Average Error: 30.7 → 16.8
Time: 3.1s
Precision: 64
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -2.103637907474162 \cdot 10^{+73}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le 1.8737997944449135 \cdot 10^{+98}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]
\log \left(\sqrt{re \cdot re + im \cdot im}\right)
↓
\begin{array}{l}
\mathbf{if}\;re \le -2.103637907474162 \cdot 10^{+73}:\\
\;\;\;\;\log \left(-re\right)\\

\mathbf{elif}\;re \le 1.8737997944449135 \cdot 10^{+98}:\\
\;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\

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

\end{array}
double f(double re, double im) {
        double r1371858 = re;
        double r1371859 = r1371858 * r1371858;
        double r1371860 = im;
        double r1371861 = r1371860 * r1371860;
        double r1371862 = r1371859 + r1371861;
        double r1371863 = sqrt(r1371862);
        double r1371864 = log(r1371863);
        return r1371864;
}
↓
double f(double re, double im) {
        double r1371865 = re;
        double r1371866 = -2.103637907474162e+73;
        bool r1371867 = r1371865 <= r1371866;
        double r1371868 = -r1371865;
        double r1371869 = log(r1371868);
        double r1371870 = 1.8737997944449135e+98;
        bool r1371871 = r1371865 <= r1371870;
        double r1371872 = im;
        double r1371873 = r1371872 * r1371872;
        double r1371874 = r1371865 * r1371865;
        double r1371875 = r1371873 + r1371874;
        double r1371876 = sqrt(r1371875);
        double r1371877 = log(r1371876);
        double r1371878 = log(r1371865);
        double r1371879 = r1371871 ? r1371877 : r1371878;
        double r1371880 = r1371867 ? r1371869 : r1371879;
        return r1371880;
}

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 < -2.103637907474162e+73

    1. Initial program 46.1

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

    2. Taylor expanded around -inf 9.7

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

    3. Simplified9.7

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

    if -2.103637907474162e+73 < re < 1.8737997944449135e+98

    1. Initial program 20.9

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

    if 1.8737997944449135e+98 < re

    1. Initial program 49.1

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

    2. Taylor expanded around inf 9.6

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

  4. Final simplification16.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -2.103637907474162 \cdot 10^{+73}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le 1.8737997944449135 \cdot 10^{+98}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Reproduce

herbie shell --seed 2019121 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  (log (sqrt (+ (* re re) (* im im)))))