Average Error: 33.1 → 3.7
Time: 22.6s
Precision: 64
\[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]
↓
\[\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, \left(\left(\sqrt[3]{y.re} \cdot \sqrt[3]{y.re}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \sqrt[3]{y.re}\right)\right)\right)\right) \cdot e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\]
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)
↓
\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, \left(\left(\sqrt[3]{y.re} \cdot \sqrt[3]{y.re}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \sqrt[3]{y.re}\right)\right)\right)\right) \cdot e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}
double f(double x_re, double x_im, double y_re, double y_im) {
        double r25715 = x_re;
        double r25716 = r25715 * r25715;
        double r25717 = x_im;
        double r25718 = r25717 * r25717;
        double r25719 = r25716 + r25718;
        double r25720 = sqrt(r25719);
        double r25721 = log(r25720);
        double r25722 = y_re;
        double r25723 = r25721 * r25722;
        double r25724 = atan2(r25717, r25715);
        double r25725 = y_im;
        double r25726 = r25724 * r25725;
        double r25727 = r25723 - r25726;
        double r25728 = exp(r25727);
        double r25729 = r25721 * r25725;
        double r25730 = r25724 * r25722;
        double r25731 = r25729 + r25730;
        double r25732 = sin(r25731);
        double r25733 = r25728 * r25732;
        return r25733;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r25734 = x_re;
        double r25735 = x_im;
        double r25736 = hypot(r25734, r25735);
        double r25737 = log(r25736);
        double r25738 = y_im;
        double r25739 = y_re;
        double r25740 = cbrt(r25739);
        double r25741 = r25740 * r25740;
        double r25742 = atan2(r25735, r25734);
        double r25743 = r25741 * r25742;
        double r25744 = r25743 * r25740;
        double r25745 = fma(r25737, r25738, r25744);
        double r25746 = sin(r25745);
        double r25747 = log1p(r25746);
        double r25748 = expm1(r25747);
        double r25749 = r25737 * r25739;
        double r25750 = r25742 * r25738;
        double r25751 = r25749 - r25750;
        double r25752 = exp(r25751);
        double r25753 = r25748 * r25752;
        return r25753;
}

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 33.1

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

  2. Simplified8.6

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

  4. Applied add-exp-log8.6

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

  5. Applied pow-exp8.6

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

  6. Applied div-exp3.5

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

  8. Applied expm1-log1p-u3.5

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

  10. Applied add-cube-cbrt3.7

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

  11. Applied associate-*r*3.7

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

  12. Simplified3.7

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

  13. Final simplification3.7

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

Reproduce

herbie shell --seed 2019209 +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)))))