  The relativistic motion of a compact binary system moving in circular orbit
is investigated using the post-Newtonian (PN) approximation and the
perturbative self-force (SF) formalism. A particular gauge-invariant observable
quantity is computed as a function of the binary's orbital frequency. The
conservative effect induced by the gravitational SF is obtained numerically
with high precision, and compared to the PN prediction developed to high order.
The PN calculation involves the computation of the 3PN regularized metric at
the location of the particle. Its divergent self-field is regularized by means
of dimensional regularization. The poles proportional to 1/(d-3) which occur
within dimensional regularization at the 3PN order disappear from the final
gauge-invariant result. The leading 4PN and next-to-leading 5PN conservative
logarithmic contributions originating from gravitational-wave tails are also
obtained. Making use of these exact PN results, some previously unknown PN
coefficients are measured up to the very high 7PN order by fitting to the
numerical self-force data. Using just the 2PN and new logarithmic terms, the
value of the 3PN coefficient is also confirmed numerically with very high
precision. The consistency of this cross-cultural comparison provides a crucial
test of the very different regularization methods used in both SF and PN
formalisms, and illustrates the complementarity of these approximation schemes
when modelling compact binary systems.
