Question

I need to compute an expression which looks like:
A*B - C*D, where their types are: signed long long int A, B, C, D;
Each number can be really big (not overflowing its type). While A*B could cause overflow, at same time expression A*B - C*D can be really small. How can I compute it correctly?

For example: MAX * MAX - (MAX - 1) * (MAX + 1) == 1, where MAX = LLONG_MAX - n and n – some natural number.

Answer 1

This seems too trivial I guess.
But A*B is the one that could overflow.

You could do the following, without losing precision

A*B - C*D = A(D+E) - (A+F)D
          = AD + AE - AD - DF
          = AE - DF
             ^smaller quantities E & F

E = B - D (hence, far smaller than B)
F = C - A (hence, far smaller than C)

This decomposition can be done further.
As @Gian pointed out, care might need to be taken during the subtraction operation if the type is unsigned long long.

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For example, with the case you have in the question, it takes just one iteration,

 MAX * MAX - (MAX - 1) * (MAX + 1)
  A     B       C           D

E = B - D = -1
F = C - A = -1

AE - DF = {MAX * -1} - {(MAX + 1) * -1} = -MAX + MAX + 1 = 1