Floating point operations are fast, but require continuous effort on the part
of the user in order to ensure that the results are correct. This burden can be
shifted away from the user by providing a library of exact analysis in which
the computer handles the error estimates. Previously, we provided
[arXiv:1105.2751v1] a fast implementation of the exact real numbers in the Coq
proof assistant. Our implementation improved on an earlier implementation by
O'Connor by using type classes to describe an abstract specification of the
underlying dense set from which the real numbers are built. In particular, we
used dyadic rationals built from Coq's machine integers to obtain a 100 times
speed up of the basic operations already.

In this article, we discuss various extensions of the implementation. First,
we implement and verify the sine and cosine function. Secondly, we create an
additional implementation of the dense set based on Coq's fast rational
numbers. Thirdly, we extend the hierarchy to capture order on undecidable
structures, while it was limited to decidable structures before. This
hierarchy, based on type classes, allows us to share theory on the naturals,
integers, rationals, dyadics, and reals in a convenient way. Finally, we obtain
another dramatic speed-up by avoiding evaluation of termination proofs at
runtime.