Derivation

Split input into 2 regimes
if beta < 8.34052549037273e+155
1. Initial program 1.2
  \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]
2. Simplified1.2
  \[\leadsto \color{blue}{\frac{\frac{\frac{1.0 + (\beta \cdot \alpha + \left(\beta + \alpha\right))_*}{2 + \left(\beta + \alpha\right)}}{2 + \left(\beta + \alpha\right)}}{\left(1.0 + \left(\beta + \alpha\right)\right) + 2}}\]
3. Using strategy rm
4. Applied *-un-lft-identity1.2
  \[\leadsto \frac{\frac{\frac{1.0 + (\beta \cdot \alpha + \left(\beta + \alpha\right))_*}{\color{blue}{1 \cdot \left(2 + \left(\beta + \alpha\right)\right)}}}{2 + \left(\beta + \alpha\right)}}{\left(1.0 + \left(\beta + \alpha\right)\right) + 2}\]
5. Applied *-un-lft-identity1.2
  \[\leadsto \frac{\frac{\frac{\color{blue}{1 \cdot \left(1.0 + (\beta \cdot \alpha + \left(\beta + \alpha\right))_*\right)}}{1 \cdot \left(2 + \left(\beta + \alpha\right)\right)}}{2 + \left(\beta + \alpha\right)}}{\left(1.0 + \left(\beta + \alpha\right)\right) + 2}\]
6. Applied times-frac1.2
  \[\leadsto \frac{\frac{\color{blue}{\frac{1}{1} \cdot \frac{1.0 + (\beta \cdot \alpha + \left(\beta + \alpha\right))_*}{2 + \left(\beta + \alpha\right)}}}{2 + \left(\beta + \alpha\right)}}{\left(1.0 + \left(\beta + \alpha\right)\right) + 2}\]
7. Applied associate-/l*1.2
  \[\leadsto \frac{\color{blue}{\frac{\frac{1}{1}}{\frac{2 + \left(\beta + \alpha\right)}{\frac{1.0 + (\beta \cdot \alpha + \left(\beta + \alpha\right))_*}{2 + \left(\beta + \alpha\right)}}}}}{\left(1.0 + \left(\beta + \alpha\right)\right) + 2}\]
8. Simplified1.2
  \[\leadsto \frac{\frac{\color{blue}{1}}{\frac{2 + \left(\beta + \alpha\right)}{\frac{1.0 + (\beta \cdot \alpha + \left(\beta + \alpha\right))_*}{2 + \left(\beta + \alpha\right)}}}}{\left(1.0 + \left(\beta + \alpha\right)\right) + 2}\]
9. Using strategy rm
10. Applied div-inv1.3
  \[\leadsto \color{blue}{\frac{1}{\frac{2 + \left(\beta + \alpha\right)}{\frac{1.0 + (\beta \cdot \alpha + \left(\beta + \alpha\right))_*}{2 + \left(\beta + \alpha\right)}}} \cdot \frac{1}{\left(1.0 + \left(\beta + \alpha\right)\right) + 2}}\]
11. Using strategy rm
12. Applied pow11.3
  \[\leadsto \frac{1}{\frac{2 + \left(\beta + \alpha\right)}{\frac{1.0 + (\beta \cdot \alpha + \left(\beta + \alpha\right))_*}{2 + \left(\beta + \alpha\right)}}} \cdot \color{blue}{{\left(\frac{1}{\left(1.0 + \left(\beta + \alpha\right)\right) + 2}\right)}^{1}}\]
13. Applied pow11.3
  \[\leadsto \color{blue}{{\left(\frac{1}{\frac{2 + \left(\beta + \alpha\right)}{\frac{1.0 + (\beta \cdot \alpha + \left(\beta + \alpha\right))_*}{2 + \left(\beta + \alpha\right)}}}\right)}^{1}} \cdot {\left(\frac{1}{\left(1.0 + \left(\beta + \alpha\right)\right) + 2}\right)}^{1}\]
14. Applied pow-prod-down1.3
  \[\leadsto \color{blue}{{\left(\frac{1}{\frac{2 + \left(\beta + \alpha\right)}{\frac{1.0 + (\beta \cdot \alpha + \left(\beta + \alpha\right))_*}{2 + \left(\beta + \alpha\right)}}} \cdot \frac{1}{\left(1.0 + \left(\beta + \alpha\right)\right) + 2}\right)}^{1}}\]
15. Simplified1.2
  \[\leadsto {\color{blue}{\left(\frac{\frac{\frac{(\beta \cdot \alpha + \left(\alpha + \beta\right))_* + 1.0}{\left(2 + \left(\alpha + \beta\right)\right) + 1.0}}{2 + \left(\alpha + \beta\right)}}{2 + \left(\alpha + \beta\right)}\right)}}^{1}\]
16. Using strategy rm
17. Applied add-sqr-sqrt1.3
  \[\leadsto {\left(\frac{\frac{\color{blue}{\sqrt{\frac{(\beta \cdot \alpha + \left(\alpha + \beta\right))_* + 1.0}{\left(2 + \left(\alpha + \beta\right)\right) + 1.0}} \cdot \sqrt{\frac{(\beta \cdot \alpha + \left(\alpha + \beta\right))_* + 1.0}{\left(2 + \left(\alpha + \beta\right)\right) + 1.0}}}}{2 + \left(\alpha + \beta\right)}}{2 + \left(\alpha + \beta\right)}\right)}^{1}\]
if 8.34052549037273e+155 < beta
1. Initial program 15.8
  \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]
2. Simplified15.8
  \[\leadsto \color{blue}{\frac{\frac{\frac{1.0 + (\beta \cdot \alpha + \left(\beta + \alpha\right))_*}{2 + \left(\beta + \alpha\right)}}{2 + \left(\beta + \alpha\right)}}{\left(1.0 + \left(\beta + \alpha\right)\right) + 2}}\]
3. Using strategy rm
4. Applied *-un-lft-identity15.8
  \[\leadsto \frac{\frac{\frac{1.0 + (\beta \cdot \alpha + \left(\beta + \alpha\right))_*}{\color{blue}{1 \cdot \left(2 + \left(\beta + \alpha\right)\right)}}}{2 + \left(\beta + \alpha\right)}}{\left(1.0 + \left(\beta + \alpha\right)\right) + 2}\]
5. Applied *-un-lft-identity15.8
  \[\leadsto \frac{\frac{\frac{\color{blue}{1 \cdot \left(1.0 + (\beta \cdot \alpha + \left(\beta + \alpha\right))_*\right)}}{1 \cdot \left(2 + \left(\beta + \alpha\right)\right)}}{2 + \left(\beta + \alpha\right)}}{\left(1.0 + \left(\beta + \alpha\right)\right) + 2}\]
6. Applied times-frac15.8
  \[\leadsto \frac{\frac{\color{blue}{\frac{1}{1} \cdot \frac{1.0 + (\beta \cdot \alpha + \left(\beta + \alpha\right))_*}{2 + \left(\beta + \alpha\right)}}}{2 + \left(\beta + \alpha\right)}}{\left(1.0 + \left(\beta + \alpha\right)\right) + 2}\]
7. Applied associate-/l*15.8
  \[\leadsto \frac{\color{blue}{\frac{\frac{1}{1}}{\frac{2 + \left(\beta + \alpha\right)}{\frac{1.0 + (\beta \cdot \alpha + \left(\beta + \alpha\right))_*}{2 + \left(\beta + \alpha\right)}}}}}{\left(1.0 + \left(\beta + \alpha\right)\right) + 2}\]
8. Simplified15.8
  \[\leadsto \frac{\frac{\color{blue}{1}}{\frac{2 + \left(\beta + \alpha\right)}{\frac{1.0 + (\beta \cdot \alpha + \left(\beta + \alpha\right))_*}{2 + \left(\beta + \alpha\right)}}}}{\left(1.0 + \left(\beta + \alpha\right)\right) + 2}\]
9. Using strategy rm
10. Applied div-inv15.8
  \[\leadsto \color{blue}{\frac{1}{\frac{2 + \left(\beta + \alpha\right)}{\frac{1.0 + (\beta \cdot \alpha + \left(\beta + \alpha\right))_*}{2 + \left(\beta + \alpha\right)}}} \cdot \frac{1}{\left(1.0 + \left(\beta + \alpha\right)\right) + 2}}\]
11. Taylor expanded around -inf 0.6
  \[\leadsto \frac{1}{\color{blue}{\frac{\beta}{\alpha} + \left(2 + \frac{\alpha}{\beta}\right)}} \cdot \frac{1}{\left(1.0 + \left(\beta + \alpha\right)\right) + 2}\]
Recombined 2 regimes into one program.
Final simplification1.2
\[\leadsto \begin{array}{l} \mathbf{if}\;\beta \le 8.34052549037273 \cdot 10^{+155}:\\ \;\;\;\;\frac{\frac{\sqrt{\frac{(\beta \cdot \alpha + \left(\alpha + \beta\right))_* + 1.0}{1.0 + \left(2 + \left(\alpha + \beta\right)\right)}} \cdot \sqrt{\frac{(\beta \cdot \alpha + \left(\alpha + \beta\right))_* + 1.0}{1.0 + \left(2 + \left(\alpha + \beta\right)\right)}}}{2 + \left(\alpha + \beta\right)}}{2 + \left(\alpha + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + \left(\left(\alpha + \beta\right) + 1.0\right)} \cdot \frac{1}{\frac{\beta}{\alpha} + \left(\frac{\alpha}{\beta} + 2\right)}\\ \end{array}\]

Error

Derivation

`if beta < 8.34052549037273e+155`

`if 8.34052549037273e+155 < beta`

Reproduce