Kahan p9 Example

Average Error: 20.8 → 5.3

Time: 2.5s

Precision: binary64

\[0 < x \land x < 1 \land y < 1\]

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \leq -1.3746297857855241 \cdot 10^{+154}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq -1.5477292189177258 \cdot 10^{-162} \lor \neg \left(y \leq 1.5914560295974384 \cdot 10^{-162}\right):\\ \;\;\;\;\log \left(e^{\frac{x \cdot x - y \cdot y}{x \cdot x + y \cdot y}}\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array}\]

FPCore

(FPCore (x y) :precision binary64 (/ (* (- x y) (+ x y)) (+ (* x x) (* y y))))

↓

(FPCore (x y)
 :precision binary64
 (if (<= y -1.3746297857855241e+154)
   -1.0
   (if (or (<= y -1.5477292189177258e-162)
           (not (<= y 1.5914560295974384e-162)))
     (log (exp (/ (- (* x x) (* y y)) (+ (* x x) (* y y)))))
     1.0)))

C

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

↓

double code(double x, double y) {
	double tmp;
	if ((y <= -1.3746297857855241e+154)) {
		tmp = -1.0;
	} else {
		double tmp_1;
		if (((y <= -1.5477292189177258e-162) || !(y <= 1.5914560295974384e-162))) {
			tmp_1 = ((double) log(((double) exp((((double) (((double) (x * x)) - ((double) (y * y)))) / ((double) (((double) (x * x)) + ((double) (y * y)))))))));
		} else {
			tmp_1 = 1.0;
		}
		tmp = tmp_1;
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Target

Original	20.8
Target	0.1
Herbie	5.3

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

Derivation

1. Split input into 3 regimes

2. if y < -1.37462978578552415e154

   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 -1.37462978578552415e154 < y < -1.54772921891772581e-162 or 1.59145602959743844e-162 < 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 add-log-exp_binary64 0.1

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

   4. Simplified 0.1

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

   if -1.54772921891772581e-162 < y < 1.59145602959743844e-162

   1. Initial program 31.5

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

   2. Taylor expanded around inf 16.3

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

4. Final simplification 5.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.3746297857855241 \cdot 10^{+154}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq -1.5477292189177258 \cdot 10^{-162} \lor \neg \left(y \leq 1.5914560295974384 \cdot 10^{-162}\right):\\ \;\;\;\;\log \left(e^{\frac{x \cdot x - y \cdot y}{x \cdot x + y \cdot y}}\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array}\]

Reproduce

herbie shell --seed 2020210 
(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))))