Average Error: 22.2 → 0.2
Time: 13.6s
Precision: 64
\[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -198843285.268944234 \lor \neg \left(y \le 212488777.898407\right):\\ \;\;\;\;\left(\frac{1}{y} + x\right) - 1 \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\\ \end{array}\]
1 - \frac{\left(1 - x\right) \cdot y}{y + 1}
↓
\begin{array}{l}
\mathbf{if}\;y \le -198843285.268944234 \lor \neg \left(y \le 212488777.898407\right):\\
\;\;\;\;\left(\frac{1}{y} + x\right) - 1 \cdot \frac{x}{y}\\

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

\end{array}
double f(double x, double y) {
        double r724799 = 1.0;
        double r724800 = x;
        double r724801 = r724799 - r724800;
        double r724802 = y;
        double r724803 = r724801 * r724802;
        double r724804 = r724802 + r724799;
        double r724805 = r724803 / r724804;
        double r724806 = r724799 - r724805;
        return r724806;
}
↓
double f(double x, double y) {
        double r724807 = y;
        double r724808 = -198843285.26894423;
        bool r724809 = r724807 <= r724808;
        double r724810 = 212488777.89840698;
        bool r724811 = r724807 <= r724810;
        double r724812 = !r724811;
        bool r724813 = r724809 || r724812;
        double r724814 = 1.0;
        double r724815 = r724814 / r724807;
        double r724816 = x;
        double r724817 = r724815 + r724816;
        double r724818 = r724816 / r724807;
        double r724819 = r724814 * r724818;
        double r724820 = r724817 - r724819;
        double r724821 = r724814 - r724816;
        double r724822 = r724821 * r724807;
        double r724823 = r724807 + r724814;
        double r724824 = r724822 / r724823;
        double r724825 = r724814 - r724824;
        double r724826 = r724813 ? r724820 : r724825;
        return r724826;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original22.2
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 < -198843285.26894423 or 212488777.89840698 < y

    1. Initial program 45.7

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

    2. Taylor expanded around inf 0.1

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

    3. Simplified0.1

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

    if -198843285.26894423 < y < 212488777.89840698

    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

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -198843285.268944234 \lor \neg \left(y \le 212488777.898407\right):\\ \;\;\;\;\left(\frac{1}{y} + x\right) - 1 \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 +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))))