Average Error: 30.0 → 17.6
Time: 10.4s
Precision: 64
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -1.3951279250903065 \cdot 10^{+116}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le -1.2552454715818077 \cdot 10^{-176}:\\ \;\;\;\;\log \left(im \cdot im + re \cdot re\right) \cdot \frac{1}{2}\\ \mathbf{elif}\;re \le 8.33451305802747 \cdot 10^{-256}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 8235632211164311.0:\\ \;\;\;\;\log \left(im \cdot im + re \cdot re\right) \cdot \frac{1}{2}\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]
\log \left(\sqrt{re \cdot re + im \cdot im}\right)
↓
\begin{array}{l}
\mathbf{if}\;re \le -1.3951279250903065 \cdot 10^{+116}:\\
\;\;\;\;\log \left(-re\right)\\

\mathbf{elif}\;re \le -1.2552454715818077 \cdot 10^{-176}:\\
\;\;\;\;\log \left(im \cdot im + re \cdot re\right) \cdot \frac{1}{2}\\

\mathbf{elif}\;re \le 8.33451305802747 \cdot 10^{-256}:\\
\;\;\;\;\log im\\

\mathbf{elif}\;re \le 8235632211164311.0:\\
\;\;\;\;\log \left(im \cdot im + re \cdot re\right) \cdot \frac{1}{2}\\

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

\end{array}
double f(double re, double im) {
        double r2070615 = re;
        double r2070616 = r2070615 * r2070615;
        double r2070617 = im;
        double r2070618 = r2070617 * r2070617;
        double r2070619 = r2070616 + r2070618;
        double r2070620 = sqrt(r2070619);
        double r2070621 = log(r2070620);
        return r2070621;
}
↓
double f(double re, double im) {
        double r2070622 = re;
        double r2070623 = -1.3951279250903065e+116;
        bool r2070624 = r2070622 <= r2070623;
        double r2070625 = -r2070622;
        double r2070626 = log(r2070625);
        double r2070627 = -1.2552454715818077e-176;
        bool r2070628 = r2070622 <= r2070627;
        double r2070629 = im;
        double r2070630 = r2070629 * r2070629;
        double r2070631 = r2070622 * r2070622;
        double r2070632 = r2070630 + r2070631;
        double r2070633 = log(r2070632);
        double r2070634 = 0.5;
        double r2070635 = r2070633 * r2070634;
        double r2070636 = 8.33451305802747e-256;
        bool r2070637 = r2070622 <= r2070636;
        double r2070638 = log(r2070629);
        double r2070639 = 8235632211164311.0;
        bool r2070640 = r2070622 <= r2070639;
        double r2070641 = log(r2070622);
        double r2070642 = r2070640 ? r2070635 : r2070641;
        double r2070643 = r2070637 ? r2070638 : r2070642;
        double r2070644 = r2070628 ? r2070635 : r2070643;
        double r2070645 = r2070624 ? r2070626 : r2070644;
        return r2070645;
}

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 < -1.3951279250903065e+116

    1. Initial program 52.7

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

    2. Taylor expanded around -inf 7.5

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

    3. Simplified7.5

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

    if -1.3951279250903065e+116 < re < -1.2552454715818077e-176 or 8.33451305802747e-256 < re < 8235632211164311.0

    1. Initial program 18.0

      \[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
    2. Using strategy rm

    3. Applied pow1/218.0

      \[\leadsto \log \color{blue}{\left({\left(re \cdot re + im \cdot im\right)}^{\frac{1}{2}}\right)}\]

    4. Applied log-pow18.0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \log \left(re \cdot re + im \cdot im\right)}\]

    if -1.2552454715818077e-176 < re < 8.33451305802747e-256

    1. Initial program 30.2

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

    2. Taylor expanded around 0 32.5

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

    if 8235632211164311.0 < re

    1. Initial program 39.5

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

    2. Taylor expanded around inf 12.8

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

  4. Final simplification17.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.3951279250903065 \cdot 10^{+116}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le -1.2552454715818077 \cdot 10^{-176}:\\ \;\;\;\;\log \left(im \cdot im + re \cdot re\right) \cdot \frac{1}{2}\\ \mathbf{elif}\;re \le 8.33451305802747 \cdot 10^{-256}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 8235632211164311.0:\\ \;\;\;\;\log \left(im \cdot im + re \cdot re\right) \cdot \frac{1}{2}\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Reproduce

herbie shell --seed 2019158 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  (log (sqrt (+ (* re re) (* im im)))))