Unequal equivalence

When is a number not a number?

In the world of floating-point numbers there is commonly a notion of a special value that represents something that is not a number, NaN for short. A NaN value has some interesting properties: it does not order against any other value, and does not even compare equal to itself. Alas, one problem with having "interesting properties" is that, left to its own devices, a NaN rather messes up the idea of a total ordering that can be used for sorting.

Because Java's type system is not fully object oriented there is a divorce between what can be done with primitive types, such as double and float, and its class-based type system, rooted in Object. Floating-point numbers do not implement the Comparable interface used for ordering objects and it is not possible to hold floating-point numbers directly within a collection. To do this the programmer works instead in terms of the wrapper types, Float and Double, which are proper object types that implement Comparable and wrap their corresponding primitive type. The implementation of compareTo ensures that a NaN will compare equal to another NaN and greater than any other value. As an aside, J2SE 5.0 offers a number of improvements that make working with wrappers less tedious for the programmer, but the underlying model remains unchanged.

C#, by contrast, has a more regular type system, one that incorporates primitive types as first class types in its object system. Wrapper types are not needed, so double and float are synonyms rather than metonyms of the System.Double and System.Single types. They implement the IComparable interface to define a total ordering for floating-point values, including NaN. However, unlike Java they choose the slightly more intuitive ordering of considering all proper values to be greater than a NaN. This policy is also consistent with the idea of treating any object as comparing greater than null (which is, in effect, "not an object").

In C++ ordering is defined, by default, in terms of the < operator rather than via a separate operation granted by an inherited class. This effectively offers the same behaviour as the corresponding operator in C# and Java, which means that because of NaN it doesn't offer a total ordering on all floating-point values.

To understand what this means in practice, consider holding a set of double values. A std::set orders its elements using the defaulted std::less function object type, which corresponds to the < operator in this case, and ensures uniqueness of keys by applying the equivalence outlined earlier — when !(a < b) && !(b < a) evaluates to true then a and b are considered equivalent. If you insert a NaN as the first element in a set then you will never be able to insert anything else, because all values will appear equivalent to NaN. The presence of a NaN pollutes the data set and therefore the container. And, for the same reason, if you insert something else first you will then never be able to insert a NaN!

However, all is not lost. You can impose your own preferred ordering criterion through a function object type, i.e. a class whose instances look like functions because they can be invoked with the function-call operator. The following type imposes an ordering so that a NaN is considered lower than any proper value:

struct total_less
{
    template<typename numeric_type>
    bool operator()(numeric_type lhs, numeric_type rhs) const
    {
        return is_nan(lhs) ? !is_nan(rhs) : lhs < rhs;
    }
};

To use this with a std::setyou replace the defaulted std::less comparator with total_less, i.e. declare the type as std::set<double, total_less> rather than as std::set<double>.

The total_less function object type is reasonably straightforward in that it is written in terms of a simple helper, is_nan, and the ordering option is made explicit. Because of what can only be considered an oversight (OK, a cock-up) in the design of the numeric features of C++ standard library, you can use std::numeric_limits to determine whether or not a numeric type has a NaN value, and what it is, but you cannot actually test a value to see whether or not it is a NaN — remember that NaN does not compare equal to itself, so you can't use == to check! However, assuming that infinity compares reflexively, we can take advantage of that very property to define our own predicate:

template<typename numeric_type>
bool is_nan(numeric_type value)
{
    return !(value == value);
}

This is generic and works for any numeric type, so consequently total_less will also work for any numeric type. ®

This article originally appeared in Application Development Advisor.

Kevlin Henney is an independent software development consultant and trainer. He can be reached at http://www.curbralan.com.