Average Error: 30.5 → 16.5
Time: 3.5s
Precision: 64
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -1.354527584030358 \cdot 10^{+95}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le 1.2298023334030224 \cdot 10^{+93}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]
\log \left(\sqrt{re \cdot re + im \cdot im}\right)
↓
\begin{array}{l}
\mathbf{if}\;re \le -1.354527584030358 \cdot 10^{+95}:\\
\;\;\;\;\log \left(-re\right)\\

\mathbf{elif}\;re \le 1.2298023334030224 \cdot 10^{+93}:\\
\;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\

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

\end{array}
double f(double re, double im) {
        double r2095454 = re;
        double r2095455 = r2095454 * r2095454;
        double r2095456 = im;
        double r2095457 = r2095456 * r2095456;
        double r2095458 = r2095455 + r2095457;
        double r2095459 = sqrt(r2095458);
        double r2095460 = log(r2095459);
        return r2095460;
}
↓
double f(double re, double im) {
        double r2095461 = re;
        double r2095462 = -1.354527584030358e+95;
        bool r2095463 = r2095461 <= r2095462;
        double r2095464 = -r2095461;
        double r2095465 = log(r2095464);
        double r2095466 = 1.2298023334030224e+93;
        bool r2095467 = r2095461 <= r2095466;
        double r2095468 = im;
        double r2095469 = r2095468 * r2095468;
        double r2095470 = r2095461 * r2095461;
        double r2095471 = r2095469 + r2095470;
        double r2095472 = sqrt(r2095471);
        double r2095473 = log(r2095472);
        double r2095474 = log(r2095461);
        double r2095475 = r2095467 ? r2095473 : r2095474;
        double r2095476 = r2095463 ? r2095465 : r2095475;
        return r2095476;
}

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 < -1.354527584030358e+95

    1. Initial program 49.6

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

    2. Taylor expanded around -inf 8.7

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

    3. Simplified8.7

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

    if -1.354527584030358e+95 < re < 1.2298023334030224e+93

    1. Initial program 20.8

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

    if 1.2298023334030224e+93 < re

    1. Initial program 48.4

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

    2. Taylor expanded around inf 8.0

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

  4. Final simplification16.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.354527584030358 \cdot 10^{+95}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le 1.2298023334030224 \cdot 10^{+93}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Reproduce

herbie shell --seed 2019164 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  (log (sqrt (+ (* re re) (* im im)))))