Average Error: 30.6 → 16.8
Time: 1.2s
Precision: 64
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -1.79671734719750555 \cdot 10^{106}:\\ \;\;\;\;\log \left(-1 \cdot re\right)\\ \mathbf{elif}\;re \le 4.55718954958977214 \cdot 10^{-298}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \le 4.0892765348340891 \cdot 10^{-181}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 5.18276077735685444 \cdot 10^{91}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]
\log \left(\sqrt{re \cdot re + im \cdot im}\right)
↓
\begin{array}{l}
\mathbf{if}\;re \le -1.79671734719750555 \cdot 10^{106}:\\
\;\;\;\;\log \left(-1 \cdot re\right)\\

\mathbf{elif}\;re \le 4.55718954958977214 \cdot 10^{-298}:\\
\;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\

\mathbf{elif}\;re \le 4.0892765348340891 \cdot 10^{-181}:\\
\;\;\;\;\log im\\

\mathbf{elif}\;re \le 5.18276077735685444 \cdot 10^{91}:\\
\;\;\;\;\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 <= -1.7967173471975055e+106)) {
		VAR = log((-1.0 * re));
	} else {
		double VAR_1;
		if ((re <= 4.557189549589772e-298)) {
			VAR_1 = log(sqrt(((re * re) + (im * im))));
		} else {
			double VAR_2;
			if ((re <= 4.089276534834089e-181)) {
				VAR_2 = log(im);
			} else {
				double VAR_3;
				if ((re <= 5.1827607773568544e+91)) {
					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 < -1.7967173471975055e+106

    1. Initial program 52.6

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

    2. Taylor expanded around -inf 9.1

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

    if -1.7967173471975055e+106 < re < 4.557189549589772e-298 or 4.089276534834089e-181 < re < 5.1827607773568544e+91

    1. Initial program 18.6

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

    if 4.557189549589772e-298 < re < 4.089276534834089e-181

    1. Initial program 30.8

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

    2. Taylor expanded around 0 35.6

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

    if 5.1827607773568544e+91 < re

    1. Initial program 50.0

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

    2. Taylor expanded around inf 8.1

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

  4. Final simplification16.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.79671734719750555 \cdot 10^{106}:\\ \;\;\;\;\log \left(-1 \cdot re\right)\\ \mathbf{elif}\;re \le 4.55718954958977214 \cdot 10^{-298}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \le 4.0892765348340891 \cdot 10^{-181}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 5.18276077735685444 \cdot 10^{91}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Reproduce

herbie shell --seed 2020092 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  :precision binary64
  (log (sqrt (+ (* re re) (* im im)))))