Error

Derivation

1. Initial program 23.4

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

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 add-cube-cbrt12.5

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

5. Applied add-cube-cbrt12.2

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

6. Applied times-frac12.2

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

8. Applied fma-udef12.2

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

10. Applied flip3-+12.3

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

11. Final simplification12.3

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

Runtime

Time bar (total: 37.2s)Debug logProfile

herbie shell --seed 2018340 +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))