Average Error: 22.2 → 0.2
Time: 2.9s
Precision: binary64
\[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \leq -198093045.25882047 \lor \neg \left(y \leq 153067620.3959889\right):\\ \;\;\;\;\left(x + \frac{1}{y}\right) - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{1 - x}{\frac{y + 1}{y}}\\ \end{array}\]
1 - \frac{\left(1 - x\right) \cdot y}{y + 1}
↓
\begin{array}{l}
\mathbf{if}\;y \leq -198093045.25882047 \lor \neg \left(y \leq 153067620.3959889\right):\\
\;\;\;\;\left(x + \frac{1}{y}\right) - \frac{x}{y}\\

\mathbf{else}:\\
\;\;\;\;1 - \frac{1 - x}{\frac{y + 1}{y}}\\

\end{array}
(FPCore (x y) :precision binary64 (- 1.0 (/ (* (- 1.0 x) y) (+ y 1.0))))
↓
(FPCore (x y)
 :precision binary64
 (if (or (<= y -198093045.25882047) (not (<= y 153067620.3959889)))
   (- (+ x (/ 1.0 y)) (/ x y))
   (- 1.0 (/ (- 1.0 x) (/ (+ y 1.0) y)))))
double code(double x, double y) {
	return 1.0 - (((1.0 - x) * y) / (y + 1.0));
}
↓
double code(double x, double y) {
	double tmp;
	if ((y <= -198093045.25882047) || !(y <= 153067620.3959889)) {
		tmp = (x + (1.0 / y)) - (x / y);
	} else {
		tmp = 1.0 - ((1.0 - x) / ((y + 1.0) / y));
	}
	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 < -198093045.25882047 or 153067620.395988911 < y

    1. Initial program 46.5

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

    2. Taylor expanded around inf 0.2

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

    if -198093045.25882047 < y < 153067620.395988911

    1. Initial program 0.2

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

    3. Applied associate-/l*_binary640.2

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

    4. Simplified0.2

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -198093045.25882047 \lor \neg \left(y \leq 153067620.3959889\right):\\ \;\;\;\;\left(x + \frac{1}{y}\right) - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{1 - x}{\frac{y + 1}{y}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020219 
(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))))