Error

Derivation

1. Initial program 23.5

   \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]

2. Initial simplification12.1

   \[\leadsto \frac{(\left(\frac{\beta - \alpha}{\left(\beta + 2.0\right) + (2 \cdot i + \alpha)_*}\right) \cdot \left(\frac{\beta + \alpha}{(2 \cdot i + \alpha)_* + \beta}\right) + 1.0)_*}{2.0}\]
3. Using strategy rm

4. Applied fma-udef12.1

   \[\leadsto \frac{\color{blue}{\frac{\beta - \alpha}{\left(\beta + 2.0\right) + (2 \cdot i + \alpha)_*} \cdot \frac{\beta + \alpha}{(2 \cdot i + \alpha)_* + \beta} + 1.0}}{2.0}\]
5. Using strategy rm

6. Applied add-cbrt-cube12.1

   \[\leadsto \frac{\color{blue}{\sqrt[3]{\left(\frac{\beta - \alpha}{\left(\beta + 2.0\right) + (2 \cdot i + \alpha)_*} \cdot \frac{\beta - \alpha}{\left(\beta + 2.0\right) + (2 \cdot i + \alpha)_*}\right) \cdot \frac{\beta - \alpha}{\left(\beta + 2.0\right) + (2 \cdot i + \alpha)_*}}} \cdot \frac{\beta + \alpha}{(2 \cdot i + \alpha)_* + \beta} + 1.0}{2.0}\]
7. Using strategy rm

8. Applied *-un-lft-identity12.1

   \[\leadsto \frac{\sqrt[3]{\left(\frac{\beta - \alpha}{\left(\beta + 2.0\right) + (2 \cdot i + \alpha)_*} \cdot \frac{\beta - \alpha}{\color{blue}{1 \cdot \left(\left(\beta + 2.0\right) + (2 \cdot i + \alpha)_*\right)}}\right) \cdot \frac{\beta - \alpha}{\left(\beta + 2.0\right) + (2 \cdot i + \alpha)_*}} \cdot \frac{\beta + \alpha}{(2 \cdot i + \alpha)_* + \beta} + 1.0}{2.0}\]

9. Applied add-cube-cbrt12.1

   \[\leadsto \frac{\sqrt[3]{\left(\frac{\beta - \alpha}{\left(\beta + 2.0\right) + (2 \cdot i + \alpha)_*} \cdot \frac{\color{blue}{\left(\sqrt[3]{\beta - \alpha} \cdot \sqrt[3]{\beta - \alpha}\right) \cdot \sqrt[3]{\beta - \alpha}}}{1 \cdot \left(\left(\beta + 2.0\right) + (2 \cdot i + \alpha)_*\right)}\right) \cdot \frac{\beta - \alpha}{\left(\beta + 2.0\right) + (2 \cdot i + \alpha)_*}} \cdot \frac{\beta + \alpha}{(2 \cdot i + \alpha)_* + \beta} + 1.0}{2.0}\]

10. Applied times-frac12.1

    \[\leadsto \frac{\sqrt[3]{\left(\frac{\beta - \alpha}{\left(\beta + 2.0\right) + (2 \cdot i + \alpha)_*} \cdot \color{blue}{\left(\frac{\sqrt[3]{\beta - \alpha} \cdot \sqrt[3]{\beta - \alpha}}{1} \cdot \frac{\sqrt[3]{\beta - \alpha}}{\left(\beta + 2.0\right) + (2 \cdot i + \alpha)_*}\right)}\right) \cdot \frac{\beta - \alpha}{\left(\beta + 2.0\right) + (2 \cdot i + \alpha)_*}} \cdot \frac{\beta + \alpha}{(2 \cdot i + \alpha)_* + \beta} + 1.0}{2.0}\]

11. Applied associate-*r*12.1

    \[\leadsto \frac{\sqrt[3]{\color{blue}{\left(\left(\frac{\beta - \alpha}{\left(\beta + 2.0\right) + (2 \cdot i + \alpha)_*} \cdot \frac{\sqrt[3]{\beta - \alpha} \cdot \sqrt[3]{\beta - \alpha}}{1}\right) \cdot \frac{\sqrt[3]{\beta - \alpha}}{\left(\beta + 2.0\right) + (2 \cdot i + \alpha)_*}\right)} \cdot \frac{\beta - \alpha}{\left(\beta + 2.0\right) + (2 \cdot i + \alpha)_*}} \cdot \frac{\beta + \alpha}{(2 \cdot i + \alpha)_* + \beta} + 1.0}{2.0}\]

12. Final simplification12.1

    \[\leadsto \frac{1.0 + \sqrt[3]{\frac{\beta - \alpha}{(2 \cdot i + \alpha)_* + \left(2.0 + \beta\right)} \cdot \left(\frac{\sqrt[3]{\beta - \alpha}}{(2 \cdot i + \alpha)_* + \left(2.0 + \beta\right)} \cdot \left(\frac{\beta - \alpha}{(2 \cdot i + \alpha)_* + \left(2.0 + \beta\right)} \cdot \left(\sqrt[3]{\beta - \alpha} \cdot \sqrt[3]{\beta - \alpha}\right)\right)\right)} \cdot \frac{\alpha + \beta}{\beta + (2 \cdot i + \alpha)_*}}{2.0}\]

Runtime

Time bar (total: 3.6m)Debug logProfile

herbie shell --seed 2018274 +o rules:numerics
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/2"
  :pre (and (> alpha -1) (> beta -1) (> i 0))
  (/ (+ (/ (/ (* (+ alpha beta) (- beta alpha)) (+ (+ alpha beta) (* 2 i))) (+ (+ (+ alpha beta) (* 2 i)) 2.0)) 1.0) 2.0))