Average Error: 32.2 → 17.8
Time: 1.4s
Precision: 64
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -1.886444123297446 \cdot 10^{126}:\\ \;\;\;\;\log \left(-1 \cdot re\right)\\ \mathbf{elif}\;re \le -2.30867664875349023 \cdot 10^{-209}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \le 1.2217527835098556 \cdot 10^{-266}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 4.9319226435479606 \cdot 10^{132}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]
\log \left(\sqrt{re \cdot re + im \cdot im}\right)
↓
\begin{array}{l}
\mathbf{if}\;re \le -1.886444123297446 \cdot 10^{126}:\\
\;\;\;\;\log \left(-1 \cdot re\right)\\

\mathbf{elif}\;re \le -2.30867664875349023 \cdot 10^{-209}:\\
\;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\

\mathbf{elif}\;re \le 1.2217527835098556 \cdot 10^{-266}:\\
\;\;\;\;\log im\\

\mathbf{elif}\;re \le 4.9319226435479606 \cdot 10^{132}:\\
\;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\

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

\end{array}
double f(double re, double im) {
        double r89712 = re;
        double r89713 = r89712 * r89712;
        double r89714 = im;
        double r89715 = r89714 * r89714;
        double r89716 = r89713 + r89715;
        double r89717 = sqrt(r89716);
        double r89718 = log(r89717);
        return r89718;
}
↓
double f(double re, double im) {
        double r89719 = re;
        double r89720 = -1.886444123297446e+126;
        bool r89721 = r89719 <= r89720;
        double r89722 = -1.0;
        double r89723 = r89722 * r89719;
        double r89724 = log(r89723);
        double r89725 = -2.3086766487534902e-209;
        bool r89726 = r89719 <= r89725;
        double r89727 = r89719 * r89719;
        double r89728 = im;
        double r89729 = r89728 * r89728;
        double r89730 = r89727 + r89729;
        double r89731 = sqrt(r89730);
        double r89732 = log(r89731);
        double r89733 = 1.2217527835098556e-266;
        bool r89734 = r89719 <= r89733;
        double r89735 = log(r89728);
        double r89736 = 4.9319226435479606e+132;
        bool r89737 = r89719 <= r89736;
        double r89738 = log(r89719);
        double r89739 = r89737 ? r89732 : r89738;
        double r89740 = r89734 ? r89735 : r89739;
        double r89741 = r89726 ? r89732 : r89740;
        double r89742 = r89721 ? r89724 : r89741;
        return r89742;
}

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 < -1.886444123297446e+126

    1. Initial program 57.6

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

    2. Taylor expanded around -inf 8.7

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

    if -1.886444123297446e+126 < re < -2.3086766487534902e-209 or 1.2217527835098556e-266 < re < 4.9319226435479606e+132

    1. Initial program 19.5

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

    if -2.3086766487534902e-209 < re < 1.2217527835098556e-266

    1. Initial program 31.3

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

    2. Taylor expanded around 0 35.3

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

    if 4.9319226435479606e+132 < re

    1. Initial program 58.4

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

    2. Taylor expanded around inf 6.5

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

  4. Final simplification17.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.886444123297446 \cdot 10^{126}:\\ \;\;\;\;\log \left(-1 \cdot re\right)\\ \mathbf{elif}\;re \le -2.30867664875349023 \cdot 10^{-209}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \le 1.2217527835098556 \cdot 10^{-266}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 4.9319226435479606 \cdot 10^{132}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Reproduce

herbie shell --seed 2020033 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  :precision binary64
  (log (sqrt (+ (* re re) (* im im)))))