Average Error: 15.9 → 6.7
Time: 15.2s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 24810729294817533835310823078796843810820:\\ \;\;\;\;\frac{\sqrt[3]{{\left(\frac{\beta}{\left(\alpha + \beta\right) + 2}\right)}^{3}} - \left(\sqrt[3]{{\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2}\right)}^{3}} - 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(4 \cdot \frac{1}{{\alpha}^{2}} - \left(2 \cdot \frac{1}{\alpha} + 8 \cdot \frac{1}{{\alpha}^{3}}\right)\right)}{2}\\ \end{array}\]
\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \le 24810729294817533835310823078796843810820:\\
\;\;\;\;\frac{\sqrt[3]{{\left(\frac{\beta}{\left(\alpha + \beta\right) + 2}\right)}^{3}} - \left(\sqrt[3]{{\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2}\right)}^{3}} - 1\right)}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(4 \cdot \frac{1}{{\alpha}^{2}} - \left(2 \cdot \frac{1}{\alpha} + 8 \cdot \frac{1}{{\alpha}^{3}}\right)\right)}{2}\\

\end{array}
double f(double alpha, double beta) {
        double r73789 = beta;
        double r73790 = alpha;
        double r73791 = r73789 - r73790;
        double r73792 = r73790 + r73789;
        double r73793 = 2.0;
        double r73794 = r73792 + r73793;
        double r73795 = r73791 / r73794;
        double r73796 = 1.0;
        double r73797 = r73795 + r73796;
        double r73798 = r73797 / r73793;
        return r73798;
}
↓
double f(double alpha, double beta) {
        double r73799 = alpha;
        double r73800 = 2.4810729294817534e+40;
        bool r73801 = r73799 <= r73800;
        double r73802 = beta;
        double r73803 = r73799 + r73802;
        double r73804 = 2.0;
        double r73805 = r73803 + r73804;
        double r73806 = r73802 / r73805;
        double r73807 = 3.0;
        double r73808 = pow(r73806, r73807);
        double r73809 = cbrt(r73808);
        double r73810 = r73799 / r73805;
        double r73811 = pow(r73810, r73807);
        double r73812 = cbrt(r73811);
        double r73813 = 1.0;
        double r73814 = r73812 - r73813;
        double r73815 = r73809 - r73814;
        double r73816 = r73815 / r73804;
        double r73817 = 4.0;
        double r73818 = 1.0;
        double r73819 = 2.0;
        double r73820 = pow(r73799, r73819);
        double r73821 = r73818 / r73820;
        double r73822 = r73817 * r73821;
        double r73823 = r73818 / r73799;
        double r73824 = r73804 * r73823;
        double r73825 = 8.0;
        double r73826 = pow(r73799, r73807);
        double r73827 = r73818 / r73826;
        double r73828 = r73825 * r73827;
        double r73829 = r73824 + r73828;
        double r73830 = r73822 - r73829;
        double r73831 = r73806 - r73830;
        double r73832 = r73831 / r73804;
        double r73833 = r73801 ? r73816 : r73832;
        return r73833;
}

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 < 2.4810729294817534e+40

    1. Initial program 1.7

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

    3. Applied div-sub1.7

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

    4. Applied associate-+l-1.7

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

    6. Applied add-cbrt-cube1.7

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

    7. Applied add-cbrt-cube1.7

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

    8. Applied cbrt-undiv1.7

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

    9. Simplified1.7

      \[\leadsto \frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\sqrt[3]{\color{blue}{{\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2}\right)}^{3}}} - 1\right)}{2}\]
    10. Using strategy rm

    11. Applied add-cbrt-cube12.9

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

    12. Applied add-cbrt-cube15.5

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

    13. Applied cbrt-undiv15.5

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

    14. Simplified1.7

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

    if 2.4810729294817534e+40 < alpha

    1. Initial program 51.1

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

    3. Applied div-sub51.1

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

    4. Applied associate-+l-49.2

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

    5. Taylor expanded around inf 19.1

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

  4. Final simplification6.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 24810729294817533835310823078796843810820:\\ \;\;\;\;\frac{\sqrt[3]{{\left(\frac{\beta}{\left(\alpha + \beta\right) + 2}\right)}^{3}} - \left(\sqrt[3]{{\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2}\right)}^{3}} - 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(4 \cdot \frac{1}{{\alpha}^{2}} - \left(2 \cdot \frac{1}{\alpha} + 8 \cdot \frac{1}{{\alpha}^{3}}\right)\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019298 
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/1"
  :precision binary64
  :pre (and (> alpha -1) (> beta -1))
  (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2)) 1) 2))