Average Error: 33.2 → 6.0
Time: 19.9s
Precision: 64
\[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \le -0.0:\\ \;\;\;\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re} \cdot \left(\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im\right)\right)\right) \cdot \cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) + \cos \left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im\right) \cdot \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}\\ \end{array}\]
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}\;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) \le -0.0:\\
\;\;\;\;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)\\

\mathbf{else}:\\
\;\;\;\;\frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re} \cdot \left(\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im\right)\right)\right) \cdot \cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) + \cos \left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im\right) \cdot \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}\\

\end{array}
double f(double x_re, double x_im, double y_re, double y_im) {
        double r19819 = x_re;
        double r19820 = r19819 * r19819;
        double r19821 = x_im;
        double r19822 = r19821 * r19821;
        double r19823 = r19820 + r19822;
        double r19824 = sqrt(r19823);
        double r19825 = log(r19824);
        double r19826 = y_re;
        double r19827 = r19825 * r19826;
        double r19828 = atan2(r19821, r19819);
        double r19829 = y_im;
        double r19830 = r19828 * r19829;
        double r19831 = r19827 - r19830;
        double r19832 = exp(r19831);
        double r19833 = r19825 * r19829;
        double r19834 = r19828 * r19826;
        double r19835 = r19833 + r19834;
        double r19836 = sin(r19835);
        double r19837 = r19832 * r19836;
        return r19837;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r19838 = x_re;
        double r19839 = r19838 * r19838;
        double r19840 = x_im;
        double r19841 = r19840 * r19840;
        double r19842 = r19839 + r19841;
        double r19843 = sqrt(r19842);
        double r19844 = log(r19843);
        double r19845 = y_re;
        double r19846 = r19844 * r19845;
        double r19847 = atan2(r19840, r19838);
        double r19848 = y_im;
        double r19849 = r19847 * r19848;
        double r19850 = r19846 - r19849;
        double r19851 = exp(r19850);
        double r19852 = r19844 * r19848;
        double r19853 = r19847 * r19845;
        double r19854 = r19852 + r19853;
        double r19855 = sin(r19854);
        double r19856 = r19851 * r19855;
        double r19857 = -0.0;
        bool r19858 = r19856 <= r19857;
        double r19859 = hypot(r19838, r19840);
        double r19860 = pow(r19859, r19845);
        double r19861 = log(r19859);
        double r19862 = r19861 * r19848;
        double r19863 = sin(r19862);
        double r19864 = expm1(r19863);
        double r19865 = log1p(r19864);
        double r19866 = cos(r19853);
        double r19867 = r19865 * r19866;
        double r19868 = cos(r19862);
        double r19869 = sin(r19853);
        double r19870 = r19868 * r19869;
        double r19871 = r19867 + r19870;
        double r19872 = r19860 * r19871;
        double r19873 = exp(r19849);
        double r19874 = r19872 / r19873;
        double r19875 = r19858 ? r19856 : r19874;
        return r19875;
}

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 (* (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)))) < -0.0

    1. Initial program 2.0

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

    if -0.0 < (* (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))))

    1. Initial program 53.6

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

    2. Simplified8.6

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

    4. Applied fma-udef8.6

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

    5. Applied sin-sum8.6

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

    7. Applied log1p-expm1-u8.6

      \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re} \cdot \left(\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im\right)\right)\right)} \cdot \cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) + \cos \left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im\right) \cdot \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification6.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \le -0.0:\\ \;\;\;\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re} \cdot \left(\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im\right)\right)\right) \cdot \cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) + \cos \left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im\right) \cdot \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020045 +o rules:numerics
(FPCore (x.re x.im y.re y.im)
  :name "powComplex, imaginary part"
  :precision binary64
  (* (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)))))