Kahan p9 Example

Average Error: 20.4 → 6.1

Time: 1.6s

Precision: 64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \le -3.35638251884482862 \cdot 10^{-43}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le -1.2372662811059696 \cdot 10^{-152}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{x + y}{{x}^{2} + {y}^{2}}\\ \mathbf{elif}\;y \le -1.27383376714252331 \cdot 10^{-209}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le 1.11546702965031499 \cdot 10^{-163}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{x + y}{{x}^{2} + {y}^{2}}\\ \end{array}\]

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 <= -3.3563825188448286e-43)) {
		VAR = -1.0;
	} else {
		double VAR_1;
		if ((y <= -1.2372662811059696e-152)) {
			VAR_1 = ((double) (((double) (x - y)) * ((double) (((double) (x + y)) / ((double) (((double) pow(x, 2.0)) + ((double) pow(y, 2.0))))))));
		} else {
			double VAR_2;
			if ((y <= -1.2738337671425233e-209)) {
				VAR_2 = -1.0;
			} else {
				double VAR_3;
				if ((y <= 1.115467029650315e-163)) {
					VAR_3 = 1.0;
				} else {
					VAR_3 = ((double) (((double) (x - y)) * ((double) (((double) (x + y)) / ((double) (((double) pow(x, 2.0)) + ((double) pow(y, 2.0))))))));
				}
				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	6.1

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

Derivation

1. Split input into 3 regimes

2. if y < -3.3563825188448286e-43 or -1.2372662811059696e-152 < y < -1.2738337671425233e-209

   1. Initial program 28.5

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

   2. Taylor expanded around 0 5.4

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

   if -3.3563825188448286e-43 < y < -1.2372662811059696e-152 or 1.115467029650315e-163 < 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 *-un-lft-identity 0.1

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

   4. Applied times-frac 0.7

      \[\leadsto \color{blue}{\frac{x - y}{1} \cdot \frac{x + y}{x \cdot x + y \cdot y}}\]

   5. Simplified 0.7

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

   6. Simplified 0.7

      \[\leadsto \left(x - y\right) \cdot \color{blue}{\frac{x + y}{{x}^{2} + {y}^{2}}}\]

   if -1.2738337671425233e-209 < y < 1.115467029650315e-163

   1. Initial program 29.4

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

   2. Taylor expanded around inf 13.2

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

4. Final simplification 6.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -3.35638251884482862 \cdot 10^{-43}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le -1.2372662811059696 \cdot 10^{-152}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{x + y}{{x}^{2} + {y}^{2}}\\ \mathbf{elif}\;y \le -1.27383376714252331 \cdot 10^{-209}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le 1.11546702965031499 \cdot 10^{-163}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{x + y}{{x}^{2} + {y}^{2}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020121 
(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))))