Average Error: 16.0 → 6.3
Time: 27.4s
Precision: 64
Internal Precision: 128
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 162608019.50338015:\\ \;\;\;\;\frac{\frac{\beta}{2.0 + \left(\beta + \alpha\right)} - \frac{\sqrt[3]{{\left(\frac{\alpha}{2.0 + \left(\beta + \alpha\right)}\right)}^{3} \cdot \left({\left(\frac{\alpha}{2.0 + \left(\beta + \alpha\right)}\right)}^{3} \cdot {\left(\frac{\alpha}{2.0 + \left(\beta + \alpha\right)}\right)}^{3}\right)} - {1.0}^{3}}{\frac{\alpha}{2.0 + \left(\beta + \alpha\right)} \cdot \frac{\alpha}{2.0 + \left(\beta + \alpha\right)} + \left(1.0 \cdot \frac{\alpha}{2.0 + \left(\beta + \alpha\right)} + 1.0 \cdot 1.0\right)}}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{\frac{\beta}{2.0 + \left(\beta + \alpha\right)}} \cdot \left(\sqrt[3]{\frac{\beta}{2.0 + \left(\beta + \alpha\right)}} \cdot \sqrt[3]{\frac{\beta}{2.0 + \left(\beta + \alpha\right)}}\right) - \frac{\left(\frac{4.0}{\alpha} - 2.0\right) - \frac{8.0}{\alpha \cdot \alpha}}{\alpha}}{2.0}\\ \end{array}\]

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. Split input into 2 regimes

  2. if alpha < 162608019.50338015

    1. Initial program 0.1

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

    3. Applied div-sub0.1

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

    4. Applied associate-+l-0.1

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

    6. Applied flip3--0.1

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

    8. Applied add-cbrt-cube0.1

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

    if 162608019.50338015 < alpha

    1. Initial program 48.8

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

    3. Applied div-sub48.8

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

    4. Applied associate-+l-47.2

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

    6. Applied add-cube-cbrt47.2

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

    7. Taylor expanded around -inf 19.0

      \[\leadsto \frac{\left(\sqrt[3]{\frac{\beta}{\left(\alpha + \beta\right) + 2.0}} \cdot \sqrt[3]{\frac{\beta}{\left(\alpha + \beta\right) + 2.0}}\right) \cdot \sqrt[3]{\frac{\beta}{\left(\alpha + \beta\right) + 2.0}} - \color{blue}{\left(4.0 \cdot \frac{1}{{\alpha}^{2}} - \left(2.0 \cdot \frac{1}{\alpha} + 8.0 \cdot \frac{1}{{\alpha}^{3}}\right)\right)}}{2.0}\]

    8. Simplified19.0

      \[\leadsto \frac{\left(\sqrt[3]{\frac{\beta}{\left(\alpha + \beta\right) + 2.0}} \cdot \sqrt[3]{\frac{\beta}{\left(\alpha + \beta\right) + 2.0}}\right) \cdot \sqrt[3]{\frac{\beta}{\left(\alpha + \beta\right) + 2.0}} - \color{blue}{\frac{\left(\frac{4.0}{\alpha} - 2.0\right) - \frac{8.0}{\alpha \cdot \alpha}}{\alpha}}}{2.0}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification6.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 162608019.50338015:\\ \;\;\;\;\frac{\frac{\beta}{2.0 + \left(\beta + \alpha\right)} - \frac{\sqrt[3]{{\left(\frac{\alpha}{2.0 + \left(\beta + \alpha\right)}\right)}^{3} \cdot \left({\left(\frac{\alpha}{2.0 + \left(\beta + \alpha\right)}\right)}^{3} \cdot {\left(\frac{\alpha}{2.0 + \left(\beta + \alpha\right)}\right)}^{3}\right)} - {1.0}^{3}}{\frac{\alpha}{2.0 + \left(\beta + \alpha\right)} \cdot \frac{\alpha}{2.0 + \left(\beta + \alpha\right)} + \left(1.0 \cdot \frac{\alpha}{2.0 + \left(\beta + \alpha\right)} + 1.0 \cdot 1.0\right)}}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{\frac{\beta}{2.0 + \left(\beta + \alpha\right)}} \cdot \left(\sqrt[3]{\frac{\beta}{2.0 + \left(\beta + \alpha\right)}} \cdot \sqrt[3]{\frac{\beta}{2.0 + \left(\beta + \alpha\right)}}\right) - \frac{\left(\frac{4.0}{\alpha} - 2.0\right) - \frac{8.0}{\alpha \cdot \alpha}}{\alpha}}{2.0}\\ \end{array}\]

Runtime

Time bar (total: 27.4s)Debug logProfile

BaselineHerbieOracleSpan%
Regimes15.56.33.112.574.1%
herbie shell --seed 2018353 
(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))