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 simplification12.8

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

4. Applied add-exp-log12.8

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

6. Applied div-sub12.8

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

8. Applied add-cbrt-cube12.8

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

10. Applied add-log-exp12.8

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

11. Final simplification12.8

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

Runtime

Time bar (total: 30.1s)Debug logProfile

herbie shell --seed 2018297 +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))