math.abs on complex

Average Error: 31.8 → 18.0

Time: 2.6s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \le -1.1200584461854231 \cdot 10^{85}:\\ \;\;\;\;-1 \cdot re\\ \mathbf{elif}\;re \le -9.40683055972494263 \cdot 10^{-234}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{elif}\;re \le 4.73210160738898182 \cdot 10^{-297}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 1.211139330221451 \cdot 10^{66}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

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 <= -1.120058446185423e+85)) {
		VAR = ((double) (-1.0 * re));
	} else {
		double VAR_1;
		if ((re <= -9.406830559724943e-234)) {
			VAR_1 = ((double) sqrt(((double) (((double) (re * re)) + ((double) (im * im))))));
		} else {
			double VAR_2;
			if ((re <= 4.732101607388982e-297)) {
				VAR_2 = im;
			} else {
				double VAR_3;
				if ((re <= 1.211139330221451e+66)) {
					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 < -1.1200584461854231e85

   1. Initial program 49.9

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

   2. Taylor expanded around -inf 10.8

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

   if -1.1200584461854231e85 < re < -9.40683055972494263e-234 or 4.73210160738898182e-297 < re < 1.211139330221451e66

   1. Initial program 20.4

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

   if -9.40683055972494263e-234 < re < 4.73210160738898182e-297

   1. Initial program 32.9

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

   2. Taylor expanded around 0 34.3

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

   if 1.211139330221451e66 < re

   1. Initial program 47.0

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

   2. Taylor expanded around inf 12.1

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

4. Final simplification 18.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.1200584461854231 \cdot 10^{85}:\\ \;\;\;\;-1 \cdot re\\ \mathbf{elif}\;re \le -9.40683055972494263 \cdot 10^{-234}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{elif}\;re \le 4.73210160738898182 \cdot 10^{-297}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 1.211139330221451 \cdot 10^{66}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Reproduce

herbie shell --seed 2020152 
(FPCore (re im)
  :name "math.abs on complex"
  :precision binary64
  (sqrt (+ (* re re) (* im im))))