Average Error: 31.8 → 17.2
Time: 1.7s
Precision: binary64
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;re \leq -1.5678980394422787 \cdot 10^{+88}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \leq -3.3370596695886217 \cdot 10^{-236}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \leq -1.7053324499790462 \cdot 10^{-305}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \leq 1.7501362194659408 \cdot 10^{+112}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]
\log \left(\sqrt{re \cdot re + im \cdot im}\right)
↓
\begin{array}{l}
\mathbf{if}\;re \leq -1.5678980394422787 \cdot 10^{+88}:\\
\;\;\;\;\log \left(-re\right)\\

\mathbf{elif}\;re \leq -3.3370596695886217 \cdot 10^{-236}:\\
\;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\

\mathbf{elif}\;re \leq -1.7053324499790462 \cdot 10^{-305}:\\
\;\;\;\;\log im\\

\mathbf{elif}\;re \leq 1.7501362194659408 \cdot 10^{+112}:\\
\;\;\;\;\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 <= -1.5678980394422787e+88)) {
		VAR = ((double) log(((double) -(re))));
	} else {
		double VAR_1;
		if ((re <= -3.3370596695886217e-236)) {
			VAR_1 = ((double) log(((double) sqrt(((double) (((double) (re * re)) + ((double) (im * im))))))));
		} else {
			double VAR_2;
			if ((re <= -1.7053324499790462e-305)) {
				VAR_2 = ((double) log(im));
			} else {
				double VAR_3;
				if ((re <= 1.7501362194659408e+112)) {
					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 < -1.56789803944227867e88

    1. Initial program 50.0

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

    2. Taylor expanded around -inf 9.2

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

    3. Simplified9.2

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

    if -1.56789803944227867e88 < re < -3.33705966958862171e-236 or -1.7053324499790462e-305 < re < 1.75013621946594082e112

    1. Initial program 20.6

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

    if -3.33705966958862171e-236 < re < -1.7053324499790462e-305

    1. Initial program 32.8

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

    2. Taylor expanded around 0 32.1

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

    if 1.75013621946594082e112 < re

    1. Initial program 53.8

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

    2. Taylor expanded around inf 7.4

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

  4. Final simplification17.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.5678980394422787 \cdot 10^{+88}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \leq -3.3370596695886217 \cdot 10^{-236}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \leq -1.7053324499790462 \cdot 10^{-305}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \leq 1.7501362194659408 \cdot 10^{+112}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Reproduce

herbie shell --seed 2020199 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  :precision binary64
  (log (sqrt (+ (* re re) (* im im)))))