Average Error: 31.8 → 17.3
Time: 1.2s
Precision: 64
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -9.4385800746292579 \cdot 10^{52}:\\ \;\;\;\;\log \left(-1 \cdot re\right)\\ \mathbf{elif}\;re \le -5.96995186281869427 \cdot 10^{-267}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \le 9.67584874540543507 \cdot 10^{-216}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 3.7629938528630668 \cdot 10^{58}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]
\log \left(\sqrt{re \cdot re + im \cdot im}\right)
↓
\begin{array}{l}
\mathbf{if}\;re \le -9.4385800746292579 \cdot 10^{52}:\\
\;\;\;\;\log \left(-1 \cdot re\right)\\

\mathbf{elif}\;re \le -5.96995186281869427 \cdot 10^{-267}:\\
\;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\

\mathbf{elif}\;re \le 9.67584874540543507 \cdot 10^{-216}:\\
\;\;\;\;\log im\\

\mathbf{elif}\;re \le 3.7629938528630668 \cdot 10^{58}:\\
\;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\

\mathbf{else}:\\
\;\;\;\;\log re\\

\end{array}
double code(double re, double im) {
	return log(sqrt(((re * re) + (im * im))));
}
↓
double code(double re, double im) {
	double temp;
	if ((re <= -9.438580074629258e+52)) {
		temp = log((-1.0 * re));
	} else {
		double temp_1;
		if ((re <= -5.969951862818694e-267)) {
			temp_1 = log(sqrt(((re * re) + (im * im))));
		} else {
			double temp_2;
			if ((re <= 9.675848745405435e-216)) {
				temp_2 = log(im);
			} else {
				double temp_3;
				if ((re <= 3.762993852863067e+58)) {
					temp_3 = log(sqrt(((re * re) + (im * im))));
				} else {
					temp_3 = log(re);
				}
				temp_2 = temp_3;
			}
			temp_1 = temp_2;
		}
		temp = temp_1;
	}
	return temp;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 4 regimes

  2. if re < -9.438580074629258e+52

    1. Initial program 45.8

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

    2. Taylor expanded around -inf 10.7

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

    if -9.438580074629258e+52 < re < -5.969951862818694e-267 or 9.675848745405435e-216 < re < 3.762993852863067e+58

    1. Initial program 19.8

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

    if -5.969951862818694e-267 < re < 9.675848745405435e-216

    1. Initial program 31.3

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

    2. Taylor expanded around 0 31.9

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

    if 3.762993852863067e+58 < re

    1. Initial program 45.8

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

    2. Taylor expanded around inf 10.5

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

  4. Final simplification17.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -9.4385800746292579 \cdot 10^{52}:\\ \;\;\;\;\log \left(-1 \cdot re\right)\\ \mathbf{elif}\;re \le -5.96995186281869427 \cdot 10^{-267}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \le 9.67584874540543507 \cdot 10^{-216}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 3.7629938528630668 \cdot 10^{58}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Reproduce

herbie shell --seed 2020058 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  :precision binary64
  (log (sqrt (+ (* re re) (* im im)))))