math.abs on complex

Average Error: 31.7 → 17.8

Time: 898.0ms

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \le -1.3108523098680432 \cdot 10^{59}:\\ \;\;\;\;-1 \cdot re\\ \mathbf{elif}\;re \le 3.7629938528630668 \cdot 10^{58}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

C

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

↓

double code(double re, double im) {
	double temp;
	if ((re <= -1.3108523098680432e+59)) {
		temp = (-1.0 * re);
	} else {
		double temp_1;
		if ((re <= 3.762993852863067e+58)) {
			temp_1 = sqrt(((re * re) + (im * im)));
		} else {
			temp_1 = re;
		}
		temp = temp_1;
	}
	return temp;
}

Error

Bits error versus re

Bits error versus im

Derivation

1. Split input into 3 regimes

2. if re < -1.3108523098680432e+59

   1. Initial program 46.3

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

   2. Taylor expanded around -inf 12.2

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

   if -1.3108523098680432e+59 < re < 3.762993852863067e+58

   1. Initial program 21.6

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

   if 3.762993852863067e+58 < re

   1. Initial program 45.8

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

   2. Taylor expanded around inf 12.3

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

4. Final simplification 17.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.3108523098680432 \cdot 10^{59}:\\ \;\;\;\;-1 \cdot re\\ \mathbf{elif}\;re \le 3.7629938528630668 \cdot 10^{58}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Reproduce

herbie shell --seed 2020058 
(FPCore (re im)
  :name "math.abs on complex"
  :precision binary64
  (sqrt (+ (* re re) (* im im))))