Average Error: 16.4 → 6.4
Time: 27.7s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 9.97332166790088 \cdot 10^{+32}:\\ \;\;\;\;\frac{\frac{1}{\left(\beta + \alpha\right) + 2.0} \cdot \beta - \mathsf{fma}\left(\frac{\alpha}{\left(\beta + \alpha\right) \cdot \left(\beta + \alpha\right) - 2.0 \cdot 2.0}, \left(\beta + \alpha\right) - 2.0, -1.0\right)}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\beta + \alpha\right) + 2.0} - \left(\left(\frac{4.0}{\alpha \cdot \alpha} - \frac{2.0}{\alpha}\right) - \frac{8.0}{\alpha \cdot \left(\alpha \cdot \alpha\right)}\right)}{2.0}\\ \end{array}\]
\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \le 9.97332166790088 \cdot 10^{+32}:\\
\;\;\;\;\frac{\frac{1}{\left(\beta + \alpha\right) + 2.0} \cdot \beta - \mathsf{fma}\left(\frac{\alpha}{\left(\beta + \alpha\right) \cdot \left(\beta + \alpha\right) - 2.0 \cdot 2.0}, \left(\beta + \alpha\right) - 2.0, -1.0\right)}{2.0}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{\beta}{\left(\beta + \alpha\right) + 2.0} - \left(\left(\frac{4.0}{\alpha \cdot \alpha} - \frac{2.0}{\alpha}\right) - \frac{8.0}{\alpha \cdot \left(\alpha \cdot \alpha\right)}\right)}{2.0}\\

\end{array}
double f(double alpha, double beta) {
        double r2394730 = beta;
        double r2394731 = alpha;
        double r2394732 = r2394730 - r2394731;
        double r2394733 = r2394731 + r2394730;
        double r2394734 = 2.0;
        double r2394735 = r2394733 + r2394734;
        double r2394736 = r2394732 / r2394735;
        double r2394737 = 1.0;
        double r2394738 = r2394736 + r2394737;
        double r2394739 = r2394738 / r2394734;
        return r2394739;
}
↓
double f(double alpha, double beta) {
        double r2394740 = alpha;
        double r2394741 = 9.97332166790088e+32;
        bool r2394742 = r2394740 <= r2394741;
        double r2394743 = 1.0;
        double r2394744 = beta;
        double r2394745 = r2394744 + r2394740;
        double r2394746 = 2.0;
        double r2394747 = r2394745 + r2394746;
        double r2394748 = r2394743 / r2394747;
        double r2394749 = r2394748 * r2394744;
        double r2394750 = r2394745 * r2394745;
        double r2394751 = r2394746 * r2394746;
        double r2394752 = r2394750 - r2394751;
        double r2394753 = r2394740 / r2394752;
        double r2394754 = r2394745 - r2394746;
        double r2394755 = 1.0;
        double r2394756 = -r2394755;
        double r2394757 = fma(r2394753, r2394754, r2394756);
        double r2394758 = r2394749 - r2394757;
        double r2394759 = r2394758 / r2394746;
        double r2394760 = r2394744 / r2394747;
        double r2394761 = 4.0;
        double r2394762 = r2394740 * r2394740;
        double r2394763 = r2394761 / r2394762;
        double r2394764 = r2394746 / r2394740;
        double r2394765 = r2394763 - r2394764;
        double r2394766 = 8.0;
        double r2394767 = r2394740 * r2394762;
        double r2394768 = r2394766 / r2394767;
        double r2394769 = r2394765 - r2394768;
        double r2394770 = r2394760 - r2394769;
        double r2394771 = r2394770 / r2394746;
        double r2394772 = r2394742 ? r2394759 : r2394771;
        return r2394772;
}

Error

Bits error versus alpha

Bits error versus beta

Derivation

  1. Split input into 2 regimes

  2. if alpha < 9.97332166790088e+32

    1. Initial program 1.4

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

    3. Applied div-sub1.4

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

    4. Applied associate-+l-1.4

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

    6. Applied div-inv1.4

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

    8. Applied flip-+1.4

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

    9. Applied associate-/r/1.4

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

    10. Applied fma-neg1.4

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

    if 9.97332166790088e+32 < alpha

    1. Initial program 51.2

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

    3. Applied div-sub51.1

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

    4. Applied associate-+l-49.5

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

    5. Taylor expanded around -inf 18.0

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

    6. Simplified18.0

      \[\leadsto \frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \color{blue}{\left(\left(\frac{4.0}{\alpha \cdot \alpha} - \frac{2.0}{\alpha}\right) - \frac{8.0}{\alpha \cdot \left(\alpha \cdot \alpha\right)}\right)}}{2.0}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification6.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 9.97332166790088 \cdot 10^{+32}:\\ \;\;\;\;\frac{\frac{1}{\left(\beta + \alpha\right) + 2.0} \cdot \beta - \mathsf{fma}\left(\frac{\alpha}{\left(\beta + \alpha\right) \cdot \left(\beta + \alpha\right) - 2.0 \cdot 2.0}, \left(\beta + \alpha\right) - 2.0, -1.0\right)}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\beta + \alpha\right) + 2.0} - \left(\left(\frac{4.0}{\alpha \cdot \alpha} - \frac{2.0}{\alpha}\right) - \frac{8.0}{\alpha \cdot \left(\alpha \cdot \alpha\right)}\right)}{2.0}\\ \end{array}\]

Reproduce

herbie shell --seed 2019138 +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))