Average Error: 15.8 → 3.5
Time: 24.4s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} \le -0.9999999999946434:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\left(\frac{4}{{\alpha}^{2}} - \frac{8}{{\alpha}^{3}}\right) - \frac{2}{\alpha}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\sqrt[3]{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1} \cdot \sqrt[3]{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}\right) \cdot \sqrt[3]{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}}{2}\\ \end{array}\]
\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 -0.9999999999946434:\\
\;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\left(\frac{4}{{\alpha}^{2}} - \frac{8}{{\alpha}^{3}}\right) - \frac{2}{\alpha}\right)}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\left(\sqrt[3]{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1} \cdot \sqrt[3]{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}\right) \cdot \sqrt[3]{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}}{2}\\

\end{array}
double f(double alpha, double beta) {
        double r103672 = beta;
        double r103673 = alpha;
        double r103674 = r103672 - r103673;
        double r103675 = r103673 + r103672;
        double r103676 = 2.0;
        double r103677 = r103675 + r103676;
        double r103678 = r103674 / r103677;
        double r103679 = 1.0;
        double r103680 = r103678 + r103679;
        double r103681 = r103680 / r103676;
        return r103681;
}
↓
double f(double alpha, double beta) {
        double r103682 = beta;
        double r103683 = alpha;
        double r103684 = r103682 - r103683;
        double r103685 = r103683 + r103682;
        double r103686 = 2.0;
        double r103687 = r103685 + r103686;
        double r103688 = r103684 / r103687;
        double r103689 = -0.9999999999946434;
        bool r103690 = r103688 <= r103689;
        double r103691 = r103682 / r103687;
        double r103692 = 4.0;
        double r103693 = 2.0;
        double r103694 = pow(r103683, r103693);
        double r103695 = r103692 / r103694;
        double r103696 = 8.0;
        double r103697 = 3.0;
        double r103698 = pow(r103683, r103697);
        double r103699 = r103696 / r103698;
        double r103700 = r103695 - r103699;
        double r103701 = r103686 / r103683;
        double r103702 = r103700 - r103701;
        double r103703 = r103691 - r103702;
        double r103704 = r103703 / r103686;
        double r103705 = 1.0;
        double r103706 = r103688 + r103705;
        double r103707 = cbrt(r103706);
        double r103708 = r103707 * r103707;
        double r103709 = r103708 * r103707;
        double r103710 = r103709 / r103686;
        double r103711 = r103690 ? r103704 : r103710;
        return r103711;
}

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)) < -0.9999999999946434

    1. Initial program 60.3

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

    3. Applied div-sub60.3

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

    4. Applied associate-+l-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 10.7

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

    6. Simplified10.7

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

    if -0.9999999999946434 < (/ (- beta alpha) (+ (+ alpha beta) 2.0))

    1. Initial program 0.3

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

    3. Applied add-cube-cbrt0.9

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

  4. Final simplification3.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} \le -0.9999999999946434:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\left(\frac{4}{{\alpha}^{2}} - \frac{8}{{\alpha}^{3}}\right) - \frac{2}{\alpha}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\sqrt[3]{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1} \cdot \sqrt[3]{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}\right) \cdot \sqrt[3]{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019198 
(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))