powComplex, imaginary part

Average Error: 32.5 → 3.7

Time: 38.7s

Precision: 64

Math

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

↓

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

C

double f(double x_re, double x_im, double y_re, double y_im) {
        double r1108884 = x_re;
        double r1108885 = r1108884 * r1108884;
        double r1108886 = x_im;
        double r1108887 = r1108886 * r1108886;
        double r1108888 = r1108885 + r1108887;
        double r1108889 = sqrt(r1108888);
        double r1108890 = log(r1108889);
        double r1108891 = y_re;
        double r1108892 = r1108890 * r1108891;
        double r1108893 = atan2(r1108886, r1108884);
        double r1108894 = y_im;
        double r1108895 = r1108893 * r1108894;
        double r1108896 = r1108892 - r1108895;
        double r1108897 = exp(r1108896);
        double r1108898 = r1108890 * r1108894;
        double r1108899 = r1108893 * r1108891;
        double r1108900 = r1108898 + r1108899;
        double r1108901 = sin(r1108900);
        double r1108902 = r1108897 * r1108901;
        return r1108902;
}

↓

double f(double x_re, double x_im, double y_re, double y_im) {
        double r1108903 = x_im;
        double r1108904 = x_re;
        double r1108905 = atan2(r1108903, r1108904);
        double r1108906 = y_re;
        double r1108907 = r1108905 * r1108906;
        double r1108908 = cos(r1108907);
        double r1108909 = y_im;
        double r1108910 = hypot(r1108904, r1108903);
        double r1108911 = cbrt(r1108910);
        double r1108912 = log(r1108911);
        double r1108913 = r1108909 * r1108912;
        double r1108914 = sin(r1108913);
        double r1108915 = r1108911 * r1108911;
        double r1108916 = log(r1108915);
        double r1108917 = r1108909 * r1108916;
        double r1108918 = cos(r1108917);
        double r1108919 = r1108914 * r1108918;
        double r1108920 = sin(r1108917);
        double r1108921 = cbrt(r1108911);
        double r1108922 = r1108921 * r1108921;
        double r1108923 = log(r1108922);
        double r1108924 = r1108923 * r1108909;
        double r1108925 = cos(r1108924);
        double r1108926 = log(r1108921);
        double r1108927 = r1108909 * r1108926;
        double r1108928 = cos(r1108927);
        double r1108929 = r1108925 * r1108928;
        double r1108930 = sin(r1108924);
        double r1108931 = /* ERROR: no posit support in C */;
        double r1108932 = /* ERROR: no posit support in C */;
        double r1108933 = sin(r1108927);
        double r1108934 = r1108932 * r1108933;
        double r1108935 = r1108929 - r1108934;
        double r1108936 = r1108920 * r1108935;
        double r1108937 = r1108919 + r1108936;
        double r1108938 = r1108908 * r1108937;
        double r1108939 = sin(r1108907);
        double r1108940 = log(r1108910);
        double r1108941 = r1108940 * r1108909;
        double r1108942 = cos(r1108941);
        double r1108943 = r1108939 * r1108942;
        double r1108944 = r1108938 + r1108943;
        double r1108945 = r1108905 * r1108909;
        double r1108946 = r1108906 * r1108940;
        double r1108947 = r1108945 - r1108946;
        double r1108948 = exp(r1108947);
        double r1108949 = r1108944 / r1108948;
        return r1108949;
}

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 32.5

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

2. Simplified 3.3

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

4. Applied fma-udef 3.4

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

5. Applied sin-sum 3.4

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

7. Applied add-cube-cbrt 3.3

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

8. Applied log-prod 3.4

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

9. Applied distribute-lft-in 3.4

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

10. Applied sin-sum 3.4

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

12. Applied add-cube-cbrt 3.4

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

13. Applied log-prod 3.4

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

14. Applied distribute-rgt-in 3.4

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

15. Applied cos-sum 3.4

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

17. Applied insert-posit16 3.7

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

18. Final simplification 3.7

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

Reproduce

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