math.abs on complex

Average Error: 31.6 → 18.7

Time: 955.0ms

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \le -3.8460535119133569 \cdot 10^{74}:\\ \;\;\;\;-1 \cdot re\\ \mathbf{elif}\;re \le -1.3504253849915568 \cdot 10^{-194}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{elif}\;re \le -2.968956980813959 \cdot 10^{-266}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 1.19099635470288769 \cdot 10^{-228}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{elif}\;re \le 1.08574889376971239 \cdot 10^{-190}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 3.98422560465703889 \cdot 10^{39}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

C

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

↓

double code(double re, double im) {
	double temp;
	if ((re <= -3.846053511913357e+74)) {
		temp = (-1.0 * re);
	} else {
		double temp_1;
		if ((re <= -1.3504253849915568e-194)) {
			temp_1 = sqrt(((re * re) + (im * im)));
		} else {
			double temp_2;
			if ((re <= -2.968956980813959e-266)) {
				temp_2 = im;
			} else {
				double temp_3;
				if ((re <= 1.1909963547028877e-228)) {
					temp_3 = sqrt(((re * re) + (im * im)));
				} else {
					double temp_4;
					if ((re <= 1.0857488937697124e-190)) {
						temp_4 = im;
					} else {
						double temp_5;
						if ((re <= 3.984225604657039e+39)) {
							temp_5 = sqrt(((re * re) + (im * im)));
						} else {
							temp_5 = re;
						}
						temp_4 = temp_5;
					}
					temp_3 = temp_4;
				}
				temp_2 = temp_3;
			}
			temp_1 = temp_2;
		}
		temp = temp_1;
	}
	return temp;
}

Error

Bits error versus re

Bits error versus im

Derivation

1. Split input into 4 regimes

2. if re < -3.846053511913357e+74

   1. Initial program 47.2

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

   2. Taylor expanded around -inf 10.7

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

   if -3.846053511913357e+74 < re < -1.3504253849915568e-194 or -2.968956980813959e-266 < re < 1.1909963547028877e-228 or 1.0857488937697124e-190 < re < 3.984225604657039e+39

   1. Initial program 20.9

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

   if -1.3504253849915568e-194 < re < -2.968956980813959e-266 or 1.1909963547028877e-228 < re < 1.0857488937697124e-190

   1. Initial program 31.4

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

   2. Taylor expanded around 0 34.8

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

   if 3.984225604657039e+39 < re

   1. Initial program 43.8

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

   2. Taylor expanded around inf 13.6

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

4. Final simplification 18.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -3.8460535119133569 \cdot 10^{74}:\\ \;\;\;\;-1 \cdot re\\ \mathbf{elif}\;re \le -1.3504253849915568 \cdot 10^{-194}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{elif}\;re \le -2.968956980813959 \cdot 10^{-266}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 1.19099635470288769 \cdot 10^{-228}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{elif}\;re \le 1.08574889376971239 \cdot 10^{-190}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 3.98422560465703889 \cdot 10^{39}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Reproduce

herbie shell --seed 2020057 
(FPCore (re im)
  :name "math.abs on complex"
  :precision binary64
  (sqrt (+ (* re re) (* im im))))