Actually, ignore that statement. You only need two integers for a0 + 1/a1, in general you need more. For example, if you look at the test program (basic.cpp in the archive), it provides some examples (where the floating-point output is float, double, long double):
3/2 + 1/4 = [ 1 1 3] 1.75 1.75 1.75
1/7 + 1/15 = [ 0 4 1 3 2 2] 0.209524 0.20952380952381 0.209523809523809524
There are other continued fraction representations where any rational can be represented with a fixed number of integers (a0 + b0/(a1 + ...)), but the algorithms are a little more complicated, and I fear the compile times would increase.
I was looking for something very similar to that, altough I believe compile times are to long for realistic problems. Especially the example with sine and cosine took approximately 10 minutes on my PC. But who knows? Maybe compilers are fast enough in some years... I have also two little questions: 1. Can one express for example 1.25e-10 in an easy manner, for example by simply specifying the exponent? 2. Can one use different representations of rational number, maybe boost::ratio? I guess this could be difficult to work with the elementary operations.