math.abs on complex

Average Error: 32.0 → 18.2

Time: 2.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \le -3.08310863609937876 \cdot 10^{138}:\\ \;\;\;\;-1 \cdot re\\ \mathbf{elif}\;re \le -1.1813831187355925 \cdot 10^{-160}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{elif}\;re \le -1.4034448373480268 \cdot 10^{-212}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 6.33411590852480896 \cdot 10^{-304}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{elif}\;re \le 5.55126095100350983 \cdot 10^{-260}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 9.23378212571996135 \cdot 10^{120}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

C

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

↓

double code(double re, double im) {
	double VAR;
	if ((re <= -3.083108636099379e+138)) {
		VAR = (-1.0 * re);
	} else {
		double VAR_1;
		if ((re <= -1.1813831187355925e-160)) {
			VAR_1 = sqrt(((re * re) + (im * im)));
		} else {
			double VAR_2;
			if ((re <= -1.4034448373480268e-212)) {
				VAR_2 = im;
			} else {
				double VAR_3;
				if ((re <= 6.334115908524809e-304)) {
					VAR_3 = sqrt(((re * re) + (im * im)));
				} else {
					double VAR_4;
					if ((re <= 5.55126095100351e-260)) {
						VAR_4 = im;
					} else {
						double VAR_5;
						if ((re <= 9.233782125719961e+120)) {
							VAR_5 = sqrt(((re * re) + (im * im)));
						} else {
							VAR_5 = re;
						}
						VAR_4 = VAR_5;
					}
					VAR_3 = VAR_4;
				}
				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 < -3.083108636099379e+138

   1. Initial program 60.4

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

   2. Taylor expanded around -inf 9.5

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

   if -3.083108636099379e+138 < re < -1.1813831187355925e-160 or -1.4034448373480268e-212 < re < 6.334115908524809e-304 or 5.55126095100351e-260 < re < 9.233782125719961e+120

   1. Initial program 20.2

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

   if -1.1813831187355925e-160 < re < -1.4034448373480268e-212 or 6.334115908524809e-304 < re < 5.55126095100351e-260

   1. Initial program 31.1

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

   2. Taylor expanded around 0 34.5

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

   if 9.233782125719961e+120 < re

   1. Initial program 56.0

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

   2. Taylor expanded around inf 9.8

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

4. Final simplification 18.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -3.08310863609937876 \cdot 10^{138}:\\ \;\;\;\;-1 \cdot re\\ \mathbf{elif}\;re \le -1.1813831187355925 \cdot 10^{-160}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{elif}\;re \le -1.4034448373480268 \cdot 10^{-212}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 6.33411590852480896 \cdot 10^{-304}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{elif}\;re \le 5.55126095100350983 \cdot 10^{-260}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 9.23378212571996135 \cdot 10^{120}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Reproduce

herbie shell --seed 2020078 
(FPCore (re im)
  :name "math.abs on complex"
  :precision binary64
  (sqrt (+ (* re re) (* im im))))