Average Error: 16.4 → 3.0
Time: 5.5s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} \le -0.9999999998847824977943332669383380562067:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\frac{\frac{4}{\alpha}}{\alpha} - \left(\frac{2}{\alpha} - \frac{-8}{{\alpha}^{3}}\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\log \left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1\right)}}{2}\\ \end{array}\]
\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 -0.9999999998847824977943332669383380562067:\\
\;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\frac{\frac{4}{\alpha}}{\alpha} - \left(\frac{2}{\alpha} - \frac{-8}{{\alpha}^{3}}\right)\right)}{2}\\

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

\end{array}
double f(double alpha, double beta) {
        double r171894 = beta;
        double r171895 = alpha;
        double r171896 = r171894 - r171895;
        double r171897 = r171895 + r171894;
        double r171898 = 2.0;
        double r171899 = r171897 + r171898;
        double r171900 = r171896 / r171899;
        double r171901 = 1.0;
        double r171902 = r171900 + r171901;
        double r171903 = r171902 / r171898;
        return r171903;
}
↓
double f(double alpha, double beta) {
        double r171904 = beta;
        double r171905 = alpha;
        double r171906 = r171904 - r171905;
        double r171907 = r171905 + r171904;
        double r171908 = 2.0;
        double r171909 = r171907 + r171908;
        double r171910 = r171906 / r171909;
        double r171911 = -0.9999999998847825;
        bool r171912 = r171910 <= r171911;
        double r171913 = r171904 / r171909;
        double r171914 = 4.0;
        double r171915 = r171914 / r171905;
        double r171916 = r171915 / r171905;
        double r171917 = r171908 / r171905;
        double r171918 = 8.0;
        double r171919 = -r171918;
        double r171920 = 3.0;
        double r171921 = pow(r171905, r171920);
        double r171922 = r171919 / r171921;
        double r171923 = r171917 - r171922;
        double r171924 = r171916 - r171923;
        double r171925 = r171913 - r171924;
        double r171926 = r171925 / r171908;
        double r171927 = 1.0;
        double r171928 = r171910 + r171927;
        double r171929 = log(r171928);
        double r171930 = exp(r171929);
        double r171931 = r171930 / r171908;
        double r171932 = r171912 ? r171926 : r171931;
        return r171932;
}

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)) < -0.9999999998847825

    1. Initial program 60.2

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

    3. Applied div-sub60.2

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

    4. Applied associate-+l-58.4

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

    5. Taylor expanded around inf 10.5

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

    6. Simplified10.5

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

    if -0.9999999998847825 < (/ (- beta alpha) (+ (+ alpha beta) 2.0))

    1. Initial program 0.2

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

    3. Applied add-exp-log0.2

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

  4. Final simplification3.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} \le -0.9999999998847824977943332669383380562067:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\frac{\frac{4}{\alpha}}{\alpha} - \left(\frac{2}{\alpha} - \frac{-8}{{\alpha}^{3}}\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\log \left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1\right)}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019354 
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/1"
  :precision binary64
  :pre (and (> alpha -1) (> beta -1))
  (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2)) 1) 2))