It is true that cancellation errors are often the source of wrong results.

However, computations involving IEEE floating point arithmetic can go horribly wrong even if there is no addition and no subtraction whatsoever, no divisions, and only a very small number of rounding errors. Here is one such example:

http://people.eecs.berkeley.edu/~wkahan/WrongR.pdf

Even professional users working in this domain will have a hard time to locate the cause of such problems involving IEEE floating point arithmetic. The format is extremely error-prone to use for casual users and experts alike.