Average Error: 31.3 → 18.3
Time: 1.7s
Precision: binary64
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -6.1304780829195806 \cdot 10^{150}:\\ \;\;\;\;\log \left(-1 \cdot re\right)\\ \mathbf{elif}\;re \le -4.13283536992341118 \cdot 10^{-166}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \le 3.54800120663930112 \cdot 10^{-302}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 2.22373343428675784 \cdot 10^{-15}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]
\log \left(\sqrt{re \cdot re + im \cdot im}\right)
↓
\begin{array}{l}
\mathbf{if}\;re \le -6.1304780829195806 \cdot 10^{150}:\\
\;\;\;\;\log \left(-1 \cdot re\right)\\

\mathbf{elif}\;re \le -4.13283536992341118 \cdot 10^{-166}:\\
\;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\

\mathbf{elif}\;re \le 3.54800120663930112 \cdot 10^{-302}:\\
\;\;\;\;\log im\\

\mathbf{elif}\;re \le 2.22373343428675784 \cdot 10^{-15}:\\
\;\;\;\;\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 <= -6.1304780829195806e+150)) {
		VAR = ((double) log(((double) (-1.0 * re))));
	} else {
		double VAR_1;
		if ((re <= -4.132835369923411e-166)) {
			VAR_1 = ((double) log(((double) sqrt(((double) (((double) (re * re)) + ((double) (im * im))))))));
		} else {
			double VAR_2;
			if ((re <= 3.548001206639301e-302)) {
				VAR_2 = ((double) log(im));
			} else {
				double VAR_3;
				if ((re <= 2.223733434286758e-15)) {
					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 < -6.1304780829195806e150

    1. Initial program 62.7

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

    2. Taylor expanded around -inf 7.1

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

    if -6.1304780829195806e150 < re < -4.13283536992341118e-166 or 3.54800120663930112e-302 < re < 2.22373343428675784e-15

    1. Initial program 19.7

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

    if -4.13283536992341118e-166 < re < 3.54800120663930112e-302

    1. Initial program 31.9

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

    2. Taylor expanded around 0 34.6

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

    if 2.22373343428675784e-15 < re

    1. Initial program 38.0

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

    2. Taylor expanded around inf 13.2

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

  4. Final simplification18.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -6.1304780829195806 \cdot 10^{150}:\\ \;\;\;\;\log \left(-1 \cdot re\right)\\ \mathbf{elif}\;re \le -4.13283536992341118 \cdot 10^{-166}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \le 3.54800120663930112 \cdot 10^{-302}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 2.22373343428675784 \cdot 10^{-15}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Reproduce

herbie shell --seed 2020175 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  :precision binary64
  (log (sqrt (+ (* re re) (* im im)))))