math.abs on complex

Average Error: 31.6 → 17.9

Time: 2.6s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \le -9.340803523077166 \cdot 10^{+139}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le -4.786656859853452 \cdot 10^{-266}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{elif}\;re \le -2.7978321376001454 \cdot 10^{-306}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 1.2882041961296455 \cdot 10^{+92}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

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 <= -9.340803523077166e+139)) {
		VAR = ((double) -(re));
	} else {
		double VAR_1;
		if ((re <= -4.786656859853452e-266)) {
			VAR_1 = ((double) sqrt(((double) (((double) (re * re)) + ((double) (im * im))))));
		} else {
			double VAR_2;
			if ((re <= -2.7978321376001454e-306)) {
				VAR_2 = im;
			} else {
				double VAR_3;
				if ((re <= 1.2882041961296455e+92)) {
					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 < -9.3408035230771657e139

   1. Initial program 60.5

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

   2. Taylor expanded around -inf 8.4

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

   3. Simplified 8.4

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

   if -9.3408035230771657e139 < re < -4.7866568598534521e-266 or -2.79783213760014545e-306 < re < 1.28820419612964551e92

   1. Initial program 20.8

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

   if -4.7866568598534521e-266 < re < -2.79783213760014545e-306

   1. Initial program 31.2

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

   2. Taylor expanded around 0 31.5

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

   if 1.28820419612964551e92 < re

   1. Initial program 50.9

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

   2. Taylor expanded around inf 11.5

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

4. Final simplification 17.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -9.340803523077166 \cdot 10^{+139}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le -4.786656859853452 \cdot 10^{-266}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{elif}\;re \le -2.7978321376001454 \cdot 10^{-306}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 1.2882041961296455 \cdot 10^{+92}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Reproduce

herbie shell --seed 2020184 
(FPCore (re im)
  :name "math.abs on complex"
  :precision binary64
  (sqrt (+ (* re re) (* im im))))