math.abs on complex

Average Error: 31.8 → 18.2

Time: 1.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \le -3.47329147792522594 \cdot 10^{143}:\\ \;\;\;\;-1 \cdot re\\ \mathbf{elif}\;re \le 2.3500620374848598 \cdot 10^{-289}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{elif}\;re \le 2.25831678575537869 \cdot 10^{-213}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 8.3961642544830301 \cdot 10^{66}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

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 <= -3.473291477925226e+143)) {
		VAR = ((double) (-1.0 * re));
	} else {
		double VAR_1;
		if ((re <= 2.3500620374848598e-289)) {
			VAR_1 = ((double) sqrt(((double) (((double) (re * re)) + ((double) (im * im))))));
		} else {
			double VAR_2;
			if ((re <= 2.2583167857553787e-213)) {
				VAR_2 = im;
			} else {
				double VAR_3;
				if ((re <= 8.39616425448303e+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 < -3.473291477925226e+143

   1. Initial program 60.9

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

   2. Taylor expanded around -inf 8.3

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

   if -3.473291477925226e+143 < re < 2.3500620374848598e-289 or 2.2583167857553787e-213 < re < 8.39616425448303e+66

   1. Initial program 20.9

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

   if 2.3500620374848598e-289 < re < 2.2583167857553787e-213

   1. Initial program 30.2

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

   2. Taylor expanded around 0 33.3

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

   if 8.39616425448303e+66 < re

   1. Initial program 47.1

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

   2. Taylor expanded around inf 11.9

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

4. Final simplification 18.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -3.47329147792522594 \cdot 10^{143}:\\ \;\;\;\;-1 \cdot re\\ \mathbf{elif}\;re \le 2.3500620374848598 \cdot 10^{-289}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{elif}\;re \le 2.25831678575537869 \cdot 10^{-213}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 8.3961642544830301 \cdot 10^{66}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Reproduce

herbie shell --seed 2020126 
(FPCore (re im)
  :name "math.abs on complex"
  :precision binary64
  (sqrt (+ (* re re) (* im im))))