Average Error: 32.4 → 18.0
Time: 1.2s
Precision: 64
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -4.75759962206180014 \cdot 10^{138}:\\ \;\;\;\;\log \left(-1 \cdot re\right)\\ \mathbf{elif}\;re \le -3.5543765182763856 \cdot 10^{-161}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \le 4.5607039117785637 \cdot 10^{-251}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 3.2663661678364143 \cdot 10^{95}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]
\log \left(\sqrt{re \cdot re + im \cdot im}\right)
↓
\begin{array}{l}
\mathbf{if}\;re \le -4.75759962206180014 \cdot 10^{138}:\\
\;\;\;\;\log \left(-1 \cdot re\right)\\

\mathbf{elif}\;re \le -3.5543765182763856 \cdot 10^{-161}:\\
\;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\

\mathbf{elif}\;re \le 4.5607039117785637 \cdot 10^{-251}:\\
\;\;\;\;\log im\\

\mathbf{elif}\;re \le 3.2663661678364143 \cdot 10^{95}:\\
\;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\

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

\end{array}
double f(double re, double im) {
        double r82820 = re;
        double r82821 = r82820 * r82820;
        double r82822 = im;
        double r82823 = r82822 * r82822;
        double r82824 = r82821 + r82823;
        double r82825 = sqrt(r82824);
        double r82826 = log(r82825);
        return r82826;
}
↓
double f(double re, double im) {
        double r82827 = re;
        double r82828 = -4.7575996220618e+138;
        bool r82829 = r82827 <= r82828;
        double r82830 = -1.0;
        double r82831 = r82830 * r82827;
        double r82832 = log(r82831);
        double r82833 = -3.5543765182763856e-161;
        bool r82834 = r82827 <= r82833;
        double r82835 = r82827 * r82827;
        double r82836 = im;
        double r82837 = r82836 * r82836;
        double r82838 = r82835 + r82837;
        double r82839 = sqrt(r82838);
        double r82840 = log(r82839);
        double r82841 = 4.560703911778564e-251;
        bool r82842 = r82827 <= r82841;
        double r82843 = log(r82836);
        double r82844 = 3.266366167836414e+95;
        bool r82845 = r82827 <= r82844;
        double r82846 = log(r82827);
        double r82847 = r82845 ? r82840 : r82846;
        double r82848 = r82842 ? r82843 : r82847;
        double r82849 = r82834 ? r82840 : r82848;
        double r82850 = r82829 ? r82832 : r82849;
        return r82850;
}

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 < -4.7575996220618e+138

    1. Initial program 59.1

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

    2. Taylor expanded around -inf 7.1

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

    if -4.7575996220618e+138 < re < -3.5543765182763856e-161 or 4.560703911778564e-251 < re < 3.266366167836414e+95

    1. Initial program 19.0

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

    if -3.5543765182763856e-161 < re < 4.560703911778564e-251

    1. Initial program 32.9

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

    2. Taylor expanded around 0 33.8

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

    if 3.266366167836414e+95 < re

    1. Initial program 51.1

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

    2. Taylor expanded around inf 9.2

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

  4. Final simplification18.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -4.75759962206180014 \cdot 10^{138}:\\ \;\;\;\;\log \left(-1 \cdot re\right)\\ \mathbf{elif}\;re \le -3.5543765182763856 \cdot 10^{-161}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \le 4.5607039117785637 \cdot 10^{-251}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 3.2663661678364143 \cdot 10^{95}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  :precision binary64
  (log (sqrt (+ (* re re) (* im im)))))