Average Error: 31.8 → 17.1
Time: 1.1s
Precision: 64
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -6.3372341303149501 \cdot 10^{130}:\\ \;\;\;\;\log \left(-1 \cdot re\right)\\ \mathbf{elif}\;re \le 1.3249567362868981 \cdot 10^{86}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]
\log \left(\sqrt{re \cdot re + im \cdot im}\right)
↓
\begin{array}{l}
\mathbf{if}\;re \le -6.3372341303149501 \cdot 10^{130}:\\
\;\;\;\;\log \left(-1 \cdot re\right)\\

\mathbf{elif}\;re \le 1.3249567362868981 \cdot 10^{86}:\\
\;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\

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

\end{array}
double f(double re, double im) {
        double r22372 = re;
        double r22373 = r22372 * r22372;
        double r22374 = im;
        double r22375 = r22374 * r22374;
        double r22376 = r22373 + r22375;
        double r22377 = sqrt(r22376);
        double r22378 = log(r22377);
        return r22378;
}
↓
double f(double re, double im) {
        double r22379 = re;
        double r22380 = -6.33723413031495e+130;
        bool r22381 = r22379 <= r22380;
        double r22382 = -1.0;
        double r22383 = r22382 * r22379;
        double r22384 = log(r22383);
        double r22385 = 1.3249567362868981e+86;
        bool r22386 = r22379 <= r22385;
        double r22387 = r22379 * r22379;
        double r22388 = im;
        double r22389 = r22388 * r22388;
        double r22390 = r22387 + r22389;
        double r22391 = sqrt(r22390);
        double r22392 = log(r22391);
        double r22393 = log(r22379);
        double r22394 = r22386 ? r22392 : r22393;
        double r22395 = r22381 ? r22384 : r22394;
        return r22395;
}

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 < -6.33723413031495e+130

    1. Initial program 58.7

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

    2. Taylor expanded around -inf 7.4

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

    if -6.33723413031495e+130 < re < 1.3249567362868981e+86

    1. Initial program 21.3

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

    if 1.3249567362868981e+86 < re

    1. Initial program 50.0

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

    2. Taylor expanded around inf 9.2

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

  4. Final simplification17.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -6.3372341303149501 \cdot 10^{130}:\\ \;\;\;\;\log \left(-1 \cdot re\right)\\ \mathbf{elif}\;re \le 1.3249567362868981 \cdot 10^{86}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Reproduce

herbie shell --seed 2020027 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  :precision binary64
  (log (sqrt (+ (* re re) (* im im)))))