Kahan p9 Example

Average Error: 20.4 → 5.5

Time: 2.5s

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \le -1.3396749332313781 \cdot 10^{154}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le -7.35822598373593517 \cdot 10^{-156}:\\ \;\;\;\;\sqrt[3]{\sqrt[3]{{\left({\left(\frac{y + x}{x \cdot x + y \cdot y} \cdot \left(x - y\right)\right)}^{3}\right)}^{3}}}\\ \mathbf{elif}\;y \le 6.6501588139279482 \cdot 10^{-174}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \le 2.47259879968152251 \cdot 10^{-162}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(\frac{y + x}{x \cdot x + y \cdot y} \cdot \left(x - y\right)\right)}^{3}}\\ \end{array}\]

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 <= -1.3396749332313781e+154)) {
		VAR = -1.0;
	} else {
		double VAR_1;
		if ((y <= -7.358225983735935e-156)) {
			VAR_1 = ((double) cbrt(((double) cbrt(((double) pow(((double) pow(((double) ((((double) (y + x)) / ((double) (((double) (x * x)) + ((double) (y * y))))) * ((double) (x - y)))), 3.0)), 3.0))))));
		} else {
			double VAR_2;
			if ((y <= 6.650158813927948e-174)) {
				VAR_2 = 1.0;
			} else {
				double VAR_3;
				if ((y <= 2.4725987996815225e-162)) {
					VAR_3 = -1.0;
				} else {
					VAR_3 = ((double) cbrt(((double) pow(((double) ((((double) (y + x)) / ((double) (((double) (x * x)) + ((double) (y * y))))) * ((double) (x - y)))), 3.0))));
				}
				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	0.0
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 4 regimes

2. if y < -1.3396749332313781e154 or 6.6501588139279482e-174 < y < 2.47259879968152251e-162

   1. Initial program 61.3

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

   2. Taylor expanded around 0 2.7

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

   if -1.3396749332313781e154 < y < -7.35822598373593517e-156

   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-cbrt-cube 38.3

      \[\leadsto \frac{\left(x - y\right) \cdot \left(x + y\right)}{\color{blue}{\sqrt[3]{\left(\left(x \cdot x + y \cdot y\right) \cdot \left(x \cdot x + y \cdot y\right)\right) \cdot \left(x \cdot x + y \cdot y\right)}}}\]

   4. Applied add-cbrt-cube 38.7

      \[\leadsto \frac{\left(x - y\right) \cdot \color{blue}{\sqrt[3]{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + y\right)}}}{\sqrt[3]{\left(\left(x \cdot x + y \cdot y\right) \cdot \left(x \cdot x + y \cdot y\right)\right) \cdot \left(x \cdot x + y \cdot y\right)}}\]

   5. Applied add-cbrt-cube 38.9

      \[\leadsto \frac{\color{blue}{\sqrt[3]{\left(\left(x - y\right) \cdot \left(x - y\right)\right) \cdot \left(x - y\right)}} \cdot \sqrt[3]{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + y\right)}}{\sqrt[3]{\left(\left(x \cdot x + y \cdot y\right) \cdot \left(x \cdot x + y \cdot y\right)\right) \cdot \left(x \cdot x + y \cdot y\right)}}\]

   6. Applied cbrt-unprod 38.8

      \[\leadsto \frac{\color{blue}{\sqrt[3]{\left(\left(\left(x - y\right) \cdot \left(x - y\right)\right) \cdot \left(x - y\right)\right) \cdot \left(\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + y\right)\right)}}}{\sqrt[3]{\left(\left(x \cdot x + y \cdot y\right) \cdot \left(x \cdot x + y \cdot y\right)\right) \cdot \left(x \cdot x + y \cdot y\right)}}\]

   7. Applied cbrt-undiv 38.7

      \[\leadsto \color{blue}{\sqrt[3]{\frac{\left(\left(\left(x - y\right) \cdot \left(x - y\right)\right) \cdot \left(x - y\right)\right) \cdot \left(\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + y\right)\right)}{\left(\left(x \cdot x + y \cdot y\right) \cdot \left(x \cdot x + y \cdot y\right)\right) \cdot \left(x \cdot x + y \cdot y\right)}}}\]

   8. Simplified 0.0

      \[\leadsto \sqrt[3]{\color{blue}{{\left(\frac{x + y}{x \cdot x + y \cdot y} \cdot \left(x - y\right)\right)}^{3}}}\]
   9. Using strategy rm

   10. Applied add-cbrt-cube 0.0

       \[\leadsto \sqrt[3]{\color{blue}{\sqrt[3]{\left({\left(\frac{x + y}{x \cdot x + y \cdot y} \cdot \left(x - y\right)\right)}^{3} \cdot {\left(\frac{x + y}{x \cdot x + y \cdot y} \cdot \left(x - y\right)\right)}^{3}\right) \cdot {\left(\frac{x + y}{x \cdot x + y \cdot y} \cdot \left(x - y\right)\right)}^{3}}}}\]

   11. Simplified 0.0

       \[\leadsto \sqrt[3]{\sqrt[3]{\color{blue}{{\left({\left(\frac{x + y}{x \cdot x + y \cdot y} \cdot \left(x - y\right)\right)}^{3}\right)}^{3}}}}\]

   if -7.35822598373593517e-156 < y < 6.6501588139279482e-174

   1. Initial program 29.5

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

   2. Taylor expanded around inf 15.7

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

   if 2.47259879968152251e-162 < 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-cbrt-cube 35.6

      \[\leadsto \frac{\left(x - y\right) \cdot \left(x + y\right)}{\color{blue}{\sqrt[3]{\left(\left(x \cdot x + y \cdot y\right) \cdot \left(x \cdot x + y \cdot y\right)\right) \cdot \left(x \cdot x + y \cdot y\right)}}}\]

   4. Applied add-cbrt-cube 35.7

      \[\leadsto \frac{\left(x - y\right) \cdot \color{blue}{\sqrt[3]{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + y\right)}}}{\sqrt[3]{\left(\left(x \cdot x + y \cdot y\right) \cdot \left(x \cdot x + y \cdot y\right)\right) \cdot \left(x \cdot x + y \cdot y\right)}}\]

   5. Applied add-cbrt-cube 35.8

      \[\leadsto \frac{\color{blue}{\sqrt[3]{\left(\left(x - y\right) \cdot \left(x - y\right)\right) \cdot \left(x - y\right)}} \cdot \sqrt[3]{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + y\right)}}{\sqrt[3]{\left(\left(x \cdot x + y \cdot y\right) \cdot \left(x \cdot x + y \cdot y\right)\right) \cdot \left(x \cdot x + y \cdot y\right)}}\]

   6. Applied cbrt-unprod 35.1

      \[\leadsto \frac{\color{blue}{\sqrt[3]{\left(\left(\left(x - y\right) \cdot \left(x - y\right)\right) \cdot \left(x - y\right)\right) \cdot \left(\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + y\right)\right)}}}{\sqrt[3]{\left(\left(x \cdot x + y \cdot y\right) \cdot \left(x \cdot x + y \cdot y\right)\right) \cdot \left(x \cdot x + y \cdot y\right)}}\]

   7. Applied cbrt-undiv 35.0

      \[\leadsto \color{blue}{\sqrt[3]{\frac{\left(\left(\left(x - y\right) \cdot \left(x - y\right)\right) \cdot \left(x - y\right)\right) \cdot \left(\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + y\right)\right)}{\left(\left(x \cdot x + y \cdot y\right) \cdot \left(x \cdot x + y \cdot y\right)\right) \cdot \left(x \cdot x + y \cdot y\right)}}}\]

   8. Simplified 0.8

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

4. Final simplification 5.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -1.3396749332313781 \cdot 10^{154}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le -7.35822598373593517 \cdot 10^{-156}:\\ \;\;\;\;\sqrt[3]{\sqrt[3]{{\left({\left(\frac{y + x}{x \cdot x + y \cdot y} \cdot \left(x - y\right)\right)}^{3}\right)}^{3}}}\\ \mathbf{elif}\;y \le 6.6501588139279482 \cdot 10^{-174}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \le 2.47259879968152251 \cdot 10^{-162}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(\frac{y + x}{x \cdot x + y \cdot y} \cdot \left(x - y\right)\right)}^{3}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020182 
(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))))