You can use floats to accelerate interval arithmetic (which is "exact" in the sense of constructive real numbers) but that requires setting the correct rounding modes, and being aware of quirks in existing hardware floating point implementations, some of which may e.g. introduce non-exact outputs in several of the least significant digits, or even flush "small" (for unclear definitions of "small", not always restricted to FP-denormal numbers) results to zero.

Equality is not computable in the general case, but apartness can be stated exactly. For some practical cases, one may also be able to prove that two real numbers are indeed equal.