Average Error: 22.2 → 0.2
Time: 13.6s
Precision: 64
\[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -104628043.80156818 \lor \neg \left(y \le 212488777.898407\right):\\ \;\;\;\;\left(\frac{1}{y} - 1 \cdot \frac{x}{y}\right) + x\\ \mathbf{else}:\\ \;\;\;\;1 - \left(1 - x\right) \cdot \frac{y}{y + 1}\\ \end{array}\]
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):\\
\;\;\;\;\left(\frac{1}{y} - 1 \cdot \frac{x}{y}\right) + x\\

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

\end{array}
double f(double x, double y) {
        double r788963 = 1.0;
        double r788964 = x;
        double r788965 = r788963 - r788964;
        double r788966 = y;
        double r788967 = r788965 * r788966;
        double r788968 = r788966 + r788963;
        double r788969 = r788967 / r788968;
        double r788970 = r788963 - r788969;
        return r788970;
}
↓
double f(double x, double y) {
        double r788971 = y;
        double r788972 = -104628043.80156818;
        bool r788973 = r788971 <= r788972;
        double r788974 = 212488777.89840698;
        bool r788975 = r788971 <= r788974;
        double r788976 = !r788975;
        bool r788977 = r788973 || r788976;
        double r788978 = 1.0;
        double r788979 = r788978 / r788971;
        double r788980 = x;
        double r788981 = r788980 / r788971;
        double r788982 = r788978 * r788981;
        double r788983 = r788979 - r788982;
        double r788984 = r788983 + r788980;
        double r788985 = r788978 - r788980;
        double r788986 = r788971 + r788978;
        double r788987 = r788971 / r788986;
        double r788988 = r788985 * r788987;
        double r788989 = r788978 - r788988;
        double r788990 = r788977 ? r788984 : r788989;
        return r788990;
}

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. Taylor expanded around inf 0.1

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

    3. Simplified0.1

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

    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

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

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))))