math.log/1 on complex, real part

Average Error: 31.8 → 18.0

Time: 1.9s

Precision: binary64

Math

\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]

↓

\[\begin{array}{l} \mathbf{if}\;re \le -2.6123365819456626 \cdot 10^{118}:\\ \;\;\;\;\log \left(-1 \cdot re\right)\\ \mathbf{elif}\;re \le -2.263966836247413 \cdot 10^{-261}:\\ \;\;\;\;\frac{1}{2} \cdot \log \left(re \cdot re + im \cdot im\right)\\ \mathbf{elif}\;re \le 9.11467363518384614 \cdot 10^{-161}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 1.5670414452839892 \cdot 10^{131}:\\ \;\;\;\;\frac{1}{2} \cdot \log \left(re \cdot re + im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

C

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

↓

double code(double re, double im) {
	double VAR;
	if ((re <= -2.6123365819456626e+118)) {
		VAR = ((double) log(((double) (-1.0 * re))));
	} else {
		double VAR_1;
		if ((re <= -2.263966836247413e-261)) {
			VAR_1 = ((double) (0.5 * ((double) log(((double) (((double) (re * re)) + ((double) (im * im))))))));
		} else {
			double VAR_2;
			if ((re <= 9.114673635183846e-161)) {
				VAR_2 = ((double) log(im));
			} else {
				double VAR_3;
				if ((re <= 1.5670414452839892e+131)) {
					VAR_3 = ((double) (0.5 * ((double) log(((double) (((double) (re * re)) + ((double) (im * im))))))));
				} else {
					VAR_3 = ((double) log(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 < -2.6123365819456626e118

   1. Initial program 55.7

      \[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]

   2. Taylor expanded around -inf 7.7

      \[\leadsto \log \color{blue}{\left(-1 \cdot re\right)}\]

   if -2.6123365819456626e118 < re < -2.263966836247413e-261 or 9.11467363518384614e-161 < re < 1.5670414452839892e131

   1. Initial program 18.2

      \[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
   2. Using strategy rm

   3. Applied pow1/2 18.2

      \[\leadsto \log \color{blue}{\left({\left(re \cdot re + im \cdot im\right)}^{\frac{1}{2}}\right)}\]

   4. Applied log-pow 18.2

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \log \left(re \cdot re + im \cdot im\right)}\]

   if -2.263966836247413e-261 < re < 9.11467363518384614e-161

   1. Initial program 31.3

      \[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]

   2. Taylor expanded around 0 35.9

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

   if 1.5670414452839892e131 < re

   1. Initial program 57.6

      \[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]

   2. Taylor expanded around inf 8.6

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

4. Final simplification 18.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -2.6123365819456626 \cdot 10^{118}:\\ \;\;\;\;\log \left(-1 \cdot re\right)\\ \mathbf{elif}\;re \le -2.263966836247413 \cdot 10^{-261}:\\ \;\;\;\;\frac{1}{2} \cdot \log \left(re \cdot re + im \cdot im\right)\\ \mathbf{elif}\;re \le 9.11467363518384614 \cdot 10^{-161}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 1.5670414452839892 \cdot 10^{131}:\\ \;\;\;\;\frac{1}{2} \cdot \log \left(re \cdot re + im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Reproduce

herbie shell --seed 2020161 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  :precision binary64
  (log (sqrt (+ (* re re) (* im im)))))