math.abs on complex

Average Error: 32.0 → 17.9

Time: 3.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \le -1.3830219520310334 \cdot 10^{128}:\\ \;\;\;\;-1 \cdot re\\ \mathbf{elif}\;re \le 6.0975882896256911 \cdot 10^{71}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

C

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

↓

double code(double re, double im) {
	double VAR;
	if ((re <= -1.3830219520310334e+128)) {
		VAR = (-1.0 * re);
	} else {
		double VAR_1;
		if ((re <= 6.097588289625691e+71)) {
			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 < -1.3830219520310334e+128

   1. Initial program 57.5

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

   2. Taylor expanded around -inf 9.6

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

   if -1.3830219520310334e+128 < re < 6.097588289625691e+71

   1. Initial program 21.6

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

   if 6.097588289625691e+71 < re

   1. Initial program 47.2

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

   2. Taylor expanded around inf 11.8

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

4. Final simplification 17.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.3830219520310334 \cdot 10^{128}:\\ \;\;\;\;-1 \cdot re\\ \mathbf{elif}\;re \le 6.0975882896256911 \cdot 10^{71}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Reproduce

herbie shell --seed 2020103 
(FPCore (re im)
  :name "math.abs on complex"
  :precision binary64
  (sqrt (+ (* re re) (* im im))))