Average Error: 31.6 → 18.5
Time: 1.8s
Precision: binary64
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -1.1485069911579876 \cdot 10^{83}:\\ \;\;\;\;\log \left(-1 \cdot re\right)\\ \mathbf{elif}\;re \le -7.4947647920515576 \cdot 10^{-306}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \le 1.49320751385885506 \cdot 10^{-117}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 2.09629540198423465 \cdot 10^{123}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]
\log \left(\sqrt{re \cdot re + im \cdot im}\right)
↓
\begin{array}{l}
\mathbf{if}\;re \le -1.1485069911579876 \cdot 10^{83}:\\
\;\;\;\;\log \left(-1 \cdot re\right)\\

\mathbf{elif}\;re \le -7.4947647920515576 \cdot 10^{-306}:\\
\;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\

\mathbf{elif}\;re \le 1.49320751385885506 \cdot 10^{-117}:\\
\;\;\;\;\log im\\

\mathbf{elif}\;re \le 2.09629540198423465 \cdot 10^{123}:\\
\;\;\;\;\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 <= -1.1485069911579876e+83)) {
		VAR = ((double) log(((double) (-1.0 * re))));
	} else {
		double VAR_1;
		if ((re <= -7.494764792051558e-306)) {
			VAR_1 = ((double) log(((double) sqrt(((double) (((double) (re * re)) + ((double) (im * im))))))));
		} else {
			double VAR_2;
			if ((re <= 1.493207513858855e-117)) {
				VAR_2 = ((double) log(im));
			} else {
				double VAR_3;
				if ((re <= 2.0962954019842346e+123)) {
					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 < -1.1485069911579876e+83

    1. Initial program 48.5

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

    2. Taylor expanded around -inf 9.2

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

    if -1.1485069911579876e+83 < re < -7.494764792051558e-306 or 1.493207513858855e-117 < re < 2.0962954019842346e+123

    1. Initial program 19.5

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

    if -7.494764792051558e-306 < re < 1.493207513858855e-117

    1. Initial program 27.1

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

    2. Taylor expanded around 0 36.4

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

    if 2.0962954019842346e+123 < re

    1. Initial program 55.4

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

    2. Taylor expanded around inf 7.3

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

  4. Final simplification18.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.1485069911579876 \cdot 10^{83}:\\ \;\;\;\;\log \left(-1 \cdot re\right)\\ \mathbf{elif}\;re \le -7.4947647920515576 \cdot 10^{-306}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \le 1.49320751385885506 \cdot 10^{-117}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 2.09629540198423465 \cdot 10^{123}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Reproduce

herbie shell --seed 2020148 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  :precision binary64
  (log (sqrt (+ (* re re) (* im im)))))