math.abs on complex

Average Error: 31.2 → 18.0

Time: 2.5s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \leq -7.520886186082616 \cdot 10^{+119}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \leq -5.274722593747874 \cdot 10^{-207}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{elif}\;re \leq 1.259166188322571 \cdot 10^{-253}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \leq 1.5992908826155528 \cdot 10^{+115}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

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 <= -7.520886186082616e+119)) {
		VAR = ((double) -(re));
	} else {
		double VAR_1;
		if ((re <= -5.274722593747874e-207)) {
			VAR_1 = ((double) sqrt(((double) (((double) (re * re)) + ((double) (im * im))))));
		} else {
			double VAR_2;
			if ((re <= 1.259166188322571e-253)) {
				VAR_2 = im;
			} else {
				double VAR_3;
				if ((re <= 1.5992908826155528e+115)) {
					VAR_3 = ((double) sqrt(((double) (((double) (re * re)) + ((double) (im * im))))));
				} else {
					VAR_3 = re;
				}
				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 < -7.5208861860826161e119

   1. Initial program Error: 54.0 bits

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

   2. Taylor expanded around -inf Error: 8.9 bits

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

   3. Simplified Error: 8.9 bits

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

   if -7.5208861860826161e119 < re < -5.27472259374787428e-207 or 1.25916618832257105e-253 < re < 1.59929088261555279e115

   1. Initial program Error: 19.4 bits

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

   if -5.27472259374787428e-207 < re < 1.25916618832257105e-253

   1. Initial program Error: 29.8 bits

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

   2. Taylor expanded around 0 Error: 33.6 bits

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

   if 1.59929088261555279e115 < re

   1. Initial program Error: 53.9 bits

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

   2. Taylor expanded around inf Error: 9.1 bits

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

4. Final simplification Error: 18.0 bits

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -7.520886186082616 \cdot 10^{+119}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \leq -5.274722593747874 \cdot 10^{-207}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{elif}\;re \leq 1.259166188322571 \cdot 10^{-253}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \leq 1.5992908826155528 \cdot 10^{+115}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Reproduce

herbie shell --seed 2020200 
(FPCore (re im)
  :name "math.abs on complex"
  :precision binary64
  (sqrt (+ (* re re) (* im im))))