Average Error: 32.9 → 3.4
Time: 8.3s
Precision: 64
\[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]
↓
\[e^{\log \left(e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)\right)\]
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(e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sin \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)\right)\right)
double f(double x_re, double x_im, double y_re, double y_im) {
        double r13825 = x_re;
        double r13826 = r13825 * r13825;
        double r13827 = x_im;
        double r13828 = r13827 * r13827;
        double r13829 = r13826 + r13828;
        double r13830 = sqrt(r13829);
        double r13831 = log(r13830);
        double r13832 = y_re;
        double r13833 = r13831 * r13832;
        double r13834 = atan2(r13827, r13825);
        double r13835 = y_im;
        double r13836 = r13834 * r13835;
        double r13837 = r13833 - r13836;
        double r13838 = exp(r13837);
        double r13839 = r13831 * r13835;
        double r13840 = r13834 * r13832;
        double r13841 = r13839 + r13840;
        double r13842 = sin(r13841);
        double r13843 = r13838 * r13842;
        return r13843;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r13844 = x_re;
        double r13845 = x_im;
        double r13846 = hypot(r13844, r13845);
        double r13847 = log(r13846);
        double r13848 = exp(r13847);
        double r13849 = log(r13848);
        double r13850 = y_re;
        double r13851 = r13849 * r13850;
        double r13852 = atan2(r13845, r13844);
        double r13853 = y_im;
        double r13854 = r13852 * r13853;
        double r13855 = r13851 - r13854;
        double r13856 = exp(r13855);
        double r13857 = r13847 * r13853;
        double r13858 = r13852 * r13850;
        double r13859 = r13857 + r13858;
        double r13860 = sin(r13859);
        double r13861 = log1p(r13860);
        double r13862 = expm1(r13861);
        double r13863 = r13856 * r13862;
        return r13863;
}

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. Initial program 32.9

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

  3. Applied hypot-def19.1

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

  5. Applied add-exp-log19.1

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

  6. Simplified3.4

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

  8. Applied expm1-log1p-u3.4

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

  9. Final simplification3.4

    \[\leadsto e^{\log \left(e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)\right)\]

Reproduce

herbie shell --seed 2020062 +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)))))