Average Error: 30.5 → 16.2
Time: 3.1s
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 r1110101 = re;
        double r1110102 = r1110101 * r1110101;
        double r1110103 = im;
        double r1110104 = r1110103 * r1110103;
        double r1110105 = r1110102 + r1110104;
        double r1110106 = sqrt(r1110105);
        double r1110107 = log(r1110106);
        return r1110107;
}
↓
double f(double re, double im) {
        double r1110108 = re;
        double r1110109 = -1.2548154926707317e+121;
        bool r1110110 = r1110108 <= r1110109;
        double r1110111 = -r1110108;
        double r1110112 = log(r1110111);
        double r1110113 = 3.721264707224919e+102;
        bool r1110114 = r1110108 <= r1110113;
        double r1110115 = im;
        double r1110116 = r1110115 * r1110115;
        double r1110117 = r1110108 * r1110108;
        double r1110118 = r1110116 + r1110117;
        double r1110119 = sqrt(r1110118);
        double r1110120 = log(r1110119);
        double r1110121 = log(r1110108);
        double r1110122 = r1110114 ? r1110120 : r1110121;
        double r1110123 = r1110110 ? r1110112 : r1110122;
        return r1110123;
}

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)))))