Average Error: 32.0 → 18.0
Time: 1.3s
Precision: 64
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -9.31413148065956627 \cdot 10^{128}:\\ \;\;\;\;\log \left(-1 \cdot re\right)\\ \mathbf{elif}\;re \le -2.02255659983759551 \cdot 10^{-162}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \le -1.800726825408113 \cdot 10^{-303}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 6.7060011203754328 \cdot 10^{101}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]
\log \left(\sqrt{re \cdot re + im \cdot im}\right)
↓
\begin{array}{l}
\mathbf{if}\;re \le -9.31413148065956627 \cdot 10^{128}:\\
\;\;\;\;\log \left(-1 \cdot re\right)\\

\mathbf{elif}\;re \le -2.02255659983759551 \cdot 10^{-162}:\\
\;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\

\mathbf{elif}\;re \le -1.800726825408113 \cdot 10^{-303}:\\
\;\;\;\;\log im\\

\mathbf{elif}\;re \le 6.7060011203754328 \cdot 10^{101}:\\
\;\;\;\;\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 <= -9.314131480659566e+128)) {
		VAR = log((-1.0 * re));
	} else {
		double VAR_1;
		if ((re <= -2.0225565998375955e-162)) {
			VAR_1 = log(sqrt(((re * re) + (im * im))));
		} else {
			double VAR_2;
			if ((re <= -1.8007268254081134e-303)) {
				VAR_2 = log(im);
			} else {
				double VAR_3;
				if ((re <= 6.706001120375433e+101)) {
					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 < -9.314131480659566e+128

    1. Initial program 58.2

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

    2. Taylor expanded around -inf 7.9

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

    if -9.314131480659566e+128 < re < -2.0225565998375955e-162 or -1.8007268254081134e-303 < re < 6.706001120375433e+101

    1. Initial program 19.5

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

    if -2.0225565998375955e-162 < re < -1.8007268254081134e-303

    1. Initial program 31.5

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

    2. Taylor expanded around 0 35.2

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

    if 6.706001120375433e+101 < re

    1. Initial program 52.2

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

    2. Taylor expanded around inf 8.9

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

  4. Final simplification18.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -9.31413148065956627 \cdot 10^{128}:\\ \;\;\;\;\log \left(-1 \cdot re\right)\\ \mathbf{elif}\;re \le -2.02255659983759551 \cdot 10^{-162}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \le -1.800726825408113 \cdot 10^{-303}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 6.7060011203754328 \cdot 10^{101}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Reproduce

herbie shell --seed 2020102 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  :precision binary64
  (log (sqrt (+ (* re re) (* im im)))))