Average Error: 32.9 → 12.5
Time: 25.9s
Precision: 64
\[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x.re \le -1.333413978679484291411721441128121739411 \cdot 10^{-11}:\\ \;\;\;\;\frac{\sqrt{e^{-\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}}{{\left(\sqrt[3]{\frac{-1}{x.re}} \cdot \sqrt[3]{\frac{-1}{x.re}}\right)}^{y.re}} \cdot \frac{\sqrt{e^{-\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}}{{\left(\sqrt[3]{\frac{-1}{x.re}}\right)}^{y.re}}\\ \mathbf{elif}\;x.re \le -1.7597595967955697899862524366054126632 \cdot 10^{-130}:\\ \;\;\;\;\left(\sqrt[3]{\frac{{\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}^{y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}} \cdot \sqrt[3]{\frac{{\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}^{y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}}\right) \cdot \sqrt[3]{\frac{{\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}^{y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}}\\ \mathbf{elif}\;x.re \le -1.496550506260112750208681223280832003923 \cdot 10^{-235}:\\ \;\;\;\;\frac{\sqrt{e^{-\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}}{{\left(\sqrt[3]{\frac{-1}{x.re}} \cdot \sqrt[3]{\frac{-1}{x.re}}\right)}^{y.re}} \cdot \frac{\sqrt{e^{-\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}}{{\left(\sqrt[3]{\frac{-1}{x.re}}\right)}^{y.re}}\\ \mathbf{elif}\;x.re \le -3.11880136992072568735074703758730827833 \cdot 10^{-271}:\\ \;\;\;\;\left(\sqrt[3]{\frac{{\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}^{y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}} \cdot \sqrt[3]{\frac{{\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}^{y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}}\right) \cdot \sqrt[3]{\frac{{\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}^{y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}}\\ \mathbf{elif}\;x.re \le -1.295559869530605943313641336988999330937 \cdot 10^{-310}:\\ \;\;\;\;\frac{\sqrt{e^{-\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}}{{\left(\sqrt[3]{\frac{-1}{x.re}} \cdot \sqrt[3]{\frac{-1}{x.re}}\right)}^{y.re}} \cdot \frac{\sqrt{e^{-\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}}{{\left(\sqrt[3]{\frac{-1}{x.re}}\right)}^{y.re}}\\ \mathbf{elif}\;x.re \le 1.205799840270675840371277195994203367027 \cdot 10^{-200} \lor \neg \left(x.re \le 1.346727696157754127498599035215473163681 \cdot 10^{-171}\right):\\ \;\;\;\;e^{\left(-\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\right) + y.re \cdot \log x.re}\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log x.im \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ \end{array}\]
e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)
↓
\begin{array}{l}
\mathbf{if}\;x.re \le -1.333413978679484291411721441128121739411 \cdot 10^{-11}:\\
\;\;\;\;\frac{\sqrt{e^{-\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}}{{\left(\sqrt[3]{\frac{-1}{x.re}} \cdot \sqrt[3]{\frac{-1}{x.re}}\right)}^{y.re}} \cdot \frac{\sqrt{e^{-\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}}{{\left(\sqrt[3]{\frac{-1}{x.re}}\right)}^{y.re}}\\

\mathbf{elif}\;x.re \le -1.7597595967955697899862524366054126632 \cdot 10^{-130}:\\
\;\;\;\;\left(\sqrt[3]{\frac{{\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}^{y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}} \cdot \sqrt[3]{\frac{{\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}^{y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}}\right) \cdot \sqrt[3]{\frac{{\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}^{y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}}\\

\mathbf{elif}\;x.re \le -1.496550506260112750208681223280832003923 \cdot 10^{-235}:\\
\;\;\;\;\frac{\sqrt{e^{-\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}}{{\left(\sqrt[3]{\frac{-1}{x.re}} \cdot \sqrt[3]{\frac{-1}{x.re}}\right)}^{y.re}} \cdot \frac{\sqrt{e^{-\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}}{{\left(\sqrt[3]{\frac{-1}{x.re}}\right)}^{y.re}}\\

\mathbf{elif}\;x.re \le -3.11880136992072568735074703758730827833 \cdot 10^{-271}:\\
\;\;\;\;\left(\sqrt[3]{\frac{{\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}^{y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}} \cdot \sqrt[3]{\frac{{\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}^{y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}}\right) \cdot \sqrt[3]{\frac{{\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}^{y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}}\\

\mathbf{elif}\;x.re \le -1.295559869530605943313641336988999330937 \cdot 10^{-310}:\\
\;\;\;\;\frac{\sqrt{e^{-\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}}{{\left(\sqrt[3]{\frac{-1}{x.re}} \cdot \sqrt[3]{\frac{-1}{x.re}}\right)}^{y.re}} \cdot \frac{\sqrt{e^{-\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}}{{\left(\sqrt[3]{\frac{-1}{x.re}}\right)}^{y.re}}\\

\mathbf{elif}\;x.re \le 1.205799840270675840371277195994203367027 \cdot 10^{-200} \lor \neg \left(x.re \le 1.346727696157754127498599035215473163681 \cdot 10^{-171}\right):\\
\;\;\;\;e^{\left(-\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\right) + y.re \cdot \log x.re}\\

\mathbf{else}:\\
\;\;\;\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log x.im \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\

\end{array}
double f(double x_re, double x_im, double y_re, double y_im) {
        double r24586 = x_re;
        double r24587 = r24586 * r24586;
        double r24588 = x_im;
        double r24589 = r24588 * r24588;
        double r24590 = r24587 + r24589;
        double r24591 = sqrt(r24590);
        double r24592 = log(r24591);
        double r24593 = y_re;
        double r24594 = r24592 * r24593;
        double r24595 = atan2(r24588, r24586);
        double r24596 = y_im;
        double r24597 = r24595 * r24596;
        double r24598 = r24594 - r24597;
        double r24599 = exp(r24598);
        double r24600 = r24592 * r24596;
        double r24601 = r24595 * r24593;
        double r24602 = r24600 + r24601;
        double r24603 = cos(r24602);
        double r24604 = r24599 * r24603;
        return r24604;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r24605 = x_re;
        double r24606 = -1.3334139786794843e-11;
        bool r24607 = r24605 <= r24606;
        double r24608 = x_im;
        double r24609 = atan2(r24608, r24605);
        double r24610 = y_im;
        double r24611 = r24609 * r24610;
        double r24612 = -r24611;
        double r24613 = exp(r24612);
        double r24614 = sqrt(r24613);
        double r24615 = -1.0;
        double r24616 = r24615 / r24605;
        double r24617 = cbrt(r24616);
        double r24618 = r24617 * r24617;
        double r24619 = y_re;
        double r24620 = pow(r24618, r24619);
        double r24621 = r24614 / r24620;
        double r24622 = pow(r24617, r24619);
        double r24623 = r24614 / r24622;
        double r24624 = r24621 * r24623;
        double r24625 = -1.7597595967955698e-130;
        bool r24626 = r24605 <= r24625;
        double r24627 = r24605 * r24605;
        double r24628 = r24608 * r24608;
        double r24629 = r24627 + r24628;
        double r24630 = sqrt(r24629);
        double r24631 = pow(r24630, r24619);
        double r24632 = exp(r24611);
        double r24633 = r24631 / r24632;
        double r24634 = cbrt(r24633);
        double r24635 = r24634 * r24634;
        double r24636 = r24635 * r24634;
        double r24637 = -1.4965505062601128e-235;
        bool r24638 = r24605 <= r24637;
        double r24639 = -3.1188013699207257e-271;
        bool r24640 = r24605 <= r24639;
        double r24641 = -1.2955598695306e-310;
        bool r24642 = r24605 <= r24641;
        double r24643 = 1.2057998402706758e-200;
        bool r24644 = r24605 <= r24643;
        double r24645 = 1.3467276961577541e-171;
        bool r24646 = r24605 <= r24645;
        double r24647 = !r24646;
        bool r24648 = r24644 || r24647;
        double r24649 = log(r24605);
        double r24650 = r24619 * r24649;
        double r24651 = r24612 + r24650;
        double r24652 = exp(r24651);
        double r24653 = log(r24630);
        double r24654 = r24653 * r24619;
        double r24655 = r24654 - r24611;
        double r24656 = exp(r24655);
        double r24657 = log(r24608);
        double r24658 = r24657 * r24610;
        double r24659 = r24609 * r24619;
        double r24660 = r24658 + r24659;
        double r24661 = cos(r24660);
        double r24662 = r24656 * r24661;
        double r24663 = r24648 ? r24652 : r24662;
        double r24664 = r24642 ? r24624 : r24663;
        double r24665 = r24640 ? r24636 : r24664;
        double r24666 = r24638 ? r24624 : r24665;
        double r24667 = r24626 ? r24636 : r24666;
        double r24668 = r24607 ? r24624 : r24667;
        return r24668;
}

Error

Bits error versus x.re

Bits error versus x.im

Bits error versus y.re

Bits error versus y.im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 4 regimes

  2. if x.re < -1.3334139786794843e-11 or -1.7597595967955698e-130 < x.re < -1.4965505062601128e-235 or -3.1188013699207257e-271 < x.re < -1.2955598695306e-310

    1. Initial program 36.2

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]

    2. Taylor expanded around 0 19.7

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{1}\]

    3. Taylor expanded around -inf 4.5

      \[\leadsto \color{blue}{e^{-\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im + y.re \cdot \log \left(\frac{-1}{x.re}\right)\right)}} \cdot 1\]

    4. Simplified10.6

      \[\leadsto \color{blue}{\frac{e^{-\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}{{\left(\frac{-1}{x.re}\right)}^{y.re}}} \cdot 1\]
    5. Using strategy rm

    6. Applied add-cube-cbrt10.6

      \[\leadsto \frac{e^{-\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}{{\color{blue}{\left(\left(\sqrt[3]{\frac{-1}{x.re}} \cdot \sqrt[3]{\frac{-1}{x.re}}\right) \cdot \sqrt[3]{\frac{-1}{x.re}}\right)}}^{y.re}} \cdot 1\]

    7. Applied unpow-prod-down10.6

      \[\leadsto \frac{e^{-\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}{\color{blue}{{\left(\sqrt[3]{\frac{-1}{x.re}} \cdot \sqrt[3]{\frac{-1}{x.re}}\right)}^{y.re} \cdot {\left(\sqrt[3]{\frac{-1}{x.re}}\right)}^{y.re}}} \cdot 1\]

    8. Applied add-sqr-sqrt10.6

      \[\leadsto \frac{\color{blue}{\sqrt{e^{-\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sqrt{e^{-\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}}}{{\left(\sqrt[3]{\frac{-1}{x.re}} \cdot \sqrt[3]{\frac{-1}{x.re}}\right)}^{y.re} \cdot {\left(\sqrt[3]{\frac{-1}{x.re}}\right)}^{y.re}} \cdot 1\]

    9. Applied times-frac10.6

      \[\leadsto \color{blue}{\left(\frac{\sqrt{e^{-\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}}{{\left(\sqrt[3]{\frac{-1}{x.re}} \cdot \sqrt[3]{\frac{-1}{x.re}}\right)}^{y.re}} \cdot \frac{\sqrt{e^{-\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}}{{\left(\sqrt[3]{\frac{-1}{x.re}}\right)}^{y.re}}\right)} \cdot 1\]

    if -1.3334139786794843e-11 < x.re < -1.7597595967955698e-130 or -1.4965505062601128e-235 < x.re < -3.1188013699207257e-271

    1. Initial program 18.8

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]

    2. Taylor expanded around 0 11.0

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{1}\]
    3. Using strategy rm

    4. Applied add-cube-cbrt11.0

      \[\leadsto \color{blue}{\left(\left(\sqrt[3]{e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sqrt[3]{e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}\right) \cdot \sqrt[3]{e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}\right)} \cdot 1\]

    5. Simplified16.0

      \[\leadsto \left(\color{blue}{\left(\sqrt[3]{\frac{{\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}^{y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}} \cdot \sqrt[3]{\frac{{\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}^{y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}}\right)} \cdot \sqrt[3]{e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}\right) \cdot 1\]

    6. Simplified16.0

      \[\leadsto \left(\left(\sqrt[3]{\frac{{\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}^{y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}} \cdot \sqrt[3]{\frac{{\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}^{y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}}\right) \cdot \color{blue}{\sqrt[3]{\frac{{\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}^{y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}}}\right) \cdot 1\]

    if -1.2955598695306e-310 < x.re < 1.2057998402706758e-200 or 1.3467276961577541e-171 < x.re

    1. Initial program 34.2

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]

    2. Taylor expanded around 0 21.7

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{1}\]

    3. Taylor expanded around inf 11.6

      \[\leadsto \color{blue}{e^{-\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im + y.re \cdot \log \left(\frac{1}{x.re}\right)\right)}} \cdot 1\]

    4. Simplified11.6

      \[\leadsto \color{blue}{e^{\left(-\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\right) + y.re \cdot \log x.re}} \cdot 1\]

    if 1.2057998402706758e-200 < x.re < 1.3467276961577541e-171

    1. Initial program 30.8

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]

    2. Taylor expanded around 0 37.8

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \color{blue}{x.im} \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]
  3. Recombined 4 regimes into one program.

  4. Final simplification12.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \le -1.333413978679484291411721441128121739411 \cdot 10^{-11}:\\ \;\;\;\;\frac{\sqrt{e^{-\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}}{{\left(\sqrt[3]{\frac{-1}{x.re}} \cdot \sqrt[3]{\frac{-1}{x.re}}\right)}^{y.re}} \cdot \frac{\sqrt{e^{-\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}}{{\left(\sqrt[3]{\frac{-1}{x.re}}\right)}^{y.re}}\\ \mathbf{elif}\;x.re \le -1.7597595967955697899862524366054126632 \cdot 10^{-130}:\\ \;\;\;\;\left(\sqrt[3]{\frac{{\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}^{y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}} \cdot \sqrt[3]{\frac{{\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}^{y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}}\right) \cdot \sqrt[3]{\frac{{\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}^{y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}}\\ \mathbf{elif}\;x.re \le -1.496550506260112750208681223280832003923 \cdot 10^{-235}:\\ \;\;\;\;\frac{\sqrt{e^{-\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}}{{\left(\sqrt[3]{\frac{-1}{x.re}} \cdot \sqrt[3]{\frac{-1}{x.re}}\right)}^{y.re}} \cdot \frac{\sqrt{e^{-\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}}{{\left(\sqrt[3]{\frac{-1}{x.re}}\right)}^{y.re}}\\ \mathbf{elif}\;x.re \le -3.11880136992072568735074703758730827833 \cdot 10^{-271}:\\ \;\;\;\;\left(\sqrt[3]{\frac{{\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}^{y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}} \cdot \sqrt[3]{\frac{{\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}^{y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}}\right) \cdot \sqrt[3]{\frac{{\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}^{y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}}\\ \mathbf{elif}\;x.re \le -1.295559869530605943313641336988999330937 \cdot 10^{-310}:\\ \;\;\;\;\frac{\sqrt{e^{-\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}}{{\left(\sqrt[3]{\frac{-1}{x.re}} \cdot \sqrt[3]{\frac{-1}{x.re}}\right)}^{y.re}} \cdot \frac{\sqrt{e^{-\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}}{{\left(\sqrt[3]{\frac{-1}{x.re}}\right)}^{y.re}}\\ \mathbf{elif}\;x.re \le 1.205799840270675840371277195994203367027 \cdot 10^{-200} \lor \neg \left(x.re \le 1.346727696157754127498599035215473163681 \cdot 10^{-171}\right):\\ \;\;\;\;e^{\left(-\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\right) + y.re \cdot \log x.re}\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log x.im \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019208 
(FPCore (x.re x.im y.re y.im)
  :name "powComplex, real part"
  :precision binary64
  (* (exp (- (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.re) (* (atan2 x.im x.re) y.im))) (cos (+ (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.im) (* (atan2 x.im x.re) y.re)))))