Average Error: 22.3 → 0.2
Time: 3.4s
Precision: 64
\[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -146664422.27513087 \lor \neg \left(y \le 154919718.94598407\right):\\ \;\;\;\;1 \cdot \left(\frac{1}{y} - \frac{x}{y}\right) + x\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{1 \cdot \left(\left(1 - x\right) \cdot y\right)}{y + 1}\\ \end{array}\]
1 - \frac{\left(1 - x\right) \cdot y}{y + 1}
↓
\begin{array}{l}
\mathbf{if}\;y \le -146664422.27513087 \lor \neg \left(y \le 154919718.94598407\right):\\
\;\;\;\;1 \cdot \left(\frac{1}{y} - \frac{x}{y}\right) + x\\

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

\end{array}
double f(double x, double y) {
        double r600860 = 1.0;
        double r600861 = x;
        double r600862 = r600860 - r600861;
        double r600863 = y;
        double r600864 = r600862 * r600863;
        double r600865 = r600863 + r600860;
        double r600866 = r600864 / r600865;
        double r600867 = r600860 - r600866;
        return r600867;
}
↓
double f(double x, double y) {
        double r600868 = y;
        double r600869 = -146664422.27513087;
        bool r600870 = r600868 <= r600869;
        double r600871 = 154919718.94598407;
        bool r600872 = r600868 <= r600871;
        double r600873 = !r600872;
        bool r600874 = r600870 || r600873;
        double r600875 = 1.0;
        double r600876 = 1.0;
        double r600877 = r600876 / r600868;
        double r600878 = x;
        double r600879 = r600878 / r600868;
        double r600880 = r600877 - r600879;
        double r600881 = r600875 * r600880;
        double r600882 = r600881 + r600878;
        double r600883 = r600875 - r600878;
        double r600884 = r600883 * r600868;
        double r600885 = r600876 * r600884;
        double r600886 = r600868 + r600875;
        double r600887 = r600885 / r600886;
        double r600888 = r600875 - r600887;
        double r600889 = r600874 ? r600882 : r600888;
        return r600889;
}

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
Target0.2
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 < -146664422.27513087 or 154919718.94598407 < y

    1. Initial program 46.0

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

    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 -146664422.27513087 < y < 154919718.94598407

    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

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -146664422.27513087 \lor \neg \left(y \le 154919718.94598407\right):\\ \;\;\;\;1 \cdot \left(\frac{1}{y} - \frac{x}{y}\right) + x\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{1 \cdot \left(\left(1 - x\right) \cdot y\right)}{y + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020100 
(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))))