Average Error: 22.1 → 0.2
Time: 18.2s
Precision: 64
\[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -166852326.3746969401836395263671875 \lor \neg \left(y \le 7066667.03937393240630626678466796875\right):\\ \;\;\;\;\left(\frac{1}{y} - 1 \cdot \frac{x}{y}\right) + x\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\left(1 - x\right) \cdot y}{y \cdot y - 1 \cdot 1} \cdot \left(y - 1\right)\\ \end{array}\]
1 - \frac{\left(1 - x\right) \cdot y}{y + 1}
↓
\begin{array}{l}
\mathbf{if}\;y \le -166852326.3746969401836395263671875 \lor \neg \left(y \le 7066667.03937393240630626678466796875\right):\\
\;\;\;\;\left(\frac{1}{y} - 1 \cdot \frac{x}{y}\right) + x\\

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

\end{array}
double f(double x, double y) {
        double r429600 = 1.0;
        double r429601 = x;
        double r429602 = r429600 - r429601;
        double r429603 = y;
        double r429604 = r429602 * r429603;
        double r429605 = r429603 + r429600;
        double r429606 = r429604 / r429605;
        double r429607 = r429600 - r429606;
        return r429607;
}
↓
double f(double x, double y) {
        double r429608 = y;
        double r429609 = -166852326.37469694;
        bool r429610 = r429608 <= r429609;
        double r429611 = 7066667.039373932;
        bool r429612 = r429608 <= r429611;
        double r429613 = !r429612;
        bool r429614 = r429610 || r429613;
        double r429615 = 1.0;
        double r429616 = r429615 / r429608;
        double r429617 = x;
        double r429618 = r429617 / r429608;
        double r429619 = r429615 * r429618;
        double r429620 = r429616 - r429619;
        double r429621 = r429620 + r429617;
        double r429622 = r429615 - r429617;
        double r429623 = r429622 * r429608;
        double r429624 = r429608 * r429608;
        double r429625 = r429615 * r429615;
        double r429626 = r429624 - r429625;
        double r429627 = r429623 / r429626;
        double r429628 = r429608 - r429615;
        double r429629 = r429627 * r429628;
        double r429630 = r429615 - r429629;
        double r429631 = r429614 ? r429621 : r429630;
        return r429631;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original22.1
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 < -166852326.37469694 or 7066667.039373932 < y

    1. Initial program 45.0

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

    3. Applied flip-+49.9

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

    4. Applied associate-/r/49.9

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

    5. Taylor expanded around inf 0.2

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

    6. Simplified0.2

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

    if -166852326.37469694 < y < 7066667.039373932

    1. Initial program 0.2

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

    3. Applied flip-+0.2

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

    4. Applied associate-/r/0.2

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -166852326.3746969401836395263671875 \lor \neg \left(y \le 7066667.03937393240630626678466796875\right):\\ \;\;\;\;\left(\frac{1}{y} - 1 \cdot \frac{x}{y}\right) + x\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\left(1 - x\right) \cdot y}{y \cdot y - 1 \cdot 1} \cdot \left(y - 1\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019306 
(FPCore (x y)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, D"
  :precision binary64

  :herbie-target
  (if (< y -3693.84827882972468) (- (/ 1 y) (- (/ x y) x)) (if (< y 6799310503.41891003) (- 1 (/ (* (- 1 x) y) (+ y 1))) (- (/ 1 y) (- (/ x y) x))))

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