Is there a way to avoid BigInteger/BigDecimal?

I need to compute something like this (pseudocode):

// a, b, x, y are long, x,y <= 10^12

long i = (a - n)/(x*y)

and

long j = (b - n)/(x*y) - ceiling

Sometimes x*y doesn't fit in long. I would like to avoid BigDecimal/BigInteger usage as it is too costly and not needed anywhere else. Is there a smart math solution e.g. with two longs or smth like that?

Thank you!

UPDATE: Sorry, guys, one more constraint: I also have a variable computed as following (maybe it can be rewritten as well):

sum += x*y

I need to recalculate it to compare with another variable to stop the cycle.

Solution

I've noticed that the value I'm comparing the sum with is long. So I've solved the problem in the following way:

I'm calculating i like this: (long) (((a - n) / (double) x) / y))
Here goes j: (long) Math.ceil((b - n) / (double) x*y)

The case above overflows of course. But I prevent overflowing doing the following try-catch before:

        try {
                xy = Math.multiplyExact(x, y);
                ...
        } catch (ArithmeticException ex) {
                // some handling
                break;
        }

This trick made by code work faster enough.

Hope it helps someone!