Average Error: 22.0 → 0.2
Time: 16.7s
Precision: 64
\[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -162929032.461607038974761962890625 \lor \neg \left(y \le 194765081.4755440056324005126953125\right):\\ \;\;\;\;\left(x + \frac{1}{y}\right) - 1 \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;1 - \left(1 - x\right) \cdot \frac{y}{1 + y}\\ \end{array}\]
1 - \frac{\left(1 - x\right) \cdot y}{y + 1}
↓
\begin{array}{l}
\mathbf{if}\;y \le -162929032.461607038974761962890625 \lor \neg \left(y \le 194765081.4755440056324005126953125\right):\\
\;\;\;\;\left(x + \frac{1}{y}\right) - 1 \cdot \frac{x}{y}\\

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

\end{array}
double f(double x, double y) {
        double r459765 = 1.0;
        double r459766 = x;
        double r459767 = r459765 - r459766;
        double r459768 = y;
        double r459769 = r459767 * r459768;
        double r459770 = r459768 + r459765;
        double r459771 = r459769 / r459770;
        double r459772 = r459765 - r459771;
        return r459772;
}
↓
double f(double x, double y) {
        double r459773 = y;
        double r459774 = -162929032.46160704;
        bool r459775 = r459773 <= r459774;
        double r459776 = 194765081.475544;
        bool r459777 = r459773 <= r459776;
        double r459778 = !r459777;
        bool r459779 = r459775 || r459778;
        double r459780 = x;
        double r459781 = 1.0;
        double r459782 = r459781 / r459773;
        double r459783 = r459780 + r459782;
        double r459784 = r459780 / r459773;
        double r459785 = r459781 * r459784;
        double r459786 = r459783 - r459785;
        double r459787 = r459781 - r459780;
        double r459788 = r459781 + r459773;
        double r459789 = r459773 / r459788;
        double r459790 = r459787 * r459789;
        double r459791 = r459781 - r459790;
        double r459792 = r459779 ? r459786 : r459791;
        return r459792;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original22.0
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 < -162929032.46160704 or 194765081.475544 < y

    1. Initial program 44.6

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

    2. Simplified28.3

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

    3. Taylor expanded around inf 0.2

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

    4. Simplified0.2

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

    if -162929032.46160704 < y < 194765081.475544

    1. Initial program 0.2

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

    2. Simplified0.2

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -162929032.461607038974761962890625 \lor \neg \left(y \le 194765081.4755440056324005126953125\right):\\ \;\;\;\;\left(x + \frac{1}{y}\right) - 1 \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;1 - \left(1 - x\right) \cdot \frac{y}{1 + y}\\ \end{array}\]

Reproduce

herbie shell --seed 2019179 
(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))))