Kahan p9 Example

Average Error: 20.5 → 6.9

Time: 1.6s

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \le -3.7792399360998879 \cdot 10^{140}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le -1.65334731372902754 \cdot 10^{-160}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \mathbf{elif}\;y \le -4.7482528442321971 \cdot 10^{-194}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le -4.81788606400663115 \cdot 10^{-217}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \le -4.1677431246892909 \cdot 10^{-249}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le 2.9821042112063788 \cdot 10^{-204}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \le 7.8558315202935171 \cdot 10^{-170}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \end{array}\]

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 <= -3.779239936099888e+140)) {
		VAR = -1.0;
	} else {
		double VAR_1;
		if ((y <= -1.6533473137290275e-160)) {
			VAR_1 = ((double) (((double) (((double) (x - y)) * ((double) (x + y)))) / ((double) (((double) (x * x)) + ((double) (y * y))))));
		} else {
			double VAR_2;
			if ((y <= -4.748252844232197e-194)) {
				VAR_2 = -1.0;
			} else {
				double VAR_3;
				if ((y <= -4.817886064006631e-217)) {
					VAR_3 = 1.0;
				} else {
					double VAR_4;
					if ((y <= -4.167743124689291e-249)) {
						VAR_4 = -1.0;
					} else {
						double VAR_5;
						if ((y <= 2.9821042112063788e-204)) {
							VAR_5 = 1.0;
						} else {
							double VAR_6;
							if ((y <= 7.855831520293517e-170)) {
								VAR_6 = -1.0;
							} else {
								VAR_6 = ((double) (((double) (((double) (x - y)) * ((double) (x + y)))) / ((double) (((double) (x * x)) + ((double) (y * y))))));
							}
							VAR_5 = VAR_6;
						}
						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	20.5
Target	0.1
Herbie	6.9

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

Derivation

1. Split input into 3 regimes

2. if y < -3.7792399360998879e140 or -1.65334731372902754e-160 < y < -4.7482528442321971e-194 or -4.81788606400663115e-217 < y < -4.1677431246892909e-249 or 2.9821042112063788e-204 < y < 7.8558315202935171e-170

   1. Initial program 48.4

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

   2. Taylor expanded around 0 15.0

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

   if -3.7792399360998879e140 < y < -1.65334731372902754e-160 or 7.8558315202935171e-170 < y

   1. Initial program 0.4

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

   if -4.7482528442321971e-194 < y < -4.81788606400663115e-217 or -4.1677431246892909e-249 < y < 2.9821042112063788e-204

   1. Initial program 31.2

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

   2. Taylor expanded around inf 11.2

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

4. Final simplification 6.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -3.7792399360998879 \cdot 10^{140}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le -1.65334731372902754 \cdot 10^{-160}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \mathbf{elif}\;y \le -4.7482528442321971 \cdot 10^{-194}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le -4.81788606400663115 \cdot 10^{-217}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \le -4.1677431246892909 \cdot 10^{-249}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le 2.9821042112063788 \cdot 10^{-204}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \le 7.8558315202935171 \cdot 10^{-170}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \end{array}\]

Reproduce

herbie shell --seed 2020156 
(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))))