Average Error: 30.4 → 0.0
Time: 2.4m
Precision: 64
\[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]
↓
\[\cos \left(\mathsf{fma}\left(y.im, \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right) \cdot e^{y.re \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\]
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 \cos \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)
↓
\cos \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) \cdot e^{y.re \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) - \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 r486905 = x_re;
        double r486906 = r486905 * r486905;
        double r486907 = x_im;
        double r486908 = r486907 * r486907;
        double r486909 = r486906 + r486908;
        double r486910 = sqrt(r486909);
        double r486911 = log(r486910);
        double r486912 = y_re;
        double r486913 = r486911 * r486912;
        double r486914 = atan2(r486907, r486905);
        double r486915 = y_im;
        double r486916 = r486914 * r486915;
        double r486917 = r486913 - r486916;
        double r486918 = exp(r486917);
        double r486919 = r486911 * r486915;
        double r486920 = r486914 * r486912;
        double r486921 = r486919 + r486920;
        double r486922 = cos(r486921);
        double r486923 = r486918 * r486922;
        return r486923;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r486924 = y_im;
        double r486925 = x_re;
        double r486926 = x_im;
        double r486927 = hypot(r486925, r486926);
        double r486928 = log(r486927);
        double r486929 = atan2(r486926, r486925);
        double r486930 = y_re;
        double r486931 = r486929 * r486930;
        double r486932 = fma(r486924, r486928, r486931);
        double r486933 = cos(r486932);
        double r486934 = r486930 * r486928;
        double r486935 = r486929 * r486924;
        double r486936 = r486934 - r486935;
        double r486937 = exp(r486936);
        double r486938 = r486933 * r486937;
        return r486938;
}

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 30.4

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

  2. Simplified0.0

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

  3. Final simplification0.0

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

Reproduce

herbie shell --seed 2019134 +o rules:numerics
(FPCore (x.re x.im y.re y.im)
  :name "powComplex, real part"
  (* (exp (- (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.re) (* (atan2 x.im x.re) y.im))) (cos (+ (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.im) (* (atan2 x.im x.re) y.re)))))