Average Error: 31.3 → 17.0
Time: 1.1s
Precision: 64
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -3.8249499853926235 \cdot 10^{103}:\\ \;\;\;\;\log \left(-1 \cdot re\right)\\ \mathbf{elif}\;re \le 2.56439878537218745 \cdot 10^{-244}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \le 2.9654757473955426 \cdot 10^{-173}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 1.556390230884587 \cdot 10^{144}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]
\log \left(\sqrt{re \cdot re + im \cdot im}\right)
↓
\begin{array}{l}
\mathbf{if}\;re \le -3.8249499853926235 \cdot 10^{103}:\\
\;\;\;\;\log \left(-1 \cdot re\right)\\

\mathbf{elif}\;re \le 2.56439878537218745 \cdot 10^{-244}:\\
\;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\

\mathbf{elif}\;re \le 2.9654757473955426 \cdot 10^{-173}:\\
\;\;\;\;\log im\\

\mathbf{elif}\;re \le 1.556390230884587 \cdot 10^{144}:\\
\;\;\;\;\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 <= -3.8249499853926235e+103)) {
		VAR = log((-1.0 * re));
	} else {
		double VAR_1;
		if ((re <= 2.5643987853721874e-244)) {
			VAR_1 = log(sqrt(((re * re) + (im * im))));
		} else {
			double VAR_2;
			if ((re <= 2.9654757473955426e-173)) {
				VAR_2 = log(im);
			} else {
				double VAR_3;
				if ((re <= 1.5563902308845874e+144)) {
					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 < -3.8249499853926235e+103

    1. Initial program 52.4

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

    2. Taylor expanded around -inf 8.9

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

    if -3.8249499853926235e+103 < re < 2.5643987853721874e-244 or 2.9654757473955426e-173 < re < 1.5563902308845874e+144

    1. Initial program 19.8

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

    if 2.5643987853721874e-244 < re < 2.9654757473955426e-173

    1. Initial program 32.1

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

    2. Taylor expanded around 0 35.2

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

    if 1.5563902308845874e+144 < re

    1. Initial program 61.2

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

    2. Taylor expanded around inf 5.2

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

  4. Final simplification17.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -3.8249499853926235 \cdot 10^{103}:\\ \;\;\;\;\log \left(-1 \cdot re\right)\\ \mathbf{elif}\;re \le 2.56439878537218745 \cdot 10^{-244}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \le 2.9654757473955426 \cdot 10^{-173}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 1.556390230884587 \cdot 10^{144}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Reproduce

herbie shell --seed 2020075 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  :precision binary64
  (log (sqrt (+ (* re re) (* im im)))))