Average Error: 22.2 → 0.2
Time: 2.9s
Precision: binary64
\[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \leq -1080471527719.7518 \lor \neg \left(y \leq 29063706.880189914\right):\\ \;\;\;\;x + \left(\frac{1}{y} - 1 \cdot \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{y \cdot \left(x - 1\right)}{y + 1}\\ \end{array}\]
1 - \frac{\left(1 - x\right) \cdot y}{y + 1}
↓
\begin{array}{l}
\mathbf{if}\;y \leq -1080471527719.7518 \lor \neg \left(y \leq 29063706.880189914\right):\\
\;\;\;\;x + \left(\frac{1}{y} - 1 \cdot \frac{x}{y}\right)\\

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

\end{array}
(FPCore (x y) :precision binary64 (- 1.0 (/ (* (- 1.0 x) y) (+ y 1.0))))
↓
(FPCore (x y)
 :precision binary64
 (if (or (<= y -1080471527719.7518) (not (<= y 29063706.880189914)))
   (+ x (- (/ 1.0 y) (* 1.0 (/ x y))))
   (+ 1.0 (/ (* y (- x 1.0)) (+ y 1.0)))))
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 tmp;
	if (((y <= -1080471527719.7518) || !(y <= 29063706.880189914))) {
		tmp = ((double) (x + ((double) ((1.0 / y) - ((double) (1.0 * (x / y)))))));
	} else {
		tmp = ((double) (1.0 + (((double) (y * ((double) (x - 1.0)))) / ((double) (y + 1.0)))));
	}
	return tmp;
}

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
Target0.2
Herbie0.2
\[\begin{array}{l} \mathbf{if}\;y < -3693.8482788297247:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \mathbf{elif}\;y < 6799310503.41891:\\ \;\;\;\;1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -1080471527719.75183 or 29063706.8801899143 < y

    1. Initial program Error: 45.8 bits

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

    2. Taylor expanded around inf Error: 0.1 bits

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

    3. SimplifiedError: 0.1 bits

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

    if -1080471527719.75183 < y < 29063706.8801899143

    1. Initial program Error: 0.3 bits

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

  4. Final simplificationError: 0.2 bits

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1080471527719.7518 \lor \neg \left(y \leq 29063706.880189914\right):\\ \;\;\;\;x + \left(\frac{1}{y} - 1 \cdot \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{y \cdot \left(x - 1\right)}{y + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020204 
(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))))