Average Error: 22.4 → 0.1
Time: 8.4s
Precision: binary64
\[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \leq -21493326675.95675 \lor \neg \left(y \leq 357233.1889861889\right):\\ \;\;\;\;\left(x + \left(\frac{1}{y} + \frac{x}{y \cdot y}\right)\right) - \left(\frac{x}{y} + {y}^{-2}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{y \cdot x}{y + 1}\right) - \frac{y}{y + 1}\\ \end{array}\]
1 - \frac{\left(1 - x\right) \cdot y}{y + 1}
↓
\begin{array}{l}
\mathbf{if}\;y \leq -21493326675.95675 \lor \neg \left(y \leq 357233.1889861889\right):\\
\;\;\;\;\left(x + \left(\frac{1}{y} + \frac{x}{y \cdot y}\right)\right) - \left(\frac{x}{y} + {y}^{-2}\right)\\

\mathbf{else}:\\
\;\;\;\;\left(1 + \frac{y \cdot x}{y + 1}\right) - \frac{y}{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 -21493326675.95675) (not (<= y 357233.1889861889)))
   (- (+ x (+ (/ 1.0 y) (/ x (* y y)))) (+ (/ x y) (pow y -2.0)))
   (- (+ 1.0 (/ (* y x) (+ y 1.0))) (/ y (+ y 1.0)))))
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 <= -21493326675.95675) || !(y <= 357233.1889861889)) {
		tmp = (x + ((1.0 / y) + (x / (y * y)))) - ((x / y) + pow(y, -2.0));
	} else {
		tmp = (1.0 + ((y * x) / (y + 1.0))) - (y / (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

Original22.4
Target0.2
Herbie0.1
\[\begin{array}{l} \mathbf{if}\;y < -3693.8482788297247:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \mathbf{elif}\;y < 6799310503.41891:\\ \;\;\;\;1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -21493326675.956749 or 357233.188986188907 < y

    1. Initial program 46.0

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

    2. Taylor expanded around inf 0.0

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

    3. Simplified0.0

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

    5. Applied pow2_binary640.0

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

    6. Applied pow-flip_binary640.0

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

    if -21493326675.956749 < y < 357233.188986188907

    1. Initial program 0.1

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

    2. Taylor expanded around 0 0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -21493326675.95675 \lor \neg \left(y \leq 357233.1889861889\right):\\ \;\;\;\;\left(x + \left(\frac{1}{y} + \frac{x}{y \cdot y}\right)\right) - \left(\frac{x}{y} + {y}^{-2}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{y \cdot x}{y + 1}\right) - \frac{y}{y + 1}\\ \end{array}\]

Alternatives

Reproduce

herbie shell --seed 2021113 
(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))))