Average Error: 31.5 → 18.5
Time: 5.3s
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 r27517 = re;
        double r27518 = r27517 * r27517;
        double r27519 = im;
        double r27520 = r27519 * r27519;
        double r27521 = r27518 + r27520;
        double r27522 = sqrt(r27521);
        double r27523 = log(r27522);
        return r27523;
}
↓
double f(double re, double im) {
        double r27524 = re;
        double r27525 = -5.330091552844717e+114;
        bool r27526 = r27524 <= r27525;
        double r27527 = -r27524;
        double r27528 = log(r27527);
        double r27529 = -4.2156616274993736e-144;
        bool r27530 = r27524 <= r27529;
        double r27531 = r27524 * r27524;
        double r27532 = im;
        double r27533 = r27532 * r27532;
        double r27534 = r27531 + r27533;
        double r27535 = sqrt(r27534);
        double r27536 = log(r27535);
        double r27537 = 3.482912996481695e-250;
        bool r27538 = r27524 <= r27537;
        double r27539 = log(r27532);
        double r27540 = 6.509770177249077e+55;
        bool r27541 = r27524 <= r27540;
        double r27542 = log(r27524);
        double r27543 = r27541 ? r27536 : r27542;
        double r27544 = r27538 ? r27539 : r27543;
        double r27545 = r27530 ? r27536 : r27544;
        double r27546 = r27526 ? r27528 : r27545;
        return r27546;
}

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)))))