  This paper presents a boundary element method (BEM) for computing the energy
transmittance of a singly-periodic grating in 2D for a wide frequency band,
which is of engineering interest in various fields with possible applications
to acoustic metamaterial design. The proposed method is based on the Pad\'e
approximants of the response. The high-order frequency derivatives of the sound
pressure necessary to evaluate the approximants are evaluated by a novel fast
BEM accelerated by the fast-multipole and hierarchical matrix methods combined
with the automatic differentiation. The target frequency band is divided
adaptively, and the Pad\'e approximation is used in each subband so as to
accurately estimate the transmittance for a wide frequency range. Through some
numerical examples, we confirm that the proposed method can efficiently and
accurately give the transmittance even when some anomalies and stopband exist
in the target band.
