Average Error: 31.8 → 17.1
Time: 3.1s
Precision: 64
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -1.300812438992646141617859246198844532718 \cdot 10^{100}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le 5.714402214507161350041984173167312711037 \cdot 10^{91}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]
\log \left(\sqrt{re \cdot re + im \cdot im}\right)
↓
\begin{array}{l}
\mathbf{if}\;re \le -1.300812438992646141617859246198844532718 \cdot 10^{100}:\\
\;\;\;\;\log \left(-re\right)\\

\mathbf{elif}\;re \le 5.714402214507161350041984173167312711037 \cdot 10^{91}:\\
\;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\

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

\end{array}
double f(double re, double im) {
        double r1502695 = re;
        double r1502696 = r1502695 * r1502695;
        double r1502697 = im;
        double r1502698 = r1502697 * r1502697;
        double r1502699 = r1502696 + r1502698;
        double r1502700 = sqrt(r1502699);
        double r1502701 = log(r1502700);
        return r1502701;
}
↓
double f(double re, double im) {
        double r1502702 = re;
        double r1502703 = -1.3008124389926461e+100;
        bool r1502704 = r1502702 <= r1502703;
        double r1502705 = -r1502702;
        double r1502706 = log(r1502705);
        double r1502707 = 5.714402214507161e+91;
        bool r1502708 = r1502702 <= r1502707;
        double r1502709 = im;
        double r1502710 = r1502709 * r1502709;
        double r1502711 = r1502702 * r1502702;
        double r1502712 = r1502710 + r1502711;
        double r1502713 = sqrt(r1502712);
        double r1502714 = log(r1502713);
        double r1502715 = log(r1502702);
        double r1502716 = r1502708 ? r1502714 : r1502715;
        double r1502717 = r1502704 ? r1502706 : r1502716;
        return r1502717;
}

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 < -1.3008124389926461e+100

    1. Initial program 51.8

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

    2. Taylor expanded around -inf 8.9

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

    3. Simplified8.9

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

    if -1.3008124389926461e+100 < re < 5.714402214507161e+91

    1. Initial program 21.6

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

    if 5.714402214507161e+91 < re

    1. Initial program 49.8

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

    2. Taylor expanded around inf 8.6

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

  4. Final simplification17.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.300812438992646141617859246198844532718 \cdot 10^{100}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le 5.714402214507161350041984173167312711037 \cdot 10^{91}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Reproduce

herbie shell --seed 2019171 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  (log (sqrt (+ (* re re) (* im im)))))