* First thing to note: the documentation makes this library less approachable than it could be. In order to even begin, one has to bury onesself in terminology (the "Definitions" section). A super-quick "hello, world" example and a very short tutorial at the beginning would help a *lot*. numeric_cast is arguably the most generally-useful interface of the whole library and yet it's not even discussed until the end of the docs. * It's not clear what the purpose of abt(x) is. In expressions, "x" usually means its value. When you introduce a mechanism like this, it would be a good idea to explain why it's needed. Most places, the use of "abstract" just seems to complicate things: "A ulp is a quantity whose abstract magnitude" what's wrong with "A ulp is a quantity whose magnitude" ?? The end of the section on "Range and Precision" begins to explain why you want "abstract," but it takes the reader a long time to get to it. * are next() and prev() always well-defined? Seems to me that an unlimited precision rational number type might not have them. Is that type not numeric? * How do size, width, and density relate to unlimited precision rationals? I imagine: size = infinity width = infinity density = ? * This is slightly ambiguous: "This definition allows for a concise definition of subranged as given in the last section." "final section" would be clearer. A link would help too. * This is unclear: "Abstract intervals are easier to compare and overlap since only boundary values need to be considered." an example would help. How can one kind of thing be "easier to overlap" than another? * Precision has a very non-generic description, being different for types with density 1 and other types. It's hard to see what use such a concept has, and as far as Google can tell it is not used anywhere outside the Definitions section * Description of unsigned types, I think, is wrong: Notice that a numeric type, such as a C++ unsigned type, can define that any V does not overflow by always representing not V itself but the abstract value U = [ V % (abt(h)+1) ], which is always in range. Seems to me that the abstract values represented by an unsigned integer type are not the set of whole numbers, but the set of whole numbers modulo whatever-the-max+1-is. That's a well-defined mathematical abstraction, isn't it? Is there a problem with that, such as multiplication not working properly (I don't think so)? * There seems to be a contradiction here: If V >= abt(l) and V <= abt(h), V is representable (can be represented) in the type T ... Given an abstract value V represented in the type T as v, the roundoff error of the representation is the abstract difference: ( abt(v)-V). If V is not representable in type T, can you write formulae using its representation in type T? Given what I read later, maybe you should have just said "exactly representable" in the first phrase. * s/at most/less than/?: Because a correctly rounded representation is always one of adjacents of the abstract value being represented, the roundoff is guaranteed to be at most 1ulp. * s/cannot be represented/cannot be exactly represented/ ?: The integer number -1 can be exactly represented in Int, Real and Whole, but cannot be represented in Cardinal, yielding negative overflow. * Uh-oh: Floating to Floating conversions are defined only if N is representable; if it is not, the conversion has undefined behavior. in many places you've used "representable" to mean "exactly representable," but I can't believe you mean that here; it would imply most double=>float conversions have undefined behavior. What does this mean, anyway? If an abstract value can be "inexactly representable," aren't all abstract values representable if there's at least one abstract value that's exactly representable? Just convert to that and you're done. I think you need to give some serious thought to the definition of "representable." * s/then/so/: R(X) & R(Y) == R(X), then X->Y is not subranged. Thus, all values of type X are representable in the type Y. (b) Y->X: R(Y) & R(X) == R(Y), then Y->X is not subranged. Thus, all * What's the difference between a policy and a "parameter that defines the conversion" in It can optionally take some policies which can be used to customize its behavior. The Traits parameter is not a policy but the parameter that defines the conversion. ?? -- Dave Abrahams BoostPro Computing Software Development Training http://www.boostpro.com Clang/LLVM/EDG Compilers C++ Boost