Average Error: 31.7 → 17.4
Time: 4.7s
Precision: 64
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -1.359664342997624889944602942441258789071 \cdot 10^{126}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le -4.900395837730481841635385837042199386348 \cdot 10^{-259}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \le 1.122991184894528950330980756169030360872 \cdot 10^{-257}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 8.839357053658903549788359163479770769181 \cdot 10^{103}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]
\log \left(\sqrt{re \cdot re + im \cdot im}\right)
↓
\begin{array}{l}
\mathbf{if}\;re \le -1.359664342997624889944602942441258789071 \cdot 10^{126}:\\
\;\;\;\;\log \left(-re\right)\\

\mathbf{elif}\;re \le -4.900395837730481841635385837042199386348 \cdot 10^{-259}:\\
\;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\

\mathbf{elif}\;re \le 1.122991184894528950330980756169030360872 \cdot 10^{-257}:\\
\;\;\;\;\log im\\

\mathbf{elif}\;re \le 8.839357053658903549788359163479770769181 \cdot 10^{103}:\\
\;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\

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

\end{array}
double f(double re, double im) {
        double r28690 = re;
        double r28691 = r28690 * r28690;
        double r28692 = im;
        double r28693 = r28692 * r28692;
        double r28694 = r28691 + r28693;
        double r28695 = sqrt(r28694);
        double r28696 = log(r28695);
        return r28696;
}
↓
double f(double re, double im) {
        double r28697 = re;
        double r28698 = -1.359664342997625e+126;
        bool r28699 = r28697 <= r28698;
        double r28700 = -r28697;
        double r28701 = log(r28700);
        double r28702 = -4.900395837730482e-259;
        bool r28703 = r28697 <= r28702;
        double r28704 = r28697 * r28697;
        double r28705 = im;
        double r28706 = r28705 * r28705;
        double r28707 = r28704 + r28706;
        double r28708 = sqrt(r28707);
        double r28709 = log(r28708);
        double r28710 = 1.122991184894529e-257;
        bool r28711 = r28697 <= r28710;
        double r28712 = log(r28705);
        double r28713 = 8.839357053658904e+103;
        bool r28714 = r28697 <= r28713;
        double r28715 = log(r28697);
        double r28716 = r28714 ? r28709 : r28715;
        double r28717 = r28711 ? r28712 : r28716;
        double r28718 = r28703 ? r28709 : r28717;
        double r28719 = r28699 ? r28701 : r28718;
        return r28719;
}

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 < -1.359664342997625e+126

    1. Initial program 56.9

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

    2. Taylor expanded around -inf 7.4

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

    3. Simplified7.4

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

    if -1.359664342997625e+126 < re < -4.900395837730482e-259 or 1.122991184894529e-257 < re < 8.839357053658904e+103

    1. Initial program 20.3

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

    if -4.900395837730482e-259 < re < 1.122991184894529e-257

    1. Initial program 31.1

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

    2. Taylor expanded around 0 32.5

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

    if 8.839357053658904e+103 < re

    1. Initial program 51.3

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

    2. Taylor expanded around inf 8.3

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

  4. Final simplification17.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.359664342997624889944602942441258789071 \cdot 10^{126}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le -4.900395837730481841635385837042199386348 \cdot 10^{-259}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \le 1.122991184894528950330980756169030360872 \cdot 10^{-257}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 8.839357053658903549788359163479770769181 \cdot 10^{103}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Reproduce

herbie shell --seed 2019306 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  :precision binary64
  (log (sqrt (+ (* re re) (* im im)))))