Kahan p9 Example

Average Error: 20.6 → 5.1

Time: 3.7s

Precision: 64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \le -1.2400159933984513 \cdot 10^{152}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le -1.3530299398450909 \cdot 10^{-158}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot x + \left(x - y\right) \cdot y}{x \cdot x + y \cdot y}\\ \mathbf{elif}\;y \le 2.0406396431233693 \cdot 10^{-163}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}} \cdot \sqrt[3]{\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}}\right) \cdot \sqrt[3]{\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}}\\ \end{array}\]

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 <= -1.2400159933984513e+152)) {
		VAR = -1.0;
	} else {
		double VAR_1;
		if ((y <= -1.3530299398450909e-158)) {
			VAR_1 = ((double) (((double) (((double) (((double) (x - y)) * x)) + ((double) (((double) (x - y)) * y)))) / ((double) (((double) (x * x)) + ((double) (y * y))))));
		} else {
			double VAR_2;
			if ((y <= 2.0406396431233693e-163)) {
				VAR_2 = 1.0;
			} else {
				VAR_2 = ((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_1 = VAR_2;
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Target

Original	20.6
Target	0.1
Herbie	5.1

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

Derivation

1. Split input into 4 regimes

2. if y < -1.2400159933984513e+152

   1. Initial program 63.6

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

   2. Taylor expanded around 0 0

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

   if -1.2400159933984513e+152 < y < -1.3530299398450909e-158

   1. Initial program 0.0

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

   3. Applied distribute-lft-in 0.0

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

   if -1.3530299398450909e-158 < y < 2.0406396431233693e-163

   1. Initial program 30.1

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

   2. Taylor expanded around inf 15.7

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

   if 2.0406396431233693e-163 < y

   1. Initial program 0.0

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

   3. Applied add-cube-cbrt 0.1

      \[\leadsto \color{blue}{\left(\sqrt[3]{\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}} \cdot \sqrt[3]{\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}}\right) \cdot \sqrt[3]{\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}}}\]
3. Recombined 4 regimes into one program.

4. Final simplification 5.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -1.2400159933984513 \cdot 10^{152}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le -1.3530299398450909 \cdot 10^{-158}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot x + \left(x - y\right) \cdot y}{x \cdot x + y \cdot y}\\ \mathbf{elif}\;y \le 2.0406396431233693 \cdot 10^{-163}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}} \cdot \sqrt[3]{\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}}\right) \cdot \sqrt[3]{\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020123 
(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))))