# Header <boost/operators.hpp>

The header <boost/operators.hpp> supplies several sets of class templates (in namespace `boost`). These templates define operators at namespace scope in terms of a minimal number of fundamental operators provided by the class.

## Contents

- Contents
- Rationale
  - Summary of Template Semantics
  - Use of concepts
- Usage
  - Two-Argument Template Forms
    - General Considerations
    - Mixed arithmetics
  - Base Class Chaining and Object Size
  - Separate, Explicit Instantiation
  - Requirement Portability
- Example
- Arithmetic operators
  - Simple Arithmetic Operators
    - Ordering Note
    - Symmetry Note
  - Grouped Arithmetic Operators
  - Example Templates
  - Arithmetic Operators Demonstration and Test Program
- Dereference Operators and Iterator Helpers
  - Dereference operators
  - Grouped Iterator Operators
  - Iterator Helpers
    - Iterator Helper Notes
  - Iterator Demonstration and Test Program
- Contributors
- Note for Users of Older Versions

## Rationale

Overloaded operators for class types typically occur in groups. If you can write `x + y`, you probably also want to be able to write `x += y`. If you can write `x < y,` you also want `x > y, x >= y,` and `x <= y`. Moreover, unless your class has really surprising behavior, some of these related operators can be defined in terms of others (e.g. `x >= y <=> !(x < y)`). Replicating this boilerplate for multiple classes is both tedious and error-prone. The boost/operators.hpp templates help by generating operators for you at namespace scope based on other operators you've defined in your class.

If, for example, you declare a class like this:

>     class MyInt
>         : boost::operators<MyInt>
>     {
>         bool operator<(const MyInt& x) const;
>         bool operator==(const MyInt& x) const;
>         MyInt& operator+=(const MyInt& x);
>         MyInt& operator-=(const MyInt& x);
>         MyInt& operator*=(const MyInt& x);
>         MyInt& operator/=(const MyInt& x);
>         MyInt& operator%=(const MyInt& x);
>         MyInt& operator|=(const MyInt& x);
>         MyInt& operator&=(const MyInt& x);
>         MyInt& operator^=(const MyInt& x);
>         MyInt& operator++();
>         MyInt& operator--();
>     };

then the `operators<>` template adds more than a dozen additional operators, such as `operator>`, `<=`, `>=`, and (binary) `+`. Two-argument forms of the templates are also provided to allow interaction with other types.

### Summary of Template Semantics

1. Each operator template completes the concept(s) it describes by defining overloaded operators for its target class.
2. The name of an operator class template indicates the concept that its target class will model.
3. Usually, the target class uses an instantation of the operator class template as a base class. Some operator templates support an alternate method.
4. The concept can be compound, i.e. it may represent a common combination of other, simpler concepts.
5. Most operator templates require their target class to support operations related to the operators supplied by the template. In accordance with widely accepted coding style recommendations, the target class is often required to supply the assignment counterpart operator of the concept's "main operator." For example, the `addable` template requires `operator+=(T const&)` and in turn supplies `operator+(T const&, T const&)`.

### Use of concepts

The discussed concepts are not necessarily the standard library's concepts (CopyConstructible, etc.), although some of them could be; they are what we call concepts with a small 'c'. In particular, they are different from the former ones in that they do not describe precise semantics of the operators they require to be defined, except the requirements that (a) the semantics of the operators grouped in one concept should be consistent (e.g. effects of evaluating of `a += b` and `a = a + b` expressions should be the same), and (b) that the return types of the operators should follow semantics of return types of corresponding operators for built-in types (e.g. `operator<` should return a type convertible to `bool`, and `T::operator-=` should return type convertible to `T`). Such "loose" requirements make operators library applicable to broader set of target classes from different domains, i.e. eventually more useful.

## Usage

### Two-Argument Template Forms

#### General Considerations

The arguments to a binary operator commonly have identical types, but it is not unusual to want to define operators which combine different types. For example, one might want to multiply a mathematical vector by a scalar. The two-argument template forms of the arithmetic operator templates are supplied for this purpose. When applying the two-argument form of a template, the desired return type of the operators typically determines which of the two types in question should be derived from the operator template. For example, if the result of `T + U` is of type `T`, then `T` (not `U`) should be derived from `addable<T, U>`. The comparison templates (`less_than_comparable<T, U>`, `equality_comparable<T, U>`, `equivalent<T, U>`, and `partially_ordered<T, U>`) are exceptions to this guideline, since the return type of the operators they define is `bool`.

On compilers which do not support partial specialization, the two-argument forms must be specified by using the names shown below with the trailing `'2'`. The single-argument forms with the trailing `'1'` are provided for symmetry and to enable certain applications of the base class chaining technique.

#### Mixed Arithmetics

Another application of the two-argument template forms is for mixed arithmetics between a type `T` and a type `U` that is convertible to `T`. In this case there are two ways where the two-argument template forms are helpful: one is to provide the respective signatures for operator overloading, the second is performance.

With respect to the operator overloading assume e.g. that `U` is `int`, that `T` is an user-defined unlimited integer type, and that `double operator-(double, const T&)` exists. If one wants to compute `int - T` and does not provide `T operator-(int, const T&)`, the compiler will consider `double operator-(double, const T&)` to be a better match than `T operator-(const T&, const T&)`, which will probably be different from the user's intention. To define a complete set of operator signatures, additional 'left' forms of the two-argument template forms are provided (`subtractable2_left<T, U>`, `dividable2_left<T, U>`, `modable2_left<T, U>`) that define the signatures for non-commutative operators where `U` appears on the left hand side (`operator-(const U&, const T&)`, `operator/(const U&, const T&)`, `operator%(const U&, const T&)`).

With respect to the performance observe that when one uses the single type binary operator for mixed type arithmetics, the type `U` argument has to be converted to type `T`. In practice, however, there are often more efficient implementations of, say `T::operator-=(const U&)` that avoid unnecessary conversions from `U` to `T`. The two-argument template forms of the arithmetic operator create additional operator interfaces that use these more efficient implementations. There is, however, no performance gain in the 'left' forms: they still need a conversion from `U` to `T` and have an implementation equivalent to the code that would be automatically created by the compiler if it considered the single type binary operator to be the best match.

### Base Class Chaining and Object Size

Every operator class template, except the arithmetic examples and the iterator helpers, has an additional, but optional, template type parameter `B`. This parameter will be a publicly-derived base class of the instantiated template. This means it must be a class type. It can be used to avoid the bloating of object sizes that is commonly associated with multiple-inheritance from several empty base classes (see the note for users of older versions for more details). To provide support for a group of operators, use the `B` parameter to chain operator templates into a single-base class hierarchy, demostrated in the usage example. The technique is also used by the composite operator templates to group operator definitions. If a chain becomes too long for the compiler to support, try replacing some of the operator templates with a single grouped operator template that chains the old templates together; the length limit only applies to the number of templates directly in the chain, not those hidden in group templates.

Caveat: to chain to a base class which is not a Boost operator template when using the single-argument form of a Boost operator template, you must specify the operator template with the trailing `'1'` in its name. Otherwise the library will assume you mean to define a binary operation combining the class you intend to use as a base class and the class you're deriving.

### Separate, Explicit Instantiation

On some compilers (e.g. Borland, GCC) even single-inheritance seems to cause an increase in object size in some cases. If you are not defining a class template, you may get better object-size performance by avoiding derivation altogether, and instead explicitly instantiating the operator template as follows:

>         class myclass // lose the inheritance...
>         {
>             //...
>         };

>         // explicitly instantiate the operators I need.
>         template struct less_than_comparable<myclass>;
>         template struct equality_comparable<myclass>;
>         template struct incrementable<myclass>;
>         template struct decrementable<myclass>;
>         template struct addable<myclass,long>;
>         template struct subtractable<myclass,long>;

Note that some operator templates cannot use this workaround and must be a base class of their primary operand type. Those templates define operators which must be member functions, and the workaround needs the operators to be independent friend functions. The relevant templates are:

- `dereferenceable<>`
- `indexable<>`
- Any composite operator template that includes at least one of the above

As Daniel Krügler pointed out, this technique violates 14.6.5/2 and is thus non-portable. The reasoning is, that the operators injected by the instantiation of e.g. `less_than_comparable<myclass>` can not be found by ADL according to the rules given by 3.4.2/2, since myclass is not an associated class of `less_than_comparable<myclass>`. Thus only use this technique if all else fails.

### Requirement Portability

Many compilers (e.g. MSVC 6.3, GCC 2.95.2) will not enforce the requirements in the operator template tables unless the operations which depend on them are actually used. This is not standard-conforming behavior. In particular, although it would be convenient to derive all your classes which need binary operators from the `operators<>` and `operators2<>` templates, regardless of whether they implement all the requirements of those templates, this shortcut is not portable. Even if this currently works with your compiler, it may not work later.

## Example

This example shows how some of the arithmetic operator templates can be used with a geometric point class (template).

    template <class T>
    class point    // note: private inheritance is OK here!
        : boost::addable< point<T>          // point + point
        , boost::subtractable< point<T>     // point - point
        , boost::dividable2< point<T>, T    // point / T
        , boost::multipliable2< point<T>, T // point * T, T * point
          > > > >
    {
    public:
        point(T, T);
        T x() const;
        T y() const;

        point operator+=(const point&);
        // point operator+(point, const point&) automatically
        // generated by addable.

        point operator-=(const point&);
        // point operator-(point, const point&) automatically
        // generated by subtractable.

        point operator*=(T);
        // point operator*(point, const T&) and
        // point operator*(const T&, point) auto-generated
        // by multipliable.

        point operator/=(T);
        // point operator/(point, const T&) auto-generated
        // by dividable.
    private:
        T x_;
        T y_;
    };

    // now use the point<> class:

    template <class T>
    T length(const point<T> p)
    {
        return sqrt(p.x()*p.x() + p.y()*p.y());
    }

    const point<float> right(0, 1);
    const point<float> up(1, 0);
    const point<float> pi_over_4 = up + right;
    const point<float> pi_over_4_normalized = pi_over_4 / length(pi_over_4);

## Arithmetic Operators

The arithmetic operator templates ease the task of creating a custom numeric type. Given a core set of operators, the templates add related operators to the numeric class. These operations are like the ones the standard arithmetic types have, and may include comparisons, adding, incrementing, logical and bitwise manipulations, etc. Further, since most numeric types need more than one of these operators, some templates are provided to combine several of the basic operator templates in one declaration.

The requirements for the types used to instantiate the simple operator templates are specified in terms of expressions which must be valid and the expression's return type. The composite operator templates only list what other templates they use. The supplied operations and requirements of the composite operator templates can be inferred from the operations and requirements of the listed components.

### Simple Arithmetic Operators

These templates are "simple" since they provide operators based on a single operation the base type has to provide. They have an additional optional template parameter `B`, which is not shown, for the base class chaining technique.

Simple Arithmetic Operator Template Classes
Key
`T`: primary operand type | `U`: alternate operand type
`t`, `t1`: values of type `T` | `u`: value of type `U`
Template | Supplied Operations | Requirements
--- | --- | ---
`less_than_comparable<T>` `less_than_comparable1<T>` | `bool operator>(const T&, const T&)` `bool operator<=(const T&, const T&)` `bool operator>=(const T&, const T&)` | `t < t1`. Return convertible to `bool`. See the Ordering Note.
`less_than_comparable<T, U>` `less_than_comparable2<T, U>` | `bool operator<=(const T&, const U&)` `bool operator>=(const T&, const U&)` `bool operator>(const U&, const T&)` `bool operator<(const U&, const T&)` `bool operator<=(const U&, const T&)` `bool operator>=(const U&, const T&)` | `t < u`. `t > u`. Returns convertible to `bool`. See the Ordering Note.
`equality_comparable<T>` `equality_comparable1<T>` | `bool operator!=(const T&, const T&)` | `t == t1`. Return convertible to `bool`.
`equality_comparable<T, U>` `equality_comparable2<T, U>` | `bool operator==(const U&, const T&)` `bool operator!=(const U&, const T&)` `bool operator!=(const T&, const U&)` | `t == u`. Return convertible to `bool`.
`addable<T>` `addable1<T>` | `T operator+(const T&, const T&)` | `T temp(t); temp += t1`. Return convertible to `T`. See the Symmetry Note.
`addable<T, U>` `addable2<T, U>` | `T operator+(const T&, const U&)` `T operator+(const U&, const T& )` | `T temp(t); temp += u`. Return convertible to `T`. See the Symmetry Note.
`subtractable<T>` `subtractable1<T>` | `T operator-(const T&, const T&)` | `T temp(t); temp -= t1`. Return convertible to `T`. See the Symmetry Note.
`subtractable<T, U>` `subtractable2<T, U>` | `T operator-(const T&, const U&)` | `T temp(t); temp -= u`. Return convertible to `T`. See the Symmetry Note.
`subtractable2_left<T, U>` | `T operator-(const U&, const T&)` | `T temp(u); temp -= t`. Return convertible to `T`.
`multipliable<T>` `multipliable1<T>` | `T operator*(const T&, const T&)` | `T temp(t); temp *= t1`. Return convertible to `T`. See the Symmetry Note.
`multipliable<T, U>` `multipliable2<T, U>` | `T operator*(const T&, const U&)` `T operator*(const U&, const T&)` | `T temp(t); temp *= u`. Return convertible to `T`. See the Symmetry Note.
`dividable<T>` `dividable1<T>` | `T operator/(const T&, const T&)` | `T temp(t); temp /= t1`. Return convertible to `T`. See the Symmetry Note.
`dividable<T, U>` `dividable2<T, U>` | `T operator/(const T&, const U&)` | `T temp(t); temp /= u`. Return convertible to `T`. See the Symmetry Note.
`dividable2_left<T, U>` | `T operator/(const U&, const T&)` | `T temp(u); temp /= t`. Return convertible to `T`.
`modable<T>` `modable1<T>` | `T operator%(const T&, const T&)` | `T temp(t); temp %= t1`. Return convertible to `T`. See the Symmetry Note.
`modable<T, U>` `modable2<T, U>` | `T operator%(const T&, const U&)` | `T temp(t); temp %= u`. Return convertible to `T`. See the Symmetry Note.
`modable2_left<T, U>` | `T operator%(const U&, const T&)` | `T temp(u); temp %= t`. Return convertible to `T`.
`orable<T>` `orable1<T>` | `T operator|(const T&, const T&)` | `T temp(t); temp |= t1`. Return convertible to `T`. See the Symmetry Note.
`orable<T, U>` `orable2<T, U>` | `T operator|(const T&, const U&)` `T operator|(const U&, const T&)` | `T temp(t); temp |= u`. Return convertible to `T`. See the Symmetry Note.
`andable<T>` `andable1<T>` | `T operator&(const T&, const T&)` | `T temp(t); temp &= t1`. Return convertible to `T`. See the Symmetry Note.
`andable<T, U>` `andable2<T, U>` | `T operator&(const T&, const U&)` `T operator&(const U&, const T&)` | `T temp(t); temp &= u`. Return convertible to `T`. See the Symmetry Note.
`xorable<T>` `xorable1<T>` | `T operator^(const T&, const T&)` | `T temp(t); temp ^= t1`. Return convertible to `T`. See the Symmetry Note.
`xorable<T, U>` `xorable2<T, U>` | `T operator^(const T&, const U&)` `T operator^(const U&, const T&)` | `T temp(t); temp ^= u`. Return convertible to `T`. See the Symmetry Note.
`incrementable<T>` | `T operator++(T&, int)` | `T temp(t); ++t` Return convertible to `T`.
`decrementable<T>` | `T operator--(T&, int)` | `T temp(t); --t;` Return convertible to `T`.
`left_shiftable<T>` `left_shiftable1<T>` | `T operator<<(const T&, const T&)` | `T temp(t); temp <<= t1`. Return convertible to `T`. See the Symmetry Note.
`left_shiftable<T, U>` `left_shiftable2<T, U>` | `T operator<<(const T&, const U&)` | `T temp(t); temp <<= u`. Return convertible to `T`. See the Symmetry Note.
`right_shiftable<T>` `right_shiftable1<T>` | `T operator>>(const T&, const T&)` | `T temp(t); temp >>= t1`. Return convertible to `T`. See the Symmetry Note.
`right_shiftable<T, U>` `right_shiftable2<T, U>` | `T operator>>(const T&, const U&)` | `T temp(t); temp >>= u`. Return convertible to `T`. See the Symmetry Note.
`equivalent<T>` `equivalent1<T>` | `bool operator==(const T&, const T&)` | `t < t1`. Return convertible to `bool`. See the Ordering Note.
`equivalent<T, U>` `equivalent2<T, U>` | `bool operator==(const T&, const U&)` | `t < u`. `t > u`. Returns convertible to `bool`. See the Ordering Note.
`partially_ordered<T>` `partially_ordered1<T>` | `bool operator>(const T&, const T&)` `bool operator<=(const T&, const T&)` `bool operator>=(const T&, const T&)` | `t < t1`. `t == t1`. Returns convertible to `bool`. See the Ordering Note.
`partially_ordered<T, U>` `partially_ordered2<T, U>` | `bool operator<=(const T&, const U&)` `bool operator>=(const T&, const U&)` `bool operator>(const U&, const T&)` `bool operator<(const U&, const T&)` `bool operator<=(const U&, const T&)` `bool operator>=(const U&, const T&)` | `t < u`. `t > u`. `t == u`. Returns convertible to `bool`. See the Ordering Note.

#### Ordering Note

The `less_than_comparable<T>` and `partially_ordered<T>` templates provide the same set of operations. However, the workings of `less_than_comparable<T>` assume that all values of type `T` can be placed in a total order. If that is not true (e.g. Not-a-Number values in IEEE floating point arithmetic), then `partially_ordered<T>` should be used. The `partially_ordered<T>` template can be used for a totally-ordered type, but it is not as efficient as `less_than_comparable<T>`. This rule also applies for `less_than_comparable<T, U>` and `partially_ordered<T, U>` with respect to the ordering of all `T` and `U` values, and for both versions of `equivalent<>`. The solution for `equivalent<>` is to write a custom `operator==` for the target class.

#### Symmetry Note

Before talking about symmetry, we need to talk about optimizations to understand the reasons for the different implementation styles of operators. Let's have a look at `operator+` for a class `T` as an example:

    T operator+( const T& lhs, const T& rhs )
    {
       return T( lhs ) += rhs;
    }

This would be a normal implementation of `operator+`, but it is not an efficient one. An unnamed local copy of `lhs` is created, `operator+=` is called on it and it is copied to the function return value (which is another unnamed object of type `T`). The standard doesn't generally allow the intermediate object to be optimized away:

> 3.7.2/2: Automatic storage duration
> If a named automatic object has initialization or a destructor with side effects, it shall not be destroyed before the end of its block, nor shall it be eliminated as an optimization even if it appears to be unused, except that a class object or its copy may be eliminated as specified in 12.8.

The reference to 12.8 is important for us:

> 12.8/15: Copying class objects
> ...
> For a function with a class return type, if the expression in the return statement is the name of a local object, and the cv-unqualified type of the local object is the same as the function return type, an implementation is permitted to omit creating the temporary object to hold the function return value, even if the class copy constructor or destructor has side effects.

This optimization is known as the named return value optimization (NRVO), which leads us to the following implementation for `operator+`:

    T operator+( const T& lhs, const T& rhs )
    {
       T nrv( lhs );
       nrv += rhs;
       return nrv;
    }

Given this implementation, the compiler is allowed to remove the intermediate object. Sadly, not all compiler implement the NRVO, some even implement it in an incorrect way which makes it useless here. Without the NRVO, the NRVO-friendly code is no worse than the original code showed above, but there is another possible implementation, which has some very special properties:

    T operator+( T lhs, const T& rhs )
    {
       return lhs += rhs;
    }

The difference to the first implementation is that `lhs` is not taken as a constant reference used to create a copy; instead, `lhs` is a by-value parameter, thus it is already the copy needed. This allows another optimization (12.2/2) for some cases. Consider `a + b + c` where the result of `a + b` is not copied when used as `lhs` when adding `c`. This is more efficient than the original code, but not as efficient as a compiler using the NRVO. For most people, it is still preferable for compilers that don't implement the NRVO, but the `operator+` now has a different function signature. Also, the number of objects created differs for `(a + b ) + c` and `a + ( b + c )`. Most probably, this won't be a problem for you, but if your code relies on the function signature or a strict symmetric behaviour, you should set `BOOST_FORCE_SYMMETRIC_OPERATORS` in your user-config. This will force the NRVO-friendly implementation to be used even for compilers that don't implement the NRVO.

### Grouped Arithmetic Operators

The following templates provide common groups of related operations. For example, since a type which is addable is usually also subractable, the `additive` template provides the combined operators of both. The grouped operator templates have an additional optional template parameter `B`, which is not shown, for the base class chaining technique.

Grouped Arithmetic Operator Template Classes
Key
`T`: primary operand type | `U`: alternate operand type
Template | Component Operator Templates
--- | ---
`totally_ordered<T>` `totally_ordered1<T>` | `less_than_comparable<T>` `equality_comparable<T>`
`totally_ordered<T, U>` `totally_ordered2<T, U>` | `less_than_comparable<T, U>` `equality_comparable<T, U>`
`additive<T>` `additive1<T>` | `addable<T>` `subtractable<T>`
`additive<T, U>` `additive2<T, U>` | `addable<T, U>` `subtractable<T, U>`
`multiplicative<T>` `multiplicative1<T>` | `multipliable<T>` `dividable<T>`
`multiplicative<T, U>` `multiplicative2<T, U>` | `multipliable<T, U>` `dividable<T, U>`
`integer_multiplicative<T>` `integer_multiplicative1<T>` | `multiplicative<T>` `modable<T>`
`integer_multiplicative<T, U>` `integer_multiplicative2<T, U>` | `multiplicative<T, U>` `modable<T, U>`
`arithmetic<T>` `arithmetic1<T>` | `additive<T>` `multiplicative<T>`
`arithmetic<T, U>` `arithmetic2<T, U>` | `additive<T, U>` `multiplicative<T, U>`
`integer_arithmetic<T>` `integer_arithmetic1<T>` | `additive<T>` `integer_multiplicative<T>`
`integer_arithmetic<T, U>` `integer_arithmetic2<T, U>` | `additive<T, U>` `integer_multiplicative<T, U>`
`bitwise<T>` `bitwise1<T>` | `xorable<T>` `andable<T>` `orable<T>`
`bitwise<T, U>` `bitwise2<T, U>` | `xorable<T, U>` `andable<T, U>` `orable<T, U>`
`unit_steppable<T>` | `incrementable<T>` `decrementable<T>`
`shiftable<T>` `shiftable1<T>` | `left_shiftable<T>` `right_shiftable<T>`
`shiftable<T, U>` `shiftable2<T, U>` | `left_shiftable<T, U>` `right_shiftable<T, U>`
`ring_operators<T>` `ring_operators1<T>` | `additive<T>` `multipliable<T>`
`ring_operators<T, U>` `ring_operators2<T, U>` | `additive<T, U>` `subtractable2_left<T, U>` `multipliable<T, U>`
`ordered_ring_operators<T>` `ordered_ring_operators1<T>` | `ring_operators<T>` `totally_ordered<T>`
`ordered_ring_operators<T, U>` `ordered_ring_operators2<T, U>` | `ring_operators<T, U>` `totally_ordered<T, U>`
`field_operators<T>` `field_operators1<T>` | `ring_operators<T>` `dividable<T>`
`field_operators<T, U>` `field_operators2<T, U>` | `ring_operators<T, U>` `dividable<T, U>` `dividable2_left<T, U>`
`ordered_field_operators<T>` `ordered_field_operators1<T>` | `field_operators<T>` `totally_ordered<T>`
`ordered_field_operators<T, U>` `ordered_field_operators2<T, U>` | `field_operators<T, U>` `totally_ordered<T, U>`
`euclidian_ring_operators<T>` `euclidian_ring_operators1<T>` | `ring_operators<T>` `dividable<T>` `modable<T>`
`euclidian_ring_operators<T, U>` `euclidian_ring_operators2<T, U>` | `ring_operators<T, U>` `dividable<T, U>` `dividable2_left<T, U>` `modable<T, U>` `modable2_left<T, U>`
`ordered_euclidian_ring_operators<T>` `ordered_euclidian_ring_operators1<T>` | `euclidian_ring_operators<T>` `totally_ordered<T>`
`ordered_euclidian_ring_operators<T, U>` `ordered_euclidian_ring_operators2<T, U>` | `euclidian_ring_operators<T, U>` `totally_ordered<T, U>`

### Example Templates

The arithmetic operator class templates `operators<>` and `operators2<>` are examples of non-extensible operator grouping classes. These legacy class templates, from previous versions of the header, cannot be used for base class chaining.

Final Arithmetic Operator Template Classes
Key
`T`: primary operand type | `U`: alternate operand type
Template | Component Operator Templates
--- | ---
`operators<T>` | `totally_ordered<T>` `integer_arithmetic<T>` `bitwise<T>` `unit_steppable<T>`
`operators<T, U>` `operators2<T, U>` | `totally_ordered<T, U>` `integer_arithmetic<T, U>` `bitwise<T, U>`

### Arithmetic Operators Demonstration and Test Program

The operators_test.cpp program demonstrates the use of the arithmetic operator templates, and can also be used to verify correct operation. Check the compiler status report for the test results with selected platforms.

## Dereference Operators and Iterator Helpers

The iterator helper templates ease the task of creating a custom iterator. Similar to arithmetic types, a complete iterator has many operators that are "redundant" and can be implemented in terms of the core set of operators.

The dereference operators were motivated by the iterator helpers, but are often useful in non-iterator contexts as well. Many of the redundant iterator operators are also arithmetic operators, so the iterator helper classes borrow many of the operators defined above. In fact, only two new operators need to be defined (the pointer-to-member `operator->` and the subscript `operator[]`)!

The requirements for the types used to instantiate the dereference operators are specified in terms of expressions which must be valid and their return type. The composite operator templates list their component templates, which the instantiating type must support, and possibly other requirements.

### Dereference Operators

All the dereference operator templates in this table accept an optional template parameter (not shown) to be used for base class chaining.

Dereference Operator Template Classes
Key
`T`: operand type | `P`: `pointer` type
`D`: `difference_type` | `R`: `reference` type
`i`: object of type `T` (an iterator) | `n`: object of type `D` (an index)
Template | Supplied Operations | Requirements
--- | --- | ---
`dereferenceable<T, P>` | `P operator->() const` | `(&*i)`. Return convertible to `P`.
`indexable<T, D, R>` | `R operator[](D n) const` | `*(i + n)`. Return of type `R`.

### Grouped Iterator Operators

There are five iterator operator class templates, each for a different category of iterator. The following table shows the operator groups for any category that a custom iterator could define. These class templates have an additional optional template parameter `B`, which is not shown, to support base class chaining.

Iterator Operator Class Templates
Key
`T`: operand type | `P`: `pointer` type
`D`: `difference_type` | `R`: `reference` type
`V`: `value_type` |
Template | Component Operator Templates
--- | ---
`input_iteratable<T, P>` | `equality_comparable<T>` `incrementable<T>` `dereferenceable<T, P>`
`output_iteratable<T>` | `incrementable<T>`
`forward_iteratable<T, P>` | `input_iteratable<T, P>`
`bidirectional_iteratable<T, P>` | `forward_iteratable<T, P>` `decrementable<T>`
`random_access_iteratable<T, P, D, R>` | `bidirectional_iteratable<T, P>` `totally_ordered<T>` `additive<T, D>` `indexable<T, D, R>`

### Iterator Helpers

There are also five iterator helper class templates, each corresponding to a different iterator category. These classes cannot be used for base class chaining. The following summaries show that these class templates supply both the iterator operators from the iterator operator class templates and the iterator typedef's required by the C++ standard (`iterator_category`, `value_type`, etc.).

Iterator Helper Class Templates
Key
`T`: operand type | `P`: `pointer` type
`D`: `difference_type` | `R`: `reference` type
`V`: `value_type` | `x1, x2`: objects of type `T`
Template | Operations & Requirements
--- | ---
`input_iterator_helper<T, V, D, P, R>` | Supports the operations and has the requirements of `input_iteratable<T, P>`
`output_iterator_helper<T>` | Supports the operations and has the requirements of `output_iteratable<T>` See also [1], [2].
`forward_iterator_helper<T, V, D, P, R>` | Supports the operations and has the requirements of `forward_iteratable<T, P>`
`bidirectional_iterator_helper<T, V, D, P, R>` | Supports the operations and has the requirements of `bidirectional_iteratable<T, P>`
`random_access_iterator_helper<T, V, D, P, R>` | Supports the operations and has the requirements of `random_access_iteratable<T, P, D, R>` To satisfy RandomAccessIterator, `x1 - x2` with return convertible to `D` is also required.

#### Iterator Helper Notes

[1] Unlike other iterator helpers templates, `output_iterator_helper` takes only one template parameter - the type of its target class. Although to some it might seem like an unnecessary restriction, the standard requires `difference_type` and `value_type` of any output iterator to be `void` (24.3.1 [lib.iterator.traits]), and `output_iterator_helper` template respects this requirement. Also, output iterators in the standard have void `pointer` and `reference` types, so the `output_iterator_helper` does the same.

[2] As self-proxying is the easiest and most common way to implement output iterators (see, for example, insert [24.4.2] and stream iterators [24.5] in the standard library), `output_iterator_helper` supports the idiom by defining `operator*` and `operator++` member functions which just return a non-const reference to the iterator itself. Support for self-proxying allows us, in many cases, to reduce the task of writing an output iterator to writing just two member functions - an appropriate constructor and a copy-assignment operator. For example, here is a possible implementation of `boost::function_output_iterator` adaptor:

    template<class UnaryFunction>
    struct function_output_iterator
        : boost::output_iterator_helper< function_output_iterator<UnaryFunction> >
    {
        explicit function_output_iterator(UnaryFunction const& f = UnaryFunction())
            : func(f) {}

        template<typename T>
        function_output_iterator& operator=(T const& value)
        {
            this->func(value);
            return *this;
        }

     private:
        UnaryFunction func;
    };

Note that support for self-proxying does not prevent you from using `output_iterator_helper` to ease any other, different kind of output iterator's implementation. If `output_iterator_helper`'s target type provides its own definition of `operator*` or/and `operator++`, then these operators will get used and the ones supplied by `output_iterator_helper` will never be instantiated.

### Iterator Demonstration and Test Program

The iterators_test.cpp program demonstrates the use of the iterator templates, and can also be used to verify correct operation. The following is the custom iterator defined in the test program. It demonstrates a correct (though trivial) implementation of the core operations that must be defined in order for the iterator helpers to "fill in" the rest of the iterator operations.

>     template <class T, class R, class P>
>     struct test_iter
>       : public boost::random_access_iterator_helper<
>          test_iter<T,R,P>, T, std::ptrdiff_t, P, R>
>     {
>       typedef test_iter self;
>       typedef R Reference;
>       typedef std::ptrdiff_t Distance;

>     public:
>       explicit test_iter(T* i =0);
>       test_iter(const self& x);
>       self& operator=(const self& x);
>       Reference operator*() const;
>       self& operator++();
>       self& operator--();
>       self& operator+=(Distance n);
>       self& operator-=(Distance n);
>       bool operator==(const self& x) const;
>       bool operator<(const self& x) const;
>       friend Distance operator-(const self& x, const self& y);
>     };

Check the compiler status report for the test results with selected platforms.

## Contributors

Dave Abrahams
  Started the library and contributed the arithmetic operators in boost/operators.hpp.
Jeremy Siek
  Contributed the dereference operators and iterator helpers in boost/operators.hpp. Also contributed iterators_test.cpp.
Aleksey Gurtovoy
  Contributed the code to support base class chaining while remaining backward-compatible with old versions of the library.
Beman Dawes
  Contributed operators_test.cpp.
Daryle Walker
  Contributed classes for the shift operators, equivalence, partial ordering, and arithmetic conversions. Added the grouped operator classes. Added helper classes for input and output iterators.
Helmut Zeisel
  Contributed the 'left' operators and added some grouped operator classes.
Daniel Frey
  Contributed the NRVO-friendly and symmetric implementation of arithmetic operators.

## Note for Users of Older Versions

The changes in the library interface and recommended usage were motivated by some practical issues described below. The new version of the library is still backward-compatible with the former one (so you're not forced change any existing code), but the old usage is deprecated. Though it was arguably simpler and more intuitive than using base class chaining, it has been discovered that the old practice of deriving from multiple operator templates can cause the resulting classes to be much larger than they should be. Most modern C++ compilers significantly bloat the size of classes derived from multiple empty base classes, even though the base classes themselves have no state. For instance, the size of `point<int>` from the example above was 12-24 bytes on various compilers for the Win32 platform, instead of the expected 8 bytes.

Strictly speaking, it was not the library's fault--the language rules allow the compiler to apply the empty base class optimization in that situation. In principle an arbitrary number of empty base classes can be allocated at the same offset, provided that none of them have a common ancestor (see section 10.5 [class.derived] paragraph 5 of the standard). But the language definition also doesn't require implementations to do the optimization, and few if any of today's compilers implement it when multiple inheritance is involved. What's worse, it is very unlikely that implementors will adopt it as a future enhancement to existing compilers, because it would break binary compatibility between code generated by two different versions of the same compiler. As Matt Austern said, "One of the few times when you have the freedom to do this sort of thing is when you're targeting a new architecture...". On the other hand, many common compilers will use the empty base optimization for single inheritance hierarchies.

Given the importance of the issue for the users of the library (which aims to be useful for writing light-weight classes like `MyInt` or `point<>`), and the forces described above, we decided to change the library interface so that the object size bloat could be eliminated even on compilers that support only the simplest form of the empty base class optimization. The current library interface is the result of those changes. Though the new usage is a bit more complicated than the old one, we think it's worth it to make the library more useful in real world. Alexy Gurtovoy contributed the code which supports the new usage idiom while allowing the library remain backward-compatible.

Revised: 29 Oct 2004

Copyright © Beman Dawes, David Abrahams, 1999-2001.

Copyright © Daniel Frey, 2002-2004.

Use, modification, and distribution is subject to the Boost Software License, Version 1.0. (See accompanying file LICENSE_1_0.txt or copy at www.boost.org/LICENSE_1_0.txt)