Average Error: 15.7 → 3.0
Time: 17.7s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} \le -0.9999999999999637:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \left(\left(\frac{4.0}{\alpha \cdot \alpha} - \frac{8.0}{\left(\alpha \cdot \alpha\right) \cdot \alpha}\right) - \frac{2.0}{\alpha}\right)}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \mathsf{fma}\left(\alpha, \frac{1}{\left(\alpha + \beta\right) + 2.0}, -1.0\right)}{2.0}\\ \end{array}\]
\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 -0.9999999999999637:\\
\;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \left(\left(\frac{4.0}{\alpha \cdot \alpha} - \frac{8.0}{\left(\alpha \cdot \alpha\right) \cdot \alpha}\right) - \frac{2.0}{\alpha}\right)}{2.0}\\

\mathbf{else}:\\
\;\;\;\;\frac{\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 r1715559 = beta;
        double r1715560 = alpha;
        double r1715561 = r1715559 - r1715560;
        double r1715562 = r1715560 + r1715559;
        double r1715563 = 2.0;
        double r1715564 = r1715562 + r1715563;
        double r1715565 = r1715561 / r1715564;
        double r1715566 = 1.0;
        double r1715567 = r1715565 + r1715566;
        double r1715568 = r1715567 / r1715563;
        return r1715568;
}
↓
double f(double alpha, double beta) {
        double r1715569 = beta;
        double r1715570 = alpha;
        double r1715571 = r1715569 - r1715570;
        double r1715572 = r1715570 + r1715569;
        double r1715573 = 2.0;
        double r1715574 = r1715572 + r1715573;
        double r1715575 = r1715571 / r1715574;
        double r1715576 = -0.9999999999999637;
        bool r1715577 = r1715575 <= r1715576;
        double r1715578 = r1715569 / r1715574;
        double r1715579 = 4.0;
        double r1715580 = r1715570 * r1715570;
        double r1715581 = r1715579 / r1715580;
        double r1715582 = 8.0;
        double r1715583 = r1715580 * r1715570;
        double r1715584 = r1715582 / r1715583;
        double r1715585 = r1715581 - r1715584;
        double r1715586 = r1715573 / r1715570;
        double r1715587 = r1715585 - r1715586;
        double r1715588 = r1715578 - r1715587;
        double r1715589 = r1715588 / r1715573;
        double r1715590 = 1.0;
        double r1715591 = r1715590 / r1715574;
        double r1715592 = 1.0;
        double r1715593 = -r1715592;
        double r1715594 = fma(r1715570, r1715591, r1715593);
        double r1715595 = r1715578 - r1715594;
        double r1715596 = r1715595 / r1715573;
        double r1715597 = r1715577 ? r1715589 : r1715596;
        return r1715597;
}

Error

Bits error versus alpha

Bits error versus beta

Derivation

  1. Split input into 2 regimes

  2. if (/ (- beta alpha) (+ (+ alpha beta) 2.0)) < -0.9999999999999637

    1. Initial program 60.4

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

    3. Applied div-sub60.4

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

    4. Applied associate-+l-58.7

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

    5. Taylor expanded around inf 10.6

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

    6. Simplified10.6

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

    if -0.9999999999999637 < (/ (- beta alpha) (+ (+ alpha beta) 2.0))

    1. Initial program 0.4

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

    3. Applied div-sub0.4

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

    4. Applied associate-+l-0.4

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

    6. Applied div-inv0.4

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

    7. Applied fma-neg0.4

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

  4. Final simplification3.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} \le -0.9999999999999637:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \left(\left(\frac{4.0}{\alpha \cdot \alpha} - \frac{8.0}{\left(\alpha \cdot \alpha\right) \cdot \alpha}\right) - \frac{2.0}{\alpha}\right)}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \mathsf{fma}\left(\alpha, \frac{1}{\left(\alpha + \beta\right) + 2.0}, -1.0\right)}{2.0}\\ \end{array}\]

Reproduce

herbie shell --seed 2019153 +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))