Average Error: 30.4 → 17.0
Time: 3.9s
Precision: 64
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -2.7728325482865987 \cdot 10^{+132}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le -3.454520874704003 \cdot 10^{-249}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{elif}\;re \le 1.0730314248253915 \cdot 10^{-288}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 1.1882351325142646 \cdot 10^{+39}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]
\log \left(\sqrt{re \cdot re + im \cdot im}\right)
↓
\begin{array}{l}
\mathbf{if}\;re \le -2.7728325482865987 \cdot 10^{+132}:\\
\;\;\;\;\log \left(-re\right)\\

\mathbf{elif}\;re \le -3.454520874704003 \cdot 10^{-249}:\\
\;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\

\mathbf{elif}\;re \le 1.0730314248253915 \cdot 10^{-288}:\\
\;\;\;\;\log im\\

\mathbf{elif}\;re \le 1.1882351325142646 \cdot 10^{+39}:\\
\;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\

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

\end{array}
double f(double re, double im) {
        double r553084 = re;
        double r553085 = r553084 * r553084;
        double r553086 = im;
        double r553087 = r553086 * r553086;
        double r553088 = r553085 + r553087;
        double r553089 = sqrt(r553088);
        double r553090 = log(r553089);
        return r553090;
}
↓
double f(double re, double im) {
        double r553091 = re;
        double r553092 = -2.7728325482865987e+132;
        bool r553093 = r553091 <= r553092;
        double r553094 = -r553091;
        double r553095 = log(r553094);
        double r553096 = -3.454520874704003e-249;
        bool r553097 = r553091 <= r553096;
        double r553098 = im;
        double r553099 = r553098 * r553098;
        double r553100 = r553091 * r553091;
        double r553101 = r553099 + r553100;
        double r553102 = sqrt(r553101);
        double r553103 = log(r553102);
        double r553104 = 1.0730314248253915e-288;
        bool r553105 = r553091 <= r553104;
        double r553106 = log(r553098);
        double r553107 = 1.1882351325142646e+39;
        bool r553108 = r553091 <= r553107;
        double r553109 = log(r553091);
        double r553110 = r553108 ? r553103 : r553109;
        double r553111 = r553105 ? r553106 : r553110;
        double r553112 = r553097 ? r553103 : r553111;
        double r553113 = r553093 ? r553095 : r553112;
        return r553113;
}

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 < -2.7728325482865987e+132

    1. Initial program 56.5

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

    2. Taylor expanded around -inf 8.1

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

    3. Simplified8.1

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

    if -2.7728325482865987e+132 < re < -3.454520874704003e-249 or 1.0730314248253915e-288 < re < 1.1882351325142646e+39

    1. Initial program 19.6

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

    if -3.454520874704003e-249 < re < 1.0730314248253915e-288

    1. Initial program 30.4

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

    2. Taylor expanded around 0 30.1

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

    if 1.1882351325142646e+39 < re

    1. Initial program 41.6

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

    2. Taylor expanded around inf 12.0

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

  4. Final simplification17.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -2.7728325482865987 \cdot 10^{+132}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le -3.454520874704003 \cdot 10^{-249}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{elif}\;re \le 1.0730314248253915 \cdot 10^{-288}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 1.1882351325142646 \cdot 10^{+39}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Reproduce

herbie shell --seed 2019137 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  (log (sqrt (+ (* re re) (* im im)))))