Average Error: 16.6 → 3.2
Time: 21.9s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} \le -0.9999439615156119:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \left(\frac{4.0}{\alpha \cdot \alpha} - \mathsf{fma}\left(2.0, \left(\frac{1}{\alpha}\right), \left(8.0 \cdot \frac{1}{\alpha \cdot \left(\alpha \cdot \alpha\right)}\right)\right)\right)}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\beta - \alpha\right), \left(\frac{1}{\left(\alpha + \beta\right) + 2.0}\right), 1.0\right)}{2.0}\\ \end{array}\]
\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 -0.9999439615156119:\\
\;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \left(\frac{4.0}{\alpha \cdot \alpha} - \mathsf{fma}\left(2.0, \left(\frac{1}{\alpha}\right), \left(8.0 \cdot \frac{1}{\alpha \cdot \left(\alpha \cdot \alpha\right)}\right)\right)\right)}{2.0}\\

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

\end{array}
double f(double alpha, double beta) {
        double r1455662 = beta;
        double r1455663 = alpha;
        double r1455664 = r1455662 - r1455663;
        double r1455665 = r1455663 + r1455662;
        double r1455666 = 2.0;
        double r1455667 = r1455665 + r1455666;
        double r1455668 = r1455664 / r1455667;
        double r1455669 = 1.0;
        double r1455670 = r1455668 + r1455669;
        double r1455671 = r1455670 / r1455666;
        return r1455671;
}
↓
double f(double alpha, double beta) {
        double r1455672 = beta;
        double r1455673 = alpha;
        double r1455674 = r1455672 - r1455673;
        double r1455675 = r1455673 + r1455672;
        double r1455676 = 2.0;
        double r1455677 = r1455675 + r1455676;
        double r1455678 = r1455674 / r1455677;
        double r1455679 = -0.9999439615156119;
        bool r1455680 = r1455678 <= r1455679;
        double r1455681 = r1455672 / r1455677;
        double r1455682 = 4.0;
        double r1455683 = r1455673 * r1455673;
        double r1455684 = r1455682 / r1455683;
        double r1455685 = 1.0;
        double r1455686 = r1455685 / r1455673;
        double r1455687 = 8.0;
        double r1455688 = r1455673 * r1455683;
        double r1455689 = r1455685 / r1455688;
        double r1455690 = r1455687 * r1455689;
        double r1455691 = fma(r1455676, r1455686, r1455690);
        double r1455692 = r1455684 - r1455691;
        double r1455693 = r1455681 - r1455692;
        double r1455694 = r1455693 / r1455676;
        double r1455695 = r1455685 / r1455677;
        double r1455696 = 1.0;
        double r1455697 = fma(r1455674, r1455695, r1455696);
        double r1455698 = r1455697 / r1455676;
        double r1455699 = r1455680 ? r1455694 : r1455698;
        return r1455699;
}

Error

Bits error versus alpha

Bits error versus beta

Derivation

  1. Split input into 2 regimes

  2. if (/ (- beta alpha) (+ (+ alpha beta) 2.0)) < -0.9999439615156119

    1. Initial program 59.3

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

    3. Applied div-sub59.2

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

    4. Applied associate-+l-57.4

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

    5. Taylor expanded around -inf 11.5

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

    6. Simplified11.5

      \[\leadsto \frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \color{blue}{\left(\frac{4.0 \cdot 1}{\alpha \cdot \alpha} - \mathsf{fma}\left(2.0, \left(\frac{1}{\alpha}\right), \left(8.0 \cdot \frac{1}{\alpha \cdot \left(\alpha \cdot \alpha\right)}\right)\right)\right)}}{2.0}\]

    if -0.9999439615156119 < (/ (- beta alpha) (+ (+ alpha beta) 2.0))

    1. Initial program 0.0

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

    3. Applied div-inv0.0

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

    4. Applied fma-def0.0

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

  4. Final simplification3.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} \le -0.9999439615156119:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \left(\frac{4.0}{\alpha \cdot \alpha} - \mathsf{fma}\left(2.0, \left(\frac{1}{\alpha}\right), \left(8.0 \cdot \frac{1}{\alpha \cdot \left(\alpha \cdot \alpha\right)}\right)\right)\right)}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\beta - \alpha\right), \left(\frac{1}{\left(\alpha + \beta\right) + 2.0}\right), 1.0\right)}{2.0}\\ \end{array}\]

Reproduce

herbie shell --seed 2019128 +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))