Average Error: 31.9 → 18.0
Time: 1.6s
Precision: binary64
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;re \leq -7.300218245200566 \cdot 10^{+92}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \leq 2.36059804000147 \cdot 10^{-250}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \leq 2.4608783481769936 \cdot 10^{-204}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \leq 2.2518402642724047 \cdot 10^{+80}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]
\log \left(\sqrt{re \cdot re + im \cdot im}\right)
↓
\begin{array}{l}
\mathbf{if}\;re \leq -7.300218245200566 \cdot 10^{+92}:\\
\;\;\;\;\log \left(-re\right)\\

\mathbf{elif}\;re \leq 2.36059804000147 \cdot 10^{-250}:\\
\;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\

\mathbf{elif}\;re \leq 2.4608783481769936 \cdot 10^{-204}:\\
\;\;\;\;\log im\\

\mathbf{elif}\;re \leq 2.2518402642724047 \cdot 10^{+80}:\\
\;\;\;\;\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 <= -7.300218245200566e+92)) {
		VAR = ((double) log(((double) -(re))));
	} else {
		double VAR_1;
		if ((re <= 2.36059804000147e-250)) {
			VAR_1 = ((double) log(((double) sqrt(((double) (((double) (re * re)) + ((double) (im * im))))))));
		} else {
			double VAR_2;
			if ((re <= 2.4608783481769936e-204)) {
				VAR_2 = ((double) log(im));
			} else {
				double VAR_3;
				if ((re <= 2.2518402642724047e+80)) {
					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 < -7.30021824520056552e92

    1. Initial program 50.5

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

    2. Taylor expanded around -inf 9.6

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

    3. Simplified9.6

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

    if -7.30021824520056552e92 < re < 2.3605980400014699e-250 or 2.46087834817699364e-204 < re < 2.25184026427240471e80

    1. Initial program 21.8

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

    if 2.3605980400014699e-250 < re < 2.46087834817699364e-204

    1. Initial program 33.6

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

    2. Taylor expanded around 0 34.5

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

    if 2.25184026427240471e80 < re

    1. Initial program 47.2

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

    2. Taylor expanded around inf 9.9

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

  4. Final simplification18.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -7.300218245200566 \cdot 10^{+92}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \leq 2.36059804000147 \cdot 10^{-250}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \leq 2.4608783481769936 \cdot 10^{-204}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \leq 2.2518402642724047 \cdot 10^{+80}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Reproduce

herbie shell --seed 2020196 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  :precision binary64
  (log (sqrt (+ (* re re) (* im im)))))