Average Error: 30.8 → 17.0
Time: 4.8s
Precision: 64
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -1.427484018494741 \cdot 10^{+134}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le 1.5824798583418597 \cdot 10^{+66}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]
\log \left(\sqrt{re \cdot re + im \cdot im}\right)
↓
\begin{array}{l}
\mathbf{if}\;re \le -1.427484018494741 \cdot 10^{+134}:\\
\;\;\;\;\log \left(-re\right)\\

\mathbf{elif}\;re \le 1.5824798583418597 \cdot 10^{+66}:\\
\;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\

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

\end{array}
double f(double re, double im) {
        double r1499363 = re;
        double r1499364 = r1499363 * r1499363;
        double r1499365 = im;
        double r1499366 = r1499365 * r1499365;
        double r1499367 = r1499364 + r1499366;
        double r1499368 = sqrt(r1499367);
        double r1499369 = log(r1499368);
        return r1499369;
}
↓
double f(double re, double im) {
        double r1499370 = re;
        double r1499371 = -1.427484018494741e+134;
        bool r1499372 = r1499370 <= r1499371;
        double r1499373 = -r1499370;
        double r1499374 = log(r1499373);
        double r1499375 = 1.5824798583418597e+66;
        bool r1499376 = r1499370 <= r1499375;
        double r1499377 = im;
        double r1499378 = r1499377 * r1499377;
        double r1499379 = r1499370 * r1499370;
        double r1499380 = r1499378 + r1499379;
        double r1499381 = sqrt(r1499380);
        double r1499382 = log(r1499381);
        double r1499383 = log(r1499370);
        double r1499384 = r1499376 ? r1499382 : r1499383;
        double r1499385 = r1499372 ? r1499374 : r1499384;
        return r1499385;
}

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 < -1.427484018494741e+134

    1. Initial program 56.6

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

    2. Taylor expanded around -inf 7.3

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

    3. Simplified7.3

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

    if -1.427484018494741e+134 < re < 1.5824798583418597e+66

    1. Initial program 21.1

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

    if 1.5824798583418597e+66 < re

    1. Initial program 46.1

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

    2. Taylor expanded around inf 10.0

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

  4. Final simplification17.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.427484018494741 \cdot 10^{+134}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le 1.5824798583418597 \cdot 10^{+66}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Reproduce

herbie shell --seed 2019163 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  (log (sqrt (+ (* re re) (* im im)))))