Average Error: 31.8 → 17.1
Time: 4.3s
Precision: 64
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -1.300812438992646141617859246198844532718 \cdot 10^{100}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le 5.714402214507161350041984173167312711037 \cdot 10^{91}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]
\log \left(\sqrt{re \cdot re + im \cdot im}\right)
↓
\begin{array}{l}
\mathbf{if}\;re \le -1.300812438992646141617859246198844532718 \cdot 10^{100}:\\
\;\;\;\;\log \left(-re\right)\\

\mathbf{elif}\;re \le 5.714402214507161350041984173167312711037 \cdot 10^{91}:\\
\;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\

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

\end{array}
double f(double re, double im) {
        double r1606340 = re;
        double r1606341 = r1606340 * r1606340;
        double r1606342 = im;
        double r1606343 = r1606342 * r1606342;
        double r1606344 = r1606341 + r1606343;
        double r1606345 = sqrt(r1606344);
        double r1606346 = log(r1606345);
        return r1606346;
}
↓
double f(double re, double im) {
        double r1606347 = re;
        double r1606348 = -1.3008124389926461e+100;
        bool r1606349 = r1606347 <= r1606348;
        double r1606350 = -r1606347;
        double r1606351 = log(r1606350);
        double r1606352 = 5.714402214507161e+91;
        bool r1606353 = r1606347 <= r1606352;
        double r1606354 = im;
        double r1606355 = r1606354 * r1606354;
        double r1606356 = r1606347 * r1606347;
        double r1606357 = r1606355 + r1606356;
        double r1606358 = sqrt(r1606357);
        double r1606359 = log(r1606358);
        double r1606360 = log(r1606347);
        double r1606361 = r1606353 ? r1606359 : r1606360;
        double r1606362 = r1606349 ? r1606351 : r1606361;
        return r1606362;
}

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 < -1.3008124389926461e+100

    1. Initial program 51.8

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

    2. Taylor expanded around -inf 8.9

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

    3. Simplified8.9

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

    if -1.3008124389926461e+100 < re < 5.714402214507161e+91

    1. Initial program 21.6

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

    if 5.714402214507161e+91 < re

    1. Initial program 49.8

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

    2. Taylor expanded around inf 8.6

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

  4. Final simplification17.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.300812438992646141617859246198844532718 \cdot 10^{100}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le 5.714402214507161350041984173167312711037 \cdot 10^{91}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Reproduce

herbie shell --seed 2019171 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  (log (sqrt (+ (* re re) (* im im)))))