Average Error: 30.7 → 17.0
Time: 5.0s
Precision: 64
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -1.9173717249563345 \cdot 10^{+124}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le 3.425958633542213 \cdot 10^{+105}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]
\log \left(\sqrt{re \cdot re + im \cdot im}\right)
↓
\begin{array}{l}
\mathbf{if}\;re \le -1.9173717249563345 \cdot 10^{+124}:\\
\;\;\;\;\log \left(-re\right)\\

\mathbf{elif}\;re \le 3.425958633542213 \cdot 10^{+105}:\\
\;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\

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

\end{array}
double f(double re, double im) {
        double r639711 = re;
        double r639712 = r639711 * r639711;
        double r639713 = im;
        double r639714 = r639713 * r639713;
        double r639715 = r639712 + r639714;
        double r639716 = sqrt(r639715);
        double r639717 = log(r639716);
        return r639717;
}
↓
double f(double re, double im) {
        double r639718 = re;
        double r639719 = -1.9173717249563345e+124;
        bool r639720 = r639718 <= r639719;
        double r639721 = -r639718;
        double r639722 = log(r639721);
        double r639723 = 3.425958633542213e+105;
        bool r639724 = r639718 <= r639723;
        double r639725 = im;
        double r639726 = r639725 * r639725;
        double r639727 = r639718 * r639718;
        double r639728 = r639726 + r639727;
        double r639729 = sqrt(r639728);
        double r639730 = log(r639729);
        double r639731 = log(r639718);
        double r639732 = r639724 ? r639730 : r639731;
        double r639733 = r639720 ? r639722 : r639732;
        return r639733;
}

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 < -1.9173717249563345e+124

    1. Initial program 55.3

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

    2. Taylor expanded around -inf 7.6

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

    3. Simplified7.6

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

    if -1.9173717249563345e+124 < re < 3.425958633542213e+105

    1. Initial program 20.8

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

    if 3.425958633542213e+105 < re

    1. Initial program 50.8

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

    2. Taylor expanded around inf 8.8

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

  4. Final simplification17.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.9173717249563345 \cdot 10^{+124}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le 3.425958633542213 \cdot 10^{+105}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Reproduce

herbie shell --seed 2019134 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  (log (sqrt (+ (* re re) (* im im)))))