Average Error: 15.9 → 6.3
Time: 14.0s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 3.73166366653750566 \cdot 10^{29}:\\ \;\;\;\;e^{\log \left(\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \sqrt[3]{{\left(\mathsf{fma}\left(\frac{1}{\sqrt{\left(\alpha + \beta\right) + 2}}, \frac{\alpha}{\sqrt{\left(\alpha + \beta\right) + 2}}, -1\right) + 1 \cdot 0\right)}^{3}}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\frac{4}{\alpha \cdot \alpha} - \left(\frac{2}{\alpha} + \frac{8}{{\alpha}^{3}}\right)\right)}{2}\\ \end{array}\]
\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \le 3.73166366653750566 \cdot 10^{29}:\\
\;\;\;\;e^{\log \left(\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \sqrt[3]{{\left(\mathsf{fma}\left(\frac{1}{\sqrt{\left(\alpha + \beta\right) + 2}}, \frac{\alpha}{\sqrt{\left(\alpha + \beta\right) + 2}}, -1\right) + 1 \cdot 0\right)}^{3}}}{2}\right)}\\

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

\end{array}
double f(double alpha, double beta) {
        double r119644 = beta;
        double r119645 = alpha;
        double r119646 = r119644 - r119645;
        double r119647 = r119645 + r119644;
        double r119648 = 2.0;
        double r119649 = r119647 + r119648;
        double r119650 = r119646 / r119649;
        double r119651 = 1.0;
        double r119652 = r119650 + r119651;
        double r119653 = r119652 / r119648;
        return r119653;
}
↓
double f(double alpha, double beta) {
        double r119654 = alpha;
        double r119655 = 3.731663666537506e+29;
        bool r119656 = r119654 <= r119655;
        double r119657 = beta;
        double r119658 = r119654 + r119657;
        double r119659 = 2.0;
        double r119660 = r119658 + r119659;
        double r119661 = r119657 / r119660;
        double r119662 = 1.0;
        double r119663 = sqrt(r119660);
        double r119664 = r119662 / r119663;
        double r119665 = r119654 / r119663;
        double r119666 = 1.0;
        double r119667 = -r119666;
        double r119668 = fma(r119664, r119665, r119667);
        double r119669 = 0.0;
        double r119670 = r119666 * r119669;
        double r119671 = r119668 + r119670;
        double r119672 = 3.0;
        double r119673 = pow(r119671, r119672);
        double r119674 = cbrt(r119673);
        double r119675 = r119661 - r119674;
        double r119676 = r119675 / r119659;
        double r119677 = log(r119676);
        double r119678 = exp(r119677);
        double r119679 = 4.0;
        double r119680 = r119654 * r119654;
        double r119681 = r119679 / r119680;
        double r119682 = r119659 / r119654;
        double r119683 = 8.0;
        double r119684 = pow(r119654, r119672);
        double r119685 = r119683 / r119684;
        double r119686 = r119682 + r119685;
        double r119687 = r119681 - r119686;
        double r119688 = r119661 - r119687;
        double r119689 = r119688 / r119659;
        double r119690 = r119656 ? r119678 : r119689;
        return r119690;
}

Error

Bits error versus alpha

Bits error versus beta

Derivation

  1. Split input into 2 regimes

  2. if alpha < 3.731663666537506e+29

    1. Initial program 1.2

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

    3. Applied div-sub1.2

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

    4. Applied associate-+l-1.2

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

    6. Applied add-exp-log1.2

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

    7. Applied add-exp-log1.2

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

    8. Applied div-exp1.2

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

    9. Simplified1.2

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

    11. Applied add-cbrt-cube1.2

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

    12. Simplified1.2

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

    14. Applied add-cube-cbrt1.2

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

    15. Applied add-sqr-sqrt1.2

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

    16. Applied *-un-lft-identity1.2

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

    17. Applied times-frac1.2

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

    18. Applied prod-diff1.2

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

    19. Simplified1.2

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

    20. Simplified1.2

      \[\leadsto e^{\log \left(\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \sqrt[3]{{\left(\mathsf{fma}\left(\frac{1}{\sqrt{\left(\alpha + \beta\right) + 2}}, \frac{\alpha}{\sqrt{\left(\alpha + \beta\right) + 2}}, -1\right) + \color{blue}{1 \cdot 0}\right)}^{3}}}{2}\right)}\]

    if 3.731663666537506e+29 < alpha

    1. Initial program 50.9

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
    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-49.1

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

    5. Taylor expanded around inf 18.4

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

    6. Simplified18.4

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

  4. Final simplification6.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 3.73166366653750566 \cdot 10^{29}:\\ \;\;\;\;e^{\log \left(\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \sqrt[3]{{\left(\mathsf{fma}\left(\frac{1}{\sqrt{\left(\alpha + \beta\right) + 2}}, \frac{\alpha}{\sqrt{\left(\alpha + \beta\right) + 2}}, -1\right) + 1 \cdot 0\right)}^{3}}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\frac{4}{\alpha \cdot \alpha} - \left(\frac{2}{\alpha} + \frac{8}{{\alpha}^{3}}\right)\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020045 +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))