Error

Derivation

1. Initial program 23.8

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

2. Initial simplification12.4

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

4. Applied add-sqr-sqrt12.5

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

5. Applied *-un-lft-identity12.5

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

6. Applied times-frac12.5

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

8. Applied add-cbrt-cube12.5

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

9. Final simplification12.5

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

Runtime

Time bar (total: 48.7s)Debug logProfile

herbie shell --seed 2018215 +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))