Average Error: 16.3 → 16.4
Time: 1.3m
Precision: 64
Internal Precision: 128
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}\]
↓
\[\frac{\sqrt[3]{\left(1.0 + \frac{\beta - \alpha}{2.0 + \left(\beta + \alpha\right)}\right) \cdot \left(e^{\log \left(1.0 + \frac{\beta - \alpha}{2.0 + \left(\beta + \alpha\right)}\right)} \cdot \frac{{\left(1.0 + \frac{\beta}{2.0 + \left(\beta + \alpha\right)}\right)}^{3} - {\left(\frac{\alpha}{2.0 + \left(\beta + \alpha\right)}\right)}^{3}}{\left(\frac{\alpha}{2.0 + \left(\beta + \alpha\right)} \cdot \left(1.0 + \frac{\beta}{2.0 + \left(\beta + \alpha\right)}\right) + \frac{\alpha}{2.0 + \left(\beta + \alpha\right)} \cdot \frac{\alpha}{2.0 + \left(\beta + \alpha\right)}\right) + \left(1.0 + \frac{\beta}{2.0 + \left(\beta + \alpha\right)}\right) \cdot \left(1.0 + \frac{\beta}{2.0 + \left(\beta + \alpha\right)}\right)}\right)}}{2.0}\]

Error

Bits error versus alpha

Bits error versus beta

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 16.3

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

  2. Initial simplification16.3

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

  4. Applied add-cbrt-cube16.4

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

  6. Applied div-sub16.4

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

  7. Applied associate-+r-16.4

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

  9. Applied flip3--16.4

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

  11. Applied add-exp-log16.4

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

  12. Final simplification16.4

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

Runtime

Time bar (total: 1.3m)Debug logProfile

herbie shell --seed 2018349 
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/1"
  :pre (and (> alpha -1) (> beta -1))
  (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0))