Average Error: 32.1 → 18.1
Time: 1.2s
Precision: 64
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -6.296857350600111 \cdot 10^{83}:\\ \;\;\;\;\log \left(-1 \cdot re\right)\\ \mathbf{elif}\;re \le 9.19480309029371711 \cdot 10^{-296}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \le 3.4134902923337802 \cdot 10^{-133}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 1.71884971088463661 \cdot 10^{152}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]
\log \left(\sqrt{re \cdot re + im \cdot im}\right)
↓
\begin{array}{l}
\mathbf{if}\;re \le -6.296857350600111 \cdot 10^{83}:\\
\;\;\;\;\log \left(-1 \cdot re\right)\\

\mathbf{elif}\;re \le 9.19480309029371711 \cdot 10^{-296}:\\
\;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\

\mathbf{elif}\;re \le 3.4134902923337802 \cdot 10^{-133}:\\
\;\;\;\;\log im\\

\mathbf{elif}\;re \le 1.71884971088463661 \cdot 10^{152}:\\
\;\;\;\;\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 temp;
	if ((re <= -6.296857350600111e+83)) {
		temp = log((-1.0 * re));
	} else {
		double temp_1;
		if ((re <= 9.194803090293717e-296)) {
			temp_1 = log(sqrt(((re * re) + (im * im))));
		} else {
			double temp_2;
			if ((re <= 3.4134902923337802e-133)) {
				temp_2 = log(im);
			} else {
				double temp_3;
				if ((re <= 1.7188497108846366e+152)) {
					temp_3 = log(sqrt(((re * re) + (im * im))));
				} else {
					temp_3 = log(re);
				}
				temp_2 = temp_3;
			}
			temp_1 = temp_2;
		}
		temp = temp_1;
	}
	return temp;
}

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 < -6.296857350600111e+83

    1. Initial program 49.8

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

    2. Taylor expanded around -inf 9.7

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

    if -6.296857350600111e+83 < re < 9.194803090293717e-296 or 3.4134902923337802e-133 < re < 1.7188497108846366e+152

    1. Initial program 19.4

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

    if 9.194803090293717e-296 < re < 3.4134902923337802e-133

    1. Initial program 29.7

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

    2. Taylor expanded around 0 35.1

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

    if 1.7188497108846366e+152 < re

    1. Initial program 63.5

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

    2. Taylor expanded around inf 6.3

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

  4. Final simplification18.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -6.296857350600111 \cdot 10^{83}:\\ \;\;\;\;\log \left(-1 \cdot re\right)\\ \mathbf{elif}\;re \le 9.19480309029371711 \cdot 10^{-296}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \le 3.4134902923337802 \cdot 10^{-133}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 1.71884971088463661 \cdot 10^{152}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Reproduce

herbie shell --seed 2020049 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  :precision binary64
  (log (sqrt (+ (* re re) (* im im)))))