Average Error: 22.3 → 0.2
Time: 11.0s
Precision: 64
\[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -126285643.80532795 \lor \neg \left(y \le 184531412.18592519\right):\\ \;\;\;\;\left(\frac{1}{y} + x\right) - 1 \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;1 - \left(1 - x\right) \cdot \frac{y}{y + 1}\\ \end{array}\]
1 - \frac{\left(1 - x\right) \cdot y}{y + 1}
↓
\begin{array}{l}
\mathbf{if}\;y \le -126285643.80532795 \lor \neg \left(y \le 184531412.18592519\right):\\
\;\;\;\;\left(\frac{1}{y} + x\right) - 1 \cdot \frac{x}{y}\\

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

\end{array}
double f(double x, double y) {
        double r700719 = 1.0;
        double r700720 = x;
        double r700721 = r700719 - r700720;
        double r700722 = y;
        double r700723 = r700721 * r700722;
        double r700724 = r700722 + r700719;
        double r700725 = r700723 / r700724;
        double r700726 = r700719 - r700725;
        return r700726;
}
↓
double f(double x, double y) {
        double r700727 = y;
        double r700728 = -126285643.80532795;
        bool r700729 = r700727 <= r700728;
        double r700730 = 184531412.1859252;
        bool r700731 = r700727 <= r700730;
        double r700732 = !r700731;
        bool r700733 = r700729 || r700732;
        double r700734 = 1.0;
        double r700735 = r700734 / r700727;
        double r700736 = x;
        double r700737 = r700735 + r700736;
        double r700738 = r700736 / r700727;
        double r700739 = r700734 * r700738;
        double r700740 = r700737 - r700739;
        double r700741 = r700734 - r700736;
        double r700742 = r700727 + r700734;
        double r700743 = r700727 / r700742;
        double r700744 = r700741 * r700743;
        double r700745 = r700734 - r700744;
        double r700746 = r700733 ? r700740 : r700745;
        return r700746;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original22.3
Target0.3
Herbie0.2
\[\begin{array}{l} \mathbf{if}\;y \lt -3693.84827882972468:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \mathbf{elif}\;y \lt 6799310503.41891003:\\ \;\;\;\;1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -126285643.80532795 or 184531412.1859252 < y

    1. Initial program 45.2

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

    2. Simplified29.6

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

    4. Applied flip-+45.0

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

    5. Applied associate-/r/45.1

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

    6. Taylor expanded around inf 0.3

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

    7. Simplified0.3

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

    if -126285643.80532795 < y < 184531412.1859252

    1. Initial program 0.2

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]
    2. Using strategy rm

    3. Applied *-un-lft-identity0.2

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

    4. Applied times-frac0.2

      \[\leadsto 1 - \color{blue}{\frac{1 - x}{1} \cdot \frac{y}{y + 1}}\]

    5. Simplified0.2

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -126285643.80532795 \lor \neg \left(y \le 184531412.18592519\right):\\ \;\;\;\;\left(\frac{1}{y} + x\right) - 1 \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;1 - \left(1 - x\right) \cdot \frac{y}{y + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020045 +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))))