Average Error: 31.5 → 17.0
Time: 3.6s
Precision: 64
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -1.181793183213821728908776663248811693415 \cdot 10^{151}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le 7.392440833541333777660561627276981553815 \cdot 10^{126}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]
\log \left(\sqrt{re \cdot re + im \cdot im}\right)
↓
\begin{array}{l}
\mathbf{if}\;re \le -1.181793183213821728908776663248811693415 \cdot 10^{151}:\\
\;\;\;\;\log \left(-re\right)\\

\mathbf{elif}\;re \le 7.392440833541333777660561627276981553815 \cdot 10^{126}:\\
\;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\

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

\end{array}
double f(double re, double im) {
        double r2288375 = re;
        double r2288376 = r2288375 * r2288375;
        double r2288377 = im;
        double r2288378 = r2288377 * r2288377;
        double r2288379 = r2288376 + r2288378;
        double r2288380 = sqrt(r2288379);
        double r2288381 = log(r2288380);
        return r2288381;
}
↓
double f(double re, double im) {
        double r2288382 = re;
        double r2288383 = -1.1817931832138217e+151;
        bool r2288384 = r2288382 <= r2288383;
        double r2288385 = -r2288382;
        double r2288386 = log(r2288385);
        double r2288387 = 7.392440833541334e+126;
        bool r2288388 = r2288382 <= r2288387;
        double r2288389 = r2288382 * r2288382;
        double r2288390 = im;
        double r2288391 = r2288390 * r2288390;
        double r2288392 = r2288389 + r2288391;
        double r2288393 = sqrt(r2288392);
        double r2288394 = log(r2288393);
        double r2288395 = log(r2288382);
        double r2288396 = r2288388 ? r2288394 : r2288395;
        double r2288397 = r2288384 ? r2288386 : r2288396;
        return r2288397;
}

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 < -1.1817931832138217e+151

    1. Initial program 63.1

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

    2. Taylor expanded around -inf 7.1

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

    3. Simplified7.1

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

    if -1.1817931832138217e+151 < re < 7.392440833541334e+126

    1. Initial program 20.7

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

    if 7.392440833541334e+126 < re

    1. Initial program 56.4

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

    2. Taylor expanded around inf 7.8

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

  4. Final simplification17.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.181793183213821728908776663248811693415 \cdot 10^{151}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le 7.392440833541333777660561627276981553815 \cdot 10^{126}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  (log (sqrt (+ (* re re) (* im im)))))