Average Error: 16.1 → 6.5
Time: 22.2s
Precision: 64
Internal Precision: 1344
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 3.2077468605638402 \cdot 10^{+37}:\\ \;\;\;\;\frac{\sqrt[3]{\frac{1}{\left(\beta + \alpha\right) + 2.0} \cdot \beta} \cdot \left(\sqrt[3]{\frac{1}{\left(\beta + \alpha\right) + 2.0} \cdot \beta} \cdot \sqrt[3]{\frac{1}{\left(\beta + \alpha\right) + 2.0} \cdot \beta}\right) - \log \left(e^{\frac{\alpha}{\left(\beta + \alpha\right) + 2.0} - 1.0}\right)}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\left(\beta + \alpha\right) + 2.0} \cdot \beta - \frac{\left(\frac{4.0}{\alpha} - 2.0\right) - \frac{8.000000000000002}{\alpha \cdot \alpha}}{\alpha}}{2.0}\\ \end{array}\]

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. Split input into 2 regimes

  2. if alpha < 3.2077468605638402e+37

    1. Initial program 1.6

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

    3. Applied div-sub1.6

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

    4. Applied associate-+l-1.6

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

    6. Applied add-log-exp1.6

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

    8. Applied div-inv1.6

      \[\leadsto \frac{\color{blue}{\beta \cdot \frac{1}{\left(\alpha + \beta\right) + 2.0}} - \log \left(e^{\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0}\right)}{2.0}\]
    9. Using strategy rm

    10. Applied add-cube-cbrt1.6

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\beta \cdot \frac{1}{\left(\alpha + \beta\right) + 2.0}} \cdot \sqrt[3]{\beta \cdot \frac{1}{\left(\alpha + \beta\right) + 2.0}}\right) \cdot \sqrt[3]{\beta \cdot \frac{1}{\left(\alpha + \beta\right) + 2.0}}} - \log \left(e^{\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0}\right)}{2.0}\]

    if 3.2077468605638402e+37 < alpha

    1. Initial program 50.9

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

    3. Applied div-sub50.8

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

    4. Applied associate-+l-49.2

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

    6. Applied add-log-exp49.2

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

    8. Applied div-inv49.2

      \[\leadsto \frac{\color{blue}{\beta \cdot \frac{1}{\left(\alpha + \beta\right) + 2.0}} - \log \left(e^{\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0}\right)}{2.0}\]

    9. Taylor expanded around -inf 18.2

      \[\leadsto \frac{\beta \cdot \frac{1}{\left(\alpha + \beta\right) + 2.0} - \color{blue}{\left(4.0 \cdot \frac{1}{{\alpha}^{2}} - \left(2.0 \cdot \frac{1}{\alpha} + 8.000000000000002 \cdot \frac{1}{{\alpha}^{3}}\right)\right)}}{2.0}\]

    10. Simplified18.2

      \[\leadsto \frac{\beta \cdot \frac{1}{\left(\alpha + \beta\right) + 2.0} - \color{blue}{\frac{\left(\frac{4.0}{\alpha} - 2.0\right) - \frac{8.000000000000002}{\alpha \cdot \alpha}}{\alpha}}}{2.0}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification6.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 3.2077468605638402 \cdot 10^{+37}:\\ \;\;\;\;\frac{\sqrt[3]{\frac{1}{\left(\beta + \alpha\right) + 2.0} \cdot \beta} \cdot \left(\sqrt[3]{\frac{1}{\left(\beta + \alpha\right) + 2.0} \cdot \beta} \cdot \sqrt[3]{\frac{1}{\left(\beta + \alpha\right) + 2.0} \cdot \beta}\right) - \log \left(e^{\frac{\alpha}{\left(\beta + \alpha\right) + 2.0} - 1.0}\right)}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\left(\beta + \alpha\right) + 2.0} \cdot \beta - \frac{\left(\frac{4.0}{\alpha} - 2.0\right) - \frac{8.000000000000002}{\alpha \cdot \alpha}}{\alpha}}{2.0}\\ \end{array}\]

Runtime

Time bar (total: 22.2s)Debug logProfile

herbie shell --seed 2018234 
(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))