Average Error: 31.6 → 18.0
Time: 2.2s
Precision: binary64
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -1.6998950373020774 \cdot 10^{87}:\\ \;\;\;\;\log \left(-1 \cdot re\right)\\ \mathbf{elif}\;re \le -1.1243171280702968 \cdot 10^{-257}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \le 2.4669184868707975 \cdot 10^{-173}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 9.9029961269328029 \cdot 10^{131}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]
\log \left(\sqrt{re \cdot re + im \cdot im}\right)
↓
\begin{array}{l}
\mathbf{if}\;re \le -1.6998950373020774 \cdot 10^{87}:\\
\;\;\;\;\log \left(-1 \cdot re\right)\\

\mathbf{elif}\;re \le -1.1243171280702968 \cdot 10^{-257}:\\
\;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\

\mathbf{elif}\;re \le 2.4669184868707975 \cdot 10^{-173}:\\
\;\;\;\;\log im\\

\mathbf{elif}\;re \le 9.9029961269328029 \cdot 10^{131}:\\
\;\;\;\;\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 <= -1.6998950373020774e+87)) {
		VAR = ((double) log(((double) (-1.0 * re))));
	} else {
		double VAR_1;
		if ((re <= -1.1243171280702968e-257)) {
			VAR_1 = ((double) log(((double) sqrt(((double) (((double) (re * re)) + ((double) (im * im))))))));
		} else {
			double VAR_2;
			if ((re <= 2.4669184868707975e-173)) {
				VAR_2 = ((double) log(im));
			} else {
				double VAR_3;
				if ((re <= 9.902996126932803e+131)) {
					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 < -1.6998950373020774e87

    1. Initial program 48.8

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

    2. Taylor expanded around -inf 8.9

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

    if -1.6998950373020774e87 < re < -1.1243171280702968e-257 or 2.4669184868707975e-173 < re < 9.9029961269328029e131

    1. Initial program 19.2

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

    if -1.1243171280702968e-257 < re < 2.4669184868707975e-173

    1. Initial program 31.2

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

    2. Taylor expanded around 0 34.2

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

    if 9.9029961269328029e131 < re

    1. Initial program 58.0

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

    2. Taylor expanded around inf 6.8

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

  4. Final simplification18.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.6998950373020774 \cdot 10^{87}:\\ \;\;\;\;\log \left(-1 \cdot re\right)\\ \mathbf{elif}\;re \le -1.1243171280702968 \cdot 10^{-257}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \le 2.4669184868707975 \cdot 10^{-173}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 9.9029961269328029 \cdot 10^{131}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Reproduce

herbie shell --seed 2020162 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  :precision binary64
  (log (sqrt (+ (* re re) (* im im)))))