Average Error: 31.1 → 17.5
Time: 1.8s
Precision: binary64
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -4.75834617317528473 \cdot 10^{108}:\\ \;\;\;\;\log \left(-1 \cdot re\right)\\ \mathbf{elif}\;re \le -7.8278737157546405 \cdot 10^{-179}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \le 3.80858442508732754 \cdot 10^{-279}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 1.8987858919513489 \cdot 10^{84}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]
\log \left(\sqrt{re \cdot re + im \cdot im}\right)
↓
\begin{array}{l}
\mathbf{if}\;re \le -4.75834617317528473 \cdot 10^{108}:\\
\;\;\;\;\log \left(-1 \cdot re\right)\\

\mathbf{elif}\;re \le -7.8278737157546405 \cdot 10^{-179}:\\
\;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\

\mathbf{elif}\;re \le 3.80858442508732754 \cdot 10^{-279}:\\
\;\;\;\;\log im\\

\mathbf{elif}\;re \le 1.8987858919513489 \cdot 10^{84}:\\
\;\;\;\;\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 <= -4.758346173175285e+108)) {
		VAR = ((double) log(((double) (-1.0 * re))));
	} else {
		double VAR_1;
		if ((re <= -7.82787371575464e-179)) {
			VAR_1 = ((double) log(((double) sqrt(((double) (((double) (re * re)) + ((double) (im * im))))))));
		} else {
			double VAR_2;
			if ((re <= 3.8085844250873275e-279)) {
				VAR_2 = ((double) log(im));
			} else {
				double VAR_3;
				if ((re <= 1.8987858919513489e+84)) {
					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 < -4.75834617317528473e108

    1. Initial program 53.4

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

    2. Taylor expanded around -inf 8.8

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

    if -4.75834617317528473e108 < re < -7.8278737157546405e-179 or 3.80858442508732754e-279 < re < 1.8987858919513489e84

    1. Initial program 18.3

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

    if -7.8278737157546405e-179 < re < 3.80858442508732754e-279

    1. Initial program 32.0

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

    2. Taylor expanded around 0 34.6

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

    if 1.8987858919513489e84 < re

    1. Initial program 48.8

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

    2. Taylor expanded around inf 10.2

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

  4. Final simplification17.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -4.75834617317528473 \cdot 10^{108}:\\ \;\;\;\;\log \left(-1 \cdot re\right)\\ \mathbf{elif}\;re \le -7.8278737157546405 \cdot 10^{-179}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \le 3.80858442508732754 \cdot 10^{-279}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 1.8987858919513489 \cdot 10^{84}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Reproduce

herbie shell --seed 2020163 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  :precision binary64
  (log (sqrt (+ (* re re) (* im im)))))