Kahan p9 Example

Average Error: 19.9 → 5.1

Time: 5.4s

Precision: 64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \le -3.7202401672427106 \cdot 10^{146}:\\ \;\;\;\;\log \left(e^{-1}\right)\\ \mathbf{elif}\;y \le -2.4149536997492782 \cdot 10^{-160}:\\ \;\;\;\;\log \left(e^{\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}}\right)\\ \mathbf{elif}\;y \le 3.08109527849197105 \cdot 10^{-177}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}}\right)\\ \end{array}\]

C

double code(double x, double y) {
	return ((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 <= -3.7202401672427106e+146)) {
		VAR = ((double) log(((double) exp(-1.0))));
	} else {
		double VAR_1;
		if ((y <= -2.4149536997492782e-160)) {
			VAR_1 = ((double) log(((double) exp(((double) (((double) (((double) (x - y)) * ((double) (x + y)))) / ((double) (((double) (x * x)) + ((double) (y * y))))))))));
		} else {
			double VAR_2;
			if ((y <= 3.081095278491971e-177)) {
				VAR_2 = 1.0;
			} else {
				VAR_2 = ((double) log(((double) exp(((double) (((double) (((double) (x - y)) * ((double) (x + y)))) / ((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	19.9
Target	0.1
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 3 regimes

2. if y < -3.7202401672427106e+146

   1. Initial program 61.1

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

   3. Applied add-log-exp 61.1

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

   4. Taylor expanded around 0 0

      \[\leadsto \log \left(e^{\color{blue}{-1}}\right)\]

   if -3.7202401672427106e+146 < y < -2.4149536997492782e-160 or 3.081095278491971e-177 < y

   1. Initial program 1.2

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

   3. Applied add-log-exp 1.2

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

   if -2.4149536997492782e-160 < y < 3.081095278491971e-177

   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 inf 14.9

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

4. Final simplification 5.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -3.7202401672427106 \cdot 10^{146}:\\ \;\;\;\;\log \left(e^{-1}\right)\\ \mathbf{elif}\;y \le -2.4149536997492782 \cdot 10^{-160}:\\ \;\;\;\;\log \left(e^{\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}}\right)\\ \mathbf{elif}\;y \le 3.08109527849197105 \cdot 10^{-177}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020113 
(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))))