powComplex, imaginary part

Average Error: 33.6 → 6.2

Time: 31.5s

Precision: 64

Math

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

↓

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

C

double f(double x_re, double x_im, double y_re, double y_im) {
        double r24733 = x_re;
        double r24734 = r24733 * r24733;
        double r24735 = x_im;
        double r24736 = r24735 * r24735;
        double r24737 = r24734 + r24736;
        double r24738 = sqrt(r24737);
        double r24739 = log(r24738);
        double r24740 = y_re;
        double r24741 = r24739 * r24740;
        double r24742 = atan2(r24735, r24733);
        double r24743 = y_im;
        double r24744 = r24742 * r24743;
        double r24745 = r24741 - r24744;
        double r24746 = exp(r24745);
        double r24747 = r24739 * r24743;
        double r24748 = r24742 * r24740;
        double r24749 = r24747 + r24748;
        double r24750 = sin(r24749);
        double r24751 = r24746 * r24750;
        return r24751;
}

↓

double f(double x_re, double x_im, double y_re, double y_im) {
        double r24752 = x_re;
        double r24753 = r24752 * r24752;
        double r24754 = x_im;
        double r24755 = r24754 * r24754;
        double r24756 = r24753 + r24755;
        double r24757 = sqrt(r24756);
        double r24758 = log(r24757);
        double r24759 = y_re;
        double r24760 = r24758 * r24759;
        double r24761 = atan2(r24754, r24752);
        double r24762 = y_im;
        double r24763 = r24761 * r24762;
        double r24764 = r24760 - r24763;
        double r24765 = exp(r24764);
        double r24766 = r24758 * r24762;
        double r24767 = r24761 * r24759;
        double r24768 = r24766 + r24767;
        double r24769 = sin(r24768);
        double r24770 = r24765 * r24769;
        double r24771 = 0.0;
        bool r24772 = r24770 <= r24771;
        double r24773 = hypot(r24752, r24754);
        double r24774 = log(r24773);
        double r24775 = r24774 * r24762;
        double r24776 = sin(r24775);
        double r24777 = cos(r24767);
        double r24778 = r24776 * r24777;
        double r24779 = cos(r24775);
        double r24780 = sin(r24767);
        double r24781 = r24779 * r24780;
        double r24782 = r24778 + r24781;
        double r24783 = log1p(r24782);
        double r24784 = expm1(r24783);
        double r24785 = pow(r24773, r24759);
        double r24786 = cbrt(r24761);
        double r24787 = r24786 * r24786;
        double r24788 = r24786 * r24762;
        double r24789 = r24787 * r24788;
        double r24790 = exp(r24789);
        double r24791 = r24785 / r24790;
        double r24792 = r24784 * r24791;
        double r24793 = r24772 ? r24770 : r24792;
        return r24793;
}

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 3.2

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

   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 43.8

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

   2. Simplified 9.1

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

   4. Applied add-cube-cbrt 9.1

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

   5. Applied associate-*l* 9.1

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

   7. Applied expm1-log1p-u 9.1

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)\right)\right)} \cdot \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{e^{\left(\sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}} \cdot \sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}}\right) \cdot \left(\sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}} \cdot y.im\right)}}\]
   8. Using strategy rm

   9. Applied fma-udef 9.1

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

   10. Applied sin-sum 9.1

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

4. Final simplification 6.2

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

Reproduce

herbie shell --seed 2019303 +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)))))