_{3} - 0.5 P x_{1} x_{3}^{2} \operatorname{UGA}^{2}{\left(x_{1} + x_{3} \right)} + 1.0 P x_{1} x_{3}^{2} \operatorname{UGA}{\left(x_{1} + x_{3} \right)} - 0.5 P x_{1} x_{3}^{2} + 0.5 P x_{1} x_{3} \operatorname{UGA}^{2}{\left(x_{1} + x_{3} \right)} - 1.0 P x_{1} x_{3} \operatorname{UGA}{\left(x_{1} + x_{3} \right)} + 0.5 P x_{1} x_{3} + 0.5 P x_{2} x_{3}^{2} \operatorname{UgA}^{2}{\left(x_{1} + x_{3} \right)} - 1.0 P x_{2} x_{3}^{2} \operatorname{UgA}{\left(x_{1} + x_{3} \right)} + 0.5 P x_{2} x_{3}^{2} - 2.0 R x_{1}^{3} \operatorname{UGA}^{2}{\left(x_{1} + x_{3} \right)} + 0.5 R x_{1}^{3} \operatorname{UgA}^{2}{\left(x_{1} + x_{3} \right)} - 1.0 R x_{1}^{2} x_{2} \operatorname{UGA}^{2}{\left(x_{1} + x_{3} \right)} + 0.5 R x_{1}^{2} x_{2} \operatorname{UgA}^{2}{\left(x_{1} + x_{3} \right)} - 3.0 R x_{1}^{2} x_{3} \operatorname{UGA}^{2}{\left(x_{1} + x_{3} \right)} + 2.0 R x_{1}^{2} \operatorname{UGA}^{2}{\left(x_{1} + x_{3} \right)} - 1.0 R x_{1}^{2} \operatorname{UgA}^{2}{\left(x_{1} + x_{3} \right)} - 1.5 R x_{1} x_{2} x_{3} \operatorname{UGA}^{2}{\left(x_{1} + x_{3} \right)} + 0.5 R x_{1} x_{2} x_{3} \operatorname{UgA}^{2}{\left(x_{1} + x_{3} \right)} + 1.0 R x_{1} x_{2} \operatorname{UGA}^{2}{\left(x_{1} + x_{3} \right)} - 0.5 R x_{1} x_{2} \operatorname{UgA}^{2}{\left(x_{1} + x_{3} \right)} - 1.0 R x_{1} x_{3}^{2} \operatorname{UGA}^{2}{\left(x_{1} + x_{3} \right)} - 0.5 R x_{1} x_{3}^{2} \operatorname{UgA}^{2}{\left(x_{1} + x_{3} \right)} + 1.0 R x_{1} x_{3} \operatorname{UGA}^{2}{\left(x_{1} + x_{3} \right)} + 0.5 R x_{1} \operatorname{UgA}^{2}{\left(x_{1} + x_{3} \right)} - 0.5 R x_{2} x_{3}^{2} \operatorname{UGA}^{2}{\left(x_{1} + x_{3} \right)} + 0.5 R x_{2} x_{3} \operatorname{UGA}^{2}{\left(x_{1} + x_{3} \right)} + 2.0 S x_{1}^{3} \operatorname{UGA}^{2}{\left(x_{1} + x_{3} \right)} - 2.0 S x_{1}^{3} \operatorname{UGA}{\left(x_{1} + x_{3} \right)} - 0.5 S x_{1}^{3} \operatorname{UgA}^{2}{\left(x_{1} + x_{3} \right)} + 0.5 S x_{1}^{3} \operatorname{UgA}{\left(x_{1} + x_{3} \right)} + 1.0 S x_{1}^{2} x_{2} \operatorname{UGA}^{2}{\left(x_{1} + x_{3} \right)} - 1.0 S x_{1}^{2} x_{2} \operatorname{UGA}{\left(x_{1} + x_{3} \right)} - 0.5 S x_{1}^{2} x_{2} \operatorname{UgA}^{2}{\left(x_{1} + x_{3} \right)} + 0.5 S x_{1}^{2} x_{2} \operatorname{UgA}{\left(x_{1} + x_{3} \right)} + 3.0 S x_{1}^{2} x_{3} \operatorname{UGA}^{2}{\left(x_{1} + x_{3} \right)} - 3.0 S x_{1}^{2} x_{3} \operatorname{UGA}{\left(x_{1} + x_{3} \right)} - 2.0 S x_{1}^{2} \operatorname{UGA}^{2}{\left(x_{1} + x_{3} \right)} + 2.0 S x_{1}^{2} \operatorname{UGA}{\left(x_{1} + x_{3} \right)} + 1.0 S x_{1}^{2} \operatorname{UgA}^{2}{\left(x_{1} + x_{3} \right)} - 1.0 S x_{1}^{2} \operatorname{UgA}{\left(x_{1} + x_{3} \right)} + 1.5 S x_{1} x_{2} x_{3} \operatorname{UGA}^{2}{\left(x_{1} + x_{3} \right)} - 1.5 S x_{1} x_{2} x_{3} \operatorname{UGA}{\left(x_{1} + x_{3} \right)} - 0.5 S x_{1} x_{2} x_{3} \operatorname{UgA}^{2}{\left(x_{1} + x_{3} \right)} + 0.5 S x_{1} x_{2} x_{3} \operatorname{UgA}{\left(x_{1} + x_{3} \right)} - 1.0 S x_{1} x_{2} \operatorname{UGA}^{2}{\left(x_{1} + x_{3} \right)} + 1.0 S x_{1} x_{2} \operatorname{UGA}{\left(x_{1} + x_{3} \right)} + 0.5 S x_{1} x_{2} \operatorname{UgA}^{2}{\left(x_{1} + x_{3} \right)} - 0.5 S x_{1} x_{2} \operatorname{UgA}{\left(x_{1} + x_{3} \right)} + 1.0 S x_{1} x_{3}^{2} \operatorname{UGA}^{2}{\left(x_{1} + x_{3} \right)} - 1.0 S x_{1} x_{3}^{2} \operatorname{UGA}{\left(x_{1} + x_{3} \right)} + 0.5 S x_{1} x_{3}^{2} \operatorname{UgA}^{2}{\left(x_{1} + x_{3} \right)} - 0.5 S x_{1} x_{3}^{2} \operatorname{UgA}{\left(x_{1} + x_{3} \right)} - 1.0 S x_{1} x_{3} \operatorname{UGA}^{2}{\left(x_{1} + x_{3} \right)} + 1.0 S x_{1} x_{3} \operatorname{UGA}{\left(x_{1} + x_{3} \right)} - 0.5 S x_{1} \operatorname{UgA}^{2}{\left(x_{1} + x_{3} \right)} + 0.5 S x_{1} \operatorname{UgA}{\left(x_{1} + x_{3} \right)} + 0.5 S x_{2} x_{3}^{2} \operatorname{UGA}^{2}{\left(x_{1} + x_{3} \right)} - 0.5 S x_{2} x_{3}^{2} \operatorname{UGA}{\left(x_{1} + x_{3} \right)} - 0.5 S x_{2} x_{3} \operatorname{UGA}^{2}{\left(x_{1} + x_{3} \right)} + 0.5 S x_{2} x_{3} \operatorname{UGA}{\left(x_{1} + x_{3} \right)} + 0.5 T x_{1}^{3} \operatorname{UGA}^{2}{\left(x_{1} + x_{3} \right)} - 0.5 T x_{1}^{3} \operatorname{UGA}{\left(x_{1} + x_{3} \right)} - 0.5 T x_{1}^{2} x_{2} \operatorname{UGA}^{2}{\left(x_{1} + x_{3} \right)} + 0.5 T x_{1}^{2} x_{2} \operatorname{UGA}{\left(x_{1} + x_{3} \right)} + 1.0 T x_{1}^{2} x_{3} \operatorname{UGA}^{2}{\left(x_{1} + x_{3} \right)} - 1.0 T x_{1}^{2} x_{3} \operatorname{UGA}{\left(x_{1} + x_{3} \right)} - 0.5 T x_{1}^{2} \operatorname{UGA}^{2}{\left(x_{1} + x_{3} \right)} + 0.5 T x_{1}^{2} \operatorname{UGA}{\left(x_{1} + x_{3} \right)} - 0.5 T x_{1} x_{2} x_{3} \operatorname{UGA}^{2}{\left(x_{1} + x_{3} \right)} + 0.5 T x_{1} x_{2} x_{3} \operatorname{UGA}{\left(x_{1} + x_{3} \right)} - 0.5 T x_{1} x_{2} x_{3} \operatorname{UgA}^{2}{\left(x_{1} + x_{3} \right)} + 0.5 T x_{1} x_{2} x_{3} \operatorname{UgA}{\left(x_{1} + x_{3} \right)} + 0.5 T x_{1} x_{3}^{2} \operatorname{UGA}^{2}{\left(x_{1} + x_{3} \right)} - 0.5 T x_{1} x_{3}^{2} \operatorname{UGA}{\left(x_{1} + x_{3} \right)} - 0.5 T x_{1} x_{3} \operatorname{UGA}^{2}{\left(x_{1} + x_{3} \right)} + 0.5 T x_{1} x_{3} \operatorname{UGA}{\left(x_{1} + x_{3} \right)} - 0.5 T x_{2} x_{3}^{2} \operatorname{UgA}^{2}{\left(x_{1} + x_{3} \right)} + 0.5 T x_{2} x_{3}^{2} \operatorname{UgA}{\left(x_{1} + x_{3} \right)}$

```python
F = sym.Matrix([f1, f2, f3])
```

```python

```

```python
F
```

$\displaystyle \left[\begin{matrix}- x_{1} + \frac{x_{1} \left(\frac{1.0 x_{1} \left(2 \left(1 - \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \left(- M + S\right) + 2 \left(- M + R\right) \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \operatorname{UGA}{\left(x_{1} + x_{3} \right)}}{x_{1} + x_{3}} + \frac{0.5 x_{2} \left(1 - \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \left(2 \left(1 - \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \left(- M + P\right) + 2 \left(- M + T\right) \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right)}{- x_{1} - x_{3} + 1} + \frac{0.5 x_{3} \left(2 \left(1 - \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \left(- M + S\right) + 2 \left(- M + R\right) \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \operatorname{UGA}{\left(x_{1} + x_{3} \right)}}{x_{1} + x_{3}} + \frac{0.25 \left(1 - \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \left(2 \left(1 - \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \left(- M + P\right) + 2 \left(- M + T\right) \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \left(- x_{1} - x_{2} - x_{3} + 1\right)}{- x_{1} - x_{3} + 1}\right) + \frac{0.25 x_{1} \left(2 R \operatorname{UgA}{\left(x_{1} + x_{3} \right)} + 2 S \left(1 - \operatorname{UgA}{\left(x_{1} + x_{3} \right)}\right)\right) \left(- x_{1} - x_{2} - x_{3} + 1\right) \operatorname{UgA}{\left(x_{1} + x_{3} \right)}}{x_{1} + x_{3}} + x_{2} \left(\frac{0.5 x_{1} \left(2 \left(1 - \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \left(- M + S\right) + 2 \left(- M + R\right) \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \operatorname{UGA}{\left(x_{1} + x_{3} \right)}}{x_{1} + x_{3}} + \frac{0.25 x_{3} \left(2 \left(1 - \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \left(- M + S\right) + 2 \left(- M + R\right) \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \operatorname{UGA}{\left(x_{1} + x_{3} \right)}}{x_{1} + x_{3}}\right) + x_{3} \left(\frac{0.5 x_{1} \left(2 R \operatorname{UgA}{\left(x_{1} + x_{3} \right)} + 2 S \left(1 - \operatorname{UgA}{\left(x_{1} + x_{3} \right)}\right)\right) \operatorname{UgA}{\left(x_{1} + x_{3} \right)}}{x_{1} + x_{3}} + \frac{0.25 x_{2} \left(1 - \operatorname{UgA}{\left(x_{1} + x_{3} \right)}\right) \left(2 P \left(1 - \operatorname{UgA}{\left(x_{1} + x_{3} \right)}\right) + 2 T \operatorname{UgA}{\left(x_{1} + x_{3} \right)}\right)}{- x_{1} - x_{3} + 1}\right)}{\frac{0.25 x_{1} \left(1 - \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \left(2 \left(1 - \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \left(- M + P\right) + 2 \left(- M + T\right) \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \left(- x_{1} - x_{2} - x_{3} + 1\right)}{- x_{1} - x_{3} + 1} + x_{1} \left(\frac{0.5 x_{2} \left(1 - \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \left(2 \left(1 - \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \left(- M + P\right) + 2 \left(- M + T\right) \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right)}{- x_{1} - x_{3} + 1} + \frac{0.25 \left(1 - \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \left(2 \left(1 - \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \left(- M + P\right) + 2 \left(- M + T\right) \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \left(- x_{1} - x_{2} - x_{3} + 1\right)}{- x_{1} - x_{3} + 1}\right) + x_{1} \left(\frac{0.5 x_{3} \left(2 \left(1 - \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \left(- M + S\right) + 2 \left(- M + R\right) \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \operatorname{UGA}{\left(x_{1} + x_{3} \right)}}{x_{1} + x_{3}} + \frac{0.25 \left(1 - \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \left(2 \left(1 - \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \left(- M + P\right) + 2 \left(- M + T\right) \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \left(- x_{1} - x_{2} - x_{3} + 1\right)}{- x_{1} - x_{3} + 1}\right) + x_{1} \left(\frac{1.0 x_{1} \left(2 \left(1 - \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \left(- M + S\right) + 2 \left(- M + R\right) \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \operatorname{UGA}{\left(x_{1} + x_{3} \right)}}{x_{1} + x_{3}} + \frac{0.5 x_{2} \left(1 - \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \left(2 \left(1 - \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \left(- M + P\right) + 2 \left(- M + T\right) \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right)}{- x_{1} - x_{3} + 1} + \frac{0.5 x_{3} \left(2 \left(1 - \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \left(- M + S\right) + 2 \left(- M + R\right) \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \operatorname{UGA}{\left(x_{1} + x_{3} \right)}}{x_{1} + x_{3}} + \frac{0.25 \left(1 - \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \left(2 \left(1 - \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \left(- M + P\right) + 2 \left(- M + T\right) \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \left(- x_{1} - x_{2} - x_{3} + 1\right)}{- x_{1} - x_{3} + 1}\right) + \frac{0.25 x_{1} \left(2 R \operatorname{UgA}{\left(x_{1} + x_{3} \right)} + 2 S \left(1 - \operatorname{UgA}{\left(x_{1} + x_{3} \right)}\right)\right) \left(- x_{1} - x_{2} - x_{3} + 1\right) \operatorname{UgA}{\left(x_{1} + x_{3} \right)}}{x_{1} + x_{3}} + \frac{0.25 x_{2} x_{3} \left(1 - \operatorname{UgA}{\left(x_{1} + x_{3} \right)}\right) \left(2 P \left(1 - \operatorname{UgA}{\left(x_{1} + x_{3} \right)}\right) + 2 T \operatorname{UgA}{\left(x_{1} + x_{3} \right)}\right)}{- x_{1} - x_{3} + 1} + \frac{0.25 x_{2} x_{3} \left(2 \left(1 - \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \left(- M + S\right) + 2 \left(- M + R\right) \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \operatorname{UGA}{\left(x_{1} + x_{3} \right)}}{x_{1} + x_{3}} + x_{2} \left(\frac{0.5 x_{1} \left(2 \left(1 - \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \left(- M + S\right) + 2 \left(- M + R\right) \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \operatorname{UGA}{\left(x_{1} + x_{3} \right)}}{x_{1} + x_{3}} + \frac{0.25 x_{3} \left(2 \left(1 - \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \left(- M + S\right) + 2 \left(- M + R\right) \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \operatorname{UGA}{\left(x_{1} + x_{3} \right)}}{x_{1} + x_{3}}\right) + x_{2} \left(\frac{0.25 x_{3} \left(2 \left(1 - \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \left(- M + S\right) + 2 \left(- M + R\right) \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \operatorname{UGA}{\left(x_{1} + x_{3} \right)}}{x_{1} + x_{3}} + \frac{0.5 \left(1 - \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \left(2 \left(1 - \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \left(- M + P\right) + 2 \left(- M + T\right) \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \left(- x_{1} - x_{2} - x_{3} + 1\right)}{- x_{1} - x_{3} + 1}\right) + x_{2} \left(\frac{0.5 x_{1} \left(2 \left(1 - \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \left(- M + S\right) + 2 \left(- M + R\right) \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \operatorname{UGA}{\left(x_{1} + x_{3} \right)}}{x_{1} + x_{3}} + \frac{1.0 x_{2} \left(1 - \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \left(2 \left(1 - \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \left(- M + P\right) + 2 \left(- M + T\right) \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right)}{- x_{1} - x_{3} + 1} + \frac{0.25 x_{3} \left(2 \left(1 - \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \left(- M + S\right) + 2 \left(- M + R\right) \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \operatorname{UGA}{\left(x_{1} + x_{3} \right)}}{x_{1} + x_{3}} + \frac{0.5 \left(1 - \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \left(2 \left(1 - \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \left(- M + P\right) + 2 \left(- M + T\right) \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \left(- x_{1} - x_{2} - x_{3} + 1\right)}{- x_{1} - x_{3} + 1}\right) + x_{3} \left(\frac{0.5 x_{1} \left(2 R \operatorname{UgA}{\left(x_{1} + x_{3} \right)} + 2 S \left(1 - \operatorname{UgA}{\left(x_{1} + x_{3} \right)}\right)\right) \operatorname{UgA}{\left(x_{1} + x_{3} \right)}}{x_{1} + x_{3}} + \frac{0.25 x_{2} \left(1 - \operatorname{UgA}{\left(x_{1} + x_{3} \right)}\right) \left(2 P \left(1 - \operatorname{UgA}{\left(x_{1} + x_{3} \right)}\right) + 2 T \operatorname{UgA}{\left(x_{1} + x_{3} \right)}\right)}{- x_{1} - x_{3} + 1}\right) + x_{3} \left(\frac{0.25 x_{2} \left(1 - \operatorname{UgA}{\left(x_{1} + x_{3} \right)}\right) \left(2 P \left(1 - \operatorname{UgA}{\left(x_{1} + x_{3} \right)}\right) + 2 T \operatorname{UgA}{\left(x_{1} + x_{3} \right)}\right)}{- x_{1} - x_{3} + 1} + \frac{0.5 \left(1 - \operatorname{UgA}{\left(x_{1} + x_{3} \right)}\right) \left(2 P \left(1 - \operatorname{UgA}{\left(x_{1} + x_{3} \right)}\right) + 2 T \operatorname{UgA}{\left(x_{1} + x_{3} \right)}\right) \left(- x_{1} - x_{2} - x_{3} + 1\right)}{- x_{1} - x_{3} + 1}\right) + x_{3} \left(\frac{0.5 x_{1} \left(2 R \operatorname{UgA}{\left(x_{1} + x_{3} \right)} + 2 S \left(1 - \operatorname{UgA}{\left(x_{1} + x_{3} \right)}\right)\right) \operatorname{UgA}{\left(x_{1} + x_{3} \right)}}{x_{1} + x_{3}} + \frac{0.25 x_{2} \left(1 - \operatorname{UgA}{\left(x_{1} + x_{3} \right)}\right) \left(2 P \left(1 - \operatorname{UgA}{\left(x_{1} + x_{3} \right)}\right) + 2 T \operatorname{UgA}{\left(x_{1} + x_{3} \right)}\right)}{- x_{1} - x_{3} + 1} + \frac{1.0 x_{3} \left(2 R \operatorname{UgA}{\left(x_{1} + x_{3} \right)} + 2 S \left(1 - \operatorname{UgA}{\left(x_{1} + x_{3} \right)}\right)\right) \operatorname{UgA}{\left(x_{1} + x_{3} \right)}}{x_{1} + x_{3}} + \frac{0.5 \left(1 - \operatorname{UgA}{\left(x_{1} + x_{3} \right)}\right) \left(2 P \left(1 - \operatorname{UgA}{\left(x_{1} + x_{3} \right)}\right) + 2 T \operatorname{UgA}{\left(x_{1} + x_{3} \right)}\right) \left(- x_{1} - x_{2} - x_{3} + 1\right)}{- x_{1} - x_{3} + 1}\right) + \left(\frac{0.25 x_{1} \left(2 R \operatorname{UgA}{\left(x_{1} + x_{3} \right)} + 2 S \left(1 - \operatorname{UgA}{\left(x_{1} + x_{3} \right)}\right)\right) \operatorname{UgA}{\left(x_{1} + x_{3} \right)}}{x_{1} + x_{3}} + \frac{0.5 x_{2} \left(1 - \operatorname{UgA}{\left(x_{1} + x_{3} \right)}\right) \left(2 P \left(1 - \operatorname{UgA}{\left(x_{1} + x_{3} \right)}\right) + 2 T \operatorname{UgA}{\left(x_{1} + x_{3} \right)}\right)}{- x_{1} - x_{3} + 1}\right) \left(- x_{1} - x_{2} - x_{3} + 1\right) + \left(\frac{0.25 x_{1} \left(2 R \operatorname{UgA}{\left(x_{1} + x_{3} \right)} + 2 S \left(1 - \operatorname{UgA}{\left(x_{1} + x_{3} \right)}\right)\right) \operatorname{UgA}{\left(x_{1} + x_{3} \right)}}{x_{1} + x_{3}} + \frac{0.5 x_{3} \left(2 R \operatorname{UgA}{\left(x_{1} + x_{3} \right)} + 2 S \left(1 - \operatorname{UgA}{\left(x_{1} + x_{3} \right)}\right)\right) \operatorname{UgA}{\left(x_{1} + x_{3} \right)}}{x_{1} + x_{3}}\right) \left(- x_{1} - x_{2} - x_{3} + 1\right) + \left(- x_{1} - x_{2} - x_{3} + 1\right) \left(\frac{0.25 x_{1} \left(2 R \operatorname{UgA}{\left(x_{1} + x_{3} \right)} + 2 S \left(1 - \operatorname{UgA}{\left(x_{1} + x_{3} \right)}\right)\right) \operatorname{UgA}{\left(x_{1} + x_{3} \right)}}{x_{1} + x_{3}} + \frac{0.5 x_{2} \left(1 - \operatorname{UgA}{\left(x_{1} + x_{3} \right)}\right) \left(2 P \left(1 - \operatorname{UgA}{\left(x_{1} + x_{3} \right)}\right) + 2 T \operatorname{UgA}{\left(x_{1} + x_{3} \right)}\right)}{- x_{1} - x_{3} + 1} + \frac{0.5 x_{3} \left(2 R \operatorname{UgA}{\left(x_{1} + x_{3} \right)} + 2 S \left(1 - \operatorname{UgA}{\left(x_{1} + x_{3} \right)}\right)\right) \operatorname{UgA}{\left(x_{1} + x_{3} \right)}}{x_{1} + x_{3}} + \frac{1.0 \left(1 - \operatorname{UgA}{\left(x_{1} + x_{3} \right)}\right) \left(2 P \left(1 - \operatorname{UgA}{\left(x_{1} + x_{3} \right)}\right) + 2 T \operatorname{UgA}{\left(x_{1} + x_{3} \right)}\right) \left(- x_{1} - x_{2} - x_{3} + 1\right)}{- x_{1} - x_{3} + 1}\right)}\\- x_{2} + \frac{x_{1} \left(\frac{0.5 x_{2} \left(1 - \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \left(2 \left(1 - \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \left(- M + P\right) + 2 \left(- M + T\right) \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right)}{- x_{1} - x_{3} + 1} + \frac{0.25 \left(1 - \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \left(2 \left(1 - \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \left(- M + P\right) + 2 \left(- M + T\right) \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \left(- x_{1} - x_{2} - x_{3} + 1\right)}{- x_{1} - x_{3} + 1}\right) + \frac{0.25 x_{2} x_{3} \left(1 - \operatorname{UgA}{\left(x_{1} + x_{3} \right)}\right) \left(2 P \left(1 - \operatorname{UgA}{\left(x_{1} + x_{3} \right)}\right) + 2 T \operatorname{UgA}{\left(x_{1} + x_{3} \right)}\right)}{- x_{1} - x_{3} + 1} + x_{2} \left(\frac{0.5 x_{1} \left(2 \left(1 - \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \left(- M + S\right) + 2 \left(- M + R\right) \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \operatorname{UGA}{\left(x_{1} + x_{3} \right)}}{x_{1} + x_{3}} + \frac{1.0 x_{2} \left(1 - \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \left(2 \left(1 - \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \left(- M + P\right) + 2 \left(- M + T\right) \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right)}{- x_{1} - x_{3} + 1} + \frac{0.25 x_{3} \left(2 \left(1 - \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \left(- M + S\right) + 2 \left(- M + R\right) \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \operatorname{UGA}{\left(x_{1} + x_{3} \right)}}{x_{1} + x_{3}} + \frac{0.5 \left(1 - \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \left(2 \left(1 - \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \left(- M + P\right) + 2 \left(- M + T\right) \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \left(- x_{1} - x_{2} - x_{3} + 1\right)}{- x_{1} - x_{3} + 1}\right) + \left(\frac{0.25 x_{1} \left(2 R \operatorname{UgA}{\left(x_{1} + x_{3} \right)} + 2 S \left(1 - \operatorname{UgA}{\left(x_{1} + x_{3} \right)}\right)\right) \operatorname{UgA}{\left(x_{1} + x_{3} \right)}}{x_{1} + x_{3}} + \frac{0.5 x_{2} \left(1 - \operatorname{UgA}{\left(x_{1} + x_{3} \right)}\right) \left(2 P \left(1 - \operatorname{UgA}{\left(x_{1} + x_{3} \right)}\right) + 2 T \operatorname{UgA}{\left(x_{1} + x_{3} \right)}\right)}{- x_{1} - x_{3} + 1}\right) \left(- x_{1} - x_{2} - x_{3} + 1\right)}{\frac{0.25 x_{1} \left(1 - \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \left(2 \left(1 - \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \left(- M + P\right) + 2 \left(- M + T\right) \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \left(- x_{1} - x_{2} - x_{3} + 1\right)}{- x_{1} - x_{3} + 1} + x_{1} \left(\frac{0.5 x_{2} \left(1 - \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \left(2 \left(1 - \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \left(- M + P\right) + 2 \left(- M + T\right) \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right)}{- x_{1} - x_{3} + 1} + \frac{0.25 \left(1 - \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \left(2 \left(1 - \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \left(- M + P\right) + 2 \left(- M + T\right) \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \left(- x_{1} - x_{2} - x_{3} + 1\right)}{- x_{1} - x_{3} + 1}\right) + x_{1} \left(\frac{0.5 x_{3} \left(2 \left(1 - \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \left(- M + S\right) + 2 \left(- M + R\right) \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \operatorname{UGA}{\left(x_{1} + x_{3} \right)}}{x_{1} + x_{3}} + \frac{0.25 \left(1 - \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \left(2 \left(1 - \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \left(- M + P\right) + 2 \left(- M + T\right) \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \left(- x_{1} - x_{2} - x_{3} + 1\right)}{- x_{1} - x_{3} + 1}\right) + x_{1} \left(\frac{1.0 x_{1} \left(2 \left(1 - \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \left(- M + S\right) + 2 \left(- M + R\right) \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \operatorname{UGA}{\left(x_{1} + x_{3} \right)}}{x_{1} + x_{3}} + \frac{0.5 x_{2} \left(1 - \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \left(2 \left(1 - \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \left(- M + P\right) + 2 \left(- M + T\right) \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right)}{- x_{1} - x_{3} + 1} + \frac{0.5 x_{3} \left(2 \left(1 - \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \left(- M + S\right) + 2 \left(- M + R\right) \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \operatorname{UGA}{\left(x_{1} + x_{3} \right)}}{x_{1} + x_{3}} + \frac{0.25 \left(1 - \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \left(2 \left(1 - \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \left(- M + P\right) + 2 \left(- M + T\right) \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \left(- x_{1} - x_{2} - x_{3} + 1\right)}{- x_{1} - x_{3} + 1}\right) + \frac{0.25 x_{1} \left(2 R \operatorname{UgA}{\left(x_{1} + x_{3} \right)} + 2 S \left(1 - \operatorname{UgA}{\left(x_{1} + x_{3} \right)}\right)\right) \left(- x_{1} - x_{2} - x_{3} + 1\right) \operatorname{UgA}{\left(x_{1} + x_{3} \right)}}{x_{1} + x_{3}} + \frac{0.25 x_{2} x_{3} \left(1 - \operatorname{UgA}{\left(x_{1} + x_{3} \right)}\right) \left(2 P \left(1 - \operatorname{UgA}{\left(x_{1} + x_{3} \right)}\right) + 2 T \operatorname{UgA}{\left(x_{1} + x_{3} \right)}\right)}{- x_{1} - x_{3} + 1} + \frac{0.25 x_{2} x_{3} \left(2 \left(1 - \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \left(- M + S\right) + 2 \left(- M + R\right) \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \operatorname{UGA}{\left(x_{1} + x_{3} \right)}}{x_{1} + x_{3}} + x_{2} \left(\frac{0.5 x_{1} \left(2 \left(1 - \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \left(- M + S\right) + 2 \left(- M + R\right) \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \operatorname{UGA}{\left(x_{1} + x_{3} \right)}}{x_{1} + x_{3}} + \frac{0.25 x_{3} \left(2 \left(1 - \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \left(- M + S\right) + 2 \left(- M + R\right) \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \operatorname{UGA}{\left(x_{1} + x_{3} \right)}}{x_{1} + x_{3}}\right) + x_{2} \left(\frac{0.25 x_{3} \left(2 \left(1 - \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \left(- M + S\right) + 2 \left(- M + R\right) \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \operatorname{UGA}{\left(x_{1} + x_{3} \right)}}{x_{1} + x_{3}} + \frac{0.5 \left(1 - \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \left(2 \left(1 - \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \left(- M + P\right) + 2 \left(- M + T\right) \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \left(- x_{1} - x_{2} - x_{3} + 1\right)}{- x_{1} - x_{3} + 1}\right) + x_{2} \left(\frac{0.5 x_{1} \left(2 \left(1 - \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \left(- M + S\right) + 2 \left(- M + R\right) \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \operatorname{UGA}{\left(x_{1} + x_{3} \right)}}{x_{1} + x_{3}} + \frac{1.0 x_{2} \left(1 - \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \left(2 \left(1 - \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \left(- M + P\right) + 2 \left(- M + T\right) \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right)}{- x_{1} - x_{3} + 1} + \frac{0.25 x_{3} \left(2 \left(1 - \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \left(- M + S\right) + 2 \left(- M + R\right) \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \operatorname{UGA}{\left(x_{1} + x_{3} \right)}}{x_{1} + x_{3}} + \frac{0.5 \left(1 - \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \left(2 \left(1 - \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \left(- M + P\right) + 2 \left(- M + T\right) \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \left(- x_{1} - x_{2} - x_{3} + 1\right)}{- x_{1} - x_{3} + 1}\right) + x_{3} \left(\frac{0.5 x_{1} \left(2 R \operatorname{UgA}{\left(x_{1} + x_{3} \right)} + 2 S \left(1 - \operatorname{UgA}{\left(x_{1} + x_{3} \right)}\right)\right) \operatorname{UgA}{\left(x_{1} + x_{3} \right)}}{x_{1} + x_{3}} + \frac{0.25 x_{2} \left(1 - \operatorname{UgA}{\left(x_{1} + x_{3} \right)}\right) \left(2 P \left(1 - \operatorname{UgA}{\left(x_{1} + x_{3} \right)}\right) + 2 T \operatorname{UgA}{\left(x_{1} + x_{3} \right)}\right)}{- x_{1} - x_{3} + 1}\right) + x_{3} \left(\frac{0.25 x_{2} \left(1 - \operatorname{UgA}{\left(x_{1} + x_{3} \right)}\right) \left(2 P \left(1 - \operatorname{UgA}{\left(x_{1} + x_{3} \right)}\right) + 2 T \operatorname{UgA}{\left(x_{1} + x_{3} \right)}\right)}{- x_{1} - x_{3} + 1} + \frac{0.5 \left(1 - \operatorname{UgA}{\left(x_{1} + x_{3} \right)}\right) \left(2 P \left(1 - \operatorname{UgA}{\left(x_{1} + x_{3} \right)}\right) + 2 T \operatorname{UgA}{\left(x_{1} + x_{3} \right)}\right) \left(- x_{1} - x_{2} - x_{3} + 1\right)}{- x_{1} - x_{3} + 1}\right) + x_{3} \left(\frac{0.5 x_{1} \left(2 R \operatorname{UgA}{\left(x_{1} + x_{3} \right)} + 2 S \left(1 - \operatorname{UgA}{\left(x_{1} + x_{3} \right)}\right)\right) \operatorname{UgA}{\left(x_{1} + x_{3} \right)}}{x_{1} + x_{3}} + \frac{0.25 x_{2} \left(1 - \operatorname{UgA}{\left(x_{1} + x_{3} \right)}\right) \left(2 P \left(1 - \operatorname{UgA}{\left(x_{1} + x_{3} \right)}\right) + 2 T \operatorname{UgA}{\left(x_{1} + x_{3} \right)}\right)}{- x_{1} - x_{3} + 1} + \frac{1.0 x_{3} \left(2 R \operatorname{UgA}{\left(x_{1} + x_{3} \right)} + 2 S \left(1 - \operatorname{UgA}{\left(x_{1} + x_{3} \right)}\right)\right) \operatorname{UgA}{\left(x_{1} + x_{3} \right)}}{x_{1} + x_{3}} + \frac{0.5 \left(1 - \operatorname{UgA}{\left(x_{1} + x_{3} \right)}\right) \left(2 P \left(1 - \operatorname{UgA}{\left(x_{1} + x_{3} \right)}\right) + 2 T \operatorname{UgA}{\left(x_{1} + x_{3} \right)}\right) \left(- x_{1} - x_{2} - x_{3} + 1\right)}{- x_{1} - x_{3} + 1}\right) + \left(\frac{0.25 x_{1} \left(2 R \operatorname{UgA}{\left(x_{1} + x_{3} \right)} + 2 S \left(1 - \operatorname{UgA}{\left(x_{1} + x_{3} \right)}\right)\right) \operatorname{UgA}{\left(x_{1} + x_{3} \right)}}{x_{1} + x_{3}} + \frac{0.5 x_{2} \left(1 - \operatorname{UgA}{\left(x_{1} + x_{3} \right)}\right) \left(2 P \left(1 - \operatorname{UgA}{\left(x_{1} + x_{3} \right)}\right) + 2 T \operatorname{UgA}{\left(x_{1} + x_{3} \right)}\right)}{- x_{1} - x_{3} + 1}\right) \left(- x_{1} - x_{2} - x_{3} + 1\right) + \left(\frac{0.25 x_{1} \left(2 R \operatorname{UgA}{\left(x_{1} + x_{3} \right)} + 2 S \left(1 - \operatorname{UgA}{\left(x_{1} + x_{3} \right)}\right)\right) \operatorname{UgA}{\left(x_{1} + x_{3} \right)}}{x_{1} + x_{3}} + \frac{0.5 x_{3} \left(2 R \operatorname{UgA}{\left(x_{1} + x_{3} \right)} + 2 S \left(1 - \operatorname{UgA}{\left(x_{1} + x_{3} \right)}\right)\right) \operatorname{UgA}{\left(x_{1} + x_{3} \right)}}{x_{1} + x_{3}}\right) \left(- x_{1} - x_{2} - x_{3} + 1\right) + \left(- x_{1} - x_{2} - x_{3} + 1\right) \left(\frac{0.25 x_{1} \left(2 R \operatorname{UgA}{\left(x_{1} + x_{3} \right)} + 2 S \left(1 - \operatorname{UgA}{\left(x_{1} + x_{3} \right)}\right)\right) \operatorname{UgA}{\left(x_{1} + x_{3} \right)}}{x_{1} + x_{3}} + \frac{0.5 x_{2} \left(1 - \operatorname{UgA}{\left(x_{1} + x_{3} \right)}\right) \left(2 P \left(1 - \operatorname{UgA}{\left(x_{1} + x_{3} \right)}\right) + 2 T \operatorname{UgA}{\left(x_{1} + x_{3} \right)}\right)}{- x_{1} - x_{3} + 1} + \frac{0.5 x_{3} \left(2 R \operatorname{UgA}{\left(x_{1} + x_{3} \right)} + 2 S \left(1 - \operatorname{UgA}{\left(x_{1} + x_{3} \right)}\right)\right) \operatorname{UgA}{\left(x_{1} + x_{3} \right)}}{x_{1} + x_{3}} + \frac{1.0 \left(1 - \operatorname{UgA}{\left(x_{1} + x_{3} \right)}\right) \left(2 P \left(1 - \operatorname{UgA}{\left(x_{1} + x_{3} \right)}\right) + 2 T \operatorname{UgA}{\left(x_{1} + x_{3} \right)}\right) \left(- x_{1} - x_{2} - x_{3} + 1\right)}{- x_{1} - x_{3} + 1}\right)}\\- x_{3} + \frac{x_{1} \left(\frac{0.5 x_{3} \left(2 \left(1 - \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \left(- M + S\right) + 2 \left(- M + R\right) \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \operatorname{UGA}{\left(x_{1} + x_{3} \right)}}{x_{1} + x_{3}} + \frac{0.25 \left(1 - \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \left(2 \left(1 - \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \left(- M + P\right) + 2 \left(- M + T\right) \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \left(- x_{1} - x_{2} - x_{3} + 1\right)}{- x_{1} - x_{3} + 1}\right) + \frac{0.25 x_{2} x_{3} \left(2 \left(1 - \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \left(- M + S\right) + 2 \left(- M + R\right) \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \operatorname{UGA}{\left(x_{1} + x_{3} \right)}}{x_{1} + x_{3}} + x_{3} \left(\frac{0.5 x_{1} \left(2 R \operatorname{UgA}{\left(x_{1} + x_{3} \right)} + 2 S \left(1 - \operatorname{UgA}{\left(x_{1} + x_{3} \right)}\right)\right) \operatorname{UgA}{\left(x_{1} + x_{3} \right)}}{x_{1} + x_{3}} + \frac{0.25 x_{2} \left(1 - \operatorname{UgA}{\left(x_{1} + x_{3} \right)}\right) \left(2 P \left(1 - \operatorname{UgA}{\left(x_{1} + x_{3} \right)}\right) + 2 T \operatorname{UgA}{\left(x_{1} + x_{3} \right)}\right)}{- x_{1} - x_{3} + 1} + \frac{1.0 x_{3} \left(2 R \operatorname{UgA}{\left(x_{1} + x_{3} \right)} + 2 S \left(1 - \operatorname{UgA}{\left(x_{1} + x_{3} \right)}\right)\right) \operatorname{UgA}{\left(x_{1} + x_{3} \right)}}{x_{1} + x_{3}} + \frac{0.5 \left(1 - \operatorname{UgA}{\left(x_{1} + x_{3} \right)}\right) \left(2 P \left(1 - \operatorname{UgA}{\left(x_{1} + x_{3} \right)}\right) + 2 T \operatorname{UgA}{\left(x_{1} + x_{3} \right)}\right) \left(- x_{1} - x_{2} - x_{3} + 1\right)}{- x_{1} - x_{3} + 1}\right) + \left(\frac{0.25 x_{1} \left(2 R \operatorname{UgA}{\left(x_{1} + x_{3} \right)} + 2 S \left(1 - \operatorname{UgA}{\left(x_{1} + x_{3} \right)}\right)\right) \operatorname{UgA}{\left(x_{1} + x_{3} \right)}}{x_{1} + x_{3}} + \frac{0.5 x_{3} \left(2 R \operatorname{UgA}{\left(x_{1} + x_{3} \right)} + 2 S \left(1 - \operatorname{UgA}{\left(x_{1} + x_{3} \right)}\right)\right) \operatorname{UgA}{\left(x_{1} + x_{3} \right)}}{x_{1} + x_{3}}\right) \left(- x_{1} - x_{2} - x_{3} + 1\right)}{\frac{0.25 x_{1} \left(1 - \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \left(2 \left(1 - \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \left(- M + P\right) + 2 \left(- M + T\right) \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \left(- x_{1} - x_{2} - x_{3} + 1\right)}{- x_{1} - x_{3} + 1} + x_{1} \left(\frac{0.5 x_{2} \left(1 - \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \left(2 \left(1 - \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \left(- M + P\right) + 2 \left(- M + T\right) \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right)}{- x_{1} - x_{3} + 1} + \frac{0.25 \left(1 - \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \left(2 \left(1 - \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \left(- M + P\right) + 2 \left(- M + T\right) \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \left(- x_{1} - x_{2} - x_{3} + 1\right)}{- x_{1} - x_{3} + 1}\right) + x_{1} \left(\frac{0.5 x_{3} \left(2 \left(1 - \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \left(- M + S\right) + 2 \left(- M + R\right) \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \operatorname{UGA}{\left(x_{1} + x_{3} \right)}}{x_{1} + x_{3}} + \frac{0.25 \left(1 - \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \left(2 \left(1 - \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \left(- M + P\right) + 2 \left(- M + T\right) \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \left(- x_{1} - x_{2} - x_{3} + 1\right)}{- x_{1} - x_{3} + 1}\right) + x_{1} \left(\frac{1.0 x_{1} \left(2 \left(1 - \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \left(- M + S\right) + 2 \left(- M + R\right) \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \operatorname{UGA}{\left(x_{1} + x_{3} \right)}}{x_{1} + x_{3}} + \frac{0.5 x_{2} \left(1 - \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \left(2 \left(1 - \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \left(- M + P\right) + 2 \left(- M + T\right) \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right)}{- x_{1} - x_{3} + 1} + \frac{0.5 x_{3} \left(2 \left(1 - \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \left(- M + S\right) + 2 \left(- M + R\right) \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \operatorname{UGA}{\left(x_{1} + x_{3} \right)}}{x_{1} + x_{3}} + \frac{0.25 \left(1 - \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \left(2 \left(1 - \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \left(- M + P\right) + 2 \left(- M + T\right) \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \left(- x_{1} - x_{2} - x_{3} + 1\right)}{- x_{1} - x_{3} + 1}\right) + \frac{0.25 x_{1} \left(2 R \operatorname{UgA}{\left(x_{1} + x_{3} \right)} + 2 S \left(1 - \operatorname{UgA}{\left(x_{1} + x_{3} \right)}\right)\right) \left(- x_{1} - x_{2} - x_{3} + 1\right) \operatorname{UgA}{\left(x_{1} + x_{3} \right)}}{x_{1} + x_{3}} + \frac{0.25 x_{2} x_{3} \left(1 - \operatorname{UgA}{\left(x_{1} + x_{3} \right)}\right) \left(2 P \left(1 - \operatorname{UgA}{\left(x_{1} + x_{3} \right)}\right) + 2 T \operatorname{UgA}{\left(x_{1} + x_{3} \right)}\right)}{- x_{1} - x_{3} + 1} + \frac{0.25 x_{2} x_{3} \left(2 \left(1 - \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \left(- M + S\right) + 2 \left(- M + R\right) \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \operatorname{UGA}{\left(x_{1} + x_{3} \right)}}{x_{1} + x_{3}} + x_{2} \left(\frac{0.5 x_{1} \left(2 \left(1 - \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \left(- M + S\right) + 2 \left(- M + R\right) \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \operatorname{UGA}{\left(x_{1} + x_{3} \right)}}{x_{1} + x_{3}} + \frac{0.25 x_{3} \left(2 \left(1 - \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \left(- M + S\right) + 2 \left(- M + R\right) \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \operatorname{UGA}{\left(x_{1} + x_{3} \right)}}{x_{1} + x_{3}}\right) + x_{2} \left(\frac{0.25 x_{3} \left(2 \left(1 - \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \left(- M + S\right) + 2 \left(- M + R\right) \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \operatorname{UGA}{\left(x_{1} + x_{3} \right)}}{x_{1} + x_{3}} + \frac{0.5 \left(1 - \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \left(2 \left(1 - \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \left(- M + P\right) + 2 \left(- M + T\right) \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \left(- x_{1} - x_{2} - x_{3} + 1\right)}{- x_{1} - x_{3} + 1}\right) + x_{2} \left(\frac{0.5 x_{1} \left(2 \left(1 - \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \left(- M + S\right) + 2 \left(- M + R\right) \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \operatorname{UGA}{\left(x_{1} + x_{3} \right)}}{x_{1} + x_{3}} + \frac{1.0 x_{2} \left(1 - \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \left(2 \left(1 - \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \left(- M + P\right) + 2 \left(- M + T\right) \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right)}{- x_{1} - x_{3} + 1} + \frac{0.25 x_{3} \left(2 \left(1 - \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \left(- M + S\right) + 2 \left(- M + R\right) \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \operatorname{UGA}{\left(x_{1} + x_{3} \right)}}{x_{1} + x_{3}} + \frac{0.5 \left(1 - \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \left(2 \left(1 - \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \left(- M + P\right) + 2 \left(- M + T\right) \operatorname{UGA}{\left(x_{1} + x_{3} \right)}\right) \left(- x_{1} - x_{2} - x_{3} + 1\right)}{- x_{1} - x_{3} + 1}\right) + x_{3} \left(\frac{0.5 x_{1} \left(2 R \operatorname{UgA}{\left(x_{1} + x_{3} \right)} + 2 S \left(1 - \operatorname{UgA}{\left(x_{1} + x_{3} \right)}\right)\right) \operatorname{UgA}{\left(x_{1} + x_{3} \right)}}{x_{1} + x_{3}} + \frac{0.25 x_{2} \left(1 - \operatorname{UgA}{\left(x_{1} + x_{3} \right)}\right) \left(2 P \left(1 - \operatorname{UgA}{\left(x_{1} + x_{3} \right)}\right) + 2 T \operatorname{UgA}{\left(x_{1} + x_{3} \right)}\right)}{- x_{1} - x_{3} + 1}\right) + x_{3} \left(\frac{0.25 x_{2} \left(1 - \operatorname{UgA}{\left(x_{1} + x_{3} \right)}\right) \left(2 P \left(1 - \operatorname{UgA}{\left(x_{1} + x_{3} \right)}\right) + 2 T \operatorname{UgA}{\left(x_{1} + x_{3} \right)}\right)}{- x_{1} - x_{3} + 1} + \frac{0.5 \left(1 - \operatorname{UgA}{\left(x_{1} + x_{3} \right)}\right) \left(2 P \left(1 - \operatorname{UgA}{\left(x_{1} + x_{3} \right)}\right) + 2 T \operatorname{UgA}{\left(x_{1} + x_{3} \right)}\right) \left(- x_{1} - x_{2} - x_{3} + 1\right)}{- x_{1} - x_{3} + 1}\right) + x_{3} \left(\frac{0.5 x_{1} \left(2 R \operatorname{UgA}{\left(x_{1} + x_{3} \right)} + 2 S \left(1 - \operatorname{UgA}{\left(x_{1} + x_{3} \right)}\right)\right) \operatorname{UgA}{\left(x_{1} + x_{3} \right)}}{x_{1} + x_{3}} + \frac{0.25 x_{2} \left(1 - \operatorname{UgA}{\left(x_{1} + x_{3} \right)}\right) \left(2 P \left(1 - \operatorname{UgA}{\left(x_{1} + x_{3} \right)}\right) + 2 T \operatorname{UgA}{\left(x_{1} + x_{3} \right)}\right)}{- x_{1} - x_{3} + 1} + \frac{1.0 x_{3} \left(2 R \operatorname{UgA}{\left(x_{1} + x_{3} \right)} + 2 S \left(1 - \operatorname{UgA}{\left(x_{1} + x_{3} \right)}\right)\right) \operatorname{UgA}{\left(x_{1} + x_{3} \right)}}{x_{1} + x_{3}} + \frac{0.5 \left(1 - \operatorname{UgA}{\left(x_{1} + x_{3} \right)}\right) \left(2 P \left(1 - \operatorname{UgA}{\left(x_{1} + x_{3} \right)}\right) + 2 T \operatorname{UgA}{\left(x_{1} + x_{3} \right)}\right) \left(- x_{1} - x_{2} - x_{3} + 1\right)}{- x_{1} - x_{3} + 1}\right) + \left(\frac{0.25 x_{1} \left(2 R \operatorname{UgA}{\left(x_{1} + x_{3} \right)} + 2 S \left(1 - \operatorname{UgA}{\left(x_{1} + x_{3} \right)}\right)\right) \operatorname{UgA}{\left(x_{1} + x_{3} \right)}}{x_{1} + x_{3}} + \frac{0.5 x_{2} \left(1 - \operatorname{UgA}{\left(x_{1} + x_{3} \right)}\right) \left(2 P \left(1 - \operatorname{UgA}{\left(x_{1} + x_{3} \right)}\right) + 2 T \operatorname{UgA}{\left(x_{1} + x_{3} \right)}\right)}{- x_{1} - x_{3} + 1}\right) \left(- x_{1} - x_{2} - x_{3} + 1\right) + \left(\frac{0.25 x_{1} \left(2 R \operatorname{UgA}{\left(x_{1} + x_{3} \right)} + 2 S \left(1 - \operatorname{UgA}{\left(x_{1} + x_{3} \right)}\right)\right) \operatorname{UgA}{\left(x_{1} + x_{3} \right)}}{x_{1} + x_{3}} + \frac{0.5 x_{3} \left(2 R \operatorname{UgA}{\left(x_{1} + x_{3} \right)} + 2 S \left(1 - \operatorname{UgA}{\left(x_{1} + x_{3} \right)}\right)\right) \operatorname{UgA}{\left(x_{1} + x_{3} \right)}}{x_{1} + x_{3}}\right) \left(- x_{1} - x_{2} - x_{3} + 1\right) + \left(- x_{1} - x_{2} - x_{3} + 1\right) \left(\frac{0.25 x_{1} \left(2 R \operatorname{UgA}{\left(x_{1} + x_{3} \right)} + 2 S \left(1 - \operatorname{UgA}{\left(x_{1} + x_{3} \right)}\right)\right) \operatorname{UgA}{\left(x_{1} + x_{3} \right)}}{x_{1} + x_{3}} + \frac{0.5 x_{2} \left(1 - \operatorname{UgA}{\left(x_{1} + x_{3} \right)}\right) \left(2 P \left(1 - \operatorname{UgA}{\left(x_{1} + x_{3} \right)}\right) + 2 T \operatorname{UgA}{\left(x_{1} + x_{3} \right)}\right)}{- x_{1} - x_{3} + 1} + \frac{0.5 x_{3} \left(2 R \operatorname{UgA}{\left(x_{1} + x_{3} \right)} + 2 S \left(1 - \operatorname{UgA}{\left(x_{1} + x_{3} \right)}\right)\right) \operatorname{UgA}{\left(x_{1} + x_{3} \right)}}{x_{1} + x_{3}} + \frac{1.0 \left(1 - \operatorname{UgA}{\left(x_{1} + x_{3} \right)}\right) \left(2 P \left(1 - \operatorname{UgA}{\left(x_{1} + x_{3} \right)}\right) + 2 T \operatorname{UgA}{\left(x_{1} + x_{3} \right)}\right) \left(- x_{1} - x_{2} - x_{3} + 1\right)}{- x_{1} - x_{3} + 1}\right)}\end{matrix}\right]$

```python
F_jac = F.jacobian([x1, x2, x3])
```

```python
UG = sym.symbols("UG", positive=True, real=True)
Ug = sym.symbols("Ug", positive=True, real=True)
```

```python
UG = sym.symbols("UG", positive=True, real=True)
Ug = sym.symbols("Ug", positive=True, real=True)
Ug_prime = sym.symbols("Ug_prime", positive=True, real=True)

evaluated_F_jac = (F_jac.subs({x1: 0, x2: 0})
                        .subs({UGA(x3): UG, UgA(x3): Ug})
                        .doit()
                        .subs({sym.Derivative(UgA(x3), x3): Ug_prime})
                        .subs({x3: x3_star}))
```

```python
evaluated_F_jac
```

$\displaystyle \left[\begin{matrix}\frac{0.5 UG \left(2 UG \left(- M + R\right) + 2 \left(1 - UG\right) \left(- M + S\right)\right) + \frac{0.25 Ug \left(2 R Ug + 2 S \left(1 - Ug\right)\right) \left(- \frac{1.0 \left(2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug\right)}{2.0 P Ug^{2} - 4.0 P Ug + 2.0 P + 2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug - 2.0 T Ug^{2} + 2.0 T Ug} + 1\right) \left(2.0 P Ug^{2} - 4.0 P Ug + 2.0 P + 2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug - 2.0 T Ug^{2} + 2.0 T Ug\right)}{2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug} + 0.5 Ug \left(2 R Ug + 2 S \left(1 - Ug\right)\right) + 0.25 \left(1 - UG\right) \left(2 UG \left(- M + T\right) + 2 \left(1 - UG\right) \left(- M + P\right)\right)}{0.5 Ug \left(2 R Ug + 2 S \left(1 - Ug\right)\right) \left(- \frac{1.0 \left(2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug\right)}{2.0 P Ug^{2} - 4.0 P Ug + 2.0 P + 2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug - 2.0 T Ug^{2} + 2.0 T Ug} + 1\right) + \frac{0.5 \left(1 - Ug\right) \left(2 P \left(1 - Ug\right) + 2 T Ug\right) \left(2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug\right)}{2.0 P Ug^{2} - 4.0 P Ug + 2.0 P + 2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug - 2.0 T Ug^{2} + 2.0 T Ug} + \left(0.5 Ug \left(2 R Ug + 2 S \left(1 - Ug\right)\right) + 1.0 \left(1 - Ug\right) \left(2 P \left(1 - Ug\right) + 2 T Ug\right)\right) \left(- \frac{1.0 \left(2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug\right)}{2.0 P Ug^{2} - 4.0 P Ug + 2.0 P + 2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug - 2.0 T Ug^{2} + 2.0 T Ug} + 1\right) + \frac{1.0 \left(1.0 Ug \left(2 R Ug + 2 S \left(1 - Ug\right)\right) + 0.5 \left(1 - Ug\right) \left(2 P \left(1 - Ug\right) + 2 T Ug\right)\right) \left(2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug\right)}{2.0 P Ug^{2} - 4.0 P Ug + 2.0 P + 2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug - 2.0 T Ug^{2} + 2.0 T Ug}} - 1 & \frac{0.25 UG \left(2 UG \left(- M + R\right) + 2 \left(1 - UG\right) \left(- M + S\right)\right) + \frac{0.25 \left(1 - Ug\right) \left(2 P \left(1 - Ug\right) + 2 T Ug\right) \left(2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug\right)}{\left(- \frac{1.0 \left(2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug\right)}{2.0 P Ug^{2} - 4.0 P Ug + 2.0 P + 2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug - 2.0 T Ug^{2} + 2.0 T Ug} + 1\right) \left(2.0 P Ug^{2} - 4.0 P Ug + 2.0 P + 2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug - 2.0 T Ug^{2} + 2.0 T Ug\right)}}{0.5 Ug \left(2 R Ug + 2 S \left(1 - Ug\right)\right) \left(- \frac{1.0 \left(2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug\right)}{2.0 P Ug^{2} - 4.0 P Ug + 2.0 P + 2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug - 2.0 T Ug^{2} + 2.0 T Ug} + 1\right) + \frac{0.5 \left(1 - Ug\right) \left(2 P \left(1 - Ug\right) + 2 T Ug\right) \left(2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug\right)}{2.0 P Ug^{2} - 4.0 P Ug + 2.0 P + 2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug - 2.0 T Ug^{2} + 2.0 T Ug} + \left(0.5 Ug \left(2 R Ug + 2 S \left(1 - Ug\right)\right) + 1.0 \left(1 - Ug\right) \left(2 P \left(1 - Ug\right) + 2 T Ug\right)\right) \left(- \frac{1.0 \left(2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug\right)}{2.0 P Ug^{2} - 4.0 P Ug + 2.0 P + 2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug - 2.0 T Ug^{2} + 2.0 T Ug} + 1\right) + \frac{1.0 \left(1.0 Ug \left(2 R Ug + 2 S \left(1 - Ug\right)\right) + 0.5 \left(1 - Ug\right) \left(2 P \left(1 - Ug\right) + 2 T Ug\right)\right) \left(2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug\right)}{2.0 P Ug^{2} - 4.0 P Ug + 2.0 P + 2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug - 2.0 T Ug^{2} + 2.0 T Ug}} & 0\\\frac{\frac{0.25 Ug \left(2 R Ug + 2 S \left(1 - Ug\right)\right) \left(- \frac{1.0 \left(2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug\right)}{2.0 P Ug^{2} - 4.0 P Ug + 2.0 P + 2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug - 2.0 T Ug^{2} + 2.0 T Ug} + 1\right) \left(2.0 P Ug^{2} - 4.0 P Ug + 2.0 P + 2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug - 2.0 T Ug^{2} + 2.0 T Ug\right)}{2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug} + 0.25 \left(1 - UG\right) \left(2 UG \left(- M + T\right) + 2 \left(1 - UG\right) \left(- M + P\right)\right)}{0.5 Ug \left(2 R Ug + 2 S \left(1 - Ug\right)\right) \left(- \frac{1.0 \left(2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug\right)}{2.0 P Ug^{2} - 4.0 P Ug + 2.0 P + 2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug - 2.0 T Ug^{2} + 2.0 T Ug} + 1\right) + \frac{0.5 \left(1 - Ug\right) \left(2 P \left(1 - Ug\right) + 2 T Ug\right) \left(2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug\right)}{2.0 P Ug^{2} - 4.0 P Ug + 2.0 P + 2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug - 2.0 T Ug^{2} + 2.0 T Ug} + \left(0.5 Ug \left(2 R Ug + 2 S \left(1 - Ug\right)\right) + 1.0 \left(1 - Ug\right) \left(2 P \left(1 - Ug\right) + 2 T Ug\right)\right) \left(- \frac{1.0 \left(2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug\right)}{2.0 P Ug^{2} - 4.0 P Ug + 2.0 P + 2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug - 2.0 T Ug^{2} + 2.0 T Ug} + 1\right) + \frac{1.0 \left(1.0 Ug \left(2 R Ug + 2 S \left(1 - Ug\right)\right) + 0.5 \left(1 - Ug\right) \left(2 P \left(1 - Ug\right) + 2 T Ug\right)\right) \left(2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug\right)}{2.0 P Ug^{2} - 4.0 P Ug + 2.0 P + 2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug - 2.0 T Ug^{2} + 2.0 T Ug}} & \frac{0.25 UG \left(2 UG \left(- M + R\right) + 2 \left(1 - UG\right) \left(- M + S\right)\right) + 0.5 \left(1 - UG\right) \left(2 UG \left(- M + T\right) + 2 \left(1 - UG\right) \left(- M + P\right)\right) + 0.5 \left(1 - Ug\right) \left(2 P \left(1 - Ug\right) + 2 T Ug\right) + \frac{0.25 \left(1 - Ug\right) \left(2 P \left(1 - Ug\right) + 2 T Ug\right) \left(2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug\right)}{\left(- \frac{1.0 \left(2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug\right)}{2.0 P Ug^{2} - 4.0 P Ug + 2.0 P + 2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug - 2.0 T Ug^{2} + 2.0 T Ug} + 1\right) \left(2.0 P Ug^{2} - 4.0 P Ug + 2.0 P + 2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug - 2.0 T Ug^{2} + 2.0 T Ug\right)}}{0.5 Ug \left(2 R Ug + 2 S \left(1 - Ug\right)\right) \left(- \frac{1.0 \left(2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug\right)}{2.0 P Ug^{2} - 4.0 P Ug + 2.0 P + 2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug - 2.0 T Ug^{2} + 2.0 T Ug} + 1\right) + \frac{0.5 \left(1 - Ug\right) \left(2 P \left(1 - Ug\right) + 2 T Ug\right) \left(2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug\right)}{2.0 P Ug^{2} - 4.0 P Ug + 2.0 P + 2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug - 2.0 T Ug^{2} + 2.0 T Ug} + \left(0.5 Ug \left(2 R Ug + 2 S \left(1 - Ug\right)\right) + 1.0 \left(1 - Ug\right) \left(2 P \left(1 - Ug\right) + 2 T Ug\right)\right) \left(- \frac{1.0 \left(2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug\right)}{2.0 P Ug^{2} - 4.0 P Ug + 2.0 P + 2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug - 2.0 T Ug^{2} + 2.0 T Ug} + 1\right) + \frac{1.0 \left(1.0 Ug \left(2 R Ug + 2 S \left(1 - Ug\right)\right) + 0.5 \left(1 - Ug\right) \left(2 P \left(1 - Ug\right) + 2 T Ug\right)\right) \left(2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug\right)}{2.0 P Ug^{2} - 4.0 P Ug + 2.0 P + 2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug - 2.0 T Ug^{2} + 2.0 T Ug}} - 1 & 0\\\frac{\left(0.5 Ug \left(2 R Ug + 2 S \left(1 - Ug\right)\right) \left(- \frac{1.0 \left(2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug\right)}{2.0 P Ug^{2} - 4.0 P Ug + 2.0 P + 2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug - 2.0 T Ug^{2} + 2.0 T Ug} + 1\right) + \frac{1.0 \left(1.0 Ug \left(2 R Ug + 2 S \left(1 - Ug\right)\right) + 0.5 \left(1 - Ug\right) \left(2 P \left(1 - Ug\right) + 2 T Ug\right)\right) \left(2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug\right)}{2.0 P Ug^{2} - 4.0 P Ug + 2.0 P + 2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug - 2.0 T Ug^{2} + 2.0 T Ug}\right) \left(- 1.0 UG \left(2 UG \left(- M + R\right) + 2 \left(1 - UG\right) \left(- M + S\right)\right) - \frac{0.5 Ug \left(2 R Ug + 2 S \left(1 - Ug\right)\right) \left(- \frac{1.0 \left(2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug\right)}{2.0 P Ug^{2} - 4.0 P Ug + 2.0 P + 2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug - 2.0 T Ug^{2} + 2.0 T Ug} + 1\right) \left(2.0 P Ug^{2} - 4.0 P Ug + 2.0 P + 2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug - 2.0 T Ug^{2} + 2.0 T Ug\right)}{2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug} + 0.5 Ug \left(2 R Ug + 2 S \left(1 - Ug\right)\right) - 1.0 \left(1 - UG\right) \left(2 UG \left(- M + T\right) + 2 \left(1 - UG\right) \left(- M + P\right)\right) + 1.0 \left(1 - Ug\right) \left(2 P \left(1 - Ug\right) + 2 T Ug\right) - \frac{1.0 \left(- 0.5 Ug_{prime} \left(2 P \left(1 - Ug\right) + 2 T Ug\right) + 0.5 \left(1 - Ug\right) \left(- 2 P Ug_{prime} + 2 T Ug_{prime}\right)\right) \left(2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug\right)}{2.0 P Ug^{2} - 4.0 P Ug + 2.0 P + 2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug - 2.0 T Ug^{2} + 2.0 T Ug} - \left(- \frac{1.0 \left(2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug\right)}{2.0 P Ug^{2} - 4.0 P Ug + 2.0 P + 2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug - 2.0 T Ug^{2} + 2.0 T Ug} + 1\right) \left(- \frac{0.25 Ug \left(2 R Ug + 2 S \left(1 - Ug\right)\right) \left(2.0 P Ug^{2} - 4.0 P Ug + 2.0 P + 2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug - 2.0 T Ug^{2} + 2.0 T Ug\right)}{2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug} + 0.5 Ug \left(2 R Ug_{prime} - 2 S Ug_{prime}\right) + 0.5 Ug_{prime} \left(2 R Ug + 2 S \left(1 - Ug\right)\right)\right) - \left(- \frac{1.0 \left(2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug\right)}{2.0 P Ug^{2} - 4.0 P Ug + 2.0 P + 2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug - 2.0 T Ug^{2} + 2.0 T Ug} + 1\right) \left(- \frac{0.25 Ug \left(2 R Ug + 2 S \left(1 - Ug\right)\right) \left(2.0 P Ug^{2} - 4.0 P Ug + 2.0 P + 2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug - 2.0 T Ug^{2} + 2.0 T Ug\right)}{2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug} + 0.5 Ug \left(2 R Ug_{prime} - 2 S Ug_{prime}\right) - 1.0 Ug_{prime} \left(2 P \left(1 - Ug\right) + 2 T Ug\right) + 0.5 Ug_{prime} \left(2 R Ug + 2 S \left(1 - Ug\right)\right) + 1.0 \left(1 - Ug\right) \left(- 2 P Ug_{prime} + 2 T Ug_{prime}\right)\right) - \frac{1.0 \left(2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug\right) \left(- \frac{0.5 Ug \left(2 R Ug + 2 S \left(1 - Ug\right)\right) \left(2.0 P Ug^{2} - 4.0 P Ug + 2.0 P + 2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug - 2.0 T Ug^{2} + 2.0 T Ug\right)}{2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug} + 1.0 Ug \left(2 R Ug_{prime} - 2 S Ug_{prime}\right) - 0.5 Ug_{prime} \left(2 P \left(1 - Ug\right) + 2 T Ug\right) + 1.0 Ug_{prime} \left(2 R Ug + 2 S \left(1 - Ug\right)\right) + 0.5 \left(1 - Ug\right) \left(- 2 P Ug_{prime} + 2 T Ug_{prime}\right)\right)}{2.0 P Ug^{2} - 4.0 P Ug + 2.0 P + 2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug - 2.0 T Ug^{2} + 2.0 T Ug}\right)}{\left(0.5 Ug \left(2 R Ug + 2 S \left(1 - Ug\right)\right) \left(- \frac{1.0 \left(2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug\right)}{2.0 P Ug^{2} - 4.0 P Ug + 2.0 P + 2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug - 2.0 T Ug^{2} + 2.0 T Ug} + 1\right) + \frac{0.5 \left(1 - Ug\right) \left(2 P \left(1 - Ug\right) + 2 T Ug\right) \left(2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug\right)}{2.0 P Ug^{2} - 4.0 P Ug + 2.0 P + 2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug - 2.0 T Ug^{2} + 2.0 T Ug} + \left(0.5 Ug \left(2 R Ug + 2 S \left(1 - Ug\right)\right) + 1.0 \left(1 - Ug\right) \left(2 P \left(1 - Ug\right) + 2 T Ug\right)\right) \left(- \frac{1.0 \left(2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug\right)}{2.0 P Ug^{2} - 4.0 P Ug + 2.0 P + 2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug - 2.0 T Ug^{2} + 2.0 T Ug} + 1\right) + \frac{1.0 \left(1.0 Ug \left(2 R Ug + 2 S \left(1 - Ug\right)\right) + 0.5 \left(1 - Ug\right) \left(2 P \left(1 - Ug\right) + 2 T Ug\right)\right) \left(2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug\right)}{2.0 P Ug^{2} - 4.0 P Ug + 2.0 P + 2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug - 2.0 T Ug^{2} + 2.0 T Ug}\right)^{2}} + \frac{0.5 UG \left(2 UG \left(- M + R\right) + 2 \left(1 - UG\right) \left(- M + S\right)\right) - 0.5 Ug \left(2 R Ug + 2 S \left(1 - Ug\right)\right) + 0.25 \left(1 - UG\right) \left(2 UG \left(- M + T\right) + 2 \left(1 - UG\right) \left(- M + P\right)\right) + \left(- \frac{1.0 \left(2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug\right)}{2.0 P Ug^{2} - 4.0 P Ug + 2.0 P + 2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug - 2.0 T Ug^{2} + 2.0 T Ug} + 1\right) \left(- \frac{0.25 Ug \left(2 R Ug + 2 S \left(1 - Ug\right)\right) \left(2.0 P Ug^{2} - 4.0 P Ug + 2.0 P + 2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug - 2.0 T Ug^{2} + 2.0 T Ug\right)}{2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug} + 0.5 Ug \left(2 R Ug_{prime} - 2 S Ug_{prime}\right) + 0.5 Ug_{prime} \left(2 R Ug + 2 S \left(1 - Ug\right)\right)\right) + \frac{1.0 \left(2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug\right) \left(- \frac{0.5 Ug \left(2 R Ug + 2 S \left(1 - Ug\right)\right) \left(2.0 P Ug^{2} - 4.0 P Ug + 2.0 P + 2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug - 2.0 T Ug^{2} + 2.0 T Ug\right)}{2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug} + 1.0 Ug \left(2 R Ug_{prime} - 2 S Ug_{prime}\right) - 0.5 Ug_{prime} \left(2 P \left(1 - Ug\right) + 2 T Ug\right) + 1.0 Ug_{prime} \left(2 R Ug + 2 S \left(1 - Ug\right)\right) + 0.5 \left(1 - Ug\right) \left(- 2 P Ug_{prime} + 2 T Ug_{prime}\right)\right)}{2.0 P Ug^{2} - 4.0 P Ug + 2.0 P + 2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug - 2.0 T Ug^{2} + 2.0 T Ug}}{0.5 Ug \left(2 R Ug + 2 S \left(1 - Ug\right)\right) \left(- \frac{1.0 \left(2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug\right)}{2.0 P Ug^{2} - 4.0 P Ug + 2.0 P + 2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug - 2.0 T Ug^{2} + 2.0 T Ug} + 1\right) + \frac{0.5 \left(1 - Ug\right) \left(2 P \left(1 - Ug\right) + 2 T Ug\right) \left(2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug\right)}{2.0 P Ug^{2} - 4.0 P Ug + 2.0 P + 2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug - 2.0 T Ug^{2} + 2.0 T Ug} + \left(0.5 Ug \left(2 R Ug + 2 S \left(1 - Ug\right)\right) + 1.0 \left(1 - Ug\right) \left(2 P \left(1 - Ug\right) + 2 T Ug\right)\right) \left(- \frac{1.0 \left(2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug\right)}{2.0 P Ug^{2} - 4.0 P Ug + 2.0 P + 2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug - 2.0 T Ug^{2} + 2.0 T Ug} + 1\right) + \frac{1.0 \left(1.0 Ug \left(2 R Ug + 2 S \left(1 - Ug\right)\right) + 0.5 \left(1 - Ug\right) \left(2 P \left(1 - Ug\right) + 2 T Ug\right)\right) \left(2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug\right)}{2.0 P Ug^{2} - 4.0 P Ug + 2.0 P + 2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug - 2.0 T Ug^{2} + 2.0 T Ug}} & \frac{\left(0.5 Ug \left(2 R Ug + 2 S \left(1 - Ug\right)\right) \left(- \frac{1.0 \left(2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug\right)}{2.0 P Ug^{2} - 4.0 P Ug + 2.0 P + 2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug - 2.0 T Ug^{2} + 2.0 T Ug} + 1\right) + \frac{1.0 \left(1.0 Ug \left(2 R Ug + 2 S \left(1 - Ug\right)\right) + 0.5 \left(1 - Ug\right) \left(2 P \left(1 - Ug\right) + 2 T Ug\right)\right) \left(2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug\right)}{2.0 P Ug^{2} - 4.0 P Ug + 2.0 P + 2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug - 2.0 T Ug^{2} + 2.0 T Ug}\right) \left(- 1.0 UG \left(2 UG \left(- M + R\right) + 2 \left(1 - UG\right) \left(- M + S\right)\right) + 1.0 Ug \left(2 R Ug + 2 S \left(1 - Ug\right)\right) - 1.0 \left(1 - UG\right) \left(2 UG \left(- M + T\right) + 2 \left(1 - UG\right) \left(- M + P\right)\right) + 1.0 \left(1 - Ug\right) \left(2 P \left(1 - Ug\right) + 2 T Ug\right)\right)}{\left(0.5 Ug \left(2 R Ug + 2 S \left(1 - Ug\right)\right) \left(- \frac{1.0 \left(2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug\right)}{2.0 P Ug^{2} - 4.0 P Ug + 2.0 P + 2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug - 2.0 T Ug^{2} + 2.0 T Ug} + 1\right) + \frac{0.5 \left(1 - Ug\right) \left(2 P \left(1 - Ug\right) + 2 T Ug\right) \left(2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug\right)}{2.0 P Ug^{2} - 4.0 P Ug + 2.0 P + 2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug - 2.0 T Ug^{2} + 2.0 T Ug} + \left(0.5 Ug \left(2 R Ug + 2 S \left(1 - Ug\right)\right) + 1.0 \left(1 - Ug\right) \left(2 P \left(1 - Ug\right) + 2 T Ug\right)\right) \left(- \frac{1.0 \left(2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug\right)}{2.0 P Ug^{2} - 4.0 P Ug + 2.0 P + 2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug - 2.0 T Ug^{2} + 2.0 T Ug} + 1\right) + \frac{1.0 \left(1.0 Ug \left(2 R Ug + 2 S \left(1 - Ug\right)\right) + 0.5 \left(1 - Ug\right) \left(2 P \left(1 - Ug\right) + 2 T Ug\right)\right) \left(2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug\right)}{2.0 P Ug^{2} - 4.0 P Ug + 2.0 P + 2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug - 2.0 T Ug^{2} + 2.0 T Ug}\right)^{2}} + \frac{0.25 UG \left(2 UG \left(- M + R\right) + 2 \left(1 - UG\right) \left(- M + S\right)\right) - 0.5 Ug \left(2 R Ug + 2 S \left(1 - Ug\right)\right) - \frac{0.25 \left(1 - Ug\right) \left(2 P \left(1 - Ug\right) + 2 T Ug\right) \left(2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug\right)}{\left(- \frac{1.0 \left(2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug\right)}{2.0 P Ug^{2} - 4.0 P Ug + 2.0 P + 2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug - 2.0 T Ug^{2} + 2.0 T Ug} + 1\right) \left(2.0 P Ug^{2} - 4.0 P Ug + 2.0 P + 2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug - 2.0 T Ug^{2} + 2.0 T Ug\right)}}{0.5 Ug \left(2 R Ug + 2 S \left(1 - Ug\right)\right) \left(- \frac{1.0 \left(2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug\right)}{2.0 P Ug^{2} - 4.0 P Ug + 2.0 P + 2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug - 2.0 T Ug^{2} + 2.0 T Ug} + 1\right) + \frac{0.5 \left(1 - Ug\right) \left(2 P \left(1 - Ug\right) + 2 T Ug\right) \left(2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug\right)}{2.0 P Ug^{2} - 4.0 P Ug + 2.0 P + 2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug - 2.0 T Ug^{2} + 2.0 T Ug} + \left(0.5 Ug \left(2 R Ug + 2 S \left(1 - Ug\right)\right) + 1.0 \left(1 - Ug\right) \left(2 P \left(1 - Ug\right) + 2 T Ug\right)\right) \left(- \frac{1.0 \left(2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug\right)}{2.0 P Ug^{2} - 4.0 P Ug + 2.0 P + 2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug - 2.0 T Ug^{2} + 2.0 T Ug} + 1\right) + \frac{1.0 \left(1.0 Ug \left(2 R Ug + 2 S \left(1 - Ug\right)\right) + 0.5 \left(1 - Ug\right) \left(2 P \left(1 - Ug\right) + 2 T Ug\right)\right) \left(2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug\right)}{2.0 P Ug^{2} - 4.0 P Ug + 2.0 P + 2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug - 2.0 T Ug^{2} + 2.0 T Ug}} & \frac{\left(0.5 Ug \left(2 R Ug + 2 S \left(1 - Ug\right)\right) \left(- \frac{1.0 \left(2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug\right)}{2.0 P Ug^{2} - 4.0 P Ug + 2.0 P + 2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug - 2.0 T Ug^{2} + 2.0 T Ug} + 1\right) + \frac{1.0 \left(1.0 Ug \left(2 R Ug + 2 S \left(1 - Ug\right)\right) + 0.5 \left(1 - Ug\right) \left(2 P \left(1 - Ug\right) + 2 T Ug\right)\right) \left(2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug\right)}{2.0 P Ug^{2} - 4.0 P Ug + 2.0 P + 2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug - 2.0 T Ug^{2} + 2.0 T Ug}\right) \left(- \left(0.5 Ug \left(2 R Ug_{prime} - 2 S Ug_{prime}\right) + 0.5 Ug_{prime} \left(2 R Ug + 2 S \left(1 - Ug\right)\right)\right) \left(- \frac{1.0 \left(2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug\right)}{2.0 P Ug^{2} - 4.0 P Ug + 2.0 P + 2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug - 2.0 T Ug^{2} + 2.0 T Ug} + 1\right) - \frac{1.0 \left(- 0.5 Ug_{prime} \left(2 P \left(1 - Ug\right) + 2 T Ug\right) + 0.5 \left(1 - Ug\right) \left(- 2 P Ug_{prime} + 2 T Ug_{prime}\right)\right) \left(2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug\right)}{2.0 P Ug^{2} - 4.0 P Ug + 2.0 P + 2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug - 2.0 T Ug^{2} + 2.0 T Ug} - \left(- \frac{1.0 \left(2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug\right)}{2.0 P Ug^{2} - 4.0 P Ug + 2.0 P + 2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug - 2.0 T Ug^{2} + 2.0 T Ug} + 1\right) \left(0.5 Ug \left(2 R Ug_{prime} - 2 S Ug_{prime}\right) - 1.0 Ug_{prime} \left(2 P \left(1 - Ug\right) + 2 T Ug\right) + 0.5 Ug_{prime} \left(2 R Ug + 2 S \left(1 - Ug\right)\right) + 1.0 \left(1 - Ug\right) \left(- 2 P Ug_{prime} + 2 T Ug_{prime}\right)\right) - \frac{1.0 \left(2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug\right) \left(1.0 Ug \left(2 R Ug_{prime} - 2 S Ug_{prime}\right) - 0.5 Ug_{prime} \left(2 P \left(1 - Ug\right) + 2 T Ug\right) + 1.0 Ug_{prime} \left(2 R Ug + 2 S \left(1 - Ug\right)\right) + 0.5 \left(1 - Ug\right) \left(- 2 P Ug_{prime} + 2 T Ug_{prime}\right)\right)}{2.0 P Ug^{2} - 4.0 P Ug + 2.0 P + 2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug - 2.0 T Ug^{2} + 2.0 T Ug}\right)}{\left(0.5 Ug \left(2 R Ug + 2 S \left(1 - Ug\right)\right) \left(- \frac{1.0 \left(2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug\right)}{2.0 P Ug^{2} - 4.0 P Ug + 2.0 P + 2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug - 2.0 T Ug^{2} + 2.0 T Ug} + 1\right) + \frac{0.5 \left(1 - Ug\right) \left(2 P \left(1 - Ug\right) + 2 T Ug\right) \left(2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug\right)}{2.0 P Ug^{2} - 4.0 P Ug + 2.0 P + 2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug - 2.0 T Ug^{2} + 2.0 T Ug} + \left(0.5 Ug \left(2 R Ug + 2 S \left(1 - Ug\right)\right) + 1.0 \left(1 - Ug\right) \left(2 P \left(1 - Ug\right) + 2 T Ug\right)\right) \left(- \frac{1.0 \left(2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug\right)}{2.0 P Ug^{2} - 4.0 P Ug + 2.0 P + 2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug - 2.0 T Ug^{2} + 2.0 T Ug} + 1\right) + \frac{1.0 \left(1.0 Ug \left(2 R Ug + 2 S \left(1 - Ug\right)\right) + 0.5 \left(1 - Ug\right) \left(2 P \left(1 - Ug\right) + 2 T Ug\right)\right) \left(2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug\right)}{2.0 P Ug^{2} - 4.0 P Ug + 2.0 P + 2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug - 2.0 T Ug^{2} + 2.0 T Ug}\right)^{2}} + \frac{0.5 Ug \left(2 R Ug + 2 S \left(1 - Ug\right)\right) + 0.5 \left(1 - Ug\right) \left(2 P \left(1 - Ug\right) + 2 T Ug\right) + \left(0.5 Ug \left(2 R Ug_{prime} - 2 S Ug_{prime}\right) + 0.5 Ug_{prime} \left(2 R Ug + 2 S \left(1 - Ug\right)\right)\right) \left(- \frac{1.0 \left(2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug\right)}{2.0 P Ug^{2} - 4.0 P Ug + 2.0 P + 2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug - 2.0 T Ug^{2} + 2.0 T Ug} + 1\right) + \frac{1.0 \left(2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug\right) \left(1.0 Ug \left(2 R Ug_{prime} - 2 S Ug_{prime}\right) - 0.5 Ug_{prime} \left(2 P \left(1 - Ug\right) + 2 T Ug\right) + 1.0 Ug_{prime} \left(2 R Ug + 2 S \left(1 - Ug\right)\right) + 0.5 \left(1 - Ug\right) \left(- 2 P Ug_{prime} + 2 T Ug_{prime}\right)\right)}{2.0 P Ug^{2} - 4.0 P Ug + 2.0 P + 2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug - 2.0 T Ug^{2} + 2.0 T Ug}}{0.5 Ug \left(2 R Ug + 2 S \left(1 - Ug\right)\right) \left(- \frac{1.0 \left(2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug\right)}{2.0 P Ug^{2} - 4.0 P Ug + 2.0 P + 2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug - 2.0 T Ug^{2} + 2.0 T Ug} + 1\right) + \frac{0.5 \left(1 - Ug\right) \left(2 P \left(1 - Ug\right) + 2 T Ug\right) \left(2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug\right)}{2.0 P Ug^{2} - 4.0 P Ug + 2.0 P + 2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug - 2.0 T Ug^{2} + 2.0 T Ug} + \left(0.5 Ug \left(2 R Ug + 2 S \left(1 - Ug\right)\right) + 1.0 \left(1 - Ug\right) \left(2 P \left(1 - Ug\right) + 2 T Ug\right)\right) \left(- \frac{1.0 \left(2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug\right)}{2.0 P Ug^{2} - 4.0 P Ug + 2.0 P + 2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug - 2.0 T Ug^{2} + 2.0 T Ug} + 1\right) + \frac{1.0 \left(1.0 Ug \left(2 R Ug + 2 S \left(1 - Ug\right)\right) + 0.5 \left(1 - Ug\right) \left(2 P \left(1 - Ug\right) + 2 T Ug\right)\right) \left(2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug\right)}{2.0 P Ug^{2} - 4.0 P Ug + 2.0 P + 2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug - 2.0 T Ug^{2} + 2.0 T Ug}} - 1\end{matrix}\right]$

```python
simplified_F_jac = sym.zeros(3, 3)

for i in range(3):
    for j in range(3):
        simplified_F_jac[i, j] = sym.simplify(evaluated_F_jac[i, j])
        #simplified_F_jac[i, j] = sym.factor(sym.cancel(sym.together(evaluated_F_jac[i, j])))

        print("Finished with element {},{}!".format(i,j))

```

    Finished with element 0,0!
    Finished with element 0,1!
    Finished with element 0,2!
    Finished with element 1,0!
    Finished with element 1,1!
    Finished with element 1,2!
    Finished with element 2,0!
    Finished with element 2,1!
    Finished with element 2,2!

```python
simplified_F_jac
```

$\displaystyle \left[\begin{matrix}\frac{- 1.0 M UG - 1.0 M + 1.0 P UG^{2} - 2.0 P UG - 3.0 P Ug^{2} + 6.0 P Ug - 2.0 P + 2.0 R UG^{2} - 2.0 R Ug^{2} - 2.0 S UG^{2} + 2.0 S UG + 2.0 S Ug^{2} - 2.0 S Ug - 1.0 T UG^{2} + 1.0 T UG + 3.0 T Ug^{2} - 3.0 T Ug}{4.0 P Ug^{2} - 8.0 P Ug + 4.0 P + 4.0 R Ug^{2} - 4.0 S Ug^{2} + 4.0 S Ug - 4.0 T Ug^{2} + 4.0 T Ug} & \frac{- 2.0 M UG + 2.0 R UG^{2} + 2.0 R Ug^{2} - 2.0 S UG^{2} + 2.0 S UG - 2.0 S Ug^{2} + 2.0 S Ug}{8.0 P Ug^{2} - 16.0 P Ug + 8.0 P + 8.0 R Ug^{2} - 8.0 S Ug^{2} + 8.0 S Ug - 8.0 T Ug^{2} + 8.0 T Ug} & 0\\\frac{1.0 \left(1.0 M UG - 1.0 M + 1.0 P UG^{2} - 2.0 P UG + 1.0 P Ug^{2} - 2.0 P Ug + 2.0 P - 1.0 T UG^{2} + 1.0 T UG - 1.0 T Ug^{2} + 1.0 T Ug\right)}{4.0 P Ug^{2} - 8.0 P Ug + 4.0 P + 4.0 R Ug^{2} - 4.0 S Ug^{2} + 4.0 S Ug - 4.0 T Ug^{2} + 4.0 T Ug} & \frac{1.0 \left(2.0 M UG - 4.0 M + 4.0 P UG^{2} - 8.0 P UG - 4.0 P Ug^{2} + 8.0 P Ug + 2.0 R UG^{2} - 6.0 R Ug^{2} - 2.0 S UG^{2} + 2.0 S UG + 6.0 S Ug^{2} - 6.0 S Ug - 4.0 T UG^{2} + 4.0 T UG + 4.0 T Ug^{2} - 4.0 T Ug\right)}{8.0 P Ug^{2} - 16.0 P Ug + 8.0 P + 8.0 R Ug^{2} - 8.0 S Ug^{2} + 8.0 S Ug - 8.0 T Ug^{2} + 8.0 T Ug} & 0\\\frac{- 4.0 M P UG Ug^{2} + 8.0 M P UG Ug - 4.0 M P UG - 4.0 M P Ug^{2} + 8.0 M P Ug - 4.0 M P - 4.0 M R UG Ug^{2} + 12.0 M R Ug^{2} + 4.0 M S UG Ug^{2} - 4.0 M S UG Ug - 12.0 M S Ug^{2} + 12.0 M S Ug + 4.0 M T UG Ug^{2} - 4.0 M T UG Ug + 4.0 M T Ug^{2} - 4.0 M T Ug + 4.0 P^{2} UG^{2} Ug^{2} - 8.0 P^{2} UG^{2} Ug + 4.0 P^{2} UG^{2} - 8.0 P^{2} UG Ug^{2} + 16.0 P^{2} UG Ug - 8.0 P^{2} UG - 4.0 P^{2} Ug^{4} + 16.0 P^{2} Ug^{3} - 20.0 P^{2} Ug^{2} + 8.0 P^{2} Ug - 4.0 P R UG^{2} Ug^{2} - 16.0 P R UG^{2} Ug + 8.0 P R UG^{2} + 24.0 P R UG Ug^{2} - 4.0 P R Ug^{4} + 8.0 P R Ug^{3} - 16.0 P R Ug^{2} Ug_{prime} - 16.0 P R Ug^{2} + 16.0 P R Ug Ug_{prime} + 4.0 P S UG^{2} Ug^{2} + 4.0 P S UG^{2} Ug - 8.0 P S UG^{2} - 16.0 P S UG Ug^{2} + 8.0 P S UG Ug + 8.0 P S UG + 4.0 P S Ug^{4} - 12.0 P S Ug^{3} + 8.0 P S Ug^{2} Ug_{prime} + 24.0 P S Ug^{2} - 16.0 P S Ug Ug_{prime} - 16.0 P S Ug + 8.0 P S Ug_{prime} - 8.0 P T UG^{2} Ug^{2} + 12.0 P T UG^{2} Ug - 4.0 P T UG^{2} + 12.0 P T UG Ug^{2} - 16.0 P T UG Ug + 4.0 P T UG + 8.0 P T Ug^{4} - 24.0 P T Ug^{3} + 20.0 P T Ug^{2} - 4.0 P T Ug - 8.0 R^{2} UG^{2} Ug^{2} + 16.0 R S UG^{2} Ug^{2} - 8.0 R S UG^{2} Ug - 8.0 R S UG Ug^{2} + 4.0 R T UG^{2} Ug^{2} + 8.0 R T UG^{2} Ug - 12.0 R T UG Ug^{2} + 4.0 R T Ug^{4} - 4.0 R T Ug^{3} + 8.0 R T Ug^{2} Ug_{prime} - 8.0 S^{2} UG^{2} Ug^{2} + 8.0 S^{2} UG^{2} Ug + 8.0 S^{2} UG Ug^{2} - 8.0 S^{2} UG Ug - 4.0 S T UG^{2} Ug^{2} + 4.0 S T UG^{2} Ug + 4.0 S T UG Ug^{2} - 4.0 S T UG Ug - 4.0 S T Ug^{4} + 8.0 S T Ug^{3} - 4.0 S T Ug^{2} + 4.0 T^{2} UG^{2} Ug^{2} - 4.0 T^{2} UG^{2} Ug - 4.0 T^{2} UG Ug^{2} + 4.0 T^{2} UG Ug - 4.0 T^{2} Ug^{4} + 8.0 T^{2} Ug^{3} - 4.0 T^{2} Ug^{2}}{16.0 P^{2} Ug^{4} - 64.0 P^{2} Ug^{3} + 96.0 P^{2} Ug^{2} - 64.0 P^{2} Ug + 16.0 P^{2} + 32.0 P R Ug^{4} - 64.0 P R Ug^{3} + 32.0 P R Ug^{2} - 32.0 P S Ug^{4} + 96.0 P S Ug^{3} - 96.0 P S Ug^{2} + 32.0 P S Ug - 32.0 P T Ug^{4} + 96.0 P T Ug^{3} - 96.0 P T Ug^{2} + 32.0 P T Ug + 16.0 R^{2} Ug^{4} - 32.0 R S Ug^{4} + 32.0 R S Ug^{3} - 32.0 R T Ug^{4} + 32.0 R T Ug^{3} + 16.0 S^{2} Ug^{4} - 32.0 S^{2} Ug^{3} + 16.0 S^{2} Ug^{2} + 32.0 S T Ug^{4} - 64.0 S T Ug^{3} + 32.0 S T Ug^{2} + 16.0 T^{2} Ug^{4} - 32.0 T^{2} Ug^{3} + 16.0 T^{2} Ug^{2}} & \frac{- 8.0 M P UG Ug^{2} + 16.0 M P UG Ug - 8.0 M P UG - 8.0 M R UG Ug^{2} + 32.0 M R Ug^{2} + 8.0 M S UG Ug^{2} - 8.0 M S UG Ug - 32.0 M S Ug^{2} + 32.0 M S Ug + 8.0 M T UG Ug^{2} - 8.0 M T UG Ug - 24.0 P R UG^{2} Ug^{2} - 16.0 P R UG^{2} Ug + 8.0 P R UG^{2} + 64.0 P R UG Ug^{2} + 8.0 P R Ug^{4} - 16.0 P R Ug^{3} - 24.0 P R Ug^{2} + 24.0 P S UG^{2} Ug^{2} - 16.0 P S UG^{2} Ug - 8.0 P S UG^{2} - 56.0 P S UG Ug^{2} + 48.0 P S UG Ug + 8.0 P S UG - 8.0 P S Ug^{4} + 24.0 P S Ug^{3} + 8.0 P S Ug^{2} - 24.0 P S Ug - 24.0 R^{2} UG^{2} Ug^{2} + 8.0 R^{2} Ug^{4} + 48.0 R S UG^{2} Ug^{2} - 24.0 R S UG^{2} Ug - 24.0 R S UG Ug^{2} - 16.0 R S Ug^{4} + 16.0 R S Ug^{3} + 24.0 R T UG^{2} Ug^{2} + 8.0 R T UG^{2} Ug - 32.0 R T UG Ug^{2} - 8.0 R T Ug^{4} + 8.0 R T Ug^{3} - 24.0 S^{2} UG^{2} Ug^{2} + 24.0 S^{2} UG^{2} Ug + 24.0 S^{2} UG Ug^{2} - 24.0 S^{2} UG Ug + 8.0 S^{2} Ug^{4} - 16.0 S^{2} Ug^{3} + 8.0 S^{2} Ug^{2} - 24.0 S T UG^{2} Ug^{2} + 24.0 S T UG^{2} Ug + 24.0 S T UG Ug^{2} - 24.0 S T UG Ug + 8.0 S T Ug^{4} - 16.0 S T Ug^{3} + 8.0 S T Ug^{2}}{32.0 P^{2} Ug^{4} - 128.0 P^{2} Ug^{3} + 192.0 P^{2} Ug^{2} - 128.0 P^{2} Ug + 32.0 P^{2} + 64.0 P R Ug^{4} - 128.0 P R Ug^{3} + 64.0 P R Ug^{2} - 64.0 P S Ug^{4} + 192.0 P S Ug^{3} - 192.0 P S Ug^{2} + 64.0 P S Ug - 64.0 P T Ug^{4} + 192.0 P T Ug^{3} - 192.0 P T Ug^{2} + 64.0 P T Ug + 32.0 R^{2} Ug^{4} - 64.0 R S Ug^{4} + 64.0 R S Ug^{3} - 64.0 R T Ug^{4} + 64.0 R T Ug^{3} + 32.0 S^{2} Ug^{4} - 64.0 S^{2} Ug^{3} + 32.0 S^{2} Ug^{2} + 64.0 S T Ug^{4} - 128.0 S T Ug^{3} + 64.0 S T Ug^{2} + 32.0 T^{2} Ug^{4} - 64.0 T^{2} Ug^{3} + 32.0 T^{2} Ug^{2}} & \frac{- 4.0 P^{2} Ug^{4} + 16.0 P^{2} Ug^{3} - 24.0 P^{2} Ug^{2} + 16.0 P^{2} Ug - 4.0 P^{2} - 8.0 P R Ug^{4} + 16.0 P R Ug^{3} - 8.0 P R Ug^{2} Ug_{prime} - 8.0 P R Ug^{2} + 8.0 P R Ug Ug_{prime} + 8.0 P S Ug^{4} - 24.0 P S Ug^{3} + 4.0 P S Ug^{2} Ug_{prime} + 24.0 P S Ug^{2} - 8.0 P S Ug Ug_{prime} - 8.0 P S Ug + 4.0 P S Ug_{prime} + 8.0 P T Ug^{4} - 24.0 P T Ug^{3} + 24.0 P T Ug^{2} - 8.0 P T Ug - 4.0 R^{2} Ug^{4} + 8.0 R S Ug^{4} - 8.0 R S Ug^{3} + 8.0 R T Ug^{4} - 8.0 R T Ug^{3} + 4.0 R T Ug^{2} Ug_{prime} - 4.0 S^{2} Ug^{4} + 8.0 S^{2} Ug^{3} - 4.0 S^{2} Ug^{2} - 8.0 S T Ug^{4} + 16.0 S T Ug^{3} - 8.0 S T Ug^{2} - 4.0 T^{2} Ug^{4} + 8.0 T^{2} Ug^{3} - 4.0 T^{2} Ug^{2}}{8.0 P^{2} Ug^{4} - 32.0 P^{2} Ug^{3} + 48.0 P^{2} Ug^{2} - 32.0 P^{2} Ug + 8.0 P^{2} + 16.0 P R Ug^{4} - 32.0 P R Ug^{3} + 16.0 P R Ug^{2} - 16.0 P S Ug^{4} + 48.0 P S Ug^{3} - 48.0 P S Ug^{2} + 16.0 P S Ug - 16.0 P T Ug^{4} + 48.0 P T Ug^{3} - 48.0 P T Ug^{2} + 16.0 P T Ug + 8.0 R^{2} Ug^{4} - 16.0 R S Ug^{4} + 16.0 R S Ug^{3} - 16.0 R T Ug^{4} + 16.0 R T Ug^{3} + 8.0 S^{2} Ug^{4} - 16.0 S^{2} Ug^{3} + 8.0 S^{2} Ug^{2} + 16.0 S T Ug^{4} - 32.0 S T Ug^{3} + 16.0 S T Ug^{2} + 8.0 T^{2} Ug^{4} - 16.0 T^{2} Ug^{3} + 8.0 T^{2} Ug^{2}}\end{matrix}\right]$

```python
n, d = sym.fraction(simplified_F_jac[2,0])
```

```python
p = sym.poly(n, M)
```

```python
c0, c1 = p.coeffs()
```

```python
sym.factor(c0, Ug)
```

$\displaystyle - 12.0 \left(0.333333333333333 P UG + 0.333333333333333 P + Ug^{2} \left(0.333333333333333 P UG + 0.333333333333333 P + 0.333333333333333 R UG - 1.0 R - 0.333333333333333 S UG + 1.0 S - 0.333333333333333 T UG - 0.333333333333333 T\right) + Ug \left(- 0.666666666666667 P UG - 0.666666666666667 P + 0.333333333333333 S UG - 1.0 S + 0.333333333333333 T UG + 0.333333333333333 T\right)\right)$

```python
sym.factor(c1, Ug)
```

$\displaystyle - 24.0 \left(- 0.166666666666667 P^{2} UG^{2} + 0.333333333333333 P^{2} UG - 0.333333333333333 P R UG^{2} + 0.333333333333333 P S UG^{2} - 0.333333333333333 P S UG - 0.333333333333333 P S Ug_{prime} + 0.166666666666667 P T UG^{2} - 0.166666666666667 P T UG + Ug^{4} \left(0.166666666666667 P^{2} + 0.166666666666667 P R - 0.166666666666667 P S - 0.333333333333333 P T - 0.166666666666667 R T + 0.166666666666667 S T + 0.166666666666667 T^{2}\right) + Ug^{3} \left(- 0.666666666666667 P^{2} - 0.333333333333333 P R + 0.5 P S + 1.0 P T + 0.166666666666667 R T - 0.333333333333333 S T - 0.333333333333333 T^{2}\right) + Ug^{2} \left(- 0.166666666666667 P^{2} UG^{2} + 0.333333333333333 P^{2} UG + 0.833333333333333 P^{2} + 0.166666666666667 P R UG^{2} - 1.0 P R UG + 0.666666666666667 P R Ug_{prime} + 0.666666666666667 P R - 0.166666666666667 P S UG^{2} + 0.666666666666667 P S UG - 0.333333333333333 P S Ug_{prime} - 1.0 P S + 0.333333333333333 P T UG^{2} - 0.5 P T UG - 0.833333333333333 P T + 0.333333333333333 R^{2} UG^{2} - 0.666666666666667 R S UG^{2} + 0.333333333333333 R S UG - 0.166666666666667 R T UG^{2} + 0.5 R T UG - 0.333333333333333 R T Ug_{prime} + 0.333333333333333 S^{2} UG^{2} - 0.333333333333333 S^{2} UG + 0.166666666666667 S T UG^{2} - 0.166666666666667 S T UG + 0.166666666666667 S T - 0.166666666666667 T^{2} UG^{2} + 0.166666666666667 T^{2} UG + 0.166666666666667 T^{2}\right) + Ug \left(0.333333333333333 P^{2} UG^{2} - 0.666666666666667 P^{2} UG - 0.333333333333333 P^{2} + 0.666666666666667 P R UG^{2} - 0.666666666666667 P R Ug_{prime} - 0.166666666666667 P S UG^{2} - 0.333333333333333 P S UG + 0.666666666666667 P S Ug_{prime} + 0.666666666666667 P S - 0.5 P T UG^{2} + 0.666666666666667 P T UG + 0.166666666666667 P T + 0.333333333333333 R S UG^{2} - 0.333333333333333 R T UG^{2} - 0.333333333333333 S^{2} UG^{2} + 0.333333333333333 S^{2} UG - 0.166666666666667 S T UG^{2} + 0.166666666666667 S T UG + 0.166666666666667 T^{2} UG^{2} - 0.166666666666667 T^{2} UG\right)\right)$

```python
sym.factor(d, UG, Ug, Ug_prime)
```

$\displaystyle 64.0 \left(0.5 P + Ug^{2} \left(0.5 P + 0.5 R - 0.5 S - 0.5 T\right) + Ug \left(- 1.0 P + 0.5 S + 0.5 T\right)\right)^{2}$

```python
charpoly = simplified_F_jac.charpoly()
```

```python
eigenvals = simplified_F_jac.eigenvals()
```

```python
e0, e1, e2 = eigenvals.keys()
```

```python
e0
```

$\displaystyle - \frac{P^{2} Ug^{4} - 4 P^{2} Ug^{3} + 6 P^{2} Ug^{2} - 4 P^{2} Ug + P^{2} + 2 P R Ug^{4} - 4 P R Ug^{3} + 2 P R Ug^{2} Ug_{prime} + 2 P R Ug^{2} - 2 P R Ug Ug_{prime} - 2 P S Ug^{4} + 6 P S Ug^{3} - P S Ug^{2} Ug_{prime} - 6 P S Ug^{2} + 2 P S Ug Ug_{prime} + 2 P S Ug - P S Ug_{prime} - 2 P T Ug^{4} + 6 P T Ug^{3} - 6 P T Ug^{2} + 2 P T Ug + R^{2} Ug^{4} - 2 R S Ug^{4} + 2 R S Ug^{3} - 2 R T Ug^{4} + 2 R T Ug^{3} - R T Ug^{2} Ug_{prime} + S^{2} Ug^{4} - 2 S^{2} Ug^{3} + S^{2} Ug^{2} + 2 S T Ug^{4} - 4 S T Ug^{3} + 2 S T Ug^{2} + T^{2} Ug^{4} - 2 T^{2} Ug^{3} + T^{2} Ug^{2}}{2 P^{2} Ug^{4} - 8 P^{2} Ug^{3} + 12 P^{2} Ug^{2} - 8 P^{2} Ug + 2 P^{2} + 4 P R Ug^{4} - 8 P R Ug^{3} + 4 P R Ug^{2} - 4 P S Ug^{4} + 12 P S Ug^{3} - 12 P S Ug^{2} + 4 P S Ug - 4 P T Ug^{4} + 12 P T Ug^{3} - 12 P T Ug^{2} + 4 P T Ug + 2 R^{2} Ug^{4} - 4 R S Ug^{4} + 4 R S Ug^{3} - 4 R T Ug^{4} + 4 R T Ug^{3} + 2 S^{2} Ug^{4} - 4 S^{2} Ug^{3} + 2 S^{2} Ug^{2} + 4 S T Ug^{4} - 8 S T Ug^{3} + 4 S T Ug^{2} + 2 T^{2} Ug^{4} - 4 T^{2} Ug^{3} + 2 T^{2} Ug^{2}}$

```python
n, d = sym.fraction(e0)
```

```python
d
```

$\displaystyle 2 P^{2} Ug^{4} - 8 P^{2} Ug^{3} + 12 P^{2} Ug^{2} - 8 P^{2} Ug + 2 P^{2} + 4 P R Ug^{4} - 8 P R Ug^{3} + 4 P R Ug^{2} - 4 P S Ug^{4} + 12 P S Ug^{3} - 12 P S Ug^{2} + 4 P S Ug - 4 P T Ug^{4} + 12 P T Ug^{3} - 12 P T Ug^{2} + 4 P T Ug + 2 R^{2} Ug^{4} - 4 R S Ug^{4} + 4 R S Ug^{3} - 4 R T Ug^{4} + 4 R T Ug^{3} + 2 S^{2} Ug^{4} - 4 S^{2} Ug^{3} + 2 S^{2} Ug^{2} + 4 S T Ug^{4} - 8 S T Ug^{3} + 4 S T Ug^{2} + 2 T^{2} Ug^{4} - 4 T^{2} Ug^{3} + 2 T^{2} Ug^{2}$

```python
sym.factor(sym.poly(d, Ug))
```

$\displaystyle 2 \left(P Ug^{2} - 2 P Ug + P + R Ug^{2} - S Ug^{2} + S Ug - T Ug^{2} + T Ug\right)^{2}$

```python
sym.factor(n, Ug_prime)
```

$\displaystyle - P^{2} Ug^{4} + 4 P^{2} Ug^{3} - 6 P^{2} Ug^{2} + 4 P^{2} Ug - P^{2} - 2 P R Ug^{4} + 4 P R Ug^{3} - 2 P R Ug^{2} + 2 P S Ug^{4} - 6 P S Ug^{3} + 6 P S Ug^{2} - 2 P S Ug + 2 P T Ug^{4} - 6 P T Ug^{3} + 6 P T Ug^{2} - 2 P T Ug - R^{2} Ug^{4} + 2 R S Ug^{4} - 2 R S Ug^{3} + 2 R T Ug^{4} - 2 R T Ug^{3} - S^{2} Ug^{4} + 2 S^{2} Ug^{3} - S^{2} Ug^{2} - 2 S T Ug^{4} + 4 S T Ug^{3} - 2 S T Ug^{2} - T^{2} Ug^{4} + 2 T^{2} Ug^{3} - T^{2} Ug^{2} - Ug_{prime} \left(2 P R Ug^{2} - 2 P R Ug - P S Ug^{2} + 2 P S Ug - P S - R T Ug^{2}\right)$

```python
sym.factor(-P**2*Ug**4 + 4*P**2*Ug**3 - 6*P**2*Ug**2 + 4*P**2*Ug - P**2 - 2*P*R*Ug**4 + 4*P*R*Ug**3 - 2*P*R*Ug**2 + 2*P*S*Ug**4 - 6*P*S*Ug**3 + 6*P*S*Ug**2 - 2*P*S*Ug + 2*P*T*Ug**4 - 6*P*T*Ug**3 + 6*P*T*Ug**2 - 2*P*T*Ug - R**2*Ug**4 + 2*R*S*Ug**4 - 2*R*S*Ug**3 + 2*R*T*Ug**4 - 2*R*T*Ug**3 - S**2*Ug**4 + 2*S**2*Ug**3 - S**2*Ug**2 - 2*S*T*Ug**4 + 4*S*T*Ug**3 - 2*S*T*Ug**2 - T**2*Ug**4 + 2*T**2*Ug**3 - T**2*Ug**2)

```

$\displaystyle - \left(P Ug^{2} - 2 P Ug + P + R Ug^{2} - S Ug^{2} + S Ug - T Ug^{2} + T Ug\right)^{2}$

```python
sym.factor(2*P*R*Ug**2 - 2*P*R*Ug - P*S*Ug**2 + 2*P*S*Ug - P*S - R*T*Ug**2, Ug)
```

$\displaystyle - P S + Ug^{2} \left(2 P R - P S - R T\right) + Ug \left(- 2 P R + 2 P S\right)$

```python
e1
```

$\displaystyle - \frac{M - P UG^{2} + 2 P UG + P Ug^{2} - 2 P Ug - R UG^{2} + R Ug^{2} + S UG^{2} - S UG - S Ug^{2} + S Ug + T UG^{2} - T UG - T Ug^{2} + T Ug}{2 P Ug^{2} - 4 P Ug + 2 P + 2 R Ug^{2} - 2 S Ug^{2} + 2 S Ug - 2 T Ug^{2} + 2 T Ug}$

```python
e2
```

$\displaystyle - \frac{M - P UG^{2} + 2 P UG + 3 P Ug^{2} - 6 P Ug + 2 P - R UG^{2} + 3 R Ug^{2} + S UG^{2} - S UG - 3 S Ug^{2} + 3 S Ug + T UG^{2} - T UG - 3 T Ug^{2} + 3 T Ug}{4 P Ug^{2} - 8 P Ug + 4 P + 4 R Ug^{2} - 4 S Ug^{2} + 4 S Ug - 4 T Ug^{2} + 4 T Ug}$

```python
import pickle

with open('non-invadable-eigenvalues.pickle', 'wb') as handle:
    pickle.dump(eigenvals, handle, protocol=pickle.HIGHEST_PROTOCOL)
```

```python
e1, e2, e3 = eigenvals.keys()
```

```python
numerator, denominator = sym.fraction(sym.factor(e1))
```

```python
denominator
```

    2*(P*UGA_star**2 - 2*P*UGA_star + P + R*UGA_star**2 - S*UGA_star**2 + S*UGA_star - T*UGA_star**2 + T*UGA_star)**2

```python
upper_bound, = sym.solve(numerator, UGA_prime_star)
```

```python
sym.factor(upper_bound)
```

    (2*epsilon + 1)*(P*UGA_star**2 - 2*P*UGA_star + P + R*UGA_star**2 - S*UGA_star**2 + S*UGA_star - T*UGA_star**2 + T*UGA_star)**2/((2*epsilon - 1)*(2*P*R*UGA_star**2 - 2*P*R*UGA_star - P*S*UGA_star**2 + 2*P*S*UGA_star - P*S - R*T*UGA_star**2))

```python
e1.subs({UGA_star: 1, UGA_prime_star: 0})
```

    -(2*R**2*epsilon + R**2)/(2*R**2)

```python
# vectorized numeric repr for the eigenvalue
_numeric_e2 = sym.lambdify((UGA_star, UgA_star, T, R, P, S, epsilon),
                           e2.subs({x1: x1_star}),
                           modules="numpy")
```

```python
def plot_second_eigenvalue(T, R, P, S, epsilon):

    fig, ax = plt.subplots(1, 1, figsize=(10, 10))
    ax.set_ylabel(r"$U_{GA}^*$", fontsize=20, rotation="horizontal")
    ax.set_xlabel(r"$U_{gA}^{*}$", fontsize=20)
    ax.set_title(r"Eigenvalue, $e_2$", fontsize=25)
    ax.grid("off")

    equilibrium_selection_probs = np.linspace(0, 1, 100)
    UgAs, UGAs = np.meshgrid(equilibrium_selection_probs, equilibrium_selection_probs)
    Z = _numeric_e2(UGAs, UgAs, T, R, P, S, epsilon)
    cax = ax.imshow(Z, origin="lower")
    contours = ax.contour(Z, colors='w', origin='lower')
    ax.clabel(contours, contours.levels, inline=True, fontsize=10)

    # adjust the tick labels
    locs, _ = plt.xticks()
    plt.xticks(locs[1:], np.linspace(0, 1, locs.size-1))
    locs, _ = plt.yticks()
    plt.yticks(locs[1:], np.linspace(0, 1, locs.size-1))

    plt.show()
```

```python
mpld3.disable_notebook()  # don't need interactive plotting for this!

# sliders used to control the Prisoner's Dilemma Payoffs
T_slider = widgets.FloatSlider(value=10, min=0, max=100, step=0.1, description=r"$T$")
R_slider = widgets.FloatSlider(value=8, min=0, max=100, step=0.1, description=r"$R$")
P_slider = widgets.FloatSlider(value=6, min=0, max=100, step=0.1, description=r"$P$")
S_slider = widgets.FloatSlider(value=4, min=0, max=100, step=0.1, description=r"$S$")

# slider used to control the mutation rate
e_slider = widgets.FloatSlider(value=0.0, min=0.0, max=1.0, step=1e-3, description=r"$\epsilon$", readout_format=".3f")

w = widgets.interactive(plot_second_eigenvalue, T=T_slider, R=R_slider, P=P_slider, S=S_slider, epsilon=e_slider)
display(w)
```

    A Jupyter Widget

```python
evaluated = (e2.subs({x1: x1_star, UgA_star: x1_star})
               .subs({UGA_star: 1, UGA_prime_star: 0}))
together = sym.together(evaluated)
```

```python
expanded = sym.expand(together)
```

```python
expanded
```

    -4.0*P*epsilon**4/(16.0*R*epsilon**3 + 16.0*R*epsilon**2 + 4.0*R*epsilon) + 6.0*P*epsilon**3/(16.0*R*epsilon**3 + 16.0*R*epsilon**2 + 4.0*R*epsilon) - 12.0*R*epsilon**3/(16.0*R*epsilon**3 + 16.0*R*epsilon**2 + 4.0*R*epsilon) - 13.0*R*epsilon**2/(16.0*R*epsilon**3 + 16.0*R*epsilon**2 + 4.0*R*epsilon) - 1.5*R*epsilon/(16.0*R*epsilon**3 + 16.0*R*epsilon**2 + 4.0*R*epsilon) - 2.0*S*epsilon**3/(16.0*R*epsilon**3 + 16.0*R*epsilon**2 + 4.0*R*epsilon) + 3.0*S*epsilon**2/(16.0*R*epsilon**3 + 16.0*R*epsilon**2 + 4.0*R*epsilon) - 2.0*T*epsilon**3/(16.0*R*epsilon**3 + 16.0*R*epsilon**2 + 4.0*R*epsilon) + 3.0*T*epsilon**2/(16.0*R*epsilon**3 + 16.0*R*epsilon**2 + 4.0*R*epsilon) + 0.015625*sqrt(65536.0*P**2*epsilon**8 + 65536.0*P**2*epsilon**7 + 16384.0*P**2*epsilon**6 + 524288.0*P*R*epsilon**8 + 655360.0*P*R*epsilon**7 + 229376.0*P*R*epsilon**6 - 8192.0*P*R*epsilon**4 + 65536.0*P*S*epsilon**7 + 65536.0*P*S*epsilon**6 + 16384.0*P*S*epsilon**5 + 65536.0*P*T*epsilon**7 + 65536.0*P*T*epsilon**6 + 16384.0*P*T*epsilon**5 + 65536.0*R**2*epsilon**6 + 163840.0*R**2*epsilon**5 + 151552.0*R**2*epsilon**4 + 61440.0*R**2*epsilon**3 + 9216.0*R**2*epsilon**2 + 65536.0*R*S*epsilon**6 + 114688.0*R*S*epsilon**5 + 65536.0*R*S*epsilon**4 + 12288.0*R*S*epsilon**3 + 262144.0*R*T*epsilon**7 + 327680.0*R*T*epsilon**6 + 114688.0*R*T*epsilon**5 - 4096.0*R*T*epsilon**3 + 16384.0*S**2*epsilon**6 + 16384.0*S**2*epsilon**5 + 4096.0*S**2*epsilon**4 + 32768.0*S*T*epsilon**6 + 32768.0*S*T*epsilon**5 + 8192.0*S*T*epsilon**4 + 16384.0*T**2*epsilon**6 + 16384.0*T**2*epsilon**5 + 4096.0*T**2*epsilon**4)/(16.0*R*epsilon**3 + 16.0*R*epsilon**2 + 4.0*R*epsilon)

```python
tmp_expr = sym.together(expanded)
```

```python
tmp_expr
```

    (-4.0*P*epsilon**3 + 6.0*P*epsilon**2 - 12.0*R*epsilon**2 - 13.0*R*epsilon - 1.5*R - 2.0*S*epsilon**2 + 3.0*S*epsilon - 2.0*T*epsilon**2 + 3.0*T*epsilon + 0.015625*sqrt(65536.0*P**2*epsilon**6 + 65536.0*P**2*epsilon**5 + 16384.0*P**2*epsilon**4 + 524288.0*P*R*epsilon**6 + 655360.0*P*R*epsilon**5 + 229376.0*P*R*epsilon**4 - 8192.0*P*R*epsilon**2 + 65536.0*P*S*epsilon**5 + 65536.0*P*S*epsilon**4 + 16384.0*P*S*epsilon**3 + 65536.0*P*T*epsilon**5 + 65536.0*P*T*epsilon**4 + 16384.0*P*T*epsilon**3 + 65536.0*R**2*epsilon**4 + 163840.0*R**2*epsilon**3 + 151552.0*R**2*epsilon**2 + 61440.0*R**2*epsilon + 9216.0*R**2 + 65536.0*R*S*epsilon**4 + 114688.0*R*S*epsilon**3 + 65536.0*R*S*epsilon**2 + 12288.0*R*S*epsilon + 262144.0*R*T*epsilon**5 + 327680.0*R*T*epsilon**4 + 114688.0*R*T*epsilon**3 - 4096.0*R*T*epsilon + 16384.0*S**2*epsilon**4 + 16384.0*S**2*epsilon**3 + 4096.0*S**2*epsilon**2 + 32768.0*S*T*epsilon**4 + 32768.0*S*T*epsilon**3 + 8192.0*S*T*epsilon**2 + 16384.0*T**2*epsilon**4 + 16384.0*T**2*epsilon**3 + 4096.0*T**2*epsilon**2))/(R*(16.0*epsilon**2 + 16.0*epsilon + 4.0))

```python
tmp_expr.subs({epsilon**2: 0})
```

    (-13.0*R*epsilon - 1.5*R + 3.0*S*epsilon + 3.0*T*epsilon + 0.015625*sqrt(61440.0*R**2*epsilon + 9216.0*R**2 + 12288.0*R*S*epsilon - 4096.0*R*T*epsilon))/(R*(16.0*epsilon + 4.0))

\begin{align}
   % (-13.0*R*epsilon - 1.5*R + 3.0*S*epsilon + 3.0*T*epsilon + 0.015625*sqrt(61440.0*R**2*epsilon + 9216.0*R**2 + 12288.0*R*S*epsilon - 4096.0*R*T*epsilon))/(R*(16.0*epsilon + 4.0)) < 0 \\
    \frac{(-13R\epsilon - 1.5R + 3S\epsilon + 3T\epsilon + 0.015625\sqrt{61440R^2\epsilon + 9216R^2 + 12288RS\epsilon - 4096RT\epsilon}}{R(16\epsilon + 4.0)} < 0 \\
    \frac{(-13R\epsilon - \frac{3}{2}R + 3S\epsilon + 3T\epsilon + \frac{1}{64}\sqrt{61440R^2\epsilon + 9216R^2 + 12288RS\epsilon - 4096RT\epsilon}}{R(16\epsilon + 4.0)} < 0 \\
    \frac{(-13R\epsilon - \frac{3}{2}R + 3S\epsilon + 3T\epsilon + \frac{1}{2}\sqrt{60R^2\epsilon + 9R^2 + 12RS\epsilon - 4RT\epsilon}}{R(16\epsilon + 4)} < 0
\end{align}

```python
sym.together(sym.expand(sym.together(e2.subs({x1: x1_star, UgA_star: x1_star})))).subs({epsilon**2: 0})
```

    (192.0*P**5*UGA_star**10*epsilon - 1920.0*P**5*UGA_star**9*epsilon + 8544.0*P**5*UGA_star**8*epsilon - 22272.0*P**5*UGA_star**7*epsilon + 37632.0*P**5*UGA_star**6*epsilon - 43008.0*P**5*UGA_star**5*epsilon + 33600.0*P**5*UGA_star**4*epsilon - 17664.0*P**5*UGA_star**3*epsilon + 5952.0*P**5*UGA_star**2*epsilon - 1152.0*P**5*UGA_star*epsilon + 96.0*P**5*epsilon + 864.0*P**4*R*UGA_star**10*epsilon + 80.0*P**4*R*UGA_star**10 - 6912.0*P**4*R*UGA_star**9*epsilon - 640.0*P**4*R*UGA_star**9 + 24224.0*P**4*R*UGA_star**8*epsilon + 2192.0*P**4*R*UGA_star**8 - 48576.0*P**4*R*UGA_star**7*epsilon - 4192.0*P**4*R*UGA_star**7 + 60960.0*P**4*R*UGA_star**6*epsilon + 4880.0*P**4*R*UGA_star**6 - 49024.0*P**4*R*UGA_star**5*epsilon - 3520.0*P**4*R*UGA_star**5 + 24672.0*P**4*R*UGA_star**4*epsilon + 1520.0*P**4*R*UGA_star**4 - 7104.0*P**4*R*UGA_star**3*epsilon - 352.0*P**4*R*UGA_star**3 + 896.0*P**4*R*UGA_star**2*epsilon + 32.0*P**4*R*UGA_star**2 - 864.0*P**4*S*UGA_star**10*epsilon - 80.0*P**4*S*UGA_star**10 + 7776.0*P**4*S*UGA_star**9*epsilon + 720.0*P**4*S*UGA_star**9 - 31136.0*P**4*S*UGA_star**8*epsilon - 2832.0*P**4*S*UGA_star**8 + 72800.0*P**4*S*UGA_star**7*epsilon + 6384.0*P**4*S*UGA_star**7 - 109536.0*P**4*S*UGA_star**6*epsilon - 9072.0*P**4*S*UGA_star**6 + 109984.0*P**4*S*UGA_star**5*epsilon + 8400.0*P**4*S*UGA_star**5 - 73696.0*P**4*S*UGA_star**4*epsilon - 5040.0*P**4*S*UGA_star**4 + 31776.0*P**4*S*UGA_star**3*epsilon + 1872.0*P**4*S*UGA_star**3 - 8000.0*P**4*S*UGA_star**2*epsilon - 384.0*P**4*S*UGA_star**2 + 896.0*P**4*S*UGA_star*epsilon + 32.0*P**4*S*UGA_star - 960.0*P**4*T*UGA_star**10*epsilon + 8640.0*P**4*T*UGA_star**9*epsilon - 34176.0*P**4*T*UGA_star**8*epsilon + 77952.0*P**4*T*UGA_star**7*epsilon - 112896.0*P**4*T*UGA_star**6*epsilon + 107520.0*P**4*T*UGA_star**5*epsilon - 67200.0*P**4*T*UGA_star**4*epsilon + 26496.0*P**4*T*UGA_star**3*epsilon - 5952.0*P**4*T*UGA_star**2*epsilon + 576.0*P**4*T*UGA_star*epsilon + 1632.0*P**3*R**2*UGA_star**10*epsilon + 240.0*P**3*R**2*UGA_star**10 - 9792.0*P**3*R**2*UGA_star**9*epsilon - 1440.0*P**3*R**2*UGA_star**9 + 24192.0*P**3*R**2*UGA_star**8*epsilon + 3600.0*P**3*R**2*UGA_star**8 - 31488.0*P**3*R**2*UGA_star**7*epsilon - 4800.0*P**3*R**2*UGA_star**7 + 22752.0*P**3*R**2*UGA_star**6*epsilon + 3600.0*P**3*R**2*UGA_star**6 - 8640.0*P**3*R**2*UGA_star**5*epsilon - 1440.0*P**3*R**2*UGA_star**5 + 1344.0*P**3*R**2*UGA_star**4*epsilon + 240.0*P**3*R**2*UGA_star**4 - 3264.0*P**3*R*S*UGA_star**10*epsilon - 480.0*P**3*R*S*UGA_star**10 + 22848.0*P**3*R*S*UGA_star**9*epsilon + 3360.0*P**3*R*S*UGA_star**9 - 68160.0*P**3*R*S*UGA_star**8*epsilon - 10080.0*P**3*R*S*UGA_star**8 + 112512.0*P**3*R*S*UGA_star**7*epsilon + 16800.0*P**3*R*S*UGA_star**7 - 111360.0*P**3*R*S*UGA_star**6*epsilon - 16800.0*P**3*R*S*UGA_star**6 + 66624.0*P**3*R*S*UGA_star**5*epsilon + 10080.0*P**3*R*S*UGA_star**5 - 22848.0*P**3*R*S*UGA_star**4*epsilon - 3360.0*P**3*R*S*UGA_star**4 + 3840.0*P**3*R*S*UGA_star**3*epsilon + 480.0*P**3*R*S*UGA_star**3 - 192.0*P**3*R*S*UGA_star**2*epsilon - 3456.0*P**3*R*T*UGA_star**10*epsilon - 320.0*P**3*R*T*UGA_star**10 + 24192.0*P**3*R*T*UGA_star**9*epsilon + 2240.0*P**3*R*T*UGA_star**9 - 72864.0*P**3*R*T*UGA_star**8*epsilon - 6576.0*P**3*R*T*UGA_star**8 + 122592.0*P**3*R*T*UGA_star**7*epsilon + 10480.0*P**3*R*T*UGA_star**7 - 124800.0*P**3*R*T*UGA_star**6*epsilon - 9760.0*P**3*R*T*UGA_star**6 + 77376.0*P**3*R*T*UGA_star**5*epsilon + 5280.0*P**3*R*T*UGA_star**5 - 27552.0*P**3*R*T*UGA_star**4*epsilon - 1520.0*P**3*R*T*UGA_star**4 + 4704.0*P**3*R*T*UGA_star**3*epsilon + 176.0*P**3*R*T*UGA_star**3 - 192.0*P**3*R*T*UGA_star**2*epsilon + 1632.0*P**3*S**2*UGA_star**10*epsilon + 240.0*P**3*S**2*UGA_star**10 - 13056.0*P**3*S**2*UGA_star**9*epsilon - 1920.0*P**3*S**2*UGA_star**9 + 45600.0*P**3*S**2*UGA_star**8*epsilon + 6720.0*P**3*S**2*UGA_star**8 - 91008.0*P**3*S**2*UGA_star**7*epsilon - 13440.0*P**3*S**2*UGA_star**7 + 113952.0*P**3*S**2*UGA_star**6*epsilon + 16800.0*P**3*S**2*UGA_star**6 - 92352.0*P**3*S**2*UGA_star**5*epsilon - 13440.0*P**3*S**2*UGA_star**5 + 48096.0*P**3*S**2*UGA_star**4*epsilon + 6720.0*P**3*S**2*UGA_star**4 - 15360.0*P**3*S**2*UGA_star**3*epsilon - 1920.0*P**3*S**2*UGA_star**3 + 2688.0*P**3*S**2*UGA_star**2*epsilon + 240.0*P**3*S**2*UGA_star**2 - 192.0*P**3*S**2*UGA_star*epsilon + 3456.0*P**3*S*T*UGA_star**10*epsilon + 320.0*P**3*S*T*UGA_star**10 - 27648.0*P**3*S*T*UGA_star**9*epsilon - 2560.0*P**3*S*T*UGA_star**9 + 97056.0*P**3*S*T*UGA_star**8*epsilon + 8816.0*P**3*S*T*UGA_star**8 - 195456.0*P**3*S*T*UGA_star**7*epsilon - 17056.0*P**3*S*T*UGA_star**7 + 247392.0*P**3*S*T*UGA_star**6*epsilon + 20240.0*P**3*S*T*UGA_star**6 - 202176.0*P**3*S*T*UGA_star**5*epsilon - 15040.0*P**3*S*T*UGA_star**5 + 104928.0*P**3*S*T*UGA_star**4*epsilon + 6800.0*P**3*S*T*UGA_star**4 - 32256.0*P**3*S*T*UGA_star**3*epsilon - 1696.0*P**3*S*T*UGA_star**3 + 4896.0*P**3*S*T*UGA_star**2*epsilon + 176.0*P**3*S*T*UGA_star**2 - 192.0*P**3*S*T*UGA_star*epsilon + 1920.0*P**3*T**2*UGA_star**10*epsilon - 15360.0*P**3*T**2*UGA_star**9*epsilon + 53184.0*P**3*T**2*UGA_star**8*epsilon - 104064.0*P**3*T**2*UGA_star**7*epsilon + 125760.0*P**3*T**2*UGA_star**6*epsilon - 96000.0*P**3*T**2*UGA_star**5*epsilon + 45120.0*P**3*T**2*UGA_star**4*epsilon - 11904.0*P**3*T**2*UGA_star**3*epsilon + 1344.0*P**3*T**2*UGA_star**2*epsilon + 1632.0*P**2*R**3*UGA_star**10*epsilon + 240.0*P**2*R**3*UGA_star**10 - 6528.0*P**2*R**3*UGA_star**9*epsilon - 960.0*P**2*R**3*UGA_star**9 + 9504.0*P**2*R**3*UGA_star**8*epsilon + 1440.0*P**2*R**3*UGA_star**8 - 5376.0*P**2*R**3*UGA_star**7*epsilon - 960.0*P**2*R**3*UGA_star**7 - 96.0*P**2*R**3*UGA_star**6*epsilon + 240.0*P**2*R**3*UGA_star**6 + 1152.0*P**2*R**3*UGA_star**5*epsilon - 288.0*P**2*R**3*UGA_star**4*epsilon - 4896.0*P**2*R**2*S*UGA_star**10*epsilon - 720.0*P**2*R**2*S*UGA_star**10 + 24480.0*P**2*R**2*S*UGA_star**9*epsilon + 3600.0*P**2*R**2*S*UGA_star**9 - 48352.0*P**2*R**2*S*UGA_star**8*epsilon - 7248.0*P**2*R**2*S*UGA_star**8 + 45952.0*P**2*R**2*S*UGA_star**7*epsilon + 7392.0*P**2*R**2*S*UGA_star**7 - 18528.0*P**2*R**2*S*UGA_star**6*epsilon - 3888.0*P**2*R**2*S*UGA_star**6 - 992.0*P**2*R**2*S*UGA_star**5*epsilon + 912.0*P**2*R**2*S*UGA_star**5 + 2912.0*P**2*R**2*S*UGA_star**4*epsilon - 48.0*P**2*R**2*S*UGA_star**4 - 576.0*P**2*R**2*S*UGA_star**3*epsilon - 4896.0*P**2*R**2*T*UGA_star**10*epsilon - 720.0*P**2*R**2*T*UGA_star**10 + 24480.0*P**2*R**2*T*UGA_star**9*epsilon + 3600.0*P**2*R**2*T*UGA_star**9 - 48352.0*P**2*R**2*T*UGA_star**8*epsilon - 7248.0*P**2*R**2*T*UGA_star**8 + 47104.0*P**2*R**2*T*UGA_star**7*epsilon + 7392.0*P**2*R**2*T*UGA_star**7 - 22560.0*P**2*R**2*T*UGA_star**6*epsilon - 3888.0*P**2*R**2*T*UGA_star**6 + 4192.0*P**2*R**2*T*UGA_star**5*epsilon + 912.0*P**2*R**2*T*UGA_star**5 + 32.0*P**2*R**2*T*UGA_star**4*epsilon - 48.0*P**2*R**2*T*UGA_star**4 + 4896.0*P**2*R*S**2*UGA_star**10*epsilon + 720.0*P**2*R*S**2*UGA_star**10 - 29376.0*P**2*R*S**2*UGA_star**9*epsilon - 4320.0*P**2*R*S**2*UGA_star**9 + 73088.0*P**2*R*S**2*UGA_star**8*epsilon + 10896.0*P**2*R*S**2*UGA_star**8 - 95872.0*P**2*R*S**2*UGA_star**7*epsilon - 14880.0*P**2*R*S**2*UGA_star**7 + 68480.0*P**2*R*S**2*UGA_star**6*epsilon + 11760.0*P**2*R*S**2*UGA_star**6 - 22976.0*P**2*R*S**2*UGA_star**5*epsilon - 5280.0*P**2*R*S**2*UGA_star**5 + 256.0*P**2*R*S**2*UGA_star**4*epsilon + 1200.0*P**2*R*S**2*UGA_star**4 + 1792.0*P**2*R*S**2*UGA_star**3*epsilon - 96.0*P**2*R*S**2*UGA_star**3 - 288.0*P**2*R*S**2*UGA_star**2*epsilon + 9792.0*P**2*R*S*T*UGA_star**10*epsilon + 1440.0*P**2*R*S*T*UGA_star**10 - 58752.0*P**2*R*S*T*UGA_star**9*epsilon - 8640.0*P**2*R*S*T*UGA_star**9 + 146240.0*P**2*R*S*T*UGA_star**8*epsilon + 21696.0*P**2*R*S*T*UGA_star**8 - 193792.0*P**2*R*S*T*UGA_star**7*epsilon - 29280.0*P**2*R*S*T*UGA_star**7 + 145088.0*P**2*R*S*T*UGA_star**6*epsilon + 22560.0*P**2*R*S*T*UGA_star**6 - 59264.0*P**2*R*S*T*UGA_star**5*epsilon - 9600.0*P**2*R*S*T*UGA_star**5 + 11200.0*P**2*R*S*T*UGA_star**4*epsilon + 1920.0*P**2*R*S*T*UGA_star**4 - 512.0*P**2*R*S*T*UGA_star**3*epsilon - 96.0*P**2*R*S*T*UGA_star**3 + 5184.0*P**2*R*T**2*UGA_star**10*epsilon + 480.0*P**2*R*T**2*UGA_star**10 - 31104.0*P**2*R*T**2*UGA_star**9*epsilon - 2880.0*P**2*R*T**2*UGA_star**9 + 78432.0*P**2*R*T**2*UGA_star**8*epsilon + 7056.0*P**2*R*T**2*UGA_star**8 - 106944.0*P**2*R*T**2*UGA_star**7*epsilon - 9024.0*P**2*R*T**2*UGA_star**7 + 84096.0*P**2*R*T**2*UGA_star**6*epsilon + 6336.0*P**2*R*T**2*UGA_star**6 - 37248.0*P**2*R*T**2*UGA_star**5*epsilon - 2304.0*P**2*R*T**2*UGA_star**5 + 8160.0*P**2*R*T**2*UGA_star**4*epsilon + 336.0*P**2*R*T**2*UGA_star**4 - 576.0*P**2*R*T**2*UGA_star**3*epsilon - 1632.0*P**2*S**3*UGA_star**10*epsilon - 240.0*P**2*S**3*UGA_star**10 + 11424.0*P**2*S**3*UGA_star**9*epsilon + 1680.0*P**2*S**3*UGA_star**9 - 34240.0*P**2*S**3*UGA_star**8*epsilon - 5088.0*P**2*S**3*UGA_star**8 + 56928.0*P**2*S**3*UGA_star**7*epsilon + 8688.0*P**2*S**3*UGA_star**7 - 56640.0*P**2*S**3*UGA_star**6*epsilon - 9120.0*P**2*S**3*UGA_star**6 + 33632.0*P**2*S**3*UGA_star**5*epsilon + 6000.0*P**2*S**3*UGA_star**5 - 10944.0*P**2*S**3*UGA_star**4*epsilon - 2400.0*P**2*S**3*UGA_star**4 + 1440.0*P**2*S**3*UGA_star**3*epsilon + 528.0*P**2*S**3*UGA_star**3 + 32.0*P**2*S**3*UGA_star**2*epsilon - 48.0*P**2*S**3*UGA_star**2 - 4896.0*P**2*S**2*T*UGA_star**10*epsilon - 720.0*P**2*S**2*T*UGA_star**10 + 34272.0*P**2*S**2*T*UGA_star**9*epsilon + 5040.0*P**2*S**2*T*UGA_star**9 - 102784.0*P**2*S**2*T*UGA_star**8*epsilon - 15168.0*P**2*S**2*T*UGA_star**8 + 171744.0*P**2*S**2*T*UGA_star**7*epsilon + 25488.0*P**2*S**2*T*UGA_star**7 - 173760.0*P**2*S**2*T*UGA_star**6*epsilon - 25920.0*P**2*S**2*T*UGA_star**6 + 107936.0*P**2*S**2*T*UGA_star**5*epsilon + 16080.0*P**2*S**2*T*UGA_star**5 - 39552.0*P**2*S**2*T*UGA_star**4*epsilon - 5760.0*P**2*S**2*T*UGA_star**4 + 7584.0*P**2*S**2*T*UGA_star**3*epsilon + 1008.0*P**2*S**2*T*UGA_star**3 - 544.0*P**2*S**2*T*UGA_star**2*epsilon - 48.0*P**2*S**2*T*UGA_star**2 - 5184.0*P**2*S*T**2*UGA_star**10*epsilon - 480.0*P**2*S*T**2*UGA_star**10 + 36288.0*P**2*S*T**2*UGA_star**9*epsilon + 3360.0*P**2*S*T**2*UGA_star**9 - 109536.0*P**2*S*T**2*UGA_star**8*epsilon - 9936.0*P**2*S*T**2*UGA_star**8 + 185376.0*P**2*S*T**2*UGA_star**7*epsilon + 16080.0*P**2*S*T**2*UGA_star**7 - 191040.0*P**2*S*T**2*UGA_star**6*epsilon - 15360.0*P**2*S*T**2*UGA_star**6 + 121344.0*P**2*S*T**2*UGA_star**5*epsilon + 8640.0*P**2*S*T**2*UGA_star**5 - 45408.0*P**2*S*T**2*UGA_star**4*epsilon - 2640.0*P**2*S*T**2*UGA_star**4 + 8736.0*P**2*S*T**2*UGA_star**3*epsilon + 336.0*P**2*S*T**2*UGA_star**3 - 576.0*P**2*S*T**2*UGA_star**2*epsilon - 1920.0*P**2*T**3*UGA_star**10*epsilon + 13440.0*P**2*T**3*UGA_star**9*epsilon - 39936.0*P**2*T**3*UGA_star**8*epsilon + 65280.0*P**2*T**3*UGA_star**7*epsilon - 63360.0*P**2*T**3*UGA_star**6*epsilon + 36480.0*P**2*T**3*UGA_star**5*epsilon - 11520.0*P**2*T**3*UGA_star**4*epsilon + 1536.0*P**2*T**3*UGA_star**3*epsilon + 864.0*P*R**4*UGA_star**10*epsilon + 80.0*P*R**4*UGA_star**10 - 1728.0*P*R**4*UGA_star**9*epsilon - 160.0*P*R**4*UGA_star**9 + 896.0*P*R**4*UGA_star**8*epsilon + 32.0*P*R**4*UGA_star**8 - 64.0*P*R**4*UGA_star**7*epsilon + 96.0*P*R**4*UGA_star**7 + 32.0*P*R**4*UGA_star**6*epsilon - 48.0*P*R**4*UGA_star**6 - 3456.0*P*R**3*S*UGA_star**10*epsilon - 320.0*P*R**3*S*UGA_star**10 + 10368.0*P*R**3*S*UGA_star**9*epsilon + 960.0*P*R**3*S*UGA_star**9 - 10656.0*P*R**3*S*UGA_star**8*epsilon - 816.0*P*R**3*S*UGA_star**8 + 4128.0*P*R**3*S*UGA_star**7*epsilon - 112.0*P*R**3*S*UGA_star**7 - 480.0*P*R**3*S*UGA_star**6*epsilon + 432.0*P*R**3*S*UGA_star**6 + 96.0*P*R**3*S*UGA_star**5*epsilon - 144.0*P*R**3*S*UGA_star**5 - 3264.0*P*R**3*T*UGA_star**10*epsilon - 480.0*P*R**3*T*UGA_star**10 + 9792.0*P*R**3*T*UGA_star**9*epsilon + 1440.0*P*R**3*T*UGA_star**9 - 9408.0*P*R**3*T*UGA_star**8*epsilon - 1440.0*P*R**3*T*UGA_star**8 + 1920.0*P*R**3*T*UGA_star**7*epsilon + 480.0*P*R**3*T*UGA_star**7 + 1536.0*P*R**3*T*UGA_star**6*epsilon - 576.0*P*R**3*T*UGA_star**5*epsilon + 5184.0*P*R**2*S**2*UGA_star**10*epsilon + 480.0*P*R**2*S**2*UGA_star**10 - 20736.0*P*R**2*S**2*UGA_star**9*epsilon - 1920.0*P*R**2*S**2*UGA_star**9 + 31776.0*P*R**2*S**2*UGA_star**8*epsilon + 2736.0*P*R**2*S**2*UGA_star**8 - 22848.0*P*R**2*S**2*UGA_star**7*epsilon - 1344.0*P*R**2*S**2*UGA_star**7 + 7488.0*P*R**2*S**2*UGA_star**6*epsilon - 384.0*P*R**2*S**2*UGA_star**6 - 960.0*P*R**2*S**2*UGA_star**5*epsilon + 576.0*P*R**2*S**2*UGA_star**5 + 96.0*P*R**2*S**2*UGA_star**4*epsilon - 144.0*P*R**2*S**2*UGA_star**4 + 9792.0*P*R**2*S*T*UGA_star**10*epsilon + 1440.0*P*R**2*S*T*UGA_star**10 - 39168.0*P*R**2*S*T*UGA_star**9*epsilon - 5760.0*P*R**2*S*T*UGA_star**9 + 58112.0*P*R**2*S*T*UGA_star**8*epsilon + 8736.0*P*R**2*S*T*UGA_star**8 - 36096.0*P*R**2*S*T*UGA_star**7*epsilon - 6048.0*P*R**2*S*T*UGA_star**7 + 4416.0*P*R**2*S*T*UGA_star**6*epsilon + 1728.0*P*R**2*S*T*UGA_star**6 + 4096.0*P*R**2*S*T*UGA_star**5*epsilon - 96.0*P*R**2*S*T*UGA_star**5 - 1152.0*P*R**2*S*T*UGA_star**4*epsilon + 4896.0*P*R**2*T**2*UGA_star**10*epsilon + 720.0*P*R**2*T**2*UGA_star**10 - 19584.0*P*R**2*T**2*UGA_star**9*epsilon - 2880.0*P*R**2*T**2*UGA_star**9 + 29024.0*P*R**2*T**2*UGA_star**8*epsilon + 4416.0*P*R**2*T**2*UGA_star**8 - 18816.0*P*R**2*T**2*UGA_star**7*epsilon - 3168.0*P*R**2*T**2*UGA_star**7 + 4416.0*P*R**2*T**2*UGA_star**6*epsilon + 1008.0*P*R**2*T**2*UGA_star**6 + 64.0*P*R**2*T**2*UGA_star**5*epsilon - 96.0*P*R**2*T**2*UGA_star**5 - 3456.0*P*R*S**3*UGA_star**10*epsilon - 320.0*P*R*S**3*UGA_star**10 + 17280.0*P*R*S**3*UGA_star**9*epsilon + 1600.0*P*R*S**3*UGA_star**9 - 35168.0*P*R*S**3*UGA_star**8*epsilon - 3152.0*P*R*S**3*UGA_star**8 + 37024.0*P*R*S**3*UGA_star**7*epsilon + 2960.0*P*R*S**3*UGA_star**7 - 21056.0*P*R*S**3*UGA_star**6*epsilon - 1120.0*P*R*S**3*UGA_star**6 + 6080.0*P*R*S**3*UGA_star**5*epsilon - 160.0*P*R*S**3*UGA_star**5 - 736.0*P*R*S**3*UGA_star**4*epsilon + 240.0*P*R*S**3*UGA_star**4 + 32.0*P*R*S**3*UGA_star**3*epsilon - 48.0*P*R*S**3*UGA_star**3 - 9792.0*P*R*S**2*T*UGA_star**10*epsilon - 1440.0*P*R*S**2*T*UGA_star**10 + 48960.0*P*R*S**2*T*UGA_star**9*epsilon + 7200.0*P*R*S**2*T*UGA_star**9 - 97792.0*P*R*S**2*T*UGA_star**8*epsilon - 14592.0*P*R*S**2*T*UGA_star**8 + 96832.0*P*R*S**2*T*UGA_star**7*epsilon + 15168.0*P*R*S**2*T*UGA_star**7 - 45888.0*P*R*S**2*T*UGA_star**6*epsilon - 8352.0*P*R*S**2*T*UGA_star**6 + 5824.0*P*R*S**2*T*UGA_star**5*epsilon + 2208.0*P*R*S**2*T*UGA_star**5 + 2432.0*P*R*S**2*T*UGA_star**4*epsilon - 192.0*P*R*S**2*T*UGA_star**4 - 576.0*P*R*S**2*T*UGA_star**3*epsilon - 9792.0*P*R*S*T**2*UGA_star**10*epsilon - 1440.0*P*R*S*T**2*UGA_star**10 + 48960.0*P*R*S*T**2*UGA_star**9*epsilon + 7200.0*P*R*S*T**2*UGA_star**9 - 97792.0*P*R*S*T**2*UGA_star**8*epsilon - 14592.0*P*R*S*T**2*UGA_star**8 + 97984.0*P*R*S*T**2*UGA_star**7*epsilon + 15168.0*P*R*S*T**2*UGA_star**7 - 49920.0*P*R*S*T**2*UGA_star**6*epsilon - 8352.0*P*R*S*T**2*UGA_star**6 + 11008.0*P*R*S*T**2*UGA_star**5*epsilon + 2208.0*P*R*S*T**2*UGA_star**5 - 448.0*P*R*S*T**2*UGA_star**4*epsilon - 192.0*P*R*S*T**2*UGA_star**4 - 3456.0*P*R*T**3*UGA_star**10*epsilon - 320.0*P*R*T**3*UGA_star**10 + 17280.0*P*R*T**3*UGA_star**9*epsilon + 1600.0*P*R*T**3*UGA_star**9 - 35168.0*P*R*T**3*UGA_star**8*epsilon - 3152.0*P*R*T**3*UGA_star**8 + 36960.0*P*R*T**3*UGA_star**7*epsilon + 3056.0*P*R*T**3*UGA_star**7 - 20832.0*P*R*T**3*UGA_star**6*epsilon - 1456.0*P*R*T**3*UGA_star**6 + 5792.0*P*R*T**3*UGA_star**5*epsilon + 272.0*P*R*T**3*UGA_star**5 - 576.0*P*R*T**3*UGA_star**4*epsilon + 864.0*P*S**4*UGA_star**10*epsilon + 80.0*P*S**4*UGA_star**10 - 5184.0*P*S**4*UGA_star**9*epsilon - 480.0*P*S**4*UGA_star**9 + 13152.0*P*S**4*UGA_star**8*epsilon + 1200.0*P*S**4*UGA_star**8 - 18240.0*P*S**4*UGA_star**7*epsilon - 1600.0*P*S**4*UGA_star**7 + 14880.0*P*S**4*UGA_star**6*epsilon + 1200.0*P*S**4*UGA_star**6 - 7104.0*P*S**4*UGA_star**5*epsilon - 480.0*P*S**4*UGA_star**5 + 1824.0*P*S**4*UGA_star**4*epsilon + 80.0*P*S**4*UGA_star**4 - 192.0*P*S**4*UGA_star**3*epsilon + 3264.0*P*S**3*T*UGA_star**10*epsilon + 480.0*P*S**3*T*UGA_star**10 - 19584.0*P*S**3*T*UGA_star**9*epsilon - 2880.0*P*S**3*T*UGA_star**9 + 49088.0*P*S**3*T*UGA_star**8*epsilon + 7296.0*P*S**3*T*UGA_star**8 - 65920.0*P*S**3*T*UGA_star**7*epsilon - 10080.0*P*S**3*T*UGA_star**7 + 50240.0*P*S**3*T*UGA_star**6*epsilon + 8160.0*P*S**3*T*UGA_star**6 - 20864.0*P*S**3*T*UGA_star**5*epsilon - 3840.0*P*S**3*T*UGA_star**5 + 3904.0*P*S**3*T*UGA_star**4*epsilon + 960.0*P*S**3*T*UGA_star**4 - 128.0*P*S**3*T*UGA_star**3*epsilon - 96.0*P*S**3*T*UGA_star**3 + 4896.0*P*S**2*T**2*UGA_star**10*epsilon + 720.0*P*S**2*T**2*UGA_star**10 - 29376.0*P*S**2*T**2*UGA_star**9*epsilon - 4320.0*P*S**2*T**2*UGA_star**9 + 73664.0*P*S**2*T**2*UGA_star**8*epsilon + 10896.0*P*S**2*T**2*UGA_star**8 - 99328.0*P*S**2*T**2*UGA_star**7*epsilon - 14880.0*P*S**2*T**2*UGA_star**7 + 76832.0*P*S**2*T**2*UGA_star**6*epsilon + 11760.0*P*S**2*T**2*UGA_star**6 - 33344.0*P*S**2*T**2*UGA_star**5*epsilon - 5280.0*P*S**2*T**2*UGA_star**5 + 7168.0*P*S**2*T**2*UGA_star**4*epsilon + 1200.0*P*S**2*T**2*UGA_star**4 - 512.0*P*S**2*T**2*UGA_star**3*epsilon - 96.0*P*S**2*T**2*UGA_star**3 + 3456.0*P*S*T**3*UGA_star**10*epsilon + 320.0*P*S*T**3*UGA_star**10 - 20736.0*P*S*T**3*UGA_star**9*epsilon - 1920.0*P*S*T**3*UGA_star**9 + 52448.0*P*S*T**3*UGA_star**8*epsilon + 4752.0*P*S*T**3*UGA_star**8 - 72128.0*P*S*T**3*UGA_star**7*epsilon - 6208.0*P*S*T**3*UGA_star**7 + 57792.0*P*S*T**3*UGA_star**6*epsilon + 4512.0*P*S*T**3*UGA_star**6 - 26624.0*P*S*T**3*UGA_star**5*epsilon - 1728.0*P*S*T**3*UGA_star**5 + 6368.0*P*S*T**3*UGA_star**4*epsilon + 272.0*P*S*T**3*UGA_star**4 - 576.0*P*S*T**3*UGA_star**3*epsilon + 960.0*P*T**4*UGA_star**10*epsilon - 5760.0*P*T**4*UGA_star**9*epsilon + 14304.0*P*T**4*UGA_star**8*epsilon - 18816.0*P*T**4*UGA_star**7*epsilon + 13824.0*P*T**4*UGA_star**6*epsilon - 5376.0*P*T**4*UGA_star**5*epsilon + 864.0*P*T**4*UGA_star**4*epsilon + 192.0*R**5*UGA_star**10*epsilon - 96.0*R**5*UGA_star**8*epsilon - 960.0*R**4*S*UGA_star**10*epsilon + 960.0*R**4*S*UGA_star**9*epsilon + 384.0*R**4*S*UGA_star**8*epsilon - 384.0*R**4*S*UGA_star**7*epsilon - 864.0*R**4*T*UGA_star**10*epsilon - 80.0*R**4*T*UGA_star**10 + 864.0*R**4*T*UGA_star**9*epsilon + 80.0*R**4*T*UGA_star**9 - 32.0*R**4*T*UGA_star**8*epsilon + 48.0*R**4*T*UGA_star**8 + 32.0*R**4*T*UGA_star**7*epsilon - 48.0*R**4*T*UGA_star**7 + 1920.0*R**3*S**2*UGA_star**10*epsilon - 3840.0*R**3*S**2*UGA_star**9*epsilon + 1344.0*R**3*S**2*UGA_star**8*epsilon + 1152.0*R**3*S**2*UGA_star**7*epsilon - 576.0*R**3*S**2*UGA_star**6*epsilon + 3456.0*R**3*S*T*UGA_star**10*epsilon + 320.0*R**3*S*T*UGA_star**10 - 6912.0*R**3*S*T*UGA_star**9*epsilon - 640.0*R**3*S*T*UGA_star**9 + 3744.0*R**3*S*T*UGA_star**8*epsilon + 176.0*R**3*S*T*UGA_star**8 - 384.0*R**3*S*T*UGA_star**7*epsilon + 288.0*R**3*S*T*UGA_star**7 + 96.0*R**3*S*T*UGA_star**6*epsilon - 144.0*R**3*S*T*UGA_star**6 + 1632.0*R**3*T**2*UGA_star**10*epsilon + 240.0*R**3*T**2*UGA_star**10 - 3264.0*R**3*T**2*UGA_star**9*epsilon - 480.0*R**3*T**2*UGA_star**9 + 1536.0*R**3*T**2*UGA_star**8*epsilon + 240.0*R**3*T**2*UGA_star**8 + 384.0*R**3*T**2*UGA_star**7*epsilon - 288.0*R**3*T**2*UGA_star**6*epsilon - 1920.0*R**2*S**3*UGA_star**10*epsilon + 5760.0*R**2*S**3*UGA_star**9*epsilon - 5376.0*R**2*S**3*UGA_star**8*epsilon + 768.0*R**2*S**3*UGA_star**7*epsilon + 1152.0*R**2*S**3*UGA_star**6*epsilon - 384.0*R**2*S**3*UGA_star**5*epsilon - 5184.0*R**2*S**2*T*UGA_star**10*epsilon - 480.0*R**2*S**2*T*UGA_star**10 + 15552.0*R**2*S**2*T*UGA_star**9*epsilon + 1440.0*R**2*S**2*T*UGA_star**9 - 16224.0*R**2*S**2*T*UGA_star**8*epsilon - 1296.0*R**2*S**2*T*UGA_star**8 + 6624.0*R**2*S**2*T*UGA_star**7*epsilon + 48.0*R**2*S**2*T*UGA_star**7 - 864.0*R**2*S**2*T*UGA_star**6*epsilon + 432.0*R**2*S**2*T*UGA_star**6 + 96.0*R**2*S**2*T*UGA_star**5*epsilon - 144.0*R**2*S**2*T*UGA_star**5 - 4896.0*R**2*S*T**2*UGA_star**10*epsilon - 720.0*R**2*S*T**2*UGA_star**10 + 14688.0*R**2*S*T**2*UGA_star**9*epsilon + 2160.0*R**2*S*T**2*UGA_star**9 - 14656.0*R**2*S*T**2*UGA_star**8*epsilon - 2208.0*R**2*S*T**2*UGA_star**8 + 4256.0*R**2*S*T**2*UGA_star**7*epsilon + 816.0*R**2*S*T**2*UGA_star**7 + 1184.0*R**2*S*T**2*UGA_star**6*epsilon - 48.0*R**2*S*T**2*UGA_star**6 - 576.0*R**2*S*T**2*UGA_star**5*epsilon - 1632.0*R**2*T**3*UGA_star**10*epsilon - 240.0*R**2*T**3*UGA_star**10 + 4896.0*R**2*T**3*UGA_star**9*epsilon + 720.0*R**2*T**3*UGA_star**9 - 4864.0*R**2*T**3*UGA_star**8*epsilon - 768.0*R**2*T**3*UGA_star**8 + 1568.0*R**2*T**3*UGA_star**7*epsilon + 336.0*R**2*T**3*UGA_star**7 + 32.0*R**2*T**3*UGA_star**6*epsilon - 48.0*R**2*T**3*UGA_star**6 + 960.0*R*S**4*UGA_star**10*epsilon - 3840.0*R*S**4*UGA_star**9*epsilon + 5664.0*R*S**4*UGA_star**8*epsilon - 3456.0*R*S**4*UGA_star**7*epsilon + 384.0*R*S**4*UGA_star**6*epsilon + 384.0*R*S**4*UGA_star**5*epsilon - 96.0*R*S**4*UGA_star**4*epsilon + 3456.0*R*S**3*T*UGA_star**10*epsilon + 320.0*R*S**3*T*UGA_star**10 - 13824.0*R*S**3*T*UGA_star**9*epsilon - 1280.0*R*S**3*T*UGA_star**9 + 21344.0*R*S**3*T*UGA_star**8*epsilon + 1872.0*R*S**3*T*UGA_star**8 - 15680.0*R*S**3*T*UGA_star**7*epsilon - 1088.0*R*S**3*T*UGA_star**7 + 5376.0*R*S**3*T*UGA_star**6*epsilon + 32.0*R*S**3*T*UGA_star**6 - 704.0*R*S**3*T*UGA_star**5*epsilon + 192.0*R*S**3*T*UGA_star**5 + 32.0*R*S**3*T*UGA_star**4*epsilon - 48.0*R*S**3*T*UGA_star**4 + 4896.0*R*S**2*T**2*UGA_star**10*epsilon + 720.0*R*S**2*T**2*UGA_star**10 - 19584.0*R*S**2*T**2*UGA_star**9*epsilon - 2880.0*R*S**2*T**2*UGA_star**9 + 29600.0*R*S**2*T**2*UGA_star**8*epsilon + 4416.0*R*S**2*T**2*UGA_star**8 - 19968.0*R*S**2*T**2*UGA_star**7*epsilon - 3168.0*R*S**2*T**2*UGA_star**7 + 4704.0*R*S**2*T**2*UGA_star**6*epsilon + 1008.0*R*S**2*T**2*UGA_star**6 + 640.0*R*S**2*T**2*UGA_star**5*epsilon - 96.0*R*S**2*T**2*UGA_star**5 - 288.0*R*S**2*T**2*UGA_star**4*epsilon + 3264.0*R*S*T**3*UGA_star**10*epsilon + 480.0*R*S*T**3*UGA_star**10 - 13056.0*R*S*T**3*UGA_star**9*epsilon - 1920.0*R*S*T**3*UGA_star**9 + 19712.0*R*S*T**3*UGA_star**8*epsilon + 2976.0*R*S*T**3*UGA_star**8 - 13440.0*R*S*T**3*UGA_star**7*epsilon - 2208.0*R*S*T**3*UGA_star**7 + 3648.0*R*S*T**3*UGA_star**6*epsilon + 768.0*R*S*T**3*UGA_star**6 - 128.0*R*S*T**3*UGA_star**5*epsilon - 96.0*R*S*T**3*UGA_star**5 + 864.0*R*T**4*UGA_star**10*epsilon + 80.0*R*T**4*UGA_star**10 - 3456.0*R*T**4*UGA_star**9*epsilon - 320.0*R*T**4*UGA_star**9 + 5376.0*R*T**4*UGA_star**8*epsilon + 480.0*R*T**4*UGA_star**8 - 4032.0*R*T**4*UGA_star**7*epsilon - 320.0*R*T**4*UGA_star**7 + 1440.0*R*T**4*UGA_star**6*epsilon + 80.0*R*T**4*UGA_star**6 - 192.0*R*T**4*UGA_star**5*epsilon - 192.0*S**5*UGA_star**10*epsilon + 960.0*S**5*UGA_star**9*epsilon - 1920.0*S**5*UGA_star**8*epsilon + 1920.0*S**5*UGA_star**7*epsilon - 960.0*S**5*UGA_star**6*epsilon + 192.0*S**5*UGA_star**5*epsilon - 864.0*S**4*T*UGA_star**10*epsilon - 80.0*S**4*T*UGA_star**10 + 4320.0*S**4*T*UGA_star**9*epsilon + 400.0*S**4*T*UGA_star**9 - 8832.0*S**4*T*UGA_star**8*epsilon - 800.0*S**4*T*UGA_star**8 + 9408.0*S**4*T*UGA_star**7*epsilon + 800.0*S**4*T*UGA_star**7 - 5472.0*S**4*T*UGA_star**6*epsilon - 400.0*S**4*T*UGA_star**6 + 1632.0*S**4*T*UGA_star**5*epsilon + 80.0*S**4*T*UGA_star**5 - 192.0*S**4*T*UGA_star**4*epsilon - 1632.0*S**3*T**2*UGA_star**10*epsilon - 240.0*S**3*T**2*UGA_star**10 + 8160.0*S**3*T**2*UGA_star**9*epsilon + 1200.0*S**3*T**2*UGA_star**9 - 16480.0*S**3*T**2*UGA_star**8*epsilon - 2448.0*S**3*T**2*UGA_star**8 + 16960.0*S**3*T**2*UGA_star**7*epsilon + 2592.0*S**3*T**2*UGA_star**7 - 9120.0*S**3*T**2*UGA_star**6*epsilon - 1488.0*S**3*T**2*UGA_star**6 + 2272.0*S**3*T**2*UGA_star**5*epsilon + 432.0*S**3*T**2*UGA_star**5 - 160.0*S**3*T**2*UGA_star**4*epsilon - 48.0*S**3*T**2*UGA_star**4 - 1632.0*S**2*T**3*UGA_star**10*epsilon - 240.0*S**2*T**3*UGA_star**10 + 8160.0*S**2*T**3*UGA_star**9*epsilon + 1200.0*S**2*T**3*UGA_star**9 - 16480.0*S**2*T**3*UGA_star**8*epsilon - 2448.0*S**2*T**3*UGA_star**8 + 16960.0*S**2*T**3*UGA_star**7*epsilon + 2592.0*S**2*T**3*UGA_star**7 - 9120.0*S**2*T**3*UGA_star**6*epsilon - 1488.0*S**2*T**3*UGA_star**6 + 2272.0*S**2*T**3*UGA_star**5*epsilon + 432.0*S**2*T**3*UGA_star**5 - 160.0*S**2*T**3*UGA_star**4*epsilon - 48.0*S**2*T**3*UGA_star**4 - 864.0*S*T**4*UGA_star**10*epsilon - 80.0*S*T**4*UGA_star**10 + 4320.0*S*T**4*UGA_star**9*epsilon + 400.0*S*T**4*UGA_star**9 - 8832.0*S*T**4*UGA_star**8*epsilon - 800.0*S*T**4*UGA_star**8 + 9408.0*S*T**4*UGA_star**7*epsilon + 800.0*S*T**4*UGA_star**7 - 5472.0*S*T**4*UGA_star**6*epsilon - 400.0*S*T**4*UGA_star**6 + 1632.0*S*T**4*UGA_star**5*epsilon + 80.0*S*T**4*UGA_star**5 - 192.0*S*T**4*UGA_star**4*epsilon - 192.0*T**5*UGA_star**10*epsilon + 960.0*T**5*UGA_star**9*epsilon - 1920.0*T**5*UGA_star**8*epsilon + 1920.0*T**5*UGA_star**7*epsilon - 960.0*T**5*UGA_star**6*epsilon + 192.0*T**5*UGA_star**5*epsilon - sqrt(2048.0*P**9*R*UGA_star**20*epsilon - 36864.0*P**9*R*UGA_star**19*epsilon + 316416.0*P**9*R*UGA_star**18*epsilon - 1720320.0*P**9*R*UGA_star**17*epsilon + 6636544.0*P**9*R*UGA_star**16*epsilon - 19281920.0*P**9*R*UGA_star**15*epsilon + 43703296.0*P**9*R*UGA_star**14*epsilon - 78966784.0*P**9*R*UGA_star**13*epsilon + 115241984.0*P**9*R*UGA_star**12*epsilon - 136767488.0*P**9*R*UGA_star**11*epsilon + 132228096.0*P**9*R*UGA_star**10*epsilon - 103833600.0*P**9*R*UGA_star**9*epsilon + 65694720.0*P**9*R*UGA_star**8*epsilon - 33015808.0*P**9*R*UGA_star**7*epsilon + 12882944.0*P**9*R*UGA_star**6*epsilon - 3764224.0*P**9*R*UGA_star**5*epsilon + 775168.0*P**9*R*UGA_star**4*epsilon - 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26624.0*P**8*R*S*UGA_star**20*epsilon - 512.0*P**8*R*S*UGA_star**20 + 452608.0*P**8*R*S*UGA_star**19*epsilon + 8704.0*P**8*R*S*UGA_star**19 - 3651584.0*P**8*R*S*UGA_star**18*epsilon - 70656.0*P**8*R*S*UGA_star**18 + 18565120.0*P**8*R*S*UGA_star**17*epsilon + 363520.0*P**8*R*S*UGA_star**17 - 66592768.0*P**8*R*S*UGA_star**16*epsilon - 1326592.0*P**8*R*S*UGA_star**16 + 178753536.0*P**8*R*S*UGA_star**15*epsilon + 3640832.0*P**8*R*S*UGA_star**15 - 371591168.0*P**8*R*S*UGA_star**14*epsilon - 7774208.0*P**8*R*S*UGA_star**14 + 610621440.0*P**8*R*S*UGA_star**13*epsilon + 13178880.0*P**8*R*S*UGA_star**13 - 802447360.0*P**8*R*S*UGA_star**12*epsilon - 17937920.0*P**8*R*S*UGA_star**12 + 847548416.0*P**8*R*S*UGA_star**11*epsilon + 19695104.0*P**8*R*S*UGA_star**11 - 718981120.0*P**8*R*S*UGA_star**10*epsilon - 17425408.0*P**8*R*S*UGA_star**10 + 486766592.0*P**8*R*S*UGA_star**9*epsilon + 12340224.0*P**8*R*S*UGA_star**9 - 259637248.0*P**8*R*S*UGA_star**8*epsilon - 6902272.0*P**8*R*S*UGA_star**8 + 106762240.0*P**8*R*S*UGA_star**7*epsilon + 2982400.0*P**8*R*S*UGA_star**7 - 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98023424.0*P**6*R**3*T*UGA_star**10*epsilon - 4234752.0*P**6*R**3*T*UGA_star**10 + 26959872.0*P**6*R**3*T*UGA_star**9*epsilon + 1171456.0*P**6*R**3*T*UGA_star**9 - 4262912.0*P**6*R**3*T*UGA_star**8*epsilon - 184320.0*P**6*R**3*T*UGA_star**8 + 218112.0*P**6*R**3*T*UGA_star**7*epsilon + 8704.0*P**6*R**3*T*UGA_star**7 + 20480.0*P**6*R**3*T*UGA_star**6*epsilon + 1024.0*P**6*R**3*T*UGA_star**6 + 411648.0*P**6*R**2*S**2*UGA_star**20*epsilon + 23040.0*P**6*R**2*S**2*UGA_star**20 - 5763072.0*P**6*R**2*S**2*UGA_star**19*epsilon - 322560.0*P**6*R**2*S**2*UGA_star**19 + 37592064.0*P**6*R**2*S**2*UGA_star**18*epsilon + 2104320.0*P**6*R**2*S**2*UGA_star**18 - 151412736.0*P**6*R**2*S**2*UGA_star**17*epsilon - 8477184.0*P**6*R**2*S**2*UGA_star**17 + 420639744.0*P**6*R**2*S**2*UGA_star**16*epsilon + 23549952.0*P**6*R**2*S**2*UGA_star**16 - 852486144.0*P**6*R**2*S**2*UGA_star**15*epsilon - 47697408.0*P**6*R**2*S**2*UGA_star**15 + 1299538944.0*P**6*R**2*S**2*UGA_star**14*epsilon + 72568320.0*P**6*R**2*S**2*UGA_star**14 - 1513678848.0*P**6*R**2*S**2*UGA_star**13*epsilon - 84142080.0*P**6*R**2*S**2*UGA_star**13 + 1354079232.0*P**6*R**2*S**2*UGA_star**12*epsilon + 74562048.0*P**6*R**2*S**2*UGA_star**12 - 926711808.0*P**6*R**2*S**2*UGA_star**11*epsilon - 50079744.0*P**6*R**2*S**2*UGA_star**11 + 479069184.0*P**6*R**2*S**2*UGA_star**10*epsilon + 24938496.0*P**6*R**2*S**2*UGA_star**10 - 182956032.0*P**6*R**2*S**2*UGA_star**9*epsilon - 8808960.0*P**6*R**2*S**2*UGA_star**9 + 50064384.0*P**6*R**2*S**2*UGA_star**8*epsilon + 2012160.0*P**6*R**2*S**2*UGA_star**8 - 9578496.0*P**6*R**2*S**2*UGA_star**7*epsilon - 228864.0*P**6*R**2*S**2*UGA_star**7 + 1342464.0*P**6*R**2*S**2*UGA_star**6*epsilon - 4608.0*P**6*R**2*S**2*UGA_star**6 - 165888.0*P**6*R**2*S**2*UGA_star**5*epsilon + 3072.0*P**6*R**2*S**2*UGA_star**5 + 15360.0*P**6*R**2*S**2*UGA_star**4*epsilon + 817152.0*P**6*R**2*S*T*UGA_star**20*epsilon + 32256.0*P**6*R**2*S*T*UGA_star**20 - 11440128.0*P**6*R**2*S*T*UGA_star**19*epsilon - 451584.0*P**6*R**2*S*T*UGA_star**19 + 74821632.0*P**6*R**2*S*T*UGA_star**18*epsilon + 2961408.0*P**6*R**2*S*T*UGA_star**18 - 302917632.0*P**6*R**2*S*T*UGA_star**17*epsilon - 12052992.0*P**6*R**2*S*T*UGA_star**17 + 847739904.0*P**6*R**2*S*T*UGA_star**16*epsilon + 33991680.0*P**6*R**2*S*T*UGA_star**16 - 1733855232.0*P**6*R**2*S*T*UGA_star**15*epsilon - 70202880.0*P**6*R**2*S*T*UGA_star**15 + 2670560256.0*P**6*R**2*S*T*UGA_star**14*epsilon + 109367808.0*P**6*R**2*S*T*UGA_star**14 - 3143522304.0*P**6*R**2*S*T*UGA_star**13*epsilon - 130369536.0*P**6*R**2*S*T*UGA_star**13 + 2837458944.0*P**6*R**2*S*T*UGA_star**12*epsilon + 119268864.0*P**6*R**2*S*T*UGA_star**12 - 1950056448.0*P**6*R**2*S*T*UGA_star**11*epsilon - 83128320.0*P**6*R**2*S*T*UGA_star**11 + 1000593408.0*P**6*R**2*S*T*UGA_star**10*epsilon + 43287552.0*P**6*R**2*S*T*UGA_star**10 - 368867328.0*P**6*R**2*S*T*UGA_star**9*epsilon - 16218624.0*P**6*R**2*S*T*UGA_star**9 + 90627072.0*P**6*R**2*S*T*UGA_star**8*epsilon + 4067328.0*P**6*R**2*S*T*UGA_star**8 - 12429312.0*P**6*R**2*S*T*UGA_star**7*epsilon - 579072.0*P**6*R**2*S*T*UGA_star**7 + 390144.0*P**6*R**2*S*T*UGA_star**6*epsilon + 23040.0*P**6*R**2*S*T*UGA_star**6 + 79872.0*P**6*R**2*S*T*UGA_star**5*epsilon + 3072.0*P**6*R**2*S*T*UGA_star**5 + 372736.0*P**6*R**2*T**2*UGA_star**20*epsilon + 7168.0*P**6*R**2*T**2*UGA_star**20 - 5218304.0*P**6*R**2*T**2*UGA_star**19*epsilon - 100352.0*P**6*R**2*T**2*UGA_star**19 + 34212864.0*P**6*R**2*T**2*UGA_star**18*epsilon + 663040.0*P**6*R**2*T**2*UGA_star**18 - 139173888.0*P**6*R**2*T**2*UGA_star**17*epsilon - 2738176.0*P**6*R**2*T**2*UGA_star**17 + 392158208.0*P**6*R**2*T**2*UGA_star**16*epsilon + 7888640.0*P**6*R**2*T**2*UGA_star**16 - 808931328.0*P**6*R**2*T**2*UGA_star**15*epsilon - 16754176.0*P**6*R**2*T**2*UGA_star**15 + 1258170368.0*P**6*R**2*T**2*UGA_star**14*epsilon + 27020544.0*P**6*R**2*T**2*UGA_star**14 - 1496598528.0*P**6*R**2*T**2*UGA_star**13*epsilon - 33576960.0*P**6*R**2*T**2*UGA_star**13 + 1365313536.0*P**6*R**2*T**2*UGA_star**12*epsilon + 32266752.0*P**6*R**2*T**2*UGA_star**12 - 947838976.0*P**6*R**2*T**2*UGA_star**11*epsilon - 23833600.0*P**6*R**2*T**2*UGA_star**11 + 490585088.0*P**6*R**2*T**2*UGA_star**10*epsilon + 13303808.0*P**6*R**2*T**2*UGA_star**10 - 181899264.0*P**6*R**2*T**2*UGA_star**9*epsilon - 5435392.0*P**6*R**2*T**2*UGA_star**9 + 44657664.0*P**6*R**2*T**2*UGA_star**8*epsilon + 1534720.0*P**6*R**2*T**2*UGA_star**8 - 5992448.0*P**6*R**2*T**2*UGA_star**7*epsilon - 267776.0*P**6*R**2*T**2*UGA_star**7 + 135168.0*P**6*R**2*T**2*UGA_star**6*epsilon + 21760.0*P**6*R**2*T**2*UGA_star**6 + 47104.0*P**6*R**2*T**2*UGA_star**5*epsilon - 274432.0*P**6*R*S**3*UGA_star**20*epsilon - 15360.0*P**6*R*S**3*UGA_star**20 + 4116480.0*P**6*R*S**3*UGA_star**19*epsilon + 230400.0*P**6*R*S**3*UGA_star**19 - 28896256.0*P**6*R*S**3*UGA_star**18*epsilon - 1617408.0*P**6*R*S**3*UGA_star**18 + 125901824.0*P**6*R*S**3*UGA_star**17*epsilon + 7047168.0*P**6*R*S**3*UGA_star**17 - 380695552.0*P**6*R*S**3*UGA_star**16*epsilon - 21304320.0*P**6*R*S**3*UGA_star**16 + 845970432.0*P**6*R*S**3*UGA_star**15*epsilon + 47305728.0*P**6*R*S**3*UGA_star**15 - 1426680832.0*P**6*R*S**3*UGA_star**14*epsilon - 79630848.0*P**6*R*S**3*UGA_star**14 + 1858481152.0*P**6*R*S**3*UGA_star**13*epsilon + 103335936.0*P**6*R*S**3*UGA_star**13 - 1884320768.0*P**6*R*S**3*UGA_star**12*epsilon - 104011776.0*P**6*R*S**3*UGA_star**12 + 1485958144.0*P**6*R*S**3*UGA_star**11*epsilon + 80931840.0*P**6*R*S**3*UGA_star**11 - 903226368.0*P**6*R*S**3*UGA_star**10*epsilon - 48001536.0*P**6*R*S**3*UGA_star**10 + 415273984.0*P**6*R*S**3*UGA_star**9*epsilon + 21067776.0*P**6*R*S**3*UGA_star**9 - 139836416.0*P**6*R*S**3*UGA_star**8*epsilon - 6448128.0*P**6*R*S**3*UGA_star**8 + 32748544.0*P**6*R*S**3*UGA_star**7*epsilon + 1195008.0*P**6*R*S**3*UGA_star**7 - 4948992.0*P**6*R*S**3*UGA_star**6*epsilon - 69120.0*P**6*R*S**3*UGA_star**6 + 467968.0*P**6*R*S**3*UGA_star**5*epsilon - 18432.0*P**6*R*S**3*UGA_star**5 - 44032.0*P**6*R*S**3*UGA_star**4*epsilon + 3072.0*P**6*R*S**3*UGA_star**4 + 5120.0*P**6*R*S**3*UGA_star**3*epsilon - 817152.0*P**6*R*S**2*T*UGA_star**20*epsilon - 32256.0*P**6*R*S**2*T*UGA_star**20 + 12257280.0*P**6*R*S**2*T*UGA_star**19*epsilon + 483840.0*P**6*R*S**2*T*UGA_star**19 - 86261760.0*P**6*R*S**2*T*UGA_star**18*epsilon - 3412992.0*P**6*R*S**2*T*UGA_star**18 + 377739264.0*P**6*R*S**2*T*UGA_star**17*epsilon + 15014400.0*P**6*R*S**2*T*UGA_star**17 - 1150639104.0*P**6*R*S**2*T*UGA_star**16*epsilon - 46044672.0*P**6*R*S**2*T*UGA_star**16 + 2581373952.0*P**6*R*S**2*T*UGA_star**15*epsilon + 104194560.0*P**6*R*S**2*T*UGA_star**15 - 4403198976.0*P**6*R*S**2*T*UGA_star**14*epsilon - 179570688.0*P**6*R*S**2*T*UGA_star**14 + 5810027520.0*P**6*R*S**2*T*UGA_star**13*epsilon + 239737344.0*P**6*R*S**2*T*UGA_star**13 - 5971857408.0*P**6*R*S**2*T*UGA_star**12*epsilon - 249638400.0*P**6*R*S**2*T*UGA_star**12 + 4772917248.0*P**6*R*S**2*T*UGA_star**11*epsilon + 202397184.0*P**6*R*S**2*T*UGA_star**11 - 2933618688.0*P**6*R*S**2*T*UGA_star**10*epsilon - 126415872.0*P**6*R*S**2*T*UGA_star**10 + 1354862592.0*P**6*R*S**2*T*UGA_star**9*epsilon + 59506176.0*P**6*R*S**2*T*UGA_star**9 - 450370560.0*P**6*R*S**2*T*UGA_star**8*epsilon - 20285952.0*P**6*R*S**2*T*UGA_star**8 + 99001344.0*P**6*R*S**2*T*UGA_star**7*epsilon + 4646400.0*P**6*R*S**2*T*UGA_star**7 - 11602944.0*P**6*R*S**2*T*UGA_star**6*epsilon - 602112.0*P**6*R*S**2*T*UGA_star**6 + 89088.0*P**6*R*S**2*T*UGA_star**5*epsilon + 19968.0*P**6*R*S**2*T*UGA_star**5 + 98304.0*P**6*R*S**2*T*UGA_star**4*epsilon + 3072.0*P**6*R*S**2*T*UGA_star**4 - 745472.0*P**6*R*S*T**2*UGA_star**20*epsilon - 14336.0*P**6*R*S*T**2*UGA_star**20 + 11182080.0*P**6*R*S*T**2*UGA_star**19*epsilon + 215040.0*P**6*R*S*T**2*UGA_star**19 - 78862336.0*P**6*R*S*T**2*UGA_star**18*epsilon - 1526784.0*P**6*R*S*T**2*UGA_star**18 + 346773504.0*P**6*R*S*T**2*UGA_star**17*epsilon + 6802432.0*P**6*R*S*T**2*UGA_star**17 - 1062664192.0*P**6*R*S*T**2*UGA_star**16*epsilon - 21253632.0*P**6*R*S*T**2*UGA_star**16 + 2402179072.0*P**6*R*S*T**2*UGA_star**15*epsilon + 49285632.0*P**6*R*S*T**2*UGA_star**15 - 4134203392.0*P**6*R*S*T**2*UGA_star**14*epsilon - 87549440.0*P**6*R*S*T**2*UGA_star**14 + 5509537792.0*P**6*R*S*T**2*UGA_star**13*epsilon + 121195008.0*P**6*R*S*T**2*UGA_star**13 - 5723824128.0*P**6*R*S*T**2*UGA_star**12*epsilon - 131687424.0*P**6*R*S*T**2*UGA_star**12 + 4626305024.0*P**6*R*S*T**2*UGA_star**11*epsilon + 112200704.0*P**6*R*S*T**2*UGA_star**11 - 2876848128.0*P**6*R*S*T**2*UGA_star**10*epsilon - 74274816.0*P**6*R*S*T**2*UGA_star**10 + 1344968704.0*P**6*R*S*T**2*UGA_star**9*epsilon + 37478400.0*P**6*R*S*T**2*UGA_star**9 - 453113856.0*P**6*R*S*T**2*UGA_star**8*epsilon - 13940224.0*P**6*R*S*T**2*UGA_star**8 + 101300224.0*P**6*R*S*T**2*UGA_star**7*epsilon + 3604992.0*P**6*R*S*T**2*UGA_star**7 - 12255232.0*P**6*R*S*T**2*UGA_star**6*epsilon - 579072.0*P**6*R*S*T**2*UGA_star**6 + 176128.0*P**6*R*S*T**2*UGA_star**5*epsilon + 43520.0*P**6*R*S*T**2*UGA_star**5 + 94208.0*P**6*R*S*T**2*UGA_star**4*epsilon - 172032.0*P**6*R*T**3*UGA_star**20*epsilon + 2580480.0*P**6*R*T**3*UGA_star**19*epsilon - 18235392.0*P**6*R*T**3*UGA_star**18*epsilon + 80510976.0*P**6*R*T**3*UGA_star**17*epsilon - 248278016.0*P**6*R*T**3*UGA_star**16*epsilon + 566207488.0*P**6*R*T**3*UGA_star**15*epsilon - 985994240.0*P**6*R*T**3*UGA_star**14*epsilon + 1334344704.0*P**6*R*T**3*UGA_star**13*epsilon - 1414060032.0*P**6*R*T**3*UGA_star**12*epsilon + 1172785152.0*P**6*R*T**3*UGA_star**11*epsilon - 754575360.0*P**6*R*T**3*UGA_star**10*epsilon + 369653760.0*P**6*R*T**3*UGA_star**9*epsilon - 133389312.0*P**6*R*T**3*UGA_star**8*epsilon + 33453056.0*P**6*R*T**3*UGA_star**7*epsilon - 5211136.0*P**6*R*T**3*UGA_star**6*epsilon + 379904.0*P**6*R*T**3*UGA_star**5*epsilon + 68608.0*P**6*S**4*UGA_star**20*epsilon + 3840.0*P**6*S**4*UGA_star**20 - 1097728.0*P**6*S**4*UGA_star**19*epsilon - 61440.0*P**6*S**4*UGA_star**19 + 8251392.0*P**6*S**4*UGA_star**18*epsilon + 461824.0*P**6*S**4*UGA_star**18 - 38672384.0*P**6*S**4*UGA_star**17*epsilon - 2164224.0*P**6*S**4*UGA_star**17 + 126455808.0*P**6*S**4*UGA_star**16*epsilon + 7074304.0*P**6*S**4*UGA_star**16 - 305803264.0*P**6*S**4*UGA_star**15*epsilon - 17092608.0*P**6*S**4*UGA_star**15 + 565467136.0*P**6*S**4*UGA_star**14*epsilon + 31549440.0*P**6*S**4*UGA_star**14 - 815040512.0*P**6*S**4*UGA_star**13*epsilon - 45320704.0*P**6*S**4*UGA_star**13 + 924571648.0*P**6*S**4*UGA_star**12*epsilon + 51104768.0*P**6*S**4*UGA_star**12 - 827047936.0*P**6*S**4*UGA_star**11*epsilon - 45247488.0*P**6*S**4*UGA_star**11 + 580163584.0*P**6*S**4*UGA_star**10*epsilon + 31190016.0*P**6*S**4*UGA_star**10 - 314615808.0*P**6*S**4*UGA_star**9*epsilon - 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277207040.0*P**6*S**2*T**2*UGA_star**7*epsilon - 8772608.0*P**6*S**2*T**2*UGA_star**7 + 56777728.0*P**6*S**2*T**2*UGA_star**6*epsilon + 2092032.0*P**6*S**2*T**2*UGA_star**6 - 6215680.0*P**6*S**2*T**2*UGA_star**5*epsilon - 311296.0*P**6*S**2*T**2*UGA_star**5 + 40960.0*P**6*S**2*T**2*UGA_star**4*epsilon + 21760.0*P**6*S**2*T**2*UGA_star**4 + 47104.0*P**6*S**2*T**2*UGA_star**3*epsilon + 172032.0*P**6*S*T**3*UGA_star**20*epsilon - 2752512.0*P**6*S*T**3*UGA_star**19*epsilon + 20815872.0*P**6*S*T**3*UGA_star**18*epsilon - 98746368.0*P**6*S*T**3*UGA_star**17*epsilon + 328788992.0*P**6*S*T**3*UGA_star**16*epsilon - 814485504.0*P**6*S*T**3*UGA_star**15*epsilon + 1552201728.0*P**6*S*T**3*UGA_star**14*epsilon - 2320338944.0*P**6*S*T**3*UGA_star**13*epsilon + 2748404736.0*P**6*S*T**3*UGA_star**12*epsilon - 2586845184.0*P**6*S*T**3*UGA_star**11*epsilon + 1927360512.0*P**6*S*T**3*UGA_star**10*epsilon - 1124229120.0*P**6*S*T**3*UGA_star**9*epsilon + 503043072.0*P**6*S*T**3*UGA_star**8*epsilon - 166842368.0*P**6*S*T**3*UGA_star**7*epsilon + 38664192.0*P**6*S*T**3*UGA_star**6*epsilon - 5591040.0*P**6*S*T**3*UGA_star**5*epsilon + 379904.0*P**6*S*T**3*UGA_star**4*epsilon + 81920.0*P**5*R**5*UGA_star**20*epsilon + 5120.0*P**5*R**5*UGA_star**20 - 819200.0*P**5*R**5*UGA_star**19*epsilon - 51200.0*P**5*R**5*UGA_star**19 + 3704832.0*P**5*R**5*UGA_star**18*epsilon + 231424.0*P**5*R**5*UGA_star**18 - 9996288.0*P**5*R**5*UGA_star**17*epsilon - 623616.0*P**5*R**5*UGA_star**17 + 17878016.0*P**5*R**5*UGA_star**16*epsilon + 1113088.0*P**5*R**5*UGA_star**16 - 22282240.0*P**5*R**5*UGA_star**15*epsilon - 1384448.0*P**5*R**5*UGA_star**15 + 19841024.0*P**5*R**5*UGA_star**14*epsilon + 1232896.0*P**5*R**5*UGA_star**14 - 12783616.0*P**5*R**5*UGA_star**13*epsilon - 800768.0*P**5*R**5*UGA_star**13 + 6023168.0*P**5*R**5*UGA_star**12*epsilon + 388096.0*P**5*R**5*UGA_star**12 - 2113536.0*P**5*R**5*UGA_star**11*epsilon - 145408.0*P**5*R**5*UGA_star**11 + 563200.0*P**5*R**5*UGA_star**10*epsilon + 43008.0*P**5*R**5*UGA_star**10 - 108544.0*P**5*R**5*UGA_star**9*epsilon - 9216.0*P**5*R**5*UGA_star**9 + 11264.0*P**5*R**5*UGA_star**8*epsilon + 1024.0*P**5*R**5*UGA_star**8 - 409600.0*P**5*R**4*S*UGA_star**20*epsilon - 25600.0*P**5*R**4*S*UGA_star**20 + 4505600.0*P**5*R**4*S*UGA_star**19*epsilon + 281600.0*P**5*R**4*S*UGA_star**19 - 22588416.0*P**5*R**4*S*UGA_star**18*epsilon - 1411072.0*P**5*R**4*S*UGA_star**18 + 68178944.0*P**5*R**4*S*UGA_star**17*epsilon + 4254208.0*P**5*R**4*S*UGA_star**17 - 137843712.0*P**5*R**4*S*UGA_star**16*epsilon - 8585728.0*P**5*R**4*S*UGA_star**16 + 196517888.0*P**5*R**4*S*UGA_star**15*epsilon + 12214272.0*P**5*R**4*S*UGA_star**15 - 202627072.0*P**5*R**4*S*UGA_star**14*epsilon - 12576768.0*P**5*R**4*S*UGA_star**14 + 152729600.0*P**5*R**4*S*UGA_star**13*epsilon + 9501696.0*P**5*R**4*S*UGA_star**13 - 84428800.0*P**5*R**4*S*UGA_star**12*epsilon - 5320704.0*P**5*R**4*S*UGA_star**12 + 34394112.0*P**5*R**4*S*UGA_star**11*epsilon + 2249728.0*P**5*R**4*S*UGA_star**11 - 10533888.0*P**5*R**4*S*UGA_star**10*epsilon - 745472.0*P**5*R**4*S*UGA_star**10 + 2511872.0*P**5*R**4*S*UGA_star**9*epsilon + 199168.0*P**5*R**4*S*UGA_star**9 - 451584.0*P**5*R**4*S*UGA_star**8*epsilon - 39424.0*P**5*R**4*S*UGA_star**8 + 45056.0*P**5*R**4*S*UGA_star**7*epsilon + 4096.0*P**5*R**4*S*UGA_star**7 - 411648.0*P**5*R**4*T*UGA_star**20*epsilon - 23040.0*P**5*R**4*T*UGA_star**20 + 4528128.0*P**5*R**4*T*UGA_star**19*epsilon + 253440.0*P**5*R**4*T*UGA_star**19 - 22738944.0*P**5*R**4*T*UGA_star**18*epsilon - 1273344.0*P**5*R**4*T*UGA_star**18 + 68788224.0*P**5*R**4*T*UGA_star**17*epsilon + 3855360.0*P**5*R**4*T*UGA_star**17 - 139238400.0*P**5*R**4*T*UGA_star**16*epsilon - 7810560.0*P**5*R**4*T*UGA_star**16 + 197974016.0*P**5*R**4*T*UGA_star**15*epsilon + 11105280.0*P**5*R**4*T*UGA_star**15 - 201834496.0*P**5*R**4*T*UGA_star**14*epsilon - 11289600.0*P**5*R**4*T*UGA_star**14 + 147857408.0*P**5*R**4*T*UGA_star**13*epsilon + 8183808.0*P**5*R**4*T*UGA_star**13 - 76853248.0*P**5*R**4*T*UGA_star**12*epsilon - 4124160.0*P**5*R**4*T*UGA_star**12 + 27699200.0*P**5*R**4*T*UGA_star**11*epsilon + 1359360.0*P**5*R**4*T*UGA_star**11 - 6842368.0*P**5*R**4*T*UGA_star**10*epsilon - 253440.0*P**5*R**4*T*UGA_star**10 + 1261568.0*P**5*R**4*T*UGA_star**9*epsilon + 15360.0*P**5*R**4*T*UGA_star**9 - 216064.0*P**5*R**4*T*UGA_star**8*epsilon + 1536.0*P**5*R**4*T*UGA_star**8 + 26624.0*P**5*R**4*T*UGA_star**7*epsilon + 819200.0*P**5*R**3*S**2*UGA_star**20*epsilon + 51200.0*P**5*R**3*S**2*UGA_star**20 - 9830400.0*P**5*R**3*S**2*UGA_star**19*epsilon - 614400.0*P**5*R**3*S**2*UGA_star**19 + 54124544.0*P**5*R**3*S**2*UGA_star**18*epsilon + 3381248.0*P**5*R**3*S**2*UGA_star**18 - 180817920.0*P**5*R**3*S**2*UGA_star**17*epsilon - 11284480.0*P**5*R**3*S**2*UGA_star**17 + 408338432.0*P**5*R**3*S**2*UGA_star**16*epsilon + 25442304.0*P**5*R**3*S**2*UGA_star**16 - 657121280.0*P**5*R**3*S**2*UGA_star**15*epsilon - 40857600.0*P**5*R**3*S**2*UGA_star**15 + 773836800.0*P**5*R**3*S**2*UGA_star**14*epsilon + 48015360.0*P**5*R**3*S**2*UGA_star**14 - 674193408.0*P**5*R**3*S**2*UGA_star**13*epsilon - 41803776.0*P**5*R**3*S**2*UGA_star**13 + 434749440.0*P**5*R**3*S**2*UGA_star**12*epsilon + 27064320.0*P**5*R**3*S**2*UGA_star**12 - 206372864.0*P**5*R**3*S**2*UGA_star**11*epsilon - 13058048.0*P**5*R**3*S**2*UGA_star**11 + 72007680.0*P**5*R**3*S**2*UGA_star**10*epsilon + 4761600.0*P**5*R**3*S**2*UGA_star**10 - 18944000.0*P**5*R**3*S**2*UGA_star**9*epsilon - 1372160.0*P**5*R**3*S**2*UGA_star**9 + 4030464.0*P**5*R**3*S**2*UGA_star**8*epsilon + 329728.0*P**5*R**3*S**2*UGA_star**8 - 696320.0*P**5*R**3*S**2*UGA_star**7*epsilon - 61440.0*P**5*R**3*S**2*UGA_star**7 + 69632.0*P**5*R**3*S**2*UGA_star**6*epsilon + 6144.0*P**5*R**3*S**2*UGA_star**6 + 1646592.0*P**5*R**3*S*T*UGA_star**20*epsilon + 92160.0*P**5*R**3*S*T*UGA_star**20 - 19759104.0*P**5*R**3*S*T*UGA_star**19*epsilon - 1105920.0*P**5*R**3*S*T*UGA_star**19 + 109025280.0*P**5*R**3*S*T*UGA_star**18*epsilon + 6104064.0*P**5*R**3*S*T*UGA_star**18 - 365629440.0*P**5*R**3*S*T*UGA_star**17*epsilon - 20481024.0*P**5*R**3*S*T*UGA_star**17 + 829646848.0*P**5*R**3*S*T*UGA_star**16*epsilon + 46492160.0*P**5*R**3*S*T*UGA_star**16 - 1341159424.0*P**5*R**3*S*T*UGA_star**15*epsilon - 75133440.0*P**5*R**3*S*T*UGA_star**15 + 1582921728.0*P**5*R**3*S*T*UGA_star**14*epsilon + 88473600.0*P**5*R**3*S*T*UGA_star**14 - 1374044160.0*P**5*R**3*S*T*UGA_star**13*epsilon - 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3308507136.0*P**5*R*S**3*T*UGA_star**11*epsilon - 178928640.0*P**5*R*S**3*T*UGA_star**11 + 1616136192.0*P**5*R*S**3*T*UGA_star**10*epsilon + 84142080.0*P**5*R*S**3*T*UGA_star**10 - 561917952.0*P**5*R*S**3*T*UGA_star**9*epsilon - 26787840.0*P**5*R*S**3*T*UGA_star**9 + 129429504.0*P**5*R*S**3*T*UGA_star**8*epsilon + 4753920.0*P**5*R*S**3*T*UGA_star**8 - 17369088.0*P**5*R*S**3*T*UGA_star**7*epsilon - 38400.0*P**5*R*S**3*T*UGA_star**7 + 1261568.0*P**5*R*S**3*T*UGA_star**6*epsilon - 161280.0*P**5*R*S**3*T*UGA_star**6 - 194560.0*P**5*R*S**3*T*UGA_star**5*epsilon + 23040.0*P**5*R*S**3*T*UGA_star**5 + 38912.0*P**5*R*S**3*T*UGA_star**4*epsilon + 2451456.0*P**5*R*S**2*T**2*UGA_star**20*epsilon + 96768.0*P**5*R*S**2*T**2*UGA_star**20 - 34320384.0*P**5*R*S**2*T**2*UGA_star**19*epsilon - 1354752.0*P**5*R*S**2*T**2*UGA_star**19 + 224040960.0*P**5*R*S**2*T**2*UGA_star**18*epsilon + 8865792.0*P**5*R*S**2*T**2*UGA_star**18 - 903499776.0*P**5*R*S**2*T**2*UGA_star**17*epsilon - 35933184.0*P**5*R*S**2*T**2*UGA_star**17 + 2513160192.0*P**5*R*S**2*T**2*UGA_star**16*epsilon + 100700160.0*P**5*R*S**2*T**2*UGA_star**16 - 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124692480.0*P**4*R**4*S**2*UGA_star**17*epsilon - 6996480.0*P**4*R**4*S**2*UGA_star**17 + 222096384.0*P**4*R**4*S**2*UGA_star**16*epsilon + 12510976.0*P**4*R**4*S**2*UGA_star**16 - 276938752.0*P**4*R**4*S**2*UGA_star**15*epsilon - 15715328.0*P**4*R**4*S**2*UGA_star**15 + 249884672.0*P**4*R**4*S**2*UGA_star**14*epsilon + 14361088.0*P**4*R**4*S**2*UGA_star**14 - 167563264.0*P**4*R**4*S**2*UGA_star**13*epsilon - 9829376.0*P**4*R**4*S**2*UGA_star**13 + 85806080.0*P**4*R**4*S**2*UGA_star**12*epsilon + 5190400.0*P**4*R**4*S**2*UGA_star**12 - 34146304.0*P**4*R**4*S**2*UGA_star**11*epsilon - 2157056.0*P**4*R**4*S**2*UGA_star**11 + 10176512.0*P**4*R**4*S**2*UGA_star**10*epsilon + 686848.0*P**4*R**4*S**2*UGA_star**10 - 1902592.0*P**4*R**4*S**2*UGA_star**9*epsilon - 147968.0*P**4*R**4*S**2*UGA_star**9 + 103424.0*P**4*R**4*S**2*UGA_star**8*epsilon + 15616.0*P**4*R**4*S**2*UGA_star**8 + 20480.0*P**4*R**4*S**2*UGA_star**7*epsilon + 2048000.0*P**4*R**4*S*T*UGA_star**20*epsilon + 128000.0*P**4*R**4*S*T*UGA_star**20 - 20480000.0*P**4*R**4*S*T*UGA_star**19*epsilon - 1280000.0*P**4*R**4*S*T*UGA_star**19 + 92303360.0*P**4*R**4*S*T*UGA_star**18*epsilon + 5765120.0*P**4*R**4*S*T*UGA_star**18 - 247050240.0*P**4*R**4*S*T*UGA_star**17*epsilon - 15406080.0*P**4*R**4*S*T*UGA_star**17 + 435419136.0*P**4*R**4*S*T*UGA_star**16*epsilon + 27083264.0*P**4*R**4*S*T*UGA_star**16 - 529563648.0*P**4*R**4*S*T*UGA_star**15*epsilon - 32837632.0*P**4*R**4*S*T*UGA_star**15 + 453111808.0*P**4*R**4*S*T*UGA_star**14*epsilon + 28055552.0*P**4*R**4*S*T*UGA_star**14 - 273657856.0*P**4*R**4*S*T*UGA_star**13*epsilon - 17065984.0*P**4*R**4*S*T*UGA_star**13 + 116244480.0*P**4*R**4*S*T*UGA_star**12*epsilon + 7516160.0*P**4*R**4*S*T*UGA_star**12 - 34902016.0*P**4*R**4*S*T*UGA_star**11*epsilon - 2525184.0*P**4*R**4*S*T*UGA_star**11 + 7610368.0*P**4*R**4*S*T*UGA_star**10*epsilon + 704512.0*P**4*R**4*S*T*UGA_star**10 - 1138688.0*P**4*R**4*S*T*UGA_star**9*epsilon - 156672.0*P**4*R**4*S*T*UGA_star**9 + 34816.0*P**4*R**4*S*T*UGA_star**8*epsilon + 18944.0*P**4*R**4*S*T*UGA_star**8 + 20480.0*P**4*R**4*S*T*UGA_star**7*epsilon + 1029120.0*P**4*R**4*T**2*UGA_star**20*epsilon + 57600.0*P**4*R**4*T**2*UGA_star**20 - 10291200.0*P**4*R**4*T**2*UGA_star**19*epsilon - 576000.0*P**4*R**4*T**2*UGA_star**19 + 46417920.0*P**4*R**4*T**2*UGA_star**18*epsilon + 2599680.0*P**4*R**4*T**2*UGA_star**18 - 124231680.0*P**4*R**4*T**2*UGA_star**17*epsilon - 6965760.0*P**4*R**4*T**2*UGA_star**17 + 218179584.0*P**4*R**4*T**2*UGA_star**16*epsilon + 12249856.0*P**4*R**4*T**2*UGA_star**16 - 262234112.0*P**4*R**4*T**2*UGA_star**15*epsilon - 14732288.0*P**4*R**4*T**2*UGA_star**15 + 217935872.0*P**4*R**4*T**2*UGA_star**14*epsilon + 12210688.0*P**4*R**4*T**2*UGA_star**14 - 123408384.0*P**4*R**4*T**2*UGA_star**13*epsilon - 6818816.0*P**4*R**4*T**2*UGA_star**13 + 45736960.0*P**4*R**4*T**2*UGA_star**12*epsilon + 2394880.0*P**4*R**4*T**2*UGA_star**12 - 10491904.0*P**4*R**4*T**2*UGA_star**11*epsilon - 436736.0*P**4*R**4*T**2*UGA_star**11 + 1615872.0*P**4*R**4*T**2*UGA_star**10*epsilon + 11008.0*P**4*R**4*T**2*UGA_star**10 - 315392.0*P**4*R**4*T**2*UGA_star**9*epsilon + 5632.0*P**4*R**4*T**2*UGA_star**9 + 57344.0*P**4*R**4*T**2*UGA_star**8*epsilon + 256.0*P**4*R**4*T**2*UGA_star**8 - 1372160.0*P**4*R**3*S**3*UGA_star**20*epsilon - 76800.0*P**4*R**3*S**3*UGA_star**20 + 15093760.0*P**4*R**3*S**3*UGA_star**19*epsilon + 844800.0*P**4*R**3*S**3*UGA_star**19 - 75427840.0*P**4*R**3*S**3*UGA_star**18*epsilon - 4224000.0*P**4*R**3*S**3*UGA_star**18 + 226222080.0*P**4*R**3*S**3*UGA_star**17*epsilon + 12687360.0*P**4*R**3*S**3*UGA_star**17 - 453257216.0*P**4*R**3*S**3*UGA_star**16*epsilon - 25498624.0*P**4*R**3*S**3*UGA_star**16 + 639442944.0*P**4*R**3*S**3*UGA_star**15*epsilon + 36182016.0*P**4*R**3*S**3*UGA_star**15 - 653346816.0*P**4*R**3*S**3*UGA_star**14*epsilon - 37361664.0*P**4*R**3*S**3*UGA_star**14 + 491618304.0*P**4*R**3*S**3*UGA_star**13*epsilon + 28655616.0*P**4*R**3*S**3*UGA_star**13 - 276381696.0*P**4*R**3*S**3*UGA_star**12*epsilon - 16671744.0*P**4*R**3*S**3*UGA_star**12 + 118378496.0*P**4*R**3*S**3*UGA_star**11*epsilon + 7578624.0*P**4*R**3*S**3*UGA_star**11 - 39337984.0*P**4*R**3*S**3*UGA_star**10*epsilon - 2774016.0*P**4*R**3*S**3*UGA_star**10 + 9773056.0*P**4*R**3*S**3*UGA_star**9*epsilon + 804864.0*P**4*R**3*S**3*UGA_star**9 - 1419264.0*P**4*R**3*S**3*UGA_star**8*epsilon - 162816.0*P**4*R**3*S**3*UGA_star**8 - 16384.0*P**4*R**3*S**3*UGA_star**7*epsilon + 16384.0*P**4*R**3*S**3*UGA_star**7 + 30720.0*P**4*R**3*S**3*UGA_star**6*epsilon - 4096000.0*P**4*R**3*S**2*T*UGA_star**20*epsilon - 256000.0*P**4*R**3*S**2*T*UGA_star**20 + 45056000.0*P**4*R**3*S**2*T*UGA_star**19*epsilon + 2816000.0*P**4*R**3*S**2*T*UGA_star**19 - 225249280.0*P**4*R**3*S**2*T*UGA_star**18*epsilon - 14069760.0*P**4*R**3*S**2*T*UGA_star**18 + 675440640.0*P**4*R**3*S**2*T*UGA_star**17*epsilon + 42132480.0*P**4*R**3*S**2*T*UGA_star**17 - 1349699584.0*P**4*R**3*S**2*T*UGA_star**16*epsilon - 84002816.0*P**4*R**3*S**2*T*UGA_star**16 + 1887481856.0*P**4*R**3*S**2*T*UGA_star**15*epsilon + 117126144.0*P**4*R**3*S**2*T*UGA_star**15 - 1886869504.0*P**4*R**3*S**2*T*UGA_star**14*epsilon - 116760576.0*P**4*R**3*S**2*T*UGA_star**14 + 1352953856.0*P**4*R**3*S**2*T*UGA_star**13*epsilon + 83748864.0*P**4*R**3*S**2*T*UGA_star**13 - 688943104.0*P**4*R**3*S**2*T*UGA_star**12*epsilon - 43186176.0*P**4*R**3*S**2*T*UGA_star**12 + 244875264.0*P**4*R**3*S**2*T*UGA_star**11*epsilon + 16162816.0*P**4*R**3*S**2*T*UGA_star**11 - 60512256.0*P**4*R**3*S**2*T*UGA_star**10*epsilon - 4665344.0*P**4*R**3*S**2*T*UGA_star**10 + 11003904.0*P**4*R**3*S**2*T*UGA_star**9*epsilon + 1176576.0*P**4*R**3*S**2*T*UGA_star**9 - 1509376.0*P**4*R**3*S**2*T*UGA_star**8*epsilon - 251904.0*P**4*R**3*S**2*T*UGA_star**8 + 36864.0*P**4*R**3*S**2*T*UGA_star**7*epsilon + 29696.0*P**4*R**3*S**2*T*UGA_star**7 + 30720.0*P**4*R**3*S**2*T*UGA_star**6*epsilon - 4116480.0*P**4*R**3*S*T**2*UGA_star**20*epsilon - 230400.0*P**4*R**3*S*T**2*UGA_star**20 + 45281280.0*P**4*R**3*S*T**2*UGA_star**19*epsilon + 2534400.0*P**4*R**3*S*T**2*UGA_star**19 - 226728960.0*P**4*R**3*S*T**2*UGA_star**18*epsilon - 12695040.0*P**4*R**3*S*T**2*UGA_star**18 + 681507840.0*P**4*R**3*S*T**2*UGA_star**17*epsilon + 38184960.0*P**4*R**3*S*T**2*UGA_star**17 - 1364612096.0*P**4*R**3*S*T**2*UGA_star**16*epsilon - 76511744.0*P**4*R**3*S*T**2*UGA_star**16 + 1907713024.0*P**4*R**3*S*T**2*UGA_star**15*epsilon + 106966016.0*P**4*R**3*S*T**2*UGA_star**15 - 1894946816.0*P**4*R**3*S*T**2*UGA_star**14*epsilon - 106015744.0*P**4*R**3*S*T**2*UGA_star**14 + 1332189184.0*P**4*R**3*S*T**2*UGA_star**13*epsilon + 73895936.0*P**4*R**3*S*T**2*UGA_star**13 - 646117376.0*P**4*R**3*S*T**2*UGA_star**12*epsilon - 34883584.0*P**4*R**3*S*T**2*UGA_star**12 + 205060096.0*P**4*R**3*S*T**2*UGA_star**11*epsilon + 10118144.0*P**4*R**3*S*T**2*UGA_star**11 - 39455744.0*P**4*R**3*S*T**2*UGA_star**10*epsilon - 1302016.0*P**4*R**3*S*T**2*UGA_star**10 + 5057536.0*P**4*R**3*S*T**2*UGA_star**9*epsilon - 96256.0*P**4*R**3*S*T**2*UGA_star**9 - 1025024.0*P**4*R**3*S*T**2*UGA_star**8*epsilon + 34304.0*P**4*R**3*S*T**2*UGA_star**8 + 193536.0*P**4*R**3*S*T**2*UGA_star**7*epsilon + 1024.0*P**4*R**3*S*T**2*UGA_star**7 - 1361920.0*P**4*R**3*T**3*UGA_star**20*epsilon - 53760.0*P**4*R**3*T**3*UGA_star**20 + 14981120.0*P**4*R**3*T**3*UGA_star**19*epsilon + 591360.0*P**4*R**3*T**3*UGA_star**19 - 75202560.0*P**4*R**3*T**3*UGA_star**18*epsilon - 2979840.0*P**4*R**3*T**3*UGA_star**18 + 227112960.0*P**4*R**3*T**3*UGA_star**17*epsilon + 9070080.0*P**4*R**3*T**3*UGA_star**17 - 457656320.0*P**4*R**3*T**3*UGA_star**16*epsilon - 18496000.0*P**4*R**3*T**3*UGA_star**16 + 644341760.0*P**4*R**3*T**3*UGA_star**15*epsilon + 26455040.0*P**4*R**3*T**3*UGA_star**15 - 643840000.0*P**4*R**3*T**3*UGA_star**14*epsilon - 26951680.0*P**4*R**3*T**3*UGA_star**14 + 452802560.0*P**4*R**3*T**3*UGA_star**13*epsilon + 19389440.0*P**4*R**3*T**3*UGA_star**13 - 215756800.0*P**4*R**3*T**3*UGA_star**12*epsilon - 9479680.0*P**4*R**3*T**3*UGA_star**12 + 63160320.0*P**4*R**3*T**3*UGA_star**11*epsilon + 2854400.0*P**4*R**3*T**3*UGA_star**11 - 8202240.0*P**4*R**3*T**3*UGA_star**10*epsilon - 378880.0*P**4*R**3*T**3*UGA_star**10 - 593920.0*P**4*R**3*T**3*UGA_star**9*epsilon - 33280.0*P**4*R**3*T**3*UGA_star**9 + 204800.0*P**4*R**3*T**3*UGA_star**8*epsilon + 12800.0*P**4*R**3*T**3*UGA_star**8 + 10240.0*P**4*R**3*T**3*UGA_star**7*epsilon + 1029120.0*P**4*R**2*S**4*UGA_star**20*epsilon + 57600.0*P**4*R**2*S**4*UGA_star**20 - 12349440.0*P**4*R**2*S**4*UGA_star**19*epsilon - 691200.0*P**4*R**2*S**4*UGA_star**19 + 67752960.0*P**4*R**2*S**4*UGA_star**18*epsilon + 3793920.0*P**4*R**2*S**4*UGA_star**18 - 224655360.0*P**4*R**2*S**4*UGA_star**17*epsilon - 12595200.0*P**4*R**2*S**4*UGA_star**17 + 501325824.0*P**4*R**2*S**4*UGA_star**16*epsilon + 28175616.0*P**4*R**2*S**4*UGA_star**16 - 793292800.0*P**4*R**2*S**4*UGA_star**15*epsilon - 44789760.0*P**4*R**2*S**4*UGA_star**15 + 913653760.0*P**4*R**2*S**4*UGA_star**14*epsilon + 52024320.0*P**4*R**2*S**4*UGA_star**14 - 774144000.0*P**4*R**2*S**4*UGA_star**13*epsilon - 44789760.0*P**4*R**2*S**4*UGA_star**13 + 483087360.0*P**4*R**2*S**4*UGA_star**12*epsilon + 28834560.0*P**4*R**2*S**4*UGA_star**12 - 221442048.0*P**4*R**2*S**4*UGA_star**11*epsilon - 14072832.0*P**4*R**2*S**4*UGA_star**11 + 74880000.0*P**4*R**2*S**4*UGA_star**10*epsilon + 5391360.0*P**4*R**2*S**4*UGA_star**10 - 19077120.0*P**4*R**2*S**4*UGA_star**9*epsilon - 1720320.0*P**4*R**2*S**4*UGA_star**9 + 3594240.0*P**4*R**2*S**4*UGA_star**8*epsilon + 464640.0*P**4*R**2*S**4*UGA_star**8 - 307200.0*P**4*R**2*S**4*UGA_star**7*epsilon - 92160.0*P**4*R**2*S**4*UGA_star**7 - 75776.0*P**4*R**2*S**4*UGA_star**6*epsilon + 9216.0*P**4*R**2*S**4*UGA_star**6 + 20480.0*P**4*R**2*S**4*UGA_star**5*epsilon + 4096000.0*P**4*R**2*S**3*T*UGA_star**20*epsilon + 256000.0*P**4*R**2*S**3*T*UGA_star**20 - 49152000.0*P**4*R**2*S**3*T*UGA_star**19*epsilon - 3072000.0*P**4*R**2*S**3*T*UGA_star**19 + 269987840.0*P**4*R**2*S**3*T*UGA_star**18*epsilon + 16865280.0*P**4*R**2*S**3*T*UGA_star**18 - 897105920.0*P**4*R**2*S**3*T*UGA_star**17*epsilon - 55971840.0*P**4*R**2*S**3*T*UGA_star**17 + 2006644736.0*P**4*R**2*S**3*T*UGA_star**16*epsilon + 124949504.0*P**4*R**2*S**3*T*UGA_star**16 - 3179560960.0*P**4*R**2*S**3*T*UGA_star**15*epsilon - 197437440.0*P**4*R**2*S**3*T*UGA_star**15 + 3653847040.0*P**4*R**2*S**3*T*UGA_star**14*epsilon + 226145280.0*P**4*R**2*S**3*T*UGA_star**14 - 3061985280.0*P**4*R**2*S**3*T*UGA_star**13*epsilon - 188989440.0*P**4*R**2*S**3*T*UGA_star**13 + 1852600320.0*P**4*R**2*S**3*T*UGA_star**12*epsilon + 114462720.0*P**4*R**2*S**3*T*UGA_star**12 - 788119552.0*P**4*R**2*S**3*T*UGA_star**11*epsilon - 49451008.0*P**4*R**2*S**3*T*UGA_star**11 + 225607680.0*P**4*R**2*S**3*T*UGA_star**10*epsilon + 15114240.0*P**4*R**2*S**3*T*UGA_star**10 - 41738240.0*P**4*R**2*S**3*T*UGA_star**9*epsilon - 3502080.0*P**4*R**2*S**3*T*UGA_star**9 + 5642240.0*P**4*R**2*S**3*T*UGA_star**8*epsilon + 783360.0*P**4*R**2*S**3*T*UGA_star**8 - 819200.0*P**4*R**2*S**3*T*UGA_star**7*epsilon - 174080.0*P**4*R**2*S**3*T*UGA_star**7 + 34816.0*P**4*R**2*S**3*T*UGA_star**6*epsilon + 21504.0*P**4*R**2*S**3*T*UGA_star**6 + 20480.0*P**4*R**2*S**3*T*UGA_star**5*epsilon + 6174720.0*P**4*R**2*S**2*T**2*UGA_star**20*epsilon + 345600.0*P**4*R**2*S**2*T**2*UGA_star**20 - 74096640.0*P**4*R**2*S**2*T**2*UGA_star**19*epsilon - 4147200.0*P**4*R**2*S**2*T**2*UGA_star**19 + 407854080.0*P**4*R**2*S**2*T**2*UGA_star**18*epsilon + 22832640.0*P**4*R**2*S**2*T**2*UGA_star**18 - 1360558080.0*P**4*R**2*S**2*T**2*UGA_star**17*epsilon - 76193280.0*P**4*R**2*S**2*T**2*UGA_star**17 + 3059994624.0*P**4*R**2*S**2*T**2*UGA_star**16*epsilon + 171403776.0*P**4*R**2*S**2*T**2*UGA_star**16 - 4880025600.0*P**4*R**2*S**2*T**2*UGA_star**15*epsilon - 273246720.0*P**4*R**2*S**2*T**2*UGA_star**15 + 5644477440.0*P**4*R**2*S**2*T**2*UGA_star**14*epsilon + 315394560.0*P**4*R**2*S**2*T**2*UGA_star**14 - 4752322560.0*P**4*R**2*S**2*T**2*UGA_star**13*epsilon - 263900160.0*P**4*R**2*S**2*T**2*UGA_star**13 + 2871982080.0*P**4*R**2*S**2*T**2*UGA_star**12*epsilon + 156879360.0*P**4*R**2*S**2*T**2*UGA_star**12 - 1201287168.0*P**4*R**2*S**2*T**2*UGA_star**11*epsilon - 62733312.0*P**4*R**2*S**2*T**2*UGA_star**11 + 323527680.0*P**4*R**2*S**2*T**2*UGA_star**10*epsilon + 14584320.0*P**4*R**2*S**2*T**2*UGA_star**10 - 49305600.0*P**4*R**2*S**2*T**2*UGA_star**9*epsilon - 898560.0*P**4*R**2*S**2*T**2*UGA_star**9 + 4423680.0*P**4*R**2*S**2*T**2*UGA_star**8*epsilon - 391680.0*P**4*R**2*S**2*T**2*UGA_star**8 - 1075200.0*P**4*R**2*S**2*T**2*UGA_star**7*epsilon + 69120.0*P**4*R**2*S**2*T**2*UGA_star**7 + 236544.0*P**4*R**2*S**2*T**2*UGA_star**6*epsilon + 1536.0*P**4*R**2*S**2*T**2*UGA_star**6 + 4085760.0*P**4*R**2*S*T**3*UGA_star**20*epsilon + 161280.0*P**4*R**2*S*T**3*UGA_star**20 - 49029120.0*P**4*R**2*S*T**3*UGA_star**19*epsilon - 1935360.0*P**4*R**2*S*T**3*UGA_star**19 + 270551040.0*P**4*R**2*S*T**3*UGA_star**18*epsilon + 10713600.0*P**4*R**2*S*T**3*UGA_star**18 - 906946560.0*P**4*R**2*S*T**3*UGA_star**17*epsilon - 36149760.0*P**4*R**2*S*T**3*UGA_star**17 + 2054246400.0*P**4*R**2*S*T**3*UGA_star**16*epsilon + 82698240.0*P**4*R**2*S*T**3*UGA_star**16 - 3305441280.0*P**4*R**2*S*T**3*UGA_star**15*epsilon - 134853120.0*P**4*R**2*S*T**3*UGA_star**15 + 3862333440.0*P**4*R**2*S*T**3*UGA_star**14*epsilon + 160220160.0*P**4*R**2*S*T**3*UGA_star**14 - 3284766720.0*P**4*R**2*S*T**3*UGA_star**13*epsilon - 139023360.0*P**4*R**2*S*T**3*UGA_star**13 + 1997936640.0*P**4*R**2*S*T**3*UGA_star**12*epsilon + 86607360.0*P**4*R**2*S*T**3*UGA_star**12 - 829009920.0*P**4*R**2*S*T**3*UGA_star**11*epsilon - 37002240.0*P**4*R**2*S*T**3*UGA_star**11 + 208926720.0*P**4*R**2*S*T**3*UGA_star**10*epsilon + 9699840.0*P**4*R**2*S*T**3*UGA_star**10 - 20613120.0*P**4*R**2*S*T**3*UGA_star**9*epsilon - 1036800.0*P**4*R**2*S*T**3*UGA_star**9 - 2949120.0*P**4*R**2*S*T**3*UGA_star**8*epsilon - 138240.0*P**4*R**2*S*T**3*UGA_star**8 + 645120.0*P**4*R**2*S*T**3*UGA_star**7*epsilon + 38400.0*P**4*R**2*S*T**3*UGA_star**7 + 30720.0*P**4*R**2*S*T**3*UGA_star**6*epsilon + 931840.0*P**4*R**2*T**4*UGA_star**20*epsilon + 17920.0*P**4*R**2*T**4*UGA_star**20 - 11182080.0*P**4*R**2*T**4*UGA_star**19*epsilon - 215040.0*P**4*R**2*T**4*UGA_star**19 + 61895680.0*P**4*R**2*T**4*UGA_star**18*epsilon + 1200640.0*P**4*R**2*T**4*UGA_star**18 - 208803840.0*P**4*R**2*T**4*UGA_star**17*epsilon - 4121600.0*P**4*R**2*T**4*UGA_star**17 + 477521920.0*P**4*R**2*T**4*UGA_star**16*epsilon + 9680640.0*P**4*R**2*T**4*UGA_star**16 - 778444800.0*P**4*R**2*T**4*UGA_star**15*epsilon - 16373760.0*P**4*R**2*T**4*UGA_star**15 + 924825600.0*P**4*R**2*T**4*UGA_star**14*epsilon + 20428800.0*P**4*R**2*T**4*UGA_star**14 - 802959360.0*P**4*R**2*T**4*UGA_star**13*epsilon - 18923520.0*P**4*R**2*T**4*UGA_star**13 + 501258240.0*P**4*R**2*T**4*UGA_star**12*epsilon + 12902400.0*P**4*R**2*T**4*UGA_star**12 - 215326720.0*P**4*R**2*T**4*UGA_star**11*epsilon - 6307840.0*P**4*R**2*T**4*UGA_star**11 + 57308160.0*P**4*R**2*T**4*UGA_star**10*epsilon + 2096640.0*P**4*R**2*T**4*UGA_star**10 - 6584320.0*P**4*R**2*T**4*UGA_star**9*epsilon - 424960.0*P**4*R**2*T**4*UGA_star**9 - 645120.0*P**4*R**2*T**4*UGA_star**8*epsilon + 39680.0*P**4*R**2*T**4*UGA_star**8 + 204800.0*P**4*R**2*T**4*UGA_star**7*epsilon - 411648.0*P**4*R*S**5*UGA_star**20*epsilon - 23040.0*P**4*R*S**5*UGA_star**20 + 5351424.0*P**4*R*S**5*UGA_star**19*epsilon + 299520.0*P**4*R*S**5*UGA_star**19 - 31985664.0*P**4*R*S**5*UGA_star**18*epsilon - 1790976.0*P**4*R*S**5*UGA_star**18 + 116275200.0*P**4*R*S**5*UGA_star**17*epsilon + 6517248.0*P**4*R*S**5*UGA_star**17 - 286448640.0*P**4*R*S**5*UGA_star**16*epsilon - 16087552.0*P**4*R*S**5*UGA_star**16 + 504070144.0*P**4*R*S**5*UGA_star**15*epsilon + 28413440.0*P**4*R*S**5*UGA_star**15 - 650067968.0*P**4*R*S**5*UGA_star**14*epsilon - 36889600.0*P**4*R*S**5*UGA_star**14 + 619429888.0*P**4*R*S**5*UGA_star**13*epsilon + 35599872.0*P**4*R*S**5*UGA_star**13 - 433044480.0*P**4*R*S**5*UGA_star**12*epsilon - 25529856.0*P**4*R*S**5*UGA_star**12 + 216606720.0*P**4*R*S**5*UGA_star**11*epsilon + 13499904.0*P**4*R*S**5*UGA_star**11 - 73396224.0*P**4*R*S**5*UGA_star**10*epsilon - 5237760.0*P**4*R*S**5*UGA_star**10 + 14874624.0*P**4*R*S**5*UGA_star**9*epsilon + 1543680.0*P**4*R*S**5*UGA_star**9 - 1127424.0*P**4*R*S**5*UGA_star**8*epsilon - 400896.0*P**4*R*S**5*UGA_star**8 - 184320.0*P**4*R*S**5*UGA_star**7*epsilon + 107008.0*P**4*R*S**5*UGA_star**7 + 81920.0*P**4*R*S**5*UGA_star**6*epsilon - 23552.0*P**4*R*S**5*UGA_star**6 - 28672.0*P**4*R*S**5*UGA_star**5*epsilon + 2560.0*P**4*R*S**5*UGA_star**5 + 5120.0*P**4*R*S**5*UGA_star**4*epsilon - 2048000.0*P**4*R*S**4*T*UGA_star**20*epsilon - 128000.0*P**4*R*S**4*T*UGA_star**20 + 26624000.0*P**4*R*S**4*T*UGA_star**19*epsilon + 1664000.0*P**4*R*S**4*T*UGA_star**19 - 159411200.0*P**4*R*S**4*T*UGA_star**18*epsilon - 9958400.0*P**4*R*S**4*T*UGA_star**18 + 581596160.0*P**4*R*S**4*T*UGA_star**17*epsilon + 36293120.0*P**4*R*S**4*T*UGA_star**17 - 1440840704.0*P**4*R*S**4*T*UGA_star**16*epsilon - 89752576.0*P**4*R*S**4*T*UGA_star**16 + 2555101184.0*P**4*R*S**4*T*UGA_star**15*epsilon + 158754816.0*P**4*R*S**4*T*UGA_star**15 - 3327923200.0*P**4*R*S**4*T*UGA_star**14*epsilon - 206074880.0*P**4*R*S**4*T*UGA_star**14 + 3209589760.0*P**4*R*S**4*T*UGA_star**13*epsilon + 197936640.0*P**4*R*S**4*T*UGA_star**13 - 2275415040.0*P**4*R*S**4*T*UGA_star**12*epsilon - 139729920.0*P**4*R*S**4*T*UGA_star**12 + 1155227648.0*P**4*R*S**4*T*UGA_star**11*epsilon + 70771712.0*P**4*R*S**4*T*UGA_star**11 - 396489728.0*P**4*R*S**4*T*UGA_star**10*epsilon - 24476672.0*P**4*R*S**4*T*UGA_star**10 + 80583680.0*P**4*R*S**4*T*UGA_star**9*epsilon + 5281280.0*P**4*R*S**4*T*UGA_star**9 - 6210560.0*P**4*R*S**4*T*UGA_star**8*epsilon - 650240.0*P**4*R*S**4*T*UGA_star**8 - 286720.0*P**4*R*S**4*T*UGA_star**7*epsilon + 102400.0*P**4*R*S**4*T*UGA_star**7 - 134144.0*P**4*R*S**4*T*UGA_star**6*epsilon - 39936.0*P**4*R*S**4*T*UGA_star**6 + 31744.0*P**4*R*S**4*T*UGA_star**5*epsilon + 6656.0*P**4*R*S**4*T*UGA_star**5 + 5120.0*P**4*R*S**4*T*UGA_star**4*epsilon - 4116480.0*P**4*R*S**3*T**2*UGA_star**20*epsilon - 230400.0*P**4*R*S**3*T**2*UGA_star**20 + 53514240.0*P**4*R*S**3*T**2*UGA_star**19*epsilon + 2995200.0*P**4*R*S**3*T**2*UGA_star**19 - 321192960.0*P**4*R*S**3*T**2*UGA_star**18*epsilon - 17978880.0*P**4*R*S**3*T**2*UGA_star**18 + 1177635840.0*P**4*R*S**3*T**2*UGA_star**17*epsilon + 65925120.0*P**4*R*S**3*T**2*UGA_star**17 - 2939713536.0*P**4*R*S**3*T**2*UGA_star**16*epsilon - 164552704.0*P**4*R*S**3*T**2*UGA_star**16 + 5268248576.0*P**4*R*S**3*T**2*UGA_star**15*epsilon + 294692864.0*P**4*R*S**3*T**2*UGA_star**15 - 6957685760.0*P**4*R*S**3*T**2*UGA_star**14*epsilon - 388395520.0*P**4*R*S**3*T**2*UGA_star**14 + 6832604160.0*P**4*R*S**3*T**2*UGA_star**13*epsilon + 379499520.0*P**4*R*S**3*T**2*UGA_star**13 - 4960296960.0*P**4*R*S**3*T**2*UGA_star**12*epsilon - 272348160.0*P**4*R*S**3*T**2*UGA_star**12 + 2601541632.0*P**4*R*S**3*T**2*UGA_star**11*epsilon + 139035648.0*P**4*R*S**3*T**2*UGA_star**11 - 937393152.0*P**4*R*S**3*T**2*UGA_star**10*epsilon - 46668288.0*P**4*R*S**3*T**2*UGA_star**10 + 208081920.0*P**4*R*S**3*T**2*UGA_star**9*epsilon + 8017920.0*P**4*R*S**3*T**2*UGA_star**9 - 21043200.0*P**4*R*S**3*T**2*UGA_star**8*epsilon + 399360.0*P**4*R*S**3*T**2*UGA_star**8 + 30720.0*P**4*R*S**3*T**2*UGA_star**7*epsilon - 450560.0*P**4*R*S**3*T**2*UGA_star**7 - 336896.0*P**4*R*S**3*T**2*UGA_star**6*epsilon + 57856.0*P**4*R*S**3*T**2*UGA_star**6 + 121856.0*P**4*R*S**3*T**2*UGA_star**5*epsilon + 1024.0*P**4*R*S**3*T**2*UGA_star**5 - 4085760.0*P**4*R*S**2*T**3*UGA_star**20*epsilon - 161280.0*P**4*R*S**2*T**3*UGA_star**20 + 53114880.0*P**4*R*S**2*T**3*UGA_star**19*epsilon + 2096640.0*P**4*R*S**2*T**3*UGA_star**19 - 319580160.0*P**4*R*S**2*T**3*UGA_star**18*epsilon - 12648960.0*P**4*R*S**2*T**3*UGA_star**18 + 1177497600.0*P**4*R*S**2*T**3*UGA_star**17*epsilon + 46863360.0*P**4*R*S**2*T**3*UGA_star**17 - 2961131520.0*P**4*R*S**2*T**3*UGA_star**16*epsilon - 118848000.0*P**4*R*S**2*T**3*UGA_star**16 + 5359073280.0*P**4*R*S**2*T**3*UGA_star**15*epsilon + 217551360.0*P**4*R*S**2*T**3*UGA_star**15 - 7165009920.0*P**4*R*S**2*T**3*UGA_star**14*epsilon - 295073280.0*P**4*R*S**2*T**3*UGA_star**14 + 7139727360.0*P**4*R*S**2*T**3*UGA_star**13*epsilon + 299243520.0*P**4*R*S**2*T**3*UGA_star**13 - 5269800960.0*P**4*R*S**2*T**3*UGA_star**12*epsilon - 225630720.0*P**4*R*S**2*T**3*UGA_star**12 + 2811463680.0*P**4*R*S**2*T**3*UGA_star**11*epsilon + 123609600.0*P**4*R*S**2*T**3*UGA_star**11 - 1025034240.0*P**4*R*S**2*T**3*UGA_star**10*epsilon - 46702080.0*P**4*R*S**2*T**3*UGA_star**10 + 222167040.0*P**4*R*S**2*T**3*UGA_star**9*epsilon + 10736640.0*P**4*R*S**2*T**3*UGA_star**9 - 14899200.0*P**4*R*S**2*T**3*UGA_star**8*epsilon - 898560.0*P**4*R*S**2*T**3*UGA_star**8 - 4208640.0*P**4*R*S**2*T**3*UGA_star**7*epsilon - 176640.0*P**4*R*S**2*T**3*UGA_star**7 + 675840.0*P**4*R*S**2*T**3*UGA_star**6*epsilon + 38400.0*P**4*R*S**2*T**3*UGA_star**6 + 30720.0*P**4*R*S**2*T**3*UGA_star**5*epsilon - 1863680.0*P**4*R*S*T**4*UGA_star**20*epsilon - 35840.0*P**4*R*S*T**4*UGA_star**20 + 24227840.0*P**4*R*S*T**4*UGA_star**19*epsilon + 465920.0*P**4*R*S*T**4*UGA_star**19 - 146155520.0*P**4*R*S*T**4*UGA_star**18*epsilon - 2831360.0*P**4*R*S*T**4*UGA_star**18 + 541399040.0*P**4*R*S*T**4*UGA_star**17*epsilon + 10644480.0*P**4*R*S*T**4*UGA_star**17 - 1372651520.0*P**4*R*S*T**4*UGA_star**16*epsilon - 27604480.0*P**4*R*S*T**4*UGA_star**16 + 2511933440.0*P**4*R*S*T**4*UGA_star**15*epsilon + 52108800.0*P**4*R*S*T**4*UGA_star**15 - 3406540800.0*P**4*R*S*T**4*UGA_star**14*epsilon - 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6553600.0*P**3*R**5*T**2*UGA_star**19*epsilon - 409600.0*P**3*R**5*T**2*UGA_star**19 + 22994944.0*P**3*R**5*T**2*UGA_star**18*epsilon + 1435648.0*P**3*R**5*T**2*UGA_star**18 - 46313472.0*P**3*R**5*T**2*UGA_star**17*epsilon - 2883584.0*P**3*R**5*T**2*UGA_star**17 + 58855424.0*P**3*R**5*T**2*UGA_star**16*epsilon + 3647488.0*P**3*R**5*T**2*UGA_star**16 - 48930816.0*P**3*R**5*T**2*UGA_star**15*epsilon - 3018752.0*P**3*R**5*T**2*UGA_star**15 + 26849280.0*P**3*R**5*T**2*UGA_star**14*epsilon + 1669120.0*P**3*R**5*T**2*UGA_star**14 - 9723904.0*P**3*R**5*T**2*UGA_star**13*epsilon - 647168.0*P**3*R**5*T**2*UGA_star**13 + 2322432.0*P**3*R**5*T**2*UGA_star**12*epsilon + 200704.0*P**3*R**5*T**2*UGA_star**12 - 303104.0*P**3*R**5*T**2*UGA_star**11*epsilon - 53248.0*P**3*R**5*T**2*UGA_star**11 - 36864.0*P**3*R**5*T**2*UGA_star**10*epsilon + 8192.0*P**3*R**5*T**2*UGA_star**10 + 20480.0*P**3*R**5*T**2*UGA_star**9*epsilon - 1361920.0*P**3*R**4*S**3*UGA_star**20*epsilon - 53760.0*P**3*R**4*S**3*UGA_star**20 + 12257280.0*P**3*R**4*S**3*UGA_star**19*epsilon + 483840.0*P**3*R**4*S**3*UGA_star**19 - 49326080.0*P**3*R**4*S**3*UGA_star**18*epsilon - 1958400.0*P**3*R**4*S**3*UGA_star**18 + 117350400.0*P**3*R**4*S**3*UGA_star**17*epsilon + 4730880.0*P**3*R**4*S**3*UGA_star**17 - 184524800.0*P**3*R**4*S**3*UGA_star**16*epsilon - 7659520.0*P**3*R**4*S**3*UGA_star**16 + 204472320.0*P**3*R**4*S**3*UGA_star**15*epsilon + 8883200.0*P**3*R**4*S**3*UGA_star**15 - 168048640.0*P**3*R**4*S**3*UGA_star**14*epsilon - 7705600.0*P**3*R**4*S**3*UGA_star**14 + 107591680.0*P**3*R**4*S**3*UGA_star**13*epsilon + 5089280.0*P**3*R**4*S**3*UGA_star**13 - 55480320.0*P**3*R**4*S**3*UGA_star**12*epsilon - 2490880.0*P**3*R**4*S**3*UGA_star**12 + 22732800.0*P**3*R**4*S**3*UGA_star**11*epsilon + 811520.0*P**3*R**4*S**3*UGA_star**11 - 6963200.0*P**3*R**4*S**3*UGA_star**10*epsilon - 125440.0*P**3*R**4*S**3*UGA_star**10 + 1525760.0*P**3*R**4*S**3*UGA_star**9*epsilon - 10240.0*P**3*R**4*S**3*UGA_star**9 - 256000.0*P**3*R**4*S**3*UGA_star**8*epsilon + 5120.0*P**3*R**4*S**3*UGA_star**8 + 30720.0*P**3*R**4*S**3*UGA_star**7*epsilon - 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219136.0*P**3*R**4*S*T**2*UGA_star**9*epsilon + 34816.0*P**3*R**4*S*T**2*UGA_star**9 + 81920.0*P**3*R**4*S*T**2*UGA_star**8*epsilon - 1372160.0*P**3*R**4*T**3*UGA_star**20*epsilon - 76800.0*P**3*R**4*T**3*UGA_star**20 + 12349440.0*P**3*R**4*T**3*UGA_star**19*epsilon + 691200.0*P**3*R**4*T**3*UGA_star**19 - 49356800.0*P**3*R**4*T**3*UGA_star**18*epsilon - 2764800.0*P**3*R**4*T**3*UGA_star**18 + 114708480.0*P**3*R**4*T**3*UGA_star**17*epsilon + 6435840.0*P**3*R**4*T**3*UGA_star**17 - 170203136.0*P**3*R**4*T**3*UGA_star**16*epsilon - 9570304.0*P**3*R**4*T**3*UGA_star**16 + 166141952.0*P**3*R**4*T**3*UGA_star**15*epsilon + 9361408.0*P**3*R**4*T**3*UGA_star**15 - 105424896.0*P**3*R**4*T**3*UGA_star**14*epsilon - 5935104.0*P**3*R**4*T**3*UGA_star**14 + 40898560.0*P**3*R**4*T**3*UGA_star**13*epsilon + 2263040.0*P**3*R**4*T**3*UGA_star**13 - 8304640.0*P**3*R**4*T**3*UGA_star**12*epsilon - 404480.0*P**3*R**4*T**3*UGA_star**12 + 659456.0*P**3*R**4*T**3*UGA_star**11*epsilon - 9216.0*P**3*R**4*T**3*UGA_star**11 - 161792.0*P**3*R**4*T**3*UGA_star**10*epsilon + 8192.0*P**3*R**4*T**3*UGA_star**10 + 65536.0*P**3*R**4*T**3*UGA_star**9*epsilon + 1024.0*P**3*R**4*T**3*UGA_star**9 + 1361920.0*P**3*R**3*S**4*UGA_star**20*epsilon + 53760.0*P**3*R**3*S**4*UGA_star**20 - 13619200.0*P**3*R**3*S**4*UGA_star**19*epsilon - 537600.0*P**3*R**3*S**4*UGA_star**19 + 61347840.0*P**3*R**3*S**4*UGA_star**18*epsilon + 2432000.0*P**3*R**3*S**4*UGA_star**18 - 164413440.0*P**3*R**3*S**4*UGA_star**17*epsilon - 6589440.0*P**3*R**3*S**4*UGA_star**17 + 292101120.0*P**3*R**3*S**4*UGA_star**16*epsilon + 11952640.0*P**3*R**3*S**4*UGA_star**16 - 363980800.0*P**3*R**3*S**4*UGA_star**15*epsilon - 15406080.0*P**3*R**3*S**4*UGA_star**15 + 330398720.0*P**3*R**3*S**4*UGA_star**14*epsilon + 14653440.0*P**3*R**3*S**4*UGA_star**14 - 226631680.0*P**3*R**3*S**4*UGA_star**13*epsilon - 10536960.0*P**3*R**3*S**4*UGA_star**13 + 122716160.0*P**3*R**3*S**4*UGA_star**12*epsilon + 5752320.0*P**3*R**3*S**4*UGA_star**12 - 54394880.0*P**3*R**3*S**4*UGA_star**11*epsilon - 2288640.0*P**3*R**3*S**4*UGA_star**11 + 19476480.0*P**3*R**3*S**4*UGA_star**10*epsilon + 568320.0*P**3*R**3*S**4*UGA_star**10 - 5201920.0*P**3*R**3*S**4*UGA_star**9*epsilon - 35840.0*P**3*R**3*S**4*UGA_star**9 + 957440.0*P**3*R**3*S**4*UGA_star**8*epsilon - 23040.0*P**3*R**3*S**4*UGA_star**8 - 133120.0*P**3*R**3*S**4*UGA_star**7*epsilon + 5120.0*P**3*R**3*S**4*UGA_star**7 + 15360.0*P**3*R**3*S**4*UGA_star**6*epsilon + 5488640.0*P**3*R**3*S**3*T*UGA_star**20*epsilon + 307200.0*P**3*R**3*S**3*T*UGA_star**20 - 54886400.0*P**3*R**3*S**3*T*UGA_star**19*epsilon - 3072000.0*P**3*R**3*S**3*T*UGA_star**19 + 246681600.0*P**3*R**3*S**3*T*UGA_star**18*epsilon + 13813760.0*P**3*R**3*S**3*T*UGA_star**18 - 656793600.0*P**3*R**3*S**3*T*UGA_star**17*epsilon - 36833280.0*P**3*R**3*S**3*T*UGA_star**17 + 1149947904.0*P**3*R**3*S**3*T*UGA_star**16*epsilon + 64695296.0*P**3*R**3*S**3*T*UGA_star**16 - 1391153152.0*P**3*R**3*S**3*T*UGA_star**15*epsilon - 78752768.0*P**3*R**3*S**3*T*UGA_star**15 + 1192927232.0*P**3*R**3*S**3*T*UGA_star**14*epsilon + 68313088.0*P**3*R**3*S**3*T*UGA_star**14 - 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1846272.0*P**3*R**2*S**5*UGA_star**8*epsilon + 21504.0*P**3*R**2*S**5*UGA_star**8 + 259072.0*P**3*R**2*S**5*UGA_star**7*epsilon - 16384.0*P**3*R**2*S**5*UGA_star**7 - 23552.0*P**3*R**2*S**5*UGA_star**6*epsilon + 2560.0*P**3*R**2*S**5*UGA_star**6 + 3072.0*P**3*R**2*S**5*UGA_star**5*epsilon - 4116480.0*P**3*R**2*S**4*T*UGA_star**20*epsilon - 230400.0*P**3*R**2*S**4*T*UGA_star**20 + 45281280.0*P**3*R**2*S**4*T*UGA_star**19*epsilon + 2534400.0*P**3*R**2*S**4*T*UGA_star**19 - 225623040.0*P**3*R**2*S**4*T*UGA_star**18*epsilon - 12633600.0*P**3*R**2*S**4*T*UGA_star**18 + 671831040.0*P**3*R**2*S**4*T*UGA_star**17*epsilon + 37662720.0*P**3*R**2*S**4*T*UGA_star**17 - 1327722496.0*P**3*R**2*S**4*T*UGA_star**16*epsilon - 74614784.0*P**3*R**2*S**4*T*UGA_star**16 + 1828491264.0*P**3*R**2*S**4*T*UGA_star**15*epsilon + 103248896.0*P**3*R**2*S**4*T*UGA_star**15 - 1792843776.0*P**3*R**2*S**4*T*UGA_star**14*epsilon - 102173184.0*P**3*R**2*S**4*T*UGA_star**14 + 1258051584.0*P**3*R**2*S**4*T*UGA_star**13*epsilon + 73015296.0*P**3*R**2*S**4*T*UGA_star**13 - 629415936.0*P**3*R**2*S**4*T*UGA_star**12*epsilon - 37929984.0*P**3*R**2*S**4*T*UGA_star**12 + 224526336.0*P**3*R**2*S**4*T*UGA_star**11*epsilon + 14644224.0*P**3*R**2*S**4*T*UGA_star**11 - 58665984.0*P**3*R**2*S**4*T*UGA_star**10*epsilon - 4471296.0*P**3*R**2*S**4*T*UGA_star**10 + 11498496.0*P**3*R**2*S**4*T*UGA_star**9*epsilon + 1142784.0*P**3*R**2*S**4*T*UGA_star**9 - 1081344.0*P**3*R**2*S**4*T*UGA_star**8*epsilon - 208896.0*P**3*R**2*S**4*T*UGA_star**8 - 303104.0*P**3*R**2*S**4*T*UGA_star**7*epsilon + 11264.0*P**3*R**2*S**4*T*UGA_star**7 + 92160.0*P**3*R**2*S**4*T*UGA_star**6*epsilon + 2560.0*P**3*R**2*S**4*T*UGA_star**6 - 8192000.0*P**3*R**2*S**3*T**2*UGA_star**20*epsilon - 512000.0*P**3*R**2*S**3*T**2*UGA_star**20 + 90112000.0*P**3*R**2*S**3*T**2*UGA_star**19*epsilon + 5632000.0*P**3*R**2*S**3*T**2*UGA_star**19 - 449228800.0*P**3*R**2*S**3*T**2*UGA_star**18*epsilon - 28057600.0*P**3*R**2*S**3*T**2*UGA_star**18 + 1338183680.0*P**3*R**2*S**3*T**2*UGA_star**17*epsilon + 83445760.0*P**3*R**2*S**3*T**2*UGA_star**17 - 2642079744.0*P**3*R**2*S**3*T**2*UGA_star**16*epsilon - 164304896.0*P**3*R**2*S**3*T**2*UGA_star**16 + 3620970496.0*P**3*R**2*S**3*T**2*UGA_star**15*epsilon + 224292864.0*P**3*R**2*S**3*T**2*UGA_star**15 - 3500580864.0*P**3*R**2*S**3*T**2*UGA_star**14*epsilon - 215801856.0*P**3*R**2*S**3*T**2*UGA_star**14 + 2370625536.0*P**3*R**2*S**3*T**2*UGA_star**13*epsilon + 145649664.0*P**3*R**2*S**3*T**2*UGA_star**13 - 1087954944.0*P**3*R**2*S**3*T**2*UGA_star**12*epsilon - 67362816.0*P**3*R**2*S**3*T**2*UGA_star**12 + 313524224.0*P**3*R**2*S**3*T**2*UGA_star**11*epsilon + 20688896.0*P**3*R**2*S**3*T**2*UGA_star**11 - 47312896.0*P**3*R**2*S**3*T**2*UGA_star**10*epsilon - 4440064.0*P**3*R**2*S**3*T**2*UGA_star**10 + 1675264.0*P**3*R**2*S**3*T**2*UGA_star**9*epsilon + 1017856.0*P**3*R**2*S**3*T**2*UGA_star**9 + 446464.0*P**3*R**2*S**3*T**2*UGA_star**8*epsilon - 292864.0*P**3*R**2*S**3*T**2*UGA_star**8 - 270336.0*P**3*R**2*S**3*T**2*UGA_star**7*epsilon + 45056.0*P**3*R**2*S**3*T**2*UGA_star**7 + 81920.0*P**3*R**2*S**3*T**2*UGA_star**6*epsilon - 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30265344.0*P**3*R**2*S**2*T**3*UGA_star**10*epsilon - 454656.0*P**3*R**2*S**2*T**3*UGA_star**10 - 1265664.0*P**3*R**2*S**2*T**3*UGA_star**9*epsilon - 577536.0*P**3*R**2*S**2*T**3*UGA_star**9 - 651264.0*P**3*R**2*S**2*T**3*UGA_star**8*epsilon + 82944.0*P**3*R**2*S**2*T**3*UGA_star**8 + 331776.0*P**3*R**2*S**2*T**3*UGA_star**7*epsilon + 6144.0*P**3*R**2*S**2*T**3*UGA_star**7 - 4085760.0*P**3*R**2*S*T**4*UGA_star**20*epsilon - 161280.0*P**3*R**2*S*T**4*UGA_star**20 + 44943360.0*P**3*R**2*S*T**4*UGA_star**19*epsilon + 1774080.0*P**3*R**2*S*T**4*UGA_star**19 - 224901120.0*P**3*R**2*S*T**4*UGA_star**18*epsilon - 8908800.0*P**3*R**2*S*T**4*UGA_star**18 + 674703360.0*P**3*R**2*S*T**4*UGA_star**17*epsilon + 26926080.0*P**3*R**2*S*T**4*UGA_star**17 - 1344983040.0*P**3*R**2*S*T**4*UGA_star**16*epsilon - 54312960.0*P**3*R**2*S*T**4*UGA_star**16 + 1863306240.0*P**3*R**2*S*T**4*UGA_star**15*epsilon + 76508160.0*P**3*R**2*S*T**4*UGA_star**15 - 1818117120.0*P**3*R**2*S*T**4*UGA_star**14*epsilon - 76339200.0*P**3*R**2*S*T**4*UGA_star**14 + 1232931840.0*P**3*R**2*S*T**4*UGA_star**13*epsilon + 53329920.0*P**3*R**2*S*T**4*UGA_star**13 - 552115200.0*P**3*R**2*S*T**4*UGA_star**12*epsilon - 24890880.0*P**3*R**2*S*T**4*UGA_star**12 + 141143040.0*P**3*R**2*S*T**4*UGA_star**11*epsilon + 6812160.0*P**3*R**2*S*T**4*UGA_star**11 - 9262080.0*P**3*R**2*S*T**4*UGA_star**10*epsilon - 583680.0*P**3*R**2*S*T**4*UGA_star**10 - 4285440.0*P**3*R**2*S*T**4*UGA_star**9*epsilon - 199680.0*P**3*R**2*S*T**4*UGA_star**9 + 660480.0*P**3*R**2*S*T**4*UGA_star**8*epsilon + 46080.0*P**3*R**2*S*T**4*UGA_star**8 + 61440.0*P**3*R**2*S*T**4*UGA_star**7*epsilon - 745472.0*P**3*R**2*T**5*UGA_star**20*epsilon - 14336.0*P**3*R**2*T**5*UGA_star**20 + 8200192.0*P**3*R**2*T**5*UGA_star**19*epsilon + 157696.0*P**3*R**2*T**5*UGA_star**19 - 41180160.0*P**3*R**2*T**5*UGA_star**18*epsilon - 799232.0*P**3*R**2*T**5*UGA_star**18 + 124472320.0*P**3*R**2*T**5*UGA_star**17*epsilon + 2462208.0*P**3*R**2*T**5*UGA_star**17 - 251126784.0*P**3*R**2*T**5*UGA_star**16*epsilon - 5119488.0*P**3*R**2*T**5*UGA_star**16 + 353974272.0*P**3*R**2*T**5*UGA_star**15*epsilon + 7537152.0*P**3*R**2*T**5*UGA_star**15 - 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3879075840.0*P**3*R*S**3*T**3*UGA_star**13*epsilon - 215408640.0*P**3*R*S**3*T**3*UGA_star**13 + 2218844160.0*P**3*R*S**3*T**3*UGA_star**12*epsilon + 121313280.0*P**3*R*S**3*T**3*UGA_star**12 - 836722688.0*P**3*R*S**3*T**3*UGA_star**11*epsilon - 43640832.0*P**3*R*S**3*T**3*UGA_star**11 + 175411200.0*P**3*R*S**3*T**3*UGA_star**10*epsilon + 7403520.0*P**3*R*S**3*T**3*UGA_star**10 - 7331840.0*P**3*R*S**3*T**3*UGA_star**9*epsilon + 829440.0*P**3*R*S**3*T**3*UGA_star**9 - 3297280.0*P**3*R*S**3*T**3*UGA_star**8*epsilon - 599040.0*P**3*R*S**3*T**3*UGA_star**8 - 266240.0*P**3*R*S**3*T**3*UGA_star**7*epsilon + 66560.0*P**3*R*S**3*T**3*UGA_star**7 + 200704.0*P**3*R*S**3*T**3*UGA_star**6*epsilon + 4096.0*P**3*R*S**3*T**3*UGA_star**6 + 4085760.0*P**3*R*S**2*T**4*UGA_star**20*epsilon + 161280.0*P**3*R*S**2*T**4*UGA_star**20 - 49029120.0*P**3*R*S**2*T**4*UGA_star**19*epsilon - 1935360.0*P**3*R*S**2*T**4*UGA_star**19 + 269844480.0*P**3*R*S**2*T**4*UGA_star**18*epsilon + 10682880.0*P**3*R*S**2*T**4*UGA_star**18 - 899604480.0*P**3*R*S**2*T**4*UGA_star**17*epsilon - 35834880.0*P**3*R*S**2*T**4*UGA_star**17 + 2019640320.0*P**3*R*S**2*T**4*UGA_star**16*epsilon + 81239040.0*P**3*R*S**2*T**4*UGA_star**16 - 3207874560.0*P**3*R*S**2*T**4*UGA_star**15*epsilon - 130821120.0*P**3*R*S**2*T**4*UGA_star**15 + 3679764480.0*P**3*R*S**2*T**4*UGA_star**14*epsilon + 152847360.0*P**3*R*S**2*T**4*UGA_star**14 - 3047178240.0*P**3*R*S**2*T**4*UGA_star**13*epsilon - 129669120.0*P**3*R*S**2*T**4*UGA_star**13 + 1779240960.0*P**3*R*S**2*T**4*UGA_star**12*epsilon + 78220800.0*P**3*R*S**2*T**4*UGA_star**12 - 687452160.0*P**3*R*S**2*T**4*UGA_star**11*epsilon - 31703040.0*P**3*R*S**2*T**4*UGA_star**11 + 146534400.0*P**3*R*S**2*T**4*UGA_star**10*epsilon + 7395840.0*P**3*R*S**2*T**4*UGA_star**10 - 3317760.0*P**3*R*S**2*T**4*UGA_star**9*epsilon - 384000.0*P**3*R*S**2*T**4*UGA_star**9 - 5360640.0*P**3*R*S**2*T**4*UGA_star**8*epsilon - 245760.0*P**3*R*S**2*T**4*UGA_star**8 + 645120.0*P**3*R*S**2*T**4*UGA_star**7*epsilon + 46080.0*P**3*R*S**2*T**4*UGA_star**7 + 61440.0*P**3*R*S**2*T**4*UGA_star**6*epsilon + 1490944.0*P**3*R*S*T**5*UGA_star**20*epsilon + 28672.0*P**3*R*S*T**5*UGA_star**20 - 17891328.0*P**3*R*S*T**5*UGA_star**19*epsilon - 344064.0*P**3*R*S*T**5*UGA_star**19 + 98760704.0*P**3*R*S*T**5*UGA_star**18*epsilon + 1913856.0*P**3*R*S*T**5*UGA_star**18 - 331304960.0*P**3*R*S*T**5*UGA_star**17*epsilon - 6522880.0*P**3*R*S*T**5*UGA_star**17 + 751198208.0*P**3*R*S*T**5*UGA_star**16*epsilon + 15163392.0*P**3*R*S*T**5*UGA_star**16 - 1210202112.0*P**3*R*S*T**5*UGA_star**15*epsilon - 25313280.0*P**3*R*S*T**5*UGA_star**15 + 1415516160.0*P**3*R*S*T**5*UGA_star**14*epsilon + 31094784.0*P**3*R*S*T**5*UGA_star**14 - 1203879936.0*P**3*R*S*T**5*UGA_star**13*epsilon - 28299264.0*P**3*R*S*T**5*UGA_star**13 + 730275840.0*P**3*R*S*T**5*UGA_star**12*epsilon + 18923520.0*P**3*R*S*T**5*UGA_star**12 - 299880448.0*P**3*R*S*T**5*UGA_star**11*epsilon - 9060352.0*P**3*R*S*T**5*UGA_star**11 + 72726528.0*P**3*R*S*T**5*UGA_star**10*epsilon + 2946048.0*P**3*R*S*T**5*UGA_star**10 - 5353472.0*P**3*R*S*T**5*UGA_star**9*epsilon - 583680.0*P**3*R*S*T**5*UGA_star**9 - 1835008.0*P**3*R*S*T**5*UGA_star**8*epsilon + 53248.0*P**3*R*S*T**5*UGA_star**8 + 378880.0*P**3*R*S*T**5*UGA_star**7*epsilon + 172032.0*P**3*R*T**6*UGA_star**20*epsilon - 2064384.0*P**3*R*T**6*UGA_star**19*epsilon + 11440128.0*P**3*R*T**6*UGA_star**18*epsilon - 38707200.0*P**3*R*T**6*UGA_star**17*epsilon + 89033728.0*P**3*R*T**6*UGA_star**16*epsilon - 146628608.0*P**3*R*T**6*UGA_star**15*epsilon + 177221632.0*P**3*R*T**6*UGA_star**14*epsilon - 158326784.0*P**3*R*T**6*UGA_star**13*epsilon + 103720960.0*P**3*R*T**6*UGA_star**12*epsilon - 48570368.0*P**3*R*T**6*UGA_star**11*epsilon + 15425536.0*P**3*R*T**6*UGA_star**10*epsilon - 2981888.0*P**3*R*T**6*UGA_star**9*epsilon + 265216.0*P**3*R*T**6*UGA_star**8*epsilon - 38912.0*P**3*S**7*UGA_star**20*epsilon - 1536.0*P**3*S**7*UGA_star**20 + 505856.0*P**3*S**7*UGA_star**19*epsilon + 19968.0*P**3*S**7*UGA_star**19 - 3016704.0*P**3*S**7*UGA_star**18*epsilon - 119296.0*P**3*S**7*UGA_star**18 + 10907648.0*P**3*S**7*UGA_star**17*epsilon + 433152.0*P**3*S**7*UGA_star**17 - 26607616.0*P**3*S**7*UGA_star**16*epsilon - 1064448.0*P**3*S**7*UGA_star**16 + 46047232.0*P**3*S**7*UGA_star**15*epsilon + 1864192.0*P**3*S**7*UGA_star**15 - 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14336.0*P**3*S**2*T**5*UGA_star**20 + 9691136.0*P**3*S**2*T**5*UGA_star**19*epsilon + 186368.0*P**3*S**2*T**5*UGA_star**19 - 58326016.0*P**3*S**2*T**5*UGA_star**18*epsilon - 1128960.0*P**3*S**2*T**5*UGA_star**18 + 215032832.0*P**3*S**2*T**5*UGA_star**17*epsilon + 4218368.0*P**3*S**2*T**5*UGA_star**17 - 541251584.0*P**3*S**2*T**5*UGA_star**16*epsilon - 10843136.0*P**3*S**2*T**5*UGA_star**16 + 980700160.0*P**3*S**2*T**5*UGA_star**15*epsilon + 20238336.0*P**3*S**2*T**5*UGA_star**15 - 1312859136.0*P**3*S**2*T**5*UGA_star**14*epsilon - 28204032.0*P**3*S**2*T**5*UGA_star**14 + 1309698048.0*P**3*S**2*T**5*UGA_star**13*epsilon + 29697024.0*P**3*S**2*T**5*UGA_star**13 - 967077888.0*P**3*S**2*T**5*UGA_star**12*epsilon - 23611392.0*P**3*S**2*T**5*UGA_star**12 + 515078144.0*P**3*S**2*T**5*UGA_star**11*epsilon + 13991936.0*P**3*S**2*T**5*UGA_star**11 - 186303488.0*P**3*S**2*T**5*UGA_star**10*epsilon - 6003200.0*P**3*S**2*T**5*UGA_star**10 + 39040000.0*P**3*S**2*T**5*UGA_star**9*epsilon + 1764864.0*P**3*S**2*T**5*UGA_star**9 - 1759232.0*P**3*S**2*T**5*UGA_star**8*epsilon - 318464.0*P**3*S**2*T**5*UGA_star**8 - 1106944.0*P**3*S**2*T**5*UGA_star**7*epsilon + 26624.0*P**3*S**2*T**5*UGA_star**7 + 189440.0*P**3*S**2*T**5*UGA_star**6*epsilon - 172032.0*P**3*S*T**6*UGA_star**20*epsilon + 2236416.0*P**3*S*T**6*UGA_star**19*epsilon - 13504512.0*P**3*S*T**6*UGA_star**18*epsilon + 50147328.0*P**3*S*T**6*UGA_star**17*epsilon - 127740928.0*P**3*S*T**6*UGA_star**16*epsilon + 235662336.0*P**3*S*T**6*UGA_star**15*epsilon - 323850240.0*P**3*S*T**6*UGA_star**14*epsilon + 335548416.0*P**3*S*T**6*UGA_star**13*epsilon - 262047744.0*P**3*S*T**6*UGA_star**12*epsilon + 152291328.0*P**3*S*T**6*UGA_star**11*epsilon - 63995904.0*P**3*S*T**6*UGA_star**10*epsilon + 18407424.0*P**3*S*T**6*UGA_star**9*epsilon - 3247104.0*P**3*S*T**6*UGA_star**8*epsilon + 265216.0*P**3*S*T**6*UGA_star**7*epsilon + 13312.0*P**2*R**8*UGA_star**20*epsilon + 256.0*P**2*R**8*UGA_star**20 - 53248.0*P**2*R**8*UGA_star**19*epsilon - 1024.0*P**2*R**8*UGA_star**19 + 95232.0*P**2*R**8*UGA_star**18*epsilon + 2048.0*P**2*R**8*UGA_star**18 - 114688.0*P**2*R**8*UGA_star**17*epsilon - 3072.0*P**2*R**8*UGA_star**17 + 106496.0*P**2*R**8*UGA_star**16*epsilon + 3584.0*P**2*R**8*UGA_star**16 - 65536.0*P**2*R**8*UGA_star**15*epsilon - 3072.0*P**2*R**8*UGA_star**15 + 21504.0*P**2*R**8*UGA_star**14*epsilon + 2048.0*P**2*R**8*UGA_star**14 - 4096.0*P**2*R**8*UGA_star**13*epsilon - 1024.0*P**2*R**8*UGA_star**13 + 1024.0*P**2*R**8*UGA_star**12*epsilon + 256.0*P**2*R**8*UGA_star**12 - 106496.0*P**2*R**7*S*UGA_star**20*epsilon - 2048.0*P**2*R**7*S*UGA_star**20 + 532480.0*P**2*R**7*S*UGA_star**19*epsilon + 10240.0*P**2*R**7*S*UGA_star**19 - 1168384.0*P**2*R**7*S*UGA_star**18*epsilon - 24064.0*P**2*R**7*S*UGA_star**18 + 1586176.0*P**2*R**7*S*UGA_star**17*epsilon + 38400.0*P**2*R**7*S*UGA_star**17 - 1586176.0*P**2*R**7*S*UGA_star**16*epsilon - 47616.0*P**2*R**7*S*UGA_star**16 + 1183744.0*P**2*R**7*S*UGA_star**15*epsilon + 45568.0*P**2*R**7*S*UGA_star**15 - 576512.0*P**2*R**7*S*UGA_star**14*epsilon - 33280.0*P**2*R**7*S*UGA_star**14 + 156672.0*P**2*R**7*S*UGA_star**13*epsilon + 18944.0*P**2*R**7*S*UGA_star**13 - 27648.0*P**2*R**7*S*UGA_star**12*epsilon - 7680.0*P**2*R**7*S*UGA_star**12 + 6144.0*P**2*R**7*S*UGA_star**11*epsilon + 1536.0*P**2*R**7*S*UGA_star**11 - 116736.0*P**2*R**7*T*UGA_star**20*epsilon - 4608.0*P**2*R**7*T*UGA_star**20 + 583680.0*P**2*R**7*T*UGA_star**19*epsilon + 23040.0*P**2*R**7*T*UGA_star**19 - 1253376.0*P**2*R**7*T*UGA_star**18*epsilon - 50688.0*P**2*R**7*T*UGA_star**18 + 1597440.0*P**2*R**7*T*UGA_star**17*epsilon + 69120.0*P**2*R**7*T*UGA_star**17 - 1449984.0*P**2*R**7*T*UGA_star**16*epsilon - 69120.0*P**2*R**7*T*UGA_star**16 + 1010688.0*P**2*R**7*T*UGA_star**15*epsilon + 50688.0*P**2*R**7*T*UGA_star**15 - 503808.0*P**2*R**7*T*UGA_star**14*epsilon - 23040.0*P**2*R**7*T*UGA_star**14 + 165888.0*P**2*R**7*T*UGA_star**13*epsilon + 4608.0*P**2*R**7*T*UGA_star**13 - 43008.0*P**2*R**7*T*UGA_star**12*epsilon + 9216.0*P**2*R**7*T*UGA_star**11*epsilon + 372736.0*P**2*R**6*S**2*UGA_star**20*epsilon + 7168.0*P**2*R**6*S**2*UGA_star**20 - 2236416.0*P**2*R**6*S**2*UGA_star**19*epsilon - 43008.0*P**2*R**6*S**2*UGA_star**19 + 5884928.0*P**2*R**6*S**2*UGA_star**18*epsilon + 118272.0*P**2*R**6*S**2*UGA_star**18 - 9246720.0*P**2*R**6*S**2*UGA_star**17*epsilon - 207872.0*P**2*R**6*S**2*UGA_star**17 + 10139648.0*P**2*R**6*S**2*UGA_star**16*epsilon + 272640.0*P**2*R**6*S**2*UGA_star**16 - 8400896.0*P**2*R**6*S**2*UGA_star**15*epsilon - 281088.0*P**2*R**6*S**2*UGA_star**15 + 5113856.0*P**2*R**6*S**2*UGA_star**14*epsilon + 226048.0*P**2*R**6*S**2*UGA_star**14 - 2029568.0*P**2*R**6*S**2*UGA_star**13*epsilon - 141312.0*P**2*R**6*S**2*UGA_star**13 + 460800.0*P**2*R**6*S**2*UGA_star**12*epsilon + 68352.0*P**2*R**6*S**2*UGA_star**12 - 73728.0*P**2*R**6*S**2*UGA_star**11*epsilon - 23040.0*P**2*R**6*S**2*UGA_star**11 + 15360.0*P**2*R**6*S**2*UGA_star**10*epsilon + 3840.0*P**2*R**6*S**2*UGA_star**10 + 817152.0*P**2*R**6*S*T*UGA_star**20*epsilon + 32256.0*P**2*R**6*S*T*UGA_star**20 - 4902912.0*P**2*R**6*S*T*UGA_star**19*epsilon - 193536.0*P**2*R**6*S*T*UGA_star**19 + 12718080.0*P**2*R**6*S*T*UGA_star**18*epsilon + 509952.0*P**2*R**6*S*T*UGA_star**18 - 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10804224.0*P**2*R**5*S**2*T*UGA_star**12*epsilon - 359424.0*P**2*R**5*S**2*T*UGA_star**12 + 2770944.0*P**2*R**5*S**2*T*UGA_star**11*epsilon + 23040.0*P**2*R**5*S**2*T*UGA_star**11 - 568320.0*P**2*R**5*S**2*T*UGA_star**10*epsilon + 7680.0*P**2*R**5*S**2*T*UGA_star**10 + 92160.0*P**2*R**5*S**2*T*UGA_star**9*epsilon - 2469888.0*P**2*R**5*S*T**2*UGA_star**20*epsilon - 138240.0*P**2*R**5*S*T**2*UGA_star**20 + 17289216.0*P**2*R**5*S*T**2*UGA_star**19*epsilon + 967680.0*P**2*R**5*S*T**2*UGA_star**19 - 52328448.0*P**2*R**5*S*T**2*UGA_star**18*epsilon - 2930688.0*P**2*R**5*S*T**2*UGA_star**18 + 89929728.0*P**2*R**5*S*T**2*UGA_star**17*epsilon + 5050368.0*P**2*R**5*S*T**2*UGA_star**17 - 97701888.0*P**2*R**5*S*T**2*UGA_star**16*epsilon - 5521920.0*P**2*R**5*S*T**2*UGA_star**16 + 71958528.0*P**2*R**5*S*T**2*UGA_star**15*epsilon + 4101120.0*P**2*R**5*S*T**2*UGA_star**15 - 38578176.0*P**2*R**5*S*T**2*UGA_star**14*epsilon - 2188800.0*P**2*R**5*S*T**2*UGA_star**14 + 15710208.0*P**2*R**5*S*T**2*UGA_star**13*epsilon + 841728.0*P**2*R**5*S*T**2*UGA_star**13 - 4313088.0*P**2*R**5*S*T**2*UGA_star**12*epsilon - 188928.0*P**2*R**5*S*T**2*UGA_star**12 + 442368.0*P**2*R**5*S*T**2*UGA_star**11*epsilon + 61440.0*P**2*R**5*S*T**2*UGA_star**10*epsilon + 7680.0*P**2*R**5*S*T**2*UGA_star**10 - 819200.0*P**2*R**5*T**3*UGA_star**20*epsilon - 51200.0*P**2*R**5*T**3*UGA_star**20 + 5734400.0*P**2*R**5*T**3*UGA_star**19*epsilon + 358400.0*P**2*R**5*T**3*UGA_star**19 - 17197056.0*P**2*R**5*T**3*UGA_star**18*epsilon - 1073152.0*P**2*R**5*T**3*UGA_star**18 + 28690432.0*P**2*R**5*T**3*UGA_star**17*epsilon + 1782784.0*P**2*R**5*T**3*UGA_star**17 - 28932096.0*P**2*R**5*T**3*UGA_star**16*epsilon - 1783808.0*P**2*R**5*T**3*UGA_star**16 + 17981440.0*P**2*R**5*T**3*UGA_star**15*epsilon + 1100800.0*P**2*R**5*T**3*UGA_star**15 - 6799360.0*P**2*R**5*T**3*UGA_star**14*epsilon - 430080.0*P**2*R**5*T**3*UGA_star**14 + 1509376.0*P**2*R**5*T**3*UGA_star**13*epsilon + 125952.0*P**2*R**5*T**3*UGA_star**13 - 120832.0*P**2*R**5*T**3*UGA_star**12*epsilon - 36864.0*P**2*R**5*T**3*UGA_star**12 - 77824.0*P**2*R**5*T**3*UGA_star**11*epsilon + 7168.0*P**2*R**5*T**3*UGA_star**11 + 30720.0*P**2*R**5*T**3*UGA_star**10*epsilon + 931840.0*P**2*R**4*S**4*UGA_star**20*epsilon + 17920.0*P**2*R**4*S**4*UGA_star**20 - 7454720.0*P**2*R**4*S**4*UGA_star**19*epsilon - 143360.0*P**2*R**4*S**4*UGA_star**19 + 26485760.0*P**2*R**4*S**4*UGA_star**18*epsilon + 519680.0*P**2*R**4*S**4*UGA_star**18 - 55480320.0*P**2*R**4*S**4*UGA_star**17*epsilon - 1146880.0*P**2*R**4*S**4*UGA_star**17 + 77224960.0*P**2*R**4*S**4*UGA_star**16*epsilon + 1760000.0*P**2*R**4*S**4*UGA_star**16 - 76963840.0*P**2*R**4*S**4*UGA_star**15*epsilon - 2037760.0*P**2*R**4*S**4*UGA_star**15 + 57845760.0*P**2*R**4*S**4*UGA_star**14*epsilon + 1863680.0*P**2*R**4*S**4*UGA_star**14 - 33259520.0*P**2*R**4*S**4*UGA_star**13*epsilon - 1361920.0*P**2*R**4*S**4*UGA_star**13 + 13905920.0*P**2*R**4*S**4*UGA_star**12*epsilon + 788480.0*P**2*R**4*S**4*UGA_star**12 - 3727360.0*P**2*R**4*S**4*UGA_star**11*epsilon - 358400.0*P**2*R**4*S**4*UGA_star**11 + 537600.0*P**2*R**4*S**4*UGA_star**10*epsilon + 125440.0*P**2*R**4*S**4*UGA_star**10 - 61440.0*P**2*R**4*S**4*UGA_star**9*epsilon - 30720.0*P**2*R**4*S**4*UGA_star**9 + 15360.0*P**2*R**4*S**4*UGA_star**8*epsilon + 3840.0*P**2*R**4*S**4*UGA_star**8 + 4085760.0*P**2*R**4*S**3*T*UGA_star**20*epsilon + 161280.0*P**2*R**4*S**3*T*UGA_star**20 - 32686080.0*P**2*R**4*S**3*T*UGA_star**19*epsilon - 1290240.0*P**2*R**4*S**3*T*UGA_star**19 + 115292160.0*P**2*R**4*S**3*T*UGA_star**18*epsilon + 4584960.0*P**2*R**4*S**3*T*UGA_star**18 - 236759040.0*P**2*R**4*S**3*T*UGA_star**17*epsilon - 9607680.0*P**2*R**4*S**3*T*UGA_star**17 + 316753920.0*P**2*R**4*S**3*T*UGA_star**16*epsilon + 13370880.0*P**2*R**4*S**3*T*UGA_star**16 - 296171520.0*P**2*R**4*S**3*T*UGA_star**15*epsilon - 13278720.0*P**2*R**4*S**3*T*UGA_star**15 + 206254080.0*P**2*R**4*S**3*T*UGA_star**14*epsilon + 9838080.0*P**2*R**4*S**3*T*UGA_star**14 - 113080320.0*P**2*R**4*S**3*T*UGA_star**13*epsilon - 5429760.0*P**2*R**4*S**3*T*UGA_star**13 + 49059840.0*P**2*R**4*S**3*T*UGA_star**12*epsilon + 2042880.0*P**2*R**4*S**3*T*UGA_star**12 - 15697920.0*P**2*R**4*S**3*T*UGA_star**11*epsilon - 391680.0*P**2*R**4*S**3*T*UGA_star**11 + 3471360.0*P**2*R**4*S**3*T*UGA_star**10*epsilon - 15360.0*P**2*R**4*S**3*T*UGA_star**10 - 614400.0*P**2*R**4*S**3*T*UGA_star**9*epsilon + 15360.0*P**2*R**4*S**3*T*UGA_star**9 + 92160.0*P**2*R**4*S**3*T*UGA_star**8*epsilon + 6174720.0*P**2*R**4*S**2*T**2*UGA_star**20*epsilon + 345600.0*P**2*R**4*S**2*T**2*UGA_star**20 - 49397760.0*P**2*R**4*S**2*T**2*UGA_star**19*epsilon - 2764800.0*P**2*R**4*S**2*T**2*UGA_star**19 + 173214720.0*P**2*R**4*S**2*T**2*UGA_star**18*epsilon + 9699840.0*P**2*R**4*S**2*T**2*UGA_star**18 - 349470720.0*P**2*R**4*S**2*T**2*UGA_star**17*epsilon - 19607040.0*P**2*R**4*S**2*T**2*UGA_star**17 + 449055744.0*P**2*R**4*S**2*T**2*UGA_star**16*epsilon + 25307136.0*P**2*R**4*S**2*T**2*UGA_star**16 - 387201024.0*P**2*R**4*S**2*T**2*UGA_star**15*epsilon - 21997056.0*P**2*R**4*S**2*T**2*UGA_star**15 + 233856000.0*P**2*R**4*S**2*T**2*UGA_star**14*epsilon + 13409280.0*P**2*R**4*S**2*T**2*UGA_star**14 - 104448000.0*P**2*R**4*S**2*T**2*UGA_star**13*epsilon - 5967360.0*P**2*R**4*S**2*T**2*UGA_star**13 + 35527680.0*P**2*R**4*S**2*T**2*UGA_star**12*epsilon + 1927680.0*P**2*R**4*S**2*T**2*UGA_star**12 - 7870464.0*P**2*R**4*S**2*T**2*UGA_star**11*epsilon - 354816.0*P**2*R**4*S**2*T**2*UGA_star**11 + 374784.0*P**2*R**4*S**2*T**2*UGA_star**10*epsilon - 13824.0*P**2*R**4*S**2*T**2*UGA_star**10 + 184320.0*P**2*R**4*S**2*T**2*UGA_star**9*epsilon + 15360.0*P**2*R**4*S**2*T**2*UGA_star**9 + 4096000.0*P**2*R**4*S*T**3*UGA_star**20*epsilon + 256000.0*P**2*R**4*S*T**3*UGA_star**20 - 32768000.0*P**2*R**4*S*T**3*UGA_star**19*epsilon - 2048000.0*P**2*R**4*S*T**3*UGA_star**19 + 114339840.0*P**2*R**4*S*T**3*UGA_star**18*epsilon + 7137280.0*P**2*R**4*S*T**3*UGA_star**18 - 227123200.0*P**2*R**4*S*T**3*UGA_star**17*epsilon - 14131200.0*P**2*R**4*S*T**3*UGA_star**17 + 280815616.0*P**2*R**4*S*T**3*UGA_star**16*epsilon + 17368064.0*P**2*R**4*S*T**3*UGA_star**16 - 221659136.0*P**2*R**4*S*T**3*UGA_star**15*epsilon - 13604864.0*P**2*R**4*S*T**3*UGA_star**15 + 110069760.0*P**2*R**4*S*T**3*UGA_star**14*epsilon + 6773760.0*P**2*R**4*S*T**3*UGA_star**14 - 32604160.0*P**2*R**4*S*T**3*UGA_star**13*epsilon - 2191360.0*P**2*R**4*S*T**3*UGA_star**13 + 5027840.0*P**2*R**4*S*T**3*UGA_star**12*epsilon + 578560.0*P**2*R**4*S*T**3*UGA_star**12 + 98304.0*P**2*R**4*S*T**3*UGA_star**11*epsilon - 169984.0*P**2*R**4*S*T**3*UGA_star**11 - 415744.0*P**2*R**4*S*T**3*UGA_star**10*epsilon + 31744.0*P**2*R**4*S*T**3*UGA_star**10 + 122880.0*P**2*R**4*S*T**3*UGA_star**9*epsilon + 1029120.0*P**2*R**4*T**4*UGA_star**20*epsilon + 57600.0*P**2*R**4*T**4*UGA_star**20 - 8232960.0*P**2*R**4*T**4*UGA_star**19*epsilon - 460800.0*P**2*R**4*T**4*UGA_star**19 + 28646400.0*P**2*R**4*T**4*UGA_star**18*epsilon + 1605120.0*P**2*R**4*T**4*UGA_star**18 - 56340480.0*P**2*R**4*T**4*UGA_star**17*epsilon - 3164160.0*P**2*R**4*T**4*UGA_star**17 + 67866624.0*P**2*R**4*T**4*UGA_star**16*epsilon + 3826176.0*P**2*R**4*T**4*UGA_star**16 - 50253824.0*P**2*R**4*T**4*UGA_star**15*epsilon - 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233472.0*P**2*R**2*S**3*T**3*UGA_star**9 - 507904.0*P**2*R**2*S**3*T**3*UGA_star**8*epsilon + 47104.0*P**2*R**2*S**3*T**3*UGA_star**8 + 122880.0*P**2*R**2*S**3*T**3*UGA_star**7*epsilon + 6174720.0*P**2*R**2*S**2*T**4*UGA_star**20*epsilon + 345600.0*P**2*R**2*S**2*T**4*UGA_star**20 - 61747200.0*P**2*R**2*S**2*T**4*UGA_star**19*epsilon - 3456000.0*P**2*R**2*S**2*T**4*UGA_star**19 + 276526080.0*P**2*R**2*S**2*T**4*UGA_star**18*epsilon + 15482880.0*P**2*R**2*S**2*T**4*UGA_star**18 - 728248320.0*P**2*R**2*S**2*T**4*UGA_star**17*epsilon - 40803840.0*P**2*R**2*S**2*T**4*UGA_star**17 + 1243167744.0*P**2*R**2*S**2*T**4*UGA_star**16*epsilon + 69720576.0*P**2*R**2*S**2*T**4*UGA_star**16 - 1425487872.0*P**2*R**2*S**2*T**4*UGA_star**15*epsilon - 80014848.0*P**2*R**2*S**2*T**4*UGA_star**15 + 1093865472.0*P**2*R**2*S**2*T**4*UGA_star**14*epsilon + 61383168.0*P**2*R**2*S**2*T**4*UGA_star**14 - 533796864.0*P**2*R**2*S**2*T**4*UGA_star**13*epsilon - 29784576.0*P**2*R**2*S**2*T**4*UGA_star**13 + 140236800.0*P**2*R**2*S**2*T**4*UGA_star**12*epsilon + 7541760.0*P**2*R**2*S**2*T**4*UGA_star**12 - 6027264.0*P**2*R**2*S**2*T**4*UGA_star**11*epsilon - 32256.0*P**2*R**2*S**2*T**4*UGA_star**11 - 4973568.0*P**2*R**2*S**2*T**4*UGA_star**10*epsilon - 440832.0*P**2*R**2*S**2*T**4*UGA_star**10 + 43008.0*P**2*R**2*S**2*T**4*UGA_star**9*epsilon + 49152.0*P**2*R**2*S**2*T**4*UGA_star**9 + 267264.0*P**2*R**2*S**2*T**4*UGA_star**8*epsilon + 9216.0*P**2*R**2*S**2*T**4*UGA_star**8 + 2451456.0*P**2*R**2*S*T**5*UGA_star**20*epsilon + 96768.0*P**2*R**2*S*T**5*UGA_star**20 - 24514560.0*P**2*R**2*S*T**5*UGA_star**19*epsilon - 967680.0*P**2*R**2*S*T**5*UGA_star**19 + 110002176.0*P**2*R**2*S*T**5*UGA_star**18*epsilon + 4359168.0*P**2*R**2*S*T**5*UGA_star**18 - 290838528.0*P**2*R**2*S*T**5*UGA_star**17*epsilon - 11625984.0*P**2*R**2*S*T**5*UGA_star**17 + 499418112.0*P**2*R**2*S*T**5*UGA_star**16*epsilon + 20258304.0*P**2*R**2*S*T**5*UGA_star**16 - 577198080.0*P**2*R**2*S*T**5*UGA_star**15*epsilon - 23944704.0*P**2*R**2*S*T**5*UGA_star**15 + 447249408.0*P**2*R**2*S*T**5*UGA_star**14*epsilon + 19192320.0*P**2*R**2*S*T**5*UGA_star**14 - 220471296.0*P**2*R**2*S*T**5*UGA_star**13*epsilon - 9988608.0*P**2*R**2*S*T**5*UGA_star**13 + 57811968.0*P**2*R**2*S*T**5*UGA_star**12*epsilon + 2924544.0*P**2*R**2*S*T**5*UGA_star**12 - 1247232.0*P**2*R**2*S*T**5*UGA_star**11*epsilon - 198144.0*P**2*R**2*S*T**5*UGA_star**11 - 3111936.0*P**2*R**2*S*T**5*UGA_star**10*epsilon - 136704.0*P**2*R**2*S*T**5*UGA_star**10 + 387072.0*P**2*R**2*S*T**5*UGA_star**9*epsilon + 30720.0*P**2*R**2*S*T**5*UGA_star**9 + 61440.0*P**2*R**2*S*T**5*UGA_star**8*epsilon + 372736.0*P**2*R**2*T**6*UGA_star**20*epsilon + 7168.0*P**2*R**2*T**6*UGA_star**20 - 3727360.0*P**2*R**2*T**6*UGA_star**19*epsilon - 71680.0*P**2*R**2*T**6*UGA_star**19 + 16794624.0*P**2*R**2*T**6*UGA_star**18*epsilon + 326144.0*P**2*R**2*T**6*UGA_star**18 - 44814336.0*P**2*R**2*T**6*UGA_star**17*epsilon - 888832.0*P**2*R**2*T**6*UGA_star**17 + 78171136.0*P**2*R**2*T**6*UGA_star**16*epsilon + 1605888.0*P**2*R**2*T**6*UGA_star**16 - 92602368.0*P**2*R**2*T**6*UGA_star**15*epsilon - 2008576.0*P**2*R**2*T**6*UGA_star**15 + 74591232.0*P**2*R**2*T**6*UGA_star**14*epsilon + 1760000.0*P**2*R**2*T**6*UGA_star**14 - 39286784.0*P**2*R**2*T**6*UGA_star**13*epsilon - 1065984.0*P**2*R**2*T**6*UGA_star**13 + 11928576.0*P**2*R**2*T**6*UGA_star**12*epsilon + 426752.0*P**2*R**2*T**6*UGA_star**12 - 1110016.0*P**2*R**2*T**6*UGA_star**11*epsilon - 101888.0*P**2*R**2*T**6*UGA_star**11 - 421888.0*P**2*R**2*T**6*UGA_star**10*epsilon + 11008.0*P**2*R**2*T**6*UGA_star**10 + 104448.0*P**2*R**2*T**6*UGA_star**9*epsilon - 106496.0*P**2*R*S**7*UGA_star**20*epsilon - 2048.0*P**2*R*S**7*UGA_star**20 + 1171456.0*P**2*R*S**7*UGA_star**19*epsilon + 22528.0*P**2*R*S**7*UGA_star**19 - 5843968.0*P**2*R*S**7*UGA_star**18*epsilon - 113152.0*P**2*R*S**7*UGA_star**18 + 17454080.0*P**2*R*S**7*UGA_star**17*epsilon + 343552.0*P**2*R*S**7*UGA_star**17 - 34695168.0*P**2*R*S**7*UGA_star**16*epsilon - 704000.0*P**2*R*S**7*UGA_star**16 + 48264192.0*P**2*R*S**7*UGA_star**15*epsilon + 1030656.0*P**2*R*S**7*UGA_star**15 - 48110592.0*P**2*R*S**7*UGA_star**14*epsilon - 1115136.0*P**2*R*S**7*UGA_star**14 + 34646016.0*P**2*R*S**7*UGA_star**13*epsilon + 912384.0*P**2*R*S**7*UGA_star**13 - 18001920.0*P**2*R*S**7*UGA_star**12*epsilon - 574464.0*P**2*R*S**7*UGA_star**12 + 6711296.0*P**2*R*S**7*UGA_star**11*epsilon + 281600.0*P**2*R*S**7*UGA_star**11 - 1770496.0*P**2*R*S**7*UGA_star**10*epsilon - 107008.0*P**2*R*S**7*UGA_star**10 + 295936.0*P**2*R*S**7*UGA_star**9*epsilon + 30208.0*P**2*R*S**7*UGA_star**9 - 2048.0*P**2*R*S**7*UGA_star**8*epsilon - 5632.0*P**2*R*S**7*UGA_star**8 - 15360.0*P**2*R*S**7*UGA_star**7*epsilon + 512.0*P**2*R*S**7*UGA_star**7 + 3072.0*P**2*R*S**7*UGA_star**6*epsilon - 817152.0*P**2*R*S**6*T*UGA_star**20*epsilon - 32256.0*P**2*R*S**6*T*UGA_star**20 + 8988672.0*P**2*R*S**6*T*UGA_star**19*epsilon + 354816.0*P**2*R*S**6*T*UGA_star**19 - 44697600.0*P**2*R*S**6*T*UGA_star**18*epsilon - 1769472.0*P**2*R*S**6*T*UGA_star**18 + 132458496.0*P**2*R*S**6*T*UGA_star**17*epsilon + 5280768.0*P**2*R*S**6*T*UGA_star**17 - 259482624.0*P**2*R*S**6*T*UGA_star**16*epsilon - 10481664.0*P**2*R*S**6*T*UGA_star**16 + 352088064.0*P**2*R*S**6*T*UGA_star**15*epsilon + 14541312.0*P**2*R*S**6*T*UGA_star**15 - 336927744.0*P**2*R*S**6*T*UGA_star**14*epsilon - 14418432.0*P**2*R*S**6*T*UGA_star**14 + 227266560.0*P**2*R*S**6*T*UGA_star**13*epsilon + 10266624.0*P**2*R*S**6*T*UGA_star**13 - 106831872.0*P**2*R*S**6*T*UGA_star**12*epsilon - 5193216.0*P**2*R*S**6*T*UGA_star**12 + 35063808.0*P**2*R*S**6*T*UGA_star**11*epsilon + 1801728.0*P**2*R*S**6*T*UGA_star**11 - 8896512.0*P**2*R*S**6*T*UGA_star**10*epsilon - 377856.0*P**2*R*S**6*T*UGA_star**10 + 2230272.0*P**2*R*S**6*T*UGA_star**9*epsilon + 9216.0*P**2*R*S**6*T*UGA_star**9 - 476160.0*P**2*R*S**6*T*UGA_star**8*epsilon + 27648.0*P**2*R*S**6*T*UGA_star**8 + 24576.0*P**2*R*S**6*T*UGA_star**7*epsilon - 10752.0*P**2*R*S**6*T*UGA_star**7 + 9216.0*P**2*R*S**6*T*UGA_star**6*epsilon + 1536.0*P**2*R*S**6*T*UGA_star**6 - 2469888.0*P**2*R*S**5*T**2*UGA_star**20*epsilon - 138240.0*P**2*R*S**5*T**2*UGA_star**20 + 27168768.0*P**2*R*S**5*T**2*UGA_star**19*epsilon + 1520640.0*P**2*R*S**5*T**2*UGA_star**19 - 134977536.0*P**2*R*S**5*T**2*UGA_star**18*epsilon - 7557120.0*P**2*R*S**5*T**2*UGA_star**18 + 398997504.0*P**2*R*S**5*T**2*UGA_star**17*epsilon + 22358016.0*P**2*R*S**5*T**2*UGA_star**17 - 777449472.0*P**2*R*S**5*T**2*UGA_star**16*epsilon - 43642368.0*P**2*R*S**5*T**2*UGA_star**16 + 1043619840.0*P**2*R*S**5*T**2*UGA_star**15*epsilon + 58791936.0*P**2*R*S**5*T**2*UGA_star**15 - 977190912.0*P**2*R*S**5*T**2*UGA_star**14*epsilon - 55448064.0*P**2*R*S**5*T**2*UGA_star**14 + 629231616.0*P**2*R*S**5*T**2*UGA_star**13*epsilon + 36274176.0*P**2*R*S**5*T**2*UGA_star**13 - 265089024.0*P**2*R*S**5*T**2*UGA_star**12*epsilon - 15906816.0*P**2*R*S**5*T**2*UGA_star**12 + 64401408.0*P**2*R*S**5*T**2*UGA_star**11*epsilon + 4396032.0*P**2*R*S**5*T**2*UGA_star**11 - 5732352.0*P**2*R*S**5*T**2*UGA_star**10*epsilon - 746496.0*P**2*R*S**5*T**2*UGA_star**10 - 608256.0*P**2*R*S**5*T**2*UGA_star**9*epsilon + 129024.0*P**2*R*S**5*T**2*UGA_star**9 + 184320.0*P**2*R*S**5*T**2*UGA_star**8*epsilon - 32256.0*P**2*R*S**5*T**2*UGA_star**8 - 122880.0*P**2*R*S**5*T**2*UGA_star**7*epsilon + 36864.0*P**2*R*S**5*T**2*UGA_star**6*epsilon + 1536.0*P**2*R*S**5*T**2*UGA_star**6 - 4096000.0*P**2*R*S**4*T**3*UGA_star**20*epsilon - 256000.0*P**2*R*S**4*T**3*UGA_star**20 + 45056000.0*P**2*R*S**4*T**3*UGA_star**19*epsilon + 2816000.0*P**2*R*S**4*T**3*UGA_star**19 - 223979520.0*P**2*R*S**4*T**3*UGA_star**18*epsilon - 13987840.0*P**2*R*S**4*T**3*UGA_star**18 + 662743040.0*P**2*R*S**4*T**3*UGA_star**17*epsilon + 41313280.0*P**2*R*S**4*T**3*UGA_star**17 - 1292457984.0*P**2*R*S**4*T**3*UGA_star**16*epsilon - 80306176.0*P**2*R*S**4*T**3*UGA_star**16 + 1734189056.0*P**2*R*S**4*T**3*UGA_star**15*epsilon + 107203584.0*P**2*R*S**4*T**3*UGA_star**15 - 1616533504.0*P**2*R*S**4*T**3*UGA_star**14*epsilon - 99188736.0*P**2*R*S**4*T**3*UGA_star**14 + 1024372736.0*P**2*R*S**4*T**3*UGA_star**13*epsilon + 62244864.0*P**2*R*S**4*T**3*UGA_star**13 - 409391104.0*P**2*R*S**4*T**3*UGA_star**12*epsilon - 24692736.0*P**2*R*S**4*T**3*UGA_star**12 + 79601664.0*P**2*R*S**4*T**3*UGA_star**11*epsilon + 5042176.0*P**2*R*S**4*T**3*UGA_star**11 + 5228544.0*P**2*R*S**4*T**3*UGA_star**10*epsilon - 118784.0*P**2*R*S**4*T**3*UGA_star**10 - 5380096.0*P**2*R*S**4*T**3*UGA_star**9*epsilon - 11264.0*P**2*R*S**4*T**3*UGA_star**9 + 681984.0*P**2*R*S**4*T**3*UGA_star**8*epsilon - 77824.0*P**2*R*S**4*T**3*UGA_star**8 - 65536.0*P**2*R*S**4*T**3*UGA_star**7*epsilon + 19456.0*P**2*R*S**4*T**3*UGA_star**7 + 30720.0*P**2*R*S**4*T**3*UGA_star**6*epsilon - 4116480.0*P**2*R*S**3*T**4*UGA_star**20*epsilon - 230400.0*P**2*R*S**3*T**4*UGA_star**20 + 45281280.0*P**2*R*S**3*T**4*UGA_star**19*epsilon + 2534400.0*P**2*R*S**3*T**4*UGA_star**19 - 225408000.0*P**2*R*S**3*T**4*UGA_star**18*epsilon - 12618240.0*P**2*R*S**3*T**4*UGA_star**18 + 668759040.0*P**2*R*S**3*T**4*UGA_star**17*epsilon + 37447680.0*P**2*R*S**3*T**4*UGA_star**17 - 1309295616.0*P**2*R*S**3*T**4*UGA_star**16*epsilon - 73334784.0*P**2*R*S**3*T**4*UGA_star**16 + 1765612544.0*P**2*R*S**3*T**4*UGA_star**15*epsilon + 98881536.0*P**2*R*S**3*T**4*UGA_star**15 - 1655597056.0*P**2*R*S**3*T**4*UGA_star**14*epsilon - 92580864.0*P**2*R*S**3*T**4*UGA_star**14 + 1055934464.0*P**2*R*S**3*T**4*UGA_star**13*epsilon + 58699776.0*P**2*R*S**3*T**4*UGA_star**13 - 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8632320.0*P*R**6*S*T**2*UGA_star**18*epsilon - 348672.0*P*R**6*S*T**2*UGA_star**18 + 10530816.0*P*R**6*S*T**2*UGA_star**17*epsilon + 454656.0*P*R**6*S*T**2*UGA_star**17 - 8918016.0*P*R**6*S*T**2*UGA_star**16*epsilon - 427008.0*P*R**6*S*T**2*UGA_star**16 + 5815296.0*P*R**6*S*T**2*UGA_star**15*epsilon + 296448.0*P*R**6*S*T**2*UGA_star**15 - 2727936.0*P*R**6*S*T**2*UGA_star**14*epsilon - 127488.0*P*R**6*S*T**2*UGA_star**14 + 804864.0*P*R**6*S*T**2*UGA_star**13*epsilon + 21504.0*P*R**6*S*T**2*UGA_star**13 - 187392.0*P*R**6*S*T**2*UGA_star**12*epsilon + 1536.0*P*R**6*S*T**2*UGA_star**12 + 46080.0*P*R**6*S*T**2*UGA_star**11*epsilon - 274432.0*P*R**6*T**3*UGA_star**20*epsilon - 15360.0*P*R**6*T**3*UGA_star**20 + 1372160.0*P*R**6*T**3*UGA_star**19*epsilon + 76800.0*P*R**6*T**3*UGA_star**19 - 2825216.0*P*R**6*T**3*UGA_star**18*epsilon - 158208.0*P*R**6*T**3*UGA_star**18 + 3165184.0*P*R**6*T**3*UGA_star**17*epsilon + 178176.0*P*R**6*T**3*UGA_star**17 - 2245632.0*P*R**6*T**3*UGA_star**16*epsilon - 127488.0*P*R**6*T**3*UGA_star**16 + 1172480.0*P*R**6*T**3*UGA_star**15*epsilon + 64512.0*P*R**6*T**3*UGA_star**15 - 448512.0*P*R**6*T**3*UGA_star**14*epsilon - 19968.0*P*R**6*T**3*UGA_star**14 + 77824.0*P*R**6*T**3*UGA_star**13*epsilon + 6144.0*P*R**6*T**3*UGA_star**12*epsilon + 1536.0*P*R**6*T**3*UGA_star**12 + 258048.0*P*R**5*S**4*UGA_star**20*epsilon - 1548288.0*P*R**5*S**4*UGA_star**19*epsilon + 4085760.0*P*R**5*S**4*UGA_star**18*epsilon - 6451200.0*P*R**5*S**4*UGA_star**17*epsilon + 7132160.0*P*R**5*S**4*UGA_star**16*epsilon - 6064128.0*P*R**5*S**4*UGA_star**15*epsilon + 4021248.0*P*R**5*S**4*UGA_star**14*epsilon - 2007040.0*P*R**5*S**4*UGA_star**13*epsilon + 752640.0*P*R**5*S**4*UGA_star**12*epsilon - 215040.0*P*R**5*S**4*UGA_star**11*epsilon + 35840.0*P*R**5*S**4*UGA_star**10*epsilon + 1490944.0*P*R**5*S**3*T*UGA_star**20*epsilon + 28672.0*P*R**5*S**3*T*UGA_star**20 - 8945664.0*P*R**5*S**3*T*UGA_star**19*epsilon - 172032.0*P*R**5*S**3*T*UGA_star**19 + 23267328.0*P*R**5*S**3*T*UGA_star**18*epsilon + 465920.0*P*R**5*S**3*T*UGA_star**18 - 35409920.0*P*R**5*S**3*T*UGA_star**17*epsilon - 788480.0*P*R**5*S**3*T*UGA_star**17 + 36720640.0*P*R**5*S**3*T*UGA_star**16*epsilon + 977920.0*P*R**5*S**3*T*UGA_star**16 - 28514304.0*P*R**5*S**3*T*UGA_star**15*epsilon - 950272.0*P*R**5*S**3*T*UGA_star**15 + 16451584.0*P*R**5*S**3*T*UGA_star**14*epsilon + 719872.0*P*R**5*S**3*T*UGA_star**14 - 6176768.0*P*R**5*S**3*T*UGA_star**13*epsilon - 419840.0*P*R**5*S**3*T*UGA_star**13 + 1228800.0*P*R**5*S**3*T*UGA_star**12*epsilon + 189440.0*P*R**5*S**3*T*UGA_star**12 - 153600.0*P*R**5*S**3*T*UGA_star**11*epsilon - 61440.0*P*R**5*S**3*T*UGA_star**11 + 40960.0*P*R**5*S**3*T*UGA_star**10*epsilon + 10240.0*P*R**5*S**3*T*UGA_star**10 + 2451456.0*P*R**5*S**2*T**2*UGA_star**20*epsilon + 96768.0*P*R**5*S**2*T**2*UGA_star**20 - 14708736.0*P*R**5*S**2*T**2*UGA_star**19*epsilon - 580608.0*P*R**5*S**2*T**2*UGA_star**19 + 37730304.0*P*R**5*S**2*T**2*UGA_star**18*epsilon + 1511424.0*P*R**5*S**2*T**2*UGA_star**18 - 55111680.0*P*R**5*S**2*T**2*UGA_star**17*epsilon - 2304000.0*P*R**5*S**2*T**2*UGA_star**17 + 52783104.0*P*R**5*S**2*T**2*UGA_star**16*epsilon + 2388480.0*P*R**5*S**2*T**2*UGA_star**16 - 37158912.0*P*R**5*S**2*T**2*UGA_star**15*epsilon - 1832448.0*P*R**5*S**2*T**2*UGA_star**15 + 20336640.0*P*R**5*S**2*T**2*UGA_star**14*epsilon + 1010688.0*P*R**5*S**2*T**2*UGA_star**14 - 8048640.0*P*R**5*S**2*T**2*UGA_star**13*epsilon - 328704.0*P*R**5*S**2*T**2*UGA_star**13 + 2064384.0*P*R**5*S**2*T**2*UGA_star**12*epsilon + 30720.0*P*R**5*S**2*T**2*UGA_star**12 - 430080.0*P*R**5*S**2*T**2*UGA_star**11*epsilon + 7680.0*P*R**5*S**2*T**2*UGA_star**11 + 92160.0*P*R**5*S**2*T**2*UGA_star**10*epsilon + 1646592.0*P*R**5*S*T**3*UGA_star**20*epsilon + 92160.0*P*R**5*S*T**3*UGA_star**20 - 9879552.0*P*R**5*S*T**3*UGA_star**19*epsilon - 552960.0*P*R**5*S*T**3*UGA_star**19 + 24963072.0*P*R**5*S*T**3*UGA_star**18*epsilon + 1397760.0*P*R**5*S*T**3*UGA_star**18 - 34738176.0*P*R**5*S*T**3*UGA_star**17*epsilon - 1950720.0*P*R**5*S*T**3*UGA_star**17 + 29755392.0*P*R**5*S*T**3*UGA_star**16*epsilon + 1681920.0*P*R**5*S*T**3*UGA_star**16 - 17274880.0*P*R**5*S*T**3*UGA_star**15*epsilon - 975360.0*P*R**5*S*T**3*UGA_star**15 + 7542784.0*P*R**5*S*T**3*UGA_star**14*epsilon + 399360.0*P*R**5*S*T**3*UGA_star**14 - 2334720.0*P*R**5*S*T**3*UGA_star**13*epsilon - 92160.0*P*R**5*S*T**3*UGA_star**13 + 268288.0*P*R**5*S*T**3*UGA_star**12*epsilon - 7680.0*P*R**5*S*T**3*UGA_star**12 + 51200.0*P*R**5*S*T**3*UGA_star**11*epsilon + 7680.0*P*R**5*S*T**3*UGA_star**11 + 409600.0*P*R**5*T**4*UGA_star**20*epsilon + 25600.0*P*R**5*T**4*UGA_star**20 - 2457600.0*P*R**5*T**4*UGA_star**19*epsilon - 153600.0*P*R**5*T**4*UGA_star**19 + 6109184.0*P*R**5*T**4*UGA_star**18*epsilon + 380928.0*P*R**5*T**4*UGA_star**18 - 8054784.0*P*R**5*T**4*UGA_star**17*epsilon - 498688.0*P*R**5*T**4*UGA_star**17 + 5977088.0*P*R**5*T**4*UGA_star**16*epsilon + 364544.0*P*R**5*T**4*UGA_star**16 - 2445312.0*P*R**5*T**4*UGA_star**15*epsilon - 147456.0*P*R**5*T**4*UGA_star**15 + 503808.0*P*R**5*T**4*UGA_star**14*epsilon + 36864.0*P*R**5*T**4*UGA_star**14 - 14336.0*P*R**5*T**4*UGA_star**13*epsilon - 11264.0*P*R**5*T**4*UGA_star**13 - 48128.0*P*R**5*T**4*UGA_star**12*epsilon + 3072.0*P*R**5*T**4*UGA_star**12 + 20480.0*P*R**5*T**4*UGA_star**11*epsilon - 258048.0*P*R**4*S**5*UGA_star**20*epsilon + 1806336.0*P*R**4*S**5*UGA_star**19*epsilon - 5591040.0*P*R**4*S**5*UGA_star**18*epsilon + 10235904.0*P*R**4*S**5*UGA_star**17*epsilon - 12665856.0*P*R**4*S**5*UGA_star**16*epsilon + 11590656.0*P*R**4*S**5*UGA_star**15*epsilon - 8279040.0*P*R**4*S**5*UGA_star**14*epsilon + 4623360.0*P*R**4*S**5*UGA_star**13*epsilon - 1956864.0*P*R**4*S**5*UGA_star**12*epsilon + 623616.0*P*R**4*S**5*UGA_star**11*epsilon - 150528.0*P*R**4*S**5*UGA_star**10*epsilon + 21504.0*P*R**4*S**5*UGA_star**9*epsilon - 1863680.0*P*R**4*S**4*T*UGA_star**20*epsilon - 35840.0*P*R**4*S**4*T*UGA_star**20 + 13045760.0*P*R**4*S**4*T*UGA_star**19*epsilon + 250880.0*P*R**4*S**4*T*UGA_star**19 - 39925760.0*P*R**4*S**4*T*UGA_star**18*epsilon - 788480.0*P*R**4*S**4*T*UGA_star**18 + 71034880.0*P*R**4*S**4*T*UGA_star**17*epsilon + 1505280.0*P*R**4*S**4*T*UGA_star**17 - 83415040.0*P*R**4*S**4*T*UGA_star**16*epsilon - 2014720.0*P*R**4*S**4*T*UGA_star**16 + 70512640.0*P*R**4*S**4*T*UGA_star**15*epsilon + 2060800.0*P*R**4*S**4*T*UGA_star**15 - 45178880.0*P*R**4*S**4*T*UGA_star**14*epsilon - 1666560.0*P*R**4*S**4*T*UGA_star**14 + 21340160.0*P*R**4*S**4*T*UGA_star**13*epsilon + 1057280.0*P*R**4*S**4*T*UGA_star**13 - 6471680.0*P*R**4*S**4*T*UGA_star**12*epsilon - 519680.0*P*R**4*S**4*T*UGA_star**12 + 983040.0*P*R**4*S**4*T*UGA_star**11*epsilon + 197120.0*P*R**4*S**4*T*UGA_star**11 - 92160.0*P*R**4*S**4*T*UGA_star**10*epsilon - 53760.0*P*R**4*S**4*T*UGA_star**10 + 30720.0*P*R**4*S**4*T*UGA_star**9*epsilon + 7680.0*P*R**4*S**4*T*UGA_star**9 - 4085760.0*P*R**4*S**3*T**2*UGA_star**20*epsilon - 161280.0*P*R**4*S**3*T**2*UGA_star**20 + 28600320.0*P*R**4*S**3*T**2*UGA_star**19*epsilon + 1128960.0*P*R**4*S**3*T**2*UGA_star**19 - 86691840.0*P*R**4*S**3*T**2*UGA_star**18*epsilon - 3456000.0*P*R**4*S**3*T**2*UGA_star**18 + 150067200.0*P*R**4*S**3*T**2*UGA_star**17*epsilon + 6151680.0*P*R**4*S**3*T**2*UGA_star**17 - 166625280.0*P*R**4*S**3*T**2*UGA_star**16*epsilon - 7219200.0*P*R**4*S**3*T**2*UGA_star**16 + 129116160.0*P*R**4*S**3*T**2*UGA_star**15*epsilon + 6059520.0*P*R**4*S**3*T**2*UGA_star**15 - 75847680.0*P*R**4*S**3*T**2*UGA_star**14*epsilon - 3778560.0*P*R**4*S**3*T**2*UGA_star**14 + 35082240.0*P*R**4*S**3*T**2*UGA_star**13*epsilon + 1651200.0*P*R**4*S**3*T**2*UGA_star**13 - 11827200.0*P*R**4*S**3*T**2*UGA_star**12*epsilon - 391680.0*P*R**4*S**3*T**2*UGA_star**12 + 2580480.0*P*R**4*S**3*T**2*UGA_star**11*epsilon - 460800.0*P*R**4*S**3*T**2*UGA_star**10*epsilon + 15360.0*P*R**4*S**3*T**2*UGA_star**10 + 92160.0*P*R**4*S**3*T**2*UGA_star**9*epsilon - 4116480.0*P*R**4*S**2*T**3*UGA_star**20*epsilon - 230400.0*P*R**4*S**2*T**3*UGA_star**20 + 28815360.0*P*R**4*S**2*T**3*UGA_star**19*epsilon + 1612800.0*P*R**4*S**2*T**3*UGA_star**19 - 86553600.0*P*R**4*S**2*T**3*UGA_star**18*epsilon - 4846080.0*P*R**4*S**2*T**3*UGA_star**18 + 145689600.0*P*R**4*S**2*T**3*UGA_star**17*epsilon + 8171520.0*P*R**4*S**2*T**3*UGA_star**17 - 151474176.0*P*R**4*S**2*T**3*UGA_star**16*epsilon - 8533504.0*P*R**4*S**2*T**3*UGA_star**16 + 102860800.0*P*R**4*S**2*T**3*UGA_star**15*epsilon + 5826560.0*P*R**4*S**2*T**3*UGA_star**15 - 48849920.0*P*R**4*S**2*T**3*UGA_star**14*epsilon - 2736640.0*P*R**4*S**2*T**3*UGA_star**14 + 17628160.0*P*R**4*S**2*T**3*UGA_star**13*epsilon + 901120.0*P*R**4*S**2*T**3*UGA_star**13 - 4372480.0*P*R**4*S**2*T**3*UGA_star**12*epsilon - 151040.0*P*R**4*S**2*T**3*UGA_star**12 + 229376.0*P*R**4*S**2*T**3*UGA_star**11*epsilon - 29696.0*P*R**4*S**2*T**3*UGA_star**11 + 143360.0*P*R**4*S**2*T**3*UGA_star**10*epsilon + 15360.0*P*R**4*S**2*T**3*UGA_star**10 - 2048000.0*P*R**4*S*T**4*UGA_star**20*epsilon - 128000.0*P*R**4*S*T**4*UGA_star**20 + 14336000.0*P*R**4*S*T**4*UGA_star**19*epsilon + 896000.0*P*R**4*S*T**4*UGA_star**19 - 42675200.0*P*R**4*S*T**4*UGA_star**18*epsilon - 2662400.0*P*R**4*S*T**4*UGA_star**18 + 69821440.0*P*R**4*S*T**4*UGA_star**17*epsilon + 4334080.0*P*R**4*S*T**4*UGA_star**17 - 67523584.0*P*R**4*S*T**4*UGA_star**16*epsilon - 4148736.0*P*R**4*S*T**4*UGA_star**16 + 38379520.0*P*R**4*S*T**4*UGA_star**15*epsilon + 2324480.0*P*R**4*S*T**4*UGA_star**15 - 11781120.0*P*R**4*S*T**4*UGA_star**14*epsilon - 732160.0*P*R**4*S*T**4*UGA_star**14 + 1397760.0*P*R**4*S*T**4*UGA_star**13*epsilon + 156160.0*P*R**4*S*T**4*UGA_star**13 + 271360.0*P*R**4*S*T**4*UGA_star**12*epsilon - 53760.0*P*R**4*S*T**4*UGA_star**12 - 260096.0*P*R**4*S*T**4*UGA_star**11*epsilon + 14336.0*P*R**4*S*T**4*UGA_star**11 + 81920.0*P*R**4*S*T**4*UGA_star**10*epsilon - 411648.0*P*R**4*T**5*UGA_star**20*epsilon - 23040.0*P*R**4*T**5*UGA_star**20 + 2881536.0*P*R**4*T**5*UGA_star**19*epsilon + 161280.0*P*R**4*T**5*UGA_star**19 - 8521728.0*P*R**4*T**5*UGA_star**18*epsilon - 477696.0*P*R**4*T**5*UGA_star**18 + 13651968.0*P*R**4*T**5*UGA_star**17*epsilon + 768000.0*P*R**4*T**5*UGA_star**17 - 12481536.0*P*R**4*T**5*UGA_star**16*epsilon - 707584.0*P*R**4*T**5*UGA_star**16 + 6053888.0*P*R**4*T**5*UGA_star**15*epsilon + 350720.0*P*R**4*T**5*UGA_star**15 - 966656.0*P*R**4*T**5*UGA_star**14*epsilon - 64000.0*P*R**4*T**5*UGA_star**14 - 294912.0*P*R**4*T**5*UGA_star**13*epsilon - 11264.0*P*R**4*T**5*UGA_star**13 + 74752.0*P*R**4*T**5*UGA_star**12*epsilon + 2560.0*P*R**4*T**5*UGA_star**12 + 14336.0*P*R**4*T**5*UGA_star**11*epsilon + 1024.0*P*R**4*T**5*UGA_star**11 + 172032.0*P*R**3*S**6*UGA_star**20*epsilon - 1376256.0*P*R**3*S**6*UGA_star**19*epsilon + 4902912.0*P*R**3*S**6*UGA_star**18*epsilon - 10321920.0*P*R**3*S**6*UGA_star**17*epsilon + 14457856.0*P*R**3*S**6*UGA_star**16*epsilon - 14508032.0*P*R**3*S**6*UGA_star**15*epsilon + 11038720.0*P*R**3*S**6*UGA_star**14*epsilon - 6594560.0*P*R**3*S**6*UGA_star**13*epsilon + 3082240.0*P*R**3*S**6*UGA_star**12*epsilon - 1089536.0*P*R**3*S**6*UGA_star**11*epsilon + 286720.0*P*R**3*S**6*UGA_star**10*epsilon - 57344.0*P*R**3*S**6*UGA_star**9*epsilon + 7168.0*P*R**3*S**6*UGA_star**8*epsilon + 1490944.0*P*R**3*S**5*T*UGA_star**20*epsilon + 28672.0*P*R**3*S**5*T*UGA_star**20 - 11927552.0*P*R**3*S**5*T*UGA_star**19*epsilon - 229376.0*P*R**3*S**5*T*UGA_star**19 + 42104832.0*P*R**3*S**5*T*UGA_star**18*epsilon + 824320.0*P*R**3*S**5*T*UGA_star**18 - 86646784.0*P*R**3*S**5*T*UGA_star**17*epsilon - 1777664.0*P*R**3*S**5*T*UGA_star**17 + 116328448.0*P*R**3*S**5*T*UGA_star**16*epsilon + 2612224.0*P*R**3*S**5*T*UGA_star**16 - 109037568.0*P*R**3*S**5*T*UGA_star**15*epsilon - 2834432.0*P*R**3*S**5*T*UGA_star**15 + 75278336.0*P*R**3*S**5*T*UGA_star**14*epsilon + 2394112.0*P*R**3*S**5*T*UGA_star**14 - 39495680.0*P*R**3*S**5*T*UGA_star**13*epsilon - 1605632.0*P*R**3*S**5*T*UGA_star**13 + 15181824.0*P*R**3*S**5*T*UGA_star**12*epsilon + 845824.0*P*R**3*S**5*T*UGA_star**12 - 3627008.0*P*R**3*S**5*T*UGA_star**11*epsilon - 344064.0*P*R**3*S**5*T*UGA_star**11 + 344064.0*P*R**3*S**5*T*UGA_star**10*epsilon + 107520.0*P*R**3*S**5*T*UGA_star**10 - 6144.0*P*R**3*S**5*T*UGA_star**9*epsilon - 24576.0*P*R**3*S**5*T*UGA_star**9 + 12288.0*P*R**3*S**5*T*UGA_star**8*epsilon + 3072.0*P*R**3*S**5*T*UGA_star**8 + 4085760.0*P*R**3*S**4*T**2*UGA_star**20*epsilon + 161280.0*P*R**3*S**4*T**2*UGA_star**20 - 32686080.0*P*R**3*S**4*T**2*UGA_star**19*epsilon - 1290240.0*P*R**3*S**4*T**2*UGA_star**19 + 114585600.0*P*R**3*S**4*T**2*UGA_star**18*epsilon + 4554240.0*P*R**3*S**4*T**2*UGA_star**18 - 231383040.0*P*R**3*S**4*T**2*UGA_star**17*epsilon - 9369600.0*P*R**3*S**4*T**2*UGA_star**17 + 298859520.0*P*R**3*S**4*T**2*UGA_star**16*epsilon + 12564480.0*P*R**3*S**4*T**2*UGA_star**16 - 262010880.0*P*R**3*S**4*T**2*UGA_star**15*epsilon - 11719680.0*P*R**3*S**4*T**2*UGA_star**15 + 164997120.0*P*R**3*S**4*T**2*UGA_star**14*epsilon + 7956480.0*P*R**3*S**4*T**2*UGA_star**14 - 80148480.0*P*R**3*S**4*T**2*UGA_star**13*epsilon - 3978240.0*P*R**3*S**4*T**2*UGA_star**13 + 31242240.0*P*R**3*S**4*T**2*UGA_star**12*epsilon + 1344000.0*P*R**3*S**4*T**2*UGA_star**12 - 8939520.0*P*R**3*S**4*T**2*UGA_star**11*epsilon - 199680.0*P*R**3*S**4*T**2*UGA_star**11 + 1566720.0*P*R**3*S**4*T**2*UGA_star**10*epsilon - 38400.0*P*R**3*S**4*T**2*UGA_star**10 - 215040.0*P*R**3*S**4*T**2*UGA_star**9*epsilon + 15360.0*P*R**3*S**4*T**2*UGA_star**9 + 46080.0*P*R**3*S**4*T**2*UGA_star**8*epsilon + 5488640.0*P*R**3*S**3*T**3*UGA_star**20*epsilon + 307200.0*P*R**3*S**3*T**3*UGA_star**20 - 43909120.0*P*R**3*S**3*T**3*UGA_star**19*epsilon - 2457600.0*P*R**3*S**3*T**3*UGA_star**19 + 153088000.0*P*R**3*S**3*T**3*UGA_star**18*epsilon + 8570880.0*P*R**3*S**3*T**3*UGA_star**18 - 304168960.0*P*R**3*S**3*T**3*UGA_star**17*epsilon - 17049600.0*P*R**3*S**3*T**3*UGA_star**17 + 378486784.0*P*R**3*S**3*T**3*UGA_star**16*epsilon + 21277696.0*P*R**3*S**3*T**3*UGA_star**16 - 306704384.0*P*R**3*S**3*T**3*UGA_star**15*epsilon - 17335296.0*P*R**3*S**3*T**3*UGA_star**15 + 165724160.0*P*R**3*S**3*T**3*UGA_star**14*epsilon + 9415680.0*P*R**3*S**3*T**3*UGA_star**14 - 62689280.0*P*R**3*S**3*T**3*UGA_star**13*epsilon - 3507200.0*P*R**3*S**3*T**3*UGA_star**13 + 18104320.0*P*R**3*S**3*T**3*UGA_star**12*epsilon + 906240.0*P*R**3*S**3*T**3*UGA_star**12 - 3477504.0*P*R**3*S**3*T**3*UGA_star**11*epsilon - 101376.0*P*R**3*S**3*T**3*UGA_star**11 - 126976.0*P*R**3*S**3*T**3*UGA_star**10*epsilon - 41984.0*P*R**3*S**3*T**3*UGA_star**10 + 184320.0*P*R**3*S**3*T**3*UGA_star**9*epsilon + 15360.0*P*R**3*S**3*T**3*UGA_star**9 + 4096000.0*P*R**3*S**2*T**4*UGA_star**20*epsilon + 256000.0*P*R**3*S**2*T**4*UGA_star**20 - 32768000.0*P*R**3*S**2*T**4*UGA_star**19*epsilon - 2048000.0*P*R**3*S**2*T**4*UGA_star**19 + 113704960.0*P*R**3*S**2*T**4*UGA_star**18*epsilon + 7096320.0*P*R**3*S**2*T**4*UGA_star**18 - 222679040.0*P*R**3*S**2*T**4*UGA_star**17*epsilon - 13844480.0*P*R**3*S**2*T**4*UGA_star**17 + 267431936.0*P*R**3*S**2*T**4*UGA_star**16*epsilon + 16502784.0*P*R**3*S**2*T**4*UGA_star**16 - 199131136.0*P*R**3*S**2*T**4*UGA_star**15*epsilon - 12140544.0*P*R**3*S**2*T**4*UGA_star**15 + 87080960.0*P*R**3*S**2*T**4*UGA_star**14*epsilon + 5263360.0*P*R**3*S**2*T**4*UGA_star**14 - 18247680.0*P*R**3*S**2*T**4*UGA_star**13*epsilon - 1228800.0*P*R**3*S**2*T**4*UGA_star**13 - 184320.0*P*R**3*S**2*T**4*UGA_star**12*epsilon + 215040.0*P*R**3*S**2*T**4*UGA_star**12 + 1040384.0*P*R**3*S**2*T**4*UGA_star**11*epsilon - 98304.0*P*R**3*S**2*T**4*UGA_star**11 - 466944.0*P*R**3*S**2*T**4*UGA_star**10*epsilon + 26624.0*P*R**3*S**2*T**4*UGA_star**10 + 122880.0*P*R**3*S**2*T**4*UGA_star**9*epsilon + 1646592.0*P*R**3*S*T**5*UGA_star**20*epsilon + 92160.0*P*R**3*S*T**5*UGA_star**20 - 13172736.0*P*R**3*S*T**5*UGA_star**19*epsilon - 737280.0*P*R**3*S*T**5*UGA_star**19 + 45570048.0*P*R**3*S*T**5*UGA_star**18*epsilon + 2552832.0*P*R**3*S*T**5*UGA_star**18 - 88387584.0*P*R**3*S*T**5*UGA_star**17*epsilon - 4961280.0*P*R**3*S*T**5*UGA_star**17 + 103602176.0*P*R**3*S*T**5*UGA_star**16*epsilon + 5837312.0*P*R**3*S*T**5*UGA_star**16 - 72595456.0*P*R**3*S*T**5*UGA_star**15*epsilon - 4123136.0*P*R**3*S*T**5*UGA_star**15 + 26587136.0*P*R**3*S*T**5*UGA_star**14*epsilon + 1546240.0*P*R**3*S*T**5*UGA_star**14 - 1875968.0*P*R**3*S*T**5*UGA_star**13*epsilon - 141312.0*P*R**3*S*T**5*UGA_star**13 - 1673216.0*P*R**3*S*T**5*UGA_star**12*epsilon - 79360.0*P*R**3*S*T**5*UGA_star**12 + 231424.0*P*R**3*S*T**5*UGA_star**11*epsilon + 9728.0*P*R**3*S*T**5*UGA_star**11 + 67584.0*P*R**3*S*T**5*UGA_star**10*epsilon + 4096.0*P*R**3*S*T**5*UGA_star**10 + 272384.0*P*R**3*T**6*UGA_star**20*epsilon + 10752.0*P*R**3*T**6*UGA_star**20 - 2179072.0*P*R**3*T**6*UGA_star**19*epsilon - 86016.0*P*R**3*T**6*UGA_star**19 + 7544832.0*P*R**3*T**6*UGA_star**18*epsilon + 299520.0*P*R**3*T**6*UGA_star**18 - 14651392.0*P*R**3*T**6*UGA_star**17*epsilon - 589824.0*P*R**3*T**6*UGA_star**17 + 17182720.0*P*R**3*T**6*UGA_star**16*epsilon + 710656.0*P*R**3*T**6*UGA_star**16 - 12007424.0*P*R**3*T**6*UGA_star**15*epsilon - 522752.0*P*R**3*T**6*UGA_star**15 + 4319232.0*P*R**3*T**6*UGA_star**14*epsilon + 211456.0*P*R**3*T**6*UGA_star**14 - 212992.0*P*R**3*T**6*UGA_star**13*epsilon - 25600.0*P*R**3*T**6*UGA_star**13 - 329728.0*P*R**3*T**6*UGA_star**12*epsilon - 11776.0*P*R**3*T**6*UGA_star**12 + 51200.0*P*R**3*T**6*UGA_star**11*epsilon + 3584.0*P*R**3*T**6*UGA_star**11 + 10240.0*P*R**3*T**6*UGA_star**10*epsilon - 73728.0*P*R**2*S**7*UGA_star**20*epsilon + 663552.0*P*R**2*S**7*UGA_star**19*epsilon - 2678784.0*P*R**2*S**7*UGA_star**18*epsilon + 6414336.0*P*R**2*S**7*UGA_star**17*epsilon - 10175488.0*P*R**2*S**7*UGA_star**16*epsilon + 11363328.0*P*R**2*S**7*UGA_star**15*epsilon - 9326592.0*P*R**2*S**7*UGA_star**14*epsilon + 5836800.0*P*R**2*S**7*UGA_star**13*epsilon - 2856960.0*P*R**2*S**7*UGA_star**12*epsilon + 1087488.0*P*R**2*S**7*UGA_star**11*epsilon - 307200.0*P*R**2*S**7*UGA_star**10*epsilon + 61440.0*P*R**2*S**7*UGA_star**9*epsilon - 9216.0*P*R**2*S**7*UGA_star**8*epsilon + 1024.0*P*R**2*S**7*UGA_star**7*epsilon - 745472.0*P*R**2*S**6*T*UGA_star**20*epsilon - 14336.0*P*R**2*S**6*T*UGA_star**20 + 6709248.0*P*R**2*S**6*T*UGA_star**19*epsilon + 129024.0*P*R**2*S**6*T*UGA_star**19 - 26880000.0*P*R**2*S**6*T*UGA_star**18*epsilon - 523264.0*P*R**2*S**6*T*UGA_star**18 + 63178752.0*P*R**2*S**6*T*UGA_star**17*epsilon + 1268736.0*P*R**2*S**6*T*UGA_star**17 - 96819200.0*P*R**2*S**6*T*UGA_star**16*epsilon - 2064896.0*P*R**2*S**6*T*UGA_star**16 + 102082560.0*P*R**2*S**6*T*UGA_star**15*epsilon + 2413056.0*P*R**2*S**6*T*UGA_star**15 - 76713984.0*P*R**2*S**6*T*UGA_star**14*epsilon - 2125824.0*P*R**2*S**6*T*UGA_star**14 + 42405888.0*P*R**2*S**6*T*UGA_star**13*epsilon + 1462272.0*P*R**2*S**6*T*UGA_star**13 - 17633280.0*P*R**2*S**6*T*UGA_star**12*epsilon - 795648.0*P*R**2*S**6*T*UGA_star**12 + 5304320.0*P*R**2*S**6*T*UGA_star**11*epsilon + 336896.0*P*R**2*S**6*T*UGA_star**11 - 903168.0*P*R**2*S**6*T*UGA_star**10*epsilon - 107520.0*P*R**2*S**6*T*UGA_star**10 - 6144.0*P*R**2*S**6*T*UGA_star**9*epsilon + 25600.0*P*R**2*S**6*T*UGA_star**9 + 18432.0*P*R**2*S**6*T*UGA_star**8*epsilon - 4608.0*P*R**2*S**6*T*UGA_star**8 + 2048.0*P*R**2*S**6*T*UGA_star**7*epsilon + 512.0*P*R**2*S**6*T*UGA_star**7 - 2451456.0*P*R**2*S**5*T**2*UGA_star**20*epsilon - 96768.0*P*R**2*S**5*T**2*UGA_star**20 + 22063104.0*P*R**2*S**5*T**2*UGA_star**19*epsilon + 870912.0*P*R**2*S**5*T**2*UGA_star**19 - 87939072.0*P*R**2*S**5*T**2*UGA_star**18*epsilon - 3488256.0*P*R**2*S**5*T**2*UGA_star**18 + 203931648.0*P*R**2*S**5*T**2*UGA_star**17*epsilon + 8193024.0*P*R**2*S**5*T**2*UGA_star**17 - 304241664.0*P*R**2*S**5*T**2*UGA_star**16*epsilon - 12535296.0*P*R**2*S**5*T**2*UGA_star**16 + 305759232.0*P*R**2*S**5*T**2*UGA_star**15*epsilon + 13163520.0*P*R**2*S**5*T**2*UGA_star**15 - 212619264.0*P*R**2*S**5*T**2*UGA_star**14*epsilon - 9784320.0*P*R**2*S**5*T**2*UGA_star**14 + 105971712.0*P*R**2*S**5*T**2*UGA_star**13*epsilon + 5246976.0*P*R**2*S**5*T**2*UGA_star**13 - 40771584.0*P*R**2*S**5*T**2*UGA_star**12*epsilon - 2010624.0*P*R**2*S**5*T**2*UGA_star**12 + 13062144.0*P*R**2*S**5*T**2*UGA_star**11*epsilon + 477696.0*P*R**2*S**5*T**2*UGA_star**11 - 3127296.0*P*R**2*S**5*T**2*UGA_star**10*epsilon - 10752.0*P*R**2*S**5*T**2*UGA_star**10 + 368640.0*P*R**2*S**5*T**2*UGA_star**9*epsilon - 33792.0*P*R**2*S**5*T**2*UGA_star**9 - 15360.0*P*R**2*S**5*T**2*UGA_star**8*epsilon + 7680.0*P*R**2*S**5*T**2*UGA_star**8 + 9216.0*P*R**2*S**5*T**2*UGA_star**7*epsilon - 4116480.0*P*R**2*S**4*T**3*UGA_star**20*epsilon - 230400.0*P*R**2*S**4*T**3*UGA_star**20 + 37048320.0*P*R**2*S**4*T**3*UGA_star**19*epsilon + 2073600.0*P*R**2*S**4*T**3*UGA_star**19 - 147194880.0*P*R**2*S**4*T**3*UGA_star**18*epsilon - 8240640.0*P*R**2*S**4*T**3*UGA_star**18 + 338273280.0*P*R**2*S**4*T**3*UGA_star**17*epsilon + 18954240.0*P*R**2*S**4*T**3*UGA_star**17 - 494601216.0*P*R**2*S**4*T**3*UGA_star**16*epsilon - 27769344.0*P*R**2*S**4*T**3*UGA_star**16 + 476480512.0*P*R**2*S**4*T**3*UGA_star**15*epsilon + 26861568.0*P*R**2*S**4*T**3*UGA_star**15 - 303256576.0*P*R**2*S**4*T**3*UGA_star**14*epsilon - 17224704.0*P*R**2*S**4*T**3*UGA_star**14 + 125721600.0*P*R**2*S**4*T**3*UGA_star**13*epsilon + 7219200.0*P*R**2*S**4*T**3*UGA_star**13 - 34288640.0*P*R**2*S**4*T**3*UGA_star**12*epsilon - 1966080.0*P*R**2*S**4*T**3*UGA_star**12 + 7003136.0*P*R**2*S**4*T**3*UGA_star**11*epsilon + 359424.0*P*R**2*S**4*T**3*UGA_star**11 - 940032.0*P*R**2*S**4*T**3*UGA_star**10*epsilon - 19968.0*P*R**2*S**4*T**3*UGA_star**10 - 241664.0*P*R**2*S**4*T**3*UGA_star**9*epsilon - 24576.0*P*R**2*S**4*T**3*UGA_star**9 + 112640.0*P*R**2*S**4*T**3*UGA_star**8*epsilon + 7680.0*P*R**2*S**4*T**3*UGA_star**8 - 4096000.0*P*R**2*S**3*T**4*UGA_star**20*epsilon - 256000.0*P*R**2*S**3*T**4*UGA_star**20 + 36864000.0*P*R**2*S**3*T**4*UGA_star**19*epsilon + 2304000.0*P*R**2*S**3*T**4*UGA_star**19 - 146155520.0*P*R**2*S**3*T**4*UGA_star**18*epsilon - 9123840.0*P*R**2*S**3*T**4*UGA_star**18 + 333752320.0*P*R**2*S**3*T**4*UGA_star**17*epsilon + 20771840.0*P*R**2*S**3*T**4*UGA_star**17 - 480548864.0*P*R**2*S**3*T**4*UGA_star**16*epsilon - 29735936.0*P*R**2*S**3*T**4*UGA_star**16 + 446701568.0*P*R**2*S**3*T**4*UGA_star**15*epsilon + 27374592.0*P*R**2*S**3*T**4*UGA_star**15 - 260511744.0*P*R**2*S**3*T**4*UGA_star**14*epsilon - 15748096.0*P*R**2*S**3*T**4*UGA_star**14 + 84336640.0*P*R**2*S**3*T**4*UGA_star**13*epsilon + 5094400.0*P*R**2*S**3*T**4*UGA_star**13 - 7874560.0*P*R**2*S**3*T**4*UGA_star**12*epsilon - 691200.0*P*R**2*S**3*T**4*UGA_star**12 - 3424256.0*P*R**2*S**3*T**4*UGA_star**11*epsilon + 70656.0*P*R**2*S**3*T**4*UGA_star**11 + 1196032.0*P*R**2*S**3*T**4*UGA_star**10*epsilon - 84992.0*P*R**2*S**3*T**4*UGA_star**10 - 321536.0*P*R**2*S**3*T**4*UGA_star**9*epsilon + 24576.0*P*R**2*S**3*T**4*UGA_star**9 + 81920.0*P*R**2*S**3*T**4*UGA_star**8*epsilon - 2469888.0*P*R**2*S**2*T**5*UGA_star**20*epsilon - 138240.0*P*R**2*S**2*T**5*UGA_star**20 + 22228992.0*P*R**2*S**2*T**5*UGA_star**19*epsilon + 1244160.0*P*R**2*S**2*T**5*UGA_star**19 - 88049664.0*P*R**2*S**2*T**5*UGA_star**18*epsilon - 4930560.0*P*R**2*S**2*T**5*UGA_star**18 + 200411136.0*P*R**2*S**2*T**5*UGA_star**17*epsilon + 11234304.0*P*R**2*S**2*T**5*UGA_star**17 - 286126080.0*P*R**2*S**2*T**5*UGA_star**16*epsilon - 16068096.0*P*R**2*S**2*T**5*UGA_star**16 + 260579328.0*P*R**2*S**2*T**5*UGA_star**15*epsilon + 14678016.0*P*R**2*S**2*T**5*UGA_star**15 - 144211968.0*P*R**2*S**2*T**5*UGA_star**14*epsilon - 8169984.0*P*R**2*S**2*T**5*UGA_star**14 + 39235584.0*P*R**2*S**2*T**5*UGA_star**13*epsilon + 2257920.0*P*R**2*S**2*T**5*UGA_star**13 + 1204224.0*P*R**2*S**2*T**5*UGA_star**12*epsilon + 47616.0*P*R**2*S**2*T**5*UGA_star**12 - 3133440.0*P*R**2*S**2*T**5*UGA_star**11*epsilon - 175104.0*P*R**2*S**2*T**5*UGA_star**11 + 215040.0*P*R**2*S**2*T**5*UGA_star**10*epsilon + 13824.0*P*R**2*S**2*T**5*UGA_star**10 + 116736.0*P*R**2*S**2*T**5*UGA_star**9*epsilon + 6144.0*P*R**2*S**2*T**5*UGA_star**9 - 817152.0*P*R**2*S*T**6*UGA_star**20*epsilon - 32256.0*P*R**2*S*T**6*UGA_star**20 + 7354368.0*P*R**2*S*T**6*UGA_star**19*epsilon + 290304.0*P*R**2*S*T**6*UGA_star**19 - 29171712.0*P*R**2*S*T**6*UGA_star**18*epsilon - 1156608.0*P*R**2*S*T**6*UGA_star**18 + 66588672.0*P*R**2*S*T**6*UGA_star**17*epsilon + 2668032.0*P*R**2*S*T**6*UGA_star**17 - 95499264.0*P*R**2*S*T**6*UGA_star**16*epsilon - 3901440.0*P*R**2*S*T**6*UGA_star**16 + 87552000.0*P*R**2*S*T**6*UGA_star**15*epsilon + 3700224.0*P*R**2*S*T**6*UGA_star**15 - 48933888.0*P*R**2*S*T**6*UGA_star**14*epsilon - 2202624.0*P*R**2*S*T**6*UGA_star**14 + 13535232.0*P*R**2*S*T**6*UGA_star**13*epsilon + 711168.0*P*R**2*S*T**6*UGA_star**13 + 396288.0*P*R**2*S*T**6*UGA_star**12*epsilon - 41472.0*P*R**2*S*T**6*UGA_star**12 - 1161216.0*P*R**2*S*T**6*UGA_star**11*epsilon - 46080.0*P*R**2*S*T**6*UGA_star**11 + 125952.0*P*R**2*S*T**6*UGA_star**10*epsilon + 10752.0*P*R**2*S*T**6*UGA_star**10 + 30720.0*P*R**2*S*T**6*UGA_star**9*epsilon - 106496.0*P*R**2*T**7*UGA_star**20*epsilon - 2048.0*P*R**2*T**7*UGA_star**20 + 958464.0*P*R**2*T**7*UGA_star**19*epsilon + 18432.0*P*R**2*T**7*UGA_star**19 - 3820544.0*P*R**2*T**7*UGA_star**18*epsilon - 74240.0*P*R**2*T**7*UGA_star**18 + 8823808.0*P*R**2*T**7*UGA_star**17*epsilon + 175616.0*P*R**2*T**7*UGA_star**17 - 12935168.0*P*R**2*T**7*UGA_star**16*epsilon - 268800.0*P*R**2*T**7*UGA_star**16 + 12332032.0*P*R**2*T**7*UGA_star**15*epsilon + 275968.0*P*R**2*T**7*UGA_star**15 - 7430144.0*P*R**2*T**7*UGA_star**14*epsilon - 189952.0*P*R**2*T**7*UGA_star**14 + 2489344.0*P*R**2*T**7*UGA_star**13*epsilon + 84480.0*P*R**2*T**7*UGA_star**13 - 217088.0*P*R**2*T**7*UGA_star**12*epsilon - 22016.0*P*R**2*T**7*UGA_star**12 - 125952.0*P*R**2*T**7*UGA_star**11*epsilon + 2560.0*P*R**2*T**7*UGA_star**11 + 31744.0*P*R**2*T**7*UGA_star**10*epsilon + 18432.0*P*R*S**8*UGA_star**20*epsilon - 184320.0*P*R*S**8*UGA_star**19*epsilon + 832512.0*P*R*S**8*UGA_star**18*epsilon - 2242560.0*P*R*S**8*UGA_star**17*epsilon + 4008960.0*P*R*S**8*UGA_star**16*epsilon - 5013504.0*P*R*S**8*UGA_star**15*epsilon + 4515840.0*P*R*S**8*UGA_star**14*epsilon - 2985984.0*P*R*S**8*UGA_star**13*epsilon + 1474560.0*P*R*S**8*UGA_star**12*epsilon - 552960.0*P*R*S**8*UGA_star**11*epsilon + 156672.0*P*R*S**8*UGA_star**10*epsilon - 30720.0*P*R*S**8*UGA_star**9*epsilon + 3072.0*P*R*S**8*UGA_star**8*epsilon + 212992.0*P*R*S**7*T*UGA_star**20*epsilon + 4096.0*P*R*S**7*T*UGA_star**20 - 2129920.0*P*R*S**7*T*UGA_star**19*epsilon - 40960.0*P*R*S**7*T*UGA_star**19 + 9558016.0*P*R*S**7*T*UGA_star**18*epsilon + 185344.0*P*R*S**7*T*UGA_star**18 - 25350144.0*P*R*S**7*T*UGA_star**17*epsilon - 501760.0*P*R*S**7*T*UGA_star**17 + 44040192.0*P*R*S**7*T*UGA_star**16*epsilon + 906240.0*P*R*S**7*T*UGA_star**16 - 52488192.0*P*R*S**7*T*UGA_star**15*epsilon - 1155072.0*P*R*S**7*T*UGA_star**15 + 43732992.0*P*R*S**7*T*UGA_star**14*epsilon + 1075200.0*P*R*S**7*T*UGA_star**14 - 25559040.0*P*R*S**7*T*UGA_star**13*epsilon - 749568.0*P*R*S**7*T*UGA_star**13 + 10444800.0*P*R*S**7*T*UGA_star**12*epsilon + 399360.0*P*R*S**7*T*UGA_star**12 - 2977792.0*P*R*S**7*T*UGA_star**11*epsilon - 163840.0*P*R*S**7*T*UGA_star**11 + 563200.0*P*R*S**7*T*UGA_star**10*epsilon + 50176.0*P*R*S**7*T*UGA_star**10 - 28672.0*P*R*S**7*T*UGA_star**9*epsilon - 10240.0*P*R*S**7*T*UGA_star**9 - 24576.0*P*R*S**7*T*UGA_star**8*epsilon + 1024.0*P*R*S**7*T*UGA_star**8 + 6144.0*P*R*S**7*T*UGA_star**7*epsilon + 817152.0*P*R*S**6*T**2*UGA_star**20*epsilon + 32256.0*P*R*S**6*T**2*UGA_star**20 - 8171520.0*P*R*S**6*T**2*UGA_star**19*epsilon - 322560.0*P*R*S**6*T**2*UGA_star**19 + 36526080.0*P*R*S**6*T**2*UGA_star**18*epsilon + 1446912.0*P*R*S**6*T**2*UGA_star**18 - 95932416.0*P*R*S**6*T**2*UGA_star**17*epsilon - 3833856.0*P*R*S**6*T**2*UGA_star**17 + 163547136.0*P*R*S**6*T**2*UGA_star**16*epsilon + 6647808.0*P*R*S**6*T**2*UGA_star**16 - 188510208.0*P*R*S**6*T**2*UGA_star**15*epsilon - 7893504.0*P*R*S**6*T**2*UGA_star**15 + 148279296.0*P*R*S**6*T**2*UGA_star**14*epsilon + 6524928.0*P*R*S**6*T**2*UGA_star**14 - 78618624.0*P*R*S**6*T**2*UGA_star**13*epsilon - 3741696.0*P*R*S**6*T**2*UGA_star**13 + 27568128.0*P*R*S**6*T**2*UGA_star**12*epsilon + 1451520.0*P*R*S**6*T**2*UGA_star**12 - 6721536.0*P*R*S**6*T**2*UGA_star**11*epsilon - 350208.0*P*R*S**6*T**2*UGA_star**11 + 1529856.0*P*R*S**6*T**2*UGA_star**10*epsilon + 27648.0*P*R*S**6*T**2*UGA_star**10 - 331776.0*P*R*S**6*T**2*UGA_star**9*epsilon + 18432.0*P*R*S**6*T**2*UGA_star**9 + 6144.0*P*R*S**6*T**2*UGA_star**8*epsilon - 9216.0*P*R*S**6*T**2*UGA_star**8 + 12288.0*P*R*S**6*T**2*UGA_star**7*epsilon + 1536.0*P*R*S**6*T**2*UGA_star**7 + 1646592.0*P*R*S**5*T**3*UGA_star**20*epsilon + 92160.0*P*R*S**5*T**3*UGA_star**20 - 16465920.0*P*R*S**5*T**3*UGA_star**19*epsilon - 921600.0*P*R*S**5*T**3*UGA_star**19 + 73476096.0*P*R*S**5*T**3*UGA_star**18*epsilon + 4113408.0*P*R*S**5*T**3*UGA_star**18 - 192098304.0*P*R*S**5*T**3*UGA_star**17*epsilon - 10761216.0*P*R*S**5*T**3*UGA_star**17 + 324335616.0*P*R*S**5*T**3*UGA_star**16*epsilon + 18194944.0*P*R*S**5*T**3*UGA_star**16 - 366598144.0*P*R*S**5*T**3*UGA_star**15*epsilon - 20625920.0*P*R*S**5*T**3*UGA_star**15 + 276844544.0*P*R*S**5*T**3*UGA_star**14*epsilon + 15671296.0*P*R*S**5*T**3*UGA_star**14 - 133709824.0*P*R*S**5*T**3*UGA_star**13*epsilon - 7675904.0*P*R*S**5*T**3*UGA_star**13 + 36392960.0*P*R*S**5*T**3*UGA_star**12*epsilon + 2176000.0*P*R*S**5*T**3*UGA_star**12 - 3444736.0*P*R*S**5*T**3*UGA_star**11*epsilon - 257024.0*P*R*S**5*T**3*UGA_star**11 - 391168.0*P*R*S**5*T**3*UGA_star**10*epsilon - 5120.0*P*R*S**5*T**3*UGA_star**10 + 45056.0*P*R*S**5*T**3*UGA_star**9*epsilon + 1024.0*P*R*S**5*T**3*UGA_star**9 - 59392.0*P*R*S**5*T**3*UGA_star**8*epsilon - 3584.0*P*R*S**5*T**3*UGA_star**8 + 26624.0*P*R*S**5*T**3*UGA_star**7*epsilon + 1536.0*P*R*S**5*T**3*UGA_star**7 + 2048000.0*P*R*S**4*T**4*UGA_star**20*epsilon + 128000.0*P*R*S**4*T**4*UGA_star**20 - 20480000.0*P*R*S**4*T**4*UGA_star**19*epsilon - 1280000.0*P*R*S**4*T**4*UGA_star**19 + 91351040.0*P*R*S**4*T**4*UGA_star**18*epsilon + 5703680.0*P*R*S**4*T**4*UGA_star**18 - 238479360.0*P*R*S**4*T**4*UGA_star**17*epsilon - 14853120.0*P*R*S**4*T**4*UGA_star**17 + 401058816.0*P*R*S**4*T**4*UGA_star**16*epsilon + 24863744.0*P*R*S**4*T**4*UGA_star**16 - 448954368.0*P*R*S**4*T**4*UGA_star**15*epsilon - 27615232.0*P*R*S**4*T**4*UGA_star**15 + 330969088.0*P*R*S**4*T**4*UGA_star**14*epsilon + 20099072.0*P*R*S**4*T**4*UGA_star**14 - 149364736.0*P*R*S**4*T**4*UGA_star**13*epsilon - 8894464.0*P*R*S**4*T**4*UGA_star**13 + 30873600.0*P*R*S**4*T**4*UGA_star**12*epsilon + 1817600.0*P*R*S**4*T**4*UGA_star**12 + 3682304.0*P*R*S**4*T**4*UGA_star**11*epsilon + 116736.0*P*R*S**4*T**4*UGA_star**11 - 3110912.0*P*R*S**4*T**4*UGA_star**10*epsilon - 63488.0*P*R*S**4*T**4*UGA_star**10 + 428032.0*P*R*S**4*T**4*UGA_star**9*epsilon - 33792.0*P*R*S**4*T**4*UGA_star**9 - 41984.0*P*R*S**4*T**4*UGA_star**8*epsilon + 11264.0*P*R*S**4*T**4*UGA_star**8 + 20480.0*P*R*S**4*T**4*UGA_star**7*epsilon + 1646592.0*P*R*S**3*T**5*UGA_star**20*epsilon + 92160.0*P*R*S**3*T**5*UGA_star**20 - 16465920.0*P*R*S**3*T**5*UGA_star**19*epsilon - 921600.0*P*R*S**3*T**5*UGA_star**19 + 73476096.0*P*R*S**3*T**5*UGA_star**18*epsilon + 4113408.0*P*R*S**3*T**5*UGA_star**18 - 191913984.0*P*R*S**3*T**5*UGA_star**17*epsilon - 10748928.0*P*R*S**3*T**5*UGA_star**17 + 322768896.0*P*R*S**3*T**5*UGA_star**16*epsilon + 18090496.0*P*R*S**3*T**5*UGA_star**16 - 360753152.0*P*R*S**3*T**5*UGA_star**15*epsilon - 20235776.0*P*R*S**3*T**5*UGA_star**15 + 264341504.0*P*R*S**3*T**5*UGA_star**14*epsilon + 14834176.0*P*R*S**3*T**5*UGA_star**14 - 116951040.0*P*R*S**3*T**5*UGA_star**13*epsilon - 6546944.0*P*R*S**3*T**5*UGA_star**13 + 22042624.0*P*R*S**3*T**5*UGA_star**12*epsilon + 1197568.0*P*R*S**3*T**5*UGA_star**12 + 4081664.0*P*R*S**3*T**5*UGA_star**11*epsilon + 269824.0*P*R*S**3*T**5*UGA_star**11 - 2396160.0*P*R*S**3*T**5*UGA_star**10*epsilon - 157184.0*P*R*S**3*T**5*UGA_star**10 + 34816.0*P*R*S**3*T**5*UGA_star**9*epsilon + 8704.0*P*R*S**3*T**5*UGA_star**9 + 88064.0*P*R*S**3*T**5*UGA_star**8*epsilon + 4096.0*P*R*S**3*T**5*UGA_star**8 + 817152.0*P*R*S**2*T**6*UGA_star**20*epsilon + 32256.0*P*R*S**2*T**6*UGA_star**20 - 8171520.0*P*R*S**2*T**6*UGA_star**19*epsilon - 322560.0*P*R*S**2*T**6*UGA_star**19 + 36526080.0*P*R*S**2*T**6*UGA_star**18*epsilon + 1446912.0*P*R*S**2*T**6*UGA_star**18 - 95760384.0*P*R*S**2*T**6*UGA_star**17*epsilon - 3824640.0*P*R*S**2*T**6*UGA_star**17 + 162084864.0*P*R*S**2*T**6*UGA_star**16*epsilon + 6569472.0*P*R*S**2*T**6*UGA_star**16 - 183029760.0*P*R*S**2*T**6*UGA_star**15*epsilon - 7601664.0*P*R*S**2*T**6*UGA_star**15 + 136421376.0*P*R*S**2*T**6*UGA_star**14*epsilon + 5902848.0*P*R*S**2*T**6*UGA_star**14 - 62361600.0*P*R*S**2*T**6*UGA_star**13*epsilon - 2913792.0*P*R*S**2*T**6*UGA_star**13 + 13031424.0*P*R*S**2*T**6*UGA_star**12*epsilon + 752640.0*P*R*S**2*T**6*UGA_star**12 + 1622016.0*P*R*S**2*T**6*UGA_star**11*epsilon + 4608.0*P*R*S**2*T**6*UGA_star**11 - 1308672.0*P*R*S**2*T**6*UGA_star**10*epsilon - 56832.0*P*R*S**2*T**6*UGA_star**10 + 98304.0*P*R*S**2*T**6*UGA_star**9*epsilon + 10752.0*P*R*S**2*T**6*UGA_star**9 + 30720.0*P*R*S**2*T**6*UGA_star**8*epsilon + 212992.0*P*R*S*T**7*UGA_star**20*epsilon + 4096.0*P*R*S*T**7*UGA_star**20 - 2129920.0*P*R*S*T**7*UGA_star**19*epsilon - 40960.0*P*R*S*T**7*UGA_star**19 + 9558016.0*P*R*S*T**7*UGA_star**18*epsilon + 185344.0*P*R*S*T**7*UGA_star**18 - 25288704.0*P*R*S*T**7*UGA_star**17*epsilon - 499712.0*P*R*S*T**7*UGA_star**17 + 43517952.0*P*R*S*T**7*UGA_star**16*epsilon + 888832.0*P*R*S*T**7*UGA_star**16 - 50534400.0*P*R*S*T**7*UGA_star**15*epsilon - 1089536.0*P*R*S*T**7*UGA_star**15 + 39524352.0*P*R*S*T**7*UGA_star**14*epsilon + 931840.0*P*R*S*T**7*UGA_star**14 - 19838976.0*P*R*S*T**7*UGA_star**13*epsilon - 548864.0*P*R*S*T**7*UGA_star**13 + 5412864.0*P*R*S*T**7*UGA_star**12*epsilon + 212992.0*P*R*S*T**7*UGA_star**12 - 182272.0*P*R*S*T**7*UGA_star**11*epsilon - 49152.0*P*R*S*T**7*UGA_star**11 - 315392.0*P*R*S*T**7*UGA_star**10*epsilon + 5120.0*P*R*S*T**7*UGA_star**10 + 63488.0*P*R*S*T**7*UGA_star**9*epsilon + 18432.0*P*R*T**8*UGA_star**20*epsilon - 184320.0*P*R*T**8*UGA_star**19*epsilon + 832512.0*P*R*T**8*UGA_star**18*epsilon - 2236416.0*P*R*T**8*UGA_star**17*epsilon + 3956736.0*P*R*T**8*UGA_star**16*epsilon - 4816896.0*P*R*T**8*UGA_star**15*epsilon + 4085760.0*P*R*T**8*UGA_star**14*epsilon - 2383872.0*P*R*T**8*UGA_star**13*epsilon + 915456.0*P*R*T**8*UGA_star**12*epsilon - 208896.0*P*R*T**8*UGA_star**11*epsilon + 21504.0*P*R*T**8*UGA_star**10*epsilon - 2048.0*P*S**9*UGA_star**20*epsilon + 22528.0*P*S**9*UGA_star**19*epsilon - 112640.0*P*S**9*UGA_star**18*epsilon + 337920.0*P*S**9*UGA_star**17*epsilon - 675840.0*P*S**9*UGA_star**16*epsilon + 946176.0*P*S**9*UGA_star**15*epsilon - 946176.0*P*S**9*UGA_star**14*epsilon + 675840.0*P*S**9*UGA_star**13*epsilon - 337920.0*P*S**9*UGA_star**12*epsilon + 112640.0*P*S**9*UGA_star**11*epsilon - 22528.0*P*S**9*UGA_star**10*epsilon + 2048.0*P*S**9*UGA_star**9*epsilon - 26624.0*P*S**8*T*UGA_star**20*epsilon - 512.0*P*S**8*T*UGA_star**20 + 292864.0*P*S**8*T*UGA_star**19*epsilon + 5632.0*P*S**8*T*UGA_star**19 - 1456128.0*P*S**8*T*UGA_star**18*epsilon - 28160.0*P*S**8*T*UGA_star**18 + 4311040.0*P*S**8*T*UGA_star**17*epsilon + 84480.0*P*S**8*T*UGA_star**17 - 8417280.0*P*S**8*T*UGA_star**16*epsilon - 168960.0*P*S**8*T*UGA_star**16 + 11317248.0*P*S**8*T*UGA_star**15*epsilon + 236544.0*P*S**8*T*UGA_star**15 - 10579968.0*P*S**8*T*UGA_star**14*epsilon - 236544.0*P*S**8*T*UGA_star**14 + 6721536.0*P*S**8*T*UGA_star**13*epsilon + 168960.0*P*S**8*T*UGA_star**13 - 2672640.0*P*S**8*T*UGA_star**12*epsilon - 84480.0*P*S**8*T*UGA_star**12 + 481280.0*P*S**8*T*UGA_star**11*epsilon + 28160.0*P*S**8*T*UGA_star**11 + 75776.0*P*S**8*T*UGA_star**10*epsilon - 5632.0*P*S**8*T*UGA_star**10 - 55296.0*P*S**8*T*UGA_star**9*epsilon + 512.0*P*S**8*T*UGA_star**9 + 8192.0*P*S**8*T*UGA_star**8*epsilon - 116736.0*P*S**7*T**2*UGA_star**20*epsilon - 4608.0*P*S**7*T**2*UGA_star**20 + 1284096.0*P*S**7*T**2*UGA_star**19*epsilon + 50688.0*P*S**7*T**2*UGA_star**19 - 6365184.0*P*S**7*T**2*UGA_star**18*epsilon - 251904.0*P*S**7*T**2*UGA_star**18 + 18708480.0*P*S**7*T**2*UGA_star**17*epsilon + 744960.0*P*S**7*T**2*UGA_star**17 - 36040704.0*P*S**7*T**2*UGA_star**16*epsilon - 1451520.0*P*S**7*T**2*UGA_star**16 + 47351808.0*P*S**7*T**2*UGA_star**15*epsilon + 1944576.0*P*S**7*T**2*UGA_star**15 - 42541056.0*P*S**7*T**2*UGA_star**14*epsilon - 1806336.0*P*S**7*T**2*UGA_star**14 + 25104384.0*P*S**7*T**2*UGA_star**13*epsilon + 1133568.0*P*S**7*T**2*UGA_star**13 - 8423424.0*P*S**7*T**2*UGA_star**12*epsilon - 437760.0*P*S**7*T**2*UGA_star**12 + 559104.0*P*S**7*T**2*UGA_star**11*epsilon + 69120.0*P*S**7*T**2*UGA_star**11 + 688128.0*P*S**7*T**2*UGA_star**10*epsilon + 18432.0*P*S**7*T**2*UGA_star**10 - 215040.0*P*S**7*T**2*UGA_star**9*epsilon - 10752.0*P*S**7*T**2*UGA_star**9 + 1536.0*P*S**7*T**2*UGA_star**8 + 6144.0*P*S**7*T**2*UGA_star**7*epsilon - 274432.0*P*S**6*T**3*UGA_star**20*epsilon - 15360.0*P*S**6*T**3*UGA_star**20 + 3018752.0*P*S**6*T**3*UGA_star**19*epsilon + 168960.0*P*S**6*T**3*UGA_star**19 - 14953472.0*P*S**6*T**3*UGA_star**18*epsilon - 837120.0*P*S**6*T**3*UGA_star**18 + 43877376.0*P*S**6*T**3*UGA_star**17*epsilon + 2457600.0*P*S**6*T**3*UGA_star**17 - 84256768.0*P*S**6*T**3*UGA_star**16*epsilon - 4724224.0*P*S**6*T**3*UGA_star**16 + 110063616.0*P*S**6*T**3*UGA_star**15*epsilon + 6183936.0*P*S**6*T**3*UGA_star**15 - 97830912.0*P*S**6*T**3*UGA_star**14*epsilon - 5520384.0*P*S**6*T**3*UGA_star**14 + 56457216.0*P*S**6*T**3*UGA_star**13*epsilon + 3219456.0*P*S**6*T**3*UGA_star**13 - 17756160.0*P*S**6*T**3*UGA_star**12*epsilon - 1050624.0*P*S**6*T**3*UGA_star**12 + 237568.0*P*S**6*T**3*UGA_star**11*epsilon + 52224.0*P*S**6*T**3*UGA_star**11 + 1954816.0*P*S**6*T**3*UGA_star**10*epsilon + 90624.0*P*S**6*T**3*UGA_star**10 - 547840.0*P*S**6*T**3*UGA_star**9*epsilon - 24576.0*P*S**6*T**3*UGA_star**9 - 6144.0*P*S**6*T**3*UGA_star**8*epsilon - 1536.0*P*S**6*T**3*UGA_star**8 + 16384.0*P*S**6*T**3*UGA_star**7*epsilon + 1024.0*P*S**6*T**3*UGA_star**7 - 409600.0*P*S**5*T**4*UGA_star**20*epsilon - 25600.0*P*S**5*T**4*UGA_star**20 + 4505600.0*P*S**5*T**4*UGA_star**19*epsilon + 281600.0*P*S**5*T**4*UGA_star**19 - 22334464.0*P*S**5*T**4*UGA_star**18*epsilon - 1394688.0*P*S**5*T**4*UGA_star**18 + 65639424.0*P*S**5*T**4*UGA_star**17*epsilon + 4090368.0*P*S**5*T**4*UGA_star**17 - 126395392.0*P*S**5*T**4*UGA_star**16*epsilon - 7846400.0*P*S**5*T**4*UGA_star**16 + 165859328.0*P*S**5*T**4*UGA_star**15*epsilon + 10229760.0*P*S**5*T**4*UGA_star**15 - 148559872.0*P*S**5*T**4*UGA_star**14*epsilon - 9062400.0*P*S**5*T**4*UGA_star**14 + 87013376.0*P*S**5*T**4*UGA_star**13*epsilon + 5200896.0*P*S**5*T**4*UGA_star**13 - 28518400.0*P*S**5*T**4*UGA_star**12*epsilon - 1622016.0*P*S**5*T**4*UGA_star**12 + 1339392.0*P*S**5*T**4*UGA_star**11*epsilon + 25600.0*P*S**5*T**4*UGA_star**11 + 2614272.0*P*S**5*T**4*UGA_star**10*epsilon + 163840.0*P*S**5*T**4*UGA_star**10 - 764928.0*P*S**5*T**4*UGA_star**9*epsilon - 38400.0*P*S**5*T**4*UGA_star**9 - 13312.0*P*S**5*T**4*UGA_star**8*epsilon - 4608.0*P*S**5*T**4*UGA_star**8 + 24576.0*P*S**5*T**4*UGA_star**7*epsilon + 2048.0*P*S**5*T**4*UGA_star**7 - 411648.0*P*S**4*T**5*UGA_star**20*epsilon - 23040.0*P*S**4*T**5*UGA_star**20 + 4528128.0*P*S**4*T**5*UGA_star**19*epsilon + 253440.0*P*S**4*T**5*UGA_star**19 - 22474752.0*P*S**4*T**5*UGA_star**18*epsilon - 1257984.0*P*S**4*T**5*UGA_star**18 + 66238464.0*P*S**4*T**5*UGA_star**17*epsilon + 3707904.0*P*S**4*T**5*UGA_star**17 - 128175104.0*P*S**4*T**5*UGA_star**16*epsilon - 7175168.0*P*S**4*T**5*UGA_star**16 + 169553920.0*P*S**4*T**5*UGA_star**15*epsilon + 9488384.0*P*S**4*T**5*UGA_star**15 - 153964544.0*P*S**4*T**5*UGA_star**14*epsilon - 8602624.0*P*S**4*T**5*UGA_star**14 + 92606464.0*P*S**4*T**5*UGA_star**13*epsilon + 5144576.0*P*S**4*T**5*UGA_star**13 - 32583680.0*P*S**4*T**5*UGA_star**12*epsilon - 1765888.0*P*S**4*T**5*UGA_star**12 + 3364864.0*P*S**4*T**5*UGA_star**11*epsilon + 132608.0*P*S**4*T**5*UGA_star**11 + 1957888.0*P*S**4*T**5*UGA_star**10*epsilon + 144896.0*P*S**4*T**5*UGA_star**10 - 641024.0*P*S**4*T**5*UGA_star**9*epsilon - 50176.0*P*S**4*T**5*UGA_star**9 - 23552.0*P*S**4*T**5*UGA_star**8*epsilon + 2048.0*P*S**4*T**5*UGA_star**8 + 24576.0*P*S**4*T**5*UGA_star**7*epsilon + 1024.0*P*S**4*T**5*UGA_star**7 - 272384.0*P*S**3*T**6*UGA_star**20*epsilon - 10752.0*P*S**3*T**6*UGA_star**20 + 2996224.0*P*S**3*T**6*UGA_star**19*epsilon + 118272.0*P*S**3*T**6*UGA_star**19 - 14899200.0*P*S**3*T**6*UGA_star**18*epsilon - 589824.0*P*S**3*T**6*UGA_star**18 + 44095488.0*P*S**3*T**6*UGA_star**17*epsilon + 1757184.0*P*S**3*T**6*UGA_star**17 - 85947392.0*P*S**3*T**6*UGA_star**16*epsilon - 3464704.0*P*S**3*T**6*UGA_star**16 + 115030016.0*P*S**3*T**6*UGA_star**15*epsilon + 4723712.0*P*S**3*T**6*UGA_star**15 - 106455040.0*P*S**3*T**6*UGA_star**14*epsilon - 4501504.0*P*S**3*T**6*UGA_star**14 + 66203648.0*P*S**3*T**6*UGA_star**13*epsilon + 2938880.0*P*S**3*T**6*UGA_star**13 - 25059328.0*P*S**3*T**6*UGA_star**12*epsilon - 1222144.0*P*S**3*T**6*UGA_star**12 + 3745792.0*P*S**3*T**6*UGA_star**11*epsilon + 249344.0*P*S**3*T**6*UGA_star**11 + 1005568.0*P*S**3*T**6*UGA_star**10*epsilon + 20480.0*P*S**3*T**6*UGA_star**10 - 477184.0*P*S**3*T**6*UGA_star**9*epsilon - 22528.0*P*S**3*T**6*UGA_star**9 + 23552.0*P*S**3*T**6*UGA_star**8*epsilon + 3584.0*P*S**3*T**6*UGA_star**8 + 10240.0*P*S**3*T**6*UGA_star**7*epsilon - 106496.0*P*S**2*T**7*UGA_star**20*epsilon - 2048.0*P*S**2*T**7*UGA_star**20 + 1171456.0*P*S**2*T**7*UGA_star**19*epsilon + 22528.0*P*S**2*T**7*UGA_star**19 - 5843968.0*P*S**2*T**7*UGA_star**18*epsilon - 113152.0*P*S**2*T**7*UGA_star**18 + 17423360.0*P*S**2*T**7*UGA_star**17*epsilon + 342528.0*P*S**2*T**7*UGA_star**17 - 34403328.0*P*S**2*T**7*UGA_star**16*epsilon - 694272.0*P*S**2*T**7*UGA_star**16 + 47026176.0*P*S**2*T**7*UGA_star**15*epsilon + 989184.0*P*S**2*T**7*UGA_star**15 - 45029376.0*P*S**2*T**7*UGA_star**14*epsilon - 1010688.0*P*S**2*T**7*UGA_star**14 + 29681664.0*P*S**2*T**7*UGA_star**13*epsilon + 740352.0*P*S**2*T**7*UGA_star**13 - 12625920.0*P*S**2*T**7*UGA_star**12*epsilon - 380928.0*P*S**2*T**7*UGA_star**12 + 2797568.0*P*S**2*T**7*UGA_star**11*epsilon + 131072.0*P*S**2*T**7*UGA_star**11 + 66560.0*P*S**2*T**7*UGA_star**10*epsilon - 27136.0*P*S**2*T**7*UGA_star**10 - 189440.0*P*S**2*T**7*UGA_star**9*epsilon + 2560.0*P*S**2*T**7*UGA_star**9 + 31744.0*P*S**2*T**7*UGA_star**8*epsilon - 18432.0*P*S*T**8*UGA_star**20*epsilon + 202752.0*P*S*T**8*UGA_star**19*epsilon - 1016832.0*P*S*T**8*UGA_star**18*epsilon + 3068928.0*P*S*T**8*UGA_star**17*epsilon - 6193152.0*P*S*T**8*UGA_star**16*epsilon + 8773632.0*P*S*T**8*UGA_star**15*epsilon - 8902656.0*P*S*T**8*UGA_star**14*epsilon + 6469632.0*P*S*T**8*UGA_star**13*epsilon - 3299328.0*P*S*T**8*UGA_star**12*epsilon + 1124352.0*P*S*T**8*UGA_star**11*epsilon - 230400.0*P*S*T**8*UGA_star**10*epsilon + 21504.0*P*S*T**8*UGA_star**9*epsilon - 2048.0*R**9*T*UGA_star**20*epsilon + 2048.0*R**9*T*UGA_star**19*epsilon - 3072.0*R**9*T*UGA_star**18*epsilon + 3072.0*R**9*T*UGA_star**17*epsilon - 1024.0*R**9*T*UGA_star**16*epsilon + 1024.0*R**9*T*UGA_star**15*epsilon + 18432.0*R**8*S*T*UGA_star**20*epsilon - 36864.0*R**8*S*T*UGA_star**19*epsilon + 43008.0*R**8*S*T*UGA_star**18*epsilon - 49152.0*R**8*S*T*UGA_star**17*epsilon + 31744.0*R**8*S*T*UGA_star**16*epsilon - 14336.0*R**8*S*T*UGA_star**15*epsilon + 7168.0*R**8*S*T*UGA_star**14*epsilon + 13312.0*R**8*T**2*UGA_star**20*epsilon + 256.0*R**8*T**2*UGA_star**20 - 26624.0*R**8*T**2*UGA_star**19*epsilon - 512.0*R**8*T**2*UGA_star**19 + 28672.0*R**8*T**2*UGA_star**18*epsilon + 768.0*R**8*T**2*UGA_star**18 - 30720.0*R**8*T**2*UGA_star**17*epsilon - 1024.0*R**8*T**2*UGA_star**17 + 16384.0*R**8*T**2*UGA_star**16*epsilon + 768.0*R**8*T**2*UGA_star**16 - 2048.0*R**8*T**2*UGA_star**15*epsilon - 512.0*R**8*T**2*UGA_star**15 + 1024.0*R**8*T**2*UGA_star**14*epsilon + 256.0*R**8*T**2*UGA_star**14 - 73728.0*R**7*S**2*T*UGA_star**20*epsilon + 221184.0*R**7*S**2*T*UGA_star**19*epsilon - 307200.0*R**7*S**2*T*UGA_star**18*epsilon + 331776.0*R**7*S**2*T*UGA_star**17*epsilon - 279552.0*R**7*S**2*T*UGA_star**16*epsilon + 150528.0*R**7*S**2*T*UGA_star**15*epsilon - 64512.0*R**7*S**2*T*UGA_star**14*epsilon + 21504.0*R**7*S**2*T*UGA_star**13*epsilon - 106496.0*R**7*S*T**2*UGA_star**20*epsilon - 2048.0*R**7*S*T**2*UGA_star**20 + 319488.0*R**7*S*T**2*UGA_star**19*epsilon + 6144.0*R**7*S*T**2*UGA_star**19 - 422912.0*R**7*S*T**2*UGA_star**18*epsilon - 9728.0*R**7*S*T**2*UGA_star**18 + 420864.0*R**7*S*T**2*UGA_star**17*epsilon + 12800.0*R**7*S*T**2*UGA_star**17 - 321536.0*R**7*S*T**2*UGA_star**16*epsilon - 12288.0*R**7*S*T**2*UGA_star**16 + 119808.0*R**7*S*T**2*UGA_star**15*epsilon + 8192.0*R**7*S*T**2*UGA_star**15 - 15360.0*R**7*S*T**2*UGA_star**14*epsilon - 4608.0*R**7*S*T**2*UGA_star**14 + 6144.0*R**7*S*T**2*UGA_star**13*epsilon + 1536.0*R**7*S*T**2*UGA_star**13 - 38912.0*R**7*T**3*UGA_star**20*epsilon - 1536.0*R**7*T**3*UGA_star**20 + 116736.0*R**7*T**3*UGA_star**19*epsilon + 4608.0*R**7*T**3*UGA_star**19 - 145408.0*R**7*T**3*UGA_star**18*epsilon - 6144.0*R**7*T**3*UGA_star**18 + 124928.0*R**7*T**3*UGA_star**17*epsilon + 6144.0*R**7*T**3*UGA_star**17 - 86016.0*R**7*T**3*UGA_star**16*epsilon - 4608.0*R**7*T**3*UGA_star**16 + 31744.0*R**7*T**3*UGA_star**15*epsilon + 1536.0*R**7*T**3*UGA_star**15 - 6144.0*R**7*T**3*UGA_star**14*epsilon + 3072.0*R**7*T**3*UGA_star**13*epsilon + 172032.0*R**6*S**3*T*UGA_star**20*epsilon - 688128.0*R**6*S**3*T*UGA_star**19*epsilon + 1204224.0*R**6*S**3*T*UGA_star**18*epsilon - 1376256.0*R**6*S**3*T*UGA_star**17*epsilon + 1240064.0*R**6*S**3*T*UGA_star**16*epsilon - 831488.0*R**6*S**3*T*UGA_star**15*epsilon + 387072.0*R**6*S**3*T*UGA_star**14*epsilon - 143360.0*R**6*S**3*T*UGA_star**13*epsilon + 35840.0*R**6*S**3*T*UGA_star**12*epsilon + 372736.0*R**6*S**2*T**2*UGA_star**20*epsilon + 7168.0*R**6*S**2*T**2*UGA_star**20 - 1490944.0*R**6*S**2*T**2*UGA_star**19*epsilon - 28672.0*R**6*S**2*T**2*UGA_star**19 + 2530304.0*R**6*S**2*T**2*UGA_star**18*epsilon + 53760.0*R**6*S**2*T**2*UGA_star**18 - 2695168.0*R**6*S**2*T**2*UGA_star**17*epsilon - 71680.0*R**6*S**2*T**2*UGA_star**17 + 2219008.0*R**6*S**2*T**2*UGA_star**16*epsilon + 75520.0*R**6*S**2*T**2*UGA_star**16 - 1267712.0*R**6*S**2*T**2*UGA_star**15*epsilon - 58368.0*R**6*S**2*T**2*UGA_star**15 + 359424.0*R**6*S**2*T**2*UGA_star**14*epsilon + 33792.0*R**6*S**2*T**2*UGA_star**14 - 43008.0*R**6*S**2*T**2*UGA_star**13*epsilon - 15360.0*R**6*S**2*T**2*UGA_star**13 + 15360.0*R**6*S**2*T**2*UGA_star**12*epsilon + 3840.0*R**6*S**2*T**2*UGA_star**12 + 272384.0*R**6*S*T**3*UGA_star**20*epsilon + 10752.0*R**6*S*T**3*UGA_star**20 - 1089536.0*R**6*S*T**3*UGA_star**19*epsilon - 43008.0*R**6*S*T**3*UGA_star**19 + 1787904.0*R**6*S*T**3*UGA_star**18*epsilon + 73216.0*R**6*S*T**3*UGA_star**18 - 1722368.0*R**6*S*T**3*UGA_star**17*epsilon - 78336.0*R**6*S*T**3*UGA_star**17 + 1244160.0*R**6*S*T**3*UGA_star**16*epsilon + 64000.0*R**6*S*T**3*UGA_star**16 - 669696.0*R**6*S*T**3*UGA_star**15*epsilon - 34816.0*R**6*S*T**3*UGA_star**15 + 202752.0*R**6*S*T**3*UGA_star**14*epsilon + 7680.0*R**6*S*T**3*UGA_star**14 - 40960.0*R**6*S*T**3*UGA_star**13*epsilon + 512.0*R**6*S*T**3*UGA_star**13 + 15360.0*R**6*S*T**3*UGA_star**12*epsilon + 68608.0*R**6*T**4*UGA_star**20*epsilon + 3840.0*R**6*T**4*UGA_star**20 - 274432.0*R**6*T**4*UGA_star**19*epsilon - 15360.0*R**6*T**4*UGA_star**19 + 430080.0*R**6*T**4*UGA_star**18*epsilon + 24064.0*R**6*T**4*UGA_star**18 - 354304.0*R**6*T**4*UGA_star**17*epsilon - 19968.0*R**6*T**4*UGA_star**17 + 195584.0*R**6*T**4*UGA_star**16*epsilon + 11008.0*R**6*T**4*UGA_star**16 - 86016.0*R**6*T**4*UGA_star**15*epsilon - 4096.0*R**6*T**4*UGA_star**15 + 18432.0*R**6*T**4*UGA_star**14*epsilon + 2048.0*R**6*T**4*UGA_star**13*epsilon + 512.0*R**6*T**4*UGA_star**13 - 258048.0*R**5*S**4*T*UGA_star**20*epsilon + 1290240.0*R**5*S**4*T*UGA_star**19*epsilon - 2795520.0*R**5*S**4*T*UGA_star**18*epsilon + 3655680.0*R**5*S**4*T*UGA_star**17*epsilon - 3476480.0*R**5*S**4*T*UGA_star**16*epsilon + 2587648.0*R**5*S**4*T*UGA_star**15*epsilon - 1433600.0*R**5*S**4*T*UGA_star**14*epsilon + 573440.0*R**5*S**4*T*UGA_star**13*epsilon - 179200.0*R**5*S**4*T*UGA_star**12*epsilon + 35840.0*R**5*S**4*T*UGA_star**11*epsilon - 745472.0*R**5*S**3*T**2*UGA_star**20*epsilon - 14336.0*R**5*S**3*T**2*UGA_star**20 + 3727360.0*R**5*S**3*T**2*UGA_star**19*epsilon + 71680.0*R**5*S**3*T**2*UGA_star**19 - 7906304.0*R**5*S**3*T**2*UGA_star**18*epsilon - 161280.0*R**5*S**3*T**2*UGA_star**18 + 9798656.0*R**5*S**3*T**2*UGA_star**17*epsilon + 232960.0*R**5*S**3*T**2*UGA_star**17 - 8561664.0*R**5*S**3*T**2*UGA_star**16*epsilon - 256000.0*R**5*S**3*T**2*UGA_star**16 + 5695488.0*R**5*S**3*T**2*UGA_star**15*epsilon + 219136.0*R**5*S**3*T**2*UGA_star**15 - 2530304.0*R**5*S**3*T**2*UGA_star**14*epsilon - 140800.0*R**5*S**3*T**2*UGA_star**14 + 558080.0*R**5*S**3*T**2*UGA_star**13*epsilon + 69120.0*R**5*S**3*T**2*UGA_star**13 - 56320.0*R**5*S**3*T**2*UGA_star**12*epsilon - 25600.0*R**5*S**3*T**2*UGA_star**12 + 20480.0*R**5*S**3*T**2*UGA_star**11*epsilon + 5120.0*R**5*S**3*T**2*UGA_star**11 - 817152.0*R**5*S**2*T**3*UGA_star**20*epsilon - 32256.0*R**5*S**2*T**3*UGA_star**20 + 4085760.0*R**5*S**2*T**3*UGA_star**19*epsilon + 161280.0*R**5*S**2*T**3*UGA_star**19 - 8491008.0*R**5*S**2*T**3*UGA_star**18*epsilon - 342528.0*R**5*S**2*T**3*UGA_star**18 + 9879552.0*R**5*S**2*T**3*UGA_star**17*epsilon + 425472.0*R**5*S**2*T**3*UGA_star**17 - 7699456.0*R**5*S**2*T**3*UGA_star**16*epsilon - 370688.0*R**5*S**2*T**3*UGA_star**16 + 4610048.0*R**5*S**2*T**3*UGA_star**15*epsilon + 240128.0*R**5*S**2*T**3*UGA_star**15 - 2015232.0*R**5*S**2*T**3*UGA_star**14*epsilon - 96768.0*R**5*S**2*T**3*UGA_star**14 + 514048.0*R**5*S**2*T**3*UGA_star**13*epsilon + 12800.0*R**5*S**2*T**3*UGA_star**13 - 97280.0*R**5*S**2*T**3*UGA_star**12*epsilon + 2560.0*R**5*S**2*T**3*UGA_star**12 + 30720.0*R**5*S**2*T**3*UGA_star**11*epsilon - 411648.0*R**5*S*T**4*UGA_star**20*epsilon - 23040.0*R**5*S*T**4*UGA_star**20 + 2058240.0*R**5*S*T**4*UGA_star**19*epsilon + 115200.0*R**5*S*T**4*UGA_star**19 - 4171776.0*R**5*S*T**4*UGA_star**18*epsilon - 233472.0*R**5*S*T**4*UGA_star**18 + 4460544.0*R**5*S*T**4*UGA_star**17*epsilon + 250368.0*R**5*S*T**4*UGA_star**17 - 2870272.0*R**5*S*T**4*UGA_star**16*epsilon - 161792.0*R**5*S*T**4*UGA_star**16 + 1320960.0*R**5*S*T**4*UGA_star**15*epsilon + 71168.0*R**5*S*T**4*UGA_star**15 - 466944.0*R**5*S*T**4*UGA_star**14*epsilon - 18432.0*R**5*S*T**4*UGA_star**14 + 65536.0*R**5*S*T**4*UGA_star**13*epsilon - 2560.0*R**5*S*T**4*UGA_star**13 + 15360.0*R**5*S*T**4*UGA_star**12*epsilon + 2560.0*R**5*S*T**4*UGA_star**12 - 81920.0*R**5*T**5*UGA_star**20*epsilon - 5120.0*R**5*T**5*UGA_star**20 + 409600.0*R**5*T**5*UGA_star**19*epsilon + 25600.0*R**5*T**5*UGA_star**19 - 805888.0*R**5*T**5*UGA_star**18*epsilon - 50176.0*R**5*T**5*UGA_star**18 + 775168.0*R**5*T**5*UGA_star**17*epsilon + 47616.0*R**5*T**5*UGA_star**17 - 363520.0*R**5*T**5*UGA_star**16*epsilon - 21504.0*R**5*T**5*UGA_star**16 + 69632.0*R**5*T**5*UGA_star**15*epsilon + 4096.0*R**5*T**5*UGA_star**15 + 1024.0*R**5*T**5*UGA_star**14*epsilon - 1024.0*R**5*T**5*UGA_star**14 - 9216.0*R**5*T**5*UGA_star**13*epsilon + 512.0*R**5*T**5*UGA_star**13 + 5120.0*R**5*T**5*UGA_star**12*epsilon + 258048.0*R**4*S**5*T*UGA_star**20*epsilon - 1548288.0*R**4*S**5*T*UGA_star**19*epsilon + 4042752.0*R**4*S**5*T*UGA_star**18*epsilon - 6193152.0*R**4*S**5*T*UGA_star**17*epsilon + 6472704.0*R**4*S**5*T*UGA_star**16*epsilon - 5117952.0*R**4*S**5*T*UGA_star**15*epsilon + 3161088.0*R**4*S**5*T*UGA_star**14*epsilon - 1462272.0*R**4*S**5*T*UGA_star**13*epsilon + 494592.0*R**4*S**5*T*UGA_star**12*epsilon - 129024.0*R**4*S**5*T*UGA_star**11*epsilon + 21504.0*R**4*S**5*T*UGA_star**10*epsilon + 931840.0*R**4*S**4*T**2*UGA_star**20*epsilon + 17920.0*R**4*S**4*T**2*UGA_star**20 - 5591040.0*R**4*S**4*T**2*UGA_star**19*epsilon - 107520.0*R**4*S**4*T**2*UGA_star**19 + 14371840.0*R**4*S**4*T**2*UGA_star**18*epsilon + 286720.0*R**4*S**4*T**2*UGA_star**18 - 21145600.0*R**4*S**4*T**2*UGA_star**17*epsilon - 465920.0*R**4*S**4*T**2*UGA_star**17 + 20561920.0*R**4*S**4*T**2*UGA_star**16*epsilon + 541440.0*R**4*S**4*T**2*UGA_star**16 - 14694400.0*R**4*S**4*T**2*UGA_star**15*epsilon - 488960.0*R**4*S**4*T**2*UGA_star**15 + 7895040.0*R**4*S**4*T**2*UGA_star**14*epsilon + 344320.0*R**4*S**4*T**2*UGA_star**14 - 2775040.0*R**4*S**4*T**2*UGA_star**13*epsilon - 184320.0*R**4*S**4*T**2*UGA_star**13 + 460800.0*R**4*S**4*T**2*UGA_star**12*epsilon + 75520.0*R**4*S**4*T**2*UGA_star**12 - 30720.0*R**4*S**4*T**2*UGA_star**11*epsilon - 23040.0*R**4*S**4*T**2*UGA_star**11 + 15360.0*R**4*S**4*T**2*UGA_star**10*epsilon + 3840.0*R**4*S**4*T**2*UGA_star**10 + 1361920.0*R**4*S**3*T**3*UGA_star**20*epsilon + 53760.0*R**4*S**3*T**3*UGA_star**20 - 8171520.0*R**4*S**3*T**3*UGA_star**19*epsilon - 322560.0*R**4*S**3*T**3*UGA_star**19 + 20725760.0*R**4*S**3*T**3*UGA_star**18*epsilon + 829440.0*R**4*S**3*T**3*UGA_star**18 - 29296640.0*R**4*S**3*T**3*UGA_star**17*epsilon - 1221120.0*R**4*S**3*T**3*UGA_star**17 + 26224640.0*R**4*S**3*T**3*UGA_star**16*epsilon + 1185280.0*R**4*S**3*T**3*UGA_star**16 - 16691200.0*R**4*S**3*T**3*UGA_star**15*epsilon - 834560.0*R**4*S**3*T**3*UGA_star**15 + 8284160.0*R**4*S**3*T**3*UGA_star**14*epsilon + 424960.0*R**4*S**3*T**3*UGA_star**14 - 3000320.0*R**4*S**3*T**3*UGA_star**13*epsilon - 125440.0*R**4*S**3*T**3*UGA_star**13 + 634880.0*R**4*S**3*T**3*UGA_star**12*epsilon + 5120.0*R**4*S**3*T**3*UGA_star**12 - 102400.0*R**4*S**3*T**3*UGA_star**11*epsilon + 5120.0*R**4*S**3*T**3*UGA_star**11 + 30720.0*R**4*S**3*T**3*UGA_star**10*epsilon + 1029120.0*R**4*S**2*T**4*UGA_star**20*epsilon + 57600.0*R**4*S**2*T**4*UGA_star**20 - 6174720.0*R**4*S**2*T**4*UGA_star**19*epsilon - 345600.0*R**4*S**2*T**4*UGA_star**19 + 15436800.0*R**4*S**2*T**4*UGA_star**18*epsilon + 864000.0*R**4*S**2*T**4*UGA_star**18 - 20828160.0*R**4*S**2*T**4*UGA_star**17*epsilon - 1167360.0*R**4*S**2*T**4*UGA_star**17 + 16646144.0*R**4*S**2*T**4*UGA_star**16*epsilon + 935936.0*R**4*S**2*T**4*UGA_star**16 - 8513536.0*R**4*S**2*T**4*UGA_star**15*epsilon - 474624.0*R**4*S**2*T**4*UGA_star**15 + 3206144.0*R**4*S**2*T**4*UGA_star**14*epsilon + 161536.0*R**4*S**2*T**4*UGA_star**14 - 905216.0*R**4*S**2*T**4*UGA_star**13*epsilon - 26624.0*R**4*S**2*T**4*UGA_star**13 + 62464.0*R**4*S**2*T**4*UGA_star**12*epsilon - 9984.0*R**4*S**2*T**4*UGA_star**12 + 40960.0*R**4*S**2*T**4*UGA_star**11*epsilon + 5120.0*R**4*S**2*T**4*UGA_star**11 + 409600.0*R**4*S*T**5*UGA_star**20*epsilon + 25600.0*R**4*S*T**5*UGA_star**20 - 2457600.0*R**4*S*T**5*UGA_star**19*epsilon - 153600.0*R**4*S*T**5*UGA_star**19 + 6045696.0*R**4*S*T**5*UGA_star**18*epsilon + 376832.0*R**4*S*T**5*UGA_star**18 - 7737344.0*R**4*S*T**5*UGA_star**17*epsilon - 478208.0*R**4*S*T**5*UGA_star**17 + 5337088.0*R**4*S*T**5*UGA_star**16*epsilon + 323072.0*R**4*S*T**5*UGA_star**16 - 1789952.0*R**4*S*T**5*UGA_star**15*epsilon - 104448.0*R**4*S*T**5*UGA_star**15 + 155648.0*R**4*S*T**5*UGA_star**14*epsilon + 13312.0*R**4*S*T**5*UGA_star**14 + 69632.0*R**4*S*T**5*UGA_star**13*epsilon - 5120.0*R**4*S*T**5*UGA_star**13 - 53248.0*R**4*S*T**5*UGA_star**12*epsilon + 2560.0*R**4*S*T**5*UGA_star**12 + 20480.0*R**4*S*T**5*UGA_star**11*epsilon + 68608.0*R**4*T**6*UGA_star**20*epsilon + 3840.0*R**4*T**6*UGA_star**20 - 411648.0*R**4*T**6*UGA_star**19*epsilon - 23040.0*R**4*T**6*UGA_star**19 + 999424.0*R**4*T**6*UGA_star**18*epsilon + 56064.0*R**4*T**6*UGA_star**18 - 1224704.0*R**4*T**6*UGA_star**17*epsilon - 69120.0*R**4*T**6*UGA_star**17 + 738304.0*R**4*T**6*UGA_star**16*epsilon + 42496.0*R**4*T**6*UGA_star**16 - 129024.0*R**4*T**6*UGA_star**15*epsilon - 8704.0*R**4*T**6*UGA_star**15 - 63488.0*R**4*T**6*UGA_star**14*epsilon - 2304.0*R**4*T**6*UGA_star**14 + 20480.0*R**4*T**6*UGA_star**13*epsilon + 512.0*R**4*T**6*UGA_star**13 + 2048.0*R**4*T**6*UGA_star**12*epsilon + 256.0*R**4*T**6*UGA_star**12 - 172032.0*R**3*S**6*T*UGA_star**20*epsilon + 1204224.0*R**3*S**6*T*UGA_star**19*epsilon - 3698688.0*R**3*S**6*T*UGA_star**18*epsilon + 6623232.0*R**3*S**6*T*UGA_star**17*epsilon - 7834624.0*R**3*S**6*T*UGA_star**16*epsilon + 6673408.0*R**3*S**6*T*UGA_star**15*epsilon - 4365312.0*R**3*S**6*T*UGA_star**14*epsilon + 2229248.0*R**3*S**6*T*UGA_star**13*epsilon - 852992.0*R**3*S**6*T*UGA_star**12*epsilon + 236544.0*R**3*S**6*T*UGA_star**11*epsilon - 50176.0*R**3*S**6*T*UGA_star**10*epsilon + 7168.0*R**3*S**6*T*UGA_star**9*epsilon - 745472.0*R**3*S**5*T**2*UGA_star**20*epsilon - 14336.0*R**3*S**5*T**2*UGA_star**20 + 5218304.0*R**3*S**5*T**2*UGA_star**19*epsilon + 100352.0*R**3*S**5*T**2*UGA_star**19 - 15834112.0*R**3*S**5*T**2*UGA_star**18*epsilon - 311808.0*R**3*S**5*T**2*UGA_star**18 + 27489280.0*R**3*S**5*T**2*UGA_star**17*epsilon + 577024.0*R**3*S**5*T**2*UGA_star**17 - 30674944.0*R**3*S**5*T**2*UGA_star**16*epsilon - 729088.0*R**3*S**5*T**2*UGA_star**16 + 23843840.0*R**3*S**5*T**2*UGA_star**15*epsilon + 688128.0*R**3*S**5*T**2*UGA_star**15 - 13795328.0*R**3*S**5*T**2*UGA_star**14*epsilon - 508928.0*R**3*S**5*T**2*UGA_star**14 + 5952512.0*R**3*S**5*T**2*UGA_star**13*epsilon + 293888.0*R**3*S**5*T**2*UGA_star**13 - 1638400.0*R**3*S**5*T**2*UGA_star**12*epsilon - 129024.0*R**3*S**5*T**2*UGA_star**12 + 175104.0*R**3*S**5*T**2*UGA_star**11*epsilon + 43008.0*R**3*S**5*T**2*UGA_star**11 + 3072.0*R**3*S**5*T**2*UGA_star**10*epsilon - 10752.0*R**3*S**5*T**2*UGA_star**10 + 6144.0*R**3*S**5*T**2*UGA_star**9*epsilon + 1536.0*R**3*S**5*T**2*UGA_star**9 - 1361920.0*R**3*S**4*T**3*UGA_star**20*epsilon - 53760.0*R**3*S**4*T**3*UGA_star**20 + 9533440.0*R**3*S**4*T**3*UGA_star**19*epsilon + 376320.0*R**3*S**4*T**3*UGA_star**19 - 28661760.0*R**3*S**4*T**3*UGA_star**18*epsilon - 1141760.0*R**3*S**4*T**3*UGA_star**18 + 48465920.0*R**3*S**4*T**3*UGA_star**17*epsilon + 1981440.0*R**3*S**4*T**3*UGA_star**17 - 51138560.0*R**3*S**4*T**3*UGA_star**16*epsilon - 2206720.0*R**3*S**4*T**3*UGA_star**16 + 36090880.0*R**3*S**4*T**3*UGA_star**15*epsilon + 1699840.0*R**3*S**4*T**3*UGA_star**15 - 18585600.0*R**3*S**4*T**3*UGA_star**14*epsilon - 952320.0*R**3*S**4*T**3*UGA_star**14 + 7592960.0*R**3*S**4*T**3*UGA_star**13*epsilon + 373760.0*R**3*S**4*T**3*UGA_star**13 - 2283520.0*R**3*S**4*T**3*UGA_star**12*epsilon - 74240.0*R**3*S**4*T**3*UGA_star**12 + 373760.0*R**3*S**4*T**3*UGA_star**11*epsilon - 7680.0*R**3*S**4*T**3*UGA_star**11 - 40960.0*R**3*S**4*T**3*UGA_star**10*epsilon + 5120.0*R**3*S**4*T**3*UGA_star**10 + 15360.0*R**3*S**4*T**3*UGA_star**9*epsilon - 1372160.0*R**3*S**3*T**4*UGA_star**20*epsilon - 76800.0*R**3*S**3*T**4*UGA_star**20 + 9605120.0*R**3*S**3*T**4*UGA_star**19*epsilon + 537600.0*R**3*S**3*T**4*UGA_star**19 - 28631040.0*R**3*S**3*T**4*UGA_star**18*epsilon - 1602560.0*R**3*S**3*T**4*UGA_star**18 + 47165440.0*R**3*S**3*T**4*UGA_star**17*epsilon + 2641920.0*R**3*S**3*T**4*UGA_star**17 - 46729216.0*R**3*S**3*T**4*UGA_star**16*epsilon - 2622464.0*R**3*S**3*T**4*UGA_star**16 + 28733440.0*R**3*S**3*T**4*UGA_star**15*epsilon + 1612800.0*R**3*S**3*T**4*UGA_star**15 - 11438080.0*R**3*S**3*T**4*UGA_star**14*epsilon - 624640.0*R**3*S**3*T**4*UGA_star**14 + 3379200.0*R**3*S**3*T**4*UGA_star**13*epsilon + 153600.0*R**3*S**3*T**4*UGA_star**13 - 747520.0*R**3*S**3*T**4*UGA_star**12*epsilon - 10240.0*R**3*S**3*T**4*UGA_star**12 - 16384.0*R**3*S**3*T**4*UGA_star**11*epsilon - 14336.0*R**3*S**3*T**4*UGA_star**11 + 51200.0*R**3*S**3*T**4*UGA_star**10*epsilon + 5120.0*R**3*S**3*T**4*UGA_star**10 - 819200.0*R**3*S**2*T**5*UGA_star**20*epsilon - 51200.0*R**3*S**2*T**5*UGA_star**20 + 5734400.0*R**3*S**2*T**5*UGA_star**19*epsilon + 358400.0*R**3*S**2*T**5*UGA_star**19 - 16943104.0*R**3*S**2*T**5*UGA_star**18*epsilon - 1056768.0*R**3*S**2*T**5*UGA_star**18 + 27166720.0*R**3*S**2*T**5*UGA_star**17*epsilon + 1684480.0*R**3*S**2*T**5*UGA_star**17 - 25102336.0*R**3*S**2*T**5*UGA_star**16*epsilon - 1536000.0*R**3*S**2*T**5*UGA_star**16 + 12800000.0*R**3*S**2*T**5*UGA_star**15*epsilon + 762880.0*R**3*S**2*T**5*UGA_star**15 - 2785280.0*R**3*S**2*T**5*UGA_star**14*epsilon - 163840.0*R**3*S**2*T**5*UGA_star**14 - 219136.0*R**3*S**2*T**5*UGA_star**13*epsilon + 7168.0*R**3*S**2*T**5*UGA_star**13 + 235520.0*R**3*S**2*T**5*UGA_star**12*epsilon - 10240.0*R**3*S**2*T**5*UGA_star**12 - 98304.0*R**3*S**2*T**5*UGA_star**11*epsilon + 5120.0*R**3*S**2*T**5*UGA_star**11 + 30720.0*R**3*S**2*T**5*UGA_star**10*epsilon - 274432.0*R**3*S*T**6*UGA_star**20*epsilon - 15360.0*R**3*S*T**6*UGA_star**20 + 1921024.0*R**3*S*T**6*UGA_star**19*epsilon + 107520.0*R**3*S*T**6*UGA_star**19 - 5637120.0*R**3*S*T**6*UGA_star**18*epsilon - 315904.0*R**3*S*T**6*UGA_star**18 + 8852480.0*R**3*S*T**6*UGA_star**17*epsilon + 497664.0*R**3*S*T**6*UGA_star**17 - 7742464.0*R**3*S*T**6*UGA_star**16*epsilon - 438784.0*R**3*S*T**6*UGA_star**16 + 3331072.0*R**3*S*T**6*UGA_star**15*epsilon + 194560.0*R**3*S*T**6*UGA_star**15 - 175104.0*R**3*S*T**6*UGA_star**14*epsilon - 17920.0*R**3*S*T**6*UGA_star**14 - 353280.0*R**3*S*T**6*UGA_star**13*epsilon - 14336.0*R**3*S*T**6*UGA_star**13 + 66560.0*R**3*S*T**6*UGA_star**12*epsilon + 1536.0*R**3*S*T**6*UGA_star**12 + 11264.0*R**3*S*T**6*UGA_star**11*epsilon + 1024.0*R**3*S*T**6*UGA_star**11 - 38912.0*R**3*T**7*UGA_star**20*epsilon - 1536.0*R**3*T**7*UGA_star**20 + 272384.0*R**3*T**7*UGA_star**19*epsilon + 10752.0*R**3*T**7*UGA_star**19 - 798720.0*R**3*T**7*UGA_star**18*epsilon - 31744.0*R**3*T**7*UGA_star**18 + 1251328.0*R**3*T**7*UGA_star**17*epsilon + 50688.0*R**3*T**7*UGA_star**17 - 1087488.0*R**3*T**7*UGA_star**16*epsilon - 46080.0*R**3*T**7*UGA_star**16 + 458752.0*R**3*T**7*UGA_star**15*epsilon + 22016.0*R**3*T**7*UGA_star**15 - 16384.0*R**3*T**7*UGA_star**14*epsilon - 3072.0*R**3*T**7*UGA_star**14 - 51200.0*R**3*T**7*UGA_star**13*epsilon - 1536.0*R**3*T**7*UGA_star**13 + 8192.0*R**3*T**7*UGA_star**12*epsilon + 512.0*R**3*T**7*UGA_star**12 + 2048.0*R**3*T**7*UGA_star**11*epsilon + 73728.0*R**2*S**7*T*UGA_star**20*epsilon - 589824.0*R**2*S**7*T*UGA_star**19*epsilon + 2088960.0*R**2*S**7*T*UGA_star**18*epsilon - 4325376.0*R**2*S**7*T*UGA_star**17*epsilon + 5850112.0*R**2*S**7*T*UGA_star**16*epsilon - 5513216.0*R**2*S**7*T*UGA_star**15*epsilon + 3813376.0*R**2*S**7*T*UGA_star**14*epsilon - 2023424.0*R**2*S**7*T*UGA_star**13*epsilon + 833536.0*R**2*S**7*T*UGA_star**12*epsilon - 253952.0*R**2*S**7*T*UGA_star**11*epsilon + 53248.0*R**2*S**7*T*UGA_star**10*epsilon - 8192.0*R**2*S**7*T*UGA_star**9*epsilon + 1024.0*R**2*S**7*T*UGA_star**8*epsilon + 372736.0*R**2*S**6*T**2*UGA_star**20*epsilon + 7168.0*R**2*S**6*T**2*UGA_star**20 - 2981888.0*R**2*S**6*T**2*UGA_star**19*epsilon - 57344.0*R**2*S**6*T**2*UGA_star**19 + 10458112.0*R**2*S**6*T**2*UGA_star**18*epsilon + 204288.0*R**2*S**6*T**2*UGA_star**18 - 21131264.0*R**2*S**6*T**2*UGA_star**17*epsilon - 430080.0*R**2*S**6*T**2*UGA_star**17 + 27278336.0*R**2*S**6*T**2*UGA_star**16*epsilon + 602368.0*R**2*S**6*T**2*UGA_star**16 - 23762944.0*R**2*S**6*T**2*UGA_star**15*epsilon - 604160.0*R**2*S**6*T**2*UGA_star**15 + 14594048.0*R**2*S**6*T**2*UGA_star**14*epsilon + 458752.0*R**2*S**6*T**2*UGA_star**14 - 6608896.0*R**2*S**6*T**2*UGA_star**13*epsilon - 272384.0*R**2*S**6*T**2*UGA_star**13 + 2207744.0*R**2*S**6*T**2*UGA_star**12*epsilon + 125440.0*R**2*S**6*T**2*UGA_star**12 - 444416.0*R**2*S**6*T**2*UGA_star**11*epsilon - 43008.0*R**2*S**6*T**2*UGA_star**11 + 7168.0*R**2*S**6*T**2*UGA_star**10*epsilon + 10752.0*R**2*S**6*T**2*UGA_star**10 + 10240.0*R**2*S**6*T**2*UGA_star**9*epsilon - 2048.0*R**2*S**6*T**2*UGA_star**9 + 1024.0*R**2*S**6*T**2*UGA_star**8*epsilon + 256.0*R**2*S**6*T**2*UGA_star**8 + 817152.0*R**2*S**5*T**3*UGA_star**20*epsilon + 32256.0*R**2*S**5*T**3*UGA_star**20 - 6537216.0*R**2*S**5*T**3*UGA_star**19*epsilon - 258048.0*R**2*S**5*T**3*UGA_star**19 + 22775808.0*R**2*S**5*T**3*UGA_star**18*epsilon + 904704.0*R**2*S**5*T**3*UGA_star**18 - 45201408.0*R**2*S**5*T**3*UGA_star**17*epsilon - 1826304.0*R**2*S**5*T**3*UGA_star**17 + 56206336.0*R**2*S**5*T**3*UGA_star**16*epsilon + 2352128.0*R**2*S**5*T**3*UGA_star**16 - 45664256.0*R**2*S**5*T**3*UGA_star**15*epsilon - 2035712.0*R**2*S**5*T**3*UGA_star**15 + 25036800.0*R**2*S**5*T**3*UGA_star**14*epsilon + 1225728.0*R**2*S**5*T**3*UGA_star**14 - 9943040.0*R**2*S**5*T**3*UGA_star**13*epsilon - 523264.0*R**2*S**5*T**3*UGA_star**13 + 3217408.0*R**2*S**5*T**3*UGA_star**12*epsilon + 146944.0*R**2*S**5*T**3*UGA_star**12 - 792576.0*R**2*S**5*T**3*UGA_star**11*epsilon - 12288.0*R**2*S**5*T**3*UGA_star**11 + 77824.0*R**2*S**5*T**3*UGA_star**10*epsilon - 8704.0*R**2*S**5*T**3*UGA_star**10 + 4096.0*R**2*S**5*T**3*UGA_star**9*epsilon + 2560.0*R**2*S**5*T**3*UGA_star**9 + 3072.0*R**2*S**5*T**3*UGA_star**8*epsilon + 1029120.0*R**2*S**4*T**4*UGA_star**20*epsilon + 57600.0*R**2*S**4*T**4*UGA_star**20 - 8232960.0*R**2*S**4*T**4*UGA_star**19*epsilon - 460800.0*R**2*S**4*T**4*UGA_star**19 + 28538880.0*R**2*S**4*T**4*UGA_star**18*epsilon + 1597440.0*R**2*S**4*T**4*UGA_star**18 - 55818240.0*R**2*S**4*T**4*UGA_star**17*epsilon - 3125760.0*R**2*S**4*T**4*UGA_star**17 + 67108864.0*R**2*S**4*T**4*UGA_star**16*epsilon + 3762176.0*R**2*S**4*T**4*UGA_star**16 - 50601984.0*R**2*S**4*T**4*UGA_star**15*epsilon - 2840576.0*R**2*S**4*T**4*UGA_star**15 + 23500800.0*R**2*S**4*T**4*UGA_star**14*epsilon + 1313280.0*R**2*S**4*T**4*UGA_star**14 - 6594560.0*R**2*S**4*T**4*UGA_star**13*epsilon - 348160.0*R**2*S**4*T**4*UGA_star**13 + 1305600.0*R**2*S**4*T**4*UGA_star**12*epsilon + 44800.0*R**2*S**4*T**4*UGA_star**12 - 221184.0*R**2*S**4*T**4*UGA_star**11*epsilon + 6144.0*R**2*S**4*T**4*UGA_star**11 - 45056.0*R**2*S**4*T**4*UGA_star**10*epsilon - 8704.0*R**2*S**4*T**4*UGA_star**10 + 30720.0*R**2*S**4*T**4*UGA_star**9*epsilon + 2560.0*R**2*S**4*T**4*UGA_star**9 + 819200.0*R**2*S**3*T**5*UGA_star**20*epsilon + 51200.0*R**2*S**3*T**5*UGA_star**20 - 6553600.0*R**2*S**3*T**5*UGA_star**19*epsilon - 409600.0*R**2*S**3*T**5*UGA_star**19 + 22614016.0*R**2*S**3*T**5*UGA_star**18*epsilon + 1411072.0*R**2*S**3*T**5*UGA_star**18 - 43646976.0*R**2*S**3*T**5*UGA_star**17*epsilon - 2711552.0*R**2*S**3*T**5*UGA_star**17 + 50825216.0*R**2*S**3*T**5*UGA_star**16*epsilon + 3128320.0*R**2*S**3*T**5*UGA_star**16 - 35414016.0*R**2*S**3*T**5*UGA_star**15*epsilon - 2140160.0*R**2*S**3*T**5*UGA_star**15 + 13056000.0*R**2*S**3*T**5*UGA_star**14*epsilon + 762880.0*R**2*S**3*T**5*UGA_star**14 - 1110016.0*R**2*S**3*T**5*UGA_star**13*epsilon - 69632.0*R**2*S**3*T**5*UGA_star**13 - 804864.0*R**2*S**3*T**5*UGA_star**12*epsilon - 17408.0*R**2*S**3*T**5*UGA_star**12 + 262144.0*R**2*S**3*T**5*UGA_star**11*epsilon - 10240.0*R**2*S**3*T**5*UGA_star**11 - 67584.0*R**2*S**3*T**5*UGA_star**10*epsilon + 5120.0*R**2*S**3*T**5*UGA_star**10 + 20480.0*R**2*S**3*T**5*UGA_star**9*epsilon + 411648.0*R**2*S**2*T**6*UGA_star**20*epsilon + 23040.0*R**2*S**2*T**6*UGA_star**20 - 3293184.0*R**2*S**2*T**6*UGA_star**19*epsilon - 184320.0*R**2*S**2*T**6*UGA_star**19 + 11326464.0*R**2*S**2*T**6*UGA_star**18*epsilon + 634368.0*R**2*S**2*T**6*UGA_star**18 - 21657600.0*R**2*S**2*T**6*UGA_star**17*epsilon - 1214976.0*R**2*S**2*T**6*UGA_star**17 + 24662016.0*R**2*S**2*T**6*UGA_star**16*epsilon + 1388544.0*R**2*S**2*T**6*UGA_star**16 - 16238592.0*R**2*S**2*T**6*UGA_star**15*epsilon - 923136.0*R**2*S**2*T**6*UGA_star**15 + 4921344.0*R**2*S**2*T**6*UGA_star**14*epsilon + 291840.0*R**2*S**2*T**6*UGA_star**14 + 423936.0*R**2*S**2*T**6*UGA_star**13*epsilon + 10752.0*R**2*S**2*T**6*UGA_star**13 - 645120.0*R**2*S**2*T**6*UGA_star**12*epsilon - 29184.0*R**2*S**2*T**6*UGA_star**12 + 67584.0*R**2*S**2*T**6*UGA_star**11*epsilon + 1536.0*R**2*S**2*T**6*UGA_star**11 + 21504.0*R**2*S**2*T**6*UGA_star**10*epsilon + 1536.0*R**2*S**2*T**6*UGA_star**10 + 116736.0*R**2*S*T**7*UGA_star**20*epsilon + 4608.0*R**2*S*T**7*UGA_star**20 - 933888.0*R**2*S*T**7*UGA_star**19*epsilon - 36864.0*R**2*S*T**7*UGA_star**19 + 3213312.0*R**2*S*T**7*UGA_star**18*epsilon + 127488.0*R**2*S*T**7*UGA_star**18 - 6150144.0*R**2*S*T**7*UGA_star**17*epsilon - 247296.0*R**2*S*T**7*UGA_star**17 + 7016448.0*R**2*S*T**7*UGA_star**16*epsilon + 290304.0*R**2*S*T**7*UGA_star**16 - 4638720.0*R**2*S*T**7*UGA_star**15*epsilon - 204288.0*R**2*S*T**7*UGA_star**15 + 1425408.0*R**2*S*T**7*UGA_star**14*epsilon + 75264.0*R**2*S*T**7*UGA_star**14 + 104448.0*R**2*S*T**7*UGA_star**13*epsilon - 4608.0*R**2*S*T**7*UGA_star**13 - 178176.0*R**2*S*T**7*UGA_star**12*epsilon - 6144.0*R**2*S*T**7*UGA_star**12 + 18432.0*R**2*S*T**7*UGA_star**11*epsilon + 1536.0*R**2*S*T**7*UGA_star**11 + 6144.0*R**2*S*T**7*UGA_star**10*epsilon + 13312.0*R**2*T**8*UGA_star**20*epsilon + 256.0*R**2*T**8*UGA_star**20 - 106496.0*R**2*T**8*UGA_star**19*epsilon - 2048.0*R**2*T**8*UGA_star**19 + 368640.0*R**2*T**8*UGA_star**18*epsilon + 7168.0*R**2*T**8*UGA_star**18 - 716800.0*R**2*T**8*UGA_star**17*epsilon - 14336.0*R**2*T**8*UGA_star**17 + 845824.0*R**2*T**8*UGA_star**16*epsilon + 17920.0*R**2*T**8*UGA_star**16 - 602112.0*R**2*T**8*UGA_star**15*epsilon - 14336.0*R**2*T**8*UGA_star**15 + 229376.0*R**2*T**8*UGA_star**14*epsilon + 7168.0*R**2*T**8*UGA_star**14 - 20480.0*R**2*T**8*UGA_star**13*epsilon - 2048.0*R**2*T**8*UGA_star**13 - 15360.0*R**2*T**8*UGA_star**12*epsilon + 256.0*R**2*T**8*UGA_star**12 + 4096.0*R**2*T**8*UGA_star**11*epsilon - 18432.0*R*S**8*T*UGA_star**20*epsilon + 165888.0*R*S**8*T*UGA_star**19*epsilon - 666624.0*R*S**8*T*UGA_star**18*epsilon + 1575936.0*R*S**8*T*UGA_star**17*epsilon - 2433024.0*R*S**8*T*UGA_star**16*epsilon + 2580480.0*R*S**8*T*UGA_star**15*epsilon - 1935360.0*R*S**8*T*UGA_star**14*epsilon + 1050624.0*R*S**8*T*UGA_star**13*epsilon - 423936.0*R*S**8*T*UGA_star**12*epsilon + 129024.0*R*S**8*T*UGA_star**11*epsilon - 27648.0*R*S**8*T*UGA_star**10*epsilon + 3072.0*R*S**8*T*UGA_star**9*epsilon - 106496.0*R*S**7*T**2*UGA_star**20*epsilon - 2048.0*R*S**7*T**2*UGA_star**20 + 958464.0*R*S**7*T**2*UGA_star**19*epsilon + 18432.0*R*S**7*T**2*UGA_star**19 - 3820544.0*R*S**7*T**2*UGA_star**18*epsilon - 74240.0*R*S**7*T**2*UGA_star**18 + 8854528.0*R*S**7*T**2*UGA_star**17*epsilon + 176640.0*R*S**7*T**2*UGA_star**17 - 13165568.0*R*S**7*T**2*UGA_star**16*epsilon - 276480.0*R*S**7*T**2*UGA_star**16 + 13078528.0*R*S**7*T**2*UGA_star**15*epsilon + 301056.0*R*S**7*T**2*UGA_star**15 - 8787968.0*R*S**7*T**2*UGA_star**14*epsilon - 236544.0*R*S**7*T**2*UGA_star**14 + 3991552.0*R*S**7*T**2*UGA_star**13*epsilon + 138240.0*R*S**7*T**2*UGA_star**13 - 1230848.0*R*S**7*T**2*UGA_star**12*epsilon - 61440.0*R*S**7*T**2*UGA_star**12 + 258048.0*R*S**7*T**2*UGA_star**11*epsilon + 20480.0*R*S**7*T**2*UGA_star**11 - 23552.0*R*S**7*T**2*UGA_star**10*epsilon - 4608.0*R*S**7*T**2*UGA_star**10 - 9216.0*R*S**7*T**2*UGA_star**9*epsilon + 512.0*R*S**7*T**2*UGA_star**9 + 3072.0*R*S**7*T**2*UGA_star**8*epsilon - 272384.0*R*S**6*T**3*UGA_star**20*epsilon - 10752.0*R*S**6*T**3*UGA_star**20 + 2451456.0*R*S**6*T**3*UGA_star**19*epsilon + 96768.0*R*S**6*T**3*UGA_star**19 - 9723904.0*R*S**6*T**3*UGA_star**18*epsilon - 385536.0*R*S**6*T**3*UGA_star**18 + 22253568.0*R*S**6*T**3*UGA_star**17*epsilon + 892416.0*R*S**6*T**3*UGA_star**17 - 32261120.0*R*S**6*T**3*UGA_star**16*epsilon - 1323520.0*R*S**6*T**3*UGA_star**16 + 30566400.0*R*S**6*T**3*UGA_star**15*epsilon + 1307648.0*R*S**6*T**3*UGA_star**15 - 18823168.0*R*S**6*T**3*UGA_star**14*epsilon - 867328.0*R*S**6*T**3*UGA_star**14 + 7297024.0*R*S**6*T**3*UGA_star**13*epsilon + 379904.0*R*S**6*T**3*UGA_star**13 - 1763328.0*R*S**6*T**3*UGA_star**12*epsilon - 103936.0*R*S**6*T**3*UGA_star**12 + 348160.0*R*S**6*T**3*UGA_star**11*epsilon + 12800.0*R*S**6*T**3*UGA_star**11 - 75776.0*R*S**6*T**3*UGA_star**10*epsilon + 3584.0*R*S**6*T**3*UGA_star**10 - 2048.0*R*S**6*T**3*UGA_star**9*epsilon - 2560.0*R*S**6*T**3*UGA_star**9 + 5120.0*R*S**6*T**3*UGA_star**8*epsilon + 512.0*R*S**6*T**3*UGA_star**8 - 411648.0*R*S**5*T**4*UGA_star**20*epsilon - 23040.0*R*S**5*T**4*UGA_star**20 + 3704832.0*R*S**5*T**4*UGA_star**19*epsilon + 207360.0*R*S**5*T**4*UGA_star**19 - 14653440.0*R*S**5*T**4*UGA_star**18*epsilon - 820224.0*R*S**5*T**4*UGA_star**18 + 33275904.0*R*S**5*T**4*UGA_star**17*epsilon + 1863168.0*R*S**5*T**4*UGA_star**17 - 47436800.0*R*S**5*T**4*UGA_star**16*epsilon - 2657792.0*R*S**5*T**4*UGA_star**16 + 43380736.0*R*S**5*T**4*UGA_star**15*epsilon + 2433024.0*R*S**5*T**4*UGA_star**15 - 24657920.0*R*S**5*T**4*UGA_star**14*epsilon - 1383424.0*R*S**5*T**4*UGA_star**14 + 7708672.0*R*S**5*T**4*UGA_star**13*epsilon + 428032.0*R*S**5*T**4*UGA_star**13 - 794624.0*R*S**5*T**4*UGA_star**12*epsilon - 35328.0*R*S**5*T**4*UGA_star**12 - 112640.0*R*S**5*T**4*UGA_star**11*epsilon - 14848.0*R*S**5*T**4*UGA_star**11 - 2048.0*R*S**5*T**4*UGA_star**10*epsilon + 4096.0*R*S**5*T**4*UGA_star**10 - 8192.0*R*S**5*T**4*UGA_star**9*epsilon - 1536.0*R*S**5*T**4*UGA_star**9 + 7168.0*R*S**5*T**4*UGA_star**8*epsilon + 512.0*R*S**5*T**4*UGA_star**8 - 409600.0*R*S**4*T**5*UGA_star**20*epsilon - 25600.0*R*S**4*T**5*UGA_star**20 + 3686400.0*R*S**4*T**5*UGA_star**19*epsilon + 230400.0*R*S**4*T**5*UGA_star**19 - 14552064.0*R*S**4*T**5*UGA_star**18*epsilon - 908288.0*R*S**4*T**5*UGA_star**18 + 32867328.0*R*S**4*T**5*UGA_star**17*epsilon + 2044416.0*R*S**4*T**5*UGA_star**17 - 46283776.0*R*S**4*T**5*UGA_star**16*epsilon - 2859008.0*R*S**4*T**5*UGA_star**16 + 41160704.0*R*S**4*T**5*UGA_star**15*epsilon + 2508800.0*R*S**4*T**5*UGA_star**15 - 21747712.0*R*S**4*T**5*UGA_star**14*epsilon - 1290240.0*R*S**4*T**5*UGA_star**14 + 5126144.0*R*S**4*T**5*UGA_star**13*epsilon + 283648.0*R*S**4*T**5*UGA_star**13 + 714752.0*R*S**4*T**5*UGA_star**12*epsilon + 41984.0*R*S**4*T**5*UGA_star**12 - 651264.0*R*S**4*T**5*UGA_star**11*epsilon - 23552.0*R*S**4*T**5*UGA_star**11 + 91136.0*R*S**4*T**5*UGA_star**10*epsilon - 5120.0*R*S**4*T**5*UGA_star**10 - 7168.0*R*S**4*T**5*UGA_star**9*epsilon + 2560.0*R*S**4*T**5*UGA_star**9 + 5120.0*R*S**4*T**5*UGA_star**8*epsilon - 274432.0*R*S**3*T**6*UGA_star**20*epsilon - 15360.0*R*S**3*T**6*UGA_star**20 + 2469888.0*R*S**3*T**6*UGA_star**19*epsilon + 138240.0*R*S**3*T**6*UGA_star**19 - 9739264.0*R*S**3*T**6*UGA_star**18*epsilon - 545280.0*R*S**3*T**6*UGA_star**18 + 21931008.0*R*S**3*T**6*UGA_star**17*epsilon + 1228800.0*R*S**3*T**6*UGA_star**17 - 30674944.0*R*S**3*T**6*UGA_star**16*epsilon - 1721344.0*R*S**3*T**6*UGA_star**16 + 26865664.0*R*S**3*T**6*UGA_star**15*epsilon + 1512448.0*R*S**3*T**6*UGA_star**15 - 13633536.0*R*S**3*T**6*UGA_star**14*epsilon - 774144.0*R*S**3*T**6*UGA_star**14 + 2668544.0*R*S**3*T**6*UGA_star**13*epsilon + 158720.0*R*S**3*T**6*UGA_star**13 + 827392.0*R*S**3*T**6*UGA_star**12*epsilon + 40960.0*R*S**3*T**6*UGA_star**12 - 475136.0*R*S**3*T**6*UGA_star**11*epsilon - 24576.0*R*S**3*T**6*UGA_star**11 + 17408.0*R*S**3*T**6*UGA_star**10*epsilon + 512.0*R*S**3*T**6*UGA_star**10 + 17408.0*R*S**3*T**6*UGA_star**9*epsilon + 1024.0*R*S**3*T**6*UGA_star**9 - 116736.0*R*S**2*T**7*UGA_star**20*epsilon - 4608.0*R*S**2*T**7*UGA_star**20 + 1050624.0*R*S**2*T**7*UGA_star**19*epsilon + 41472.0*R*S**2*T**7*UGA_star**19 - 4147200.0*R*S**2*T**7*UGA_star**18*epsilon - 164352.0*R*S**2*T**7*UGA_star**18 + 9363456.0*R*S**2*T**7*UGA_star**17*epsilon + 374784.0*R*S**2*T**7*UGA_star**17 - 13166592.0*R*S**2*T**7*UGA_star**16*epsilon - 537600.0*R*S**2*T**7*UGA_star**16 + 11655168.0*R*S**2*T**7*UGA_star**15*epsilon + 494592.0*R*S**2*T**7*UGA_star**15 - 6064128.0*R*S**2*T**7*UGA_star**14*epsilon - 279552.0*R*S**2*T**7*UGA_star**14 + 1320960.0*R*S**2*T**7*UGA_star**13*epsilon + 79872.0*R*S**2*T**7*UGA_star**13 + 282624.0*R*S**2*T**7*UGA_star**12*epsilon + 1536.0*R*S**2*T**7*UGA_star**12 - 196608.0*R*S**2*T**7*UGA_star**11*epsilon - 7680.0*R*S**2*T**7*UGA_star**11 + 12288.0*R*S**2*T**7*UGA_star**10*epsilon + 1536.0*R*S**2*T**7*UGA_star**10 + 6144.0*R*S**2*T**7*UGA_star**9*epsilon - 26624.0*R*S*T**8*UGA_star**20*epsilon - 512.0*R*S*T**8*UGA_star**20 + 239616.0*R*S*T**8*UGA_star**19*epsilon + 4608.0*R*S*T**8*UGA_star**19 - 950272.0*R*S*T**8*UGA_star**18*epsilon - 18432.0*R*S*T**8*UGA_star**18 + 2170880.0*R*S*T**8*UGA_star**17*epsilon + 43008.0*R*S*T**8*UGA_star**17 - 3125248.0*R*S*T**8*UGA_star**16*epsilon - 64512.0*R*S*T**8*UGA_star**16 + 2895872.0*R*S*T**8*UGA_star**15*epsilon + 64512.0*R*S*T**8*UGA_star**15 - 1662976.0*R*S*T**8*UGA_star**14*epsilon - 43008.0*R*S*T**8*UGA_star**14 + 499712.0*R*S*T**8*UGA_star**13*epsilon + 18432.0*R*S*T**8*UGA_star**13 - 10240.0*R*S*T**8*UGA_star**12*epsilon - 4608.0*R*S*T**8*UGA_star**12 - 38912.0*R*S*T**8*UGA_star**11*epsilon + 512.0*R*S*T**8*UGA_star**11 + 8192.0*R*S*T**8*UGA_star**10*epsilon - 2048.0*R*T**9*UGA_star**20*epsilon + 18432.0*R*T**9*UGA_star**19*epsilon - 73728.0*R*T**9*UGA_star**18*epsilon + 172032.0*R*T**9*UGA_star**17*epsilon - 258048.0*R*T**9*UGA_star**16*epsilon + 258048.0*R*T**9*UGA_star**15*epsilon - 172032.0*R*T**9*UGA_star**14*epsilon + 73728.0*R*T**9*UGA_star**13*epsilon - 18432.0*R*T**9*UGA_star**12*epsilon + 2048.0*R*T**9*UGA_star**11*epsilon + 2048.0*S**9*T*UGA_star**20*epsilon - 20480.0*S**9*T*UGA_star**19*epsilon + 92160.0*S**9*T*UGA_star**18*epsilon - 245760.0*S**9*T*UGA_star**17*epsilon + 430080.0*S**9*T*UGA_star**16*epsilon - 516096.0*S**9*T*UGA_star**15*epsilon + 430080.0*S**9*T*UGA_star**14*epsilon - 245760.0*S**9*T*UGA_star**13*epsilon + 92160.0*S**9*T*UGA_star**12*epsilon - 20480.0*S**9*T*UGA_star**11*epsilon + 2048.0*S**9*T*UGA_star**10*epsilon + 13312.0*S**8*T**2*UGA_star**20*epsilon + 256.0*S**8*T**2*UGA_star**20 - 133120.0*S**8*T**2*UGA_star**19*epsilon - 2560.0*S**8*T**2*UGA_star**19 + 594944.0*S**8*T**2*UGA_star**18*epsilon + 11520.0*S**8*T**2*UGA_star**18 - 1560576.0*S**8*T**2*UGA_star**17*epsilon - 30720.0*S**8*T**2*UGA_star**17 + 2648064.0*S**8*T**2*UGA_star**16*epsilon + 53760.0*S**8*T**2*UGA_star**16 - 3010560.0*S**8*T**2*UGA_star**15*epsilon - 64512.0*S**8*T**2*UGA_star**15 + 2279424.0*S**8*T**2*UGA_star**14*epsilon + 53760.0*S**8*T**2*UGA_star**14 - 1081344.0*S**8*T**2*UGA_star**13*epsilon - 30720.0*S**8*T**2*UGA_star**13 + 254976.0*S**8*T**2*UGA_star**12*epsilon + 11520.0*S**8*T**2*UGA_star**12 + 14336.0*S**8*T**2*UGA_star**11*epsilon - 2560.0*S**8*T**2*UGA_star**11 - 23552.0*S**8*T**2*UGA_star**10*epsilon + 256.0*S**8*T**2*UGA_star**10 + 4096.0*S**8*T**2*UGA_star**9*epsilon + 38912.0*S**7*T**3*UGA_star**20*epsilon + 1536.0*S**7*T**3*UGA_star**20 - 389120.0*S**7*T**3*UGA_star**19*epsilon - 15360.0*S**7*T**3*UGA_star**19 + 1732608.0*S**7*T**3*UGA_star**18*epsilon + 68608.0*S**7*T**3*UGA_star**18 - 4503552.0*S**7*T**3*UGA_star**17*epsilon - 179712.0*S**7*T**3*UGA_star**17 + 7510016.0*S**7*T**3*UGA_star**16*epsilon + 304128.0*S**7*T**3*UGA_star**16 - 8273920.0*S**7*T**3*UGA_star**15*epsilon - 344064.0*S**7*T**3*UGA_star**15 + 5906432.0*S**7*T**3*UGA_star**14*epsilon + 258048.0*S**7*T**3*UGA_star**14 - 2461696.0*S**7*T**3*UGA_star**13*epsilon - 119808.0*S**7*T**3*UGA_star**13 + 346112.0*S**7*T**3*UGA_star**12*epsilon + 26112.0*S**7*T**3*UGA_star**12 + 159744.0*S**7*T**3*UGA_star**11*epsilon + 3072.0*S**7*T**3*UGA_star**11 - 69632.0*S**7*T**3*UGA_star**10*epsilon - 3072.0*S**7*T**3*UGA_star**10 + 2048.0*S**7*T**3*UGA_star**9*epsilon + 512.0*S**7*T**3*UGA_star**9 + 2048.0*S**7*T**3*UGA_star**8*epsilon + 68608.0*S**6*T**4*UGA_star**20*epsilon + 3840.0*S**6*T**4*UGA_star**20 - 686080.0*S**6*T**4*UGA_star**19*epsilon - 38400.0*S**6*T**4*UGA_star**19 + 3050496.0*S**6*T**4*UGA_star**18*epsilon + 170752.0*S**6*T**4*UGA_star**18 - 7901184.0*S**6*T**4*UGA_star**17*epsilon - 442368.0*S**6*T**4*UGA_star**17 + 13085696.0*S**6*T**4*UGA_star**16*epsilon + 732928.0*S**6*T**4*UGA_star**16 - 14233600.0*S**6*T**4*UGA_star**15*epsilon - 797696.0*S**6*T**4*UGA_star**15 + 9906176.0*S**6*T**4*UGA_star**14*epsilon + 555520.0*S**6*T**4*UGA_star**14 - 3874816.0*S**6*T**4*UGA_star**13*epsilon - 217088.0*S**6*T**4*UGA_star**13 + 349184.0*S**6*T**4*UGA_star**12*epsilon + 18688.0*S**6*T**4*UGA_star**12 + 354304.0*S**6*T**4*UGA_star**11*epsilon + 20992.0*S**6*T**4*UGA_star**11 - 119808.0*S**6*T**4*UGA_star**10*epsilon - 7424.0*S**6*T**4*UGA_star**10 - 4096.0*S**6*T**4*UGA_star**9*epsilon + 5120.0*S**6*T**4*UGA_star**8*epsilon + 256.0*S**6*T**4*UGA_star**8 + 81920.0*S**5*T**5*UGA_star**20*epsilon + 5120.0*S**5*T**5*UGA_star**20 - 819200.0*S**5*T**5*UGA_star**19*epsilon - 51200.0*S**5*T**5*UGA_star**19 + 3641344.0*S**5*T**5*UGA_star**18*epsilon + 227328.0*S**5*T**5*UGA_star**18 - 9424896.0*S**5*T**5*UGA_star**17*epsilon - 586752.0*S**5*T**5*UGA_star**17 + 15587328.0*S**5*T**5*UGA_star**16*epsilon + 965120.0*S**5*T**5*UGA_star**16 - 16908288.0*S**5*T**5*UGA_star**15*epsilon - 1036288.0*S**5*T**5*UGA_star**15 + 11698176.0*S**5*T**5*UGA_star**14*epsilon + 702464.0*S**5*T**5*UGA_star**14 - 4497408.0*S**5*T**5*UGA_star**13*epsilon - 256000.0*S**5*T**5*UGA_star**13 + 331776.0*S**5*T**5*UGA_star**12*epsilon + 8192.0*S**5*T**5*UGA_star**12 + 458752.0*S**5*T**5*UGA_star**11*epsilon + 30720.0*S**5*T**5*UGA_star**11 - 151552.0*S**5*T**5*UGA_star**10*epsilon - 8192.0*S**5*T**5*UGA_star**10 - 4096.0*S**5*T**5*UGA_star**9*epsilon - 1024.0*S**5*T**5*UGA_star**9 + 6144.0*S**5*T**5*UGA_star**8*epsilon + 512.0*S**5*T**5*UGA_star**8 + 68608.0*S**4*T**6*UGA_star**20*epsilon + 3840.0*S**4*T**6*UGA_star**20 - 686080.0*S**4*T**6*UGA_star**19*epsilon - 38400.0*S**4*T**6*UGA_star**19 + 3050496.0*S**4*T**6*UGA_star**18*epsilon + 170752.0*S**4*T**6*UGA_star**18 - 7901184.0*S**4*T**6*UGA_star**17*epsilon - 442368.0*S**4*T**6*UGA_star**17 + 13085696.0*S**4*T**6*UGA_star**16*epsilon + 732928.0*S**4*T**6*UGA_star**16 - 14233600.0*S**4*T**6*UGA_star**15*epsilon - 797696.0*S**4*T**6*UGA_star**15 + 9906176.0*S**4*T**6*UGA_star**14*epsilon + 555520.0*S**4*T**6*UGA_star**14 - 3874816.0*S**4*T**6*UGA_star**13*epsilon - 217088.0*S**4*T**6*UGA_star**13 + 349184.0*S**4*T**6*UGA_star**12*epsilon + 18688.0*S**4*T**6*UGA_star**12 + 354304.0*S**4*T**6*UGA_star**11*epsilon + 20992.0*S**4*T**6*UGA_star**11 - 119808.0*S**4*T**6*UGA_star**10*epsilon - 7424.0*S**4*T**6*UGA_star**10 - 4096.0*S**4*T**6*UGA_star**9*epsilon + 5120.0*S**4*T**6*UGA_star**8*epsilon + 256.0*S**4*T**6*UGA_star**8 + 38912.0*S**3*T**7*UGA_star**20*epsilon + 1536.0*S**3*T**7*UGA_star**20 - 389120.0*S**3*T**7*UGA_star**19*epsilon - 15360.0*S**3*T**7*UGA_star**19 + 1732608.0*S**3*T**7*UGA_star**18*epsilon + 68608.0*S**3*T**7*UGA_star**18 - 4503552.0*S**3*T**7*UGA_star**17*epsilon - 179712.0*S**3*T**7*UGA_star**17 + 7510016.0*S**3*T**7*UGA_star**16*epsilon + 304128.0*S**3*T**7*UGA_star**16 - 8273920.0*S**3*T**7*UGA_star**15*epsilon - 344064.0*S**3*T**7*UGA_star**15 + 5906432.0*S**3*T**7*UGA_star**14*epsilon + 258048.0*S**3*T**7*UGA_star**14 - 2461696.0*S**3*T**7*UGA_star**13*epsilon - 119808.0*S**3*T**7*UGA_star**13 + 346112.0*S**3*T**7*UGA_star**12*epsilon + 26112.0*S**3*T**7*UGA_star**12 + 159744.0*S**3*T**7*UGA_star**11*epsilon + 3072.0*S**3*T**7*UGA_star**11 - 69632.0*S**3*T**7*UGA_star**10*epsilon - 3072.0*S**3*T**7*UGA_star**10 + 2048.0*S**3*T**7*UGA_star**9*epsilon + 512.0*S**3*T**7*UGA_star**9 + 2048.0*S**3*T**7*UGA_star**8*epsilon + 13312.0*S**2*T**8*UGA_star**20*epsilon + 256.0*S**2*T**8*UGA_star**20 - 133120.0*S**2*T**8*UGA_star**19*epsilon - 2560.0*S**2*T**8*UGA_star**19 + 594944.0*S**2*T**8*UGA_star**18*epsilon + 11520.0*S**2*T**8*UGA_star**18 - 1560576.0*S**2*T**8*UGA_star**17*epsilon - 30720.0*S**2*T**8*UGA_star**17 + 2648064.0*S**2*T**8*UGA_star**16*epsilon + 53760.0*S**2*T**8*UGA_star**16 - 3010560.0*S**2*T**8*UGA_star**15*epsilon - 64512.0*S**2*T**8*UGA_star**15 + 2279424.0*S**2*T**8*UGA_star**14*epsilon + 53760.0*S**2*T**8*UGA_star**14 - 1081344.0*S**2*T**8*UGA_star**13*epsilon - 30720.0*S**2*T**8*UGA_star**13 + 254976.0*S**2*T**8*UGA_star**12*epsilon + 11520.0*S**2*T**8*UGA_star**12 + 14336.0*S**2*T**8*UGA_star**11*epsilon - 2560.0*S**2*T**8*UGA_star**11 - 23552.0*S**2*T**8*UGA_star**10*epsilon + 256.0*S**2*T**8*UGA_star**10 + 4096.0*S**2*T**8*UGA_star**9*epsilon + 2048.0*S*T**9*UGA_star**20*epsilon - 20480.0*S*T**9*UGA_star**19*epsilon + 92160.0*S*T**9*UGA_star**18*epsilon - 245760.0*S*T**9*UGA_star**17*epsilon + 430080.0*S*T**9*UGA_star**16*epsilon - 516096.0*S*T**9*UGA_star**15*epsilon + 430080.0*S*T**9*UGA_star**14*epsilon - 245760.0*S*T**9*UGA_star**13*epsilon + 92160.0*S*T**9*UGA_star**12*epsilon - 20480.0*S*T**9*UGA_star**11*epsilon + 2048.0*S*T**9*UGA_star**10*epsilon))/(-256.0*P**5*UGA_star**10*epsilon + 2560.0*P**5*UGA_star**9*epsilon - 11520.0*P**5*UGA_star**8*epsilon + 30720.0*P**5*UGA_star**7*epsilon - 53760.0*P**5*UGA_star**6*epsilon + 64512.0*P**5*UGA_star**5*epsilon - 53760.0*P**5*UGA_star**4*epsilon + 30720.0*P**5*UGA_star**3*epsilon - 11520.0*P**5*UGA_star**2*epsilon + 2560.0*P**5*UGA_star*epsilon - 256.0*P**5*epsilon - 1280.0*P**4*R*UGA_star**10*epsilon - 128.0*P**4*R*UGA_star**10 + 10240.0*P**4*R*UGA_star**9*epsilon + 1024.0*P**4*R*UGA_star**9 - 35840.0*P**4*R*UGA_star**8*epsilon - 3584.0*P**4*R*UGA_star**8 + 71680.0*P**4*R*UGA_star**7*epsilon + 7168.0*P**4*R*UGA_star**7 - 89600.0*P**4*R*UGA_star**6*epsilon - 8960.0*P**4*R*UGA_star**6 + 71680.0*P**4*R*UGA_star**5*epsilon + 7168.0*P**4*R*UGA_star**5 - 35840.0*P**4*R*UGA_star**4*epsilon - 3584.0*P**4*R*UGA_star**4 + 10240.0*P**4*R*UGA_star**3*epsilon + 1024.0*P**4*R*UGA_star**3 - 1280.0*P**4*R*UGA_star**2*epsilon - 128.0*P**4*R*UGA_star**2 + 1280.0*P**4*S*UGA_star**10*epsilon + 128.0*P**4*S*UGA_star**10 - 11520.0*P**4*S*UGA_star**9*epsilon - 1152.0*P**4*S*UGA_star**9 + 46080.0*P**4*S*UGA_star**8*epsilon + 4608.0*P**4*S*UGA_star**8 - 107520.0*P**4*S*UGA_star**7*epsilon - 10752.0*P**4*S*UGA_star**7 + 161280.0*P**4*S*UGA_star**6*epsilon + 16128.0*P**4*S*UGA_star**6 - 161280.0*P**4*S*UGA_star**5*epsilon - 16128.0*P**4*S*UGA_star**5 + 107520.0*P**4*S*UGA_star**4*epsilon + 10752.0*P**4*S*UGA_star**4 - 46080.0*P**4*S*UGA_star**3*epsilon - 4608.0*P**4*S*UGA_star**3 + 11520.0*P**4*S*UGA_star**2*epsilon + 1152.0*P**4*S*UGA_star**2 - 1280.0*P**4*S*UGA_star*epsilon - 128.0*P**4*S*UGA_star + 1280.0*P**4*T*UGA_star**10*epsilon - 11520.0*P**4*T*UGA_star**9*epsilon + 46080.0*P**4*T*UGA_star**8*epsilon - 107520.0*P**4*T*UGA_star**7*epsilon + 161280.0*P**4*T*UGA_star**6*epsilon - 161280.0*P**4*T*UGA_star**5*epsilon + 107520.0*P**4*T*UGA_star**4*epsilon - 46080.0*P**4*T*UGA_star**3*epsilon + 11520.0*P**4*T*UGA_star**2*epsilon - 1280.0*P**4*T*UGA_star*epsilon - 2560.0*P**3*R**2*UGA_star**10*epsilon - 384.0*P**3*R**2*UGA_star**10 + 15360.0*P**3*R**2*UGA_star**9*epsilon + 2304.0*P**3*R**2*UGA_star**9 - 38400.0*P**3*R**2*UGA_star**8*epsilon - 5760.0*P**3*R**2*UGA_star**8 + 51200.0*P**3*R**2*UGA_star**7*epsilon + 7680.0*P**3*R**2*UGA_star**7 - 38400.0*P**3*R**2*UGA_star**6*epsilon - 5760.0*P**3*R**2*UGA_star**6 + 15360.0*P**3*R**2*UGA_star**5*epsilon + 2304.0*P**3*R**2*UGA_star**5 - 2560.0*P**3*R**2*UGA_star**4*epsilon - 384.0*P**3*R**2*UGA_star**4 + 5120.0*P**3*R*S*UGA_star**10*epsilon + 768.0*P**3*R*S*UGA_star**10 - 35840.0*P**3*R*S*UGA_star**9*epsilon - 5376.0*P**3*R*S*UGA_star**9 + 107520.0*P**3*R*S*UGA_star**8*epsilon + 16128.0*P**3*R*S*UGA_star**8 - 179200.0*P**3*R*S*UGA_star**7*epsilon - 26880.0*P**3*R*S*UGA_star**7 + 179200.0*P**3*R*S*UGA_star**6*epsilon + 26880.0*P**3*R*S*UGA_star**6 - 107520.0*P**3*R*S*UGA_star**5*epsilon - 16128.0*P**3*R*S*UGA_star**5 + 35840.0*P**3*R*S*UGA_star**4*epsilon + 5376.0*P**3*R*S*UGA_star**4 - 5120.0*P**3*R*S*UGA_star**3*epsilon - 768.0*P**3*R*S*UGA_star**3 + 5120.0*P**3*R*T*UGA_star**10*epsilon + 512.0*P**3*R*T*UGA_star**10 - 35840.0*P**3*R*T*UGA_star**9*epsilon - 3584.0*P**3*R*T*UGA_star**9 + 107520.0*P**3*R*T*UGA_star**8*epsilon + 10752.0*P**3*R*T*UGA_star**8 - 179200.0*P**3*R*T*UGA_star**7*epsilon - 17920.0*P**3*R*T*UGA_star**7 + 179200.0*P**3*R*T*UGA_star**6*epsilon + 17920.0*P**3*R*T*UGA_star**6 - 107520.0*P**3*R*T*UGA_star**5*epsilon - 10752.0*P**3*R*T*UGA_star**5 + 35840.0*P**3*R*T*UGA_star**4*epsilon + 3584.0*P**3*R*T*UGA_star**4 - 5120.0*P**3*R*T*UGA_star**3*epsilon - 512.0*P**3*R*T*UGA_star**3 - 2560.0*P**3*S**2*UGA_star**10*epsilon - 384.0*P**3*S**2*UGA_star**10 + 20480.0*P**3*S**2*UGA_star**9*epsilon + 3072.0*P**3*S**2*UGA_star**9 - 71680.0*P**3*S**2*UGA_star**8*epsilon - 10752.0*P**3*S**2*UGA_star**8 + 143360.0*P**3*S**2*UGA_star**7*epsilon + 21504.0*P**3*S**2*UGA_star**7 - 179200.0*P**3*S**2*UGA_star**6*epsilon - 26880.0*P**3*S**2*UGA_star**6 + 143360.0*P**3*S**2*UGA_star**5*epsilon + 21504.0*P**3*S**2*UGA_star**5 - 71680.0*P**3*S**2*UGA_star**4*epsilon - 10752.0*P**3*S**2*UGA_star**4 + 20480.0*P**3*S**2*UGA_star**3*epsilon + 3072.0*P**3*S**2*UGA_star**3 - 2560.0*P**3*S**2*UGA_star**2*epsilon - 384.0*P**3*S**2*UGA_star**2 - 5120.0*P**3*S*T*UGA_star**10*epsilon - 512.0*P**3*S*T*UGA_star**10 + 40960.0*P**3*S*T*UGA_star**9*epsilon + 4096.0*P**3*S*T*UGA_star**9 - 143360.0*P**3*S*T*UGA_star**8*epsilon - 14336.0*P**3*S*T*UGA_star**8 + 286720.0*P**3*S*T*UGA_star**7*epsilon + 28672.0*P**3*S*T*UGA_star**7 - 358400.0*P**3*S*T*UGA_star**6*epsilon - 35840.0*P**3*S*T*UGA_star**6 + 286720.0*P**3*S*T*UGA_star**5*epsilon + 28672.0*P**3*S*T*UGA_star**5 - 143360.0*P**3*S*T*UGA_star**4*epsilon - 14336.0*P**3*S*T*UGA_star**4 + 40960.0*P**3*S*T*UGA_star**3*epsilon + 4096.0*P**3*S*T*UGA_star**3 - 5120.0*P**3*S*T*UGA_star**2*epsilon - 512.0*P**3*S*T*UGA_star**2 - 2560.0*P**3*T**2*UGA_star**10*epsilon + 20480.0*P**3*T**2*UGA_star**9*epsilon - 71680.0*P**3*T**2*UGA_star**8*epsilon + 143360.0*P**3*T**2*UGA_star**7*epsilon - 179200.0*P**3*T**2*UGA_star**6*epsilon + 143360.0*P**3*T**2*UGA_star**5*epsilon - 71680.0*P**3*T**2*UGA_star**4*epsilon + 20480.0*P**3*T**2*UGA_star**3*epsilon - 2560.0*P**3*T**2*UGA_star**2*epsilon - 2560.0*P**2*R**3*UGA_star**10*epsilon - 384.0*P**2*R**3*UGA_star**10 + 10240.0*P**2*R**3*UGA_star**9*epsilon + 1536.0*P**2*R**3*UGA_star**9 - 15360.0*P**2*R**3*UGA_star**8*epsilon - 2304.0*P**2*R**3*UGA_star**8 + 10240.0*P**2*R**3*UGA_star**7*epsilon + 1536.0*P**2*R**3*UGA_star**7 - 2560.0*P**2*R**3*UGA_star**6*epsilon - 384.0*P**2*R**3*UGA_star**6 + 7680.0*P**2*R**2*S*UGA_star**10*epsilon + 1152.0*P**2*R**2*S*UGA_star**10 - 38400.0*P**2*R**2*S*UGA_star**9*epsilon - 5760.0*P**2*R**2*S*UGA_star**9 + 76800.0*P**2*R**2*S*UGA_star**8*epsilon + 11520.0*P**2*R**2*S*UGA_star**8 - 76800.0*P**2*R**2*S*UGA_star**7*epsilon - 11520.0*P**2*R**2*S*UGA_star**7 + 38400.0*P**2*R**2*S*UGA_star**6*epsilon + 5760.0*P**2*R**2*S*UGA_star**6 - 7680.0*P**2*R**2*S*UGA_star**5*epsilon - 1152.0*P**2*R**2*S*UGA_star**5 + 7680.0*P**2*R**2*T*UGA_star**10*epsilon + 1152.0*P**2*R**2*T*UGA_star**10 - 38400.0*P**2*R**2*T*UGA_star**9*epsilon - 5760.0*P**2*R**2*T*UGA_star**9 + 76800.0*P**2*R**2*T*UGA_star**8*epsilon + 11520.0*P**2*R**2*T*UGA_star**8 - 76800.0*P**2*R**2*T*UGA_star**7*epsilon - 11520.0*P**2*R**2*T*UGA_star**7 + 38400.0*P**2*R**2*T*UGA_star**6*epsilon + 5760.0*P**2*R**2*T*UGA_star**6 - 7680.0*P**2*R**2*T*UGA_star**5*epsilon - 1152.0*P**2*R**2*T*UGA_star**5 - 7680.0*P**2*R*S**2*UGA_star**10*epsilon - 1152.0*P**2*R*S**2*UGA_star**10 + 46080.0*P**2*R*S**2*UGA_star**9*epsilon + 6912.0*P**2*R*S**2*UGA_star**9 - 115200.0*P**2*R*S**2*UGA_star**8*epsilon - 17280.0*P**2*R*S**2*UGA_star**8 + 153600.0*P**2*R*S**2*UGA_star**7*epsilon + 23040.0*P**2*R*S**2*UGA_star**7 - 115200.0*P**2*R*S**2*UGA_star**6*epsilon - 17280.0*P**2*R*S**2*UGA_star**6 + 46080.0*P**2*R*S**2*UGA_star**5*epsilon + 6912.0*P**2*R*S**2*UGA_star**5 - 7680.0*P**2*R*S**2*UGA_star**4*epsilon - 1152.0*P**2*R*S**2*UGA_star**4 - 15360.0*P**2*R*S*T*UGA_star**10*epsilon - 2304.0*P**2*R*S*T*UGA_star**10 + 92160.0*P**2*R*S*T*UGA_star**9*epsilon + 13824.0*P**2*R*S*T*UGA_star**9 - 230400.0*P**2*R*S*T*UGA_star**8*epsilon - 34560.0*P**2*R*S*T*UGA_star**8 + 307200.0*P**2*R*S*T*UGA_star**7*epsilon + 46080.0*P**2*R*S*T*UGA_star**7 - 230400.0*P**2*R*S*T*UGA_star**6*epsilon - 34560.0*P**2*R*S*T*UGA_star**6 + 92160.0*P**2*R*S*T*UGA_star**5*epsilon + 13824.0*P**2*R*S*T*UGA_star**5 - 15360.0*P**2*R*S*T*UGA_star**4*epsilon - 2304.0*P**2*R*S*T*UGA_star**4 - 7680.0*P**2*R*T**2*UGA_star**10*epsilon - 768.0*P**2*R*T**2*UGA_star**10 + 46080.0*P**2*R*T**2*UGA_star**9*epsilon + 4608.0*P**2*R*T**2*UGA_star**9 - 115200.0*P**2*R*T**2*UGA_star**8*epsilon - 11520.0*P**2*R*T**2*UGA_star**8 + 153600.0*P**2*R*T**2*UGA_star**7*epsilon + 15360.0*P**2*R*T**2*UGA_star**7 - 115200.0*P**2*R*T**2*UGA_star**6*epsilon - 11520.0*P**2*R*T**2*UGA_star**6 + 46080.0*P**2*R*T**2*UGA_star**5*epsilon + 4608.0*P**2*R*T**2*UGA_star**5 - 7680.0*P**2*R*T**2*UGA_star**4*epsilon - 768.0*P**2*R*T**2*UGA_star**4 + 2560.0*P**2*S**3*UGA_star**10*epsilon + 384.0*P**2*S**3*UGA_star**10 - 17920.0*P**2*S**3*UGA_star**9*epsilon - 2688.0*P**2*S**3*UGA_star**9 + 53760.0*P**2*S**3*UGA_star**8*epsilon + 8064.0*P**2*S**3*UGA_star**8 - 89600.0*P**2*S**3*UGA_star**7*epsilon - 13440.0*P**2*S**3*UGA_star**7 + 89600.0*P**2*S**3*UGA_star**6*epsilon + 13440.0*P**2*S**3*UGA_star**6 - 53760.0*P**2*S**3*UGA_star**5*epsilon - 8064.0*P**2*S**3*UGA_star**5 + 17920.0*P**2*S**3*UGA_star**4*epsilon + 2688.0*P**2*S**3*UGA_star**4 - 2560.0*P**2*S**3*UGA_star**3*epsilon - 384.0*P**2*S**3*UGA_star**3 + 7680.0*P**2*S**2*T*UGA_star**10*epsilon + 1152.0*P**2*S**2*T*UGA_star**10 - 53760.0*P**2*S**2*T*UGA_star**9*epsilon - 8064.0*P**2*S**2*T*UGA_star**9 + 161280.0*P**2*S**2*T*UGA_star**8*epsilon + 24192.0*P**2*S**2*T*UGA_star**8 - 268800.0*P**2*S**2*T*UGA_star**7*epsilon - 40320.0*P**2*S**2*T*UGA_star**7 + 268800.0*P**2*S**2*T*UGA_star**6*epsilon + 40320.0*P**2*S**2*T*UGA_star**6 - 161280.0*P**2*S**2*T*UGA_star**5*epsilon - 24192.0*P**2*S**2*T*UGA_star**5 + 53760.0*P**2*S**2*T*UGA_star**4*epsilon + 8064.0*P**2*S**2*T*UGA_star**4 - 7680.0*P**2*S**2*T*UGA_star**3*epsilon - 1152.0*P**2*S**2*T*UGA_star**3 + 7680.0*P**2*S*T**2*UGA_star**10*epsilon + 768.0*P**2*S*T**2*UGA_star**10 - 53760.0*P**2*S*T**2*UGA_star**9*epsilon - 5376.0*P**2*S*T**2*UGA_star**9 + 161280.0*P**2*S*T**2*UGA_star**8*epsilon + 16128.0*P**2*S*T**2*UGA_star**8 - 268800.0*P**2*S*T**2*UGA_star**7*epsilon - 26880.0*P**2*S*T**2*UGA_star**7 + 268800.0*P**2*S*T**2*UGA_star**6*epsilon + 26880.0*P**2*S*T**2*UGA_star**6 - 161280.0*P**2*S*T**2*UGA_star**5*epsilon - 16128.0*P**2*S*T**2*UGA_star**5 + 53760.0*P**2*S*T**2*UGA_star**4*epsilon + 5376.0*P**2*S*T**2*UGA_star**4 - 7680.0*P**2*S*T**2*UGA_star**3*epsilon - 768.0*P**2*S*T**2*UGA_star**3 + 2560.0*P**2*T**3*UGA_star**10*epsilon - 17920.0*P**2*T**3*UGA_star**9*epsilon + 53760.0*P**2*T**3*UGA_star**8*epsilon - 89600.0*P**2*T**3*UGA_star**7*epsilon + 89600.0*P**2*T**3*UGA_star**6*epsilon - 53760.0*P**2*T**3*UGA_star**5*epsilon + 17920.0*P**2*T**3*UGA_star**4*epsilon - 2560.0*P**2*T**3*UGA_star**3*epsilon - 1280.0*P*R**4*UGA_star**10*epsilon - 128.0*P*R**4*UGA_star**10 + 2560.0*P*R**4*UGA_star**9*epsilon + 256.0*P*R**4*UGA_star**9 - 1280.0*P*R**4*UGA_star**8*epsilon - 128.0*P*R**4*UGA_star**8 + 5120.0*P*R**3*S*UGA_star**10*epsilon + 512.0*P*R**3*S*UGA_star**10 - 15360.0*P*R**3*S*UGA_star**9*epsilon - 1536.0*P*R**3*S*UGA_star**9 + 15360.0*P*R**3*S*UGA_star**8*epsilon + 1536.0*P*R**3*S*UGA_star**8 - 5120.0*P*R**3*S*UGA_star**7*epsilon - 512.0*P*R**3*S*UGA_star**7 + 5120.0*P*R**3*T*UGA_star**10*epsilon + 768.0*P*R**3*T*UGA_star**10 - 15360.0*P*R**3*T*UGA_star**9*epsilon - 2304.0*P*R**3*T*UGA_star**9 + 15360.0*P*R**3*T*UGA_star**8*epsilon + 2304.0*P*R**3*T*UGA_star**8 - 5120.0*P*R**3*T*UGA_star**7*epsilon - 768.0*P*R**3*T*UGA_star**7 - 7680.0*P*R**2*S**2*UGA_star**10*epsilon - 768.0*P*R**2*S**2*UGA_star**10 + 30720.0*P*R**2*S**2*UGA_star**9*epsilon + 3072.0*P*R**2*S**2*UGA_star**9 - 46080.0*P*R**2*S**2*UGA_star**8*epsilon - 4608.0*P*R**2*S**2*UGA_star**8 + 30720.0*P*R**2*S**2*UGA_star**7*epsilon + 3072.0*P*R**2*S**2*UGA_star**7 - 7680.0*P*R**2*S**2*UGA_star**6*epsilon - 768.0*P*R**2*S**2*UGA_star**6 - 15360.0*P*R**2*S*T*UGA_star**10*epsilon - 2304.0*P*R**2*S*T*UGA_star**10 + 61440.0*P*R**2*S*T*UGA_star**9*epsilon + 9216.0*P*R**2*S*T*UGA_star**9 - 92160.0*P*R**2*S*T*UGA_star**8*epsilon - 13824.0*P*R**2*S*T*UGA_star**8 + 61440.0*P*R**2*S*T*UGA_star**7*epsilon + 9216.0*P*R**2*S*T*UGA_star**7 - 15360.0*P*R**2*S*T*UGA_star**6*epsilon - 2304.0*P*R**2*S*T*UGA_star**6 - 7680.0*P*R**2*T**2*UGA_star**10*epsilon - 1152.0*P*R**2*T**2*UGA_star**10 + 30720.0*P*R**2*T**2*UGA_star**9*epsilon + 4608.0*P*R**2*T**2*UGA_star**9 - 46080.0*P*R**2*T**2*UGA_star**8*epsilon - 6912.0*P*R**2*T**2*UGA_star**8 + 30720.0*P*R**2*T**2*UGA_star**7*epsilon + 4608.0*P*R**2*T**2*UGA_star**7 - 7680.0*P*R**2*T**2*UGA_star**6*epsilon - 1152.0*P*R**2*T**2*UGA_star**6 + 5120.0*P*R*S**3*UGA_star**10*epsilon + 512.0*P*R*S**3*UGA_star**10 - 25600.0*P*R*S**3*UGA_star**9*epsilon - 2560.0*P*R*S**3*UGA_star**9 + 51200.0*P*R*S**3*UGA_star**8*epsilon + 5120.0*P*R*S**3*UGA_star**8 - 51200.0*P*R*S**3*UGA_star**7*epsilon - 5120.0*P*R*S**3*UGA_star**7 + 25600.0*P*R*S**3*UGA_star**6*epsilon + 2560.0*P*R*S**3*UGA_star**6 - 5120.0*P*R*S**3*UGA_star**5*epsilon - 512.0*P*R*S**3*UGA_star**5 + 15360.0*P*R*S**2*T*UGA_star**10*epsilon + 2304.0*P*R*S**2*T*UGA_star**10 - 76800.0*P*R*S**2*T*UGA_star**9*epsilon - 11520.0*P*R*S**2*T*UGA_star**9 + 153600.0*P*R*S**2*T*UGA_star**8*epsilon + 23040.0*P*R*S**2*T*UGA_star**8 - 153600.0*P*R*S**2*T*UGA_star**7*epsilon - 23040.0*P*R*S**2*T*UGA_star**7 + 76800.0*P*R*S**2*T*UGA_star**6*epsilon + 11520.0*P*R*S**2*T*UGA_star**6 - 15360.0*P*R*S**2*T*UGA_star**5*epsilon - 2304.0*P*R*S**2*T*UGA_star**5 + 15360.0*P*R*S*T**2*UGA_star**10*epsilon + 2304.0*P*R*S*T**2*UGA_star**10 - 76800.0*P*R*S*T**2*UGA_star**9*epsilon - 11520.0*P*R*S*T**2*UGA_star**9 + 153600.0*P*R*S*T**2*UGA_star**8*epsilon + 23040.0*P*R*S*T**2*UGA_star**8 - 153600.0*P*R*S*T**2*UGA_star**7*epsilon - 23040.0*P*R*S*T**2*UGA_star**7 + 76800.0*P*R*S*T**2*UGA_star**6*epsilon + 11520.0*P*R*S*T**2*UGA_star**6 - 15360.0*P*R*S*T**2*UGA_star**5*epsilon - 2304.0*P*R*S*T**2*UGA_star**5 + 5120.0*P*R*T**3*UGA_star**10*epsilon + 512.0*P*R*T**3*UGA_star**10 - 25600.0*P*R*T**3*UGA_star**9*epsilon - 2560.0*P*R*T**3*UGA_star**9 + 51200.0*P*R*T**3*UGA_star**8*epsilon + 5120.0*P*R*T**3*UGA_star**8 - 51200.0*P*R*T**3*UGA_star**7*epsilon - 5120.0*P*R*T**3*UGA_star**7 + 25600.0*P*R*T**3*UGA_star**6*epsilon + 2560.0*P*R*T**3*UGA_star**6 - 5120.0*P*R*T**3*UGA_star**5*epsilon - 512.0*P*R*T**3*UGA_star**5 - 1280.0*P*S**4*UGA_star**10*epsilon - 128.0*P*S**4*UGA_star**10 + 7680.0*P*S**4*UGA_star**9*epsilon + 768.0*P*S**4*UGA_star**9 - 19200.0*P*S**4*UGA_star**8*epsilon - 1920.0*P*S**4*UGA_star**8 + 25600.0*P*S**4*UGA_star**7*epsilon + 2560.0*P*S**4*UGA_star**7 - 19200.0*P*S**4*UGA_star**6*epsilon - 1920.0*P*S**4*UGA_star**6 + 7680.0*P*S**4*UGA_star**5*epsilon + 768.0*P*S**4*UGA_star**5 - 1280.0*P*S**4*UGA_star**4*epsilon - 128.0*P*S**4*UGA_star**4 - 5120.0*P*S**3*T*UGA_star**10*epsilon - 768.0*P*S**3*T*UGA_star**10 + 30720.0*P*S**3*T*UGA_star**9*epsilon + 4608.0*P*S**3*T*UGA_star**9 - 76800.0*P*S**3*T*UGA_star**8*epsilon - 11520.0*P*S**3*T*UGA_star**8 + 102400.0*P*S**3*T*UGA_star**7*epsilon + 15360.0*P*S**3*T*UGA_star**7 - 76800.0*P*S**3*T*UGA_star**6*epsilon - 11520.0*P*S**3*T*UGA_star**6 + 30720.0*P*S**3*T*UGA_star**5*epsilon + 4608.0*P*S**3*T*UGA_star**5 - 5120.0*P*S**3*T*UGA_star**4*epsilon - 768.0*P*S**3*T*UGA_star**4 - 7680.0*P*S**2*T**2*UGA_star**10*epsilon - 1152.0*P*S**2*T**2*UGA_star**10 + 46080.0*P*S**2*T**2*UGA_star**9*epsilon + 6912.0*P*S**2*T**2*UGA_star**9 - 115200.0*P*S**2*T**2*UGA_star**8*epsilon - 17280.0*P*S**2*T**2*UGA_star**8 + 153600.0*P*S**2*T**2*UGA_star**7*epsilon + 23040.0*P*S**2*T**2*UGA_star**7 - 115200.0*P*S**2*T**2*UGA_star**6*epsilon - 17280.0*P*S**2*T**2*UGA_star**6 + 46080.0*P*S**2*T**2*UGA_star**5*epsilon + 6912.0*P*S**2*T**2*UGA_star**5 - 7680.0*P*S**2*T**2*UGA_star**4*epsilon - 1152.0*P*S**2*T**2*UGA_star**4 - 5120.0*P*S*T**3*UGA_star**10*epsilon - 512.0*P*S*T**3*UGA_star**10 + 30720.0*P*S*T**3*UGA_star**9*epsilon + 3072.0*P*S*T**3*UGA_star**9 - 76800.0*P*S*T**3*UGA_star**8*epsilon - 7680.0*P*S*T**3*UGA_star**8 + 102400.0*P*S*T**3*UGA_star**7*epsilon + 10240.0*P*S*T**3*UGA_star**7 - 76800.0*P*S*T**3*UGA_star**6*epsilon - 7680.0*P*S*T**3*UGA_star**6 + 30720.0*P*S*T**3*UGA_star**5*epsilon + 3072.0*P*S*T**3*UGA_star**5 - 5120.0*P*S*T**3*UGA_star**4*epsilon - 512.0*P*S*T**3*UGA_star**4 - 1280.0*P*T**4*UGA_star**10*epsilon + 7680.0*P*T**4*UGA_star**9*epsilon - 19200.0*P*T**4*UGA_star**8*epsilon + 25600.0*P*T**4*UGA_star**7*epsilon - 19200.0*P*T**4*UGA_star**6*epsilon + 7680.0*P*T**4*UGA_star**5*epsilon - 1280.0*P*T**4*UGA_star**4*epsilon - 256.0*R**5*UGA_star**10*epsilon + 1280.0*R**4*S*UGA_star**10*epsilon - 1280.0*R**4*S*UGA_star**9*epsilon + 1280.0*R**4*T*UGA_star**10*epsilon + 128.0*R**4*T*UGA_star**10 - 1280.0*R**4*T*UGA_star**9*epsilon - 128.0*R**4*T*UGA_star**9 - 2560.0*R**3*S**2*UGA_star**10*epsilon + 5120.0*R**3*S**2*UGA_star**9*epsilon - 2560.0*R**3*S**2*UGA_star**8*epsilon - 5120.0*R**3*S*T*UGA_star**10*epsilon - 512.0*R**3*S*T*UGA_star**10 + 10240.0*R**3*S*T*UGA_star**9*epsilon + 1024.0*R**3*S*T*UGA_star**9 - 5120.0*R**3*S*T*UGA_star**8*epsilon - 512.0*R**3*S*T*UGA_star**8 - 2560.0*R**3*T**2*UGA_star**10*epsilon - 384.0*R**3*T**2*UGA_star**10 + 5120.0*R**3*T**2*UGA_star**9*epsilon + 768.0*R**3*T**2*UGA_star**9 - 2560.0*R**3*T**2*UGA_star**8*epsilon - 384.0*R**3*T**2*UGA_star**8 + 2560.0*R**2*S**3*UGA_star**10*epsilon - 7680.0*R**2*S**3*UGA_star**9*epsilon + 7680.0*R**2*S**3*UGA_star**8*epsilon - 2560.0*R**2*S**3*UGA_star**7*epsilon + 7680.0*R**2*S**2*T*UGA_star**10*epsilon + 768.0*R**2*S**2*T*UGA_star**10 - 23040.0*R**2*S**2*T*UGA_star**9*epsilon - 2304.0*R**2*S**2*T*UGA_star**9 + 23040.0*R**2*S**2*T*UGA_star**8*epsilon + 2304.0*R**2*S**2*T*UGA_star**8 - 7680.0*R**2*S**2*T*UGA_star**7*epsilon - 768.0*R**2*S**2*T*UGA_star**7 + 7680.0*R**2*S*T**2*UGA_star**10*epsilon + 1152.0*R**2*S*T**2*UGA_star**10 - 23040.0*R**2*S*T**2*UGA_star**9*epsilon - 3456.0*R**2*S*T**2*UGA_star**9 + 23040.0*R**2*S*T**2*UGA_star**8*epsilon + 3456.0*R**2*S*T**2*UGA_star**8 - 7680.0*R**2*S*T**2*UGA_star**7*epsilon - 1152.0*R**2*S*T**2*UGA_star**7 + 2560.0*R**2*T**3*UGA_star**10*epsilon + 384.0*R**2*T**3*UGA_star**10 - 7680.0*R**2*T**3*UGA_star**9*epsilon - 1152.0*R**2*T**3*UGA_star**9 + 7680.0*R**2*T**3*UGA_star**8*epsilon + 1152.0*R**2*T**3*UGA_star**8 - 2560.0*R**2*T**3*UGA_star**7*epsilon - 384.0*R**2*T**3*UGA_star**7 - 1280.0*R*S**4*UGA_star**10*epsilon + 5120.0*R*S**4*UGA_star**9*epsilon - 7680.0*R*S**4*UGA_star**8*epsilon + 5120.0*R*S**4*UGA_star**7*epsilon - 1280.0*R*S**4*UGA_star**6*epsilon - 5120.0*R*S**3*T*UGA_star**10*epsilon - 512.0*R*S**3*T*UGA_star**10 + 20480.0*R*S**3*T*UGA_star**9*epsilon + 2048.0*R*S**3*T*UGA_star**9 - 30720.0*R*S**3*T*UGA_star**8*epsilon - 3072.0*R*S**3*T*UGA_star**8 + 20480.0*R*S**3*T*UGA_star**7*epsilon + 2048.0*R*S**3*T*UGA_star**7 - 5120.0*R*S**3*T*UGA_star**6*epsilon - 512.0*R*S**3*T*UGA_star**6 - 7680.0*R*S**2*T**2*UGA_star**10*epsilon - 1152.0*R*S**2*T**2*UGA_star**10 + 30720.0*R*S**2*T**2*UGA_star**9*epsilon + 4608.0*R*S**2*T**2*UGA_star**9 - 46080.0*R*S**2*T**2*UGA_star**8*epsilon - 6912.0*R*S**2*T**2*UGA_star**8 + 30720.0*R*S**2*T**2*UGA_star**7*epsilon + 4608.0*R*S**2*T**2*UGA_star**7 - 7680.0*R*S**2*T**2*UGA_star**6*epsilon - 1152.0*R*S**2*T**2*UGA_star**6 - 5120.0*R*S*T**3*UGA_star**10*epsilon - 768.0*R*S*T**3*UGA_star**10 + 20480.0*R*S*T**3*UGA_star**9*epsilon + 3072.0*R*S*T**3*UGA_star**9 - 30720.0*R*S*T**3*UGA_star**8*epsilon - 4608.0*R*S*T**3*UGA_star**8 + 20480.0*R*S*T**3*UGA_star**7*epsilon + 3072.0*R*S*T**3*UGA_star**7 - 5120.0*R*S*T**3*UGA_star**6*epsilon - 768.0*R*S*T**3*UGA_star**6 - 1280.0*R*T**4*UGA_star**10*epsilon - 128.0*R*T**4*UGA_star**10 + 5120.0*R*T**4*UGA_star**9*epsilon + 512.0*R*T**4*UGA_star**9 - 7680.0*R*T**4*UGA_star**8*epsilon - 768.0*R*T**4*UGA_star**8 + 5120.0*R*T**4*UGA_star**7*epsilon + 512.0*R*T**4*UGA_star**7 - 1280.0*R*T**4*UGA_star**6*epsilon - 128.0*R*T**4*UGA_star**6 + 256.0*S**5*UGA_star**10*epsilon - 1280.0*S**5*UGA_star**9*epsilon + 2560.0*S**5*UGA_star**8*epsilon - 2560.0*S**5*UGA_star**7*epsilon + 1280.0*S**5*UGA_star**6*epsilon - 256.0*S**5*UGA_star**5*epsilon + 1280.0*S**4*T*UGA_star**10*epsilon + 128.0*S**4*T*UGA_star**10 - 6400.0*S**4*T*UGA_star**9*epsilon - 640.0*S**4*T*UGA_star**9 + 12800.0*S**4*T*UGA_star**8*epsilon + 1280.0*S**4*T*UGA_star**8 - 12800.0*S**4*T*UGA_star**7*epsilon - 1280.0*S**4*T*UGA_star**7 + 6400.0*S**4*T*UGA_star**6*epsilon + 640.0*S**4*T*UGA_star**6 - 1280.0*S**4*T*UGA_star**5*epsilon - 128.0*S**4*T*UGA_star**5 + 2560.0*S**3*T**2*UGA_star**10*epsilon + 384.0*S**3*T**2*UGA_star**10 - 12800.0*S**3*T**2*UGA_star**9*epsilon - 1920.0*S**3*T**2*UGA_star**9 + 25600.0*S**3*T**2*UGA_star**8*epsilon + 3840.0*S**3*T**2*UGA_star**8 - 25600.0*S**3*T**2*UGA_star**7*epsilon - 3840.0*S**3*T**2*UGA_star**7 + 12800.0*S**3*T**2*UGA_star**6*epsilon + 1920.0*S**3*T**2*UGA_star**6 - 2560.0*S**3*T**2*UGA_star**5*epsilon - 384.0*S**3*T**2*UGA_star**5 + 2560.0*S**2*T**3*UGA_star**10*epsilon + 384.0*S**2*T**3*UGA_star**10 - 12800.0*S**2*T**3*UGA_star**9*epsilon - 1920.0*S**2*T**3*UGA_star**9 + 25600.0*S**2*T**3*UGA_star**8*epsilon + 3840.0*S**2*T**3*UGA_star**8 - 25600.0*S**2*T**3*UGA_star**7*epsilon - 3840.0*S**2*T**3*UGA_star**7 + 12800.0*S**2*T**3*UGA_star**6*epsilon + 1920.0*S**2*T**3*UGA_star**6 - 2560.0*S**2*T**3*UGA_star**5*epsilon - 384.0*S**2*T**3*UGA_star**5 + 1280.0*S*T**4*UGA_star**10*epsilon + 128.0*S*T**4*UGA_star**10 - 6400.0*S*T**4*UGA_star**9*epsilon - 640.0*S*T**4*UGA_star**9 + 12800.0*S*T**4*UGA_star**8*epsilon + 1280.0*S*T**4*UGA_star**8 - 12800.0*S*T**4*UGA_star**7*epsilon - 1280.0*S*T**4*UGA_star**7 + 6400.0*S*T**4*UGA_star**6*epsilon + 640.0*S*T**4*UGA_star**6 - 1280.0*S*T**4*UGA_star**5*epsilon - 128.0*S*T**4*UGA_star**5 + 256.0*T**5*UGA_star**10*epsilon - 1280.0*T**5*UGA_star**9*epsilon + 2560.0*T**5*UGA_star**8*epsilon - 2560.0*T**5*UGA_star**7*epsilon + 1280.0*T**5*UGA_star**6*epsilon - 256.0*T**5*UGA_star**5*epsilon)

```python
numerator, denominator = sym.fraction(sym.factor(e2))
```

```python
denominator
```

    8*x1*(x1 - 1)*(P*UGA_star**2 - 2*P*UGA_star + P + R*UGA_star**2 - S*UGA_star**2 + S*UGA_star - T*UGA_star**2 + T*UGA_star)

```python
# vectorized numeric repr for the eigenvalue
_numeric_e3 = sym.lambdify((UGA_star, UgA_star, T, R, P, S, epsilon),
                           e3.subs({x1: x1_star}),
                           modules="numpy")
```

```python
def plot_third_eigenvalue(T, R, P, S, epsilon):

    fig, ax = plt.subplots(1, 1, figsize=(10, 10))
    ax.set_ylabel(r"$U_{GA}^*$", fontsize=20, rotation="horizontal")
    ax.set_xlabel(r"$U_{gA}^{*}$", fontsize=20)
    ax.set_title(r"Eigenvalue, $e_3$", fontsize=25)
    ax.grid("off")

    equilibrium_selection_probs = np.linspace(0, 1, 100)
    UgAs, UGAs = np.meshgrid(equilibrium_selection_probs, equilibrium_selection_probs)
    Z = _numeric_e3(UGAs, UgAs, T, R, P, S, epsilon)
    cax = ax.imshow(Z, origin="lower")
    contours = ax.contour(Z, colors='w', origin='lower')
    ax.clabel(contours, contours.levels, inline=True, fontsize=10)

    # adjust the tick labels
    locs, _ = plt.xticks()
    plt.xticks(locs[1:], np.linspace(0, 1, locs.size-1))
    locs, _ = plt.yticks()
    plt.yticks(locs[1:], np.linspace(0, 1, locs.size-1))

    plt.show()
```

```python
mpld3.disable_notebook()  # don't need interactive plotting for this!

# sliders used to control the Prisoner's Dilemma Payoffs
T_slider = widgets.FloatSlider(value=25, min=0, max=100, step=0.1, description=r"$T$")
R_slider = widgets.FloatSlider(value=3, min=0, max=100, step=0.1, description=r"$R$")
P_slider = widgets.FloatSlider(value=2, min=0, max=100, step=0.1, description=r"$P$")
S_slider = widgets.FloatSlider(value=1, min=0, max=100, step=0.1, description=r"$S$")

# slider used to control the mutation rate
e_slider = widgets.FloatSlider(value=0.0, min=0.0, max=1.0, step=1e-3, description=r"$\epsilon$", readout_format=".3f")

w = widgets.interactive(plot_third_eigenvalue, T=T_slider, R=R_slider, P=P_slider, S=S_slider, epsilon=e_slider)
display(w)
```

    A Jupyter Widget

```python
e2 == e3
```

    False

```python
sym.simplify(e2.subs({x1: x1_star, UgA_star: x1_star}).subs({UGA_star: 1, UGA_prime_star: 0})).subs({epsilon**2: 0})
```

    (-13.0*R*epsilon - 1.5*R + 3.0*S*epsilon + 3.0*T*epsilon + 0.015625*sqrt(61440.0*R**2*epsilon + 9216.0*R**2 + 12288.0*R*S*epsilon - 4096.0*R*T*epsilon))/(R*(16.0*epsilon + 4.0))

```python
sym.simplify(e3.subs({x1: x1_star, UgA_star: x1_star}).subs({UGA_star: 1, UGA_prime_star: 0})).subs({epsilon**2: 0})
```

    (-13.0*R*epsilon - 1.5*R + 3.0*S*epsilon + 3.0*T*epsilon - 0.015625*sqrt(61440.0*R**2*epsilon + 9216.0*R**2 + 12288.0*R*S*epsilon - 4096.0*R*T*epsilon))/(R*(16.0*epsilon + 4.0))

```python
evaluated = (e3.subs({x1: x1_star, UgA_star: x1_star})
               .subs({UGA_star: 1, UGA_prime_star: 0}))
together = sym.together(evaluated)
```

```python
expanded = sym.expand(together)
```

```python
tmp_expr = sym.together(expanded)
```

```python
tmp_expr.subs({epsilon**2: 0})
```

    (-13.0*R*epsilon - 1.5*R + 3.0*S*epsilon + 3.0*T*epsilon + 0.015625*sqrt(61440.0*R**2*epsilon + 9216.0*R**2 + 12288.0*R*S*epsilon - 4096.0*R*T*epsilon))/(R*(16.0*epsilon + 4.0))

```python
(-13.0*R*epsilon - 1.5*R + 3.0*S*epsilon + 3.0*T*epsilon + 0.015625*sqrt(61440.0*R**2*epsilon + 9216.0*R**2 + 12288.0*R*S*epsilon - 4096.0*R*T*epsilon))/(R*(16.0*epsilon + 4.0))
```

```python

```

First eigenvalue can be written as a hyperbola in $U_{gA}^*,U_{GA}^*$ plane.

$$ e_1 = \left(\frac{\big((R + P) - (T + S)\big)U_{gA}^{*2} + \big((T + S) - 2P\big)U_{gA}^*}{N^*}\right) - \left(\frac{\big((R + P) - (T + S)\big)U_{GA}^{*2} + \big((T + S) - 2P\big)U_{GA}^*}{N^*}\right) $$

### Completing the square

To write this hyperbola in standard form we need to [complete the square](https://en.wikipedia.org/wiki/Completing_the_square). Completing the square for the quadratic polynomial in $U_{GA}(x^*)$ yields the following.

\begin{align}
    \big((R + P) - (T + S)\big)\left(U_{GA}^* - \bar{U}_{GA}^*\right)^2 - \frac{1}{2}\big(\bar{N}^* - 2P\big) \\
\end{align}

where $\bar{U}_{GA}^*$ is the value of $U_{GA}^*$ that maximizes total offspring $N$ derived above. Completing the square for the quadratic polynomial in $U_{gA}(x^*)$ yields a similar expression.

\begin{align}
    \big((R + P) - (T + S)\big)\left(U_{gA}^* - \bar{U}_{GA}^*\right)^2 - \frac{1}{2}\big(\bar{N}^* - 2P\big) \\
\end{align}

Substituting these results into the expression for the eigenvalue yields the following.

$$ e_1 = \left(\frac{\big(U_{gA}^* - \bar{U}_{GA}^*\big)^{2}}{\frac{N^*}{\big((R + P) - (T + S)\big)}}\right) - \left(\frac{\big(U_{GA}^* - \bar{U}_{GA}^*\big)^{2}}{\frac{N^*}{\big((R + P) - (T + S)\big)}}\right) $$

### Conditions for negative eigenvalue

Non-invadability requires that this eigenvalue is strictly negative.

$$ \left(\frac{\big(U_{gA}^* - \bar{U}_{GA}^*\big)^{2}}{\frac{N^*}{\big((R + P) - (T + S)\big)}}\right) - \left(\frac{\big(U_{GA}^* - \bar{U}_{GA}^*\big)^{2}}{\frac{N^*}{\big((R + P) - (T + S)\big)}}\right) < 0$$

#### Case: (R + P) < (T + S)
In this case the inequality simplifies to the following.

$$ \big(U_{gA}^* - \bar{U}_{GA}^*\big)^{2} - \big(U_{GA}^* - \bar{U}_{GA}^*\big)^{2} > 0 $$

This implies that we have four sub-cases to consider.

\begin{align}
    \big(U_{gA}^* - \bar{U}_{GA}^*\big) >& \big(U_{GA}^* - \bar{U}_{GA}^*\big) \\
    \big(U_{gA}^* - \bar{U}_{GA}^*\big) >& -\big(U_{GA}^* - \bar{U}_{GA}^*\big) \\
    -\big(U_{gA}^* - \bar{U}_{GA}^*\big) >& \big(U_{GA}^* - \bar{U}_{GA}^*\big) \\
    -\big(U_{gA}^* - \bar{U}_{GA}^*\big) >& -\big(U_{GA}^* - \bar{U}_{GA}^*\big)
\end{align}

These four sub-cases define the following two regions of the $U_{gA}^*,U_{GA}^*$ plane where the eigenvalue is negative.

\begin{align}
   & 2\big(\bar{U}_{GA}^* - U_{gA}^*\big) < U_{GA}^* - U_{gA}^* < 0\\
   & 0 < U_{GA}^* - U_{gA}^* < 2\big(\bar{U}_{GA}^* - U_{gA}\big)
\end{align}

#### Case: (R + P) > (T + S)
In this case the inequality simplifies to the following.

$$ \big(U_{gA}^* - \bar{U}_{GA}^*\big)^{2} - \big(U_{GA}^* - \bar{U}_{GA}^*\big)^{2} < 0 $$

This implies that we again have four sub-cases to consider.

\begin{align}
    \big(U_{gA}^* - \bar{U}_{GA}^*\big) <& \big(U_{GA}^* - \bar{U}_{GA}^*\big) \\
    \big(U_{gA}^* - \bar{U}_{GA}^*\big) <& -\big(U_{GA}^* - \bar{U}_{GA}^*\big) \\
    -\big(U_{gA}^* - \bar{U}_{GA}^*\big) <& \big(U_{GA}^* - \bar{U}_{GA}^*\big) \\
    -\big(U_{gA}^* - \bar{U}_{GA}^*\big) <& -\big(U_{GA}^* - \bar{U}_{GA}^*\big)
\end{align}

These four sub-cases define the following two regions of the $U_{gA}^*,U_{GA}^*$ plane where the eigenvalue is negative.

\begin{align}
   & 2\big(\bar{U}_{GA}^* - U_{gA}^*\big) < U_{GA}^* - U_{gA}^* < 0\\
   & 0 < U_{GA}^* - U_{gA}^* < 2\big(\bar{U}_{GA}^* - U_{gA}\big)
\end{align}

### Asymptotes

Asymptotes of this hyperbola can be defined as follows.

$$ U_{GA}^* - \bar{U}_{GA}^* = \pm\big(U_{gA}^* - \bar{U}_{GA}^*\big) $$

The above can be written as linear equations in $U_{gA}^*,U_{GA}^*$ plane.

\begin{align}
    U_{GA}^* =& U_{gA}^* \\
    U_{GA}^* =& 2\bar{U}_{GA}^* - U_{gA}^*
\end{align}

Note that the asymptotes of this hyperbola correspond to the locus of points for which the value of the first eigenvalue is exactly zero.

#### Discussion

Suppose $U_{GA}^* = \bar{U}_{GA}^*$, then both of the sets of inequalities will hold for any value of $U_{gA}^*$. This implies that the eigenvalue will be negative for any value of $U_{gA}^*$.

```python
# vectorized numeric repr for the eigenvalue
_numeric_e1 = sym.lambdify((UGA_star, UGA_prime_star, T, R, P, S, M), e1, modules="numpy")
```

```python
def plot_first_eigenvalue(T, R, P, S, M):

    fig, ax = plt.subplots(1, 1, figsize=(10, 10))
    ax.set_ylabel(r"$U_{GA}^*$", fontsize=20, rotation="horizontal")
    ax.set_xlabel(r"$U_{GA}^{'*}$", fontsize=20)
    ax.set_title(r"Eigenvalue, $e_1$", fontsize=25)
    ax.grid("off")

    equilibrium_selection_probs = np.linspace(0, 1, 100).reshape(-1, 1)
    equilibrium_selection_derivs = np.linspace(0, 10, 1000).reshape(1, -1) # sensible UGA'(0) >= 1
    Z = _numeric_e1(equilibrium_selection_probs, equilibrium_selection_derivs, T, R, P, S, M)
    cax = ax.imshow(Z, origin="lower", aspect="auto", vmin=-0.5, vmax=1.0)

    levels = np.arange(-0.25, 1.25, 0.25)
    contours = ax.contour(Z, levels=levels, colors='w', origin='lower')
    ax.clabel(contours, contours.levels, inline=True, fontsize=10)

    # adjust the tick labels
    locs, _ = plt.xticks()
    plt.xticks(locs[1:], np.linspace(0, 10, locs.size-1))
    locs, _ = plt.yticks()
    plt.yticks(locs[1:], np.linspace(0, 1, locs.size-1))

    plt.show()
```

```python
mpld3.disable_notebook()  # don't need interactive plotting for this!

# sliders used to control the Prisoner's Dilemma Payoffs
T_slider = widgets.FloatSlider(value=25, min=0, max=100, step=0.1, description=r"$T$")
R_slider = widgets.FloatSlider(value=3, min=0, max=100, step=0.1, description=r"$R$")
P_slider = widgets.FloatSlider(value=2, min=0, max=100, step=0.1, description=r"$P$")
S_slider = widgets.FloatSlider(value=1, min=0, max=100, step=0.1, description=r"$S$")

M_slider = widgets.FloatSlider(value=1, min=0, max=100, step=0.1, description=r"$M$")

w = widgets.interactive(plot_first_eigenvalue, T=T_slider, R=R_slider, P=P_slider, S=S_slider, M=M_slider)
display(w)
```

    A Jupyter Widget

```python
UGA_prime_star, = sym.solve(numerator, UGA_prime_star)
```

```python
sym.factor(UGA_prime_star)
```

    -(P*UGA_star**2 - 2*P*UGA_star + P + R*UGA_star**2 - S*UGA_star**2 + S*UGA_star - T*UGA_star**2 + T*UGA_star)**2/(2*P*R*UGA_star**2 - 2*P*R*UGA_star - P*S*UGA_star**2 + 2*P*S*UGA_star - P*S - R*T*UGA_star**2)

After a bit of fiddling, we arrive at the same stability condition derived above.

\begin{align}
   % U'_{GA}(x^*) < -\frac{(PU_{GA}(x^*)^2 - 2PU_{GA}(x^*) + P + RU_{GA}(x^*)^2 - SU_{GA}(x^*)^2 + SU_{GA}(x^*) - TU_{GA}(x^*)^2 + TU_{GA}(x^*))^2}{(2PRU_{GA}(x^*)^2 - 2PRU_{GA}(x^*) - PSU_{GA}(x^*)^2 + 2PSU_{GA}(x^*) - PS - RTU_{GA}(x^*)^2)} \\
    U'_{GA}(x^*) < \frac{\bigg(\big((R + P) - (T + S)\big)U_{GA}(x^*)^2 + \big((T + S) - 2P\big)U_{GA}(x^*) + P\bigg)^2}{\big(R(T-2P) + PS\big)U_{GA}(x^*)^2 + 2P(R - S)U_{GA}(x^*) + PS}
\end{align}

```python
e2
```

    (-3*M + 3*P*UGA_star**2 - 6*P*UGA_star - P*UgA_star**2 + 2*P*UgA_star + 2*P + 3*R*UGA_star**2 - R*UgA_star**2 - 3*S*UGA_star**2 + 3*S*UGA_star + S*UgA_star**2 - S*UgA_star - 3*T*UGA_star**2 + 3*T*UGA_star + T*UgA_star**2 - T*UgA_star + m)/(4*M - 4*P*UGA_star**2 + 8*P*UGA_star - 4*P - 4*R*UGA_star**2 + 4*S*UGA_star**2 - 4*S*UGA_star + 4*T*UGA_star**2 - 4*T*UGA_star)

$$ e_2 = \left(\frac{\big((R + P) - (T + S)\big)U_{gA}^{*2} + \big((T + S) - 2P\big)U_{gA}^* + P}{4N^*}\right) - \frac{3}{4} $$

### Completing the square

Completing the square for the quadratic polynomial in $U_{gA}(x^*)$ yields the following expression.

\begin{align}
    \big((R + P) - (T + S)\big)\left(U_{gA}^* - \bar{U}_{GA}^*\right)^2 + \frac{1}{2}\bar{N}^* \\
\end{align}

Finally we need to find conditions under which the third eigenvalue is strictly negative.

```python
# vectorized numeric repr for the eigenvalue
_numeric_e2 = sym.lambdify((UGA_star, UgA_star, T, R, P, S, M, m), e2, modules="numpy")
```

```python
def plot_second_eigenvalue(T, R, P, S, M, m):

    fig, ax = plt.subplots(1, 1, figsize=(10, 10))
    ax.set_ylabel(r"$U_{GA}^*$", fontsize=20, rotation="horizontal")
    ax.set_xlabel(r"$U_{gA}^{*}$", fontsize=20)
    ax.set_title(r"Eigenvalue, $e_2$", fontsize=25)
    ax.grid("off")

    equilibrium_selection_probs = np.linspace(0, 1, 100)
    UgAs, UGAs = np.meshgrid(equilibrium_selection_probs, equilibrium_selection_probs)
    Z = _numeric_e2(UGAs, UgAs, T, R, P, S, M, m)
    cax = ax.imshow(Z, origin="lower")
    contours = ax.contour(Z, colors='w', origin='lower')
    ax.clabel(contours, contours.levels, inline=True, fontsize=10)

    # adjust the tick labels
    locs, _ = plt.xticks()
    plt.xticks(locs[1:], np.linspace(0, 1, locs.size-1))
    locs, _ = plt.yticks()
    plt.yticks(locs[1:], np.linspace(0, 1, locs.size-1))

    plt.show()
```

```python
mpld3.disable_notebook()  # don't need interactive plotting for this!

# sliders used to control the Prisoner's Dilemma Payoffs
T_slider = widgets.FloatSlider(value=25, min=0, max=100, step=0.1, description=r"$T$")
R_slider = widgets.FloatSlider(value=3, min=0, max=100, step=0.1, description=r"$R$")
P_slider = widgets.FloatSlider(value=2, min=0, max=100, step=0.1, description=r"$P$")
S_slider = widgets.FloatSlider(value=1, min=0, max=100, step=0.1, description=r"$S$")

M_slider = widgets.FloatSlider(value=1, min=0, max=100, step=0.1, description=r"$M$")
m_slider = widgets.FloatSlider(value=0, min=0, max=100, step=0.1, description=r"$m$")

w = widgets.interactive(plot_second_eigenvalue, T=T_slider, R=R_slider, P=P_slider, S=S_slider, M=M_slider, m=m_slider)
display(w)
```

    A Jupyter Widget

```python
e3
```

    (-M + P*UGA_star**2 - 2*P*UGA_star - P*UgA_star**2 + 2*P*UgA_star + R*UGA_star**2 - R*UgA_star**2 - S*UGA_star**2 + S*UGA_star + S*UgA_star**2 - S*UgA_star - T*UGA_star**2 + T*UGA_star + T*UgA_star**2 - T*UgA_star + m)/(2*M - 2*P*UGA_star**2 + 4*P*UGA_star - 2*P - 2*R*UGA_star**2 + 2*S*UGA_star**2 - 2*S*UGA_star + 2*T*UGA_star**2 - 2*T*UGA_star)

```python
# vectorized numeric repr for the eigenvalue
_numeric_e3 = sym.lambdify((UGA_star, UgA_star, T, R, P, S, M, m), e3, modules="numpy")
```

```python
def plot_third_eigenvalue(T, R, P, S, M, m):

    fig, ax = plt.subplots(1, 1, figsize=(10, 10))
    ax.set_ylabel(r"$U_{GA}^*$", fontsize=20, rotation="horizontal")
    ax.set_xlabel(r"$U_{gA}^{*}$", fontsize=20)
    ax.set_title(r"Eigenvalue, $e_3$", fontsize=25)
    ax.grid("off")

    equilibrium_selection_probs = np.linspace(0, 1, 100)
    UgAs, UGAs = np.meshgrid(equilibrium_selection_probs, equilibrium_selection_probs)
    Z = _numeric_e3(UGAs, UgAs, T, R, P, S, M, m)
    cax = ax.imshow(Z, origin="lower")
    contours = ax.contour(Z, colors='w', origin='lower')
    ax.clabel(contours, contours.levels, inline=True, fontsize=10)

    # adjust the tick labels
    locs, _ = plt.xticks()
    plt.xticks(locs[1:], np.linspace(0, 1, locs.size-1))
    locs, _ = plt.yticks()
    plt.yticks(locs[1:], np.linspace(0, 1, locs.size-1))

    plt.show()
```

```python
mpld3.disable_notebook()  # don't need interactive plotting for this!

# sliders used to control the Prisoner's Dilemma Payoffs
T_slider = widgets.FloatSlider(value=25, min=0, max=100, step=0.1, description=r"$T$")
R_slider = widgets.FloatSlider(value=3, min=0, max=100, step=0.1, description=r"$R$")
P_slider = widgets.FloatSlider(value=2, min=0, max=100, step=0.1, description=r"$P$")
S_slider = widgets.FloatSlider(value=1, min=0, max=100, step=0.1, description=r"$S$")

M_slider = widgets.FloatSlider(value=1, min=0, max=100, step=0.1, description=r"$M$")
m_slider = widgets.FloatSlider(value=0, min=0, max=100, step=0.1, description=r"$m$")

w = widgets.interactive(plot_third_eigenvalue, T=T_slider, R=R_slider, P=P_slider, S=S_slider, M=M_slider, m=m_slider)
display(w)
```

    A Jupyter Widget

## When is the fitness-maximizing interior equilibrium un-invadable by randomista?

```python
evaluated_e3 = e3.subs({UGA_star: optimal_UGA_star, UgA_star: optimal_x1_star, m: 0})
simplified_e3 = sym.factor(sym.cancel(sym.together(evaluated_e3)))
```

```python
numerator, denominator = sym.fraction(simplified_e3)
```

```python
numerator
```

    -4.0*M**3*P**3 - 12.0*M**3*P**2*R + 12.0*M**3*P**2*S + 12.0*M**3*P**2*T - 12.0*M**3*P*R**2 + 24.0*M**3*P*R*S + 24.0*M**3*P*R*T - 12.0*M**3*P*S**2 - 24.0*M**3*P*S*T - 12.0*M**3*P*T**2 - 4.0*M**3*R**3 + 12.0*M**3*R**2*S + 12.0*M**3*R**2*T - 12.0*M**3*R*S**2 - 24.0*M**3*R*S*T - 12.0*M**3*R*T**2 + 4.0*M**3*S**3 + 12.0*M**3*S**2*T + 12.0*M**3*S*T**2 + 4.0*M**3*T**3 + 8.0*M**2*P**3*R + 16.0*M**2*P**2*R**2 - 16.0*M**2*P**2*R*S - 16.0*M**2*P**2*R*T - 2.0*M**2*P**2*S**2 - 4.0*M**2*P**2*S*T - 2.0*M**2*P**2*T**2 + 8.0*M**2*P*R**3 - 16.0*M**2*P*R**2*S - 16.0*M**2*P*R**2*T + 4.0*M**2*P*R*S**2 + 8.0*M**2*P*R*S*T + 4.0*M**2*P*R*T**2 + 4.0*M**2*P*S**3 + 12.0*M**2*P*S**2*T + 12.0*M**2*P*S*T**2 + 4.0*M**2*P*T**3 - 2.0*M**2*R**2*S**2 - 4.0*M**2*R**2*S*T - 2.0*M**2*R**2*T**2 + 4.0*M**2*R*S**3 + 12.0*M**2*R*S**2*T + 12.0*M**2*R*S*T**2 + 4.0*M**2*R*T**3 - 2.0*M**2*S**4 - 8.0*M**2*S**3*T - 12.0*M**2*S**2*T**2 - 8.0*M**2*S*T**3 - 2.0*M**2*T**4 - 4.0*M*P**3*R**2 - 4.0*M*P**2*R**3 + 4.0*M*P**2*R**2*S + 4.0*M*P**2*R**2*T + 2.0*M*P**2*R*S**2 + 4.0*M*P**2*R*S*T + 2.0*M*P**2*R*T**2 + 2.0*M*P*R**2*S**2 + 4.0*M*P*R**2*S*T + 2.0*M*P*R**2*T**2 - 2.0*M*P*R*S**3 - 6.0*M*P*R*S**2*T - 6.0*M*P*R*S*T**2 - 2.0*M*P*R*T**3 - 0.25*M*P*S**4 - 1.0*M*P*S**3*T - 1.5*M*P*S**2*T**2 - 1.0*M*P*S*T**3 - 0.25*M*P*T**4 - 0.25*M*R*S**4 - 1.0*M*R*S**3*T - 1.5*M*R*S**2*T**2 - 1.0*M*R*S*T**3 - 0.25*M*R*T**4 + 0.25*M*S**5 + 1.25*M*S**4*T + 2.5*M*S**3*T**2 + 2.5*M*S**2*T**3 + 1.25*M*S*T**4 + 0.25*M*T**5 - 1.0*P**2*R**2*S**2 + 2.0*P**2*R**2*S*T - 1.0*P**2*R**2*T**2 + 1.0*P**2*R*S**3 - 1.0*P**2*R*S**2*T - 1.0*P**2*R*S*T**2 + 1.0*P**2*R*T**3 - 0.25*P**2*S**4 + 0.5*P**2*S**2*T**2 - 0.25*P**2*T**4 + 1.0*P*R**2*S**3 - 1.0*P*R**2*S**2*T - 1.0*P*R**2*S*T**2 + 1.0*P*R**2*T**3 - 1.0*P*R*S**4 + 2.0*P*R*S**2*T**2 - 1.0*P*R*T**4 + 0.25*P*S**5 + 0.25*P*S**4*T - 0.5*P*S**3*T**2 - 0.5*P*S**2*T**3 + 0.25*P*S*T**4 + 0.25*P*T**5 - 0.25*R**2*S**4 + 0.5*R**2*S**2*T**2 - 0.25*R**2*T**4 + 0.25*R*S**5 + 0.25*R*S**4*T - 0.5*R*S**3*T**2 - 0.5*R*S**2*T**3 + 0.25*R*S*T**4 + 0.25*R*T**5 - 0.0625*S**6 - 0.125*S**5*T + 0.0625*S**4*T**2 + 0.25*S**3*T**3 + 0.0625*S**2*T**4 - 0.125*S*T**5 - 0.0625*T**6

\begin{align}
    4\bigg(-\big(P^3 + 3P^2R + 3PR^2 + R^3\big) - \big(P^3 - 3P^2S + 3PS^2 + S^3\big) + \big(-P^3 + 3P^2T - 3PT^2 + T^3\big) - \big(R^3 - 3R^2S + 3RS^2 - S^3\big) + \big(T^3 - 3RT^2 + 3R^2T - R^3\big) + 2P^3 + 2R^3 + 8PRS + 8PRT - 8PST - 8RST - 2T^3 + \big(S^3 + 3S^2T + 3ST^2 + T^3\big)\bigg)M^3 + 2\big(4P^3R + 8P^2R^2 - 8P^2RS - 8P^2RT - P^2S^2 - 2P^2ST - P^2T^2 + 4PR^3 - 8PR^2S - 8PR^2T + 2PRS^2 + 4PRST + 2PRT^2 + 2PS^3 + 6PS^2T + 6PST^2 + 2PT^3 - R^2S^2 - 2R^2ST - R^2T^2 + 2RS^3 + 6RS^2T + 6RST^2 + 2RT^3 - S^4 - 4S^3T - 6S^2T^2 - 4ST^3 - T^4\big)M^2 + \big(- 4P^3R^2 - 4P^2R^3 + 4P^2R^2S + 4P^2R^2T + 2P^2RS^2 + 4P^2RST + 2P^2RT^2 + 2PR^2S^2 + 4PR^2ST + 2PR^2T^2 - 2PRS^3 - 6PRS^2T - 6PRST^2 - 2PRT^3 - 0.25PS^4 -PS^3T - 1.5PS^2T^2 - PST^3 - 0.25PT^4 - 0.25RS^4 - RS^3T - 1.5RS^2T^2 - RST^3 - 0.25RT^4 + 0.25S^5 + 1.25S^4T + 2.5S^3T^2 + 2.5S^2T^3 + 1.25ST^4 + 0.25T^5\big)M + \big(- P^2R^2S^2 + 2P^2R^2ST - P^2R^2T^2 + P^2RS^3 - P^2RS^2T - P^2RST^2 + P^2RT^3 - 0.25P^2S^4 + 0.5P^2S^2T^2 - 0.25P^2T^4 + PR^2S^3 - PR^2S^2T - PR^2ST^2 + PR^2T^3 - PRS^4 + 2PRS^2T^2 - PRT^4 + 0.25PS^5 + 0.25PS^4T - 0.5PS^3T^2 - 0.5PS^2T^3 + 0.25PST^4 + 0.25PT^5 - 0.25R^2S^4 + 0.5R^2S^2T^2 - 0.25R^2T^4 + 0.25RS^5 + 0.25RS^4T - 0.5RS^3T^2 - 0.5RS^2T^3 + 0.25RST^4 + 0.25RT^5 - 0.0625S^6 - 0.125S^5T + 0.0625S^4T^2 + 0.25S^3T^3 + 0.0625S^2T^4 - 0.125ST^5 - 0.0625T^6\big) \\
    4\bigg(-\big(R + P\big)^3 - \big(P - S\big)^3 + \big(T - P\big)^3 - \big(R - S\big)^3 + \big(T - R\big)^3 + 2P^3 + 2R^3 + 8PRS + 8PRT - 8PST - 8RST - 2T^3 + \big(T + S\big)^3\bigg)M^3 + 2\big(4P^3R + 8P^2R^2 - 8P^2RS - 8P^2RT - P^2S^2 - 2P^2ST - P^2T^2 + 4PR^3 - 8PR^2S - 8PR^2T + 2PRS^2 + 4PRST + 2PRT^2 + 2PS^3 + 6PS^2T + 6PST^2 + 2PT^3 - R^2S^2 - 2R^2ST - R^2T^2 + 2RS^3 + 6RS^2T + 6RST^2 + 2RT^3 - S^4 - 4S^3T - 6S^2T^2 - 4ST^3 - T^4\big)M^2 + \big(- 4P^3R^2 - 4P^2R^3 + 4P^2R^2S + 4P^2R^2T + 2P^2RS^2 + 4P^2RST + 2P^2RT^2 + 2PR^2S^2 + 4PR^2ST + 2PR^2T^2 - 2PRS^3 - 6PRS^2T - 6PRST^2 - 2PRT^3 - 0.25PS^4 -PS^3T - 1.5PS^2T^2 - PST^3 - 0.25PT^4 - 0.25RS^4 - RS^3T - 1.5RS^2T^2 - RST^3 - 0.25RT^4 + 0.25S^5 + 1.25S^4T + 2.5S^3T^2 + 2.5S^2T^3 + 1.25ST^4 + 0.25T^5\big)M + \big(- P^2R^2S^2 + 2P^2R^2ST - P^2R^2T^2 + P^2RS^3 - P^2RS^2T - P^2RST^2 + P^2RT^3 - 0.25P^2S^4 + 0.5P^2S^2T^2 - 0.25P^2T^4 + PR^2S^3 - PR^2S^2T - PR^2ST^2 + PR^2T^3 - PRS^4 + 2PRS^2T^2 - PRT^4 + 0.25PS^5 + 0.25PS^4T - 0.5PS^3T^2 - 0.5PS^2T^3 + 0.25PST^4 + 0.25PT^5 - 0.25R^2S^4 + 0.5R^2S^2T^2 - 0.25R^2T^4 + 0.25RS^5 + 0.25RS^4T - 0.5RS^3T^2 - 0.5RS^2T^3 + 0.25RST^4 + 0.25RT^5 - 0.0625S^6 - 0.125S^5T + 0.0625S^4T^2 + 0.25S^3T^3 + 0.0625S^2T^4 - 0.125ST^5 - 0.0625T^6\big) \\
\end{align}

```python
(S - R)(S^2 - 2SR + R^2) = S^3 - 3RS^2 + 3SR^2 - R^3
```

```python
sym.factor(-P**3 - 3*P**2*R + 3*P**2*S + 3*P**2*T - 3*P*R**2 + 8*P*R*S + 8*P*R*T - 3*P*S**2 - 8*P*S*T - 3*P*T**2 - R**3 + 3*R**2*S + 3*R**2*T - 3*R*S**2 - 8*R*S*T - 3*R*T**2 + S**3 + 3*S**2*T + 3*S*T**2 + T**3)
```

    -P**3 - 3*P**2*R + 3*P**2*S + 3*P**2*T - 3*P*R**2 + 8*P*R*S + 8*P*R*T - 3*P*S**2 - 8*P*S*T - 3*P*T**2 - R**3 + 3*R**2*S + 3*R**2*T - 3*R*S**2 - 8*R*S*T - 3*R*T**2 + S**3 + 3*S**2*T + 3*S*T**2 + T**3

```python
r1, r2, r3 = sym.solve(numerator, M)
```

```python
sym.cancel(sym.together(r1))
```

```python
denominator
```

    (0.5*M*P + 0.5*M*R - 0.5*M*S - 0.5*M*T - 0.5*P*R + 0.125*S**2 + 0.25*S*T + 0.125*T**2)**3

Denominator will be negative if and only if...

\begin{align}
    % 4MP + 4MR - 4MS - 4MT - 4PR + S^2 + 2ST + T^2 < 0 \\
    % 4M\big((R + P) - (T + S)\big) - 4PR + (T + S)^2< 0 \\
    % 4M\big((R + P) - (T + S)\big) < 4RP - (T + S)^2 \\
    M < \frac{1}{4}\frac{4RP - (T + S)^2}{(R + P) - (T + S)} \\
    M < \frac{RP - \left(\frac{T + S}{2}\right)^2}{(R + P) - (T + S)} \\
\end{align}

...since we are looking at interior equilibrium the RHS of the above inequality will be positive.

```python
_f = sym.lambdify((T, R, P, S, M), evaluated_e3, modules = "numpy")

def plot_non_invadable_by_randomista(T, R, P, S):
    fig, ax = plt.subplots(1, 1, figsize=(10,6))
    Ms = np.linspace(0, min(0.5 * (T + S), P), 100)
    ax.plot(Ms, _f(T, R, P, S, Ms))
    ax.set_xlabel(r"$M$", fontsize=15)
    ax.set_ylabel(r"$e_3$", rotation="horizontal", fontsize=15)
    plt.show()
```

```python
# sliders used to control the Prisoner's Dilemma Payoffs
T_slider = widgets.FloatSlider(value=10, min=0, max=100, step=0.1, description=r"$T$")
R_slider = widgets.FloatSlider(value=8, min=0, max=100, step=0.1, description=r"$R$")
P_slider = widgets.FloatSlider(value=6, min=0, max=100, step=0.1, description=r"$P$")
S_slider = widgets.FloatSlider(value=4, min=0, max=100, step=0.1, description=r"$S$")

w = widgets.interactive(plot_non_invadable_by_randomista, T=T_slider, R=R_slider, P=P_slider, S=S_slider)
display(w)
```

    A Jupyter Widget

```python
solutions = sym.solve(simplified_e3, M)
```

```python

```