Data.Octree.Internal:octantDistance from Octree-0.5.4.2

Average Error: 32.0 → 18.2

Time: 1.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -3.08310863609937876 \cdot 10^{138}:\\ \;\;\;\;-1 \cdot x\\ \mathbf{elif}\;x \le -1.1813831187355925 \cdot 10^{-160}:\\ \;\;\;\;\sqrt{x \cdot x + y \cdot y}\\ \mathbf{elif}\;x \le -1.4034448373480268 \cdot 10^{-212}:\\ \;\;\;\;y\\ \mathbf{elif}\;x \le 6.33411590852480896 \cdot 10^{-304}:\\ \;\;\;\;\sqrt{x \cdot x + y \cdot y}\\ \mathbf{elif}\;x \le 5.55126095100350983 \cdot 10^{-260}:\\ \;\;\;\;y\\ \mathbf{elif}\;x \le 9.23378212571996135 \cdot 10^{120}:\\ \;\;\;\;\sqrt{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]

C

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

↓

double code(double x, double y) {
	double VAR;
	if ((x <= -3.083108636099379e+138)) {
		VAR = (-1.0 * x);
	} else {
		double VAR_1;
		if ((x <= -1.1813831187355925e-160)) {
			VAR_1 = sqrt(((x * x) + (y * y)));
		} else {
			double VAR_2;
			if ((x <= -1.4034448373480268e-212)) {
				VAR_2 = y;
			} else {
				double VAR_3;
				if ((x <= 6.334115908524809e-304)) {
					VAR_3 = sqrt(((x * x) + (y * y)));
				} else {
					double VAR_4;
					if ((x <= 5.55126095100351e-260)) {
						VAR_4 = y;
					} else {
						double VAR_5;
						if ((x <= 9.233782125719961e+120)) {
							VAR_5 = sqrt(((x * x) + (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	32.0
Target	18.1
Herbie	18.2

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

Derivation

1. Split input into 4 regimes

2. if x < -3.083108636099379e+138

   1. Initial program 60.4

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

   2. Taylor expanded around -inf 9.5

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

   if -3.083108636099379e+138 < x < -1.1813831187355925e-160 or -1.4034448373480268e-212 < x < 6.334115908524809e-304 or 5.55126095100351e-260 < x < 9.233782125719961e+120

   1. Initial program 20.2

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

   if -1.1813831187355925e-160 < x < -1.4034448373480268e-212 or 6.334115908524809e-304 < x < 5.55126095100351e-260

   1. Initial program 31.1

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

   2. Taylor expanded around 0 34.5

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

   if 9.233782125719961e+120 < x

   1. Initial program 56.0

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

   2. Taylor expanded around inf 9.8

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

4. Final simplification 18.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -3.08310863609937876 \cdot 10^{138}:\\ \;\;\;\;-1 \cdot x\\ \mathbf{elif}\;x \le -1.1813831187355925 \cdot 10^{-160}:\\ \;\;\;\;\sqrt{x \cdot x + y \cdot y}\\ \mathbf{elif}\;x \le -1.4034448373480268 \cdot 10^{-212}:\\ \;\;\;\;y\\ \mathbf{elif}\;x \le 6.33411590852480896 \cdot 10^{-304}:\\ \;\;\;\;\sqrt{x \cdot x + y \cdot y}\\ \mathbf{elif}\;x \le 5.55126095100350983 \cdot 10^{-260}:\\ \;\;\;\;y\\ \mathbf{elif}\;x \le 9.23378212571996135 \cdot 10^{120}:\\ \;\;\;\;\sqrt{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]

Reproduce

herbie shell --seed 2020078 
(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))))