Average Error: 16.6 → 3.3
Time: 6.8s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} \le -1:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \mathsf{fma}\left(4, \frac{1}{{\alpha}^{2}}, -\mathsf{fma}\left(2, \frac{1}{\alpha}, 8 \cdot \frac{1}{{\alpha}^{3}}\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(e^{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2}}\right) + 1}{2}\\ \end{array}\]
\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}
↓
\begin{array}{l}
\mathbf{if}\;\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} \le -1:\\
\;\;\;\;\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}\\

\mathbf{else}:\\
\;\;\;\;\frac{\log \left(e^{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2}}\right) + 1}{2}\\

\end{array}
double f(double alpha, double beta) {
        double r105627 = beta;
        double r105628 = alpha;
        double r105629 = r105627 - r105628;
        double r105630 = r105628 + r105627;
        double r105631 = 2.0;
        double r105632 = r105630 + r105631;
        double r105633 = r105629 / r105632;
        double r105634 = 1.0;
        double r105635 = r105633 + r105634;
        double r105636 = r105635 / r105631;
        return r105636;
}
↓
double f(double alpha, double beta) {
        double r105637 = beta;
        double r105638 = alpha;
        double r105639 = r105637 - r105638;
        double r105640 = r105638 + r105637;
        double r105641 = 2.0;
        double r105642 = r105640 + r105641;
        double r105643 = r105639 / r105642;
        double r105644 = -1.0;
        bool r105645 = r105643 <= r105644;
        double r105646 = r105637 / r105642;
        double r105647 = 4.0;
        double r105648 = 1.0;
        double r105649 = 2.0;
        double r105650 = pow(r105638, r105649);
        double r105651 = r105648 / r105650;
        double r105652 = r105648 / r105638;
        double r105653 = 8.0;
        double r105654 = 3.0;
        double r105655 = pow(r105638, r105654);
        double r105656 = r105648 / r105655;
        double r105657 = r105653 * r105656;
        double r105658 = fma(r105641, r105652, r105657);
        double r105659 = -r105658;
        double r105660 = fma(r105647, r105651, r105659);
        double r105661 = r105646 - r105660;
        double r105662 = r105661 / r105641;
        double r105663 = exp(r105643);
        double r105664 = log(r105663);
        double r105665 = 1.0;
        double r105666 = r105664 + r105665;
        double r105667 = r105666 / r105641;
        double r105668 = r105645 ? r105662 : r105667;
        return r105668;
}

Error

Bits error versus alpha

Bits error versus beta

Derivation

  1. Split input into 2 regimes

  2. if (/ (- beta alpha) (+ (+ alpha beta) 2.0)) < -1.0

    1. Initial program 60.6

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

    3. Applied div-sub60.6

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

    4. Applied associate-+l-58.7

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

    5. Taylor expanded around inf 10.6

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

    6. Simplified10.6

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

    if -1.0 < (/ (- beta alpha) (+ (+ alpha beta) 2.0))

    1. Initial program 0.6

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

    3. Applied add-log-exp0.6

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

  4. Final simplification3.3

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

Reproduce

herbie shell --seed 2020089 +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))