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

↓

\[\begin{array}{l} \mathbf{if}\;\beta \le 1.8711564538081698 \cdot 10^{+147}:\\ \;\;\;\;\frac{\sqrt{\frac{\frac{1}{\frac{\left(\alpha + \beta\right) + 2}{\left(\left(\alpha + \beta\right) + \alpha \cdot \beta\right) + 1.0}}}{\left(\alpha + \beta\right) + 2}} \cdot \sqrt{\frac{\frac{1}{\frac{\left(\alpha + \beta\right) + 2}{\left(\left(\alpha + \beta\right) + \alpha \cdot \beta\right) + 1.0}}}{\left(\alpha + \beta\right) + 2}}}{\left(1.0 + \left(\alpha + \beta\right)\right) + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{\frac{1}{\alpha} + \left(\frac{1}{\beta} - \frac{\frac{1}{\beta}}{\beta}\right)}}{\left(\alpha + \beta\right) + 2}}{\left(1.0 + \left(\alpha + \beta\right)\right) + 2}\\ \end{array}\]

Derivation

Split input into 2 regimes
if beta < 1.8711564538081698e+147
1. Initial program 1.0
  \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]
2. Simplified1.0
  \[\leadsto \color{blue}{\frac{\frac{\frac{1.0 + \left(\alpha \cdot \beta + \left(\beta + \alpha\right)\right)}{2 + \left(\beta + \alpha\right)}}{2 + \left(\beta + \alpha\right)}}{\left(1.0 + \left(\beta + \alpha\right)\right) + 2}}\]
3. Using strategy rm
4. Applied clear-num1.0
  \[\leadsto \frac{\frac{\color{blue}{\frac{1}{\frac{2 + \left(\beta + \alpha\right)}{1.0 + \left(\alpha \cdot \beta + \left(\beta + \alpha\right)\right)}}}}{2 + \left(\beta + \alpha\right)}}{\left(1.0 + \left(\beta + \alpha\right)\right) + 2}\]
5. Using strategy rm
6. Applied add-sqr-sqrt1.1
  \[\leadsto \frac{\color{blue}{\sqrt{\frac{\frac{1}{\frac{2 + \left(\beta + \alpha\right)}{1.0 + \left(\alpha \cdot \beta + \left(\beta + \alpha\right)\right)}}}{2 + \left(\beta + \alpha\right)}} \cdot \sqrt{\frac{\frac{1}{\frac{2 + \left(\beta + \alpha\right)}{1.0 + \left(\alpha \cdot \beta + \left(\beta + \alpha\right)\right)}}}{2 + \left(\beta + \alpha\right)}}}}{\left(1.0 + \left(\beta + \alpha\right)\right) + 2}\]
if 1.8711564538081698e+147 < beta
1. Initial program 14.7
  \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]
2. Simplified14.7
  \[\leadsto \color{blue}{\frac{\frac{\frac{1.0 + \left(\alpha \cdot \beta + \left(\beta + \alpha\right)\right)}{2 + \left(\beta + \alpha\right)}}{2 + \left(\beta + \alpha\right)}}{\left(1.0 + \left(\beta + \alpha\right)\right) + 2}}\]
3. Using strategy rm
4. Applied clear-num14.7
  \[\leadsto \frac{\frac{\color{blue}{\frac{1}{\frac{2 + \left(\beta + \alpha\right)}{1.0 + \left(\alpha \cdot \beta + \left(\beta + \alpha\right)\right)}}}}{2 + \left(\beta + \alpha\right)}}{\left(1.0 + \left(\beta + \alpha\right)\right) + 2}\]
5. Taylor expanded around inf 2.3
  \[\leadsto \frac{\frac{\frac{1}{\color{blue}{\left(\frac{1}{\alpha} + \frac{1}{\beta}\right) - \frac{1}{{\beta}^{2}}}}}{2 + \left(\beta + \alpha\right)}}{\left(1.0 + \left(\beta + \alpha\right)\right) + 2}\]
6. Simplified2.3
  \[\leadsto \frac{\frac{\frac{1}{\color{blue}{\left(\frac{1}{\beta} - \frac{\frac{1}{\beta}}{\beta}\right) + \frac{1}{\alpha}}}}{2 + \left(\beta + \alpha\right)}}{\left(1.0 + \left(\beta + \alpha\right)\right) + 2}\]
Recombined 2 regimes into one program.
Final simplification1.3
\[\leadsto \begin{array}{l} \mathbf{if}\;\beta \le 1.8711564538081698 \cdot 10^{+147}:\\ \;\;\;\;\frac{\sqrt{\frac{\frac{1}{\frac{\left(\alpha + \beta\right) + 2}{\left(\left(\alpha + \beta\right) + \alpha \cdot \beta\right) + 1.0}}}{\left(\alpha + \beta\right) + 2}} \cdot \sqrt{\frac{\frac{1}{\frac{\left(\alpha + \beta\right) + 2}{\left(\left(\alpha + \beta\right) + \alpha \cdot \beta\right) + 1.0}}}{\left(\alpha + \beta\right) + 2}}}{\left(1.0 + \left(\alpha + \beta\right)\right) + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{\frac{1}{\alpha} + \left(\frac{1}{\beta} - \frac{\frac{1}{\beta}}{\beta}\right)}}{\left(\alpha + \beta\right) + 2}}{\left(1.0 + \left(\alpha + \beta\right)\right) + 2}\\ \end{array}\]

Error

Derivation

`if beta < 1.8711564538081698e+147`

`if 1.8711564538081698e+147 < beta`

Reproduce