math.log/1 on complex, real part

Average Error: 31.8 → 18.1

Time: 1.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \le -1.10321569695692608 \cdot 10^{72}:\\ \;\;\;\;\log \left(-1 \cdot re\right)\\ \mathbf{elif}\;re \le -1.3504253849915568 \cdot 10^{-194}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \le -2.968956980813959 \cdot 10^{-266}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 6.80536176757501775 \cdot 10^{-229}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \le 1.30573406095301773 \cdot 10^{-191}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 5.15621950091572796 \cdot 10^{39}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

C

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

↓

double code(double re, double im) {
	double temp;
	if ((re <= -1.103215696956926e+72)) {
		temp = log((-1.0 * re));
	} else {
		double temp_1;
		if ((re <= -1.3504253849915568e-194)) {
			temp_1 = log(sqrt(((re * re) + (im * im))));
		} else {
			double temp_2;
			if ((re <= -2.968956980813959e-266)) {
				temp_2 = log(im);
			} else {
				double temp_3;
				if ((re <= 6.805361767575018e-229)) {
					temp_3 = log(sqrt(((re * re) + (im * im))));
				} else {
					double temp_4;
					if ((re <= 1.3057340609530177e-191)) {
						temp_4 = log(im);
					} else {
						double temp_5;
						if ((re <= 5.156219500915728e+39)) {
							temp_5 = log(sqrt(((re * re) + (im * im))));
						} else {
							temp_5 = log(re);
						}
						temp_4 = temp_5;
					}
					temp_3 = temp_4;
				}
				temp_2 = temp_3;
			}
			temp_1 = temp_2;
		}
		temp = temp_1;
	}
	return temp;
}

Error

Bits error versus re

Bits error versus im

Derivation

1. Split input into 4 regimes

2. if re < -1.103215696956926e+72

   1. Initial program 46.8

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

   2. Taylor expanded around -inf 9.3

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

   if -1.103215696956926e+72 < re < -1.3504253849915568e-194 or -2.968956980813959e-266 < re < 6.805361767575018e-229 or 1.3057340609530177e-191 < re < 5.156219500915728e+39

   1. Initial program 21.1

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

   if -1.3504253849915568e-194 < re < -2.968956980813959e-266 or 6.805361767575018e-229 < re < 1.3057340609530177e-191

   1. Initial program 32.2

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

   2. Taylor expanded around 0 34.8

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

   if 5.156219500915728e+39 < re

   1. Initial program 43.9

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

   2. Taylor expanded around inf 11.6

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

4. Final simplification 18.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.10321569695692608 \cdot 10^{72}:\\ \;\;\;\;\log \left(-1 \cdot re\right)\\ \mathbf{elif}\;re \le -1.3504253849915568 \cdot 10^{-194}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \le -2.968956980813959 \cdot 10^{-266}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 6.80536176757501775 \cdot 10^{-229}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \le 1.30573406095301773 \cdot 10^{-191}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 5.15621950091572796 \cdot 10^{39}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Reproduce

herbie shell --seed 2020057 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  :precision binary64
  (log (sqrt (+ (* re re) (* im im)))))