Logic constraint function for 'round' function

Hello,

I am solving an optimisation problem. And I have a constraint (x-round(x))*(y-round(y))=0, which means, at least one variable of x and y should be Integer. I would like to introduce other variables to force it. The logic is as follows: -100*b1<=x-e1<=100*b1, -100*b2<=y-e2<=100*b2, b1+b2<=1; where e1 and e2 are integers and b1 and b2 are binary variables.

If I use this for MIQP solver, it works. But it just takes a long time because of e1 and e2 have a large integer options from 1 to 10. Therefore, I would like to ask whether there are other logic constraints to describe 'round' function to reduce the size of the problem.

Best