Average Error: 19.5 → 5.1
Time: 2.4s
Precision: 64
\[0.0 \lt x \lt 1 \land y \lt 1\]
\[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -2.59481508547337103 \cdot 10^{153}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le -8.7850604012387935 \cdot 10^{-158}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \mathbf{elif}\;y \le -6.91653153357935118 \cdot 10^{-192}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le 3.08060170809400764 \cdot 10^{-173}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \end{array}\]
\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}
↓
\begin{array}{l}
\mathbf{if}\;y \le -2.59481508547337103 \cdot 10^{153}:\\
\;\;\;\;-1\\

\mathbf{elif}\;y \le -8.7850604012387935 \cdot 10^{-158}:\\
\;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\

\mathbf{elif}\;y \le -6.91653153357935118 \cdot 10^{-192}:\\
\;\;\;\;-1\\

\mathbf{elif}\;y \le 3.08060170809400764 \cdot 10^{-173}:\\
\;\;\;\;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 (((x - y) * (x + y)) / ((x * x) + (y * y)));
}
↓
double code(double x, double y) {
	double VAR;
	if ((y <= -2.594815085473371e+153)) {
		VAR = -1.0;
	} else {
		double VAR_1;
		if ((y <= -8.785060401238793e-158)) {
			VAR_1 = (((x - y) * (x + y)) / ((x * x) + (y * y)));
		} else {
			double VAR_2;
			if ((y <= -6.916531533579351e-192)) {
				VAR_2 = -1.0;
			} else {
				double VAR_3;
				if ((y <= 3.0806017080940076e-173)) {
					VAR_3 = 1.0;
				} else {
					VAR_3 = (((x - y) * (x + y)) / ((x * x) + (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

Original19.5
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 < -2.594815085473371e+153 or -8.785060401238793e-158 < y < -6.916531533579351e-192

    1. Initial program 57.1

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

    2. Taylor expanded around 0 6.4

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

    if -2.594815085473371e+153 < y < -8.785060401238793e-158 or 3.0806017080940076e-173 < y

    1. Initial program 0.8

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

    if -6.916531533579351e-192 < y < 3.0806017080940076e-173

    1. Initial program 29.1

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

    2. Taylor expanded around inf 12.8

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

  4. Final simplification5.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -2.59481508547337103 \cdot 10^{153}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le -8.7850604012387935 \cdot 10^{-158}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \mathbf{elif}\;y \le -6.91653153357935118 \cdot 10^{-192}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le 3.08060170809400764 \cdot 10^{-173}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \end{array}\]

Reproduce

herbie shell --seed 2020106 
(FPCore (x y)
  :name "Kahan p9 Example"
  :precision binary64
  :pre (and (< 0.0 x 1) (< y 1))

  :herbie-target
  (if (< 0.5 (fabs (/ x y)) 2) (/ (* (- x y) (+ x y)) (+ (* x x) (* y y))) (- 1 (/ 2 (+ 1 (* (/ x y) (/ x y))))))

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