Kahan p9 Example

Average Error: 19.8 → 5.1

Time: 2.3s

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \le -1.34010805875991645 \cdot 10^{154}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le -4.31204332737563829 \cdot 10^{-156}:\\ \;\;\;\;\left(\frac{x - y}{\sqrt{x \cdot x + y \cdot y}} \cdot \left(\sqrt[3]{\frac{x + y}{\sqrt{x \cdot x + y \cdot y}}} \cdot \sqrt[3]{\frac{x + y}{\sqrt{x \cdot x + y \cdot y}}}\right)\right) \cdot \sqrt[3]{\frac{x + y}{\sqrt{x \cdot x + y \cdot y}}}\\ \mathbf{elif}\;y \le 1.4337271120113957 \cdot 10^{-163}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y}{\sqrt{x \cdot x + y \cdot y}} \cdot \frac{x + y}{\sqrt{x \cdot x + y \cdot y}}\\ \end{array}\]

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 <= -1.3401080587599165e+154)) {
		VAR = -1.0;
	} else {
		double VAR_1;
		if ((y <= -4.3120433273756383e-156)) {
			VAR_1 = ((double) (((double) (((double) (((double) (x - y)) / ((double) sqrt(((double) (((double) (x * x)) + ((double) (y * y)))))))) * ((double) (((double) cbrt(((double) (((double) (x + y)) / ((double) sqrt(((double) (((double) (x * x)) + ((double) (y * y)))))))))) * ((double) cbrt(((double) (((double) (x + y)) / ((double) sqrt(((double) (((double) (x * x)) + ((double) (y * y)))))))))))))) * ((double) cbrt(((double) (((double) (x + y)) / ((double) sqrt(((double) (((double) (x * x)) + ((double) (y * y))))))))))));
		} else {
			double VAR_2;
			if ((y <= 1.4337271120113957e-163)) {
				VAR_2 = 1.0;
			} else {
				VAR_2 = ((double) (((double) (((double) (x - y)) / ((double) sqrt(((double) (((double) (x * x)) + ((double) (y * y)))))))) * ((double) (((double) (x + y)) / ((double) sqrt(((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	19.8
Target	0.0
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 < -1.34010805875991645e154

   1. Initial program 64.0

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

   2. Taylor expanded around 0 0

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

   if -1.34010805875991645e154 < y < -4.31204332737563829e-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-sqr-sqrt 0.0

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

   4. Applied times-frac 0.1

      \[\leadsto \color{blue}{\frac{x - y}{\sqrt{x \cdot x + y \cdot y}} \cdot \frac{x + y}{\sqrt{x \cdot x + y \cdot y}}}\]
   5. Using strategy rm

   6. Applied add-cube-cbrt 0.1

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

   7. Applied associate-*r* 0.1

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

   if -4.31204332737563829e-156 < y < 1.4337271120113957e-163

   1. Initial program 29.0

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

   2. Taylor expanded around inf 15.6

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

   if 1.4337271120113957e-163 < y

   1. Initial program 0.2

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

   3. Applied add-sqr-sqrt 0.2

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

   4. Applied times-frac 0.7

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

4. Final simplification 5.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -1.34010805875991645 \cdot 10^{154}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le -4.31204332737563829 \cdot 10^{-156}:\\ \;\;\;\;\left(\frac{x - y}{\sqrt{x \cdot x + y \cdot y}} \cdot \left(\sqrt[3]{\frac{x + y}{\sqrt{x \cdot x + y \cdot y}}} \cdot \sqrt[3]{\frac{x + y}{\sqrt{x \cdot x + y \cdot y}}}\right)\right) \cdot \sqrt[3]{\frac{x + y}{\sqrt{x \cdot x + y \cdot y}}}\\ \mathbf{elif}\;y \le 1.4337271120113957 \cdot 10^{-163}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y}{\sqrt{x \cdot x + y \cdot y}} \cdot \frac{x + y}{\sqrt{x \cdot x + y \cdot y}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020162 
(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))))