Average Error: 31.5 → 18.5
Time: 5.6s
Precision: 64
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -5.330091552844717472226479932066920744645 \cdot 10^{114}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le -4.215661627499373563855656419004671791113 \cdot 10^{-144}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \le 3.482912996481695209350075344359753892544 \cdot 10^{-250}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 6.50977017724907722738153182022955067076 \cdot 10^{55}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]
\log \left(\sqrt{re \cdot re + im \cdot im}\right)
↓
\begin{array}{l}
\mathbf{if}\;re \le -5.330091552844717472226479932066920744645 \cdot 10^{114}:\\
\;\;\;\;\log \left(-re\right)\\

\mathbf{elif}\;re \le -4.215661627499373563855656419004671791113 \cdot 10^{-144}:\\
\;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\

\mathbf{elif}\;re \le 3.482912996481695209350075344359753892544 \cdot 10^{-250}:\\
\;\;\;\;\log im\\

\mathbf{elif}\;re \le 6.50977017724907722738153182022955067076 \cdot 10^{55}:\\
\;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\

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

\end{array}
double f(double re, double im) {
        double r27642 = re;
        double r27643 = r27642 * r27642;
        double r27644 = im;
        double r27645 = r27644 * r27644;
        double r27646 = r27643 + r27645;
        double r27647 = sqrt(r27646);
        double r27648 = log(r27647);
        return r27648;
}
↓
double f(double re, double im) {
        double r27649 = re;
        double r27650 = -5.330091552844717e+114;
        bool r27651 = r27649 <= r27650;
        double r27652 = -r27649;
        double r27653 = log(r27652);
        double r27654 = -4.2156616274993736e-144;
        bool r27655 = r27649 <= r27654;
        double r27656 = r27649 * r27649;
        double r27657 = im;
        double r27658 = r27657 * r27657;
        double r27659 = r27656 + r27658;
        double r27660 = sqrt(r27659);
        double r27661 = log(r27660);
        double r27662 = 3.482912996481695e-250;
        bool r27663 = r27649 <= r27662;
        double r27664 = log(r27657);
        double r27665 = 6.509770177249077e+55;
        bool r27666 = r27649 <= r27665;
        double r27667 = log(r27649);
        double r27668 = r27666 ? r27661 : r27667;
        double r27669 = r27663 ? r27664 : r27668;
        double r27670 = r27655 ? r27661 : r27669;
        double r27671 = r27651 ? r27653 : r27670;
        return r27671;
}

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 < -5.330091552844717e+114

    1. Initial program 54.3

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

    2. Taylor expanded around -inf 7.4

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

    3. Simplified7.4

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

    if -5.330091552844717e+114 < re < -4.2156616274993736e-144 or 3.482912996481695e-250 < re < 6.509770177249077e+55

    1. Initial program 18.7

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

    if -4.2156616274993736e-144 < re < 3.482912996481695e-250

    1. Initial program 31.0

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

    2. Taylor expanded around 0 35.8

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

    if 6.509770177249077e+55 < re

    1. Initial program 44.0

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

    2. Taylor expanded around inf 11.1

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

  4. Final simplification18.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -5.330091552844717472226479932066920744645 \cdot 10^{114}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le -4.215661627499373563855656419004671791113 \cdot 10^{-144}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \le 3.482912996481695209350075344359753892544 \cdot 10^{-250}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 6.50977017724907722738153182022955067076 \cdot 10^{55}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Reproduce

herbie shell --seed 2019325 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  :precision binary64
  (log (sqrt (+ (* re re) (* im im)))))