math.abs on complex

Average Error: 31.3 → 17.2

Time: 1.8s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \le -9.71254003645233016 \cdot 10^{153}:\\ \;\;\;\;-1 \cdot re\\ \mathbf{elif}\;re \le 4.8567367746758544 \cdot 10^{114}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

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 <= -9.71254003645233e+153)) {
		VAR = ((double) (-1.0 * re));
	} else {
		double VAR_1;
		if ((re <= 4.8567367746758544e+114)) {
			VAR_1 = ((double) sqrt(((double) (((double) (re * re)) + ((double) (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 < -9.71254003645233016e153

   1. Initial program 64.0

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

   2. Taylor expanded around -inf 8.9

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

   if -9.71254003645233016e153 < re < 4.8567367746758544e114

   1. Initial program 20.4

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

   if 4.8567367746758544e114 < re

   1. Initial program 53.9

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

   2. Taylor expanded around inf 8.3

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

4. Final simplification 17.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -9.71254003645233016 \cdot 10^{153}:\\ \;\;\;\;-1 \cdot re\\ \mathbf{elif}\;re \le 4.8567367746758544 \cdot 10^{114}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Reproduce

herbie shell --seed 2020157 
(FPCore (re im)
  :name "math.abs on complex"
  :precision binary64
  (sqrt (+ (* re re) (* im im))))