Average Error: 33.2 → 3.6
Time: 39.4s
Precision: 64
\[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]
↓
\[e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(y.im, \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\]
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)
↓
e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(y.im, \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)
double f(double x_re, double x_im, double y_re, double y_im) {
        double r907713 = x_re;
        double r907714 = r907713 * r907713;
        double r907715 = x_im;
        double r907716 = r907715 * r907715;
        double r907717 = r907714 + r907716;
        double r907718 = sqrt(r907717);
        double r907719 = log(r907718);
        double r907720 = y_re;
        double r907721 = r907719 * r907720;
        double r907722 = atan2(r907715, r907713);
        double r907723 = y_im;
        double r907724 = r907722 * r907723;
        double r907725 = r907721 - r907724;
        double r907726 = exp(r907725);
        double r907727 = r907719 * r907723;
        double r907728 = r907722 * r907720;
        double r907729 = r907727 + r907728;
        double r907730 = sin(r907729);
        double r907731 = r907726 * r907730;
        return r907731;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r907732 = x_re;
        double r907733 = x_im;
        double r907734 = hypot(r907732, r907733);
        double r907735 = log(r907734);
        double r907736 = y_re;
        double r907737 = r907735 * r907736;
        double r907738 = y_im;
        double r907739 = atan2(r907733, r907732);
        double r907740 = r907738 * r907739;
        double r907741 = r907737 - r907740;
        double r907742 = exp(r907741);
        double r907743 = r907736 * r907739;
        double r907744 = fma(r907738, r907735, r907743);
        double r907745 = sin(r907744);
        double r907746 = r907742 * r907745;
        return r907746;
}

Error

Bits error versus x.re

Bits error versus x.im

Bits error versus y.re

Bits error versus y.im

Derivation

  1. Initial program 33.2

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

  2. Simplified3.6

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

  3. Taylor expanded around 0 3.6

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

  4. Final simplification3.6

    \[\leadsto e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(y.im, \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\]

Reproduce

herbie shell --seed 2019149 +o rules:numerics
(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)))))