Average Error: 22.2 → 0.2
Time: 3.3s
Precision: 64
\[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -4286551152.9063902 \lor \neg \left(y \le 360844840.908958614\right):\\ \;\;\;\;1 \cdot \left(\frac{1}{y} - \frac{x}{y}\right) + x\\ \mathbf{else}:\\ \;\;\;\;1 - \left(1 - x\right) \cdot \frac{y}{y + 1}\\ \end{array}\]
1 - \frac{\left(1 - x\right) \cdot y}{y + 1}
↓
\begin{array}{l}
\mathbf{if}\;y \le -4286551152.9063902 \lor \neg \left(y \le 360844840.908958614\right):\\
\;\;\;\;1 \cdot \left(\frac{1}{y} - \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 r1497740 = 1.0;
        double r1497741 = x;
        double r1497742 = r1497740 - r1497741;
        double r1497743 = y;
        double r1497744 = r1497742 * r1497743;
        double r1497745 = r1497743 + r1497740;
        double r1497746 = r1497744 / r1497745;
        double r1497747 = r1497740 - r1497746;
        return r1497747;
}
↓
double f(double x, double y) {
        double r1497748 = y;
        double r1497749 = -4286551152.90639;
        bool r1497750 = r1497748 <= r1497749;
        double r1497751 = 360844840.9089586;
        bool r1497752 = r1497748 <= r1497751;
        double r1497753 = !r1497752;
        bool r1497754 = r1497750 || r1497753;
        double r1497755 = 1.0;
        double r1497756 = 1.0;
        double r1497757 = r1497756 / r1497748;
        double r1497758 = x;
        double r1497759 = r1497758 / r1497748;
        double r1497760 = r1497757 - r1497759;
        double r1497761 = r1497755 * r1497760;
        double r1497762 = r1497761 + r1497758;
        double r1497763 = r1497755 - r1497758;
        double r1497764 = r1497748 + r1497755;
        double r1497765 = r1497748 / r1497764;
        double r1497766 = r1497763 * r1497765;
        double r1497767 = r1497755 - r1497766;
        double r1497768 = r1497754 ? r1497762 : r1497767;
        return r1497768;
}

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
Target0.3
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 < -4286551152.90639 or 360844840.9089586 < y

    1. Initial program 45.7

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

    2. Taylor expanded around inf 0.2

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

    3. Simplified0.2

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

    if -4286551152.90639 < y < 360844840.9089586

    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

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -4286551152.9063902 \lor \neg \left(y \le 360844840.908958614\right):\\ \;\;\;\;1 \cdot \left(\frac{1}{y} - \frac{x}{y}\right) + x\\ \mathbf{else}:\\ \;\;\;\;1 - \left(1 - x\right) \cdot \frac{y}{y + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020057 
(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))))