Average Error: 20.4 → 5.8
Time: 2.0s
Precision: binary64
\[0.0 \lt x \lt 1 \land y \lt 1\]
\[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -1.3305849395080257 \cdot 10^{154}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le -2.370600095132076 \cdot 10^{-168}:\\ \;\;\;\;\frac{x - y}{\sqrt{x \cdot x + y \cdot y}} \cdot \frac{x + y}{\sqrt{x \cdot x + y \cdot y}}\\ \mathbf{elif}\;y \le -8.0264458424328158 \cdot 10^{-205}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le -1.78962356940894908 \cdot 10^{-250}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \le -2.7974025216737748 \cdot 10^{-258}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le 3.6362331184202328 \cdot 10^{-162}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \end{array}\]
\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}
↓
\begin{array}{l}
\mathbf{if}\;y \le -1.3305849395080257 \cdot 10^{154}:\\
\;\;\;\;-1\\

\mathbf{elif}\;y \le -2.370600095132076 \cdot 10^{-168}:\\
\;\;\;\;\frac{x - y}{\sqrt{x \cdot x + y \cdot y}} \cdot \frac{x + y}{\sqrt{x \cdot x + y \cdot y}}\\

\mathbf{elif}\;y \le -8.0264458424328158 \cdot 10^{-205}:\\
\;\;\;\;-1\\

\mathbf{elif}\;y \le -1.78962356940894908 \cdot 10^{-250}:\\
\;\;\;\;1\\

\mathbf{elif}\;y \le -2.7974025216737748 \cdot 10^{-258}:\\
\;\;\;\;-1\\

\mathbf{elif}\;y \le 3.6362331184202328 \cdot 10^{-162}:\\
\;\;\;\;1\\

\mathbf{else}:\\
\;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{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 <= -1.3305849395080257e+154)) {
		VAR = -1.0;
	} else {
		double VAR_1;
		if ((y <= -2.370600095132076e-168)) {
			VAR_1 = ((double) (((double) (((double) (x - y)) / ((double) sqrt(((double) (((double) (x * x)) + ((double) (y * y)))))))) * ((double) (((double) (x + y)) / ((double) sqrt(((double) (((double) (x * x)) + ((double) (y * y))))))))));
		} else {
			double VAR_2;
			if ((y <= -8.026445842432816e-205)) {
				VAR_2 = -1.0;
			} else {
				double VAR_3;
				if ((y <= -1.789623569408949e-250)) {
					VAR_3 = 1.0;
				} else {
					double VAR_4;
					if ((y <= -2.797402521673775e-258)) {
						VAR_4 = -1.0;
					} else {
						double VAR_5;
						if ((y <= 3.636233118420233e-162)) {
							VAR_5 = 1.0;
						} else {
							VAR_5 = ((double) (((double) (((double) (x - y)) * ((double) (x + y)))) / ((double) (((double) (x * x)) + ((double) (y * y))))));
						}
						VAR_4 = VAR_5;
					}
					VAR_3 = VAR_4;
				}
				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

Original20.4
Target0.1
Herbie5.8
\[\begin{array}{l} \mathbf{if}\;0.5 \lt \left|\frac{x}{y}\right| \lt 2:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{2}{1 + \frac{x}{y} \cdot \frac{x}{y}}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if y < -1.3305849395080257e154 or -2.370600095132076e-168 < y < -8.0264458424328158e-205 or -1.78962356940894908e-250 < y < -2.7974025216737748e-258

    1. Initial program 56.4

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

    2. Taylor expanded around 0 9.4

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

    if -1.3305849395080257e154 < y < -2.370600095132076e-168

    1. Initial program 0.6

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

    3. Applied add-sqr-sqrt0.6

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

    4. Applied times-frac0.8

      \[\leadsto \color{blue}{\frac{x - y}{\sqrt{x \cdot x + y \cdot y}} \cdot \frac{x + y}{\sqrt{x \cdot x + y \cdot y}}}\]

    if -8.0264458424328158e-205 < y < -1.78962356940894908e-250 or -2.7974025216737748e-258 < y < 3.6362331184202328e-162

    1. Initial program 31.0

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

    2. Taylor expanded around inf 13.4

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

    if 3.6362331184202328e-162 < y

    1. Initial program 0.1

      \[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification5.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -1.3305849395080257 \cdot 10^{154}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le -2.370600095132076 \cdot 10^{-168}:\\ \;\;\;\;\frac{x - y}{\sqrt{x \cdot x + y \cdot y}} \cdot \frac{x + y}{\sqrt{x \cdot x + y \cdot y}}\\ \mathbf{elif}\;y \le -8.0264458424328158 \cdot 10^{-205}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le -1.78962356940894908 \cdot 10^{-250}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \le -2.7974025216737748 \cdot 10^{-258}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le 3.6362331184202328 \cdot 10^{-162}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \end{array}\]

Reproduce

herbie shell --seed 2020155 
(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))))