Average Error: 16.1 → 16.1
Time: 35.6s
Precision: 64
Internal Precision: 128
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}\]
↓
\[\frac{e^{\sqrt[3]{\log \left(1.0 + \left(\frac{\beta}{\left(\beta + \alpha\right) + 2.0} - \frac{\alpha}{\left(\beta + \alpha\right) + 2.0}\right)\right) \cdot \left(\log \left(1.0 + \left(\frac{\beta}{\left(\beta + \alpha\right) + 2.0} - \frac{\alpha}{\left(\beta + \alpha\right) + 2.0}\right)\right) \cdot \log \left(1.0 + \left(\frac{\beta}{\left(\beta + \alpha\right) + 2.0} - \frac{\alpha}{\left(\beta + \alpha\right) + 2.0}\right)\right)\right)}}}{2.0}\]

Error

Bits error versus alpha

Bits error versus beta

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 16.1

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

  2. Initial simplification16.1

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

  4. Applied add-exp-log16.1

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

  6. Applied div-sub16.1

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

  8. Applied add-cbrt-cube16.1

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

  9. Final simplification16.1

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

Runtime

Time bar (total: 35.6s)Debug logProfile

herbie shell --seed 2018346 
(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))