Kahan p9 Example

Average Error: 20.5 → 5.5

Time: 1.3s

Precision: 64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \le -1.0010895938656462 \cdot 10^{153}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le -2.27003686660070324 \cdot 10^{-163}:\\ \;\;\;\;\frac{1}{\frac{x \cdot x + y \cdot y}{\left(x - y\right) \cdot \left(x + y\right)}}\\ \mathbf{elif}\;y \le -7.956381646089189 \cdot 10^{-195}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le 1.8163835546796256 \cdot 10^{-162}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x \cdot x + y \cdot y}{\left(x - y\right) \cdot \left(x + y\right)}}\\ \end{array}\]

C

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 <= -1.0010895938656462e+153)) {
		VAR = -1.0;
	} else {
		double VAR_1;
		if ((y <= -2.2700368666007032e-163)) {
			VAR_1 = (1.0 / (((x * x) + (y * y)) / ((x - y) * (x + y))));
		} else {
			double VAR_2;
			if ((y <= -7.956381646089189e-195)) {
				VAR_2 = -1.0;
			} else {
				double VAR_3;
				if ((y <= 1.8163835546796256e-162)) {
					VAR_3 = 1.0;
				} else {
					VAR_3 = (1.0 / (((x * x) + (y * y)) / ((x - y) * (x + 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	20.5
Target	0.1
Herbie	5.5

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

Derivation

1. Split input into 3 regimes

2. if y < -1.0010895938656462e+153 or -2.2700368666007032e-163 < y < -7.956381646089189e-195

   1. Initial program 58.5

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

   2. Taylor expanded around 0 5.9

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

   if -1.0010895938656462e+153 < y < -2.2700368666007032e-163 or 1.8163835546796256e-162 < y

   1. Initial program 0.1

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

   3. Applied clear-num 0.1

      \[\leadsto \color{blue}{\frac{1}{\frac{x \cdot x + y \cdot y}{\left(x - y\right) \cdot \left(x + y\right)}}}\]

   if -7.956381646089189e-195 < y < 1.8163835546796256e-162

   1. Initial program 31.3

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

   2. Taylor expanded around inf 14.8

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

4. Final simplification 5.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -1.0010895938656462 \cdot 10^{153}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le -2.27003686660070324 \cdot 10^{-163}:\\ \;\;\;\;\frac{1}{\frac{x \cdot x + y \cdot y}{\left(x - y\right) \cdot \left(x + y\right)}}\\ \mathbf{elif}\;y \le -7.956381646089189 \cdot 10^{-195}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le 1.8163835546796256 \cdot 10^{-162}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x \cdot x + y \cdot y}{\left(x - y\right) \cdot \left(x + y\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020091 
(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))))