Average Error: 22.5 → 0.2
Time: 18.9s
Precision: 64
\[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -17413228841055.01171875:\\ \;\;\;\;\left(x - \frac{1}{\frac{y}{x}}\right) + \frac{1}{y}\\ \mathbf{elif}\;y \le 256213875.7381432950496673583984375:\\ \;\;\;\;1 - \left(1 - x\right) \cdot \frac{y}{1 + y}\\ \mathbf{else}:\\ \;\;\;\;\left(x - \frac{1}{\frac{y}{x}}\right) + \frac{1}{y}\\ \end{array}\]
1 - \frac{\left(1 - x\right) \cdot y}{y + 1}
↓
\begin{array}{l}
\mathbf{if}\;y \le -17413228841055.01171875:\\
\;\;\;\;\left(x - \frac{1}{\frac{y}{x}}\right) + \frac{1}{y}\\

\mathbf{elif}\;y \le 256213875.7381432950496673583984375:\\
\;\;\;\;1 - \left(1 - x\right) \cdot \frac{y}{1 + y}\\

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

\end{array}
double f(double x, double y) {
        double r36423603 = 1.0;
        double r36423604 = x;
        double r36423605 = r36423603 - r36423604;
        double r36423606 = y;
        double r36423607 = r36423605 * r36423606;
        double r36423608 = r36423606 + r36423603;
        double r36423609 = r36423607 / r36423608;
        double r36423610 = r36423603 - r36423609;
        return r36423610;
}
↓
double f(double x, double y) {
        double r36423611 = y;
        double r36423612 = -17413228841055.012;
        bool r36423613 = r36423611 <= r36423612;
        double r36423614 = x;
        double r36423615 = 1.0;
        double r36423616 = r36423611 / r36423614;
        double r36423617 = r36423615 / r36423616;
        double r36423618 = r36423614 - r36423617;
        double r36423619 = r36423615 / r36423611;
        double r36423620 = r36423618 + r36423619;
        double r36423621 = 256213875.7381433;
        bool r36423622 = r36423611 <= r36423621;
        double r36423623 = r36423615 - r36423614;
        double r36423624 = r36423615 + r36423611;
        double r36423625 = r36423611 / r36423624;
        double r36423626 = r36423623 * r36423625;
        double r36423627 = r36423615 - r36423626;
        double r36423628 = r36423622 ? r36423627 : r36423620;
        double r36423629 = r36423613 ? r36423620 : r36423628;
        return r36423629;
}

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

Derivation

  1. Split input into 2 regimes

  2. if y < -17413228841055.012 or 256213875.7381433 < y

    1. Initial program 46.6

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

    3. Applied *-un-lft-identity46.6

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

    4. Applied times-frac29.7

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

    5. Simplified29.7

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

    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

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

    if -17413228841055.012 < y < 256213875.7381433

    1. Initial program 0.3

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

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

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

    4. Applied times-frac0.3

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

    5. Simplified0.3

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -17413228841055.01171875:\\ \;\;\;\;\left(x - \frac{1}{\frac{y}{x}}\right) + \frac{1}{y}\\ \mathbf{elif}\;y \le 256213875.7381432950496673583984375:\\ \;\;\;\;1 - \left(1 - x\right) \cdot \frac{y}{1 + y}\\ \mathbf{else}:\\ \;\;\;\;\left(x - \frac{1}{\frac{y}{x}}\right) + \frac{1}{y}\\ \end{array}\]

Reproduce

herbie shell --seed 2019171 
(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))))