Average Error: 31.3 → 18.6
Time: 1.2s
Precision: 64
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -5.83181885032133734 \cdot 10^{55}:\\ \;\;\;\;\log \left(-1 \cdot re\right)\\ \mathbf{elif}\;re \le -3.80996693730795831 \cdot 10^{-103}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \le 2.88566167285968699 \cdot 10^{-297}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 3.32569577851814611 \cdot 10^{94}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]
\log \left(\sqrt{re \cdot re + im \cdot im}\right)
↓
\begin{array}{l}
\mathbf{if}\;re \le -5.83181885032133734 \cdot 10^{55}:\\
\;\;\;\;\log \left(-1 \cdot re\right)\\

\mathbf{elif}\;re \le -3.80996693730795831 \cdot 10^{-103}:\\
\;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\

\mathbf{elif}\;re \le 2.88566167285968699 \cdot 10^{-297}:\\
\;\;\;\;\log im\\

\mathbf{elif}\;re \le 3.32569577851814611 \cdot 10^{94}:\\
\;\;\;\;\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 <= -5.831818850321337e+55)) {
		VAR = log((-1.0 * re));
	} else {
		double VAR_1;
		if ((re <= -3.8099669373079583e-103)) {
			VAR_1 = log(sqrt(((re * re) + (im * im))));
		} else {
			double VAR_2;
			if ((re <= 2.885661672859687e-297)) {
				VAR_2 = log(im);
			} else {
				double VAR_3;
				if ((re <= 3.325695778518146e+94)) {
					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 < -5.831818850321337e+55

    1. Initial program 44.4

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

    2. Taylor expanded around -inf 10.5

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

    if -5.831818850321337e+55 < re < -3.8099669373079583e-103 or 2.885661672859687e-297 < re < 3.325695778518146e+94

    1. Initial program 19.4

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

    if -3.8099669373079583e-103 < re < 2.885661672859687e-297

    1. Initial program 28.5

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

    2. Taylor expanded around 0 35.8

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

    if 3.325695778518146e+94 < re

    1. Initial program 50.1

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

    2. Taylor expanded around inf 8.7

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

  4. Final simplification18.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -5.83181885032133734 \cdot 10^{55}:\\ \;\;\;\;\log \left(-1 \cdot re\right)\\ \mathbf{elif}\;re \le -3.80996693730795831 \cdot 10^{-103}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \le 2.88566167285968699 \cdot 10^{-297}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 3.32569577851814611 \cdot 10^{94}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Reproduce

herbie shell --seed 2020100 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  :precision binary64
  (log (sqrt (+ (* re re) (* im im)))))