Kahan p9 Example

Average Error: 20.4 → 5.7

Time: 1.9s

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \le -2.216920479818269 \cdot 10^{151}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le -2.370600095132076 \cdot 10^{-168}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \mathbf{elif}\;y \le -8.0264458424328158 \cdot 10^{-205}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le -1.78962356940894908 \cdot 10^{-250}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \le -2.7974025216737748 \cdot 10^{-258}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le 3.6362331184202328 \cdot 10^{-162}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \end{array}\]

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 <= -2.2169204798182685e+151)) {
		VAR = -1.0;
	} else {
		double VAR_1;
		if ((y <= -2.370600095132076e-168)) {
			VAR_1 = ((double) (((double) (((double) (x - y)) * ((double) (x + y)))) / ((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

Target

Original	20.4
Target	0.1
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 < -2.216920479818269e151 or -2.370600095132076e-168 < y < -8.0264458424328158e-205 or -1.78962356940894908e-250 < y < -2.7974025216737748e-258

   1. Initial program 55.5

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

   2. Taylor expanded around 0 9.2

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

   if -2.216920479818269e151 < y < -2.370600095132076e-168 or 3.6362331184202328e-162 < y

   1. Initial program 0.4

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

   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

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

4. Final simplification 5.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -2.216920479818269 \cdot 10^{151}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le -2.370600095132076 \cdot 10^{-168}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \mathbf{elif}\;y \le -8.0264458424328158 \cdot 10^{-205}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le -1.78962356940894908 \cdot 10^{-250}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \le -2.7974025216737748 \cdot 10^{-258}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le 3.6362331184202328 \cdot 10^{-162}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \end{array}\]

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))))