Average Error: 22.7 → 0.2
Time: 17.3s
Precision: 64
\[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -179455656.4260170757770538330078125:\\ \;\;\;\;\left(x + \frac{1}{y}\right) - \frac{x}{y} \cdot 1\\ \mathbf{elif}\;y \le 186709514.105293214321136474609375:\\ \;\;\;\;1 - \frac{1}{y + 1} \cdot \left(\left(1 - x\right) \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x + \frac{1}{y}\right) - \frac{x}{y} \cdot 1\\ \end{array}\]
1 - \frac{\left(1 - x\right) \cdot y}{y + 1}
↓
\begin{array}{l}
\mathbf{if}\;y \le -179455656.4260170757770538330078125:\\
\;\;\;\;\left(x + \frac{1}{y}\right) - \frac{x}{y} \cdot 1\\

\mathbf{elif}\;y \le 186709514.105293214321136474609375:\\
\;\;\;\;1 - \frac{1}{y + 1} \cdot \left(\left(1 - x\right) \cdot y\right)\\

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

\end{array}
double f(double x, double y) {
        double r34118622 = 1.0;
        double r34118623 = x;
        double r34118624 = r34118622 - r34118623;
        double r34118625 = y;
        double r34118626 = r34118624 * r34118625;
        double r34118627 = r34118625 + r34118622;
        double r34118628 = r34118626 / r34118627;
        double r34118629 = r34118622 - r34118628;
        return r34118629;
}
↓
double f(double x, double y) {
        double r34118630 = y;
        double r34118631 = -179455656.42601708;
        bool r34118632 = r34118630 <= r34118631;
        double r34118633 = x;
        double r34118634 = 1.0;
        double r34118635 = r34118634 / r34118630;
        double r34118636 = r34118633 + r34118635;
        double r34118637 = r34118633 / r34118630;
        double r34118638 = r34118637 * r34118634;
        double r34118639 = r34118636 - r34118638;
        double r34118640 = 186709514.1052932;
        bool r34118641 = r34118630 <= r34118640;
        double r34118642 = 1.0;
        double r34118643 = r34118630 + r34118634;
        double r34118644 = r34118642 / r34118643;
        double r34118645 = r34118634 - r34118633;
        double r34118646 = r34118645 * r34118630;
        double r34118647 = r34118644 * r34118646;
        double r34118648 = r34118634 - r34118647;
        double r34118649 = r34118641 ? r34118648 : r34118639;
        double r34118650 = r34118632 ? r34118639 : r34118649;
        return r34118650;
}

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
\[\begin{array}{l} \mathbf{if}\;y \lt -3693.848278829724677052581682801246643066:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \mathbf{elif}\;y \lt 6799310503.41891002655029296875:\\ \;\;\;\;1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -179455656.42601708 or 186709514.1052932 < y

    1. Initial program 46.3

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

    3. Applied associate-/l*29.5

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

    5. Applied flip--45.8

      \[\leadsto \color{blue}{\frac{1 \cdot 1 - \frac{1 - x}{\frac{y + 1}{y}} \cdot \frac{1 - x}{\frac{y + 1}{y}}}{1 + \frac{1 - x}{\frac{y + 1}{y}}}}\]

    6. Taylor expanded around inf 0.1

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

    7. Simplified0.1

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

    if -179455656.42601708 < y < 186709514.1052932

    1. Initial program 0.2

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

    3. Applied associate-/l*0.3

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

    5. Applied div-inv0.3

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

    6. Applied *-un-lft-identity0.3

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

    7. Applied times-frac0.3

      \[\leadsto 1 - \color{blue}{\frac{1}{y + 1} \cdot \frac{1 - x}{\frac{1}{y}}}\]

    8. Simplified0.2

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -179455656.4260170757770538330078125:\\ \;\;\;\;\left(x + \frac{1}{y}\right) - \frac{x}{y} \cdot 1\\ \mathbf{elif}\;y \le 186709514.105293214321136474609375:\\ \;\;\;\;1 - \frac{1}{y + 1} \cdot \left(\left(1 - x\right) \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x + \frac{1}{y}\right) - \frac{x}{y} \cdot 1\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 
(FPCore (x y)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, D"

  :herbie-target
  (if (< y -3693.8482788297247) (- (/ 1.0 y) (- (/ x y) x)) (if (< y 6799310503.41891) (- 1.0 (/ (* (- 1.0 x) y) (+ y 1.0))) (- (/ 1.0 y) (- (/ x y) x))))

  (- 1.0 (/ (* (- 1.0 x) y) (+ y 1.0))))