Average Error: 31.1 → 17.1
Time: 4.2s
Precision: 64
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -9.68163596973405975259895298385316105053 \cdot 10^{102}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le 3.545380571942664302984715356869784321431 \cdot 10^{140}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]
\log \left(\sqrt{re \cdot re + im \cdot im}\right)
↓
\begin{array}{l}
\mathbf{if}\;re \le -9.68163596973405975259895298385316105053 \cdot 10^{102}:\\
\;\;\;\;\log \left(-re\right)\\

\mathbf{elif}\;re \le 3.545380571942664302984715356869784321431 \cdot 10^{140}:\\
\;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\

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

\end{array}
double f(double re, double im) {
        double r2758808 = re;
        double r2758809 = r2758808 * r2758808;
        double r2758810 = im;
        double r2758811 = r2758810 * r2758810;
        double r2758812 = r2758809 + r2758811;
        double r2758813 = sqrt(r2758812);
        double r2758814 = log(r2758813);
        return r2758814;
}
↓
double f(double re, double im) {
        double r2758815 = re;
        double r2758816 = -9.68163596973406e+102;
        bool r2758817 = r2758815 <= r2758816;
        double r2758818 = -r2758815;
        double r2758819 = log(r2758818);
        double r2758820 = 3.5453805719426643e+140;
        bool r2758821 = r2758815 <= r2758820;
        double r2758822 = im;
        double r2758823 = r2758822 * r2758822;
        double r2758824 = r2758815 * r2758815;
        double r2758825 = r2758823 + r2758824;
        double r2758826 = sqrt(r2758825);
        double r2758827 = log(r2758826);
        double r2758828 = log(r2758815);
        double r2758829 = r2758821 ? r2758827 : r2758828;
        double r2758830 = r2758817 ? r2758819 : r2758829;
        return r2758830;
}

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 3 regimes

  2. if re < -9.68163596973406e+102

    1. Initial program 51.9

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

    2. Taylor expanded around -inf 8.5

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

    3. Simplified8.5

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

    if -9.68163596973406e+102 < re < 3.5453805719426643e+140

    1. Initial program 21.0

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

    if 3.5453805719426643e+140 < re

    1. Initial program 59.5

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

    2. Taylor expanded around inf 7.5

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

  4. Final simplification17.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -9.68163596973405975259895298385316105053 \cdot 10^{102}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le 3.545380571942664302984715356869784321431 \cdot 10^{140}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  (log (sqrt (+ (* re re) (* im im)))))