Data.Octree.Internal:octantDistance from Octree-0.5.4.2

Average Error: 31.8 → 18.3

Time: 1.6s

Precision: 64

Math

\[\sqrt{x \cdot x + y \cdot y}\]

↓

\[\begin{array}{l} \mathbf{if}\;x \le -1.20078310669033337 \cdot 10^{154}:\\ \;\;\;\;-1 \cdot x\\ \mathbf{elif}\;x \le 7.1840670810958776 \cdot 10^{-197}:\\ \;\;\;\;\sqrt{x \cdot x + y \cdot y}\\ \mathbf{elif}\;x \le 1.8702588275085934 \cdot 10^{-161}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \le 1.37839921125173732 \cdot 10^{122}:\\ \;\;\;\;\sqrt{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]

C

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

↓

double code(double x, double y) {
	double VAR;
	if ((x <= -1.2007831066903334e+154)) {
		VAR = ((double) (-1.0 * x));
	} else {
		double VAR_1;
		if ((x <= 7.184067081095878e-197)) {
			VAR_1 = ((double) sqrt(((double) (((double) (x * x)) + ((double) (y * y))))));
		} else {
			double VAR_2;
			if ((x <= 1.8702588275085934e-161)) {
				VAR_2 = x;
			} else {
				double VAR_3;
				if ((x <= 1.3783992112517373e+122)) {
					VAR_3 = ((double) sqrt(((double) (((double) (x * x)) + ((double) (y * y))))));
				} else {
					VAR_3 = x;
				}
				VAR_2 = VAR_3;
			}
			VAR_1 = VAR_2;
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Target

Original	31.8
Target	17.9
Herbie	18.3

\[\begin{array}{l} \mathbf{if}\;x \lt -1.123695082659983 \cdot 10^{145}:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \lt 1.11655762118336204 \cdot 10^{93}:\\ \;\;\;\;\sqrt{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if x < -1.2007831066903334e+154

   1. Initial program 63.9

      \[\sqrt{x \cdot x + y \cdot y}\]

   2. Taylor expanded around -inf 8.1

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

   if -1.2007831066903334e+154 < x < 7.184067081095878e-197 or 1.8702588275085934e-161 < x < 1.3783992112517373e+122

   1. Initial program 20.7

      \[\sqrt{x \cdot x + y \cdot y}\]

   if 7.184067081095878e-197 < x < 1.8702588275085934e-161 or 1.3783992112517373e+122 < x

   1. Initial program 51.1

      \[\sqrt{x \cdot x + y \cdot y}\]

   2. Taylor expanded around inf 16.3

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

4. Final simplification 18.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.20078310669033337 \cdot 10^{154}:\\ \;\;\;\;-1 \cdot x\\ \mathbf{elif}\;x \le 7.1840670810958776 \cdot 10^{-197}:\\ \;\;\;\;\sqrt{x \cdot x + y \cdot y}\\ \mathbf{elif}\;x \le 1.8702588275085934 \cdot 10^{-161}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \le 1.37839921125173732 \cdot 10^{122}:\\ \;\;\;\;\sqrt{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]

Reproduce

herbie shell --seed 2020120 
(FPCore (x y)
  :name "Data.Octree.Internal:octantDistance  from Octree-0.5.4.2"
  :precision binary64

  :herbie-target
  (if (< x -1.123695082659983e+145) (- x) (if (< x 1.116557621183362e+93) (sqrt (+ (* x x) (* y y))) x))

  (sqrt (+ (* x x) (* y y))))