math.abs on complex

Average Error: 31.7 → 18.7

Time: 2.0s

Precision: binary64

Math

\[\sqrt{re \cdot re + im \cdot im}\]

↓

\[\begin{array}{l} \mathbf{if}\;re \le -5.02179966160642987 \cdot 10^{145}:\\ \;\;\;\;-1 \cdot re\\ \mathbf{elif}\;re \le -1.5391970346231962 \cdot 10^{-190}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{elif}\;re \le -3.9573712583367902 \cdot 10^{-295}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 1.29362631726006162 \cdot 10^{74}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

C

double code(double re, double im) {
	return ((double) sqrt(((double) (((double) (re * re)) + ((double) (im * im))))));
}

↓

double code(double re, double im) {
	double VAR;
	if ((re <= -5.02179966160643e+145)) {
		VAR = ((double) (-1.0 * re));
	} else {
		double VAR_1;
		if ((re <= -1.5391970346231962e-190)) {
			VAR_1 = ((double) sqrt(((double) (((double) (re * re)) + ((double) (im * im))))));
		} else {
			double VAR_2;
			if ((re <= -3.95737125833679e-295)) {
				VAR_2 = im;
			} else {
				double VAR_3;
				if ((re <= 1.2936263172600616e+74)) {
					VAR_3 = ((double) sqrt(((double) (((double) (re * re)) + ((double) (im * im))))));
				} else {
					VAR_3 = re;
				}
				VAR_2 = VAR_3;
			}
			VAR_1 = VAR_2;
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus re

Bits error versus im

Derivation

1. Split input into 4 regimes

2. if re < -5.02179966160642987e145

   1. Initial program 62.1

      \[\sqrt{re \cdot re + im \cdot im}\]

   2. Taylor expanded around -inf 9.5

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

   if -5.02179966160642987e145 < re < -1.5391970346231962e-190 or -3.9573712583367902e-295 < re < 1.29362631726006162e74

   1. Initial program 20.3

      \[\sqrt{re \cdot re + im \cdot im}\]

   if -1.5391970346231962e-190 < re < -3.9573712583367902e-295

   1. Initial program 31.7

      \[\sqrt{re \cdot re + im \cdot im}\]

   2. Taylor expanded around 0 35.1

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

   if 1.29362631726006162e74 < re

   1. Initial program 47.2

      \[\sqrt{re \cdot re + im \cdot im}\]

   2. Taylor expanded around inf 12.9

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

4. Final simplification 18.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -5.02179966160642987 \cdot 10^{145}:\\ \;\;\;\;-1 \cdot re\\ \mathbf{elif}\;re \le -1.5391970346231962 \cdot 10^{-190}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{elif}\;re \le -3.9573712583367902 \cdot 10^{-295}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 1.29362631726006162 \cdot 10^{74}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Reproduce

herbie shell --seed 2020147 
(FPCore (re im)
  :name "math.abs on complex"
  :precision binary64
  (sqrt (+ (* re re) (* im im))))