Average Error: 32.1 → 18.1
Time: 1.2s
Precision: 64
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -6.296857350600111 \cdot 10^{83}:\\ \;\;\;\;\log \left(-1 \cdot re\right)\\ \mathbf{elif}\;re \le 9.19480309029371711 \cdot 10^{-296}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \le 3.4134902923337802 \cdot 10^{-133}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 1.71884971088463661 \cdot 10^{152}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]
\log \left(\sqrt{re \cdot re + im \cdot im}\right)
↓
\begin{array}{l}
\mathbf{if}\;re \le -6.296857350600111 \cdot 10^{83}:\\
\;\;\;\;\log \left(-1 \cdot re\right)\\

\mathbf{elif}\;re \le 9.19480309029371711 \cdot 10^{-296}:\\
\;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\

\mathbf{elif}\;re \le 3.4134902923337802 \cdot 10^{-133}:\\
\;\;\;\;\log im\\

\mathbf{elif}\;re \le 1.71884971088463661 \cdot 10^{152}:\\
\;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\

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

\end{array}
double f(double re, double im) {
        double r43436 = re;
        double r43437 = r43436 * r43436;
        double r43438 = im;
        double r43439 = r43438 * r43438;
        double r43440 = r43437 + r43439;
        double r43441 = sqrt(r43440);
        double r43442 = log(r43441);
        return r43442;
}
↓
double f(double re, double im) {
        double r43443 = re;
        double r43444 = -6.296857350600111e+83;
        bool r43445 = r43443 <= r43444;
        double r43446 = -1.0;
        double r43447 = r43446 * r43443;
        double r43448 = log(r43447);
        double r43449 = 9.194803090293717e-296;
        bool r43450 = r43443 <= r43449;
        double r43451 = r43443 * r43443;
        double r43452 = im;
        double r43453 = r43452 * r43452;
        double r43454 = r43451 + r43453;
        double r43455 = sqrt(r43454);
        double r43456 = log(r43455);
        double r43457 = 3.4134902923337802e-133;
        bool r43458 = r43443 <= r43457;
        double r43459 = log(r43452);
        double r43460 = 1.7188497108846366e+152;
        bool r43461 = r43443 <= r43460;
        double r43462 = log(r43443);
        double r43463 = r43461 ? r43456 : r43462;
        double r43464 = r43458 ? r43459 : r43463;
        double r43465 = r43450 ? r43456 : r43464;
        double r43466 = r43445 ? r43448 : r43465;
        return r43466;
}

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 < -6.296857350600111e+83

    1. Initial program 49.8

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

    2. Taylor expanded around -inf 9.7

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

    if -6.296857350600111e+83 < re < 9.194803090293717e-296 or 3.4134902923337802e-133 < re < 1.7188497108846366e+152

    1. Initial program 19.4

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

    if 9.194803090293717e-296 < re < 3.4134902923337802e-133

    1. Initial program 29.7

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

    2. Taylor expanded around 0 35.1

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

    if 1.7188497108846366e+152 < re

    1. Initial program 63.5

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

    2. Taylor expanded around inf 6.3

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

  4. Final simplification18.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -6.296857350600111 \cdot 10^{83}:\\ \;\;\;\;\log \left(-1 \cdot re\right)\\ \mathbf{elif}\;re \le 9.19480309029371711 \cdot 10^{-296}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \le 3.4134902923337802 \cdot 10^{-133}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 1.71884971088463661 \cdot 10^{152}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Reproduce

herbie shell --seed 2020049 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  :precision binary64
  (log (sqrt (+ (* re re) (* im im)))))