Average Error: 23.2 → 12.0
Time: 43.7s
Precision: 64
Internal Precision: 1344
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
↓
\[\frac{e^{\log \left({\left((\left(\frac{\beta - \alpha}{(2 \cdot i + \alpha)_* + \left(2.0 + \beta\right)}\right) \cdot \left(\left(\alpha + \beta\right) \cdot \frac{1}{\beta + (2 \cdot i + \alpha)_*}\right) + 1.0)_* \cdot \left((\left(\frac{\beta - \alpha}{(2 \cdot i + \alpha)_* + \left(2.0 + \beta\right)}\right) \cdot \left(\left(\alpha + \beta\right) \cdot \frac{1}{\beta + (2 \cdot i + \alpha)_*}\right) + 1.0)_* \cdot (\left(\frac{\beta - \alpha}{(2 \cdot i + \alpha)_* + \left(2.0 + \beta\right)}\right) \cdot \left(\left(\alpha + \beta\right) \cdot \frac{1}{\beta + (2 \cdot i + \alpha)_*}\right) + 1.0)_*\right)\right)}^{\frac{1}{3}}\right)}}{2.0}\]

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

  1. Initial program 23.2

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

  2. Initial simplification12.1

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

  4. Applied div-inv12.0

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

  6. Applied add-cbrt-cube12.1

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

  8. Applied pow1/312.0

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

  10. Applied add-exp-log12.0

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

  11. Final simplification12.0

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

Runtime

Time bar (total: 43.7s)Debug logProfile

herbie shell --seed 2018235 +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))