Average Error: 31.8 → 18.7
Time: 1.8s
Precision: binary64
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -2.52990826688309827 \cdot 10^{116}:\\ \;\;\;\;\log \left(-1 \cdot re\right)\\ \mathbf{elif}\;re \le -2.80809519100113736 \cdot 10^{-191}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \le 1.99786214231465739 \cdot 10^{-154}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 4.03768629029534308 \cdot 10^{74}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]
\log \left(\sqrt{re \cdot re + im \cdot im}\right)
↓
\begin{array}{l}
\mathbf{if}\;re \le -2.52990826688309827 \cdot 10^{116}:\\
\;\;\;\;\log \left(-1 \cdot re\right)\\

\mathbf{elif}\;re \le -2.80809519100113736 \cdot 10^{-191}:\\
\;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\

\mathbf{elif}\;re \le 1.99786214231465739 \cdot 10^{-154}:\\
\;\;\;\;\log im\\

\mathbf{elif}\;re \le 4.03768629029534308 \cdot 10^{74}:\\
\;\;\;\;\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 <= -2.5299082668830983e+116)) {
		VAR = ((double) log(((double) (-1.0 * re))));
	} else {
		double VAR_1;
		if ((re <= -2.8080951910011374e-191)) {
			VAR_1 = ((double) log(((double) sqrt(((double) (((double) (re * re)) + ((double) (im * im))))))));
		} else {
			double VAR_2;
			if ((re <= 1.9978621423146574e-154)) {
				VAR_2 = ((double) log(im));
			} else {
				double VAR_3;
				if ((re <= 4.037686290295343e+74)) {
					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 < -2.52990826688309827e116

    1. Initial program 54.3

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

    2. Taylor expanded around -inf 8.7

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

    if -2.52990826688309827e116 < re < -2.80809519100113736e-191 or 1.99786214231465739e-154 < re < 4.03768629029534308e74

    1. Initial program 16.9

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

    if -2.80809519100113736e-191 < re < 1.99786214231465739e-154

    1. Initial program 32.4

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

    2. Taylor expanded around 0 35.2

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

    if 4.03768629029534308e74 < re

    1. Initial program 48.0

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

    2. Taylor expanded around inf 11.0

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

  4. Final simplification18.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -2.52990826688309827 \cdot 10^{116}:\\ \;\;\;\;\log \left(-1 \cdot re\right)\\ \mathbf{elif}\;re \le -2.80809519100113736 \cdot 10^{-191}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \le 1.99786214231465739 \cdot 10^{-154}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 4.03768629029534308 \cdot 10^{74}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Reproduce

herbie shell --seed 2020153 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  :precision binary64
  (log (sqrt (+ (* re re) (* im im)))))