Average Error: 22.7 → 0.2
Time: 3.7s
Precision: 64
\[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -102786700.59358019 \lor \neg \left(y \le 156635002.80615079\right):\\ \;\;\;\;1 \cdot \left(\frac{1}{y} - \frac{x}{y}\right) + x\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{1 - x}{y + 1} \cdot y\\ \end{array}\]
1 - \frac{\left(1 - x\right) \cdot y}{y + 1}
↓
\begin{array}{l}
\mathbf{if}\;y \le -102786700.59358019 \lor \neg \left(y \le 156635002.80615079\right):\\
\;\;\;\;1 \cdot \left(\frac{1}{y} - \frac{x}{y}\right) + x\\

\mathbf{else}:\\
\;\;\;\;1 - \frac{1 - x}{y + 1} \cdot y\\

\end{array}
double f(double x, double y) {
        double r785672 = 1.0;
        double r785673 = x;
        double r785674 = r785672 - r785673;
        double r785675 = y;
        double r785676 = r785674 * r785675;
        double r785677 = r785675 + r785672;
        double r785678 = r785676 / r785677;
        double r785679 = r785672 - r785678;
        return r785679;
}
↓
double f(double x, double y) {
        double r785680 = y;
        double r785681 = -102786700.59358019;
        bool r785682 = r785680 <= r785681;
        double r785683 = 156635002.8061508;
        bool r785684 = r785680 <= r785683;
        double r785685 = !r785684;
        bool r785686 = r785682 || r785685;
        double r785687 = 1.0;
        double r785688 = 1.0;
        double r785689 = r785688 / r785680;
        double r785690 = x;
        double r785691 = r785690 / r785680;
        double r785692 = r785689 - r785691;
        double r785693 = r785687 * r785692;
        double r785694 = r785693 + r785690;
        double r785695 = r785687 - r785690;
        double r785696 = r785680 + r785687;
        double r785697 = r785695 / r785696;
        double r785698 = r785697 * r785680;
        double r785699 = r785687 - r785698;
        double r785700 = r785686 ? r785694 : r785699;
        return r785700;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original22.7
Target0.2
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 < -102786700.59358019 or 156635002.8061508 < y

    1. Initial program 46.2

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

    2. Taylor expanded around inf 0.2

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

    3. Simplified0.2

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

    if -102786700.59358019 < y < 156635002.8061508

    1. Initial program 0.1

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

    3. Applied associate-/l*0.2

      \[\leadsto 1 - \color{blue}{\frac{1 - x}{\frac{y + 1}{y}}}\]
    4. Using strategy rm

    5. Applied associate-/r/0.1

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -102786700.59358019 \lor \neg \left(y \le 156635002.80615079\right):\\ \;\;\;\;1 \cdot \left(\frac{1}{y} - \frac{x}{y}\right) + x\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{1 - x}{y + 1} \cdot y\\ \end{array}\]

Reproduce

herbie shell --seed 2020083 
(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))))