Average Error: 31.5 → 17.0
Time: 3.4s
Precision: 64
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -1.181793183213821728908776663248811693415 \cdot 10^{151}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le 7.392440833541333777660561627276981553815 \cdot 10^{126}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]
\log \left(\sqrt{re \cdot re + im \cdot im}\right)
↓
\begin{array}{l}
\mathbf{if}\;re \le -1.181793183213821728908776663248811693415 \cdot 10^{151}:\\
\;\;\;\;\log \left(-re\right)\\

\mathbf{elif}\;re \le 7.392440833541333777660561627276981553815 \cdot 10^{126}:\\
\;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\

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

\end{array}
double f(double re, double im) {
        double r1580855 = re;
        double r1580856 = r1580855 * r1580855;
        double r1580857 = im;
        double r1580858 = r1580857 * r1580857;
        double r1580859 = r1580856 + r1580858;
        double r1580860 = sqrt(r1580859);
        double r1580861 = log(r1580860);
        return r1580861;
}
↓
double f(double re, double im) {
        double r1580862 = re;
        double r1580863 = -1.1817931832138217e+151;
        bool r1580864 = r1580862 <= r1580863;
        double r1580865 = -r1580862;
        double r1580866 = log(r1580865);
        double r1580867 = 7.392440833541334e+126;
        bool r1580868 = r1580862 <= r1580867;
        double r1580869 = im;
        double r1580870 = r1580869 * r1580869;
        double r1580871 = r1580862 * r1580862;
        double r1580872 = r1580870 + r1580871;
        double r1580873 = sqrt(r1580872);
        double r1580874 = log(r1580873);
        double r1580875 = log(r1580862);
        double r1580876 = r1580868 ? r1580874 : r1580875;
        double r1580877 = r1580864 ? r1580866 : r1580876;
        return r1580877;
}

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 < -1.1817931832138217e+151

    1. Initial program 63.1

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

    2. Taylor expanded around -inf 7.1

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

    3. Simplified7.1

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

    if -1.1817931832138217e+151 < re < 7.392440833541334e+126

    1. Initial program 20.7

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

    if 7.392440833541334e+126 < re

    1. Initial program 56.4

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

    2. Taylor expanded around inf 7.8

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

  4. Final simplification17.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.181793183213821728908776663248811693415 \cdot 10^{151}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le 7.392440833541333777660561627276981553815 \cdot 10^{126}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  (log (sqrt (+ (* re re) (* im im)))))