Average Error: 15.8 → 3.0
Time: 22.9s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} \le -0.9999999999999994:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \left(\left(\frac{\frac{4.0}{\alpha}}{\alpha} - \frac{8.0}{\alpha \cdot \left(\alpha \cdot \alpha\right)}\right) - \frac{2.0}{\alpha}\right)}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{\left(\frac{\beta}{\left(\alpha + \beta\right) + 2.0} \cdot \frac{\beta}{\left(\alpha + \beta\right) + 2.0}\right) \cdot \frac{\beta}{\left(\alpha + \beta\right) + 2.0}} - \mathsf{fma}\left(\alpha, \frac{1}{\left(\alpha + \beta\right) + 2.0}, -1.0\right)}{2.0}\\ \end{array}\]
\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}
↓
\begin{array}{l}
\mathbf{if}\;\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} \le -0.9999999999999994:\\
\;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \left(\left(\frac{\frac{4.0}{\alpha}}{\alpha} - \frac{8.0}{\alpha \cdot \left(\alpha \cdot \alpha\right)}\right) - \frac{2.0}{\alpha}\right)}{2.0}\\

\mathbf{else}:\\
\;\;\;\;\frac{\sqrt[3]{\left(\frac{\beta}{\left(\alpha + \beta\right) + 2.0} \cdot \frac{\beta}{\left(\alpha + \beta\right) + 2.0}\right) \cdot \frac{\beta}{\left(\alpha + \beta\right) + 2.0}} - \mathsf{fma}\left(\alpha, \frac{1}{\left(\alpha + \beta\right) + 2.0}, -1.0\right)}{2.0}\\

\end{array}
double f(double alpha, double beta) {
        double r2555683 = beta;
        double r2555684 = alpha;
        double r2555685 = r2555683 - r2555684;
        double r2555686 = r2555684 + r2555683;
        double r2555687 = 2.0;
        double r2555688 = r2555686 + r2555687;
        double r2555689 = r2555685 / r2555688;
        double r2555690 = 1.0;
        double r2555691 = r2555689 + r2555690;
        double r2555692 = r2555691 / r2555687;
        return r2555692;
}
↓
double f(double alpha, double beta) {
        double r2555693 = beta;
        double r2555694 = alpha;
        double r2555695 = r2555693 - r2555694;
        double r2555696 = r2555694 + r2555693;
        double r2555697 = 2.0;
        double r2555698 = r2555696 + r2555697;
        double r2555699 = r2555695 / r2555698;
        double r2555700 = -0.9999999999999994;
        bool r2555701 = r2555699 <= r2555700;
        double r2555702 = r2555693 / r2555698;
        double r2555703 = 4.0;
        double r2555704 = r2555703 / r2555694;
        double r2555705 = r2555704 / r2555694;
        double r2555706 = 8.0;
        double r2555707 = r2555694 * r2555694;
        double r2555708 = r2555694 * r2555707;
        double r2555709 = r2555706 / r2555708;
        double r2555710 = r2555705 - r2555709;
        double r2555711 = r2555697 / r2555694;
        double r2555712 = r2555710 - r2555711;
        double r2555713 = r2555702 - r2555712;
        double r2555714 = r2555713 / r2555697;
        double r2555715 = r2555702 * r2555702;
        double r2555716 = r2555715 * r2555702;
        double r2555717 = cbrt(r2555716);
        double r2555718 = 1.0;
        double r2555719 = r2555718 / r2555698;
        double r2555720 = 1.0;
        double r2555721 = -r2555720;
        double r2555722 = fma(r2555694, r2555719, r2555721);
        double r2555723 = r2555717 - r2555722;
        double r2555724 = r2555723 / r2555697;
        double r2555725 = r2555701 ? r2555714 : r2555724;
        return r2555725;
}

Error

Bits error versus alpha

Bits error versus beta

Derivation

  1. Split input into 2 regimes

  2. if (/ (- beta alpha) (+ (+ alpha beta) 2.0)) < -0.9999999999999994

    1. Initial program 60.5

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}\]
    2. Using strategy rm

    3. Applied div-sub60.5

      \[\leadsto \frac{\color{blue}{\left(\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \frac{\alpha}{\left(\alpha + \beta\right) + 2.0}\right)} + 1.0}{2.0}\]

    4. Applied associate-+l-58.8

      \[\leadsto \frac{\color{blue}{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0\right)}}{2.0}\]

    5. Taylor expanded around inf 10.4

      \[\leadsto \frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \color{blue}{\left(4.0 \cdot \frac{1}{{\alpha}^{2}} - \left(2.0 \cdot \frac{1}{\alpha} + 8.0 \cdot \frac{1}{{\alpha}^{3}}\right)\right)}}{2.0}\]

    6. Simplified10.4

      \[\leadsto \frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \color{blue}{\left(\left(\frac{\frac{4.0}{\alpha}}{\alpha} - \frac{8.0}{\alpha \cdot \left(\alpha \cdot \alpha\right)}\right) - \frac{2.0}{\alpha}\right)}}{2.0}\]

    if -0.9999999999999994 < (/ (- beta alpha) (+ (+ alpha beta) 2.0))

    1. Initial program 0.5

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}\]
    2. Using strategy rm

    3. Applied div-sub0.5

      \[\leadsto \frac{\color{blue}{\left(\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \frac{\alpha}{\left(\alpha + \beta\right) + 2.0}\right)} + 1.0}{2.0}\]

    4. Applied associate-+l-0.5

      \[\leadsto \frac{\color{blue}{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0\right)}}{2.0}\]
    5. Using strategy rm

    6. Applied add-cbrt-cube0.5

      \[\leadsto \frac{\color{blue}{\sqrt[3]{\left(\frac{\beta}{\left(\alpha + \beta\right) + 2.0} \cdot \frac{\beta}{\left(\alpha + \beta\right) + 2.0}\right) \cdot \frac{\beta}{\left(\alpha + \beta\right) + 2.0}}} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0\right)}{2.0}\]
    7. Using strategy rm

    8. Applied div-inv0.5

      \[\leadsto \frac{\sqrt[3]{\left(\frac{\beta}{\left(\alpha + \beta\right) + 2.0} \cdot \frac{\beta}{\left(\alpha + \beta\right) + 2.0}\right) \cdot \frac{\beta}{\left(\alpha + \beta\right) + 2.0}} - \left(\color{blue}{\alpha \cdot \frac{1}{\left(\alpha + \beta\right) + 2.0}} - 1.0\right)}{2.0}\]

    9. Applied fma-neg0.5

      \[\leadsto \frac{\sqrt[3]{\left(\frac{\beta}{\left(\alpha + \beta\right) + 2.0} \cdot \frac{\beta}{\left(\alpha + \beta\right) + 2.0}\right) \cdot \frac{\beta}{\left(\alpha + \beta\right) + 2.0}} - \color{blue}{\mathsf{fma}\left(\alpha, \frac{1}{\left(\alpha + \beta\right) + 2.0}, -1.0\right)}}{2.0}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification3.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} \le -0.9999999999999994:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \left(\left(\frac{\frac{4.0}{\alpha}}{\alpha} - \frac{8.0}{\alpha \cdot \left(\alpha \cdot \alpha\right)}\right) - \frac{2.0}{\alpha}\right)}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{\left(\frac{\beta}{\left(\alpha + \beta\right) + 2.0} \cdot \frac{\beta}{\left(\alpha + \beta\right) + 2.0}\right) \cdot \frac{\beta}{\left(\alpha + \beta\right) + 2.0}} - \mathsf{fma}\left(\alpha, \frac{1}{\left(\alpha + \beta\right) + 2.0}, -1.0\right)}{2.0}\\ \end{array}\]

Reproduce

herbie shell --seed 2019143 +o rules:numerics
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/1"
  :pre (and (> alpha -1) (> beta -1))
  (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0))