Error

Derivation

1. Initial program 23.9

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

2. Initial simplification12.5

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

4. Applied add-cbrt-cube21.7

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

5. Applied add-cbrt-cube27.4

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

6. Applied cbrt-undiv27.4

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

7. Simplified12.5

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

9. Applied clear-num12.5

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

11. Applied add-exp-log12.5

    \[\leadsto \frac{\color{blue}{e^{\log \left((\left(\sqrt[3]{{\left(\frac{1}{\frac{\left(2.0 + \beta\right) + (2 \cdot i + \alpha)_*}{\beta - \alpha}}\right)}^{3}}\right) \cdot \left(\frac{\beta + \alpha}{(2 \cdot i + \alpha)_* + \beta}\right) + 1.0)_*\right)}}}{2.0}\]
12. Using strategy rm

13. Applied rem-cbrt-cube12.5

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

14. Final simplification12.5

    \[\leadsto \frac{e^{\log \left((\left(\frac{1}{\frac{\left(2.0 + \beta\right) + (2 \cdot i + \alpha)_*}{\beta - \alpha}}\right) \cdot \left(\frac{\beta + \alpha}{(2 \cdot i + \alpha)_* + \beta}\right) + 1.0)_*\right)}}{2.0}\]

Runtime

Time bar (total: 1.4m)Debug logProfile

herbie shell --seed 2018349 +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))