Average Error: 22.7 → 0.3
Time: 2.6s
Precision: 64
\[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -0.990940610756336437 \lor \neg \left(y \le 175576689.4578166\right):\\ \;\;\;\;1 \cdot \left(\frac{1}{y} - \frac{x}{y}\right) + x\\ \mathbf{else}:\\ \;\;\;\;1 - \left(1 - x\right) \cdot \left(\frac{1}{\sqrt{y + 1}} \cdot \frac{y}{\sqrt{y + 1}}\right)\\ \end{array}\]
1 - \frac{\left(1 - x\right) \cdot y}{y + 1}
↓
\begin{array}{l}
\mathbf{if}\;y \le -0.990940610756336437 \lor \neg \left(y \le 175576689.4578166\right):\\
\;\;\;\;1 \cdot \left(\frac{1}{y} - \frac{x}{y}\right) + x\\

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

\end{array}
double f(double x, double y) {
        double r787790 = 1.0;
        double r787791 = x;
        double r787792 = r787790 - r787791;
        double r787793 = y;
        double r787794 = r787792 * r787793;
        double r787795 = r787793 + r787790;
        double r787796 = r787794 / r787795;
        double r787797 = r787790 - r787796;
        return r787797;
}
↓
double f(double x, double y) {
        double r787798 = y;
        double r787799 = -0.9909406107563364;
        bool r787800 = r787798 <= r787799;
        double r787801 = 175576689.4578166;
        bool r787802 = r787798 <= r787801;
        double r787803 = !r787802;
        bool r787804 = r787800 || r787803;
        double r787805 = 1.0;
        double r787806 = 1.0;
        double r787807 = r787806 / r787798;
        double r787808 = x;
        double r787809 = r787808 / r787798;
        double r787810 = r787807 - r787809;
        double r787811 = r787805 * r787810;
        double r787812 = r787811 + r787808;
        double r787813 = r787805 - r787808;
        double r787814 = r787798 + r787805;
        double r787815 = sqrt(r787814);
        double r787816 = r787806 / r787815;
        double r787817 = r787798 / r787815;
        double r787818 = r787816 * r787817;
        double r787819 = r787813 * r787818;
        double r787820 = r787805 - r787819;
        double r787821 = r787804 ? r787812 : r787820;
        return r787821;
}

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
Herbie0.3
\[\begin{array}{l} \mathbf{if}\;y \lt -3693.84827882972468:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \mathbf{elif}\;y \lt 6799310503.41891003:\\ \;\;\;\;1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -0.9909406107563364 or 175576689.4578166 < y

    1. Initial program 45.2

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

    2. Taylor expanded around inf 0.6

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

    3. Simplified0.6

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

    if -0.9909406107563364 < y < 175576689.4578166

    1. Initial program 0.1

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

    3. Applied *-un-lft-identity0.1

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

    4. Applied times-frac0.1

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

    5. Simplified0.1

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

    7. Applied add-sqr-sqrt0.1

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

    8. Applied *-un-lft-identity0.1

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

    9. Applied times-frac0.1

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -0.990940610756336437 \lor \neg \left(y \le 175576689.4578166\right):\\ \;\;\;\;1 \cdot \left(\frac{1}{y} - \frac{x}{y}\right) + x\\ \mathbf{else}:\\ \;\;\;\;1 - \left(1 - x\right) \cdot \left(\frac{1}{\sqrt{y + 1}} \cdot \frac{y}{\sqrt{y + 1}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020065 
(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))))