Average Error: 30.8 → 17.0
Time: 4.5s
Precision: 64
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -1.427484018494741 \cdot 10^{+134}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le 1.5824798583418597 \cdot 10^{+66}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]
\log \left(\sqrt{re \cdot re + im \cdot im}\right)
↓
\begin{array}{l}
\mathbf{if}\;re \le -1.427484018494741 \cdot 10^{+134}:\\
\;\;\;\;\log \left(-re\right)\\

\mathbf{elif}\;re \le 1.5824798583418597 \cdot 10^{+66}:\\
\;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\

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

\end{array}
double f(double re, double im) {
        double r1497784 = re;
        double r1497785 = r1497784 * r1497784;
        double r1497786 = im;
        double r1497787 = r1497786 * r1497786;
        double r1497788 = r1497785 + r1497787;
        double r1497789 = sqrt(r1497788);
        double r1497790 = log(r1497789);
        return r1497790;
}
↓
double f(double re, double im) {
        double r1497791 = re;
        double r1497792 = -1.427484018494741e+134;
        bool r1497793 = r1497791 <= r1497792;
        double r1497794 = -r1497791;
        double r1497795 = log(r1497794);
        double r1497796 = 1.5824798583418597e+66;
        bool r1497797 = r1497791 <= r1497796;
        double r1497798 = im;
        double r1497799 = r1497798 * r1497798;
        double r1497800 = r1497791 * r1497791;
        double r1497801 = r1497799 + r1497800;
        double r1497802 = sqrt(r1497801);
        double r1497803 = log(r1497802);
        double r1497804 = log(r1497791);
        double r1497805 = r1497797 ? r1497803 : r1497804;
        double r1497806 = r1497793 ? r1497795 : r1497805;
        return r1497806;
}

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 < -1.427484018494741e+134

    1. Initial program 56.6

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

    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 -1.427484018494741e+134 < re < 1.5824798583418597e+66

    1. Initial program 21.1

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

    if 1.5824798583418597e+66 < re

    1. Initial program 46.1

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

    2. Taylor expanded around inf 10.0

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

  4. Final simplification17.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.427484018494741 \cdot 10^{+134}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le 1.5824798583418597 \cdot 10^{+66}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Reproduce

herbie shell --seed 2019163 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  (log (sqrt (+ (* re re) (* im im)))))