Average Error: 31.5 → 17.5
Time: 1.2s
Precision: 64
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -6.95178634438711926 \cdot 10^{80}:\\ \;\;\;\;\log \left(-1 \cdot re\right)\\ \mathbf{elif}\;re \le -8.2687035626692377 \cdot 10^{-252}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \le 1.24277237965156329 \cdot 10^{-264}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 5.18134830580201715 \cdot 10^{95}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]
\log \left(\sqrt{re \cdot re + im \cdot im}\right)
↓
\begin{array}{l}
\mathbf{if}\;re \le -6.95178634438711926 \cdot 10^{80}:\\
\;\;\;\;\log \left(-1 \cdot re\right)\\

\mathbf{elif}\;re \le -8.2687035626692377 \cdot 10^{-252}:\\
\;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\

\mathbf{elif}\;re \le 1.24277237965156329 \cdot 10^{-264}:\\
\;\;\;\;\log im\\

\mathbf{elif}\;re \le 5.18134830580201715 \cdot 10^{95}:\\
\;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\

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

\end{array}
double f(double re, double im) {
        double r33636 = re;
        double r33637 = r33636 * r33636;
        double r33638 = im;
        double r33639 = r33638 * r33638;
        double r33640 = r33637 + r33639;
        double r33641 = sqrt(r33640);
        double r33642 = log(r33641);
        return r33642;
}
↓
double f(double re, double im) {
        double r33643 = re;
        double r33644 = -6.951786344387119e+80;
        bool r33645 = r33643 <= r33644;
        double r33646 = -1.0;
        double r33647 = r33646 * r33643;
        double r33648 = log(r33647);
        double r33649 = -8.268703562669238e-252;
        bool r33650 = r33643 <= r33649;
        double r33651 = r33643 * r33643;
        double r33652 = im;
        double r33653 = r33652 * r33652;
        double r33654 = r33651 + r33653;
        double r33655 = sqrt(r33654);
        double r33656 = log(r33655);
        double r33657 = 1.2427723796515633e-264;
        bool r33658 = r33643 <= r33657;
        double r33659 = log(r33652);
        double r33660 = 5.181348305802017e+95;
        bool r33661 = r33643 <= r33660;
        double r33662 = log(r33643);
        double r33663 = r33661 ? r33656 : r33662;
        double r33664 = r33658 ? r33659 : r33663;
        double r33665 = r33650 ? r33656 : r33664;
        double r33666 = r33645 ? r33648 : r33665;
        return r33666;
}

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 4 regimes

  2. if re < -6.951786344387119e+80

    1. Initial program 47.9

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

    2. Taylor expanded around -inf 9.3

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

    if -6.951786344387119e+80 < re < -8.268703562669238e-252 or 1.2427723796515633e-264 < re < 5.181348305802017e+95

    1. Initial program 20.4

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

    if -8.268703562669238e-252 < re < 1.2427723796515633e-264

    1. Initial program 32.4

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

    2. Taylor expanded around 0 34.3

      \[\leadsto \log \color{blue}{im}\]

    if 5.181348305802017e+95 < re

    1. Initial program 50.3

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

    2. Taylor expanded around inf 8.9

      \[\leadsto \log \color{blue}{re}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification17.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -6.95178634438711926 \cdot 10^{80}:\\ \;\;\;\;\log \left(-1 \cdot re\right)\\ \mathbf{elif}\;re \le -8.2687035626692377 \cdot 10^{-252}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \le 1.24277237965156329 \cdot 10^{-264}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 5.18134830580201715 \cdot 10^{95}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Reproduce

herbie shell --seed 2020003 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  :precision binary64
  (log (sqrt (+ (* re re) (* im im)))))