Error

Derivation

1. Initial program 24.3

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

2. Applied simplify12.6

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

4. Applied add-exp-log12.6

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

6. Applied add-cube-cbrt12.8

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

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

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

8. Applied times-frac12.8

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

10. Applied add-cbrt-cube12.8

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

11. Applied simplify12.8

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

Runtime

Time bar (total: 51.5s)Debug logProfile

herbie shell --seed '#(1070258749 1877548225 2229079127 1588002776 3179087814 1886870650)' +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))