Average Error: 22.6 → 0.1
Time: 18.3s
Precision: 64
\[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -205884948.0981209278106689453125:\\ \;\;\;\;\frac{1}{y} - \left(\frac{1}{\frac{y}{x}} - x\right)\\ \mathbf{elif}\;y \le 214825219.7474500238895416259765625:\\ \;\;\;\;1 - \left(1 - x\right) \cdot \frac{y}{1 + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} - \left(\frac{1}{\frac{y}{x}} - x\right)\\ \end{array}\]
1 - \frac{\left(1 - x\right) \cdot y}{y + 1}
↓
\begin{array}{l}
\mathbf{if}\;y \le -205884948.0981209278106689453125:\\
\;\;\;\;\frac{1}{y} - \left(\frac{1}{\frac{y}{x}} - x\right)\\

\mathbf{elif}\;y \le 214825219.7474500238895416259765625:\\
\;\;\;\;1 - \left(1 - x\right) \cdot \frac{y}{1 + y}\\

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

\end{array}
double f(double x, double y) {
        double r36092741 = 1.0;
        double r36092742 = x;
        double r36092743 = r36092741 - r36092742;
        double r36092744 = y;
        double r36092745 = r36092743 * r36092744;
        double r36092746 = r36092744 + r36092741;
        double r36092747 = r36092745 / r36092746;
        double r36092748 = r36092741 - r36092747;
        return r36092748;
}
↓
double f(double x, double y) {
        double r36092749 = y;
        double r36092750 = -205884948.09812093;
        bool r36092751 = r36092749 <= r36092750;
        double r36092752 = 1.0;
        double r36092753 = r36092752 / r36092749;
        double r36092754 = x;
        double r36092755 = r36092749 / r36092754;
        double r36092756 = r36092752 / r36092755;
        double r36092757 = r36092756 - r36092754;
        double r36092758 = r36092753 - r36092757;
        double r36092759 = 214825219.74745002;
        bool r36092760 = r36092749 <= r36092759;
        double r36092761 = r36092752 - r36092754;
        double r36092762 = r36092752 + r36092749;
        double r36092763 = r36092749 / r36092762;
        double r36092764 = r36092761 * r36092763;
        double r36092765 = r36092752 - r36092764;
        double r36092766 = r36092760 ? r36092765 : r36092758;
        double r36092767 = r36092751 ? r36092758 : r36092766;
        return r36092767;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original22.6
Target0.2
Herbie0.1
\[\begin{array}{l} \mathbf{if}\;y \lt -3693.848278829724677052581682801246643066:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \mathbf{elif}\;y \lt 6799310503.41891002655029296875:\\ \;\;\;\;1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -205884948.09812093 or 214825219.74745002 < y

    1. Initial program 45.8

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

    3. Applied *-un-lft-identity45.8

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

    4. Applied times-frac29.5

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

    5. Simplified29.5

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

    6. Taylor expanded around inf 0.2

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

    7. Simplified0.2

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

    if -205884948.09812093 < y < 214825219.74745002

    1. Initial program 0.1

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

    3. Applied *-un-lft-identity0.1

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

    4. Applied times-frac0.1

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

    5. Simplified0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -205884948.0981209278106689453125:\\ \;\;\;\;\frac{1}{y} - \left(\frac{1}{\frac{y}{x}} - x\right)\\ \mathbf{elif}\;y \le 214825219.7474500238895416259765625:\\ \;\;\;\;1 - \left(1 - x\right) \cdot \frac{y}{1 + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} - \left(\frac{1}{\frac{y}{x}} - x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019170 
(FPCore (x y)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, D"

  :herbie-target
  (if (< y -3693.8482788297247) (- (/ 1.0 y) (- (/ x y) x)) (if (< y 6799310503.41891) (- 1.0 (/ (* (- 1.0 x) y) (+ y 1.0))) (- (/ 1.0 y) (- (/ x y) x))))

  (- 1.0 (/ (* (- 1.0 x) y) (+ y 1.0))))