math.abs on complex

Average Error: 31.3 → 17.8

Time: 1.5s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \leq -6.39388144865274 \cdot 10^{+123}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \leq -2.2949124501890465 \cdot 10^{-243}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{elif}\;re \leq 1.0223475346325806 \cdot 10^{-246}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \leq 2.597914161920686 \cdot 10^{+74}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

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 <= -6.39388144865274e+123)) {
		VAR = ((double) -(re));
	} else {
		double VAR_1;
		if ((re <= -2.2949124501890465e-243)) {
			VAR_1 = ((double) sqrt(((double) (((double) (re * re)) + ((double) (im * im))))));
		} else {
			double VAR_2;
			if ((re <= 1.0223475346325806e-246)) {
				VAR_2 = im;
			} else {
				double VAR_3;
				if ((re <= 2.597914161920686e+74)) {
					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 < -6.39388144865274029e123

   1. Initial program 56.7

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

   2. Taylor expanded around -inf 8.5

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

   3. Simplified 8.5

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

   if -6.39388144865274029e123 < re < -2.2949124501890465e-243 or 1.02234753463258063e-246 < re < 2.59791416192068604e74

   1. Initial program 19.7

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

   if -2.2949124501890465e-243 < re < 1.02234753463258063e-246

   1. Initial program 30.1

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

   2. Taylor expanded around 0 33.1

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

   if 2.59791416192068604e74 < re

   1. Initial program 47.0

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

   2. Taylor expanded around inf 10.9

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

4. Final simplification 17.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -6.39388144865274 \cdot 10^{+123}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \leq -2.2949124501890465 \cdot 10^{-243}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{elif}\;re \leq 1.0223475346325806 \cdot 10^{-246}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \leq 2.597914161920686 \cdot 10^{+74}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Reproduce

herbie shell --seed 2020196 
(FPCore (re im)
  :name "math.abs on complex"
  :precision binary64
  (sqrt (+ (* re re) (* im im))))