math.abs on complex

Average Error: 31.5 → 17.6

Time: 2.6s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \le -3.5104348785110151 \cdot 10^{151}:\\ \;\;\;\;-1 \cdot re\\ \mathbf{elif}\;re \le 1.6805901359658878 \cdot 10^{131}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

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 <= -3.510434878511015e+151)) {
		VAR = ((double) (-1.0 * re));
	} else {
		double VAR_1;
		if ((re <= 1.6805901359658878e+131)) {
			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 < -3.5104348785110151e151

   1. Initial program 63.2

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

   2. Taylor expanded around -inf 8.2

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

   if -3.5104348785110151e151 < re < 1.6805901359658878e131

   1. Initial program 20.9

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

   if 1.6805901359658878e131 < re

   1. Initial program 57.8

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

   2. Taylor expanded around inf 9.1

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

4. Final simplification 17.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -3.5104348785110151 \cdot 10^{151}:\\ \;\;\;\;-1 \cdot re\\ \mathbf{elif}\;re \le 1.6805901359658878 \cdot 10^{131}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Reproduce

herbie shell --seed 2020171 
(FPCore (re im)
  :name "math.abs on complex"
  :precision binary64
  (sqrt (+ (* re re) (* im im))))