Average Error: 31.6 → 17.4
Time: 3.3s
Precision: 64
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -2.273448994496035253530762983989520870583 \cdot 10^{87}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le 1.763702591686904819827254628881572839528 \cdot 10^{111}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]
\log \left(\sqrt{re \cdot re + im \cdot im}\right)
↓
\begin{array}{l}
\mathbf{if}\;re \le -2.273448994496035253530762983989520870583 \cdot 10^{87}:\\
\;\;\;\;\log \left(-re\right)\\

\mathbf{elif}\;re \le 1.763702591686904819827254628881572839528 \cdot 10^{111}:\\
\;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\

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

\end{array}
double f(double re, double im) {
        double r1517352 = re;
        double r1517353 = r1517352 * r1517352;
        double r1517354 = im;
        double r1517355 = r1517354 * r1517354;
        double r1517356 = r1517353 + r1517355;
        double r1517357 = sqrt(r1517356);
        double r1517358 = log(r1517357);
        return r1517358;
}
↓
double f(double re, double im) {
        double r1517359 = re;
        double r1517360 = -2.2734489944960353e+87;
        bool r1517361 = r1517359 <= r1517360;
        double r1517362 = -r1517359;
        double r1517363 = log(r1517362);
        double r1517364 = 1.7637025916869048e+111;
        bool r1517365 = r1517359 <= r1517364;
        double r1517366 = im;
        double r1517367 = r1517366 * r1517366;
        double r1517368 = r1517359 * r1517359;
        double r1517369 = r1517367 + r1517368;
        double r1517370 = sqrt(r1517369);
        double r1517371 = log(r1517370);
        double r1517372 = log(r1517359);
        double r1517373 = r1517365 ? r1517371 : r1517372;
        double r1517374 = r1517361 ? r1517363 : r1517373;
        return r1517374;
}

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 < -2.2734489944960353e+87

    1. Initial program 49.1

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

    2. Taylor expanded around -inf 9.4

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

    3. Simplified9.4

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

    if -2.2734489944960353e+87 < re < 1.7637025916869048e+111

    1. Initial program 21.6

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

    if 1.7637025916869048e+111 < re

    1. Initial program 53.8

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

    2. Taylor expanded around inf 8.7

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

  4. Final simplification17.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -2.273448994496035253530762983989520870583 \cdot 10^{87}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le 1.763702591686904819827254628881572839528 \cdot 10^{111}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Reproduce

herbie shell --seed 2019192 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  (log (sqrt (+ (* re re) (* im im)))))