Average Error: 31.0 → 17.1
Time: 8.2s
Precision: 64
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -3.201775397353183 \cdot 10^{+150}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le 7.407160440832304 \cdot 10^{+80}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]
\log \left(\sqrt{re \cdot re + im \cdot im}\right)
↓
\begin{array}{l}
\mathbf{if}\;re \le -3.201775397353183 \cdot 10^{+150}:\\
\;\;\;\;\log \left(-re\right)\\

\mathbf{elif}\;re \le 7.407160440832304 \cdot 10^{+80}:\\
\;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\

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

\end{array}
double f(double re, double im) {
        double r643347 = re;
        double r643348 = r643347 * r643347;
        double r643349 = im;
        double r643350 = r643349 * r643349;
        double r643351 = r643348 + r643350;
        double r643352 = sqrt(r643351);
        double r643353 = log(r643352);
        return r643353;
}
↓
double f(double re, double im) {
        double r643354 = re;
        double r643355 = -3.201775397353183e+150;
        bool r643356 = r643354 <= r643355;
        double r643357 = -r643354;
        double r643358 = log(r643357);
        double r643359 = 7.407160440832304e+80;
        bool r643360 = r643354 <= r643359;
        double r643361 = im;
        double r643362 = r643361 * r643361;
        double r643363 = r643354 * r643354;
        double r643364 = r643362 + r643363;
        double r643365 = sqrt(r643364);
        double r643366 = log(r643365);
        double r643367 = log(r643354);
        double r643368 = r643360 ? r643366 : r643367;
        double r643369 = r643356 ? r643358 : r643368;
        return r643369;
}

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 < -3.201775397353183e+150

    1. Initial program 60.5

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

    2. Taylor expanded around -inf 6.1

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

    3. Simplified6.1

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

    if -3.201775397353183e+150 < re < 7.407160440832304e+80

    1. Initial program 21.1

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

    if 7.407160440832304e+80 < re

    1. Initial program 47.3

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

    2. Taylor expanded around inf 9.9

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

  4. Final simplification17.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -3.201775397353183 \cdot 10^{+150}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le 7.407160440832304 \cdot 10^{+80}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Reproduce

herbie shell --seed 2019141 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  (log (sqrt (+ (* re re) (* im im)))))