Average Error: 21.8 → 0.0
Time: 9.7s
Precision: binary64
\[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \leq -11918.280626172522 \lor \neg \left(y \leq 10184.918629957767\right):\\ \;\;\;\;\left(\frac{x}{y \cdot y} + \left(x + \left(\frac{1}{y} + \frac{1}{{y}^{3}}\right)\right)\right) - \left(\frac{x}{{y}^{3}} + \left(\frac{x}{y} + \frac{1}{y \cdot y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{y \cdot \left(1 - x\right)}{y \cdot y - 1} \cdot \left(y - 1\right)\\ \end{array}\]
1 - \frac{\left(1 - x\right) \cdot y}{y + 1}
↓
\begin{array}{l}
\mathbf{if}\;y \leq -11918.280626172522 \lor \neg \left(y \leq 10184.918629957767\right):\\
\;\;\;\;\left(\frac{x}{y \cdot y} + \left(x + \left(\frac{1}{y} + \frac{1}{{y}^{3}}\right)\right)\right) - \left(\frac{x}{{y}^{3}} + \left(\frac{x}{y} + \frac{1}{y \cdot y}\right)\right)\\

\mathbf{else}:\\
\;\;\;\;1 - \frac{y \cdot \left(1 - x\right)}{y \cdot y - 1} \cdot \left(y - 1\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 -11918.280626172522) (not (<= y 10184.918629957767)))
   (-
    (+ (/ x (* y y)) (+ x (+ (/ 1.0 y) (/ 1.0 (pow y 3.0)))))
    (+ (/ x (pow y 3.0)) (+ (/ x y) (/ 1.0 (* y y)))))
   (- 1.0 (* (/ (* y (- 1.0 x)) (- (* y y) 1.0)) (- 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 <= -11918.280626172522) || !(y <= 10184.918629957767)) {
		tmp = ((x / (y * y)) + (x + ((1.0 / y) + (1.0 / pow(y, 3.0))))) - ((x / pow(y, 3.0)) + ((x / y) + (1.0 / (y * y))));
	} else {
		tmp = 1.0 - (((y * (1.0 - x)) / ((y * y) - 1.0)) * (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

Original21.8
Target0.3
Herbie0.0
\[\begin{array}{l} \mathbf{if}\;y < -3693.8482788297247:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \mathbf{elif}\;y < 6799310503.41891:\\ \;\;\;\;1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -11918.2806261725218 or 10184.918629957767 < y

    1. Initial program 45.2

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

    2. Taylor expanded around inf 0.0

      \[\leadsto \color{blue}{\left(\frac{x}{{y}^{2}} + \left(\frac{1}{{y}^{3}} + \left(\frac{1}{y} + x\right)\right)\right) - \left(\frac{x}{{y}^{3}} + \left(\frac{1}{{y}^{2}} + \frac{x}{y}\right)\right)}\]

    3. Simplified0.0

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

    if -11918.2806261725218 < y < 10184.918629957767

    1. Initial program 0.0

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

    3. Applied flip-+_binary64_184660.0

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

    4. Applied associate-/r/_binary64_184380.0

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -11918.280626172522 \lor \neg \left(y \leq 10184.918629957767\right):\\ \;\;\;\;\left(\frac{x}{y \cdot y} + \left(x + \left(\frac{1}{y} + \frac{1}{{y}^{3}}\right)\right)\right) - \left(\frac{x}{{y}^{3}} + \left(\frac{x}{y} + \frac{1}{y \cdot y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{y \cdot \left(1 - x\right)}{y \cdot y - 1} \cdot \left(y - 1\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2021015 
(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))))