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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&

```r += ca```

RealType&

```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

```cr + ca```

RealType

```ca + cr```

RealType

```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

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`, `modf` and `log10` are not currently used, but may be if further special functions are added.

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 libs/math/tools/ntl_rr_lanczos.hpp: 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.