Data.Octree.Internal:octantDistance from Octree-0.5.4.2

Average Error: 31.7 → 19.2

Time: 2.1s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -6.10750123063483244 \cdot 10^{116}:\\ \;\;\;\;-1 \cdot x\\ \mathbf{elif}\;x \le -2.80809519100113736 \cdot 10^{-191}:\\ \;\;\;\;\sqrt{x \cdot x + y \cdot y}\\ \mathbf{elif}\;x \le 1.99786214231465739 \cdot 10^{-154}:\\ \;\;\;\;y\\ \mathbf{elif}\;x \le 4.03768629029534308 \cdot 10^{74}:\\ \;\;\;\;\sqrt{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]

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 <= -6.1075012306348324e+116)) {
		VAR = ((double) (-1.0 * x));
	} else {
		double VAR_1;
		if ((x <= -2.8080951910011374e-191)) {
			VAR_1 = ((double) sqrt(((double) (((double) (x * x)) + ((double) (y * y))))));
		} else {
			double VAR_2;
			if ((x <= 1.9978621423146574e-154)) {
				VAR_2 = y;
			} else {
				double VAR_3;
				if ((x <= 4.037686290295343e+74)) {
					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	31.7
Target	18.3
Herbie	19.2

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

Derivation

1. Split input into 4 regimes

2. if x < -6.10750123063483244e116

   1. Initial program 54.4

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

   2. Taylor expanded around -inf 10.2

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

   if -6.10750123063483244e116 < x < -2.80809519100113736e-191 or 1.99786214231465739e-154 < x < 4.03768629029534308e74

   1. Initial program 16.9

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

   if -2.80809519100113736e-191 < x < 1.99786214231465739e-154

   1. Initial program 31.6

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

   2. Taylor expanded around 0 35.1

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

   if 4.03768629029534308e74 < x

   1. Initial program 48.0

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

   2. Taylor expanded around inf 13.0

      \[\leadsto \color{blue}{x}\]
3. Recombined 4 regimes into one program.

4. Final simplification 19.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -6.10750123063483244 \cdot 10^{116}:\\ \;\;\;\;-1 \cdot x\\ \mathbf{elif}\;x \le -2.80809519100113736 \cdot 10^{-191}:\\ \;\;\;\;\sqrt{x \cdot x + y \cdot y}\\ \mathbf{elif}\;x \le 1.99786214231465739 \cdot 10^{-154}:\\ \;\;\;\;y\\ \mathbf{elif}\;x \le 4.03768629029534308 \cdot 10^{74}:\\ \;\;\;\;\sqrt{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]

Reproduce

herbie shell --seed 2020153 
(FPCore (x y)
  :name "Data.Octree.Internal:octantDistance  from Octree-0.5.4.2"
  :precision binary64

  :herbie-target
  (if (< x -1.1236950826599826e+145) (neg x) (if (< x 1.116557621183362e+93) (sqrt (+ (* x x) (* y y))) x))

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