Average Error: 20.2 → 6.2
Time: 4.1s
Precision: 64
\[0.0 \lt x \lt 1 \land y \lt 1\]
\[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -1.3453816967374705 \cdot 10^{153}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le -1.2162082839611001 \cdot 10^{-151}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \mathbf{elif}\;y \le 1.19916786656018 \cdot 10^{-225}:\\ \;\;\;\;\sqrt{\frac{\frac{x - y}{\frac{\mathsf{hypot}\left(x, y\right)}{x + y}}}{\mathsf{hypot}\left(x, y\right)}} \cdot \sqrt{\frac{\frac{x - y}{\frac{\mathsf{hypot}\left(x, y\right)}{x + y}}}{\mathsf{hypot}\left(x, y\right)}}\\ \mathbf{elif}\;y \le 3.7703083762849001 \cdot 10^{-166}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \end{array}\]
\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}
↓
\begin{array}{l}
\mathbf{if}\;y \le -1.3453816967374705 \cdot 10^{153}:\\
\;\;\;\;-1\\

\mathbf{elif}\;y \le -1.2162082839611001 \cdot 10^{-151}:\\
\;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\

\mathbf{elif}\;y \le 1.19916786656018 \cdot 10^{-225}:\\
\;\;\;\;\sqrt{\frac{\frac{x - y}{\frac{\mathsf{hypot}\left(x, y\right)}{x + y}}}{\mathsf{hypot}\left(x, y\right)}} \cdot \sqrt{\frac{\frac{x - y}{\frac{\mathsf{hypot}\left(x, y\right)}{x + y}}}{\mathsf{hypot}\left(x, y\right)}}\\

\mathbf{elif}\;y \le 3.7703083762849001 \cdot 10^{-166}:\\
\;\;\;\;-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 <= -1.3453816967374705e+153)) {
		VAR = -1.0;
	} else {
		double VAR_1;
		if ((y <= -1.2162082839611001e-151)) {
			VAR_1 = (((x - y) * (x + y)) / ((x * x) + (y * y)));
		} else {
			double VAR_2;
			if ((y <= 1.1991678665601801e-225)) {
				VAR_2 = (sqrt((((x - y) / (hypot(x, y) / (x + y))) / hypot(x, y))) * sqrt((((x - y) / (hypot(x, y) / (x + y))) / hypot(x, y))));
			} else {
				double VAR_3;
				if ((y <= 3.7703083762849e-166)) {
					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

Original20.2
Target0.1
Herbie6.2
\[\begin{array}{l} \mathbf{if}\;0.5 \lt \left|\frac{x}{y}\right| \lt 2:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{2}{1 + \frac{x}{y} \cdot \frac{x}{y}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if y < -1.3453816967374705e+153 or 1.1991678665601801e-225 < y < 3.7703083762849e-166

    1. Initial program 55.2

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

    2. Taylor expanded around 0 10.7

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

    if -1.3453816967374705e+153 < y < -1.2162082839611001e-151 or 3.7703083762849e-166 < y

    1. Initial program 0.2

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

    if -1.2162082839611001e-151 < y < 1.1991678665601801e-225

    1. Initial program 29.1

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

    3. Applied add-sqr-sqrt29.1

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

    4. Applied associate-/r*29.2

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

    5. Simplified30.1

      \[\leadsto \frac{\color{blue}{\frac{x - y}{\frac{\mathsf{hypot}\left(x, y\right)}{x + y}}}}{\sqrt{x \cdot x + y \cdot y}}\]
    6. Using strategy rm

    7. Applied hypot-def0.1

      \[\leadsto \frac{\frac{x - y}{\frac{\mathsf{hypot}\left(x, y\right)}{x + y}}}{\color{blue}{\mathsf{hypot}\left(x, y\right)}}\]
    8. Using strategy rm

    9. Applied add-sqr-sqrt13.9

      \[\leadsto \color{blue}{\sqrt{\frac{\frac{x - y}{\frac{\mathsf{hypot}\left(x, y\right)}{x + y}}}{\mathsf{hypot}\left(x, y\right)}} \cdot \sqrt{\frac{\frac{x - y}{\frac{\mathsf{hypot}\left(x, y\right)}{x + y}}}{\mathsf{hypot}\left(x, y\right)}}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification6.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -1.3453816967374705 \cdot 10^{153}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le -1.2162082839611001 \cdot 10^{-151}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \mathbf{elif}\;y \le 1.19916786656018 \cdot 10^{-225}:\\ \;\;\;\;\sqrt{\frac{\frac{x - y}{\frac{\mathsf{hypot}\left(x, y\right)}{x + y}}}{\mathsf{hypot}\left(x, y\right)}} \cdot \sqrt{\frac{\frac{x - y}{\frac{\mathsf{hypot}\left(x, y\right)}{x + y}}}{\mathsf{hypot}\left(x, y\right)}}\\ \mathbf{elif}\;y \le 3.7703083762849001 \cdot 10^{-166}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \end{array}\]

Reproduce

herbie shell --seed 2020075 +o rules:numerics
(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))))