Average Error: 20.9 → 5.3
Time: 1.8s
Precision: binary64
\[0.0 \lt x \lt 1 \land y \lt 1\]
\[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -1.3688387676508722 \cdot 10^{154}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le -1.5984312638924005 \cdot 10^{-162}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(y + x\right)}{x \cdot x + y \cdot y}\\ \mathbf{elif}\;y \le 6.78652207301613204 \cdot 10^{-175}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \le 9.35876682712504823 \cdot 10^{-165}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y}{\sqrt{x \cdot x + y \cdot y}} \cdot \frac{y + x}{\sqrt{x \cdot x + y \cdot y}}\\ \end{array}\]
\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}
↓
\begin{array}{l}
\mathbf{if}\;y \le -1.3688387676508722 \cdot 10^{154}:\\
\;\;\;\;-1\\

\mathbf{elif}\;y \le -1.5984312638924005 \cdot 10^{-162}:\\
\;\;\;\;\frac{\left(x - y\right) \cdot \left(y + x\right)}{x \cdot x + y \cdot y}\\

\mathbf{elif}\;y \le 6.78652207301613204 \cdot 10^{-175}:\\
\;\;\;\;1\\

\mathbf{elif}\;y \le 9.35876682712504823 \cdot 10^{-165}:\\
\;\;\;\;-1\\

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

\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 <= -1.3688387676508722e+154)) {
		VAR = -1.0;
	} else {
		double VAR_1;
		if ((y <= -1.5984312638924005e-162)) {
			VAR_1 = ((double) (((double) (((double) (x - y)) * ((double) (y + x)))) / ((double) (((double) (x * x)) + ((double) (y * y))))));
		} else {
			double VAR_2;
			if ((y <= 6.786522073016132e-175)) {
				VAR_2 = 1.0;
			} else {
				double VAR_3;
				if ((y <= 9.358766827125048e-165)) {
					VAR_3 = -1.0;
				} else {
					VAR_3 = ((double) (((double) (((double) (x - y)) / ((double) sqrt(((double) (((double) (x * x)) + ((double) (y * y)))))))) * ((double) (((double) (y + x)) / ((double) sqrt(((double) (((double) (x * x)) + ((double) (y * y))))))))));
				}
				VAR_2 = VAR_3;
			}
			VAR_1 = VAR_2;
		}
		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

Original20.9
Target0.1
Herbie5.3
\[\begin{array}{l} \mathbf{if}\;0.5 \lt \left|\frac{x}{y}\right| \lt 2:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{2}{1 + \frac{x}{y} \cdot \frac{x}{y}}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if y < -1.3688387676508722e154 or 6.78652207301613204e-175 < y < 9.35876682712504823e-165

    1. Initial program 62.3

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

    2. Taylor expanded around 0 1.8

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

    if -1.3688387676508722e154 < y < -1.5984312638924005e-162

    1. Initial program 0.1

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

    if -1.5984312638924005e-162 < y < 6.78652207301613204e-175

    1. Initial program 31.6

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

    2. Taylor expanded around inf 15.9

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

    if 9.35876682712504823e-165 < y

    1. Initial program 0.3

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

    3. Applied add-sqr-sqrt0.4

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

    4. Applied times-frac0.8

      \[\leadsto \color{blue}{\frac{x - y}{\sqrt{x \cdot x + y \cdot y}} \cdot \frac{x + y}{\sqrt{x \cdot x + y \cdot y}}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification5.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -1.3688387676508722 \cdot 10^{154}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le -1.5984312638924005 \cdot 10^{-162}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(y + x\right)}{x \cdot x + y \cdot y}\\ \mathbf{elif}\;y \le 6.78652207301613204 \cdot 10^{-175}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \le 9.35876682712504823 \cdot 10^{-165}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y}{\sqrt{x \cdot x + y \cdot y}} \cdot \frac{y + x}{\sqrt{x \cdot x + y \cdot y}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020185 
(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))))