Average Error: 31.1 → 17.0
Time: 1.4s
Precision: 64
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -2.0900224301544303 \cdot 10^{81}:\\ \;\;\;\;\log \left(-1 \cdot re\right)\\ \mathbf{elif}\;re \le 4.3762682966534536 \cdot 10^{-289}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \le 5.84329385744146969 \cdot 10^{-170}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 2.90313000711163415 \cdot 10^{125}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]
\log \left(\sqrt{re \cdot re + im \cdot im}\right)
↓
\begin{array}{l}
\mathbf{if}\;re \le -2.0900224301544303 \cdot 10^{81}:\\
\;\;\;\;\log \left(-1 \cdot re\right)\\

\mathbf{elif}\;re \le 4.3762682966534536 \cdot 10^{-289}:\\
\;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\

\mathbf{elif}\;re \le 5.84329385744146969 \cdot 10^{-170}:\\
\;\;\;\;\log im\\

\mathbf{elif}\;re \le 2.90313000711163415 \cdot 10^{125}:\\
\;\;\;\;\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 VAR;
	if ((re <= -2.0900224301544303e+81)) {
		VAR = log((-1.0 * re));
	} else {
		double VAR_1;
		if ((re <= 4.376268296653454e-289)) {
			VAR_1 = log(sqrt(((re * re) + (im * im))));
		} else {
			double VAR_2;
			if ((re <= 5.84329385744147e-170)) {
				VAR_2 = log(im);
			} else {
				double VAR_3;
				if ((re <= 2.903130007111634e+125)) {
					VAR_3 = log(sqrt(((re * re) + (im * im))));
				} else {
					VAR_3 = log(re);
				}
				VAR_2 = VAR_3;
			}
			VAR_1 = VAR_2;
		}
		VAR = VAR_1;
	}
	return VAR;
}

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 < -2.0900224301544303e+81

    1. Initial program 47.5

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

    2. Taylor expanded around -inf 9.1

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

    if -2.0900224301544303e+81 < re < 4.376268296653454e-289 or 5.84329385744147e-170 < re < 2.903130007111634e+125

    1. Initial program 19.3

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

    if 4.376268296653454e-289 < re < 5.84329385744147e-170

    1. Initial program 30.3

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

    2. Taylor expanded around 0 34.8

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

    if 2.903130007111634e+125 < re

    1. Initial program 57.0

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

    2. Taylor expanded around inf 7.0

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

  4. Final simplification17.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -2.0900224301544303 \cdot 10^{81}:\\ \;\;\;\;\log \left(-1 \cdot re\right)\\ \mathbf{elif}\;re \le 4.3762682966534536 \cdot 10^{-289}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \le 5.84329385744146969 \cdot 10^{-170}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 2.90313000711163415 \cdot 10^{125}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Reproduce

herbie shell --seed 2020106 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  :precision binary64
  (log (sqrt (+ (* re re) (* im im)))))