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Derivation

1. Split input into 2 regimes

2. if (/ (- beta alpha) (+ (+ alpha beta) 2.0)) < -0.9999999999943141

   1. Initial program 60.2

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

   3. Applied div-sub60.2

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

   4. Applied associate-+l-58.4

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

   6. Applied clear-num58.4

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

   7. Taylor expanded around inf 10.8

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

   8. Simplified10.8

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

   if -0.9999999999943141 < (/ (- beta alpha) (+ (+ alpha beta) 2.0))

   1. Initial program 0.3

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

   3. Applied add-log-exp0.3

      \[\leadsto \frac{\color{blue}{\log \left(e^{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}\right)}}{2.0}\]
3. Recombined 2 regimes into one program.

4. Final simplification3.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} \le -0.9999999999943141:\\ \;\;\;\;\frac{\frac{1}{\frac{\left(\alpha + \beta\right) + 2.0}{\beta}} - \frac{\left(\frac{4.0}{\alpha} - 2.0\right) - \frac{8.0}{\alpha \cdot \alpha}}{\alpha}}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(e^{1.0 + \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0}}\right)}{2.0}\\ \end{array}\]

Reproduce

herbie shell --seed 2019005 
(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: 15.3s)Debug log