Average Error: 32.1 → 17.8
Time: 1.5s
Precision: binary64
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;re \leq -1.8014590457402938 \cdot 10^{+51}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \leq 8.860843829360954 \cdot 10^{+93}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]
\log \left(\sqrt{re \cdot re + im \cdot im}\right)
↓
\begin{array}{l}
\mathbf{if}\;re \leq -1.8014590457402938 \cdot 10^{+51}:\\
\;\;\;\;\log \left(-re\right)\\

\mathbf{elif}\;re \leq 8.860843829360954 \cdot 10^{+93}:\\
\;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\

\mathbf{else}:\\
\;\;\;\;\log re\\

\end{array}
(FPCore (re im) :precision binary64 (log (sqrt (+ (* re re) (* im im)))))
↓
(FPCore (re im)
 :precision binary64
 (if (<= re -1.8014590457402938e+51)
   (log (- re))
   (if (<= re 8.860843829360954e+93)
     (log (sqrt (+ (* re re) (* im im))))
     (log re))))
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 tmp;
	if ((re <= -1.8014590457402938e+51)) {
		tmp = ((double) log(((double) -(re))));
	} else {
		double tmp_1;
		if ((re <= 8.860843829360954e+93)) {
			tmp_1 = ((double) log(((double) sqrt(((double) (((double) (re * re)) + ((double) (im * im))))))));
		} else {
			tmp_1 = ((double) log(re));
		}
		tmp = tmp_1;
	}
	return tmp;
}

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 3 regimes

  2. if re < -1.8014590457402938e51

    1. Initial program 45.8

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

    2. Taylor expanded around -inf 10.9

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

    3. Simplified10.9

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

    if -1.8014590457402938e51 < re < 8.86084382936095357e93

    1. Initial program 22.4

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

    if 8.86084382936095357e93 < re

    1. Initial program 50.6

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

    2. Taylor expanded around inf 9.2

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

  4. Final simplification17.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.8014590457402938 \cdot 10^{+51}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \leq 8.860843829360954 \cdot 10^{+93}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Reproduce

herbie shell --seed 2020205 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  :precision binary64
  (log (sqrt (+ (* re re) (* im im)))))