Average Error: 16.7 → 6.3
Time: 19.2s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 439115944925299008:\\ \;\;\;\;\frac{\mathsf{fma}\left(\beta - \alpha, \frac{1}{\left(\alpha + \beta\right) + 2}, 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\sqrt[3]{\frac{\beta}{\left(\alpha + \beta\right) + 2}}\right)}^{3} - \left(\left(\frac{4}{\alpha \cdot \alpha} - \frac{8}{{\alpha}^{3}}\right) - \frac{2}{\alpha}\right)}{2}\\ \end{array}\]
\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \le 439115944925299008:\\
\;\;\;\;\frac{\mathsf{fma}\left(\beta - \alpha, \frac{1}{\left(\alpha + \beta\right) + 2}, 1\right)}{2}\\

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

\end{array}
double f(double alpha, double beta) {
        double r71574 = beta;
        double r71575 = alpha;
        double r71576 = r71574 - r71575;
        double r71577 = r71575 + r71574;
        double r71578 = 2.0;
        double r71579 = r71577 + r71578;
        double r71580 = r71576 / r71579;
        double r71581 = 1.0;
        double r71582 = r71580 + r71581;
        double r71583 = r71582 / r71578;
        return r71583;
}
↓
double f(double alpha, double beta) {
        double r71584 = alpha;
        double r71585 = 4.39115944925299e+17;
        bool r71586 = r71584 <= r71585;
        double r71587 = beta;
        double r71588 = r71587 - r71584;
        double r71589 = 1.0;
        double r71590 = r71584 + r71587;
        double r71591 = 2.0;
        double r71592 = r71590 + r71591;
        double r71593 = r71589 / r71592;
        double r71594 = 1.0;
        double r71595 = fma(r71588, r71593, r71594);
        double r71596 = r71595 / r71591;
        double r71597 = r71587 / r71592;
        double r71598 = cbrt(r71597);
        double r71599 = 3.0;
        double r71600 = pow(r71598, r71599);
        double r71601 = 4.0;
        double r71602 = r71584 * r71584;
        double r71603 = r71601 / r71602;
        double r71604 = 8.0;
        double r71605 = pow(r71584, r71599);
        double r71606 = r71604 / r71605;
        double r71607 = r71603 - r71606;
        double r71608 = r71591 / r71584;
        double r71609 = r71607 - r71608;
        double r71610 = r71600 - r71609;
        double r71611 = r71610 / r71591;
        double r71612 = r71586 ? r71596 : r71611;
        return r71612;
}

Error

Bits error versus alpha

Bits error versus beta

Derivation

  1. Split input into 2 regimes

  2. if alpha < 4.39115944925299e+17

    1. Initial program 0.5

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

    3. Applied div-inv0.5

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

    4. Applied fma-def0.5

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

    if 4.39115944925299e+17 < alpha

    1. Initial program 50.5

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

    3. Applied div-sub50.5

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

    4. Applied associate-+l-48.9

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

    6. Applied add-cube-cbrt48.9

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

    7. Applied add-cube-cbrt48.9

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

    8. Applied prod-diff48.9

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

    9. Simplified48.9

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

    10. Simplified48.9

      \[\leadsto \frac{\left({\left(\sqrt[3]{\frac{\beta}{\left(\alpha + \beta\right) + 2}}\right)}^{3} - \left(\frac{\alpha}{2 + \left(\alpha + \beta\right)} - 1\right)\right) + \color{blue}{0}}{2}\]

    11. Taylor expanded around inf 18.3

      \[\leadsto \frac{\left({\left(\sqrt[3]{\frac{\beta}{\left(\alpha + \beta\right) + 2}}\right)}^{3} - \color{blue}{\left(4 \cdot \frac{1}{{\alpha}^{2}} - \left(2 \cdot \frac{1}{\alpha} + 8 \cdot \frac{1}{{\alpha}^{3}}\right)\right)}\right) + 0}{2}\]

    12. Simplified18.3

      \[\leadsto \frac{\left({\left(\sqrt[3]{\frac{\beta}{\left(\alpha + \beta\right) + 2}}\right)}^{3} - \color{blue}{\left(\left(\frac{4}{\alpha \cdot \alpha} - \frac{8}{{\alpha}^{3}}\right) - \frac{2}{\alpha}\right)}\right) + 0}{2}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification6.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 439115944925299008:\\ \;\;\;\;\frac{\mathsf{fma}\left(\beta - \alpha, \frac{1}{\left(\alpha + \beta\right) + 2}, 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\sqrt[3]{\frac{\beta}{\left(\alpha + \beta\right) + 2}}\right)}^{3} - \left(\left(\frac{4}{\alpha \cdot \alpha} - \frac{8}{{\alpha}^{3}}\right) - \frac{2}{\alpha}\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019306 +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))