Average Error: 16.4 → 5.9
Time: 22.1s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 460518623.3564523:\\ \;\;\;\;e^{\log \left(\frac{\frac{1}{\frac{\left(\alpha + \beta\right) + 2.0}{\beta}} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0\right)}{2.0}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \left(\frac{4.0}{\alpha \cdot \alpha} - \left(\frac{2.0}{\alpha} + \frac{\frac{8.0}{\alpha}}{\alpha \cdot \alpha}\right)\right)}{2.0}\\ \end{array}\]
\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \le 460518623.3564523:\\
\;\;\;\;e^{\log \left(\frac{\frac{1}{\frac{\left(\alpha + \beta\right) + 2.0}{\beta}} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0\right)}{2.0}\right)}\\

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

\end{array}
double f(double alpha, double beta) {
        double r3614793 = beta;
        double r3614794 = alpha;
        double r3614795 = r3614793 - r3614794;
        double r3614796 = r3614794 + r3614793;
        double r3614797 = 2.0;
        double r3614798 = r3614796 + r3614797;
        double r3614799 = r3614795 / r3614798;
        double r3614800 = 1.0;
        double r3614801 = r3614799 + r3614800;
        double r3614802 = r3614801 / r3614797;
        return r3614802;
}
↓
double f(double alpha, double beta) {
        double r3614803 = alpha;
        double r3614804 = 460518623.3564523;
        bool r3614805 = r3614803 <= r3614804;
        double r3614806 = 1.0;
        double r3614807 = beta;
        double r3614808 = r3614803 + r3614807;
        double r3614809 = 2.0;
        double r3614810 = r3614808 + r3614809;
        double r3614811 = r3614810 / r3614807;
        double r3614812 = r3614806 / r3614811;
        double r3614813 = r3614803 / r3614810;
        double r3614814 = 1.0;
        double r3614815 = r3614813 - r3614814;
        double r3614816 = r3614812 - r3614815;
        double r3614817 = r3614816 / r3614809;
        double r3614818 = log(r3614817);
        double r3614819 = exp(r3614818);
        double r3614820 = r3614807 / r3614810;
        double r3614821 = 4.0;
        double r3614822 = r3614803 * r3614803;
        double r3614823 = r3614821 / r3614822;
        double r3614824 = r3614809 / r3614803;
        double r3614825 = 8.0;
        double r3614826 = r3614825 / r3614803;
        double r3614827 = r3614826 / r3614822;
        double r3614828 = r3614824 + r3614827;
        double r3614829 = r3614823 - r3614828;
        double r3614830 = r3614820 - r3614829;
        double r3614831 = r3614830 / r3614809;
        double r3614832 = r3614805 ? r3614819 : r3614831;
        return r3614832;
}

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 < 460518623.3564523

    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 add-exp-log0.1

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

    8. Applied clear-num0.1

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

    if 460518623.3564523 < alpha

    1. Initial program 50.0

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

    3. Applied div-sub50.0

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

    4. Applied associate-+l-48.5

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

    5. Taylor expanded around inf 17.9

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

    6. Simplified17.9

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

  4. Final simplification5.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 460518623.3564523:\\ \;\;\;\;e^{\log \left(\frac{\frac{1}{\frac{\left(\alpha + \beta\right) + 2.0}{\beta}} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0\right)}{2.0}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \left(\frac{4.0}{\alpha \cdot \alpha} - \left(\frac{2.0}{\alpha} + \frac{\frac{8.0}{\alpha}}{\alpha \cdot \alpha}\right)\right)}{2.0}\\ \end{array}\]

Reproduce

herbie shell --seed 2019162 +o rules:numerics
(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))