Sounds like a job for an arbitrary precision decimal library and an estimation of the precision the irrationals need to minimize the error at N.
Do you know of any attempts to overcome the error of computing with irrational numbers?
I will check out some of these implementations.
http://cs.stackexchange.com/questions/7145/calculating-binet...
This post suggests that the matrix solution is desirable.
I wonder if the complexity of performing exponentiation on high precision values would ever overtake a logarithmic solution.
If continued-fraction algorithms have developed to the point where you can do arbitrary exponentiations of a continued fraction, then you can just calculate the result to one decimal place and round and you'll get the right result.