math.abs on complex

Average Error: 31.4 → 17.1

Time: 838.0ms

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \le -1.42313293869710733 \cdot 10^{141}:\\ \;\;\;\;-1 \cdot re\\ \mathbf{elif}\;re \le 2.55702876926703994 \cdot 10^{101}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

C

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

↓

double code(double re, double im) {
	double temp;
	if ((re <= -1.4231329386971073e+141)) {
		temp = (-1.0 * re);
	} else {
		double temp_1;
		if ((re <= 2.55702876926704e+101)) {
			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 < -1.4231329386971073e+141

   1. Initial program 60.5

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

   2. Taylor expanded around -inf 8.0

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

   if -1.4231329386971073e+141 < re < 2.55702876926704e+101

   1. Initial program 20.8

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

   if 2.55702876926704e+101 < re

   1. Initial program 51.6

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

   2. Taylor expanded around inf 9.2

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

4. Final simplification 17.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.42313293869710733 \cdot 10^{141}:\\ \;\;\;\;-1 \cdot re\\ \mathbf{elif}\;re \le 2.55702876926703994 \cdot 10^{101}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Reproduce

herbie shell --seed 2020066 
(FPCore (re im)
  :name "math.abs on complex"
  :precision binary64
  (sqrt (+ (* re re) (* im im))))