Average Error: 31.3 → 17.2
Time: 2.8s
Precision: 64
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -6.981440569710891526213709442760433333215 \cdot 10^{152}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le 264502897229656192:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]
\log \left(\sqrt{re \cdot re + im \cdot im}\right)
↓
\begin{array}{l}
\mathbf{if}\;re \le -6.981440569710891526213709442760433333215 \cdot 10^{152}:\\
\;\;\;\;\log \left(-re\right)\\

\mathbf{elif}\;re \le 264502897229656192:\\
\;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\

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

\end{array}
double f(double re, double im) {
        double r25851 = re;
        double r25852 = r25851 * r25851;
        double r25853 = im;
        double r25854 = r25853 * r25853;
        double r25855 = r25852 + r25854;
        double r25856 = sqrt(r25855);
        double r25857 = log(r25856);
        return r25857;
}
↓
double f(double re, double im) {
        double r25858 = re;
        double r25859 = -6.9814405697108915e+152;
        bool r25860 = r25858 <= r25859;
        double r25861 = -r25858;
        double r25862 = log(r25861);
        double r25863 = 2.645028972296562e+17;
        bool r25864 = r25858 <= r25863;
        double r25865 = r25858 * r25858;
        double r25866 = im;
        double r25867 = r25866 * r25866;
        double r25868 = r25865 + r25867;
        double r25869 = sqrt(r25868);
        double r25870 = log(r25869);
        double r25871 = log(r25858);
        double r25872 = r25864 ? r25870 : r25871;
        double r25873 = r25860 ? r25862 : r25872;
        return r25873;
}

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 < -6.9814405697108915e+152

    1. Initial program 63.7

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

    2. Taylor expanded around -inf 6.3

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

    3. Simplified6.3

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

    if -6.9814405697108915e+152 < re < 2.645028972296562e+17

    1. Initial program 21.3

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

    if 2.645028972296562e+17 < re

    1. Initial program 41.2

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

    2. Taylor expanded around inf 11.8

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

  4. Final simplification17.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -6.981440569710891526213709442760433333215 \cdot 10^{152}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le 264502897229656192:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Reproduce

herbie shell --seed 2019350 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  :precision binary64
  (log (sqrt (+ (* re re) (* im im)))))