Average Error: 16.4 → 6.0
Time: 5.4s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 6769387663364106240:\\ \;\;\;\;\frac{e^{\log \left(\frac{1}{\frac{\left(\alpha + \beta\right) + 2}{\beta}} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\sqrt[3]{\frac{\beta}{\left(\alpha + \beta\right) + 2}} \cdot \sqrt[3]{\frac{\beta}{\left(\alpha + \beta\right) + 2}}\right) \cdot \sqrt[3]{\frac{\beta}{\left(\alpha + \beta\right) + 2}} - \left(\frac{1}{{\alpha}^{2}} \cdot \left(4 - \frac{8}{\alpha}\right) + \frac{-2}{\alpha}\right)}{2}\\ \end{array}\]
\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \le 6769387663364106240:\\
\;\;\;\;\frac{e^{\log \left(\frac{1}{\frac{\left(\alpha + \beta\right) + 2}{\beta}} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)\right)}}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\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{1}{{\alpha}^{2}} \cdot \left(4 - \frac{8}{\alpha}\right) + \frac{-2}{\alpha}\right)}{2}\\

\end{array}
double f(double alpha, double beta) {
        double r61993 = beta;
        double r61994 = alpha;
        double r61995 = r61993 - r61994;
        double r61996 = r61994 + r61993;
        double r61997 = 2.0;
        double r61998 = r61996 + r61997;
        double r61999 = r61995 / r61998;
        double r62000 = 1.0;
        double r62001 = r61999 + r62000;
        double r62002 = r62001 / r61997;
        return r62002;
}
↓
double f(double alpha, double beta) {
        double r62003 = alpha;
        double r62004 = 6.769387663364106e+18;
        bool r62005 = r62003 <= r62004;
        double r62006 = 1.0;
        double r62007 = beta;
        double r62008 = r62003 + r62007;
        double r62009 = 2.0;
        double r62010 = r62008 + r62009;
        double r62011 = r62010 / r62007;
        double r62012 = r62006 / r62011;
        double r62013 = r62003 / r62010;
        double r62014 = 1.0;
        double r62015 = r62013 - r62014;
        double r62016 = r62012 - r62015;
        double r62017 = log(r62016);
        double r62018 = exp(r62017);
        double r62019 = r62018 / r62009;
        double r62020 = r62007 / r62010;
        double r62021 = cbrt(r62020);
        double r62022 = r62021 * r62021;
        double r62023 = r62022 * r62021;
        double r62024 = 2.0;
        double r62025 = pow(r62003, r62024);
        double r62026 = r62006 / r62025;
        double r62027 = 4.0;
        double r62028 = 8.0;
        double r62029 = r62028 / r62003;
        double r62030 = r62027 - r62029;
        double r62031 = r62026 * r62030;
        double r62032 = -r62009;
        double r62033 = r62032 / r62003;
        double r62034 = r62031 + r62033;
        double r62035 = r62023 - r62034;
        double r62036 = r62035 / r62009;
        double r62037 = r62005 ? r62019 : r62036;
        return r62037;
}

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 < 6.769387663364106e+18

    1. Initial program 0.7

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

    3. Applied div-sub0.7

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

    4. Applied associate-+l-0.7

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

    6. Applied clear-num0.7

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

    8. Applied add-exp-log0.7

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

    if 6.769387663364106e+18 < alpha

    1. Initial program 51.0

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

    3. Applied div-sub50.9

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

    4. Applied associate-+l-49.3

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

    6. Applied add-cube-cbrt49.4

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

    7. Taylor expanded around inf 17.6

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

    8. Simplified17.6

      \[\leadsto \frac{\left(\sqrt[3]{\frac{\beta}{\left(\alpha + \beta\right) + 2}} \cdot \sqrt[3]{\frac{\beta}{\left(\alpha + \beta\right) + 2}}\right) \cdot \sqrt[3]{\frac{\beta}{\left(\alpha + \beta\right) + 2}} - \color{blue}{\left(\frac{1}{{\alpha}^{2}} \cdot \left(4 - \frac{8}{\alpha}\right) + \frac{-2}{\alpha}\right)}}{2}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification6.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 6769387663364106240:\\ \;\;\;\;\frac{e^{\log \left(\frac{1}{\frac{\left(\alpha + \beta\right) + 2}{\beta}} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\sqrt[3]{\frac{\beta}{\left(\alpha + \beta\right) + 2}} \cdot \sqrt[3]{\frac{\beta}{\left(\alpha + \beta\right) + 2}}\right) \cdot \sqrt[3]{\frac{\beta}{\left(\alpha + \beta\right) + 2}} - \left(\frac{1}{{\alpha}^{2}} \cdot \left(4 - \frac{8}{\alpha}\right) + \frac{-2}{\alpha}\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019352 
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/1"
  :precision binary64
  :pre (and (> alpha -1) (> beta -1))
  (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2)) 1) 2))