Average Error: 30.9 → 17.8
Time: 3.4s
Precision: 64
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -8.019718531677109 \cdot 10^{+83}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le -4.41122201426305 \cdot 10^{-167}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{elif}\;re \le 3.772349416931329 \cdot 10^{-236}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 5.362757127991631 \cdot 10^{+122}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]
\log \left(\sqrt{re \cdot re + im \cdot im}\right)
↓
\begin{array}{l}
\mathbf{if}\;re \le -8.019718531677109 \cdot 10^{+83}:\\
\;\;\;\;\log \left(-re\right)\\

\mathbf{elif}\;re \le -4.41122201426305 \cdot 10^{-167}:\\
\;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\

\mathbf{elif}\;re \le 3.772349416931329 \cdot 10^{-236}:\\
\;\;\;\;\log im\\

\mathbf{elif}\;re \le 5.362757127991631 \cdot 10^{+122}:\\
\;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\

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

\end{array}
double f(double re, double im) {
        double r1184671 = re;
        double r1184672 = r1184671 * r1184671;
        double r1184673 = im;
        double r1184674 = r1184673 * r1184673;
        double r1184675 = r1184672 + r1184674;
        double r1184676 = sqrt(r1184675);
        double r1184677 = log(r1184676);
        return r1184677;
}
↓
double f(double re, double im) {
        double r1184678 = re;
        double r1184679 = -8.019718531677109e+83;
        bool r1184680 = r1184678 <= r1184679;
        double r1184681 = -r1184678;
        double r1184682 = log(r1184681);
        double r1184683 = -4.41122201426305e-167;
        bool r1184684 = r1184678 <= r1184683;
        double r1184685 = im;
        double r1184686 = r1184685 * r1184685;
        double r1184687 = r1184678 * r1184678;
        double r1184688 = r1184686 + r1184687;
        double r1184689 = sqrt(r1184688);
        double r1184690 = log(r1184689);
        double r1184691 = 3.772349416931329e-236;
        bool r1184692 = r1184678 <= r1184691;
        double r1184693 = log(r1184685);
        double r1184694 = 5.362757127991631e+122;
        bool r1184695 = r1184678 <= r1184694;
        double r1184696 = log(r1184678);
        double r1184697 = r1184695 ? r1184690 : r1184696;
        double r1184698 = r1184692 ? r1184693 : r1184697;
        double r1184699 = r1184684 ? r1184690 : r1184698;
        double r1184700 = r1184680 ? r1184682 : r1184699;
        return r1184700;
}

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 < -8.019718531677109e+83

    1. Initial program 47.6

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

    2. Taylor expanded around -inf 10.4

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

    3. Simplified10.4

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

    if -8.019718531677109e+83 < re < -4.41122201426305e-167 or 3.772349416931329e-236 < re < 5.362757127991631e+122

    1. Initial program 18.1

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

    if -4.41122201426305e-167 < re < 3.772349416931329e-236

    1. Initial program 29.8

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

    2. Taylor expanded around 0 33.3

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

    if 5.362757127991631e+122 < re

    1. Initial program 53.1

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

    2. Taylor expanded around inf 7.8

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

  4. Final simplification17.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -8.019718531677109 \cdot 10^{+83}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le -4.41122201426305 \cdot 10^{-167}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{elif}\;re \le 3.772349416931329 \cdot 10^{-236}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 5.362757127991631 \cdot 10^{+122}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Reproduce

herbie shell --seed 2019146 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  (log (sqrt (+ (* re re) (* im im)))))