Average Error: 19.6 → 5.1
Time: 1.9s
Precision: binary64
\[0.0 \lt x \lt 1 \land y \lt 1\]
\[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -7.95073760504717886 \cdot 10^{-28}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le -1.6002416138314979 \cdot 10^{-162} \lor \neg \left(y \le 9.42443533501087923 \cdot 10^{-165}\right):\\ \;\;\;\;\left(x - y\right) \cdot \frac{y + x}{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array}\]
\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}
↓
\begin{array}{l}
\mathbf{if}\;y \le -7.95073760504717886 \cdot 10^{-28}:\\
\;\;\;\;-1\\

\mathbf{elif}\;y \le -1.6002416138314979 \cdot 10^{-162} \lor \neg \left(y \le 9.42443533501087923 \cdot 10^{-165}\right):\\
\;\;\;\;\left(x - y\right) \cdot \frac{y + x}{x \cdot x + y \cdot y}\\

\mathbf{else}:\\
\;\;\;\;1\\

\end{array}
double code(double x, double y) {
	return ((double) (((double) (((double) (x - y)) * ((double) (x + y)))) / ((double) (((double) (x * x)) + ((double) (y * y))))));
}
↓
double code(double x, double y) {
	double VAR;
	if ((y <= -7.950737605047179e-28)) {
		VAR = -1.0;
	} else {
		double VAR_1;
		if (((y <= -1.600241613831498e-162) || !(y <= 9.42443533501088e-165))) {
			VAR_1 = ((double) (((double) (x - y)) * ((double) (((double) (y + x)) / ((double) (((double) (x * x)) + ((double) (y * y))))))));
		} else {
			VAR_1 = 1.0;
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original19.6
Target0.1
Herbie5.1
\[\begin{array}{l} \mathbf{if}\;0.5 \lt \left|\frac{x}{y}\right| \lt 2:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{2}{1 + \frac{x}{y} \cdot \frac{x}{y}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if y < -7.95073760504717886e-28

    1. Initial program 29.2

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

    2. Taylor expanded around 0 0.1

      \[\leadsto \color{blue}{-1}\]

    if -7.95073760504717886e-28 < y < -1.6002416138314979e-162 or 9.42443533501087923e-165 < y

    1. Initial program 0.4

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

    2. Simplified1.2

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

    if -1.6002416138314979e-162 < y < 9.42443533501087923e-165

    1. Initial program 28.7

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

    2. Taylor expanded around inf 14.7

      \[\leadsto \color{blue}{1}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification5.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -7.95073760504717886 \cdot 10^{-28}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le -1.6002416138314979 \cdot 10^{-162} \lor \neg \left(y \le 9.42443533501087923 \cdot 10^{-165}\right):\\ \;\;\;\;\left(x - y\right) \cdot \frac{y + x}{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array}\]

Reproduce

herbie shell --seed 2020184 
(FPCore (x y)
  :name "Kahan p9 Example"
  :precision binary64
  :pre (and (< 0.0 x 1.0) (< y 1.0))

  :herbie-target
  (if (< 0.5 (fabs (/ x y)) 2.0) (/ (* (- x y) (+ x y)) (+ (* x x) (* y y))) (- 1.0 (/ 2.0 (+ 1.0 (* (/ x y) (/ x y))))))

  (/ (* (- x y) (+ x y)) (+ (* x x) (* y y))))