math.log/1 on complex, real part

Average Error: 32.0 → 18.6

Time: 1.4s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \le -8633833088570918:\\ \;\;\;\;\log \left(-1 \cdot re\right)\\ \mathbf{elif}\;re \le -2.53084796457160171 \cdot 10^{-161}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \le -3.05336089420992703 \cdot 10^{-237}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 1.67464630382278954 \cdot 10^{-295}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \le 4.27713989863704614 \cdot 10^{-212}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 2.1568457595562577 \cdot 10^{61}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

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 <= -8633833088570918.0)) {
		VAR = ((double) log(((double) (-1.0 * re))));
	} else {
		double VAR_1;
		if ((re <= -2.5308479645716017e-161)) {
			VAR_1 = ((double) log(((double) sqrt(((double) (((double) (re * re)) + ((double) (im * im))))))));
		} else {
			double VAR_2;
			if ((re <= -3.053360894209927e-237)) {
				VAR_2 = ((double) log(im));
			} else {
				double VAR_3;
				if ((re <= 1.6746463038227895e-295)) {
					VAR_3 = ((double) log(((double) sqrt(((double) (((double) (re * re)) + ((double) (im * im))))))));
				} else {
					double VAR_4;
					if ((re <= 4.277139898637046e-212)) {
						VAR_4 = ((double) log(im));
					} else {
						double VAR_5;
						if ((re <= 2.1568457595562577e+61)) {
							VAR_5 = ((double) log(((double) sqrt(((double) (((double) (re * re)) + ((double) (im * im))))))));
						} else {
							VAR_5 = ((double) 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 < -8633833088570918.0

   1. Initial program 41.4

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

   2. Taylor expanded around -inf 12.2

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

   if -8633833088570918.0 < re < -2.5308479645716017e-161 or -3.053360894209927e-237 < re < 1.6746463038227895e-295 or 4.277139898637046e-212 < re < 2.1568457595562577e+61

   1. Initial program 20.8

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

   if -2.5308479645716017e-161 < re < -3.053360894209927e-237 or 1.6746463038227895e-295 < re < 4.277139898637046e-212

   1. Initial program 31.1

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

   2. Taylor expanded around 0 35.0

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

   if 2.1568457595562577e+61 < re

   1. Initial program 46.3

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

   2. Taylor expanded around inf 10.5

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

4. Final simplification 18.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -8633833088570918:\\ \;\;\;\;\log \left(-1 \cdot re\right)\\ \mathbf{elif}\;re \le -2.53084796457160171 \cdot 10^{-161}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \le -3.05336089420992703 \cdot 10^{-237}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 1.67464630382278954 \cdot 10^{-295}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \le 4.27713989863704614 \cdot 10^{-212}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 2.1568457595562577 \cdot 10^{61}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Reproduce

herbie shell --seed 2020126 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  :precision binary64
  (log (sqrt (+ (* re re) (* im im)))))