You can use spigot as a practical implementation of exact/constructive real numbers. It's in the Debian software repo, or you can find it at https://www.chiark.greenend.org.uk/~sgtatham/spigot/

The main weirdness about exact reals is that "trivial" computations like sin(asin(0.12345)) will actually hang after outputing 0.1234, since spigot can't know whether the result is 0.12344999..99something or 0.12345000..00something; the "something" part that might break the tie never appears. There are cases where an 'exact' computation can't even decide whether the result is exactly zero or some number other than zero.