powComplex, real part

Average Error: 32.2 → 3.2

Time: 39.0s

Precision: 64

Math

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

↓

\[\frac{\cos \left(\mathsf{fma}\left(y.im, \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), \left(\left(\left(\left(\left(\sqrt[3]{\sqrt{\sqrt[3]{\sqrt[3]{y.re} \cdot \sqrt[3]{y.re}}}} \cdot \sqrt[3]{\sqrt{\sqrt[3]{\sqrt[3]{y.re} \cdot \sqrt[3]{y.re}}}}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\sqrt[3]{\sqrt{\sqrt[3]{\sqrt[3]{y.re} \cdot \sqrt[3]{y.re}}}} \cdot \sqrt[3]{\sqrt{\sqrt[3]{\sqrt[3]{y.re} \cdot \sqrt[3]{y.re}}}}\right)\right) \cdot \left(\sqrt[3]{\sqrt[3]{\sqrt[3]{y.re}} \cdot \sqrt[3]{y.re}} \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{y.re}} \cdot \sqrt[3]{y.re}}\right)\right) \cdot \sqrt[3]{\sqrt[3]{y.re} \cdot \sqrt[3]{y.re}}\right) \cdot \sqrt[3]{y.re}\right)\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im - y.re \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}}\]

C

double f(double x_re, double x_im, double y_re, double y_im) {
        double r1512908 = x_re;
        double r1512909 = r1512908 * r1512908;
        double r1512910 = x_im;
        double r1512911 = r1512910 * r1512910;
        double r1512912 = r1512909 + r1512911;
        double r1512913 = sqrt(r1512912);
        double r1512914 = log(r1512913);
        double r1512915 = y_re;
        double r1512916 = r1512914 * r1512915;
        double r1512917 = atan2(r1512910, r1512908);
        double r1512918 = y_im;
        double r1512919 = r1512917 * r1512918;
        double r1512920 = r1512916 - r1512919;
        double r1512921 = exp(r1512920);
        double r1512922 = r1512914 * r1512918;
        double r1512923 = r1512917 * r1512915;
        double r1512924 = r1512922 + r1512923;
        double r1512925 = cos(r1512924);
        double r1512926 = r1512921 * r1512925;
        return r1512926;
}

↓

double f(double x_re, double x_im, double y_re, double y_im) {
        double r1512927 = y_im;
        double r1512928 = x_re;
        double r1512929 = x_im;
        double r1512930 = hypot(r1512928, r1512929);
        double r1512931 = log(r1512930);
        double r1512932 = y_re;
        double r1512933 = cbrt(r1512932);
        double r1512934 = r1512933 * r1512933;
        double r1512935 = cbrt(r1512934);
        double r1512936 = sqrt(r1512935);
        double r1512937 = cbrt(r1512936);
        double r1512938 = r1512937 * r1512937;
        double r1512939 = atan2(r1512929, r1512928);
        double r1512940 = r1512938 * r1512939;
        double r1512941 = r1512940 * r1512938;
        double r1512942 = cbrt(r1512933);
        double r1512943 = r1512942 * r1512933;
        double r1512944 = cbrt(r1512943);
        double r1512945 = r1512944 * r1512944;
        double r1512946 = r1512941 * r1512945;
        double r1512947 = r1512946 * r1512935;
        double r1512948 = r1512947 * r1512933;
        double r1512949 = fma(r1512927, r1512931, r1512948);
        double r1512950 = cos(r1512949);
        double r1512951 = r1512939 * r1512927;
        double r1512952 = r1512932 * r1512931;
        double r1512953 = r1512951 - r1512952;
        double r1512954 = exp(r1512953);
        double r1512955 = r1512950 / r1512954;
        return r1512955;
}

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 32.2

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

2. Simplified 3.2

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

4. Applied add-cube-cbrt 3.2

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

5. Applied associate-*r* 3.2

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

7. Applied add-cube-cbrt 3.2

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

8. Applied associate-*r* 3.2

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

10. Applied add-cube-cbrt 3.2

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

11. Applied cbrt-prod 3.2

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

12. Applied associate-*l* 3.2

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

13. Applied cbrt-prod 3.2

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

14. Applied add-cube-cbrt 3.2

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

15. Applied cbrt-prod 3.2

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

16. Applied associate-*l* 3.2

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

17. Applied cbrt-prod 3.2

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

18. Applied swap-sqr 3.2

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

19. Applied associate-*r* 3.2

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

21. Applied add-sqr-sqrt 3.2

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

22. Applied cbrt-prod 3.2

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

23. Applied add-sqr-sqrt 3.2

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

24. Applied cbrt-prod 3.2

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

25. Applied swap-sqr 3.2

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

26. Applied associate-*r* 3.2

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

27. Final simplification 3.2

    \[\leadsto \frac{\cos \left(\mathsf{fma}\left(y.im, \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), \left(\left(\left(\left(\left(\sqrt[3]{\sqrt{\sqrt[3]{\sqrt[3]{y.re} \cdot \sqrt[3]{y.re}}}} \cdot \sqrt[3]{\sqrt{\sqrt[3]{\sqrt[3]{y.re} \cdot \sqrt[3]{y.re}}}}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\sqrt[3]{\sqrt{\sqrt[3]{\sqrt[3]{y.re} \cdot \sqrt[3]{y.re}}}} \cdot \sqrt[3]{\sqrt{\sqrt[3]{\sqrt[3]{y.re} \cdot \sqrt[3]{y.re}}}}\right)\right) \cdot \left(\sqrt[3]{\sqrt[3]{\sqrt[3]{y.re}} \cdot \sqrt[3]{y.re}} \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{y.re}} \cdot \sqrt[3]{y.re}}\right)\right) \cdot \sqrt[3]{\sqrt[3]{y.re} \cdot \sqrt[3]{y.re}}\right) \cdot \sqrt[3]{y.re}\right)\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im - y.re \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}}\]

Reproduce

herbie shell --seed 2019158 +o rules:numerics
(FPCore (x.re x.im y.re y.im)
  :name "powComplex, real part"
  (* (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)))))