Average Error: 16.5 → 16.5
Time: 5.5m
Precision: 64
Internal Precision: 128
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}\]
↓
\[\frac{\frac{\frac{e^{\log \left(\sqrt[3]{\left({\left({\left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0}\right)}^{3}\right)}^{3} \cdot {\left({\left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0}\right)}^{3}\right)}^{3}\right) \cdot {\left({\left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0}\right)}^{3}\right)}^{3}} + {\left({1.0}^{3}\right)}^{3}\right)}}{\left({\left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0}\right)}^{3} \cdot {\left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0}\right)}^{3} - {\left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0}\right)}^{3} \cdot {1.0}^{3}\right) + {1.0}^{3} \cdot {1.0}^{3}}}{1.0 \cdot 1.0 + \left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} \cdot \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0 \cdot \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0}\right)}}{2.0}\]

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 16.5

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

  2. Initial simplification16.5

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

  4. Applied flip3-+16.5

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

  6. Applied flip3-+16.5

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

  8. Applied add-cbrt-cube16.5

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

  10. Applied add-exp-log16.5

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

  11. Final simplification16.5

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

Runtime

Time bar (total: 5.5m)Debug logProfile

herbie shell --seed 2018277 
(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))