Average Error: 30.8 → 17.1
Time: 3.3s
Precision: 64
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -8.203187278539673 \cdot 10^{+152}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le 1.426377138919928 \cdot 10^{+124}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]
\log \left(\sqrt{re \cdot re + im \cdot im}\right)
↓
\begin{array}{l}
\mathbf{if}\;re \le -8.203187278539673 \cdot 10^{+152}:\\
\;\;\;\;\log \left(-re\right)\\

\mathbf{elif}\;re \le 1.426377138919928 \cdot 10^{+124}:\\
\;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\

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

\end{array}
double f(double re, double im) {
        double r787785 = re;
        double r787786 = r787785 * r787785;
        double r787787 = im;
        double r787788 = r787787 * r787787;
        double r787789 = r787786 + r787788;
        double r787790 = sqrt(r787789);
        double r787791 = log(r787790);
        return r787791;
}
↓
double f(double re, double im) {
        double r787792 = re;
        double r787793 = -8.203187278539673e+152;
        bool r787794 = r787792 <= r787793;
        double r787795 = -r787792;
        double r787796 = log(r787795);
        double r787797 = 1.426377138919928e+124;
        bool r787798 = r787792 <= r787797;
        double r787799 = im;
        double r787800 = r787799 * r787799;
        double r787801 = r787792 * r787792;
        double r787802 = r787800 + r787801;
        double r787803 = sqrt(r787802);
        double r787804 = log(r787803);
        double r787805 = log(r787792);
        double r787806 = r787798 ? r787804 : r787805;
        double r787807 = r787794 ? r787796 : r787806;
        return r787807;
}

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 < -8.203187278539673e+152

    1. Initial program 62.0

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

    2. Taylor expanded around -inf 7.4

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

    3. Simplified7.4

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

    if -8.203187278539673e+152 < re < 1.426377138919928e+124

    1. Initial program 20.6

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

    if 1.426377138919928e+124 < re

    1. Initial program 54.7

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

    2. Taylor expanded around inf 8.2

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

  4. Final simplification17.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -8.203187278539673 \cdot 10^{+152}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le 1.426377138919928 \cdot 10^{+124}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Reproduce

herbie shell --seed 2019155 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  (log (sqrt (+ (* re re) (* im im)))))