Kahan p9 Example

Average Error: 20.5 → 5.2

Time: 2.5s

Precision: 64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \le -1.10115633480067138 \cdot 10^{153}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le -4.6546003190309958 \cdot 10^{-157}:\\ \;\;\;\;\log \left(e^{\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}}\right)\\ \mathbf{elif}\;y \le 3.32161448822691038 \cdot 10^{-168}:\\ \;\;\;\;\log \left(e^{1}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}}\right)\\ \end{array}\]

C

double code(double x, double y) {
	return (((x - y) * (x + y)) / ((x * x) + (y * y)));
}

↓

double code(double x, double y) {
	double VAR;
	if ((y <= -1.1011563348006714e+153)) {
		VAR = -1.0;
	} else {
		double VAR_1;
		if ((y <= -4.654600319030996e-157)) {
			VAR_1 = log(exp((((x - y) * (x + y)) / ((x * x) + (y * y)))));
		} else {
			double VAR_2;
			if ((y <= 3.3216144882269104e-168)) {
				VAR_2 = log(exp(1.0));
			} else {
				VAR_2 = log(exp((((x - y) * (x + y)) / ((x * x) + (y * y)))));
			}
			VAR_1 = VAR_2;
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Target

Original	20.5
Target	0.1
Herbie	5.2

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

Derivation

1. Split input into 3 regimes

2. if y < -1.1011563348006714e+153

   1. Initial program 63.7

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

   2. Taylor expanded around 0 0

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

   if -1.1011563348006714e+153 < y < -4.654600319030996e-157 or 3.3216144882269104e-168 < y

   1. Initial program 0.3

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

   3. Applied add-log-exp 0.3

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

   if -4.654600319030996e-157 < y < 3.3216144882269104e-168

   1. Initial program 30.2

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

   3. Applied add-log-exp 30.2

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

   4. Taylor expanded around inf 16.1

      \[\leadsto \log \left(e^{\color{blue}{1}}\right)\]
3. Recombined 3 regimes into one program.

4. Final simplification 5.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -1.10115633480067138 \cdot 10^{153}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le -4.6546003190309958 \cdot 10^{-157}:\\ \;\;\;\;\log \left(e^{\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}}\right)\\ \mathbf{elif}\;y \le 3.32161448822691038 \cdot 10^{-168}:\\ \;\;\;\;\log \left(e^{1}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020078 
(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))))