Conceptual Requirements for Real Number Types

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### Conceptual Requirements for Real Number Types

The functions, and statistical distributions in this library can be used with any type RealType that meets the conceptual requirements given below. All the built in floating point types will meet these requirements. User defined types that meet the requirements can also be used. For example, with a thin wrapper class one of the types provided with NTL (RR) can be used. Submissions of binding to other extended precision types would also be most welcome!

The guiding principal behind these requirements, is that a RealType behaves just like a built in floating point type.

##### Basic Arithmetic Requirements

These requirements are common to all of the functions in this library.

In the following table r is an object of type `RealType`, cr and cr2 are objects of type `const RealType`, and ca is an object of type `const arithmetic-type` (arithmetic types include all the built in integers and floating point types).

Expression | Result Type | Notes
--- | --- | ---
`RealType(cr)` | RealType | RealType is copy constructible.
`RealType(ca)` | RealType | RealType is copy constructible from the arithmetic types.
`r = cr` | RealType& | Assignment operator.
`r = ca` | RealType& | Assignment operator from the arithmetic types.
`r += cr` | RealType& | Adds cr to r.
`r += ca` | RealType& | Adds ar to r.
`r -= cr` | RealType& | Subtracts cr from r.
`r -= ca` | RealType& | Subtracts ca from r.
`r *= cr` | RealType& | Multiplies r by cr.
`r *= ca` | RealType& | Multiplies r by ca.
`r /= cr` | RealType& | Divides r by cr.
`r /= ca` | RealType& | Divides r by ca.
`-r` | RealType | Unary Negation.
`+r` | RealType& | Identity Operation.
`cr + cr2` | RealType | Binary Addition
`cr + ca` | RealType | Binary Addition
`ca + cr` | RealType | Binary Addition
`cr - cr2` | RealType | Binary Subtraction
`cr - ca` | RealType | Binary Subtraction
`ca - cr` | RealType | Binary Subtraction
`cr * cr2` | RealType | Binary Multiplication
`cr * ca` | RealType | Binary Multiplication
`ca * cr` | RealType | Binary Multiplication
`cr / cr2` | RealType | Binary Subtraction
`cr / ca` | RealType | Binary Subtraction
`ca / cr` | RealType | Binary Subtraction
`cr == cr2` | bool | Equality Comparison
`cr == ca` | bool | Equality Comparison
`ca == cr` | bool | Equality Comparison
`cr != cr2` | bool | Inequality Comparison
`cr != ca` | bool | Inequality Comparison
`ca != cr` | bool | Inequality Comparison
`cr <= cr2` | bool | Less than equal to.
`cr <= ca` | bool | Less than equal to.
`ca <= cr` | bool | Less than equal to.
`cr >= cr2` | bool | Greater than equal to.
`cr >= ca` | bool | Greater than equal to.
`ca >= cr` | bool | Greater than equal to.
`cr < cr2` | bool | Less than comparison.
`cr < ca` | bool | Less than comparison.
`ca < cr` | bool | Less than comparison.
`cr > cr2` | bool | Greater than comparison.
`cr > ca` | bool | Greater than comparison.
`ca > cr` | bool | Greater than comparison.
`boost::math::tools::digits<RealType>()` | int | The number of digits in the significand of RealType.
`boost::math::tools::max_value<RealType>()` | RealType | The largest representable number by type RealType.
`boost::math::tools::min_value<RealType>()` | RealType | The smallest representable number by type RealType.
`boost::math::tools::log_max_value<RealType>()` | RealType | The natural logarithm of the largest representable number by type RealType.
`boost::math::tools::log_min_value<RealType>()` | RealType | The natural logarithm of the smallest representable number by type RealType.
`boost::math::tools::epsilon<RealType>()` | RealType | The machine epsilon of RealType.

Note that:

1. The functions `log_max_value` and `log_min_value` can be synthesised from the others, and so no explicit specialisation is required.
2. The function `epsilon` can be synthesised from the others, so no explicit specialisation is required provided the precision of RealType does not vary at runtime (see the header boost/math/bindings/rr.hpp for an example where the precision does vary at runtime).
3. The functions `digits`, `max_value` and `min_value`, all get synthesised automatically from `std::numeric_limits`. However, if `numeric_limits` is not specialised for type RealType, then you will get a compiler error when code tries to use these functions, unless you explicitly specialise them. For example if the precision of RealType varies at runtime, then `numeric_limits` support may not be appropriate, see boost/math/bindings/rr.hpp for examples.

| Warning

If `std::numeric_limits<>` is not specialized for type RealType then the default float precision of 6 decimal digits will be used by other Boost programs including:

Boost.Test: giving misleading error messages like

"difference between {9.79796} and {9.79796} exceeds 5.42101e-19%".

Boost.LexicalCast and Boost.Serialization when converting the number to a string, causing potentially serious loss of accuracy on output.

Although it might seem obvious that RealType should require `std::numeric_limits` to be specialized, this is not sensible for `NTL::RR` and similar classes where the number of digits is a runtime parameter (where as for `numeric_limits` it has to be fixed at compile time).

##### Standard Library Support Requirements

Many (though not all) of the functions in this library make calls to standard library functions, the following table summarises the requirements. Note that most of the functions in this library will only call a small subset of the functions listed here, so if in doubt whether a user defined type has enough standard library support to be useable the best advise is to try it and see!

In the following table r is an object of type `RealType`, cr1 and cr2 are objects of type `const RealType`, and i is an object of type `int`.

Expression | Result Type
--- | ---
`fabs(cr1)` | RealType
`abs(cr1)` | RealType
`ceil(cr1)` | RealType
`floor(cr1)` | RealType
`exp(cr1)` | RealType
`pow(cr1, cr2)` | RealType
`sqrt(cr1)` | RealType
`log(cr1)` | RealType
`frexp(cr1, &i)` | RealType
`ldexp(cr1, i)` | RealType
`cos(cr1)` | RealType
`sin(cr1)` | RealType
`asin(cr1)` | RealType
`tan(cr1)` | RealType
`atan(cr1)` | RealType
`fmod(cr1)` | RealType
`round(cr1)` | RealType
`iround(cr1)` | int
`trunc(cr1)` | RealType
`itrunc(cr1)` | int

Note that the table above lists only those standard library functions known to be used (or likely to be used in the near future) by this library. The following functions: `acos`, `atan2`, `fmod`, `cosh`, `sinh`, `tanh`, `log10`, `lround`, `llround`, ltrunc`,` lltrunc`and` modf` are not currently used, but may be if further special functions are added.

Note that the `round`, `trunc` and `modf` functions are not part of the current C++ standard: they are part of the additions added to C99 which will likely be in the next C++ standard. There are Boost versions of these provided as a backup, and the functions are always called unqualified so that argument-dependent-lookup can take place.

In addition, for efficient and accurate results, a Lanczos approximation is highly desirable. You may be able to adapt an existing approximation from boost/math/special_functions/lanczos.hpp or boost/math/bindings/detail/big_lanczos.hpp: in the former case you will need change static_cast's to lexical_cast's, and the constants to strings (in order to ensure the coefficients aren't truncated to long double) and then specialise `lanczos_traits` for type T. Otherwise you may have to hack libs/math/tools/lanczos_generator.cpp to find a suitable approximation for your RealType. The code will still compile if you don't do this, but both accuracy and efficiency will be greatly compromised in any function that makes use of the gamma/beta/erf family of functions.