Average Error: 30.5 → 16.2
Time: 3.3s
Precision: 64
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -1.2548154926707317 \cdot 10^{+121}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le 3.721264707224919 \cdot 10^{+102}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]
\log \left(\sqrt{re \cdot re + im \cdot im}\right)
↓
\begin{array}{l}
\mathbf{if}\;re \le -1.2548154926707317 \cdot 10^{+121}:\\
\;\;\;\;\log \left(-re\right)\\

\mathbf{elif}\;re \le 3.721264707224919 \cdot 10^{+102}:\\
\;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\

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

\end{array}
double f(double re, double im) {
        double r1148837 = re;
        double r1148838 = r1148837 * r1148837;
        double r1148839 = im;
        double r1148840 = r1148839 * r1148839;
        double r1148841 = r1148838 + r1148840;
        double r1148842 = sqrt(r1148841);
        double r1148843 = log(r1148842);
        return r1148843;
}
↓
double f(double re, double im) {
        double r1148844 = re;
        double r1148845 = -1.2548154926707317e+121;
        bool r1148846 = r1148844 <= r1148845;
        double r1148847 = -r1148844;
        double r1148848 = log(r1148847);
        double r1148849 = 3.721264707224919e+102;
        bool r1148850 = r1148844 <= r1148849;
        double r1148851 = im;
        double r1148852 = r1148851 * r1148851;
        double r1148853 = r1148844 * r1148844;
        double r1148854 = r1148852 + r1148853;
        double r1148855 = sqrt(r1148854);
        double r1148856 = log(r1148855);
        double r1148857 = log(r1148844);
        double r1148858 = r1148850 ? r1148856 : r1148857;
        double r1148859 = r1148846 ? r1148848 : r1148858;
        return r1148859;
}

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 < -1.2548154926707317e+121

    1. Initial program 53.8

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

    2. Taylor expanded around -inf 7.3

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

    3. Simplified7.3

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

    if -1.2548154926707317e+121 < re < 3.721264707224919e+102

    1. Initial program 20.1

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

    if 3.721264707224919e+102 < re

    1. Initial program 51.4

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

    2. Taylor expanded around inf 8.3

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

  4. Final simplification16.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.2548154926707317 \cdot 10^{+121}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le 3.721264707224919 \cdot 10^{+102}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Reproduce

herbie shell --seed 2019152 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  (log (sqrt (+ (* re re) (* im im)))))