Bohr sommerfeld quantiztion rule and Gutzwiller trace

Question

assuming we can evaluate the eigenvalue staircase $ N(E) $ in both manners with the Bohr-Sommerfeld quantization rule

$ N(E)2\pi \hbar = \oint _{C}p.dq $

and using the Gutzwiller trace $ N(E)= N_{smooth} (E) + \delta N_{osc} (E) $

here the oscillating part is given by a sum over orbits in the phase space

are the two formulations equivalent ??