Average Error: 22.2 → 0.2
Time: 12.7s
Precision: 64
\[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -104628043.80156818 \lor \neg \left(y \le 212488777.898407\right):\\ \;\;\;\;\frac{1}{y} \cdot \left(1 - x\right) + x\\ \mathbf{else}:\\ \;\;\;\;1 - \left(1 - x\right) \cdot \frac{y}{y + 1}\\ \end{array}\]
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):\\
\;\;\;\;\frac{1}{y} \cdot \left(1 - x\right) + x\\

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

\end{array}
double f(double x, double y) {
        double r814996 = 1.0;
        double r814997 = x;
        double r814998 = r814996 - r814997;
        double r814999 = y;
        double r815000 = r814998 * r814999;
        double r815001 = r814999 + r814996;
        double r815002 = r815000 / r815001;
        double r815003 = r814996 - r815002;
        return r815003;
}
↓
double f(double x, double y) {
        double r815004 = y;
        double r815005 = -104628043.80156818;
        bool r815006 = r815004 <= r815005;
        double r815007 = 212488777.89840698;
        bool r815008 = r815004 <= r815007;
        double r815009 = !r815008;
        bool r815010 = r815006 || r815009;
        double r815011 = 1.0;
        double r815012 = r815011 / r815004;
        double r815013 = 1.0;
        double r815014 = x;
        double r815015 = r815013 - r815014;
        double r815016 = r815012 * r815015;
        double r815017 = r815016 + r815014;
        double r815018 = r815011 - r815014;
        double r815019 = r815004 + r815011;
        double r815020 = r815004 / r815019;
        double r815021 = r815018 * r815020;
        double r815022 = r815011 - r815021;
        double r815023 = r815010 ? r815017 : r815022;
        return r815023;
}

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. 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

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

    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

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

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))))