Average Error: 23.6 → 0.1
Time: 41.2s
Precision: 64
\[1.0 - \frac{\left(1.0 - x\right) \cdot y}{y + 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -124597878.5240411:\\ \;\;\;\;\left(x + \frac{1.0}{y}\right) - 1.0 \cdot \frac{x}{y}\\ \mathbf{elif}\;y \le 204123916.41808376:\\ \;\;\;\;1.0 - \left(1.0 - x\right) \cdot \frac{y}{1.0 + y}\\ \mathbf{else}:\\ \;\;\;\;\left(x + \frac{1.0}{y}\right) - 1.0 \cdot \frac{x}{y}\\ \end{array}\]
1.0 - \frac{\left(1.0 - x\right) \cdot y}{y + 1.0}
↓
\begin{array}{l}
\mathbf{if}\;y \le -124597878.5240411:\\
\;\;\;\;\left(x + \frac{1.0}{y}\right) - 1.0 \cdot \frac{x}{y}\\

\mathbf{elif}\;y \le 204123916.41808376:\\
\;\;\;\;1.0 - \left(1.0 - x\right) \cdot \frac{y}{1.0 + y}\\

\mathbf{else}:\\
\;\;\;\;\left(x + \frac{1.0}{y}\right) - 1.0 \cdot \frac{x}{y}\\

\end{array}
double f(double x, double y) {
        double r31548868 = 1.0;
        double r31548869 = x;
        double r31548870 = r31548868 - r31548869;
        double r31548871 = y;
        double r31548872 = r31548870 * r31548871;
        double r31548873 = r31548871 + r31548868;
        double r31548874 = r31548872 / r31548873;
        double r31548875 = r31548868 - r31548874;
        return r31548875;
}
↓
double f(double x, double y) {
        double r31548876 = y;
        double r31548877 = -124597878.5240411;
        bool r31548878 = r31548876 <= r31548877;
        double r31548879 = x;
        double r31548880 = 1.0;
        double r31548881 = r31548880 / r31548876;
        double r31548882 = r31548879 + r31548881;
        double r31548883 = r31548879 / r31548876;
        double r31548884 = r31548880 * r31548883;
        double r31548885 = r31548882 - r31548884;
        double r31548886 = 204123916.41808376;
        bool r31548887 = r31548876 <= r31548886;
        double r31548888 = r31548880 - r31548879;
        double r31548889 = r31548880 + r31548876;
        double r31548890 = r31548876 / r31548889;
        double r31548891 = r31548888 * r31548890;
        double r31548892 = r31548880 - r31548891;
        double r31548893 = r31548887 ? r31548892 : r31548885;
        double r31548894 = r31548878 ? r31548885 : r31548893;
        return r31548894;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original23.6
Target0.2
Herbie0.1
\[\begin{array}{l} \mathbf{if}\;y \lt -3693.8482788297247:\\ \;\;\;\;\frac{1.0}{y} - \left(\frac{x}{y} - x\right)\\ \mathbf{elif}\;y \lt 6799310503.41891:\\ \;\;\;\;1.0 - \frac{\left(1.0 - x\right) \cdot y}{y + 1.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{1.0}{y} - \left(\frac{x}{y} - x\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -124597878.5240411 or 204123916.41808376 < y

    1. Initial program 46.8

      \[1.0 - \frac{\left(1.0 - x\right) \cdot y}{y + 1.0}\]
    2. Using strategy rm

    3. Applied *-un-lft-identity46.8

      \[\leadsto 1.0 - \frac{\left(1.0 - x\right) \cdot y}{\color{blue}{1 \cdot \left(y + 1.0\right)}}\]

    4. Applied times-frac30.1

      \[\leadsto 1.0 - \color{blue}{\frac{1.0 - x}{1} \cdot \frac{y}{y + 1.0}}\]

    5. Simplified30.1

      \[\leadsto 1.0 - \color{blue}{\left(1.0 - x\right)} \cdot \frac{y}{y + 1.0}\]

    6. Taylor expanded around inf 0.1

      \[\leadsto \color{blue}{\left(x + 1.0 \cdot \frac{1}{y}\right) - 1.0 \cdot \frac{x}{y}}\]

    7. Simplified0.1

      \[\leadsto \color{blue}{\left(\frac{1.0}{y} + x\right) - 1.0 \cdot \frac{x}{y}}\]

    if -124597878.5240411 < y < 204123916.41808376

    1. Initial program 0.2

      \[1.0 - \frac{\left(1.0 - x\right) \cdot y}{y + 1.0}\]
    2. Using strategy rm

    3. Applied *-un-lft-identity0.2

      \[\leadsto 1.0 - \frac{\left(1.0 - x\right) \cdot y}{\color{blue}{1 \cdot \left(y + 1.0\right)}}\]

    4. Applied times-frac0.2

      \[\leadsto 1.0 - \color{blue}{\frac{1.0 - x}{1} \cdot \frac{y}{y + 1.0}}\]

    5. Simplified0.2

      \[\leadsto 1.0 - \color{blue}{\left(1.0 - x\right)} \cdot \frac{y}{y + 1.0}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -124597878.5240411:\\ \;\;\;\;\left(x + \frac{1.0}{y}\right) - 1.0 \cdot \frac{x}{y}\\ \mathbf{elif}\;y \le 204123916.41808376:\\ \;\;\;\;1.0 - \left(1.0 - x\right) \cdot \frac{y}{1.0 + y}\\ \mathbf{else}:\\ \;\;\;\;\left(x + \frac{1.0}{y}\right) - 1.0 \cdot \frac{x}{y}\\ \end{array}\]

Reproduce

herbie shell --seed 2019165 
(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))))