Error

Derivation

1. Split input into 2 regimes

2. if alpha < 4.334882831224304e+18 or 2.941801276354749e+52 < alpha < 1.061009875984785e+69

   1. Initial program 1.3

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

   3. Applied div-sub1.3

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

   4. Applied associate-+l-1.3

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

   6. Applied clear-num1.3

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

   8. Applied add-cube-cbrt1.3

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\frac{1}{\frac{\left(\alpha + \beta\right) + 2.0}{\beta}}} \cdot \sqrt[3]{\frac{1}{\frac{\left(\alpha + \beta\right) + 2.0}{\beta}}}\right) \cdot \sqrt[3]{\frac{1}{\frac{\left(\alpha + \beta\right) + 2.0}{\beta}}}} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0\right)}{2.0}\]
   9. Using strategy rm

   10. Applied add-log-exp1.3

       \[\leadsto \frac{\left(\sqrt[3]{\frac{1}{\frac{\left(\alpha + \beta\right) + 2.0}{\beta}}} \cdot \sqrt[3]{\frac{1}{\frac{\left(\alpha + \beta\right) + 2.0}{\beta}}}\right) \cdot \color{blue}{\log \left(e^{\sqrt[3]{\frac{1}{\frac{\left(\alpha + \beta\right) + 2.0}{\beta}}}}\right)} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0\right)}{2.0}\]

   if 4.334882831224304e+18 < alpha < 2.941801276354749e+52 or 1.061009875984785e+69 < 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-49.2

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

   5. Taylor expanded around inf 17.7

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

   6. Simplified17.7

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

4. Final simplification6.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 4.334882831224304 \cdot 10^{+18} \lor \neg \left(\alpha \le 2.941801276354749 \cdot 10^{+52}\right) \land \alpha \le 1.061009875984785 \cdot 10^{+69}:\\ \;\;\;\;\frac{\left(\sqrt[3]{\frac{1}{\frac{\left(\alpha + \beta\right) + 2.0}{\beta}}} \cdot \sqrt[3]{\frac{1}{\frac{\left(\alpha + \beta\right) + 2.0}{\beta}}}\right) \cdot \log \left(e^{\sqrt[3]{\frac{1}{\frac{\left(\alpha + \beta\right) + 2.0}{\beta}}}}\right) - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0\right)}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - (\left(\frac{1}{\alpha \cdot \alpha}\right) \cdot \left(4.0 - \frac{8.0}{\alpha}\right) + \left(\frac{-2.0}{\alpha}\right))_*}{2.0}\\ \end{array}\]

Runtime

Time bar (total: 33.1s)Debug logProfile

herbie shell --seed 2018286 +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))