powComplex, imaginary part

Average Error: 33.3 → 5.8

Time: 16.8s

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 r23789 = x_re;
        double r23790 = r23789 * r23789;
        double r23791 = x_im;
        double r23792 = r23791 * r23791;
        double r23793 = r23790 + r23792;
        double r23794 = sqrt(r23793);
        double r23795 = log(r23794);
        double r23796 = y_re;
        double r23797 = r23795 * r23796;
        double r23798 = atan2(r23791, r23789);
        double r23799 = y_im;
        double r23800 = r23798 * r23799;
        double r23801 = r23797 - r23800;
        double r23802 = exp(r23801);
        double r23803 = r23795 * r23799;
        double r23804 = r23798 * r23796;
        double r23805 = r23803 + r23804;
        double r23806 = sin(r23805);
        double r23807 = r23802 * r23806;
        return r23807;
}

↓

double f(double x_re, double x_im, double y_re, double y_im) {
        double r23808 = x_re;
        double r23809 = r23808 * r23808;
        double r23810 = x_im;
        double r23811 = r23810 * r23810;
        double r23812 = r23809 + r23811;
        double r23813 = sqrt(r23812);
        double r23814 = log(r23813);
        double r23815 = y_re;
        double r23816 = r23814 * r23815;
        double r23817 = atan2(r23810, r23808);
        double r23818 = y_im;
        double r23819 = r23817 * r23818;
        double r23820 = r23816 - r23819;
        double r23821 = exp(r23820);
        double r23822 = r23814 * r23818;
        double r23823 = r23817 * r23815;
        double r23824 = r23822 + r23823;
        double r23825 = sin(r23824);
        double r23826 = r23821 * r23825;
        double r23827 = -0.0;
        bool r23828 = r23826 <= r23827;
        double r23829 = hypot(r23808, r23810);
        double r23830 = pow(r23829, r23815);
        double r23831 = log(r23829);
        double r23832 = r23831 * r23818;
        double r23833 = sin(r23832);
        double r23834 = log1p(r23833);
        double r23835 = expm1(r23834);
        double r23836 = cos(r23823);
        double r23837 = r23835 * r23836;
        double r23838 = cos(r23832);
        double r23839 = sin(r23823);
        double r23840 = r23838 * r23839;
        double r23841 = r23837 + r23840;
        double r23842 = r23830 * r23841;
        double r23843 = cbrt(r23818);
        double r23844 = cbrt(r23843);
        double r23845 = r23844 * r23844;
        double r23846 = r23845 * r23844;
        double r23847 = r23843 * r23846;
        double r23848 = r23817 * r23847;
        double r23849 = r23848 * r23843;
        double r23850 = exp(r23849);
        double r23851 = r23842 / r23850;
        double r23852 = r23828 ? r23826 : r23851;
        return r23852;
}

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)))))