Average Error: 31.8 → 18.3
Time: 1.7s
Precision: binary64
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -1.0565365445657685 \cdot 10^{+139}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le -8.395017692797379 \cdot 10^{-165}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \le -2.5556997558112552 \cdot 10^{-228}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le -8.989235959462165 \cdot 10^{-268}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \le -6.972847979648137 \cdot 10^{-305}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 1.0407504367576015 \cdot 10^{+92}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]
\log \left(\sqrt{re \cdot re + im \cdot im}\right)
↓
\begin{array}{l}
\mathbf{if}\;re \le -1.0565365445657685 \cdot 10^{+139}:\\
\;\;\;\;\log \left(-re\right)\\

\mathbf{elif}\;re \le -8.395017692797379 \cdot 10^{-165}:\\
\;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\

\mathbf{elif}\;re \le -2.5556997558112552 \cdot 10^{-228}:\\
\;\;\;\;\log \left(-re\right)\\

\mathbf{elif}\;re \le -8.989235959462165 \cdot 10^{-268}:\\
\;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\

\mathbf{elif}\;re \le -6.972847979648137 \cdot 10^{-305}:\\
\;\;\;\;\log im\\

\mathbf{elif}\;re \le 1.0407504367576015 \cdot 10^{+92}:\\
\;\;\;\;\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 <= -1.0565365445657685e+139)) {
		VAR = ((double) log(((double) -(re))));
	} else {
		double VAR_1;
		if ((re <= -8.395017692797379e-165)) {
			VAR_1 = ((double) log(((double) sqrt(((double) (((double) (re * re)) + ((double) (im * im))))))));
		} else {
			double VAR_2;
			if ((re <= -2.5556997558112552e-228)) {
				VAR_2 = ((double) log(((double) -(re))));
			} else {
				double VAR_3;
				if ((re <= -8.989235959462165e-268)) {
					VAR_3 = ((double) log(((double) sqrt(((double) (((double) (re * re)) + ((double) (im * im))))))));
				} else {
					double VAR_4;
					if ((re <= -6.972847979648137e-305)) {
						VAR_4 = ((double) log(im));
					} else {
						double VAR_5;
						if ((re <= 1.0407504367576015e+92)) {
							VAR_5 = ((double) log(((double) sqrt(((double) (((double) (re * re)) + ((double) (im * im))))))));
						} else {
							VAR_5 = ((double) log(re));
						}
						VAR_4 = VAR_5;
					}
					VAR_3 = VAR_4;
				}
				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 < -1.05653654456576854e139 or -8.39501769279737873e-165 < re < -2.5556997558112552e-228

    1. Initial program 52.3

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

    2. Taylor expanded around -inf 18.6

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

    3. Simplified18.6

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

    if -1.05653654456576854e139 < re < -8.39501769279737873e-165 or -2.5556997558112552e-228 < re < -8.98923595946216538e-268 or -6.9728479796481374e-305 < re < 1.0407504367576015e92

    1. Initial program 20.0

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

    if -8.98923595946216538e-268 < re < -6.9728479796481374e-305

    1. Initial program 31.3

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

    2. Taylor expanded around 0 32.9

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

    if 1.0407504367576015e92 < re

    1. Initial program 50.9

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

    2. Taylor expanded around inf 9.7

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

  4. Final simplification18.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.0565365445657685 \cdot 10^{+139}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le -8.395017692797379 \cdot 10^{-165}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \le -2.5556997558112552 \cdot 10^{-228}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le -8.989235959462165 \cdot 10^{-268}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \le -6.972847979648137 \cdot 10^{-305}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 1.0407504367576015 \cdot 10^{+92}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Reproduce

herbie shell --seed 2020184 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  :precision binary64
  (log (sqrt (+ (* re re) (* im im)))))