Average Error: 22.9 → 7.6
Time: 3.5s
Precision: 64
\[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -22026595796721872 \lor \neg \left(y \le 14000563147.5999813\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, \frac{1}{y} - 1, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{y + 1}, x - 1, 1\right)\\ \end{array}\]
1 - \frac{\left(1 - x\right) \cdot y}{y + 1}
↓
\begin{array}{l}
\mathbf{if}\;y \le -22026595796721872 \lor \neg \left(y \le 14000563147.5999813\right):\\
\;\;\;\;\mathsf{fma}\left(\frac{x}{y}, \frac{1}{y} - 1, x\right)\\

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

\end{array}
double f(double x, double y) {
        double r904589 = 1.0;
        double r904590 = x;
        double r904591 = r904589 - r904590;
        double r904592 = y;
        double r904593 = r904591 * r904592;
        double r904594 = r904592 + r904589;
        double r904595 = r904593 / r904594;
        double r904596 = r904589 - r904595;
        return r904596;
}
↓
double f(double x, double y) {
        double r904597 = y;
        double r904598 = -2.202659579672187e+16;
        bool r904599 = r904597 <= r904598;
        double r904600 = 14000563147.599981;
        bool r904601 = r904597 <= r904600;
        double r904602 = !r904601;
        bool r904603 = r904599 || r904602;
        double r904604 = x;
        double r904605 = r904604 / r904597;
        double r904606 = 1.0;
        double r904607 = r904606 / r904597;
        double r904608 = r904607 - r904606;
        double r904609 = fma(r904605, r904608, r904604);
        double r904610 = r904597 + r904606;
        double r904611 = r904597 / r904610;
        double r904612 = r904604 - r904606;
        double r904613 = fma(r904611, r904612, r904606);
        double r904614 = r904603 ? r904609 : r904613;
        return r904614;
}

Error

Bits error versus x

Bits error versus y

Target

Original22.9
Target0.2
Herbie7.6
\[\begin{array}{l} \mathbf{if}\;y \lt -3693.84827882972468:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \mathbf{elif}\;y \lt 6799310503.41891003:\\ \;\;\;\;1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -2.202659579672187e+16 or 14000563147.599981 < y

    1. Initial program 46.4

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

    2. Simplified29.3

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

    3. Taylor expanded around inf 15.1

      \[\leadsto \color{blue}{\left(x + 1 \cdot \frac{x}{{y}^{2}}\right) - 1 \cdot \frac{x}{y}}\]

    4. Simplified15.1

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

    if -2.202659579672187e+16 < y < 14000563147.599981

    1. Initial program 0.5

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

    2. Simplified0.5

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

  4. Final simplification7.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -22026595796721872 \lor \neg \left(y \le 14000563147.5999813\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, \frac{1}{y} - 1, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{y + 1}, x - 1, 1\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020060 +o rules:numerics
(FPCore (x y)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, D"
  :precision binary64

  :herbie-target
  (if (< y -3693.8482788297247) (- (/ 1 y) (- (/ x y) x)) (if (< y 6799310503.41891) (- 1 (/ (* (- 1 x) y) (+ y 1))) (- (/ 1 y) (- (/ x y) x))))

  (- 1 (/ (* (- 1 x) y) (+ y 1))))