Average Error: 32.2 → 3.4
Time: 34.4s
Precision: 64
\[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]
↓
\[\frac{\cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot \cos \left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im\right) - \sin \left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im\right) \cdot \left(\left(\sqrt[3]{\sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)} \cdot \sqrt[3]{\sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}\right) \cdot \sqrt[3]{\sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im - \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re}}\]
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)
↓
\frac{\cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot \cos \left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im\right) - \sin \left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im\right) \cdot \left(\left(\sqrt[3]{\sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)} \cdot \sqrt[3]{\sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}\right) \cdot \sqrt[3]{\sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im - \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re}}
double f(double x_re, double x_im, double y_re, double y_im) {
        double r1138833 = x_re;
        double r1138834 = r1138833 * r1138833;
        double r1138835 = x_im;
        double r1138836 = r1138835 * r1138835;
        double r1138837 = r1138834 + r1138836;
        double r1138838 = sqrt(r1138837);
        double r1138839 = log(r1138838);
        double r1138840 = y_re;
        double r1138841 = r1138839 * r1138840;
        double r1138842 = atan2(r1138835, r1138833);
        double r1138843 = y_im;
        double r1138844 = r1138842 * r1138843;
        double r1138845 = r1138841 - r1138844;
        double r1138846 = exp(r1138845);
        double r1138847 = r1138839 * r1138843;
        double r1138848 = r1138842 * r1138840;
        double r1138849 = r1138847 + r1138848;
        double r1138850 = cos(r1138849);
        double r1138851 = r1138846 * r1138850;
        return r1138851;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r1138852 = x_im;
        double r1138853 = x_re;
        double r1138854 = atan2(r1138852, r1138853);
        double r1138855 = y_re;
        double r1138856 = r1138854 * r1138855;
        double r1138857 = cos(r1138856);
        double r1138858 = hypot(r1138853, r1138852);
        double r1138859 = log(r1138858);
        double r1138860 = y_im;
        double r1138861 = r1138859 * r1138860;
        double r1138862 = cos(r1138861);
        double r1138863 = r1138857 * r1138862;
        double r1138864 = sin(r1138861);
        double r1138865 = sin(r1138856);
        double r1138866 = cbrt(r1138865);
        double r1138867 = r1138866 * r1138866;
        double r1138868 = r1138867 * r1138866;
        double r1138869 = r1138864 * r1138868;
        double r1138870 = r1138863 - r1138869;
        double r1138871 = r1138854 * r1138860;
        double r1138872 = r1138859 * r1138855;
        double r1138873 = r1138871 - r1138872;
        double r1138874 = exp(r1138873);
        double r1138875 = r1138870 / r1138874;
        return r1138875;
}

Error

Bits error versus x.re

Bits error versus x.im

Bits error versus y.re

Bits error versus y.im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 32.2

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

  2. Simplified3.4

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

  4. Applied fma-udef3.4

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

  5. Applied cos-sum3.4

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

  7. Applied add-cube-cbrt3.4

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

  8. Final simplification3.4

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

Reproduce

herbie shell --seed 2019162 +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)))))