math.abs on complex

Average Error: 31.3 → 18.4

Time: 1.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \le -2.30805639812844361 \cdot 10^{37}:\\ \;\;\;\;-1 \cdot re\\ \mathbf{elif}\;re \le -3.4890999084961649 \cdot 10^{-238}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{elif}\;re \le 1.0888489179947855 \cdot 10^{-216}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 3.48696801776793644 \cdot 10^{141}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

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 <= -2.3080563981284436e+37)) {
		VAR = ((double) (-1.0 * re));
	} else {
		double VAR_1;
		if ((re <= -3.489099908496165e-238)) {
			VAR_1 = ((double) sqrt(((double) (((double) (re * re)) + ((double) (im * im))))));
		} else {
			double VAR_2;
			if ((re <= 1.0888489179947855e-216)) {
				VAR_2 = im;
			} else {
				double VAR_3;
				if ((re <= 3.4869680177679364e+141)) {
					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 < -2.3080563981284436e+37

   1. Initial program 42.2

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

   2. Taylor expanded around -inf 13.5

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

   if -2.3080563981284436e+37 < re < -3.489099908496165e-238 or 1.0888489179947855e-216 < re < 3.4869680177679364e+141

   1. Initial program 19.5

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

   if -3.489099908496165e-238 < re < 1.0888489179947855e-216

   1. Initial program 31.0

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

   2. Taylor expanded around 0 33.3

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

   if 3.4869680177679364e+141 < re

   1. Initial program 59.6

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

   2. Taylor expanded around inf 8.1

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

4. Final simplification 18.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -2.30805639812844361 \cdot 10^{37}:\\ \;\;\;\;-1 \cdot re\\ \mathbf{elif}\;re \le -3.4890999084961649 \cdot 10^{-238}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{elif}\;re \le 1.0888489179947855 \cdot 10^{-216}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 3.48696801776793644 \cdot 10^{141}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Reproduce

herbie shell --seed 2020113 
(FPCore (re im)
  :name "math.abs on complex"
  :precision binary64
  (sqrt (+ (* re re) (* im im))))