Average Error: 16.0 → 3.1
Time: 16.2s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} \le -1:\\ \;\;\;\;\frac{\left(\sqrt[3]{\frac{\beta}{\left(\alpha + \beta\right) + 2}} \cdot \sqrt[3]{\frac{\beta}{\left(\alpha + \beta\right) + 2}}\right) \cdot \sqrt[3]{\frac{\beta}{\left(\alpha + \beta\right) + 2}} - \left(\frac{1}{{\alpha}^{2}} \cdot \left(4 - \frac{8}{\alpha}\right) + \frac{-2}{\alpha}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(e^{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}\right)}{2}\\ \end{array}\]
\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 -1:\\
\;\;\;\;\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}\\

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

\end{array}
double f(double alpha, double beta) {
        double r63619 = beta;
        double r63620 = alpha;
        double r63621 = r63619 - r63620;
        double r63622 = r63620 + r63619;
        double r63623 = 2.0;
        double r63624 = r63622 + r63623;
        double r63625 = r63621 / r63624;
        double r63626 = 1.0;
        double r63627 = r63625 + r63626;
        double r63628 = r63627 / r63623;
        return r63628;
}
↓
double f(double alpha, double beta) {
        double r63629 = beta;
        double r63630 = alpha;
        double r63631 = r63629 - r63630;
        double r63632 = r63630 + r63629;
        double r63633 = 2.0;
        double r63634 = r63632 + r63633;
        double r63635 = r63631 / r63634;
        double r63636 = -1.0;
        bool r63637 = r63635 <= r63636;
        double r63638 = r63629 / r63634;
        double r63639 = cbrt(r63638);
        double r63640 = r63639 * r63639;
        double r63641 = r63640 * r63639;
        double r63642 = 1.0;
        double r63643 = 2.0;
        double r63644 = pow(r63630, r63643);
        double r63645 = r63642 / r63644;
        double r63646 = 4.0;
        double r63647 = 8.0;
        double r63648 = r63647 / r63630;
        double r63649 = r63646 - r63648;
        double r63650 = r63645 * r63649;
        double r63651 = -r63633;
        double r63652 = r63651 / r63630;
        double r63653 = r63650 + r63652;
        double r63654 = r63641 - r63653;
        double r63655 = r63654 / r63633;
        double r63656 = 1.0;
        double r63657 = r63635 + r63656;
        double r63658 = exp(r63657);
        double r63659 = log(r63658);
        double r63660 = r63659 / r63633;
        double r63661 = r63637 ? r63655 : r63660;
        return r63661;
}

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-58.7

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

    6. Applied add-cube-cbrt58.7

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

    7. Taylor expanded around inf 10.6

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

    8. Simplified10.6

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

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

    1. Initial program 0.5

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

    3. Applied add-log-exp0.5

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

    4. Applied add-log-exp0.5

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

    5. Applied sum-log0.6

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

    6. Simplified0.6

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

  4. Final simplification3.1

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

Reproduce

herbie shell --seed 2019291 
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/1"
  :precision binary64
  :pre (and (> alpha -1) (> beta -1))
  (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2)) 1) 2))