Average Error: 23.4 → 12.4
Time: 6.8m
Precision: 64
Internal Precision: 1344
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
↓
\[\frac{\left(\frac{\frac{\beta + \alpha}{(2 \cdot i + \beta)_* + \left(2.0 + \alpha\right)}}{\sqrt[3]{(i \cdot 2 + \left(\beta + \alpha\right))_*}} \cdot \frac{\sqrt[3]{\beta - \alpha} \cdot \sqrt[3]{\beta - \alpha}}{\sqrt[3]{(i \cdot 2 + \left(\beta + \alpha\right))_*}}\right) \cdot \frac{\sqrt[3]{\beta - \alpha}}{\sqrt[3]{(2 \cdot i + \left(\beta + \alpha\right))_*}} + 1.0}{2.0}\]

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Try it out

  1. Inputs

  2. Original Output:

    Herbie Output:

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. Applied simplify12.3

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

  4. Applied fma-udef12.3

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

  6. Applied add-cube-cbrt12.6

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

  7. Applied *-un-lft-identity12.6

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

  8. Applied times-frac12.6

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

  9. Applied associate-*r*12.6

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

  10. Applied simplify12.6

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

  12. Applied *-un-lft-identity12.6

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

  13. Applied add-cube-cbrt12.4

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

  14. Applied times-frac12.4

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

  15. Applied associate-*r*12.4

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

  16. Applied simplify12.4

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

Runtime

Time bar (total: 6.8m)Debug logProfile

herbie shell --seed '#(1072361757 3390613284 2339397988 1175251238 145061547 3101881848)' +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))