Average Error: 20.7 → 5.4
Time: 3.5s
Precision: 64
\[0.0 \lt x \lt 1 \land y \lt 1\]
\[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -3.0487862136205886 \cdot 10^{-28}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le -6.0144793476533727 \cdot 10^{-152}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{x + y}{x \cdot x + y \cdot y}\\ \mathbf{elif}\;y \le 4.92974356624905665 \cdot 10^{-160}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{x + y}{x \cdot x + y \cdot y}\\ \end{array}\]
\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}
↓
\begin{array}{l}
\mathbf{if}\;y \le -3.0487862136205886 \cdot 10^{-28}:\\
\;\;\;\;-1\\

\mathbf{elif}\;y \le -6.0144793476533727 \cdot 10^{-152}:\\
\;\;\;\;\left(x - y\right) \cdot \frac{x + y}{x \cdot x + y \cdot y}\\

\mathbf{elif}\;y \le 4.92974356624905665 \cdot 10^{-160}:\\
\;\;\;\;1\\

\mathbf{else}:\\
\;\;\;\;\left(x - y\right) \cdot \frac{x + y}{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 <= -3.0487862136205886e-28)) {
		VAR = -1.0;
	} else {
		double VAR_1;
		if ((y <= -6.014479347653373e-152)) {
			VAR_1 = ((x - y) * ((x + y) / ((x * x) + (y * y))));
		} else {
			double VAR_2;
			if ((y <= 4.9297435662490567e-160)) {
				VAR_2 = 1.0;
			} else {
				VAR_2 = ((x - y) * ((x + y) / ((x * x) + (y * y))));
			}
			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

Original20.7
Target0.1
Herbie5.4
\[\begin{array}{l} \mathbf{if}\;0.5 \lt \left|\frac{x}{y}\right| \lt 2:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{2}{1 + \frac{x}{y} \cdot \frac{x}{y}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if y < -3.0487862136205886e-28

    1. Initial program 30.1

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

    2. Taylor expanded around 0 0.3

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

    if -3.0487862136205886e-28 < y < -6.014479347653373e-152 or 4.9297435662490567e-160 < y

    1. Initial program 0.1

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

    3. Applied *-un-lft-identity0.1

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

    4. Applied times-frac0.5

      \[\leadsto \color{blue}{\frac{x - y}{1} \cdot \frac{x + y}{x \cdot x + y \cdot y}}\]

    5. Simplified0.5

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

    if -6.014479347653373e-152 < y < 4.9297435662490567e-160

    1. Initial program 28.9

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

    2. Taylor expanded around inf 15.6

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

  4. Final simplification5.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -3.0487862136205886 \cdot 10^{-28}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le -6.0144793476533727 \cdot 10^{-152}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{x + y}{x \cdot x + y \cdot y}\\ \mathbf{elif}\;y \le 4.92974356624905665 \cdot 10^{-160}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{x + y}{x \cdot x + y \cdot y}\\ \end{array}\]

Reproduce

herbie shell --seed 2020103 
(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))))