Kahan p9 Example

Average Error: 20.1 → 5.7

Time: 2.5s

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \le -1.2570093409768951 \cdot 10^{-25}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le -1.45404192359399537 \cdot 10^{-159}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{x + y}{{x}^{2} + {y}^{2}}\\ \mathbf{elif}\;y \le 1.8047328055627691 \cdot 10^{-146}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{x + y}{{x}^{2} + {y}^{2}}\\ \end{array}\]

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 <= -1.2570093409768951e-25)) {
		VAR = -1.0;
	} else {
		double VAR_1;
		if ((y <= -1.4540419235939954e-159)) {
			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 <= 1.804732805562769e-146)) {
				VAR_2 = 1.0;
			} else {
				VAR_2 = ((double) (((double) (x - y)) * ((double) (((double) (x + y)) / ((double) (((double) pow(x, 2.0)) + ((double) pow(y, 2.0))))))));
			}
			VAR_1 = VAR_2;
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Target

Original	20.1
Target	0.1
Herbie	5.7

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

Derivation

1. Split input into 3 regimes

2. if y < -1.2570093409768951e-25

   1. Initial program 30.0

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

   2. Taylor expanded around 0 0.3

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

   if -1.2570093409768951e-25 < y < -1.45404192359399537e-159 or 1.8047328055627691e-146 < 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 *-un-lft-identity 0.0

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

   4. Applied times-frac 0.4

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

   5. Simplified 0.4

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

   6. Simplified 0.4

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

   if -1.45404192359399537e-159 < y < 1.8047328055627691e-146

   1. Initial program 28.2

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

   2. Taylor expanded around inf 16.3

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

4. Final simplification 5.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -1.2570093409768951 \cdot 10^{-25}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le -1.45404192359399537 \cdot 10^{-159}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{x + y}{{x}^{2} + {y}^{2}}\\ \mathbf{elif}\;y \le 1.8047328055627691 \cdot 10^{-146}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{x + y}{{x}^{2} + {y}^{2}}\\ \end{array}\]

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))))