Error

Derivation

1. Initial program 23.3

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

2. Initial simplification12.2

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

4. Applied div-inv12.2

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

6. Applied add-cbrt-cube12.2

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

8. Applied pow1/312.2

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

10. Applied add-exp-log12.2

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

11. Applied add-exp-log12.2

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

12. Applied add-exp-log12.2

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

13. Applied prod-exp12.2

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

14. Applied prod-exp12.4

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

15. Applied pow-exp12.4

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

16. Simplified12.2

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

17. Final simplification12.2

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

Runtime

Time bar (total: 28.2s)Debug logProfile

herbie shell --seed 2018339 +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))