  We develop a complexity theory for approximate real computations. We first
produce a theory for exact computations but with condition numbers. The input
size depends on a condition number, which is not assumed known by the machine.
The theory admits deterministic and nondeterministic polynomial time
recognizable problems. We prove that P is not NP in this theory if and only if
P is not NP in the BSS theory over the reals.
  Then we develop a theory with weak and strong approximate computations. This
theory is intended to model actual numerical computations that are usually
performed in floating point arithmetic. It admits classes P and NP and also an
NP-complete problem. We relate the P vs NP question in this new theory to the
classical P vs NP problem.
