Kahan p9 Example

Average Error: 20.7 → 6.4

Time: 2.0s

Precision: binary64

\[0 < x \land x < 1 \land y < 1\]

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \leq -1.3285261522622341 \cdot 10^{+154}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq -2.5696995781342912 \cdot 10^{-169}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(y + x\right)}{x \cdot x + y \cdot y}\\ \mathbf{elif}\;y \leq -1.2939862339758965 \cdot 10^{-233}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 1.406937817525643 \cdot 10^{-164}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(y + x\right)}{x \cdot x + y \cdot y}\\ \end{array}\]

C

double code(double x, double y) {
	return (((double) (((double) (x - y)) * ((double) (x + y)))) / ((double) (((double) (x * x)) + ((double) (y * y)))));
}

↓

double code(double x, double y) {
	double VAR;
	if ((y <= -1.3285261522622341e+154)) {
		VAR = -1.0;
	} else {
		double VAR_1;
		if ((y <= -2.5696995781342912e-169)) {
			VAR_1 = (((double) (((double) (x - y)) * ((double) (y + x)))) / ((double) (((double) (x * x)) + ((double) (y * y)))));
		} else {
			double VAR_2;
			if ((y <= -1.2939862339758965e-233)) {
				VAR_2 = -1.0;
			} else {
				double VAR_3;
				if ((y <= 1.406937817525643e-164)) {
					VAR_3 = 1.0;
				} else {
					VAR_3 = (((double) (((double) (x - y)) * ((double) (y + x)))) / ((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	20.7
Target	0.0
Herbie	6.4

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

Derivation

1. Split input into 3 regimes

2. if y < -1.32852615226223415e154 or -2.5696995781342912e-169 < y < -1.2939862339758965e-233

   1. Initial program Error: 54.6 bits

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

   2. Taylor expanded around 0 Error: 11.7 bits

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

   if -1.32852615226223415e154 < y < -2.5696995781342912e-169 or 1.4069378175256429e-164 < y

   1. Initial program Error: 0.7 bits

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

   if -1.2939862339758965e-233 < y < 1.4069378175256429e-164

   1. Initial program Error: 30.9 bits

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

   2. Taylor expanded around inf Error: 13.7 bits

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

4. Final simplification Error: 6.4 bits

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.3285261522622341 \cdot 10^{+154}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq -2.5696995781342912 \cdot 10^{-169}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(y + x\right)}{x \cdot x + y \cdot y}\\ \mathbf{elif}\;y \leq -1.2939862339758965 \cdot 10^{-233}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 1.406937817525643 \cdot 10^{-164}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(y + x\right)}{x \cdot x + y \cdot y}\\ \end{array}\]

Reproduce

herbie shell --seed 2020200 
(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))))