Average Error: 22.9 → 11.9
Time: 4.1m
Precision: 64
Internal Precision: 1344
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
↓
\[\frac{\log \left(e^{\left(\left(\beta + \alpha\right) \cdot \left(\sqrt[3]{\frac{\beta - \alpha}{\beta + (i \cdot 2 + \alpha)_*}} \cdot \sqrt[3]{\frac{\beta - \alpha}{\beta + (i \cdot 2 + \alpha)_*}}\right)\right) \cdot \frac{\sqrt[3]{\frac{\beta - \alpha}{\beta + (i \cdot 2 + \alpha)_*}}}{2.0 + \left(\beta + (i \cdot 2 + \alpha)_*\right)} + 1.0}\right)}{2.0}\]

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

  1. Initial program 22.9

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

  2. Initial simplification22.7

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

  4. Applied *-un-lft-identity22.7

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

  5. Applied times-frac18.9

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

  6. Simplified18.9

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

  7. Simplified18.9

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

  9. Applied associate-/r*11.8

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

  11. Applied add-log-exp11.8

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

  13. Applied *-un-lft-identity11.8

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

  14. Applied add-cube-cbrt11.9

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

  15. Applied times-frac11.9

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

  16. Applied associate-*r*11.9

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

  17. Final simplification11.9

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

Runtime

Time bar (total: 4.1m)Debug logProfile

herbie shell --seed 2018234 +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))