Average Error: 22.5 → 0.2
Time: 3.3s
Precision: 64
\[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -132294310.463103905 \lor \neg \left(y \le 94033634.6152255684\right):\\ \;\;\;\;1 \cdot \left(\frac{1}{y} - \frac{x}{y}\right) + x\\ \mathbf{else}:\\ \;\;\;\;\left(1 - y \cdot \left(\left(1 - x\right) \cdot \frac{y}{y \cdot y - 1 \cdot 1}\right)\right) - \left(-1\right) \cdot \left(\left(1 - x\right) \cdot \frac{y}{y \cdot y - 1 \cdot 1}\right)\\ \end{array}\]
1 - \frac{\left(1 - x\right) \cdot y}{y + 1}
↓
\begin{array}{l}
\mathbf{if}\;y \le -132294310.463103905 \lor \neg \left(y \le 94033634.6152255684\right):\\
\;\;\;\;1 \cdot \left(\frac{1}{y} - \frac{x}{y}\right) + x\\

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

\end{array}
double f(double x, double y) {
        double r716860 = 1.0;
        double r716861 = x;
        double r716862 = r716860 - r716861;
        double r716863 = y;
        double r716864 = r716862 * r716863;
        double r716865 = r716863 + r716860;
        double r716866 = r716864 / r716865;
        double r716867 = r716860 - r716866;
        return r716867;
}
↓
double f(double x, double y) {
        double r716868 = y;
        double r716869 = -132294310.4631039;
        bool r716870 = r716868 <= r716869;
        double r716871 = 94033634.61522557;
        bool r716872 = r716868 <= r716871;
        double r716873 = !r716872;
        bool r716874 = r716870 || r716873;
        double r716875 = 1.0;
        double r716876 = 1.0;
        double r716877 = r716876 / r716868;
        double r716878 = x;
        double r716879 = r716878 / r716868;
        double r716880 = r716877 - r716879;
        double r716881 = r716875 * r716880;
        double r716882 = r716881 + r716878;
        double r716883 = r716875 - r716878;
        double r716884 = r716868 * r716868;
        double r716885 = r716875 * r716875;
        double r716886 = r716884 - r716885;
        double r716887 = r716868 / r716886;
        double r716888 = r716883 * r716887;
        double r716889 = r716868 * r716888;
        double r716890 = r716875 - r716889;
        double r716891 = -r716875;
        double r716892 = r716891 * r716888;
        double r716893 = r716890 - r716892;
        double r716894 = r716874 ? r716882 : r716893;
        return r716894;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original22.5
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 < -132294310.4631039 or 94033634.61522557 < y

    1. Initial program 45.7

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

    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 -132294310.4631039 < y < 94033634.61522557

    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

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

    7. Applied flip-+0.2

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

    8. Applied associate-/r/0.2

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

    9. Applied associate-*r*0.2

      \[\leadsto 1 - \color{blue}{\left(\left(1 - x\right) \cdot \frac{y}{y \cdot y - 1 \cdot 1}\right) \cdot \left(y - 1\right)}\]
    10. Using strategy rm

    11. Applied sub-neg0.2

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

    12. Applied distribute-rgt-in0.2

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

    13. Applied associate--r+0.2

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -132294310.463103905 \lor \neg \left(y \le 94033634.6152255684\right):\\ \;\;\;\;1 \cdot \left(\frac{1}{y} - \frac{x}{y}\right) + x\\ \mathbf{else}:\\ \;\;\;\;\left(1 - y \cdot \left(\left(1 - x\right) \cdot \frac{y}{y \cdot y - 1 \cdot 1}\right)\right) - \left(-1\right) \cdot \left(\left(1 - x\right) \cdot \frac{y}{y \cdot y - 1 \cdot 1}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020003 
(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))))