math.abs on complex

Average Error: 31.6 → 18.6

Time: 871.0ms

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \le -3.217573693982948 \cdot 10^{127}:\\ \;\;\;\;-1 \cdot re\\ \mathbf{elif}\;re \le 3.1105424028317862 \cdot 10^{46}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

C

double code(double re, double im) {
	return sqrt(((re * re) + (im * im)));
}

↓

double code(double re, double im) {
	double VAR;
	if ((re <= -3.217573693982948e+127)) {
		VAR = (-1.0 * re);
	} else {
		double VAR_1;
		if ((re <= 3.110542402831786e+46)) {
			VAR_1 = sqrt(((re * re) + (im * im)));
		} else {
			VAR_1 = re;
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus re

Bits error versus im

Derivation

1. Split input into 3 regimes

2. if re < -3.217573693982948e+127

   1. Initial program 56.7

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

   2. Taylor expanded around -inf 8.8

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

   if -3.217573693982948e+127 < re < 3.110542402831786e+46

   1. Initial program 22.2

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

   if 3.110542402831786e+46 < re

   1. Initial program 43.8

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

   2. Taylor expanded around inf 13.9

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

4. Final simplification 18.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -3.217573693982948 \cdot 10^{127}:\\ \;\;\;\;-1 \cdot re\\ \mathbf{elif}\;re \le 3.1105424028317862 \cdot 10^{46}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Reproduce

herbie shell --seed 2020071 
(FPCore (re im)
  :name "math.abs on complex"
  :precision binary64
  (sqrt (+ (* re re) (* im im))))