Average Error: 16.5 → 6.1
Time: 19.0s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 12052311324865314.0:\\ \;\;\;\;\frac{\sqrt[3]{\frac{\beta}{2.0 + \left(\beta + \alpha\right)} \cdot \left(\frac{\beta}{2.0 + \left(\beta + \alpha\right)} \cdot \frac{\beta}{2.0 + \left(\beta + \alpha\right)}\right)} - \left(\frac{\alpha}{2.0 + \left(\beta + \alpha\right)} - 1.0\right)}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{2.0 + \left(\beta + \alpha\right)} - \left(\frac{4.0}{\alpha \cdot \alpha} - \left(\frac{\frac{8.0}{\alpha \cdot \alpha}}{\alpha} + \frac{2.0}{\alpha}\right)\right)}{2.0}\\ \end{array}\]
\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \le 12052311324865314.0:\\
\;\;\;\;\frac{\sqrt[3]{\frac{\beta}{2.0 + \left(\beta + \alpha\right)} \cdot \left(\frac{\beta}{2.0 + \left(\beta + \alpha\right)} \cdot \frac{\beta}{2.0 + \left(\beta + \alpha\right)}\right)} - \left(\frac{\alpha}{2.0 + \left(\beta + \alpha\right)} - 1.0\right)}{2.0}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{\beta}{2.0 + \left(\beta + \alpha\right)} - \left(\frac{4.0}{\alpha \cdot \alpha} - \left(\frac{\frac{8.0}{\alpha \cdot \alpha}}{\alpha} + \frac{2.0}{\alpha}\right)\right)}{2.0}\\

\end{array}
double f(double alpha, double beta) {
        double r3014842 = beta;
        double r3014843 = alpha;
        double r3014844 = r3014842 - r3014843;
        double r3014845 = r3014843 + r3014842;
        double r3014846 = 2.0;
        double r3014847 = r3014845 + r3014846;
        double r3014848 = r3014844 / r3014847;
        double r3014849 = 1.0;
        double r3014850 = r3014848 + r3014849;
        double r3014851 = r3014850 / r3014846;
        return r3014851;
}
↓
double f(double alpha, double beta) {
        double r3014852 = alpha;
        double r3014853 = 12052311324865314.0;
        bool r3014854 = r3014852 <= r3014853;
        double r3014855 = beta;
        double r3014856 = 2.0;
        double r3014857 = r3014855 + r3014852;
        double r3014858 = r3014856 + r3014857;
        double r3014859 = r3014855 / r3014858;
        double r3014860 = r3014859 * r3014859;
        double r3014861 = r3014859 * r3014860;
        double r3014862 = cbrt(r3014861);
        double r3014863 = r3014852 / r3014858;
        double r3014864 = 1.0;
        double r3014865 = r3014863 - r3014864;
        double r3014866 = r3014862 - r3014865;
        double r3014867 = r3014866 / r3014856;
        double r3014868 = 4.0;
        double r3014869 = r3014852 * r3014852;
        double r3014870 = r3014868 / r3014869;
        double r3014871 = 8.0;
        double r3014872 = r3014871 / r3014869;
        double r3014873 = r3014872 / r3014852;
        double r3014874 = r3014856 / r3014852;
        double r3014875 = r3014873 + r3014874;
        double r3014876 = r3014870 - r3014875;
        double r3014877 = r3014859 - r3014876;
        double r3014878 = r3014877 / r3014856;
        double r3014879 = r3014854 ? r3014867 : r3014878;
        return r3014879;
}

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 < 12052311324865314.0

    1. Initial program 0.4

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

    3. Applied div-sub0.4

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

    4. Applied associate-+l-0.4

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

    6. Applied add-cbrt-cube0.4

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

    if 12052311324865314.0 < alpha

    1. Initial program 50.3

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

    3. Applied div-sub50.3

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

    4. Applied associate-+l-48.8

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

    5. Taylor expanded around inf 18.0

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

    6. Simplified18.0

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

  4. Final simplification6.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 12052311324865314.0:\\ \;\;\;\;\frac{\sqrt[3]{\frac{\beta}{2.0 + \left(\beta + \alpha\right)} \cdot \left(\frac{\beta}{2.0 + \left(\beta + \alpha\right)} \cdot \frac{\beta}{2.0 + \left(\beta + \alpha\right)}\right)} - \left(\frac{\alpha}{2.0 + \left(\beta + \alpha\right)} - 1.0\right)}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{2.0 + \left(\beta + \alpha\right)} - \left(\frac{4.0}{\alpha \cdot \alpha} - \left(\frac{\frac{8.0}{\alpha \cdot \alpha}}{\alpha} + \frac{2.0}{\alpha}\right)\right)}{2.0}\\ \end{array}\]

Reproduce

herbie shell --seed 2019146 
(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))