How to avoid overflow in expr. A * B - C * D

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

Answer 1

The simplest and most general solution is to use a representation that can't overflow, either by using a long integer library (e.g. http://gmplib.org/) or representing using a struct or array and implementing a kind of long multiplication (i.e. separating each number to two 32bit halves and performing the multiplication as below:

(R1 + R2 * 2^32 + R3 * 2^64 + R4 * 2^96) = R = A*B = (A1 + A2 * 2^32) * (B1 + B2 * 2^32) 
R1 = (A1*B1) % 2^32
R2 = ((A1*B1) / 2^32 + (A1*B2) % 2^32 + (A2*B1) % 2^32) % 2^32
R3 = (((A1*B1) / 2^32 + (A1*B2) % 2^32 + (A2*B1) % 2^32) / 2^32 + (A1*B2) / 2^32 + (A2*B1) / 2^32 + (A2*B2) % 2^32) %2^32
R4 = ((((A1*B1) / 2^32 + (A1*B2) % 2^32 + (A2*B1) % 2^32) / 2^32 + (A1*B2) / 2^32 + (A2*B1) / 2^32 + (A2*B2) % 2^32) / 2^32) + (A2*B2) / 2^32

Assuming the end result fits in 64 bits you actually don't really need most bits of R3 and none of R4