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Derivation

1. Split input into 2 regimes

2. if alpha < 2.352416918038036e+32

   1. Initial program 1.5

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}\]
   2. Using strategy rm

   3. Applied div-sub1.5

      \[\leadsto \frac{\color{blue}{\left(\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \frac{\alpha}{\left(\alpha + \beta\right) + 2.0}\right)} + 1.0}{2.0}\]

   4. Applied associate-+l-1.5

      \[\leadsto \frac{\color{blue}{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0\right)}}{2.0}\]
   5. Using strategy rm

   6. Applied flip3--1.5

      \[\leadsto \frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \color{blue}{\frac{{\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0}\right)}^{3} - {1.0}^{3}}{\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} \cdot \frac{\alpha}{\left(\alpha + \beta\right) + 2.0} + \left(1.0 \cdot 1.0 + \frac{\alpha}{\left(\alpha + \beta\right) + 2.0} \cdot 1.0\right)}}}{2.0}\]
   7. Using strategy rm

   8. Applied add-log-exp1.5

      \[\leadsto \frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \frac{{\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0}\right)}^{3} - \color{blue}{\log \left(e^{{1.0}^{3}}\right)}}{\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} \cdot \frac{\alpha}{\left(\alpha + \beta\right) + 2.0} + \left(1.0 \cdot 1.0 + \frac{\alpha}{\left(\alpha + \beta\right) + 2.0} \cdot 1.0\right)}}{2.0}\]

   9. Applied add-log-exp1.6

      \[\leadsto \frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \frac{\color{blue}{\log \left(e^{{\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0}\right)}^{3}}\right)} - \log \left(e^{{1.0}^{3}}\right)}{\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} \cdot \frac{\alpha}{\left(\alpha + \beta\right) + 2.0} + \left(1.0 \cdot 1.0 + \frac{\alpha}{\left(\alpha + \beta\right) + 2.0} \cdot 1.0\right)}}{2.0}\]

   10. Applied diff-log1.6

       \[\leadsto \frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \frac{\color{blue}{\log \left(\frac{e^{{\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0}\right)}^{3}}}{e^{{1.0}^{3}}}\right)}}{\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} \cdot \frac{\alpha}{\left(\alpha + \beta\right) + 2.0} + \left(1.0 \cdot 1.0 + \frac{\alpha}{\left(\alpha + \beta\right) + 2.0} \cdot 1.0\right)}}{2.0}\]
   11. Using strategy rm

   12. Applied flip-+1.6

       \[\leadsto \frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \frac{\log \left(\frac{e^{{\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0}\right)}^{3}}}{e^{{1.0}^{3}}}\right)}{\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} \cdot \frac{\alpha}{\left(\alpha + \beta\right) + 2.0} + \color{blue}{\frac{\left(1.0 \cdot 1.0\right) \cdot \left(1.0 \cdot 1.0\right) - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} \cdot 1.0\right) \cdot \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} \cdot 1.0\right)}{1.0 \cdot 1.0 - \frac{\alpha}{\left(\alpha + \beta\right) + 2.0} \cdot 1.0}}}}{2.0}\]

   13. Applied associate-*r/1.6

       \[\leadsto \frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \frac{\log \left(\frac{e^{{\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0}\right)}^{3}}}{e^{{1.0}^{3}}}\right)}{\color{blue}{\frac{\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} \cdot \alpha}{\left(\alpha + \beta\right) + 2.0}} + \frac{\left(1.0 \cdot 1.0\right) \cdot \left(1.0 \cdot 1.0\right) - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} \cdot 1.0\right) \cdot \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} \cdot 1.0\right)}{1.0 \cdot 1.0 - \frac{\alpha}{\left(\alpha + \beta\right) + 2.0} \cdot 1.0}}}{2.0}\]

   14. Applied frac-add1.6

       \[\leadsto \frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \frac{\log \left(\frac{e^{{\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0}\right)}^{3}}}{e^{{1.0}^{3}}}\right)}{\color{blue}{\frac{\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} \cdot \alpha\right) \cdot \left(1.0 \cdot 1.0 - \frac{\alpha}{\left(\alpha + \beta\right) + 2.0} \cdot 1.0\right) + \left(\left(\alpha + \beta\right) + 2.0\right) \cdot \left(\left(1.0 \cdot 1.0\right) \cdot \left(1.0 \cdot 1.0\right) - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} \cdot 1.0\right) \cdot \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} \cdot 1.0\right)\right)}{\left(\left(\alpha + \beta\right) + 2.0\right) \cdot \left(1.0 \cdot 1.0 - \frac{\alpha}{\left(\alpha + \beta\right) + 2.0} \cdot 1.0\right)}}}}{2.0}\]

   15. Applied associate-/r/1.6

       \[\leadsto \frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \color{blue}{\frac{\log \left(\frac{e^{{\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0}\right)}^{3}}}{e^{{1.0}^{3}}}\right)}{\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} \cdot \alpha\right) \cdot \left(1.0 \cdot 1.0 - \frac{\alpha}{\left(\alpha + \beta\right) + 2.0} \cdot 1.0\right) + \left(\left(\alpha + \beta\right) + 2.0\right) \cdot \left(\left(1.0 \cdot 1.0\right) \cdot \left(1.0 \cdot 1.0\right) - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} \cdot 1.0\right) \cdot \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} \cdot 1.0\right)\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2.0\right) \cdot \left(1.0 \cdot 1.0 - \frac{\alpha}{\left(\alpha + \beta\right) + 2.0} \cdot 1.0\right)\right)}}{2.0}\]

   16. Simplified1.6

       \[\leadsto \frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \frac{\log \left(\frac{e^{{\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0}\right)}^{3}}}{e^{{1.0}^{3}}}\right)}{\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} \cdot \alpha\right) \cdot \left(1.0 \cdot 1.0 - \frac{\alpha}{\left(\alpha + \beta\right) + 2.0} \cdot 1.0\right) + \left(\left(\alpha + \beta\right) + 2.0\right) \cdot \left(\left(1.0 \cdot 1.0\right) \cdot \left(1.0 \cdot 1.0\right) - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} \cdot 1.0\right) \cdot \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} \cdot 1.0\right)\right)} \cdot \color{blue}{\left(\left(1.0 - \frac{\alpha}{2.0 + \left(\beta + \alpha\right)}\right) \cdot \left(\left(2.0 + \left(\beta + \alpha\right)\right) \cdot 1.0\right)\right)}}{2.0}\]

   if 2.352416918038036e+32 < alpha

   1. Initial program 50.8

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}\]
   2. Using strategy rm

   3. Applied div-sub50.8

      \[\leadsto \frac{\color{blue}{\left(\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \frac{\alpha}{\left(\alpha + \beta\right) + 2.0}\right)} + 1.0}{2.0}\]

   4. Applied associate-+l-49.1

      \[\leadsto \frac{\color{blue}{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0\right)}}{2.0}\]
   5. Using strategy rm

   6. Applied flip3--49.1

      \[\leadsto \frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \color{blue}{\frac{{\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0}\right)}^{3} - {1.0}^{3}}{\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} \cdot \frac{\alpha}{\left(\alpha + \beta\right) + 2.0} + \left(1.0 \cdot 1.0 + \frac{\alpha}{\left(\alpha + \beta\right) + 2.0} \cdot 1.0\right)}}}{2.0}\]

   7. Taylor expanded around inf 18.3

      \[\leadsto \frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \color{blue}{\left(4.0 \cdot \frac{1}{{\alpha}^{2}} - \left(2.0 \cdot \frac{1}{\alpha} + 8.0 \cdot \frac{1}{{\alpha}^{3}}\right)\right)}}{2.0}\]

   8. Simplified18.3

      \[\leadsto \frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \color{blue}{\frac{\left(\frac{4.0}{\alpha} - 2.0\right) - \frac{8.0}{\alpha \cdot \alpha}}{\alpha}}}{2.0}\]
3. Recombined 2 regimes into one program.

4. Final simplification6.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 2.352416918038036 \cdot 10^{+32}:\\ \;\;\;\;\frac{\frac{\beta}{2.0 + \left(\beta + \alpha\right)} - \frac{\log \left(\frac{e^{{\left(\frac{\alpha}{2.0 + \left(\beta + \alpha\right)}\right)}^{3}}}{e^{{1.0}^{3}}}\right)}{\left(1.0 \cdot 1.0 - \frac{\alpha}{2.0 + \left(\beta + \alpha\right)} \cdot 1.0\right) \cdot \left(\alpha \cdot \frac{\alpha}{2.0 + \left(\beta + \alpha\right)}\right) + \left(\left(1.0 \cdot 1.0\right) \cdot \left(1.0 \cdot 1.0\right) - \left(\frac{\alpha}{2.0 + \left(\beta + \alpha\right)} \cdot 1.0\right) \cdot \left(\frac{\alpha}{2.0 + \left(\beta + \alpha\right)} \cdot 1.0\right)\right) \cdot \left(2.0 + \left(\beta + \alpha\right)\right)} \cdot \left(\left(1.0 - \frac{\alpha}{2.0 + \left(\beta + \alpha\right)}\right) \cdot \left(\left(2.0 + \left(\beta + \alpha\right)\right) \cdot 1.0\right)\right)}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{2.0 + \left(\beta + \alpha\right)} - \frac{\left(\frac{4.0}{\alpha} - 2.0\right) - \frac{8.0}{\alpha \cdot \alpha}}{\alpha}}{2.0}\\ \end{array}\]

Reproduce

herbie shell --seed 2019004 
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/1"
  :pre (and (> alpha -1) (> beta -1))
  (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0))

Details

Time bar (total: 50.2s)Debug log