Data.Octree.Internal:octantDistance from Octree-0.5.4.2

Average Error: 31.7 → 19.0

Time: 2.1s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -2.16131408325038182 \cdot 10^{112}:\\ \;\;\;\;-1 \cdot x\\ \mathbf{elif}\;x \le -1.1454886226922574 \cdot 10^{-300}:\\ \;\;\;\;\sqrt{x \cdot x + y \cdot y}\\ \mathbf{elif}\;x \le 1.8543864403264038 \cdot 10^{-255}:\\ \;\;\;\;y\\ \mathbf{elif}\;x \le 2.601933874540376 \cdot 10^{-42}:\\ \;\;\;\;\sqrt{x \cdot x + y \cdot y}\\ \mathbf{elif}\;x \le 548.395158744075729:\\ \;\;\;\;y\\ \mathbf{elif}\;x \le 1.9954821974516343 \cdot 10^{99}:\\ \;\;\;\;\sqrt{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]

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 <= -2.161314083250382e+112)) {
		VAR = ((double) (-1.0 * x));
	} else {
		double VAR_1;
		if ((x <= -1.1454886226922574e-300)) {
			VAR_1 = ((double) sqrt(((double) (((double) (x * x)) + ((double) (y * y))))));
		} else {
			double VAR_2;
			if ((x <= 1.8543864403264038e-255)) {
				VAR_2 = y;
			} else {
				double VAR_3;
				if ((x <= 2.601933874540376e-42)) {
					VAR_3 = ((double) sqrt(((double) (((double) (x * x)) + ((double) (y * y))))));
				} else {
					double VAR_4;
					if ((x <= 548.3951587440757)) {
						VAR_4 = y;
					} else {
						double VAR_5;
						if ((x <= 1.9954821974516343e+99)) {
							VAR_5 = ((double) sqrt(((double) (((double) (x * x)) + ((double) (y * y))))));
						} else {
							VAR_5 = x;
						}
						VAR_4 = VAR_5;
					}
					VAR_3 = VAR_4;
				}
				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	17.5
Herbie	19.0

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

Derivation

1. Split input into 4 regimes

2. if x < -2.16131408325038182e112

   1. Initial program 54.0

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

   2. Taylor expanded around -inf 9.0

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

   if -2.16131408325038182e112 < x < -1.1454886226922574e-300 or 1.8543864403264038e-255 < x < 2.601933874540376e-42 or 548.395158744075729 < x < 1.9954821974516343e99

   1. Initial program 21.1

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

   if -1.1454886226922574e-300 < x < 1.8543864403264038e-255 or 2.601933874540376e-42 < x < 548.395158744075729

   1. Initial program 22.9

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

   2. Taylor expanded around 0 39.3

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

   if 1.9954821974516343e99 < x

   1. Initial program 51.5

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

   2. Taylor expanded around inf 10.9

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

4. Final simplification 19.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -2.16131408325038182 \cdot 10^{112}:\\ \;\;\;\;-1 \cdot x\\ \mathbf{elif}\;x \le -1.1454886226922574 \cdot 10^{-300}:\\ \;\;\;\;\sqrt{x \cdot x + y \cdot y}\\ \mathbf{elif}\;x \le 1.8543864403264038 \cdot 10^{-255}:\\ \;\;\;\;y\\ \mathbf{elif}\;x \le 2.601933874540376 \cdot 10^{-42}:\\ \;\;\;\;\sqrt{x \cdot x + y \cdot y}\\ \mathbf{elif}\;x \le 548.395158744075729:\\ \;\;\;\;y\\ \mathbf{elif}\;x \le 1.9954821974516343 \cdot 10^{99}:\\ \;\;\;\;\sqrt{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]

Reproduce

herbie shell --seed 2020175 
(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))))