Average Error: 32.4 → 3.5
Time: 39.7s
Precision: 64
\[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]
↓
\[e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \left(\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\right)\right)\right) \cdot \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) + \cos \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\right) \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\]
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)
↓
e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \left(\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\right)\right)\right) \cdot \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) + \cos \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\right) \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)
double f(double x_re, double x_im, double y_re, double y_im) {
        double r748802 = x_re;
        double r748803 = r748802 * r748802;
        double r748804 = x_im;
        double r748805 = r748804 * r748804;
        double r748806 = r748803 + r748805;
        double r748807 = sqrt(r748806);
        double r748808 = log(r748807);
        double r748809 = y_re;
        double r748810 = r748808 * r748809;
        double r748811 = atan2(r748804, r748802);
        double r748812 = y_im;
        double r748813 = r748811 * r748812;
        double r748814 = r748810 - r748813;
        double r748815 = exp(r748814);
        double r748816 = r748808 * r748812;
        double r748817 = r748811 * r748809;
        double r748818 = r748816 + r748817;
        double r748819 = sin(r748818);
        double r748820 = r748815 * r748819;
        return r748820;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r748821 = x_re;
        double r748822 = x_im;
        double r748823 = hypot(r748821, r748822);
        double r748824 = log(r748823);
        double r748825 = y_re;
        double r748826 = r748824 * r748825;
        double r748827 = y_im;
        double r748828 = atan2(r748822, r748821);
        double r748829 = r748827 * r748828;
        double r748830 = r748826 - r748829;
        double r748831 = exp(r748830);
        double r748832 = r748827 * r748824;
        double r748833 = sin(r748832);
        double r748834 = expm1(r748833);
        double r748835 = log1p(r748834);
        double r748836 = r748825 * r748828;
        double r748837 = cos(r748836);
        double r748838 = r748835 * r748837;
        double r748839 = cos(r748832);
        double r748840 = sin(r748836);
        double r748841 = r748839 * r748840;
        double r748842 = r748838 + r748841;
        double r748843 = r748831 * r748842;
        return r748843;
}

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 32.4

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

  2. Simplified3.5

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

  4. Applied fma-udef3.5

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

  5. Applied sin-sum3.5

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

  7. Applied log1p-expm1-u3.5

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

  8. Final simplification3.5

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

Reproduce

herbie shell --seed 2019151 +o rules:numerics
(FPCore (x.re x.im y.re y.im)
  :name "powComplex, imaginary part"
  (* (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)))))