math.abs on complex

Average Error: 31.8 → 18.3

Time: 1.6s

Precision: 64

Math

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

↓

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

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 <= -1.2007831066903334e+154)) {
		VAR = ((double) (-1.0 * re));
	} else {
		double VAR_1;
		if ((re <= 7.184067081095878e-197)) {
			VAR_1 = ((double) sqrt(((double) (((double) (re * re)) + ((double) (im * im))))));
		} else {
			double VAR_2;
			if ((re <= 1.8702588275085934e-161)) {
				VAR_2 = re;
			} else {
				double VAR_3;
				if ((re <= 1.3783992112517373e+122)) {
					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 3 regimes

2. if re < -1.2007831066903334e+154

   1. Initial program 63.9

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

   2. Taylor expanded around -inf 8.1

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

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

   1. Initial program 20.7

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

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

   1. Initial program 51.1

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

   2. Taylor expanded around inf 16.3

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

4. Final simplification 18.3

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

Reproduce

herbie shell --seed 2020120 
(FPCore (re im)
  :name "math.abs on complex"
  :precision binary64
  (sqrt (+ (* re re) (* im im))))