powComplex, imaginary part

Average Error: 33.3 → 5.8

Time: 21.6s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \le -0.0:\\ \;\;\;\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re} \cdot \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im\right)\right)\right) \cdot \cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) + \cos \left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im\right) \cdot \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)}{e^{\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(\sqrt[3]{y.im} \cdot \left(\left(\sqrt[3]{\sqrt[3]{y.im}} \cdot \sqrt[3]{\sqrt[3]{y.im}}\right) \cdot \sqrt[3]{\sqrt[3]{y.im}}\right)\right)\right) \cdot \sqrt[3]{y.im}}}\\ \end{array}\]

C

double f(double x_re, double x_im, double y_re, double y_im) {
        double r22877 = x_re;
        double r22878 = r22877 * r22877;
        double r22879 = x_im;
        double r22880 = r22879 * r22879;
        double r22881 = r22878 + r22880;
        double r22882 = sqrt(r22881);
        double r22883 = log(r22882);
        double r22884 = y_re;
        double r22885 = r22883 * r22884;
        double r22886 = atan2(r22879, r22877);
        double r22887 = y_im;
        double r22888 = r22886 * r22887;
        double r22889 = r22885 - r22888;
        double r22890 = exp(r22889);
        double r22891 = r22883 * r22887;
        double r22892 = r22886 * r22884;
        double r22893 = r22891 + r22892;
        double r22894 = sin(r22893);
        double r22895 = r22890 * r22894;
        return r22895;
}

↓

double f(double x_re, double x_im, double y_re, double y_im) {
        double r22896 = x_re;
        double r22897 = r22896 * r22896;
        double r22898 = x_im;
        double r22899 = r22898 * r22898;
        double r22900 = r22897 + r22899;
        double r22901 = sqrt(r22900);
        double r22902 = log(r22901);
        double r22903 = y_re;
        double r22904 = r22902 * r22903;
        double r22905 = atan2(r22898, r22896);
        double r22906 = y_im;
        double r22907 = r22905 * r22906;
        double r22908 = r22904 - r22907;
        double r22909 = exp(r22908);
        double r22910 = r22902 * r22906;
        double r22911 = r22905 * r22903;
        double r22912 = r22910 + r22911;
        double r22913 = sin(r22912);
        double r22914 = r22909 * r22913;
        double r22915 = -0.0;
        bool r22916 = r22914 <= r22915;
        double r22917 = hypot(r22896, r22898);
        double r22918 = pow(r22917, r22903);
        double r22919 = log(r22917);
        double r22920 = r22919 * r22906;
        double r22921 = sin(r22920);
        double r22922 = log1p(r22921);
        double r22923 = expm1(r22922);
        double r22924 = cos(r22911);
        double r22925 = r22923 * r22924;
        double r22926 = cos(r22920);
        double r22927 = sin(r22911);
        double r22928 = r22926 * r22927;
        double r22929 = r22925 + r22928;
        double r22930 = r22918 * r22929;
        double r22931 = cbrt(r22906);
        double r22932 = cbrt(r22931);
        double r22933 = r22932 * r22932;
        double r22934 = r22933 * r22932;
        double r22935 = r22931 * r22934;
        double r22936 = r22905 * r22935;
        double r22937 = r22936 * r22931;
        double r22938 = exp(r22937);
        double r22939 = r22930 / r22938;
        double r22940 = r22916 ? r22914 : r22939;
        return r22940;
}

Error

Bits error versus x.re

Bits error versus x.im

Bits error versus y.re

Bits error versus y.im

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 1.7

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

   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. Simplified 8.4

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

   4. Applied add-cube-cbrt 8.4

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

   5. Applied associate-*r* 8.4

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

   7. Applied add-cube-cbrt 8.4

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

   9. Applied fma-udef 8.4

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

   10. Applied sin-sum 8.4

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

   12. Applied expm1-log1p-u 8.4

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

4. Final simplification 5.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \le -0.0:\\ \;\;\;\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re} \cdot \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im\right)\right)\right) \cdot \cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) + \cos \left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im\right) \cdot \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)}{e^{\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(\sqrt[3]{y.im} \cdot \left(\left(\sqrt[3]{\sqrt[3]{y.im}} \cdot \sqrt[3]{\sqrt[3]{y.im}}\right) \cdot \sqrt[3]{\sqrt[3]{y.im}}\right)\right)\right) \cdot \sqrt[3]{y.im}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 +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)))))