Average Error: 16.4 → 6.3
Time: 17.9s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 337398417361397.5625:\\ \;\;\;\;\frac{\log \left(e^{\sqrt[3]{\frac{\beta}{2 + \left(\beta + \alpha\right)}}}\right) \cdot \left(\sqrt[3]{\frac{\beta}{2 + \left(\beta + \alpha\right)}} \cdot \sqrt[3]{\frac{\beta}{2 + \left(\beta + \alpha\right)}}\right) - \left(\frac{\alpha}{2 + \left(\beta + \alpha\right)} - 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{2 + \left(\beta + \alpha\right)} - \left(\frac{4}{\alpha \cdot \alpha} - \left(\frac{2}{\alpha} + \frac{\frac{8}{\alpha}}{\alpha \cdot \alpha}\right)\right)}{2}\\ \end{array}\]
\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \le 337398417361397.5625:\\
\;\;\;\;\frac{\log \left(e^{\sqrt[3]{\frac{\beta}{2 + \left(\beta + \alpha\right)}}}\right) \cdot \left(\sqrt[3]{\frac{\beta}{2 + \left(\beta + \alpha\right)}} \cdot \sqrt[3]{\frac{\beta}{2 + \left(\beta + \alpha\right)}}\right) - \left(\frac{\alpha}{2 + \left(\beta + \alpha\right)} - 1\right)}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{\beta}{2 + \left(\beta + \alpha\right)} - \left(\frac{4}{\alpha \cdot \alpha} - \left(\frac{2}{\alpha} + \frac{\frac{8}{\alpha}}{\alpha \cdot \alpha}\right)\right)}{2}\\

\end{array}
double f(double alpha, double beta) {
        double r4446831 = beta;
        double r4446832 = alpha;
        double r4446833 = r4446831 - r4446832;
        double r4446834 = r4446832 + r4446831;
        double r4446835 = 2.0;
        double r4446836 = r4446834 + r4446835;
        double r4446837 = r4446833 / r4446836;
        double r4446838 = 1.0;
        double r4446839 = r4446837 + r4446838;
        double r4446840 = r4446839 / r4446835;
        return r4446840;
}
↓
double f(double alpha, double beta) {
        double r4446841 = alpha;
        double r4446842 = 337398417361397.56;
        bool r4446843 = r4446841 <= r4446842;
        double r4446844 = beta;
        double r4446845 = 2.0;
        double r4446846 = r4446844 + r4446841;
        double r4446847 = r4446845 + r4446846;
        double r4446848 = r4446844 / r4446847;
        double r4446849 = cbrt(r4446848);
        double r4446850 = exp(r4446849);
        double r4446851 = log(r4446850);
        double r4446852 = r4446849 * r4446849;
        double r4446853 = r4446851 * r4446852;
        double r4446854 = r4446841 / r4446847;
        double r4446855 = 1.0;
        double r4446856 = r4446854 - r4446855;
        double r4446857 = r4446853 - r4446856;
        double r4446858 = r4446857 / r4446845;
        double r4446859 = 4.0;
        double r4446860 = r4446841 * r4446841;
        double r4446861 = r4446859 / r4446860;
        double r4446862 = r4446845 / r4446841;
        double r4446863 = 8.0;
        double r4446864 = r4446863 / r4446841;
        double r4446865 = r4446864 / r4446860;
        double r4446866 = r4446862 + r4446865;
        double r4446867 = r4446861 - r4446866;
        double r4446868 = r4446848 - r4446867;
        double r4446869 = r4446868 / r4446845;
        double r4446870 = r4446843 ? r4446858 : r4446869;
        return r4446870;
}

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 < 337398417361397.56

    1. Initial program 0.3

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
    2. Using strategy rm

    3. Applied div-sub0.3

      \[\leadsto \frac{\color{blue}{\left(\frac{\beta}{\left(\alpha + \beta\right) + 2} - \frac{\alpha}{\left(\alpha + \beta\right) + 2}\right)} + 1}{2}\]

    4. Applied associate-+l-0.3

      \[\leadsto \frac{\color{blue}{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)}}{2}\]
    5. Using strategy rm

    6. Applied add-cube-cbrt0.3

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\frac{\beta}{\left(\alpha + \beta\right) + 2}} \cdot \sqrt[3]{\frac{\beta}{\left(\alpha + \beta\right) + 2}}\right) \cdot \sqrt[3]{\frac{\beta}{\left(\alpha + \beta\right) + 2}}} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)}{2}\]
    7. Using strategy rm

    8. Applied add-log-exp0.3

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

    if 337398417361397.56 < alpha

    1. Initial program 50.1

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
    2. Using strategy rm

    3. Applied div-sub50.1

      \[\leadsto \frac{\color{blue}{\left(\frac{\beta}{\left(\alpha + \beta\right) + 2} - \frac{\alpha}{\left(\alpha + \beta\right) + 2}\right)} + 1}{2}\]

    4. Applied associate-+l-48.4

      \[\leadsto \frac{\color{blue}{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)}}{2}\]

    5. Taylor expanded around inf 18.7

      \[\leadsto \frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \color{blue}{\left(4 \cdot \frac{1}{{\alpha}^{2}} - \left(2 \cdot \frac{1}{\alpha} + 8 \cdot \frac{1}{{\alpha}^{3}}\right)\right)}}{2}\]

    6. Simplified18.7

      \[\leadsto \frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \color{blue}{\left(\frac{4}{\alpha \cdot \alpha} - \left(\frac{\frac{8}{\alpha}}{\alpha \cdot \alpha} + \frac{2}{\alpha}\right)\right)}}{2}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification6.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 337398417361397.5625:\\ \;\;\;\;\frac{\log \left(e^{\sqrt[3]{\frac{\beta}{2 + \left(\beta + \alpha\right)}}}\right) \cdot \left(\sqrt[3]{\frac{\beta}{2 + \left(\beta + \alpha\right)}} \cdot \sqrt[3]{\frac{\beta}{2 + \left(\beta + \alpha\right)}}\right) - \left(\frac{\alpha}{2 + \left(\beta + \alpha\right)} - 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{2 + \left(\beta + \alpha\right)} - \left(\frac{4}{\alpha \cdot \alpha} - \left(\frac{2}{\alpha} + \frac{\frac{8}{\alpha}}{\alpha \cdot \alpha}\right)\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019192 
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/1"
  :pre (and (> alpha -1.0) (> beta -1.0))
  (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0))