Error

Derivation

1. Split input into 2 regimes

2. if alpha < 31224578.285005733

   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 div-inv0.1

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

   7. Applied fma-neg0.1

      \[\leadsto \frac{\color{blue}{(\beta \cdot \left(\frac{1}{\left(\alpha + \beta\right) + 2.0}\right) + \left(-\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0\right)\right))_*}}{2.0}\]
   8. Using strategy rm

   9. Applied add-log-exp0.1

      \[\leadsto \frac{\color{blue}{\log \left(e^{(\beta \cdot \left(\frac{1}{\left(\alpha + \beta\right) + 2.0}\right) + \left(-\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0\right)\right))_*}\right)}}{2.0}\]

   if 31224578.285005733 < alpha

   1. Initial program 49.5

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

   3. Applied div-sub49.5

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

   4. Applied associate-+l-47.9

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

   6. Applied div-inv47.9

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

   7. Applied fma-neg47.9

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

   8. Taylor expanded around inf 18.7

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

   9. Simplified18.7

      \[\leadsto \frac{(\beta \cdot \left(\frac{1}{\left(\alpha + \beta\right) + 2.0}\right) + \left(-\color{blue}{(\left(\frac{1}{\alpha \cdot \alpha}\right) \cdot \left(4.0 - \frac{8.0}{\alpha}\right) + \left(-\frac{2.0}{\alpha}\right))_*}\right))_*}{2.0}\]
3. Recombined 2 regimes into one program.

4. Final simplification6.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 31224578.285005733:\\ \;\;\;\;\frac{\log \left(e^{(\beta \cdot \left(\frac{1}{\left(\beta + \alpha\right) + 2.0}\right) + \left(-\left(\frac{\alpha}{\left(\beta + \alpha\right) + 2.0} - 1.0\right)\right))_*}\right)}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{(\beta \cdot \left(\frac{1}{\left(\beta + \alpha\right) + 2.0}\right) + \left(-(\left(\frac{1}{\alpha \cdot \alpha}\right) \cdot \left(4.0 - \frac{8.0}{\alpha}\right) + \left(-\frac{2.0}{\alpha}\right))_*\right))_*}{2.0}\\ \end{array}\]

Reproduce

herbie shell --seed 2019007 +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))

Details

Time bar (total: 54.1s)Debug log