Average Error: 24.0 → 12.6
Time: 2.9m
Precision: 64
Internal Precision: 1344
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
↓
\[\frac{\sqrt[3]{\left(1.0 + \left(\sqrt[3]{\alpha + \beta} \cdot \sqrt[3]{\alpha + \beta}\right) \cdot \left(\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{2.0 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \cdot \sqrt[3]{\alpha + \beta}\right)\right) \cdot \left(\left(1.0 + \left(\sqrt[3]{\alpha + \beta} \cdot \sqrt[3]{\alpha + \beta}\right) \cdot \left(\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{2.0 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \cdot \sqrt[3]{\alpha + \beta}\right)\right) \cdot \left(1.0 + \left(\sqrt[3]{\alpha + \beta} \cdot \sqrt[3]{\alpha + \beta}\right) \cdot \left(\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{2.0 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \cdot \sqrt[3]{\alpha + \beta}\right)\right)\right)}}{2.0}\]

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 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. Using strategy rm

  3. Applied *-un-lft-identity24.0

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

  4. Applied *-un-lft-identity24.0

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

  5. Applied times-frac12.4

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

  6. Applied times-frac12.4

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

  7. Simplified12.4

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

  9. Applied add-cube-cbrt12.7

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

  10. Applied associate-*l*12.7

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

  12. Applied add-cbrt-cube12.6

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

  13. Final simplification12.6

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

Runtime

Time bar (total: 2.9m)Debug logProfile

herbie shell --seed 2018225 
(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))