math.abs on complex

Average Error: 31.7 → 19.0

Time: 2.2s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \le -2.16131408325038182 \cdot 10^{112}:\\ \;\;\;\;-1 \cdot re\\ \mathbf{elif}\;re \le -1.1454886226922574 \cdot 10^{-300}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{elif}\;re \le 1.8543864403264038 \cdot 10^{-255}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 2.601933874540376 \cdot 10^{-42}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{elif}\;re \le 548.395158744075729:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 1.9954821974516343 \cdot 10^{99}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

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 <= -2.161314083250382e+112)) {
		VAR = ((double) (-1.0 * re));
	} else {
		double VAR_1;
		if ((re <= -1.1454886226922574e-300)) {
			VAR_1 = ((double) sqrt(((double) (((double) (re * re)) + ((double) (im * im))))));
		} else {
			double VAR_2;
			if ((re <= 1.8543864403264038e-255)) {
				VAR_2 = im;
			} else {
				double VAR_3;
				if ((re <= 2.601933874540376e-42)) {
					VAR_3 = ((double) sqrt(((double) (((double) (re * re)) + ((double) (im * im))))));
				} else {
					double VAR_4;
					if ((re <= 548.3951587440757)) {
						VAR_4 = im;
					} else {
						double VAR_5;
						if ((re <= 1.9954821974516343e+99)) {
							VAR_5 = ((double) sqrt(((double) (((double) (re * re)) + ((double) (im * im))))));
						} else {
							VAR_5 = re;
						}
						VAR_4 = VAR_5;
					}
					VAR_3 = VAR_4;
				}
				VAR_2 = VAR_3;
			}
			VAR_1 = VAR_2;
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus re

Bits error versus im

Derivation

1. Split input into 4 regimes

2. if re < -2.16131408325038182e112

   1. Initial program 54.0

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

   2. Taylor expanded around -inf 9.0

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

   if -2.16131408325038182e112 < re < -1.1454886226922574e-300 or 1.8543864403264038e-255 < re < 2.601933874540376e-42 or 548.395158744075729 < re < 1.9954821974516343e99

   1. Initial program 21.1

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

   if -1.1454886226922574e-300 < re < 1.8543864403264038e-255 or 2.601933874540376e-42 < re < 548.395158744075729

   1. Initial program 22.9

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

   2. Taylor expanded around 0 39.3

      \[\leadsto \color{blue}{im}\]

   if 1.9954821974516343e99 < re

   1. Initial program 51.5

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

   2. Taylor expanded around inf 10.9

      \[\leadsto \color{blue}{re}\]
3. Recombined 4 regimes into one program.

4. Final simplification 19.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -2.16131408325038182 \cdot 10^{112}:\\ \;\;\;\;-1 \cdot re\\ \mathbf{elif}\;re \le -1.1454886226922574 \cdot 10^{-300}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{elif}\;re \le 1.8543864403264038 \cdot 10^{-255}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 2.601933874540376 \cdot 10^{-42}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{elif}\;re \le 548.395158744075729:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 1.9954821974516343 \cdot 10^{99}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Reproduce

herbie shell --seed 2020175 
(FPCore (re im)
  :name "math.abs on complex"
  :precision binary64
  (sqrt (+ (* re re) (* im im))))