Average Error: 24.0 → 12.4
Time: 8.0m
Precision: 64
Internal Precision: 1344
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
↓
\[\frac{e^{\sqrt[3]{\left(\log \left((\left(\sqrt[3]{\left(\left(\frac{\alpha + \beta}{\sqrt{\left(\beta + 2.0\right) + (2 \cdot i + \alpha)_*}} \cdot \frac{1}{\sqrt{\left(\beta + 2.0\right) + (2 \cdot i + \alpha)_*}}\right) \cdot \left(\frac{\alpha + \beta}{\sqrt{\left(\beta + 2.0\right) + (2 \cdot i + \alpha)_*}} \cdot \frac{1}{\sqrt{\left(\beta + 2.0\right) + (2 \cdot i + \alpha)_*}}\right)\right) \cdot \left(\frac{\alpha + \beta}{\sqrt{\left(\beta + 2.0\right) + (2 \cdot i + \alpha)_*}} \cdot \frac{1}{\sqrt{\left(\beta + 2.0\right) + (2 \cdot i + \alpha)_*}}\right)}\right) \cdot \left(\frac{\beta - \alpha}{\beta + (2 \cdot i + \alpha)_*}\right) + 1.0)_*\right) \cdot \log \left((\left(\sqrt[3]{\left(\left(\frac{\alpha + \beta}{\sqrt{\left(\beta + 2.0\right) + (2 \cdot i + \alpha)_*}} \cdot \frac{1}{\sqrt{\left(\beta + 2.0\right) + (2 \cdot i + \alpha)_*}}\right) \cdot \left(\frac{\alpha + \beta}{\sqrt{\left(\beta + 2.0\right) + (2 \cdot i + \alpha)_*}} \cdot \frac{1}{\sqrt{\left(\beta + 2.0\right) + (2 \cdot i + \alpha)_*}}\right)\right) \cdot \left(\frac{\alpha + \beta}{\sqrt{\left(\beta + 2.0\right) + (2 \cdot i + \alpha)_*}} \cdot \frac{1}{\sqrt{\left(\beta + 2.0\right) + (2 \cdot i + \alpha)_*}}\right)}\right) \cdot \left(\frac{\beta - \alpha}{\beta + (2 \cdot i + \alpha)_*}\right) + 1.0)_*\right)\right) \cdot \log \left((\left(\sqrt[3]{\left(\left(\frac{\alpha + \beta}{\sqrt{\left(\beta + 2.0\right) + (2 \cdot i + \alpha)_*}} \cdot \frac{1}{\sqrt{\left(\beta + 2.0\right) + (2 \cdot i + \alpha)_*}}\right) \cdot \left(\frac{\alpha + \beta}{\sqrt{\left(\beta + 2.0\right) + (2 \cdot i + \alpha)_*}} \cdot \frac{1}{\sqrt{\left(\beta + 2.0\right) + (2 \cdot i + \alpha)_*}}\right)\right) \cdot \left(\frac{\alpha + \beta}{\sqrt{\left(\beta + 2.0\right) + (2 \cdot i + \alpha)_*}} \cdot \frac{1}{\sqrt{\left(\beta + 2.0\right) + (2 \cdot i + \alpha)_*}}\right)}\right) \cdot \left(\frac{\beta - \alpha}{\beta + (2 \cdot i + \alpha)_*}\right) + 1.0)_*\right)}}}{2.0}\]

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

  1. Initial program 24.0

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

  2. Initial simplification12.4

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

  4. Applied add-sqr-sqrt12.5

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

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

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

  6. Applied times-frac12.4

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

  8. Applied add-cbrt-cube12.4

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

  10. Applied add-exp-log12.4

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

  12. Applied add-cbrt-cube12.4

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

  13. Final simplification12.4

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

Runtime

Time bar (total: 8.0m)Debug logProfile

herbie shell --seed 2018225 +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))