c++ - Overflow issues when implementing math formulas

Question

I heard that, when computing mean value, start+(end-start)/2 differs from (start+end)/2 because the latter can cause overflow. I do not quite understand why this second one can cause overflow while the first one does not. What are the generic rule to implement a math formula that can avoid overflow.

Answer 1

Suppose you are using a computer where the maximum integer value is 10 and you want to compute the average of 5 and 7.

The first method (begin + (end-begin)/2) gives

5 + (7-5)/2 == 5 + 2/2 == 6

The second method (begin + end)/2 gives an overflow, since the intermediate 12 value is over the maximum value of 10 that we accept and "wraps over" to something else (if you are using unsigned numbers its usual to wrap back to zero but if your numbers are signed you could get a negative number!).

12/2 => overflow occurs => 2/2 == 1

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Of course, in real computers integers overflow at a large value like 2^32 instead of 10, but the idea is the same. Unfortunately, there is no "general" way to get rid of overflow that I know of, and it greatly depends on what particular algorithm you are using. And event then, things get more complicated. You can get different behaviour depending on what number type you are using under the hood and there are other kinds of numerical errors to worry about in addition to over and underflow.