Average Error: 22.7 → 0.2
Time: 19.0s
Precision: 64
\[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -179455656.4260170757770538330078125:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{1}{y} - \frac{x}{y}, x\right)\\ \mathbf{elif}\;y \le 192804976.65096271038055419921875:\\ \;\;\;\;1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{1}{y} - \frac{x}{y}, x\right)\\ \end{array}\]
1 - \frac{\left(1 - x\right) \cdot y}{y + 1}
↓
\begin{array}{l}
\mathbf{if}\;y \le -179455656.4260170757770538330078125:\\
\;\;\;\;\mathsf{fma}\left(1, \frac{1}{y} - \frac{x}{y}, x\right)\\

\mathbf{elif}\;y \le 192804976.65096271038055419921875:\\
\;\;\;\;1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(1, \frac{1}{y} - \frac{x}{y}, x\right)\\

\end{array}
double f(double x, double y) {
        double r41806775 = 1.0;
        double r41806776 = x;
        double r41806777 = r41806775 - r41806776;
        double r41806778 = y;
        double r41806779 = r41806777 * r41806778;
        double r41806780 = r41806778 + r41806775;
        double r41806781 = r41806779 / r41806780;
        double r41806782 = r41806775 - r41806781;
        return r41806782;
}
↓
double f(double x, double y) {
        double r41806783 = y;
        double r41806784 = -179455656.42601708;
        bool r41806785 = r41806783 <= r41806784;
        double r41806786 = 1.0;
        double r41806787 = 1.0;
        double r41806788 = r41806787 / r41806783;
        double r41806789 = x;
        double r41806790 = r41806789 / r41806783;
        double r41806791 = r41806788 - r41806790;
        double r41806792 = fma(r41806786, r41806791, r41806789);
        double r41806793 = 192804976.6509627;
        bool r41806794 = r41806783 <= r41806793;
        double r41806795 = r41806786 - r41806789;
        double r41806796 = r41806795 * r41806783;
        double r41806797 = r41806783 + r41806786;
        double r41806798 = r41806796 / r41806797;
        double r41806799 = r41806786 - r41806798;
        double r41806800 = r41806794 ? r41806799 : r41806792;
        double r41806801 = r41806785 ? r41806792 : r41806800;
        return r41806801;
}

Error

Bits error versus x

Bits error versus y

Target

Original22.7
Target0.2
Herbie0.2
\[\begin{array}{l} \mathbf{if}\;y \lt -3693.848278829724677052581682801246643066:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \mathbf{elif}\;y \lt 6799310503.41891002655029296875:\\ \;\;\;\;1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -179455656.42601708 or 192804976.6509627 < y

    1. Initial program 46.3

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]

    2. Simplified29.6

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x - 1}{y + 1}, y, 1\right)}\]

    3. Taylor expanded around inf 0.1

      \[\leadsto \color{blue}{\left(x + 1 \cdot \frac{1}{y}\right) - 1 \cdot \frac{x}{y}}\]

    4. Simplified0.1

      \[\leadsto \color{blue}{\mathsf{fma}\left(1, \frac{1}{y} - \frac{x}{y}, x\right)}\]

    if -179455656.42601708 < y < 192804976.6509627

    1. Initial program 0.2

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -179455656.4260170757770538330078125:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{1}{y} - \frac{x}{y}, x\right)\\ \mathbf{elif}\;y \le 192804976.65096271038055419921875:\\ \;\;\;\;1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{1}{y} - \frac{x}{y}, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 +o rules:numerics
(FPCore (x y)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, D"

  :herbie-target
  (if (< y -3693.8482788297247) (- (/ 1.0 y) (- (/ x y) x)) (if (< y 6799310503.41891) (- 1.0 (/ (* (- 1.0 x) y) (+ y 1.0))) (- (/ 1.0 y) (- (/ x y) x))))

  (- 1.0 (/ (* (- 1.0 x) y) (+ y 1.0))))