Average Error: 16.6 → 16.6
Time: 3.3m
Precision: 64
Internal Precision: 128
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}\]
↓
\[\frac{\frac{\frac{{\left(\log \left(e^{{\left(\frac{\beta}{2.0 + \left(\alpha + \beta\right)} - \frac{\alpha}{2.0 + \left(\alpha + \beta\right)}\right)}^{3}}\right)\right)}^{3} + {\left({1.0}^{3}\right)}^{3}}{{1.0}^{3} \cdot {1.0}^{3} + \left(\log \left(e^{{\left(\frac{\beta}{2.0 + \left(\alpha + \beta\right)} - \frac{\alpha}{2.0 + \left(\alpha + \beta\right)}\right)}^{3}}\right) \cdot \log \left(e^{{\left(\frac{\beta}{2.0 + \left(\alpha + \beta\right)} - \frac{\alpha}{2.0 + \left(\alpha + \beta\right)}\right)}^{3}}\right) - {1.0}^{3} \cdot \log \left(e^{{\left(\frac{\beta}{2.0 + \left(\alpha + \beta\right)} - \frac{\alpha}{2.0 + \left(\alpha + \beta\right)}\right)}^{3}}\right)\right)}}{\left(\left(\frac{\beta}{2.0 + \left(\alpha + \beta\right)} - \frac{\alpha}{2.0 + \left(\alpha + \beta\right)}\right) \cdot \left(\frac{\beta}{2.0 + \left(\alpha + \beta\right)} - \frac{\alpha}{2.0 + \left(\alpha + \beta\right)}\right) - \left(\frac{\beta}{2.0 + \left(\alpha + \beta\right)} - \frac{\alpha}{2.0 + \left(\alpha + \beta\right)}\right) \cdot 1.0\right) + 1.0 \cdot 1.0}}{2.0}\]

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 16.6

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

  2. Initial simplification16.6

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

  4. Applied div-sub16.6

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

  6. Applied flip3-+16.6

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

  8. Applied add-log-exp16.6

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

  10. Applied flip3-+16.6

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

  11. Final simplification16.6

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

Runtime

Time bar (total: 3.3m)Debug logProfile

BaselineHerbieOracleSpan%
Regimes16.616.616.30.30%
herbie shell --seed 2018354 
(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))