Average Error: 31.9 → 17.8
Time: 1.2s
Precision: 64
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -1.989101613628458 \cdot 10^{44}:\\ \;\;\;\;\log \left(-1 \cdot re\right)\\ \mathbf{elif}\;re \le 7.9423972447061974 \cdot 10^{-271}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \le 2.744576864806113 \cdot 10^{-224}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 1.1417370863594925 \cdot 10^{101}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]
\log \left(\sqrt{re \cdot re + im \cdot im}\right)
↓
\begin{array}{l}
\mathbf{if}\;re \le -1.989101613628458 \cdot 10^{44}:\\
\;\;\;\;\log \left(-1 \cdot re\right)\\

\mathbf{elif}\;re \le 7.9423972447061974 \cdot 10^{-271}:\\
\;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\

\mathbf{elif}\;re \le 2.744576864806113 \cdot 10^{-224}:\\
\;\;\;\;\log im\\

\mathbf{elif}\;re \le 1.1417370863594925 \cdot 10^{101}:\\
\;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\

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

\end{array}
double f(double re, double im) {
        double r34677 = re;
        double r34678 = r34677 * r34677;
        double r34679 = im;
        double r34680 = r34679 * r34679;
        double r34681 = r34678 + r34680;
        double r34682 = sqrt(r34681);
        double r34683 = log(r34682);
        return r34683;
}
↓
double f(double re, double im) {
        double r34684 = re;
        double r34685 = -1.989101613628458e+44;
        bool r34686 = r34684 <= r34685;
        double r34687 = -1.0;
        double r34688 = r34687 * r34684;
        double r34689 = log(r34688);
        double r34690 = 7.942397244706197e-271;
        bool r34691 = r34684 <= r34690;
        double r34692 = r34684 * r34684;
        double r34693 = im;
        double r34694 = r34693 * r34693;
        double r34695 = r34692 + r34694;
        double r34696 = sqrt(r34695);
        double r34697 = log(r34696);
        double r34698 = 2.744576864806113e-224;
        bool r34699 = r34684 <= r34698;
        double r34700 = log(r34693);
        double r34701 = 1.1417370863594925e+101;
        bool r34702 = r34684 <= r34701;
        double r34703 = log(r34684);
        double r34704 = r34702 ? r34697 : r34703;
        double r34705 = r34699 ? r34700 : r34704;
        double r34706 = r34691 ? r34697 : r34705;
        double r34707 = r34686 ? r34689 : r34706;
        return r34707;
}

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 < -1.989101613628458e+44

    1. Initial program 43.3

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

    2. Taylor expanded around -inf 12.1

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

    if -1.989101613628458e+44 < re < 7.942397244706197e-271 or 2.744576864806113e-224 < re < 1.1417370863594925e+101

    1. Initial program 21.6

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

    if 7.942397244706197e-271 < re < 2.744576864806113e-224

    1. Initial program 29.7

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

    2. Taylor expanded around 0 31.9

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

    if 1.1417370863594925e+101 < re

    1. Initial program 52.5

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

    2. Taylor expanded around inf 8.8

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

  4. Final simplification17.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.989101613628458 \cdot 10^{44}:\\ \;\;\;\;\log \left(-1 \cdot re\right)\\ \mathbf{elif}\;re \le 7.9423972447061974 \cdot 10^{-271}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \le 2.744576864806113 \cdot 10^{-224}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 1.1417370863594925 \cdot 10^{101}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Reproduce

herbie shell --seed 2020060 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  :precision binary64
  (log (sqrt (+ (* re re) (* im im)))))