\[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]

↓

\[\begin{array}{l} \mathbf{if}\;\beta \le 1.620987782235559 \cdot 10^{+142}:\\ \;\;\;\;\frac{\frac{\frac{1.0 + (\beta \cdot \alpha + \left(\beta + \alpha\right))_*}{\left(\beta + \alpha\right) + 2}}{\left(\beta + \alpha\right) + 2}}{\left(1.0 + \left(\beta + \alpha\right)\right) + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\frac{2.0}{\beta} - 1.0}{\beta} + 1}{\left(\beta + \alpha\right) + 2}}{\left(1.0 + \left(\beta + \alpha\right)\right) + 2}\\ \end{array}\]

Derivation

Split input into 2 regimes
if beta < 1.620987782235559e+142
1. Initial program 0.9
  \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]
2. Simplified0.9
  \[\leadsto \color{blue}{\frac{\frac{\frac{1.0 + (\beta \cdot \alpha + \left(\beta + \alpha\right))_*}{2 + \left(\beta + \alpha\right)}}{2 + \left(\beta + \alpha\right)}}{\left(1.0 + \left(\beta + \alpha\right)\right) + 2}}\]
3. Using strategy rm
4. Applied +-commutative0.9
  \[\leadsto \frac{\frac{\frac{1.0 + (\beta \cdot \alpha + \left(\beta + \alpha\right))_*}{2 + \left(\beta + \alpha\right)}}{2 + \left(\beta + \alpha\right)}}{\color{blue}{2 + \left(1.0 + \left(\beta + \alpha\right)\right)}}\]
if 1.620987782235559e+142 < beta
1. Initial program 15.3
  \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]
2. Simplified15.3
  \[\leadsto \color{blue}{\frac{\frac{\frac{1.0 + (\beta \cdot \alpha + \left(\beta + \alpha\right))_*}{2 + \left(\beta + \alpha\right)}}{2 + \left(\beta + \alpha\right)}}{\left(1.0 + \left(\beta + \alpha\right)\right) + 2}}\]
3. Using strategy rm
4. Applied +-commutative15.3
  \[\leadsto \frac{\frac{\frac{1.0 + (\beta \cdot \alpha + \left(\beta + \alpha\right))_*}{2 + \left(\beta + \alpha\right)}}{2 + \left(\beta + \alpha\right)}}{\color{blue}{2 + \left(1.0 + \left(\beta + \alpha\right)\right)}}\]
5. Taylor expanded around -inf 8.3
  \[\leadsto \frac{\frac{\color{blue}{\left(2.0 \cdot \frac{1}{{\beta}^{2}} + 1\right) - 1.0 \cdot \frac{1}{\beta}}}{2 + \left(\beta + \alpha\right)}}{2 + \left(1.0 + \left(\beta + \alpha\right)\right)}\]
6. Simplified8.3
  \[\leadsto \frac{\frac{\color{blue}{1 + \frac{\frac{2.0}{\beta} - 1.0}{\beta}}}{2 + \left(\beta + \alpha\right)}}{2 + \left(1.0 + \left(\beta + \alpha\right)\right)}\]
Recombined 2 regimes into one program.
Final simplification2.2
\[\leadsto \begin{array}{l} \mathbf{if}\;\beta \le 1.620987782235559 \cdot 10^{+142}:\\ \;\;\;\;\frac{\frac{\frac{1.0 + (\beta \cdot \alpha + \left(\beta + \alpha\right))_*}{\left(\beta + \alpha\right) + 2}}{\left(\beta + \alpha\right) + 2}}{\left(1.0 + \left(\beta + \alpha\right)\right) + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\frac{2.0}{\beta} - 1.0}{\beta} + 1}{\left(\beta + \alpha\right) + 2}}{\left(1.0 + \left(\beta + \alpha\right)\right) + 2}\\ \end{array}\]

Error

Derivation

`if beta < 1.620987782235559e+142`

`if 1.620987782235559e+142 < beta`

Reproduce