math.abs on complex

Average Error: 31.4 → 18.4

Time: 2.0s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \le -1.6998950373020774 \cdot 10^{87}:\\ \;\;\;\;-1 \cdot re\\ \mathbf{elif}\;re \le -1.1243171280702968 \cdot 10^{-257}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{elif}\;re \le 2.4669184868707975 \cdot 10^{-173}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 1.127182207522478 \cdot 10^{132}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

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 <= -1.6998950373020774e+87)) {
		VAR = ((double) (-1.0 * re));
	} else {
		double VAR_1;
		if ((re <= -1.1243171280702968e-257)) {
			VAR_1 = ((double) sqrt(((double) (((double) (re * re)) + ((double) (im * im))))));
		} else {
			double VAR_2;
			if ((re <= 2.4669184868707975e-173)) {
				VAR_2 = im;
			} else {
				double VAR_3;
				if ((re <= 1.127182207522478e+132)) {
					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 < -1.6998950373020774e87

   1. Initial program 48.8

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

   2. Taylor expanded around -inf 10.5

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

   if -1.6998950373020774e87 < re < -1.1243171280702968e-257 or 2.4669184868707975e-173 < re < 1.127182207522478e132

   1. Initial program 19.1

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

   if -1.1243171280702968e-257 < re < 2.4669184868707975e-173

   1. Initial program 30.4

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

   2. Taylor expanded around 0 34.0

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

   if 1.127182207522478e132 < re

   1. Initial program 58.0

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

   2. Taylor expanded around inf 8.1

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

4. Final simplification 18.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.6998950373020774 \cdot 10^{87}:\\ \;\;\;\;-1 \cdot re\\ \mathbf{elif}\;re \le -1.1243171280702968 \cdot 10^{-257}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{elif}\;re \le 2.4669184868707975 \cdot 10^{-173}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 1.127182207522478 \cdot 10^{132}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Reproduce

herbie shell --seed 2020162 
(FPCore (re im)
  :name "math.abs on complex"
  :precision binary64
  (sqrt (+ (* re re) (* im im))))