From: Moore, Paul (paul.moore_at_[hidden])
Date: 2002-03-12 03:55:54
From: "Chuck Allison" <cda_at_[hidden]>
> Please pardon my naievete, but isn't the underlying numeric
> type to rational a template parameter (I don't have the
> class handy)? Isn't it equipped therefore to use an
> instantiation based on unlimited precision integers? If
> not, could you please explain why not. (I assume that an
> unlimited precision integer package would overload
> the pertinent operators).
The rational<> class will work fine with an unlimited-precision type. The
problem arises when the rational<> type is used with a *limited* precision
integer type. Here, overflow issues raise their ugly heads.
Imagine, for example, a single-digit decimal type I. Now consider
rational<I> r1(1,4); // No problem - one quarter
rational<I> r2(1,3); // Again, no problem, one third
// Now watch: The correct answer is seven twelfths. But the
// denominator (12) is not representable as a value of type I!
// We can't rely on overflow like this raising an exception,
// as the built-in int type doesn't. So what do we do? Silently
// give the wrong answer? Try to detect overflow manually, which
// is expensive, hurting "normal" performance for the benefit of
// an extreme case?
rational<I> r3 = r1 + r2;
Worse still is the fact that for "real" 32-bit integers, this problem isn't
going to hit often, making it even nastier when it does. And there are
(potentially) worse cases, where the result is in the representable range,
but intermediate results overflow, unless you take extreme measures to avoid
it (and I'm not sure it's theoretically possible to avoid intermediate
overflow in all cases). As a (trivial) example here, try 1/6 + 1/2. The
result is 4/6, but the naive algorithm works via twelfths.
These sorts of problems plague floating point arithmetic. People are likely
to use rationals as a way of avoiding the problems perceived with floating
point. If rationals merely exchange a set of commonly known problems for a
set of equally subtle but less well-known problems, they haven't actually
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