Average Error: 32.0 → 18.6
Time: 1.4s
Precision: 64
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -6.8846914672875272 \cdot 10^{72}:\\ \;\;\;\;\log \left(-1 \cdot re\right)\\ \mathbf{elif}\;re \le -6.10111052612686754 \cdot 10^{-93}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \le -3.3124292600034948 \cdot 10^{-125}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 3661266352.2377081:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]
\log \left(\sqrt{re \cdot re + im \cdot im}\right)
↓
\begin{array}{l}
\mathbf{if}\;re \le -6.8846914672875272 \cdot 10^{72}:\\
\;\;\;\;\log \left(-1 \cdot re\right)\\

\mathbf{elif}\;re \le -6.10111052612686754 \cdot 10^{-93}:\\
\;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\

\mathbf{elif}\;re \le -3.3124292600034948 \cdot 10^{-125}:\\
\;\;\;\;\log im\\

\mathbf{elif}\;re \le 3661266352.2377081:\\
\;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\

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

\end{array}
double f(double re, double im) {
        double r19855 = re;
        double r19856 = r19855 * r19855;
        double r19857 = im;
        double r19858 = r19857 * r19857;
        double r19859 = r19856 + r19858;
        double r19860 = sqrt(r19859);
        double r19861 = log(r19860);
        return r19861;
}
↓
double f(double re, double im) {
        double r19862 = re;
        double r19863 = -6.884691467287527e+72;
        bool r19864 = r19862 <= r19863;
        double r19865 = -1.0;
        double r19866 = r19865 * r19862;
        double r19867 = log(r19866);
        double r19868 = -6.1011105261268675e-93;
        bool r19869 = r19862 <= r19868;
        double r19870 = r19862 * r19862;
        double r19871 = im;
        double r19872 = r19871 * r19871;
        double r19873 = r19870 + r19872;
        double r19874 = sqrt(r19873);
        double r19875 = log(r19874);
        double r19876 = -3.312429260003495e-125;
        bool r19877 = r19862 <= r19876;
        double r19878 = log(r19871);
        double r19879 = 3661266352.237708;
        bool r19880 = r19862 <= r19879;
        double r19881 = log(r19862);
        double r19882 = r19880 ? r19875 : r19881;
        double r19883 = r19877 ? r19878 : r19882;
        double r19884 = r19869 ? r19875 : r19883;
        double r19885 = r19864 ? r19867 : r19884;
        return r19885;
}

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 4 regimes

  2. if re < -6.884691467287527e+72

    1. Initial program 47.3

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

    2. Taylor expanded around -inf 10.8

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

    if -6.884691467287527e+72 < re < -6.1011105261268675e-93 or -3.312429260003495e-125 < re < 3661266352.237708

    1. Initial program 23.1

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

    if -6.1011105261268675e-93 < re < -3.312429260003495e-125

    1. Initial program 17.7

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

    2. Taylor expanded around 0 38.3

      \[\leadsto \log \color{blue}{im}\]

    if 3661266352.237708 < re

    1. Initial program 41.6

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

    2. Taylor expanded around inf 12.9

      \[\leadsto \log \color{blue}{re}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification18.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -6.8846914672875272 \cdot 10^{72}:\\ \;\;\;\;\log \left(-1 \cdot re\right)\\ \mathbf{elif}\;re \le -6.10111052612686754 \cdot 10^{-93}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \le -3.3124292600034948 \cdot 10^{-125}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 3661266352.2377081:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Reproduce

herbie shell --seed 2020083 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  :precision binary64
  (log (sqrt (+ (* re re) (* im im)))))