Kahan p9 Example

Average Error: 20.9 → 8.4

Time: 2.2s

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \leq -2.5805430812011672 \cdot 10^{-160}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 1.1265125259841753 \cdot 10^{-197}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 3.60390796991782 \cdot 10^{-163}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(\frac{x \cdot x - y \cdot y}{x \cdot x + y \cdot y}\right)}^{3}}\\ \end{array}\]

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (if (<= y -2.5805430812011672e-160)
   -1.0
   (if (<= y 1.1265125259841753e-197)
     1.0
     (if (<= y 3.60390796991782e-163)
       -1.0
       (cbrt (pow (/ (- (* x x) (* y y)) (+ (* x x) (* y y))) 3.0))))))

C

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

↓

double code(double x, double y) {
	double tmp;
	if (y <= -2.5805430812011672e-160) {
		tmp = -1.0;
	} else if (y <= 1.1265125259841753e-197) {
		tmp = 1.0;
	} else if (y <= 3.60390796991782e-163) {
		tmp = -1.0;
	} else {
		tmp = cbrt(pow((((x * x) - (y * y)) / ((x * x) + (y * y))), 3.0));
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Target

Original	20.9
Target	0.1
Herbie	8.4

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

Derivation

1. Split input into 3 regimes

2. if y < -2.5805430812011672e-160 or 1.1265125259841753e-197 < y < 3.60390796991782017e-163

   1. Initial program 22.2

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

   2. Taylor expanded around 0 7.4

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

   if -2.5805430812011672e-160 < y < 1.1265125259841753e-197

   1. Initial program 30.1

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

   2. Taylor expanded around inf 14.7

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

   if 3.60390796991782017e-163 < y

   1. Initial program 0.2

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

   3. Applied add-cbrt-cube_binary64_1147 0.2

      \[\leadsto \color{blue}{\sqrt[3]{\left(\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y} \cdot \frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\right) \cdot \frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}}}\]

   4. Simplified 0.2

      \[\leadsto \sqrt[3]{\color{blue}{{\left(\frac{x \cdot x - y \cdot y}{x \cdot x + y \cdot y}\right)}^{3}}}\]
3. Recombined 3 regimes into one program.

4. Final simplification 8.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.5805430812011672 \cdot 10^{-160}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 1.1265125259841753 \cdot 10^{-197}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 3.60390796991782 \cdot 10^{-163}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(\frac{x \cdot x - y \cdot y}{x \cdot x + y \cdot y}\right)}^{3}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020288 
(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))))