  The slow-roll inflation for a single scalar field that couples to the
Gauss-Bonnet (GB) term represents an important higher-order curvature
correction inspired by string theory. With the arrival of the era of precision
cosmology, it is expected that the high-order corrections become more and more
important. In this paper we study the observational predictions of the
slow-roll inflation with the GB term by using the third-order uniform
asymptotic approximation method. We calculate explicitly the primordial power
spectra, spectral indices, running of the spectral indices for both scalar and
tensor perturbations, and the ratio between tensor and scalar spectra. These
expressions are all written in terms of the Hubble and GB coupling flow
parameters and expanded up to the next-to-leading order in the slow-roll
expansions. The upper bounds of errors of the approximations at the third-order
are $0.15\%$, so they represent the most accurate results obtained so far in
the literature. We expect that the understanding of the GB corrections in the
primordial spectra and their constraints by forthcoming observational data will
provide clues for the UV complete theory of quantum gravity, such as the
string/M-theory.
