Error

Derivation

1. Split input into 2 regimes

2. if (/ (- beta alpha) (+ (+ alpha beta) 2.0)) < -0.999999987917626

   1. Initial program 59.7

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

   3. Applied div-sub59.7

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

   4. Applied associate-+l-57.8

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

   6. Applied add-cube-cbrt57.8

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

   7. Taylor expanded around -inf 11.5

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

   8. Simplified11.5

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

   if -0.999999987917626 < (/ (- beta alpha) (+ (+ alpha beta) 2.0))

   1. Initial program 0.1

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

   3. Applied div-sub0.1

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

   4. Applied associate-+l-0.1

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

   6. Applied add-log-exp0.2

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

4. Final simplification3.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} \le -0.999999987917626:\\ \;\;\;\;\frac{\sqrt[3]{\frac{\beta}{\left(\alpha + \beta\right) + 2.0}} \cdot \left(\sqrt[3]{\frac{\beta}{\left(\alpha + \beta\right) + 2.0}} \cdot \sqrt[3]{\frac{\beta}{\left(\alpha + \beta\right) + 2.0}}\right) - (\left(\frac{1}{\alpha \cdot \alpha}\right) \cdot \left(4.0 - \frac{8.0}{\alpha}\right) + \left(\frac{-2.0}{\alpha}\right))_*}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(e^{\frac{\beta}{\left(\alpha + \beta\right) + 2.0}}\right) - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0\right)}{2.0}\\ \end{array}\]

Runtime

Time bar (total: 36.9s)Debug logProfile

herbie shell --seed 2018353 +o rules:numerics
(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))