doc/concepts/concepts.qbk

[section:high_precision Using Boost.Math with High-Precision Floating-Point Libraries]

The special functions, distributions, constants and tools in this library
can be used with a number of high-precision libraries, including:

* __multiprecision
* __e_float
* __NTL
* __GMP
* __MPFR
* __float128

The last four have some license restrictions;
only __multiprecision when using the `cpp_float` backend
can provide an unrestricted [@http://www.boost.org/LICENSE_1_0.txt Boost] license.

At present, the price of a free license is slightly lower speed.

Of course, the main cost of higher precision is very much decreased
(usually at least hundred-fold) computation speed, and big increases in memory use.

Some libraries offer true
[@http://en.wikipedia.org/wiki/Arbitrary-precision_arithmetic arbitrary precision arithmetic]
where the precision is limited only by avilable memory and compute time, but most are used
at some arbitrarily-fixed precision, say 100 decimal digits.

__multiprecision can operate in both ways, but the most popular choice is likely to be about a hundred
decimal digits, though examples of computing tens of thousands of digits have been demonstrated.

[section:why_high_precision  Why use a high-precision library rather than built-in floating-point types?]

For nearly all applications, the built-in floating-point types, `double`
(and `long double` if this offers higher precision than `double`)
offer enough precision, typically a dozen decimal digits.

Some reasons why one would want to use a higher precision:

* A much more precise result (many more digits) is just a requirement.
* The range of the computed value exceeds the range of the type: factorials are the textbook example.
* Using double is (or may be) too inaccurate.
* Using long double (or may be) is too inaccurate.
* Using an extended precision type implemented in software as
[@http://en.wikipedia.org/wiki/Double-double_(arithmetic)#Double-double_arithmetic double-double]
([@http://en.wikipedia.org/wiki/Darwin_(operating_system) Darwin]) is sometimes unpredictably inaccurate.
* Loss of precision or inaccuracy caused by extreme arguments or cancellation error.
* An accuracy as good as possible for a chosen built-in floating-point type is required.
* As a reference value, for example, to determine the inaccuracy
of a value computed with a built-in floating point type,
(perhaps even using some quick'n'dirty algorithm).
The accuracy of many functions and distributions in Boost.Math has been measured in this way
from tables of very high precision (up to 1000 decimal digits).

Many functions and distributions have differences from exact values
that are only a few least significant bits - computation noise.
Others, often those for which analytical solutions are not available,
require approximations and iteration:
these may lose several decimal digits of precision.

Much larger loss of precision can occur for [@http://en.wikipedia.org/wiki/Boundary_case boundary]
or [@http://en.wikipedia.org/wiki/Corner_case corner cases],
often caused by [@http://en.wikipedia.org/wiki/Loss_of_significance cancellation errors].

(Some of the worst and most common examples of
[@http://en.wikipedia.org/wiki/Loss_of_significance cancellation error or loss of significance]
can be avoided by using __complements: see __why_complements).

If you require a value which is as accurate as can be represented in the floating-point type,
and is thus the closest representable value and has an error less than 1/2 a
[@http://en.wikipedia.org/wiki/Least_significant_bit least significant bit] or
[@http://en.wikipedia.org/wiki/Unit_in_the_last_place ulp]
it may be useful to use a higher-precision type,
for example, `cpp_dec_float_50`, to generate this value.
Conversion of this value to a built-in floating-point type ('float', `double` or `long double`)
will not cause any further loss of precision.
A decimal digit string will also be 'read' precisely by the compiler
into a built-in floating-point type to the nearest representable value.

[note In contrast, reading a value from an `std::istream` into a built-in floating-point type
is [*not guaranteed] by the C++ Standard to give the nearest representable value.]

William Kahan coined the term
[@http://en.wikipedia.org/wiki/Rounding#The_table-maker.27s_dilemma Table-Maker's Dilemma]
for the problem of correctly rounding functions.
Using a much higher precision (50 or 100 decimal digits)
is a practical way of generating (almost always) correctly rounded values.

[endsect] [/section:why_high_precision  Why use a high-precision library rather than built-in floating-point types?]

[section:use_multiprecision Using Boost.Multiprecision]

[*All new projects are recommended to use __multiprecision.]

[import ../../example/big_seventh.cpp]

[big_seventh_example_1]

[import ../../example/fft_sines_table.cpp]

[fft_sines_table_example_1]

The table output is:

[fft_sines_table_example_output]

[fft_sines_table_example_check]


[/TODO another example needed here]

[/import ../../example/ibeta_mp_example.cpp]

[/ibeta_mp_example_1]

[/The program output is:]

[/ibeta_mp_output_1]

[endsect] [/section:use_multiprecision Using Boost.Multiprecision]

[section:float128 Using with GCC's __float128 datatype]

At present support for GCC's native `__float128` datatype is extremely limited: the numeric constants
will all work with that type, and that's about it.  If you want to use the distributions or special
functions then you will need to provide your own wrapper header that:

* Provides std::numeric_limits<__float128> support.
* Provides overloads of the standard library math function for type `__float128`and which forward to the libquadmath equivalents.

Ultimately these facilities should be provided by GCC and libstdc++.

[endsect]

[section:use_mpfr Using With MPFR or GMP - High-Precision Floating-Point Library]

The special functions and tools in this library can be used with
[@http://www.mpfr.org MPFR] (an arbitrary precision number type based on the __GMP),
either via the bindings in [@../../../../boost/math/bindings/mpfr.hpp boost/math/bindings/mpfr.hpp],
or via [@../../../../boost/math/bindings/mpfr.hpp boost/math/bindings/mpreal.hpp].

[*New projects are recommended to use __multiprecision with GMP/MPFR backend instead.]

In order to use these bindings you will need to have installed [@http://www.mpfr.org MPFR]
plus its dependency the [@http://gmplib.org GMP library].  You will also need one of the
two supported C++ wrappers for MPFR:
[@http://math.berkeley.edu/~wilken/code/gmpfrxx/ gmpfrxx (or mpfr_class)],
or [@http://www.holoborodko.com/pavel/mpfr/ mpfr-C++ (mpreal)].

Unfortunately neither `mpfr_class` nor `mpreal` quite satisfy our conceptual requirements,
so there is a very thin set of additional interfaces and some helper traits defined in
[@../../../../boost/math/bindings/mpfr.hpp boost/math/bindings/mpfr.hpp] and
[@../../../../boost/math/bindings/mpreal.hpp boost/math/bindings/mpreal.hpp]
that you should use in place of including 'gmpfrxx.h' or 'mpreal.h' directly.
The classes `mpfr_class` or `mpreal` are
then usable unchanged once this header is included, so for example `mpfr_class`'s
performance-enhancing expression templates are preserved and fully supported by this library:

   #include <boost/math/bindings/mpfr.hpp>
   #include <boost/math/special_functions/gamma.hpp>

   int main()
   {
      mpfr_class::set_dprec(500); // 500 bit precision
      //
      // Note that the argument to tgamma is
      // an expression template - that's just fine here.
      //
      mpfr_class v = boost::math::tgamma(sqrt(mpfr_class(2)));
      std::cout << std::setprecision(50) << v << std::endl;
   }

Alternatively use with `mpreal` would look like:

   #include <boost/math/bindings/mpreal.hpp>
   #include <boost/math/special_functions/gamma.hpp>

   int main()
   {
      mpfr::mpreal::set_precision(500); // 500 bit precision
      mpfr::mpreal v = boost::math::tgamma(sqrt(mpfr::mpreal(2)));
      std::cout << std::setprecision(50) << v << std::endl;
   }

For those functions that are based upon the __lanczos, the bindings
defines a series of approximations with up to 61 terms and accuracy
up to approximately 3e-113.  This therefore sets the upper limit for accuracy
to the majority of functions defined this library when used with either `mpfr_class` or `mpreal`.

There is a concept checking test program for mpfr support
[@../../../../libs/math/test/mpfr_concept_check.cpp here] and
[@../../../../libs/math/test/mpreal_concept_check.cpp here].

[endsect] [/section:use_mpfr Using With MPFR / GMP - a High-Precision Floating-Point Library]

[section:e_float Using e_float Library]

__multiprecision was a development from the __e_float library by Christopher Kormanyos.

e_float can still be used with Boost.Math library via the header:

   <boost/math/bindings/e_float.hpp>

And the type `boost::math::ef::e_float`:
this type is a thin wrapper class around ::e_float which provides the necessary
syntactic sugar to make everything "just work".

There is also a concept checking test program for e_float support
[@../../../../libs/math/test/e_float_concept_check.cpp here].

[*New projects are recommended to use __multiprecision with `cpp_float` backend instead.]

[endsect] [/section:e_float Using e_float Library]

[section:use_ntl Using NTL Library]

[@http://shoup.net/ntl/doc/RR.txt NTL::RR]
(an arbitrarily-fixed precision floating-point number type),
can be used via the bindings in
[@../../../../boost/math/bindings/rr.hpp boost/math/bindings/rr.hpp].
For details, see [@http://shoup.net/ntl/ NTL: A Library for doing Number Theory by
Victor Shoup].

[*New projects are recommended to use __multiprecision instead.]

Unfortunately `NTL::RR` doesn't quite satisfy our conceptual requirements,
so there is a very thin wrapper class `boost::math::ntl::RR` defined in
[@../../../../boost/math/bindings/rr.hpp boost/math/bindings/rr.hpp] that you
should use in place of `NTL::RR`.  The class is intended to be a drop-in
replacement for the "real" NTL::RR that adds some syntactic sugar to keep
this library happy, plus some of the standard library functions not implemented
in NTL.

For those functions that are based upon the __lanczos, the bindings
defines a series of approximations with up to 61 terms and accuracy
up to approximately 3e-113.  This therefore sets the upper limit for accuracy
to the majority of functions defined this library when used with `NTL::RR`.

There is a concept checking test program for NTL support
[@../../../../libs/math/test/ntl_concept_check.cpp here].


[endsect] [/section:use_ntl Using With NTL - a High-Precision Floating-Point Library]

[section:using_test Using without expression templates for Boost.Test and others]

As noted in the __multiprecision documentation, certain program constructs will not compile
when using expression templates.  One example that many users may encounter
is Boost.Test (1.54 and earlier) when using macro BOOST_CHECK_CLOSE and BOOST_CHECK_CLOSE_FRACTION.

If, for example, you wish to use any multiprecision type like `cpp_dec_float_50`
in place of `double` to give more precision,
you will need to override the default `boost::multiprecision::et_on` with
`boost::multiprecision::et_off`.

[import ../../example/test_cpp_float_close_fraction.cpp]

[expression_template_1]

A full example code is at [@../../example/test_cpp_float_close_fraction.cpp test_cpp_float_close_fraction.cpp]

[endsect] [/section:using_test Using without expression templates for Boost.Test and others]

[endsect] [/section:high_precision Using With High-Precision Floating-Point Libraries]


[section:concepts 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 [link math_toolkit.high_precision.use_ntl a thin wrapper class]
one of the types provided with [@http://shoup.net/ntl/ NTL (RR)] can be used.
But now that __multiprecision library is available,
this has become the reference real number type.

Submissions of binding to other extended precision types would also still be welcome.

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

[h4 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).

[table
[[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:

# The functions `log_max_value` and `log_min_value` can be
synthesised from the others, and so no explicit specialisation is required.
# 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 boost/math/bindings/rr.hpp]
for an example where the precision does vary at runtime).
# 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 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).
]

[h4 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`.

[table
[[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 is highly desirable.
You may be able to adapt an existing approximation from
[@../../../../boost/math/special_functions/lanczos.hpp
boost/math/special_functions/lanczos.hpp] or
[@../../../../boost/math/bindings/detail/big_lanczos.hpp
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
[@../../tools/lanczos_generator.cpp
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.

[endsect] [/section: ]

[section:dist_concept Conceptual Requirements for Distribution Types]

A /DistributionType/ is a type that implements the following conceptual
requirements, and encapsulates a statistical distribution.

Please note that this documentation should not be used as a substitute
for the
[link math_toolkit.dist_ref reference documentation], and
[link math_toolkit.stat_tut tutorial] of the statistical
distributions.

In the following table, /d/ is an object of type `DistributionType`,
/cd/ is an object of type `const DistributionType` and /cr/ is an
object of a type convertible to `RealType`.

[table
[[Expression][Result Type][Notes]]
[[DistributionType::value_type][RealType]
      [The real-number type /RealType/ upon which the distribution operates.]]
[[DistributionType::policy_type][RealType]
      [The __Policy to use when evaluating functions that depend on this distribution.]]
[[d = cd][Distribution&][Distribution types are assignable.]]
[[Distribution(cd)][Distribution][Distribution types are copy constructible.]]
[[pdf(cd, cr)][RealType][Returns the PDF of the distribution.]]
[[cdf(cd, cr)][RealType][Returns the CDF of the distribution.]]
[[cdf(complement(cd, cr))][RealType]
      [Returns the complement of the CDF of the distribution,
      the same as: `1-cdf(cd, cr)`]]
[[quantile(cd, cr)][RealType][Returns the quantile (or percentile) of the distribution.]]
[[quantile(complement(cd, cr))][RealType]
      [Returns the quantile (or percentile) of the distribution, starting from
      the complement of the probability, the same as: `quantile(cd, 1-cr)`]]
[[chf(cd, cr)][RealType][Returns the cumulative hazard function of the distribution.]]
[[hazard(cd, cr)][RealType][Returns the hazard function of the distribution.]]
[[kurtosis(cd)][RealType][Returns the kurtosis of the distribution.]]
[[kurtosis_excess(cd)][RealType][Returns the kurtosis excess of the distribution.]]
[[mean(cd)][RealType][Returns the mean of the distribution.]]
[[mode(cd)][RealType][Returns the mode of the distribution.]]
[[skewness(cd)][RealType][Returns the skewness of the distribution.]]
[[standard_deviation(cd)][RealType][Returns the standard deviation of the distribution.]]
[[variance(cd)][RealType][Returns the variance of the distribution.]]
]

[endsect]

[section:archetypes Conceptual Archetypes for Reals and Distributions]

There are a few concept archetypes available:

* Real concept for floating-point types.
* distribution Concept for statistical distributions.

[h5 Real concept]

`std_real_concept` is an archetype for theReal types,
including the built-in float, double, long double.

``#include <boost/concepts/std_real_concept.hpp>``

   namespace boost{
   namespace math{
   namespace concepts
   {
     class std_real_concept;
   }
   }} // namespaces


The main purpose in providing this type is to verify
that standard library functions are found via a using declaration -
bringing those functions into the current scope -
and not just because they happen to be in global scope.

In order to ensure that a call to say `pow` can be found
either via argument dependent lookup, or failing that then
in the std namespace: all calls to standard library functions
are unqualified, with the std:: versions found via a `using` declaration
to make them visible in the current scope.  Unfortunately it's all
to easy to forget the `using` declaration, and call the double version of
the function that happens to be in the global scope by mistake.

For example if the code calls ::pow rather than std::pow,
the code will cleanly compile, but truncation of long doubles to
double will cause a significant loss of precision.
In contrast a template instantiated with std_real_concept will *only*
compile if the all the standard library functions used have
been brought into the current scope with a using declaration.

[h6 Testing the real concept]

There is a test program
[@../../test/std_real_concept_check.cpp libs/math/test/std_real_concept_check.cpp]
that instantiates every template in this library with type
`std_real_concept` to verify its usage of standard library functions.

``#include <boost/math/concepts/real_concept.hpp>``

   namespace boost{
   namespace math{
   namespace concepts{

   class real_concept;

   }}} // namespaces

`real_concept` is an archetype for
[link math_toolkit.concepts user defined real types],
it declares its standard library functions in its own
namespace: these will only be found if they are called unqualified
allowing argument dependent lookup to locate them.  In addition
this type is useable at runtime:
this allows code that would not otherwise be exercised by the built-in
floating point types to be tested.  There is no std::numeric_limits<>
support for this type, since numeric_limits is not a conceptual requirement
for [link math_toolkit.concepts RealType]s.

NTL RR is an example of a type meeting the requirements that this type
models, but note that use of a thin wrapper class is required: refer to
[link math_toolkit.high_precision.use_ntl "Using With NTL - a High-Precision Floating-Point Library"].

There is no specific test case for type `real_concept`, instead, since this
type is usable at runtime, each individual test case as well as testing
`float`, `double` and `long double`, also tests `real_concept`.

[h6 Distribution Concept]

Distribution Concept models statistical distributions.

``#include <boost/math/concepts/distribution.hpp>``

   namespace boost{
   namespace math{
   namespace concepts
   {
     template <class RealType>
     class distribution_archetype;

     template <class Distribution>
     struct DistributionConcept;

   }}} // namespaces

The class template `distribution_archetype` is a model of the
[link math_toolkit.dist_concept Distribution concept].

The class template `DistributionConcept` is a
[@../../../../libs/concept_check/index.html concept checking class]
for distribution types.

[h6 Testing the distribution concept]

The test program
[@../../test/compile_test/distribution_concept_check.cpp distribution_concept_check.cpp]
is responsible for using `DistributionConcept` to verify that all the
distributions in this library conform to the
[link math_toolkit.dist_concept Distribution concept].

The class template `DistributionConcept` verifies the existence
(but not proper function) of the non-member accessors
required by the [link math_toolkit.dist_concept Distribution concept].
These are checked by calls like

v = pdf(dist, x); // (Result v is ignored).

And in addition, those that accept two arguments do the right thing when the
arguments are of different types (the result type is always the same as the
distribution's value_type).  (This is implemented by some additional
forwarding-functions in derived_accessors.hpp, so that there is no need for
any code changes.  Likewise boilerplate versions of the
hazard\/chf\/coefficient_of_variation functions are implemented in
there too.)

[endsect] [/section:archetypes Conceptual Archetypes for Reals and Distributions]
[/
  Copyright 2006, 2010, 2012 John Maddock and Paul A. Bristow.
  Distributed under the Boost Software License, Version 1.0.
  (See accompanying file LICENSE_1_0.txt or copy at
  http://www.boost.org/LICENSE_1_0.txt).
]