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Derivation

1. Split input into 2 regimes

2. if alpha < 3.2376133163816985e+43

   1. Initial program 2.1

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

   3. Applied div-sub2.1

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

   4. Applied associate-+l-2.1

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

   6. Applied flip3--2.1

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

   7. Applied frac-sub2.1

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

   8. Applied associate-/l/2.1

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

   10. Applied add-cube-cbrt2.2

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

   11. Applied unpow-prod-down2.2

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

   12. Simplified2.2

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

   if 3.2376133163816985e+43 < alpha

   1. Initial program 52.0

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

   3. Applied div-sub52.0

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

   4. Applied associate-+l-50.3

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

   6. Applied clear-num50.4

      \[\leadsto \frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \left(\color{blue}{\frac{1}{\frac{\left(\alpha + \beta\right) + 2.0}{\alpha}}} - 1.0\right)}{2.0}\]

   7. Taylor expanded around inf 17.2

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

   8. Simplified17.2

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

4. Final simplification6.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 3.2376133163816985 \cdot 10^{+43}:\\ \;\;\;\;\frac{\beta \cdot \left(\left(1.0 \cdot \frac{\alpha}{\left(\beta + \alpha\right) + 2.0} + 1.0 \cdot 1.0\right) + \frac{\alpha}{\left(\beta + \alpha\right) + 2.0} \cdot \frac{\alpha}{\left(\beta + \alpha\right) + 2.0}\right) - \left(\left(\beta + \alpha\right) + 2.0\right) \cdot \left(\left(\frac{\alpha}{\left(\beta + \alpha\right) + 2.0} \cdot \frac{\alpha}{\left(\beta + \alpha\right) + 2.0}\right) \cdot {\left(\sqrt[3]{\frac{\alpha}{\left(\beta + \alpha\right) + 2.0}}\right)}^{3} - {1.0}^{3}\right)}{\left(\left(\left(1.0 \cdot \frac{\alpha}{\left(\beta + \alpha\right) + 2.0} + 1.0 \cdot 1.0\right) + \frac{\alpha}{\left(\beta + \alpha\right) + 2.0} \cdot \frac{\alpha}{\left(\beta + \alpha\right) + 2.0}\right) \cdot \left(\left(\beta + \alpha\right) + 2.0\right)\right) \cdot 2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\beta + \alpha\right) + 2.0} - \frac{\left(\frac{4.0}{\alpha} - 2.0\right) - \frac{8.0}{\alpha \cdot \alpha}}{\alpha}}{2.0}\\ \end{array}\]

Reproduce

herbie shell --seed 2019002 
(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

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