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Users can extend the library with various new types whose concepts are defined here.
An Axis maps input values to indices. It holds state specific to that axis, like the number of bins and any metadata. Must be CopyConstructible, CopyAssignable, and nothrow MoveAssignable.
A is a type meeting the
            requirements of Axis
          a is a value of type
            A
          I is an alias for boost::histogram::axis::index_type
          Table 1.1. Valid expressions
| Expression | Returns | Semantics, Pre/Post-conditions | 
|---|---|---|
| 
                   | 
                   | 
                  Const member function which returns the number of bins of the axis.
                  All indices from  | 
| 
                   | 
                   | 
                  Const member function which maps a value  | 
| 
                   | 
                   | 
                  Const member function which returns the allocator  | 
A is a type meeting the
            requirements of Axis
          a and b
            are values of type A
          i and j
            are indices of type boost::histogram::axis::index_type
          M is a metadata type
            that is CopyConstructible
            and CopyAssignable
            and nothrow MoveAssignable.
          Table 1.2. Valid expressions
| Expression | Returns | Semantics, Pre/Post-conditions | 
|---|---|---|
| 
                   | 
                   | 
                  Non-const member function which maps a value to an index (first
                  argument of the returned pair) and offset (second argument of the
                  returned pair). If the value is not covered by the axis, this method
                  may grow the current axis size ( | 
| 
                   | 
                  Special constructor used by the reduce algorithm.  | |
| 
                   | 
                   | 
                  Static constexpr member function which returns the  | 
| 
                   | 
                   | Static constexpr member function which returns true, if the axis has a bin for every possible input value, and false otherwise. Faster code can be generated if all axes types in a histogram are inclusive. An axis with underflow and overflow bins is always inclusive. An axis may be inclusive even if underflow or overflow bins are missing. For example, a category axis is inclusive if it has an overflow bin or if it is growing. | 
| 
                   | 
                   | Const member function which returns a const reference to the metadata associated with the axis (usually a string). | 
| 
                   | 
                   | Likewise, but non-const member function which returns a reference to the metadata. If this member function exists, also the previous one must exist. | 
| 
                   | 
                   | 
                  Returns  | 
| 
                   | 
                   | 
                  Must be implemented if  | 
| 
                   | 
                   | 
                   | 
| 
                   | 
                   | 
A DiscreteAxis is one of two optional refinements of the Axis concept, the other one is the IntervalAxis. This concept is for values that do not form intervals, and for axes with intervals that contain exactly one value.
A is a type meeting the
            requirements of DiscreteAxis
          a is a value of type
            A
          V is the type accepted
            for conversion into an index
          v is a value of type
            V
          i is a value of type
            boost::histogram::axis::index_type
          AxisIter is an RandomAccessIterator
            over the bins of A
          ReAxisIter is a reverse
            RandomAccessIterator over the bins of A
          Table 1.3. Valid expressions
| Expression | Returns | Semantics, Pre/Post-conditions | 
|---|---|---|
| 
                   | 
                   | 
                  Const member function which maps an index to a value. The mapping
                  must be injective: each index must be uniquely mapped to one value.
                  The effect must be exactly the inverse of  | 
| 
                   | 
                   | 
                  Must have the same effect as  | 
| 
                   | 
                   | 
                  Const member function which return an iterator to the bin with
                  index  | 
| 
                   | 
                   | 
                  Const member function which returns an iterator to the bin with
                  index  | 
| 
                   | 
                   | 
                  Const member function which return a reverse iterator to the bin
                  with index  | 
| 
                   | 
                   | 
                  Const member function which returns an iterator to the bin with
                  index  | 
| ![[Tip]](../../../../../doc/src/images/tip.png) | Tip | 
|---|---|
| 
          The complete iterator interface can be added to a user-defined axis which
          implements  | 
boost::histogram::axis::category
          boost::histogram::axis::integer,
            if first template parameter is boost::histogram::index_type
          A IntervalAxis is one of two optional refinements of the Axis concept, the other one is the DiscreteAxis. It is for ordered values that form intervals with a well-defined lower and upper edge, and a center. Each bin represents an interval of values.
A is a type meeting the
            requirements of IntervalAxis
          a is a value of type
            A
          V is the type accepted
            for conversion into an index
          B is the type that represents
            the bin interval
          v is a value of type
            V
          i is a value of type
            boost::histogram::axis::index_type
          j is a value of type
            boost::histogram::axis::real_index_type
          AxisIter is an RandomAccessIterator
            over the bins of A
          ReAxisIter is a reverse
            RandomAccessIterator over the bins of A
          Table 1.4. Valid expressions
| Expression | Returns | Semantics, Pre/Post-conditions | 
|---|---|---|
| 
                   | 
                   | 
                  Const member function which maps an index to a value. The mapping
                  must be injective: each index must be uniquely mapped to one value.
                  The result of  | 
| 
                   | 
                   | 
                  Const member function which returns an instance that represents
                  the current bin. Nothing about the type is required, but it is
                  recommended that the type has the methods  | 
| 
                   | 
                   | 
                  Const member function which return an iterator to the bin with
                  index  | 
| 
                   | 
                   | 
                  Const member function which returns an iterator to the bin with
                  index  | 
| 
                   | 
                   | 
                  Const member function which return a reverse iterator to the bin
                  with index  | 
| 
                   | 
                   | 
                  Const member function which returns an iterator to the bin with
                  index  | 
| ![[Tip]](../../../../../doc/src/images/tip.png) | Tip | 
|---|---|
| 
          The complete iterator interface can be added to a user-defined axis which
          implements  | 
boost::histogram::axis::regular
          boost::histogram::axis::variable
          boost::histogram::axis::integer,
            if first template parameter is a floating point type
          
        A Transform implements a monotonic mapping
        between two real-valued domains, external and internal. It is used to extend
        the regular axis.
        The bins in the internal domain are of equal width, while the bins in the
        external domain are non-equal width. Must be DefaultConstructible,
        CopyConstructible,
        and CopyAssignable.
      
T is a type meeting the
            requirements of Transform
          t is a value of type
            T
          X is a type with the
            semantics of a floating-point type
          x is a value of type
            X
          Y is a floating-point
            type
          y is a value of type
            Y
          Table 1.5. Valid expressions
| Expression | Returns | Semantics, Pre/Post-conditions | 
|---|---|---|
| 
                   | 
                   | 
                  Const or static member function which maps the external value to
                  the corresponding internal value. The return type  | 
| 
                   | 
                   | 
                  Const or static member function which maps the internal value to
                  the corresponding external value. The result of  | 
| 
                   | 
                   | 
                   | 
T is a type meeting the
            requirements of Transform
          t is a value of type
            T
          ar is a value of an archive
            with Boost.Serialization semantics
          Table 1.6. Valid expressions
| Expression | Returns | Semantics, Pre/Post-conditions | 
|---|---|---|
| 
                   | 
                   | 
A Storage handles memory for the bin counters and provides a uniform vector-like interface for accessing cell values for reading and writing. Must be DefaultConstructible, CopyConstructible, and CopyAssignable.
S is a type meeting the
            requirements of Storage
          s is a value of types
            S
          i and n
            are values of type std::size_t
          Table 1.7. Valid expressions
| Expression | Returns | Semantics, Pre/Post-conditions | 
|---|---|---|
| 
                   | Cell element type, may be either an integral type, floating-point type, or a type meeting the requirements of Accumulator. | |
| 
                   | 
                   | |
| 
                   | 
                   | |
| 
                   | 
                  Returns an STL-compliant iterator type which dereferences to  | |
| 
                   | 
                  Returns an STL-compliant iterator type which dereferences to  | |
| 
                   | bool | Static constexpr member. True, if storage supports parallel read-write access to all cells. False, if such parallel access would either cause data corruption or require synchronization so that effectively only one cell can be accessed at a time, making cell-access single-threaded. | 
| 
                   | 
                   | Const member function which returns the current number of cells in the storage. | 
| 
                   | 
                  Non-const member function which discards current cell values, changes
                  storage size to  | |
| 
                   | 
                   | Non-const member function which returns the iterator to the first storage cell. | 
| 
                   | 
                   | Likewise, but a const member function which returns the const_iterator. | 
| 
                   | 
                   | Member function which returns the iterator to the cell after the last valid storage cell. | 
| 
                   | 
                   | Likewise, but a const member function which returns the const_iterator. | 
| 
                   | 
                   | 
                  Member function which returns a reference to the cell which is
                  addressed by  | 
| 
                   | 
                   | Likewise, but a const member function which returns a const reference. | 
| 
                   | 
                   | 
                   | 
| 
                   | 
                   | 
                  Const member function which returns the allocator  | 
S is a type meeting the
            requirements of Storage
          s is a value of types
            S
          x is convertible to
            double
          ar is a value of an archive
            with Boost.Serialization semantics
          Table 1.8. Valid expressions
| Expression | Returns | Semantics, Pre/Post-conditions | 
|---|---|---|
| 
                   | 
                   | 
                  Scales all cell values by the factor  | 
| 
                   | 
                   | 
An Accumulator is a functor which consumes the argument to update some internal state. The state can be read with member functions or free functions. Must be DefaultConstructible, CopyConstructible, and CopyAssignable.
A is a type meeting the
            requirements of Accumulator
          a and b
            are values of type A
          ts...
            is a pack of values of arbitrary types
          Table 1.9. Valid expressions
| Expression | Returns | Semantics, Pre/Post-conditions | 
|---|---|---|
| 
                   | Either a call operator accepting a fixed number of arguments must be implemented, or the pre-increment operator. The call operator may not be templated and not overloaded, except to support weights as described under optional features. | |
| 
                   | 
                   | 
                  Returns  | 
| 
                   | 
                   | 
                  Must be implemented if  | 
A is a type meeting the
            requirements of Accumulator
          a and b
            are values of type A
          w is a value of type
            boost::histogram::weight_type,
            where T is a number type
          ts...
            is a pack of values of arbitrary types
          v is a number value (integral
            or floating point)
          Table 1.10. Valid expressions
| Expression | Return | Semantics, Pre/Post-conditions | 
|---|---|---|
| 
                   | 
                  Does a weighted fill of the accumulator. Use this to implement
                  weight support for an accumulator that is normally filled with
                   | |
| 
                   | 
                   | 
                  Adds a second accumulator  | 
| 
                   | 
                   | 
                  Scales the accumulator state by the real value  | 
| 
                   | 
                   | 
                   | 
| 
                   | 
                   |