Average Error: 22.5 → 7.3
Time: 4.7s
Precision: binary64
\[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \leq -53035177634780.61 \lor \neg \left(y \leq 2.9017319871335257 \cdot 10^{+23}\right):\\ \;\;\;\;x + \frac{x}{y} \cdot \left(\frac{1}{y} - 1\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{y \cdot \left(x - 1\right)}{y + 1}\\ \end{array}\]
1 - \frac{\left(1 - x\right) \cdot y}{y + 1}
↓
\begin{array}{l}
\mathbf{if}\;y \leq -53035177634780.61 \lor \neg \left(y \leq 2.9017319871335257 \cdot 10^{+23}\right):\\
\;\;\;\;x + \frac{x}{y} \cdot \left(\frac{1}{y} - 1\right)\\

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

\end{array}
double code(double x, double y) {
	return ((double) (1.0 - (((double) (((double) (1.0 - x)) * y)) / ((double) (y + 1.0)))));
}
↓
double code(double x, double y) {
	double VAR;
	if (((y <= -53035177634780.61) || !(y <= 2.9017319871335257e+23))) {
		VAR = ((double) (x + ((double) ((x / y) * ((double) ((1.0 / y) - 1.0))))));
	} else {
		VAR = ((double) (1.0 + (((double) (y * ((double) (x - 1.0)))) / ((double) (y + 1.0)))));
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original22.5
Target0.2
Herbie7.3
\[\begin{array}{l} \mathbf{if}\;y < -3693.8482788297247:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \mathbf{elif}\;y < 6799310503.41891:\\ \;\;\;\;1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -53035177634780.609 or 2.9017319871335257e23 < y

    1. Initial program 46.7

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

    2. Simplified29.9

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

    3. Taylor expanded around inf 14.6

      \[\leadsto \color{blue}{\left(x + 1 \cdot \frac{x}{{y}^{2}}\right) - 1 \cdot \frac{x}{y}}\]

    4. Simplified14.6

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

    if -53035177634780.609 < y < 2.9017319871335257e23

    1. Initial program 0.9

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

  4. Final simplification7.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -53035177634780.61 \lor \neg \left(y \leq 2.9017319871335257 \cdot 10^{+23}\right):\\ \;\;\;\;x + \frac{x}{y} \cdot \left(\frac{1}{y} - 1\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{y \cdot \left(x - 1\right)}{y + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020196 
(FPCore (x y)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, D"
  :precision binary64

  :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))))