Average Error: 16.4 → 16.4
Time: 1.4m
Precision: 64
Internal Precision: 1344
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}\]
↓
\[\frac{e^{\log \left(\frac{\log \left(\sqrt{e^{{\left(\frac{\beta - \alpha}{2.0 + \left(\beta + \alpha\right)}\right)}^{3} + {1.0}^{3}}}\right) + \log \left(\sqrt{e^{{\left(\frac{\beta - \alpha}{2.0 + \left(\beta + \alpha\right)}\right)}^{3} + {1.0}^{3}}}\right)}{1.0 \cdot 1.0 + \left(\frac{\beta - \alpha}{2.0 + \left(\beta + \alpha\right)} \cdot \frac{\beta - \alpha}{2.0 + \left(\beta + \alpha\right)} - \frac{\beta - \alpha}{2.0 + \left(\beta + \alpha\right)} \cdot 1.0\right)}\right)}}{2.0}\]

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 16.4

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

  2. Initial simplification16.4

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

  4. Applied add-exp-log16.4

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

  6. Applied flip3-+16.4

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

  8. Applied add-log-exp16.4

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

  10. Applied add-sqr-sqrt16.5

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

  11. Applied log-prod16.4

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

  12. Final simplification16.4

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

Runtime

Time bar (total: 1.4m)Debug logProfile

herbie shell --seed 2018256 
(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))