Average Error: 22.9 → 11.9
Time: 2.6m
Precision: 64
Internal Precision: 1344
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
↓
\[\frac{\log \left(e^{1.0 + \left(\frac{\sqrt[3]{\frac{\beta - \alpha}{2 \cdot i + \left(\alpha + \beta\right)}}}{2.0 + \left(2 \cdot i + \left(\alpha + \beta\right)\right)} \cdot \left(\sqrt[3]{\frac{\beta - \alpha}{2 \cdot i + \left(\alpha + \beta\right)}} \cdot \sqrt[3]{\frac{\beta - \alpha}{2 \cdot i + \left(\alpha + \beta\right)}}\right)\right) \cdot \left(\alpha + \beta\right)}\right)}{2.0}\]

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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. Using strategy rm

  3. Applied *-un-lft-identity22.9

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

  4. Applied *-un-lft-identity22.9

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

  5. Applied times-frac11.8

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

  6. Applied times-frac11.8

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

  7. Simplified11.8

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

  9. Applied add-log-exp11.8

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

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

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

  12. Applied add-cube-cbrt11.9

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

  13. Applied times-frac11.9

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

  14. Simplified11.9

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

  15. Final simplification11.9

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

Runtime

Time bar (total: 2.6m)Debug logProfile

herbie shell --seed 2018234 
(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))