Average Error: 22.0 → 0.2
Time: 6.5s
Precision: binary64
\[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \leq -1169275214.8150542 \lor \neg \left(y \leq 89866444.08949669\right):\\ \;\;\;\;\left(x + \frac{1}{y}\right) - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{y \cdot \left(1 - x\right)}{1 + {y}^{3}} \cdot \left(y \cdot y + \left(1 - y\right)\right)\\ \end{array}\]
1 - \frac{\left(1 - x\right) \cdot y}{y + 1}
↓
\begin{array}{l}
\mathbf{if}\;y \leq -1169275214.8150542 \lor \neg \left(y \leq 89866444.08949669\right):\\
\;\;\;\;\left(x + \frac{1}{y}\right) - \frac{x}{y}\\

\mathbf{else}:\\
\;\;\;\;1 - \frac{y \cdot \left(1 - x\right)}{1 + {y}^{3}} \cdot \left(y \cdot y + \left(1 - y\right)\right)\\

\end{array}
(FPCore (x y) :precision binary64 (- 1.0 (/ (* (- 1.0 x) y) (+ y 1.0))))
↓
(FPCore (x y)
 :precision binary64
 (if (or (<= y -1169275214.8150542) (not (<= y 89866444.08949669)))
   (- (+ x (/ 1.0 y)) (/ x y))
   (- 1.0 (* (/ (* y (- 1.0 x)) (+ 1.0 (pow y 3.0))) (+ (* y 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 <= -1169275214.8150542) || !(y <= 89866444.08949669)) {
		tmp = (x + (1.0 / y)) - (x / y);
	} else {
		tmp = 1.0 - (((y * (1.0 - x)) / (1.0 + pow(y, 3.0))) * ((y * 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

Original22.0
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 < -1169275214.81505418 or 89866444.0894966871 < y

    1. Initial program 45.9

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

    2. Taylor expanded around inf 0.1

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

    if -1169275214.81505418 < y < 89866444.0894966871

    1. Initial program 0.2

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

    3. Applied flip3-+_binary64_164490.2

      \[\leadsto 1 - \frac{\left(1 - x\right) \cdot y}{\color{blue}{\frac{{y}^{3} + {1}^{3}}{y \cdot y + \left(1 \cdot 1 - y \cdot 1\right)}}}\]

    4. Applied associate-/r/_binary64_163920.2

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1169275214.8150542 \lor \neg \left(y \leq 89866444.08949669\right):\\ \;\;\;\;\left(x + \frac{1}{y}\right) - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{y \cdot \left(1 - x\right)}{1 + {y}^{3}} \cdot \left(y \cdot y + \left(1 - y\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2021044 
(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))))