Average Error: 33.0 → 22.1
Time: 27.3s
Precision: 64
\[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x.re \le -5.189251590720128421046416075251281086703 \cdot 10^{-310}:\\ \;\;\;\;\sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re + \log \left(-x.re\right) \cdot y.im\right) \cdot e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \sqrt[3]{y.im} \cdot \left(\left(\sqrt[3]{y.im} \cdot \sqrt[3]{y.im}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\log x.re \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \sqrt[3]{y.im} \cdot \left(\left(\sqrt[3]{y.im} \cdot \sqrt[3]{y.im}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\\ \end{array}\]
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 \sin \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 -5.189251590720128421046416075251281086703 \cdot 10^{-310}:\\
\;\;\;\;\sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re + \log \left(-x.re\right) \cdot y.im\right) \cdot e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \sqrt[3]{y.im} \cdot \left(\left(\sqrt[3]{y.im} \cdot \sqrt[3]{y.im}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\\

\mathbf{else}:\\
\;\;\;\;\sin \left(\log x.re \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \sqrt[3]{y.im} \cdot \left(\left(\sqrt[3]{y.im} \cdot \sqrt[3]{y.im}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\\

\end{array}
double f(double x_re, double x_im, double y_re, double y_im) {
        double r29671 = x_re;
        double r29672 = r29671 * r29671;
        double r29673 = x_im;
        double r29674 = r29673 * r29673;
        double r29675 = r29672 + r29674;
        double r29676 = sqrt(r29675);
        double r29677 = log(r29676);
        double r29678 = y_re;
        double r29679 = r29677 * r29678;
        double r29680 = atan2(r29673, r29671);
        double r29681 = y_im;
        double r29682 = r29680 * r29681;
        double r29683 = r29679 - r29682;
        double r29684 = exp(r29683);
        double r29685 = r29677 * r29681;
        double r29686 = r29680 * r29678;
        double r29687 = r29685 + r29686;
        double r29688 = sin(r29687);
        double r29689 = r29684 * r29688;
        return r29689;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r29690 = x_re;
        double r29691 = -5.18925159072013e-310;
        bool r29692 = r29690 <= r29691;
        double r29693 = x_im;
        double r29694 = atan2(r29693, r29690);
        double r29695 = y_re;
        double r29696 = r29694 * r29695;
        double r29697 = -r29690;
        double r29698 = log(r29697);
        double r29699 = y_im;
        double r29700 = r29698 * r29699;
        double r29701 = r29696 + r29700;
        double r29702 = sin(r29701);
        double r29703 = r29690 * r29690;
        double r29704 = r29693 * r29693;
        double r29705 = r29703 + r29704;
        double r29706 = sqrt(r29705);
        double r29707 = log(r29706);
        double r29708 = r29695 * r29707;
        double r29709 = cbrt(r29699);
        double r29710 = r29709 * r29709;
        double r29711 = r29710 * r29694;
        double r29712 = r29709 * r29711;
        double r29713 = r29708 - r29712;
        double r29714 = exp(r29713);
        double r29715 = r29702 * r29714;
        double r29716 = log(r29690);
        double r29717 = r29716 * r29699;
        double r29718 = r29717 + r29696;
        double r29719 = sin(r29718);
        double r29720 = r29719 * r29714;
        double r29721 = r29692 ? r29715 : r29720;
        return r29721;
}

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 2 regimes

  2. if x.re < -5.18925159072013e-310

    1. Initial program 31.7

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

    3. Applied add-cube-cbrt31.7

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

    4. Applied associate-*r*31.7

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

    5. Taylor expanded around -inf 20.8

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

    6. Simplified20.8

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

    if -5.18925159072013e-310 < x.re

    1. Initial program 34.3

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

    3. Applied add-cube-cbrt34.3

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

    4. Applied associate-*r*34.3

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

    5. Taylor expanded around inf 23.4

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

    6. Simplified23.4

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(\sqrt[3]{y.im} \cdot \sqrt[3]{y.im}\right)\right) \cdot \sqrt[3]{y.im}} \cdot \color{blue}{\sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re + y.im \cdot \log x.re\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification22.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \le -5.189251590720128421046416075251281086703 \cdot 10^{-310}:\\ \;\;\;\;\sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re + \log \left(-x.re\right) \cdot y.im\right) \cdot e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \sqrt[3]{y.im} \cdot \left(\left(\sqrt[3]{y.im} \cdot \sqrt[3]{y.im}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\log x.re \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \sqrt[3]{y.im} \cdot \left(\left(\sqrt[3]{y.im} \cdot \sqrt[3]{y.im}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 
(FPCore (x.re x.im y.re y.im)
  :name "powComplex, imaginary part"
  (* (exp (- (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.re) (* (atan2 x.im x.re) y.im))) (sin (+ (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.im) (* (atan2 x.im x.re) y.re)))))