Average Error: 30.8 → 16.9
Time: 4.0s
Precision: 64
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -3.510504290224585 \cdot 10^{+110}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le 2.9670775722691427 \cdot 10^{+83}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]
\log \left(\sqrt{re \cdot re + im \cdot im}\right)
↓
\begin{array}{l}
\mathbf{if}\;re \le -3.510504290224585 \cdot 10^{+110}:\\
\;\;\;\;\log \left(-re\right)\\

\mathbf{elif}\;re \le 2.9670775722691427 \cdot 10^{+83}:\\
\;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\

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

\end{array}
double f(double re, double im) {
        double r1671356 = re;
        double r1671357 = r1671356 * r1671356;
        double r1671358 = im;
        double r1671359 = r1671358 * r1671358;
        double r1671360 = r1671357 + r1671359;
        double r1671361 = sqrt(r1671360);
        double r1671362 = log(r1671361);
        return r1671362;
}
↓
double f(double re, double im) {
        double r1671363 = re;
        double r1671364 = -3.510504290224585e+110;
        bool r1671365 = r1671363 <= r1671364;
        double r1671366 = -r1671363;
        double r1671367 = log(r1671366);
        double r1671368 = 2.9670775722691427e+83;
        bool r1671369 = r1671363 <= r1671368;
        double r1671370 = im;
        double r1671371 = r1671370 * r1671370;
        double r1671372 = r1671363 * r1671363;
        double r1671373 = r1671371 + r1671372;
        double r1671374 = sqrt(r1671373);
        double r1671375 = log(r1671374);
        double r1671376 = log(r1671363);
        double r1671377 = r1671369 ? r1671375 : r1671376;
        double r1671378 = r1671365 ? r1671367 : r1671377;
        return r1671378;
}

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 < -3.510504290224585e+110

    1. Initial program 51.3

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

    2. Taylor expanded around -inf 7.5

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

    3. Simplified7.5

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

    if -3.510504290224585e+110 < re < 2.9670775722691427e+83

    1. Initial program 21.2

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

    if 2.9670775722691427e+83 < re

    1. Initial program 46.4

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

    2. Taylor expanded around inf 9.8

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

  4. Final simplification16.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -3.510504290224585 \cdot 10^{+110}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le 2.9670775722691427 \cdot 10^{+83}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Reproduce

herbie shell --seed 2019168 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  (log (sqrt (+ (* re re) (* im im)))))