Average Error: 31.4 → 17.2
Time: 1.6s
Precision: binary64
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;re \leq -1.2212465663556388 \cdot 10^{+84}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \leq -1.1094194130263304 \cdot 10^{-267}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \leq 7.114272991027333 \cdot 10^{-197}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \leq 1.1202921247401748 \cdot 10^{+30}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]
\log \left(\sqrt{re \cdot re + im \cdot im}\right)
↓
\begin{array}{l}
\mathbf{if}\;re \leq -1.2212465663556388 \cdot 10^{+84}:\\
\;\;\;\;\log \left(-re\right)\\

\mathbf{elif}\;re \leq -1.1094194130263304 \cdot 10^{-267}:\\
\;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\

\mathbf{elif}\;re \leq 7.114272991027333 \cdot 10^{-197}:\\
\;\;\;\;\log im\\

\mathbf{elif}\;re \leq 1.1202921247401748 \cdot 10^{+30}:\\
\;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\

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

\end{array}
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 <= -1.2212465663556388e+84)) {
		VAR = ((double) log(((double) -(re))));
	} else {
		double VAR_1;
		if ((re <= -1.1094194130263304e-267)) {
			VAR_1 = ((double) log(((double) sqrt(((double) (((double) (re * re)) + ((double) (im * im))))))));
		} else {
			double VAR_2;
			if ((re <= 7.114272991027333e-197)) {
				VAR_2 = ((double) log(im));
			} else {
				double VAR_3;
				if ((re <= 1.1202921247401748e+30)) {
					VAR_3 = ((double) log(((double) sqrt(((double) (((double) (re * re)) + ((double) (im * im))))))));
				} else {
					VAR_3 = ((double) 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 < -1.22124656635563882e84

    1. Initial program 48.6

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

    2. Taylor expanded around -inf 9.8

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

    3. Simplified9.8

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

    if -1.22124656635563882e84 < re < -1.1094194130263304e-267 or 7.11427299102733342e-197 < re < 1.120292124740175e30

    1. Initial program 19.0

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

    if -1.1094194130263304e-267 < re < 7.11427299102733342e-197

    1. Initial program 30.1

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

    2. Taylor expanded around 0 32.6

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

    if 1.120292124740175e30 < re

    1. Initial program 43.5

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

    2. Taylor expanded around inf 11.2

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

  4. Final simplification17.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.2212465663556388 \cdot 10^{+84}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \leq -1.1094194130263304 \cdot 10^{-267}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \leq 7.114272991027333 \cdot 10^{-197}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \leq 1.1202921247401748 \cdot 10^{+30}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Reproduce

herbie shell --seed 2020196 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  :precision binary64
  (log (sqrt (+ (* re re) (* im im)))))