Average Error: 16.8 → 3.2
Time: 9.4s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} \le -1:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\left(\frac{4}{\alpha \cdot \alpha} - \frac{2}{\alpha}\right) - \frac{8}{{\alpha}^{3}}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \log \left(e^{\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1}\right)}{2}\\ \end{array}\]
\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}
↓
\begin{array}{l}
\mathbf{if}\;\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} \le -1:\\
\;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\left(\frac{4}{\alpha \cdot \alpha} - \frac{2}{\alpha}\right) - \frac{8}{{\alpha}^{3}}\right)}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \log \left(e^{\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1}\right)}{2}\\

\end{array}
double f(double alpha, double beta) {
        double r107858 = beta;
        double r107859 = alpha;
        double r107860 = r107858 - r107859;
        double r107861 = r107859 + r107858;
        double r107862 = 2.0;
        double r107863 = r107861 + r107862;
        double r107864 = r107860 / r107863;
        double r107865 = 1.0;
        double r107866 = r107864 + r107865;
        double r107867 = r107866 / r107862;
        return r107867;
}
↓
double f(double alpha, double beta) {
        double r107868 = beta;
        double r107869 = alpha;
        double r107870 = r107868 - r107869;
        double r107871 = r107869 + r107868;
        double r107872 = 2.0;
        double r107873 = r107871 + r107872;
        double r107874 = r107870 / r107873;
        double r107875 = -1.0;
        bool r107876 = r107874 <= r107875;
        double r107877 = r107868 / r107873;
        double r107878 = 4.0;
        double r107879 = r107869 * r107869;
        double r107880 = r107878 / r107879;
        double r107881 = r107872 / r107869;
        double r107882 = r107880 - r107881;
        double r107883 = 8.0;
        double r107884 = 3.0;
        double r107885 = pow(r107869, r107884);
        double r107886 = r107883 / r107885;
        double r107887 = r107882 - r107886;
        double r107888 = r107877 - r107887;
        double r107889 = r107888 / r107872;
        double r107890 = r107869 / r107873;
        double r107891 = 1.0;
        double r107892 = r107890 - r107891;
        double r107893 = exp(r107892);
        double r107894 = log(r107893);
        double r107895 = r107877 - r107894;
        double r107896 = r107895 / r107872;
        double r107897 = r107876 ? r107889 : r107896;
        return r107897;
}

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 (/ (- beta alpha) (+ (+ alpha beta) 2.0)) < -1.0

    1. Initial program 60.6

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

    3. Applied div-sub60.6

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

    4. Applied associate-+l-58.8

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

    5. Taylor expanded around inf 10.4

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

    6. Simplified10.4

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

    if -1.0 < (/ (- beta alpha) (+ (+ alpha beta) 2.0))

    1. Initial program 0.6

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

    3. Applied div-sub0.6

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

    4. Applied associate-+l-0.6

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

    6. Applied add-log-exp0.6

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

    7. Applied add-log-exp0.6

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

    8. Applied diff-log0.6

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

    9. Simplified0.6

      \[\leadsto \frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \log \color{blue}{\left(e^{\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1}\right)}}{2}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification3.2

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

Reproduce

herbie shell --seed 2020046 
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/1"
  :precision binary64
  :pre (and (> alpha -1) (> beta -1))
  (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2)) 1) 2))