math.abs on complex

Average Error: 31.7 → 18.3

Time: 1.5s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \le -2.7914972225733451 \cdot 10^{118}:\\ \;\;\;\;-1 \cdot re\\ \mathbf{elif}\;re \le -2.88672912953644354 \cdot 10^{-292}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{elif}\;re \le 5.4203859696291379 \cdot 10^{-163}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 1.18445552989457458 \cdot 10^{131}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

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 <= -2.791497222573345e+118)) {
		VAR = ((double) (-1.0 * re));
	} else {
		double VAR_1;
		if ((re <= -2.8867291295364435e-292)) {
			VAR_1 = ((double) sqrt(((double) (((double) (re * re)) + ((double) (im * im))))));
		} else {
			double VAR_2;
			if ((re <= 5.420385969629138e-163)) {
				VAR_2 = im;
			} else {
				double VAR_3;
				if ((re <= 1.1844555298945746e+131)) {
					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 < -2.7914972225733451e118

   1. Initial program 55.7

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

   2. Taylor expanded around -inf 9.1

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

   if -2.7914972225733451e118 < re < -2.88672912953644354e-292 or 5.4203859696291379e-163 < re < 1.18445552989457458e131

   1. Initial program 18.7

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

   if -2.88672912953644354e-292 < re < 5.4203859696291379e-163

   1. Initial program 31.1

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

   2. Taylor expanded around 0 36.3

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

   if 1.18445552989457458e131 < re

   1. Initial program 57.6

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

   2. Taylor expanded around inf 10.1

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

4. Final simplification 18.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -2.7914972225733451 \cdot 10^{118}:\\ \;\;\;\;-1 \cdot re\\ \mathbf{elif}\;re \le -2.88672912953644354 \cdot 10^{-292}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{elif}\;re \le 5.4203859696291379 \cdot 10^{-163}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 1.18445552989457458 \cdot 10^{131}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Reproduce

herbie shell --seed 2020161 
(FPCore (re im)
  :name "math.abs on complex"
  :precision binary64
  (sqrt (+ (* re re) (* im im))))