Average Error: 32.9 → 3.7
Time: 8.6s
Precision: 64
\[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]
↓
\[e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(\sin \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot y.im\right) \cdot \cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) + \cos \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot y.im\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)\right)\right)\]
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 - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(\sin \left(\log \left(\mathsf{hypot}\left(x.im, x.re\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.im, x.re\right)\right) \cdot y.im\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)\right)\right)
double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return ((double) (((double) exp(((double) (((double) (((double) log(((double) sqrt(((double) (((double) (x_46_re * x_46_re)) + ((double) (x_46_im * x_46_im)))))))) * y_46_re)) - ((double) (((double) atan2(x_46_im, x_46_re)) * y_46_im)))))) * ((double) sin(((double) (((double) (((double) log(((double) sqrt(((double) (((double) (x_46_re * x_46_re)) + ((double) (x_46_im * x_46_im)))))))) * y_46_im)) + ((double) (((double) atan2(x_46_im, x_46_re)) * y_46_re))))))));
}
↓
double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return ((double) (((double) exp(((double) (((double) (((double) log(((double) hypot(x_46_re, x_46_im)))) * y_46_re)) - ((double) (((double) atan2(x_46_im, x_46_re)) * y_46_im)))))) * ((double) (((double) (((double) sin(((double) (((double) log(((double) hypot(x_46_im, x_46_re)))) * y_46_im)))) * ((double) cos(((double) (((double) atan2(x_46_im, x_46_re)) * y_46_re)))))) + ((double) (((double) cos(((double) (((double) log(((double) hypot(x_46_im, x_46_re)))) * y_46_im)))) * ((double) expm1(((double) log1p(((double) sin(((double) (((double) atan2(x_46_im, x_46_re)) * y_46_re))))))))))))));
}

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 32.9

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

  3. Applied +-commutative32.9

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

  4. Applied hypot-def18.8

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

  6. Applied hypot-def3.7

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

  8. Applied sin-sum3.7

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

  10. Applied expm1-log1p-u3.7

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

  11. Final simplification3.7

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

Reproduce

herbie shell --seed 2020114 +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)))))