Kahan p9 Example

Average Error: 19.9 → 5.7

Time: 2.8s

Precision: binary64

\[0.0 \lt x \lt 1 \land y \lt 1\]

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \le -1.79743655938017033 \cdot 10^{152}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le -2.4992551874327323 \cdot 10^{-162}:\\ \;\;\;\;\left(\sqrt[3]{\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}} \cdot \sqrt[3]{\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}}\right) \cdot \sqrt[3]{\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}}\\ \mathbf{elif}\;y \le 2.41279850469291811 \cdot 10^{-218}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \le 9.34726976867030345 \cdot 10^{-181}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}} \cdot \sqrt[3]{\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}}\right) \cdot \sqrt[3]{\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}}\\ \end{array}\]

C

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 <= -1.7974365593801703e+152)) {
		VAR = -1.0;
	} else {
		double VAR_1;
		if ((y <= -2.4992551874327323e-162)) {
			VAR_1 = ((double) (((double) (((double) cbrt(((double) (((double) (((double) (x - y)) * ((double) (x + y)))) / ((double) (((double) (x * x)) + ((double) (y * y)))))))) * ((double) cbrt(((double) (((double) (((double) (x - y)) * ((double) (x + y)))) / ((double) (((double) (x * x)) + ((double) (y * y)))))))))) * ((double) cbrt(((double) (((double) (((double) (x - y)) * ((double) (x + y)))) / ((double) (((double) (x * x)) + ((double) (y * y))))))))));
		} else {
			double VAR_2;
			if ((y <= 2.412798504692918e-218)) {
				VAR_2 = 1.0;
			} else {
				double VAR_3;
				if ((y <= 9.347269768670303e-181)) {
					VAR_3 = -1.0;
				} else {
					VAR_3 = ((double) (((double) (((double) cbrt(((double) (((double) (((double) (x - y)) * ((double) (x + y)))) / ((double) (((double) (x * x)) + ((double) (y * y)))))))) * ((double) cbrt(((double) (((double) (((double) (x - y)) * ((double) (x + y)))) / ((double) (((double) (x * x)) + ((double) (y * y)))))))))) * ((double) cbrt(((double) (((double) (((double) (x - y)) * ((double) (x + y)))) / ((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

Target

Original	19.9
Target	0.0
Herbie	5.7

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

Derivation

1. Split input into 3 regimes

2. if y < -1.79743655938017033e152 or 2.41279850469291811e-218 < y < 9.34726976867030345e-181

   1. Initial program 57.3

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

   2. Taylor expanded around 0 8.0

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

   if -1.79743655938017033e152 < y < -2.4992551874327323e-162 or 9.34726976867030345e-181 < y

   1. Initial program 1.3

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

   3. Applied add-cube-cbrt 1.4

      \[\leadsto \color{blue}{\left(\sqrt[3]{\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}} \cdot \sqrt[3]{\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}}\right) \cdot \sqrt[3]{\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}}}\]

   if -2.4992551874327323e-162 < y < 2.41279850469291811e-218

   1. Initial program 29.5

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

   2. Taylor expanded around inf 13.0

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

4. Final simplification 5.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -1.79743655938017033 \cdot 10^{152}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le -2.4992551874327323 \cdot 10^{-162}:\\ \;\;\;\;\left(\sqrt[3]{\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}} \cdot \sqrt[3]{\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}}\right) \cdot \sqrt[3]{\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}}\\ \mathbf{elif}\;y \le 2.41279850469291811 \cdot 10^{-218}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \le 9.34726976867030345 \cdot 10^{-181}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}} \cdot \sqrt[3]{\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}}\right) \cdot \sqrt[3]{\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020157 
(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))))