Average Error: 22.2 → 0.2
Time: 3.3s
Precision: 64
\[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -104628043.80156818 \lor \neg \left(y \le 212488777.898407\right):\\ \;\;\;\;\left(x + 1 \cdot \frac{1}{y}\right) - 1 \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;1 - \left(1 - x\right) \cdot \frac{y}{y + 1}\\ \end{array}\]
1 - \frac{\left(1 - x\right) \cdot y}{y + 1}
↓
\begin{array}{l}
\mathbf{if}\;y \le -104628043.80156818 \lor \neg \left(y \le 212488777.898407\right):\\
\;\;\;\;\left(x + 1 \cdot \frac{1}{y}\right) - 1 \cdot \frac{x}{y}\\

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

\end{array}
double f(double x, double y) {
        double r718565 = 1.0;
        double r718566 = x;
        double r718567 = r718565 - r718566;
        double r718568 = y;
        double r718569 = r718567 * r718568;
        double r718570 = r718568 + r718565;
        double r718571 = r718569 / r718570;
        double r718572 = r718565 - r718571;
        return r718572;
}
↓
double f(double x, double y) {
        double r718573 = y;
        double r718574 = -104628043.80156818;
        bool r718575 = r718573 <= r718574;
        double r718576 = 212488777.89840698;
        bool r718577 = r718573 <= r718576;
        double r718578 = !r718577;
        bool r718579 = r718575 || r718578;
        double r718580 = x;
        double r718581 = 1.0;
        double r718582 = 1.0;
        double r718583 = r718582 / r718573;
        double r718584 = r718581 * r718583;
        double r718585 = r718580 + r718584;
        double r718586 = r718580 / r718573;
        double r718587 = r718581 * r718586;
        double r718588 = r718585 - r718587;
        double r718589 = r718581 - r718580;
        double r718590 = r718573 + r718581;
        double r718591 = r718573 / r718590;
        double r718592 = r718589 * r718591;
        double r718593 = r718581 - r718592;
        double r718594 = r718579 ? r718588 : r718593;
        return r718594;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original22.2
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 < -104628043.80156818 or 212488777.89840698 < y

    1. Initial program 45.6

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

    3. Applied *-un-lft-identity45.6

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

    4. Applied times-frac29.1

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

    5. Simplified29.1

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

    7. Applied flip3--50.1

      \[\leadsto \color{blue}{\frac{{1}^{3} - {\left(\left(1 - x\right) \cdot \frac{y}{y + 1}\right)}^{3}}{1 \cdot 1 + \left(\left(\left(1 - x\right) \cdot \frac{y}{y + 1}\right) \cdot \left(\left(1 - x\right) \cdot \frac{y}{y + 1}\right) + 1 \cdot \left(\left(1 - x\right) \cdot \frac{y}{y + 1}\right)\right)}}\]

    8. Simplified50.1

      \[\leadsto \frac{{1}^{3} - {\left(\left(1 - x\right) \cdot \frac{y}{y + 1}\right)}^{3}}{\color{blue}{\left(\left(1 - x\right) \cdot \frac{y}{y + 1}\right) \cdot \left(\left(1 - x\right) \cdot \frac{y}{y + 1} + 1\right) + 1 \cdot 1}}\]

    9. Taylor expanded around inf 0.1

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

    if -104628043.80156818 < y < 212488777.89840698

    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

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -104628043.80156818 \lor \neg \left(y \le 212488777.898407\right):\\ \;\;\;\;\left(x + 1 \cdot \frac{1}{y}\right) - 1 \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;1 - \left(1 - x\right) \cdot \frac{y}{y + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 
(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))))