math.abs on complex

Average Error: 31.4 → 17.1

Time: 847.0ms

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \le -2.77502973126125612 \cdot 10^{119}:\\ \;\;\;\;-1 \cdot re\\ \mathbf{elif}\;re \le 7.57619582189923613 \cdot 10^{130}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

C

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

↓

double code(double re, double im) {
	double VAR;
	if ((re <= -2.775029731261256e+119)) {
		VAR = (-1.0 * re);
	} else {
		double VAR_1;
		if ((re <= 7.576195821899236e+130)) {
			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 < -2.775029731261256e+119

   1. Initial program 55.4

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

   2. Taylor expanded around -inf 9.0

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

   if -2.775029731261256e+119 < re < 7.576195821899236e+130

   1. Initial program 20.4

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

   if 7.576195821899236e+130 < re

   1. Initial program 57.8

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

   2. Taylor expanded around inf 9.9

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

4. Final simplification 17.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -2.77502973126125612 \cdot 10^{119}:\\ \;\;\;\;-1 \cdot re\\ \mathbf{elif}\;re \le 7.57619582189923613 \cdot 10^{130}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Reproduce

herbie shell --seed 2020091 
(FPCore (re im)
  :name "math.abs on complex"
  :precision binary64
  (sqrt (+ (* re re) (* im im))))