math.log/1 on complex, real part

Average Error: 32.1 → 18.2

Time: 1.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \le -6.6680059753335194 \cdot 10^{135}:\\ \;\;\;\;\log \left(-1 \cdot re\right)\\ \mathbf{elif}\;re \le -1.0951361490029777 \cdot 10^{-160}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \le -1.4034448373480268 \cdot 10^{-212}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 5.0001842566281268 \cdot 10^{-304}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \le 7.54212455848276206 \cdot 10^{-167}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 5.111282847825261 \cdot 10^{121}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

C

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

↓

double code(double re, double im) {
	double VAR;
	if ((re <= -6.66800597533352e+135)) {
		VAR = log((-1.0 * re));
	} else {
		double VAR_1;
		if ((re <= -1.0951361490029777e-160)) {
			VAR_1 = log(sqrt(((re * re) + (im * im))));
		} else {
			double VAR_2;
			if ((re <= -1.4034448373480268e-212)) {
				VAR_2 = log(im);
			} else {
				double VAR_3;
				if ((re <= 5.000184256628127e-304)) {
					VAR_3 = log(sqrt(((re * re) + (im * im))));
				} else {
					double VAR_4;
					if ((re <= 7.542124558482762e-167)) {
						VAR_4 = log(im);
					} else {
						double VAR_5;
						if ((re <= 5.111282847825261e+121)) {
							VAR_5 = log(sqrt(((re * re) + (im * im))));
						} else {
							VAR_5 = log(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 < -6.66800597533352e+135

   1. Initial program 59.8

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

   2. Taylor expanded around -inf 8.1

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

   if -6.66800597533352e+135 < re < -1.0951361490029777e-160 or -1.4034448373480268e-212 < re < 5.000184256628127e-304 or 7.542124558482762e-167 < re < 5.111282847825261e+121

   1. Initial program 18.8

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

   if -1.0951361490029777e-160 < re < -1.4034448373480268e-212 or 5.000184256628127e-304 < re < 7.542124558482762e-167

   1. Initial program 31.4

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

   2. Taylor expanded around 0 34.1

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

   if 5.111282847825261e+121 < re

   1. Initial program 56.3

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

   2. Taylor expanded around inf 8.2

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

4. Final simplification 18.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -6.6680059753335194 \cdot 10^{135}:\\ \;\;\;\;\log \left(-1 \cdot re\right)\\ \mathbf{elif}\;re \le -1.0951361490029777 \cdot 10^{-160}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \le -1.4034448373480268 \cdot 10^{-212}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 5.0001842566281268 \cdot 10^{-304}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \le 7.54212455848276206 \cdot 10^{-167}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 5.111282847825261 \cdot 10^{121}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Reproduce

herbie shell --seed 2020078 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  :precision binary64
  (log (sqrt (+ (* re re) (* im im)))))