Average Error: 22.4 → 0.2
Time: 11.8s
Precision: 64
\[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -193411162.0749186575412750244140625 \lor \neg \left(y \le 216523043.9852226078510284423828125\right):\\ \;\;\;\;\left(\frac{1}{y} - 1 \cdot \frac{x}{y}\right) + x\\ \mathbf{else}:\\ \;\;\;\;1 - \left(1 - x\right) \cdot \left(y \cdot \frac{1}{y + 1}\right)\\ \end{array}\]
1 - \frac{\left(1 - x\right) \cdot y}{y + 1}
↓
\begin{array}{l}
\mathbf{if}\;y \le -193411162.0749186575412750244140625 \lor \neg \left(y \le 216523043.9852226078510284423828125\right):\\
\;\;\;\;\left(\frac{1}{y} - 1 \cdot \frac{x}{y}\right) + x\\

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

\end{array}
double f(double x, double y) {
        double r460839 = 1.0;
        double r460840 = x;
        double r460841 = r460839 - r460840;
        double r460842 = y;
        double r460843 = r460841 * r460842;
        double r460844 = r460842 + r460839;
        double r460845 = r460843 / r460844;
        double r460846 = r460839 - r460845;
        return r460846;
}
↓
double f(double x, double y) {
        double r460847 = y;
        double r460848 = -193411162.07491866;
        bool r460849 = r460847 <= r460848;
        double r460850 = 216523043.9852226;
        bool r460851 = r460847 <= r460850;
        double r460852 = !r460851;
        bool r460853 = r460849 || r460852;
        double r460854 = 1.0;
        double r460855 = r460854 / r460847;
        double r460856 = x;
        double r460857 = r460856 / r460847;
        double r460858 = r460854 * r460857;
        double r460859 = r460855 - r460858;
        double r460860 = r460859 + r460856;
        double r460861 = r460854 - r460856;
        double r460862 = 1.0;
        double r460863 = r460847 + r460854;
        double r460864 = r460862 / r460863;
        double r460865 = r460847 * r460864;
        double r460866 = r460861 * r460865;
        double r460867 = r460854 - r460866;
        double r460868 = r460853 ? r460860 : r460867;
        return r460868;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original22.4
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 < -193411162.07491866 or 216523043.9852226 < y

    1. Initial program 46.0

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

    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 -193411162.07491866 < y < 216523043.9852226

    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

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

    7. Applied div-inv0.2

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -193411162.0749186575412750244140625 \lor \neg \left(y \le 216523043.9852226078510284423828125\right):\\ \;\;\;\;\left(\frac{1}{y} - 1 \cdot \frac{x}{y}\right) + x\\ \mathbf{else}:\\ \;\;\;\;1 - \left(1 - x\right) \cdot \left(y \cdot \frac{1}{y + 1}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019305 
(FPCore (x y)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, D"
  :precision binary64

  :herbie-target
  (if (< y -3693.84827882972468) (- (/ 1 y) (- (/ x y) x)) (if (< y 6799310503.41891003) (- 1 (/ (* (- 1 x) y) (+ y 1))) (- (/ 1 y) (- (/ x y) x))))

  (- 1 (/ (* (- 1 x) y) (+ y 1))))