Romberg or composite Simpson? I'm trying to integrate an output variable from a very complex algorithm.
I extensively checked my implementation of the two integration algorithms.
When I use the two methods for my problem, I see a discrepancy: the accuracy is 10-6 and the results are:
Simpson: -1.16844182693551
Romberg: -1.16842500337272

The difference is 1.68 · 10-5, which is greater than the desired accuracy.
Is there any way to know which result is the best?
 A: No, given the available information it is not possible to determine which of the two results is the most accurate one. Indeed there is reason to suspect that the error estimate used internally by your routine is faulty or you are outside the range of stepsizes where it is reliable.
However, given a sequence of approximations, say, $A_h, A_{2h}, A_{4h}, \dotsc$ computed using a fixed method and stepsize $h, 2h, 4h, \dotsc,$ we can determine the (effective) order of the method, estimates of the truncation error, as well as the range of stepsizes $h$ for which the error estimates are accurate and the truncation error is larger than the accumulated rounding error. The technique is known as Richardson extrapolation.
These answers to other questions discuss Richardson extrapolation in the context of differentiation, integration and solving ordinary differential equations. The last answer provide the best illustration of what you can do if you compute multiple approximations as it contains actual numerical results.
It is possible to say much more about this, but this is perhaps a subject for another question?
