Average Error: 16.6 → 7.0
Time: 8.5s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 3.98853083409438599 \cdot 10^{48}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\beta, \mathsf{fma}\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2}, \frac{\alpha}{\left(\alpha + \beta\right) + 2}, \mathsf{fma}\left(1, 1, \frac{\alpha}{\left(\alpha + \beta\right) + 2} \cdot 1\right)\right), -\left(\left(\alpha + \beta\right) + 2\right) \cdot \left(\sqrt[3]{{\left({\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2}\right)}^{3}\right)}^{3}} - {1}^{3}\right)\right)}{\mathsf{fma}\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2}, \frac{\alpha}{\left(\alpha + \beta\right) + 2}, \mathsf{fma}\left(1, 1, \frac{\alpha}{\left(\alpha + \beta\right) + 2} \cdot 1\right)\right) \cdot \left(\left(\alpha + \beta\right) + 2\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \mathsf{fma}\left(4, \frac{1}{{\alpha}^{2}}, -\mathsf{fma}\left(2, \frac{1}{\alpha}, 8 \cdot \frac{1}{{\alpha}^{3}}\right)\right)}{2}\\ \end{array}\]
\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \le 3.98853083409438599 \cdot 10^{48}:\\
\;\;\;\;\frac{\frac{\mathsf{fma}\left(\beta, \mathsf{fma}\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2}, \frac{\alpha}{\left(\alpha + \beta\right) + 2}, \mathsf{fma}\left(1, 1, \frac{\alpha}{\left(\alpha + \beta\right) + 2} \cdot 1\right)\right), -\left(\left(\alpha + \beta\right) + 2\right) \cdot \left(\sqrt[3]{{\left({\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2}\right)}^{3}\right)}^{3}} - {1}^{3}\right)\right)}{\mathsf{fma}\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2}, \frac{\alpha}{\left(\alpha + \beta\right) + 2}, \mathsf{fma}\left(1, 1, \frac{\alpha}{\left(\alpha + \beta\right) + 2} \cdot 1\right)\right) \cdot \left(\left(\alpha + \beta\right) + 2\right)}}{2}\\

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

\end{array}
double f(double alpha, double beta) {
        double r158755 = beta;
        double r158756 = alpha;
        double r158757 = r158755 - r158756;
        double r158758 = r158756 + r158755;
        double r158759 = 2.0;
        double r158760 = r158758 + r158759;
        double r158761 = r158757 / r158760;
        double r158762 = 1.0;
        double r158763 = r158761 + r158762;
        double r158764 = r158763 / r158759;
        return r158764;
}
↓
double f(double alpha, double beta) {
        double r158765 = alpha;
        double r158766 = 3.988530834094386e+48;
        bool r158767 = r158765 <= r158766;
        double r158768 = beta;
        double r158769 = r158765 + r158768;
        double r158770 = 2.0;
        double r158771 = r158769 + r158770;
        double r158772 = r158765 / r158771;
        double r158773 = 1.0;
        double r158774 = r158772 * r158773;
        double r158775 = fma(r158773, r158773, r158774);
        double r158776 = fma(r158772, r158772, r158775);
        double r158777 = 3.0;
        double r158778 = pow(r158772, r158777);
        double r158779 = pow(r158778, r158777);
        double r158780 = cbrt(r158779);
        double r158781 = pow(r158773, r158777);
        double r158782 = r158780 - r158781;
        double r158783 = r158771 * r158782;
        double r158784 = -r158783;
        double r158785 = fma(r158768, r158776, r158784);
        double r158786 = r158776 * r158771;
        double r158787 = r158785 / r158786;
        double r158788 = r158787 / r158770;
        double r158789 = r158768 / r158771;
        double r158790 = 4.0;
        double r158791 = 1.0;
        double r158792 = 2.0;
        double r158793 = pow(r158765, r158792);
        double r158794 = r158791 / r158793;
        double r158795 = r158791 / r158765;
        double r158796 = 8.0;
        double r158797 = pow(r158765, r158777);
        double r158798 = r158791 / r158797;
        double r158799 = r158796 * r158798;
        double r158800 = fma(r158770, r158795, r158799);
        double r158801 = -r158800;
        double r158802 = fma(r158790, r158794, r158801);
        double r158803 = r158789 - r158802;
        double r158804 = r158803 / r158770;
        double r158805 = r158767 ? r158788 : r158804;
        return r158805;
}

Error

Bits error versus alpha

Bits error versus beta

Derivation

  1. Split input into 2 regimes

  2. if alpha < 3.988530834094386e+48

    1. Initial program 2.4

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

    3. Applied div-sub2.4

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

    4. Applied associate-+l-2.4

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

    6. Applied flip3--2.4

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

    7. Applied frac-sub2.4

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

    8. Simplified2.4

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

    9. Simplified2.4

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

    11. Applied add-cbrt-cube2.4

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

    12. Simplified2.4

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

    if 3.988530834094386e+48 < alpha

    1. Initial program 51.6

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

    3. Applied div-sub51.5

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

    4. Applied associate-+l-49.8

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

    5. Taylor expanded around inf 18.2

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

    6. Simplified18.2

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

  4. Final simplification7.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 3.98853083409438599 \cdot 10^{48}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\beta, \mathsf{fma}\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2}, \frac{\alpha}{\left(\alpha + \beta\right) + 2}, \mathsf{fma}\left(1, 1, \frac{\alpha}{\left(\alpha + \beta\right) + 2} \cdot 1\right)\right), -\left(\left(\alpha + \beta\right) + 2\right) \cdot \left(\sqrt[3]{{\left({\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2}\right)}^{3}\right)}^{3}} - {1}^{3}\right)\right)}{\mathsf{fma}\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2}, \frac{\alpha}{\left(\alpha + \beta\right) + 2}, \mathsf{fma}\left(1, 1, \frac{\alpha}{\left(\alpha + \beta\right) + 2} \cdot 1\right)\right) \cdot \left(\left(\alpha + \beta\right) + 2\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \mathsf{fma}\left(4, \frac{1}{{\alpha}^{2}}, -\mathsf{fma}\left(2, \frac{1}{\alpha}, 8 \cdot \frac{1}{{\alpha}^{3}}\right)\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020035 +o rules:numerics
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/1"
  :precision binary64
  :pre (and (> alpha -1) (> beta -1))
  (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2)) 1) 2))