[mathpart constants..Mathematical Constants]

[section:constants_intro Introduction]

Boost.Math provides a collection of mathematical constants.

[h4 Why use Boost.Math mathematical constants?]

* Readable. For the very many jobs just using built-in like `double`, you can just write expressions like
``double area = pi * r * r;``
(If that's all you want, jump direct to [link math_toolkit.tutorial.non_templ use in non-template code]!)
* Effortless - avoiding a search of reference sources.
* Usable with both builtin floating point types, and user-defined, possibly extended precision, types such as
NTL, MPFR/GMP, mp_float: in the latter case the constants are computed to the necessary precision and then cached.
* Accurate - ensuring that the values are as accurate as possible for the
chosen floating-point type
  * No loss of accuracy from repeated rounding of intermediate computations.
  * Result is computed with higher precision and only rounded once.
  * Less risk of inaccurate result from functions pow, trig and log at [@http://en.wikipedia.org/wiki/Corner_case corner cases].
  * Less risk of [@http://docs.oracle.com/cd/E19957-01/806-3568/ncg_goldberg.html cancellation error].
* Portable - as possible between different systems using different floating-point precisions:
see [link math_toolkit.tutorial.templ use in template code].
* Tested - by comparison with other published sources, or separately computed at long double precision.
* Faster - can avoid (re-)calculation at runtime.
  * If the value returned is a builtin type then it's returned by value as a `constexpr` (C++11 feature, if available).
  * If the value is computed and cached (or constructed from a string representation and cached), then it's returned by constant reference.

  This can be 
significant if:
    * Functions pow, trig or log are used.
    * Inside an inner loop.
    * Using a high-precision UDT like __multiprecision.
    * Compiler optimizations possible with built-in types, especially `double`, are not available.

[endsect] [/section:intro Introduction]

[section:tutorial Tutorial]

[section:non_templ Use in non-template code]

When using the math constants at your chosen fixed precision in non-template code,
you can simply add a `using namespace` declaration, for example,
`using namespace boost::math::double_constants`,
to make the constants of the correct precision for your code
visible in the current scope, and then use each constant ['as a simple variable - sans brackets]:

   #include <boost/math/constants/constants.hpp>

   double area(double r)
   {
      using namespace boost::math::double_constants;
      return pi * r * r;
   }

Had our function been written as taking a `float` rather than a `double`,
we could have written instead:

   #include <boost/math/constants/constants.hpp>

   float area(float r)
   {
      using namespace boost::math::float_constants;
      return pi * r * r;
   }

Likewise, constants that are suitable for use at `long double` precision
are available in the namespace `boost::math::long_double_constants`.

You can see the full list of available constants at [link math_toolkit.constants].

Some examples of using constants are at [@../../example/constants_eg1.cpp constants_eg1].

[endsect] [/section:non_templ Use in non-template code]

[section:templ Use in template code]

When using the constants inside a function template, we need to ensure that
we use a constant of the correct precision for our template parameters.
We can do this by calling the function-template versions, `pi<FPType>()`, of the constants
like this:

   #include <boost/math/constants/constants.hpp>

   template <class Real>
   Real area(Real r)
   {
      using namespace boost::math::constants;
      return pi<Real>() * r * r;
   }

Although this syntax is a little less "cute" than the non-template version,
the code is no less efficient
(at least for the built-in types `float`, `double` and `long double`) :
the function template versions of the constants are simple inline functions that
return a constant of the correct precision for the type used. In addition, these
functions are declared `constexp` for those compilers that support this, allowing
the result to be used in constant-expressions provided the template argument is a literal type.

[tip Keep in mind the difference between the variable version,
just `pi`, and the template-function version:
the template-function requires both a <[~floating-point-type]>
and function call `()` brackets, for example: `pi<double>()`.
You cannot write `double p = pi<>()`, nor `double p = pi()`.]

[note You can always use [*both] variable and template-function versions
[*provided calls are fully qualified], for example:
``
double my_pi1 = boost::math::constants::pi<double>();
double my_pi2 = boost::math::double_constants::pi;
``
]

[warning It may be tempting to simply define
``
using namespace boost::math::double_constants;
using namespace boost::math::constants;
``
but if you do define two namespaces, this will, of course, create ambiguity!
``
double my_pi = pi(); // error C2872: 'pi' : ambiguous symbol
double my_pi2 = pi; // Context does not allow for disambiguation of overloaded function
``
Although the  mistake above is fairly obvious,
it is also not too difficult to do this accidentally, or worse, create it in someone elses code.

Therefore is it prudent to avoid this risk by [*localising the scope of such definitions], as shown above.]

[tip Be very careful with the type provided as parameter.
For example, providing an [*integer] instead of a floating-point type can be disastrous (a C++ feature).

``cout << "Area = " << area(2) << endl; // Area = 12!!!``

You should get a compiler warning
[pre
warning : 'return' : conversion from 'double' to 'int', possible loss of data
] [/pre]
Failure to heed this warning can lead to very wrong answers!

You can also avoid this by being explicit about the type of `Area`.
``cout << "Area = " << area<double>(2) << endl; // Area = 12.566371``
]

[endsect] [/section:templ Use in template code]

[section:user_def Use With User-Defined Types]

The most common example of a high-precision user-defined type will probably be __multiprecision.

The syntax for using the function-call constants with user-defined types is the same
as it is in the template class, which is to say we use:

   #include <boost/math/constants/constants.hpp>

   boost::math::constants::pi<UserDefinedType>();

For example:

  boost::math::constants::pi<boost::multiprecision::cpp_dec_float_50>();

giving [pi] with a precision of 50 decimal digits.

However, since the precision of the user-defined type may be much greater than that
of the built-in floating point types, how the value returned is created is as follows:

* If the precision of the type is known at compile time:
   * If the precision is less than or equal to that of a `float` and the type is constructable from a `float`
     then our code returns a `float` literal.  If the user-defined type is a literal type
     then the function call that returns the constant will be a `constexp`.
   * If the precision is less than or equal to that of a `double` and the type is constructable from a `double`
     then our code returns a `double` literal.  If the user-defined type is a literal type
     then the function call that returns the constant will be a `constexp`.
   * If the precision is less than or equal to that of a `long double` and the type is constructable from a `long double`
     then our code returns a `long double` literal.  If the user-defined type is a literal type
     then the function call that returns the constant will be a `constexp`.
   * If the precision is less than or equal to that of a `__float128` (and the compiler supports such a type)
     and the type is constructable from a `__float128`
     then our code returns a `__float128` literal.  If the user-defined type is a literal type
     then the function call that returns the constant will be a `constexp`.
   * If the precision is less than 100 decimal digits, then the constant will be constructed
      (just the once, then cached in a thread-safe manner) from a string representation of the constant.
      In this case the value is returned as a const reference to the cached value.
   * Otherwise the value is computed (just once, then cached in a thread-safe manner).
     In this case the value is returned as a const reference to the cached value.
* If the precision is unknown at compile time then:
   * If the runtime precision (obtained from a call to `boost::math::tools::digits<T>()`) is
     less than 100 decimal digits, then the constant is constructed "on the fly" from the string
     representation of the constant.
   * Otherwise the value is constructed "on the fly" by calculating then value of the constant
     using the current default precision of the type.  Note that this can make use of the constants
     rather expensive.

In addition, it is possible to pass a `Policy` type as a second template argument, and use this to control
the precision:

   #include <boost/math/constants/constants.hpp>

   typedef boost::math::policies::policy<boost::math::policies::digits2<80> > my_policy_type;
   boost::math::constants::pi<MyType, my_policy_type>();

[note Boost.Math doesn't know how to control the internal precision of `MyType`, the policy
just controls how the selection process above is carried out, and the calculation precision
if the result is computed.]

It is also possible to control which method is used to construct the constant by specialising
the traits class `construction_traits`:

   namespace boost{ namespace math{ namespace constant{

   template <class T, class Policy>
   struct construction_traits
   {
      typedef std::integral_constant<int, N> type;
   };

   }}} // namespaces

Where ['N] takes one of the following values:

[table
[[['N]][Meaning]]
[[0][The precision is unavailable at compile time;
either construct from a decimal digit string or calculate on the fly depending upon the runtime precision.]]
[[1][Return a float precision constant.]]
[[2][Return a double precision constant.]]
[[3][Return a long double precision constant.]]
[[4][Construct the result from the string representation, and cache the result.]]
[[Any other value ['N]][Sets the compile time precision to ['N] bits.]]
]

[h5 Custom Specializing a constant]

In addition, for user-defined types that need special handling, it's possible to partially-specialize
the internal structure used by each constant.  For example, suppose we're using the C++ wrapper around MPFR
`mpfr_class`: this has its own representation of Pi which we may well wish to use in place of the above
mechanism.  We can achieve this by specialising the class template `boost::math::constants::detail::constant_pi`:

   namespace boost{ namespace math{ namespace constants{ namespace detail{

   template<>
   struct constant_pi<mpfr_class>
   {
      template<int N>
      static mpfr_class get(const std::integral_constant<int, N>&)
      {
         // The template param N is one of the values in the table above,
         // we can either handle all cases in one as is the case here,
         // or overload "get" for the different options.
         mpfr_class result;
         mpfr_const_pi(result.get_mpfr_t(), GMP_RNDN);
         return result;
      }
   };

   }}}} // namespaces

[h5 Diagnosing what meta-programmed code is doing]

Finally, since it can be tricky to diagnose what meta-programmed code is doing, there is a
diagnostic routine that prints information about how this library will handle a specific type,
it can be used like this:

   #include <boost/math/constants/info.hpp>

   int main()
   {
      boost::math::constants::print_info_on_type<MyType>();
   }

If you wish, you can also pass an optional std::ostream argument to the `print_info_on_type` function.
Typical output for a user-defined type looks like this:

[pre
Information on the Implementation and Handling of
Mathematical Constants for Type class boost::math::concepts::real_concept

Checking for std::numeric_limits<class boost::math::concepts::real_concept> specialisation: no
boost::math::policies::precision<class boost::math::concepts::real_concept, Policy>
reports that there is no compile type precision available.
boost::math::tools::digits<class boost::math::concepts::real_concept>()
reports that the current runtime precision is
53 binary digits.
No compile time precision is available, the construction method
will be decided at runtime and results will not be cached
- this may lead to poor runtime performance.
Current runtime precision indicates that
the constant will be constructed from a string on each call.
]

[endsect] [/section:user_def Use With User Defined Types]

[endsect] [/section:tutorial Tutorial]

[section:constants The Mathematical Constants]

This section lists the mathematical constants, their use(s) (and sometimes rationale for their inclusion).
[table Mathematical Constants
[[name] [formula] [Value (6 decimals)] [Uses and Rationale]]
[[[*Rational fractions]] [] [] [] ]
[[half] [1/2] [0.5] [] ]
[[third] [1/3] [0.333333] [] ]
[[two_thirds] [2/3] [0.66667] [] ]
[[three_quarters] [3/4] [0.75] [] ]

[[[*two and related]] [] [] [] ]
[[root_two] [[radic]2] [1.41421] [Equivalent to POSIX constant M_SQRT2] ]
[[root_three] [[radic]3] [1.73205] [] ]
[[half_root_two] [[radic]2 /2] [0.707106] [] ]
[[ln_two] [ln(2)] [0.693147] [Equivalent to POSIX constant M_LN2] ]
[[ln_ten] [ln(10)] [2.30258] [Equivalent to POSIX constant M_LN10] ]
[[ln_ln_two] [ln(ln(2))] [-0.366512] [Gumbel distribution median] ]
[[root_ln_four] [[radic]ln(4)] [1.177410] [] ]
[[one_div_root_two] [1/[radic]2] [0.707106] [Equivalent to POSIX constant M_SQRT1_2] ]

[[[*[pi] and related]] [] [] [] ]
[[pi] [[pi]] [3.14159] [Ubiquitous. Archimedes constant [@http://en.wikipedia.org/wiki/Pi [pi]].  Equivalent to POSIX constant M_PI]]
[[half_pi] [[pi]/2] [1.570796] [Equivalent to POSIX constant M_PI2] ]
[[third_pi] [[pi]/3] [1.04719] [] ]
[[quarter_pi] [[pi]/4] [0.78539816] [Equivalent to POSIX constant M_PI_4] ]
[[sixth_pi] [[pi]/6] [0.523598] [] ]
[[two_pi] [2[pi]] [6.28318] [Many uses, most simply, circumference of a circle]]
[[tau] [[tau]] [6.28318] [@https://en.wikipedia.org/wiki/Turn_(angle)#Tau_proposals Many uses, most simply, circumference of a circle. Equal to two_pi.]]
[[two_thirds_pi] [2/3 [pi]] [2.09439] [[@http://en.wikipedia.org/wiki/Sphere#Volume_of_a_sphere volume of a hemi-sphere] = 4/3 [pi] r[cubed]]]
[[three_quarters_pi] [3/4 [pi]] [2.35619] [ = 3/4 [pi] ]]
[[four_thirds_pi] [4/3 [pi]] [4.18879] [[@http://en.wikipedia.org/wiki/Sphere#Volume_of_a_sphere volume of a sphere] = 4/3 [pi] r[cubed]]]
[[one_div_two_pi] [1/(2[pi])] [1.59155] [Widely used]]
[[root_pi] [[radic][pi]][1.77245] [Widely used]]
[[root_half_pi] [[radic] [pi]/2] [1.25331] [Widely used]]
[[root_two_pi][[radic] [pi]*2] [2.50662] [Widely used]]
[[one_div_pi] [1/[pi]] [0.31830988] [Equivalent to POSIX constant M_1_PI] ]
[[two_div_pi] [2/[pi]] [0.63661977] [Equivalent to POSIX constant M_2_PI] ]
[[one_div_root_pi] [1/[radic][pi]] [0.564189] [] ]
[[two_div_root_pi] [2/[radic][pi]] [1.128379] [Equivalent to POSIX constant M_2_SQRTPI] ]
[[one_div_root_two_pi] [1/[radic](2[pi])] [0.398942] [] ]
[[root_one_div_pi] [[radic](1/[pi]] [0.564189] [] ]
[[pi_minus_three] [[pi]-3] [0.141593] [] ]
[[four_minus_pi] [4 -[pi]] [0.858407] [] ]
[[pi_pow_e] [[pi][super e]] [22.4591] [] ]

[[pi_sqr] [[pi][super 2]] [9.86960] [] ]
[[pi_sqr_div_six] [[pi][super 2]/6] [1.64493] [] ]
[[pi_cubed] [[pi][super 3]] [31.00627] [] ]
[[cbrt_pi] [[radic][super 3] [pi]] [1.46459] [] ]
[[one_div_cbrt_pi] [1/[radic][super 3] [pi]] [0.682784] [] ]

[[[*Euler's e and related]] [] [] [] ]
[[e] [e] [2.71828] [[@http://en.wikipedia.org/wiki/E_(mathematical_constant) Euler's constant e], equivalent to POSIX constant M_E] ]
[[exp_minus_half] [e [super -1/2]] [0.606530] [] ]
[[e_pow_pi] [e [super [pi]]] [23.14069] [] ]
[[root_e] [[radic] e] [1.64872] [] ]
[[log10_e] [log10(e)] [0.434294] [Equivalent to POSIX constant M_LOG10E] ]
[[one_div_log10_e] [1/log10(e)] [2.30258] [] ]
[[log2_e] [log[sub 2](e)] [1.442695] [This is the same as 1/ln(2) and is equivalent to POSIX constant M_LOG2E] ]

[[[*Trigonometric]] [] [] [] ]
[[degree] [radians = [pi] / 180] [0.017453] [] ]
[[radian] [degrees = 180 / [pi]] [57.2957] [] ]
[[sin_one] [sin(1)] [0.841470] [] ]
[[cos_one] [cos(1)] [0.54030] [] ]
[[sinh_one] [sinh(1)] [1.17520] [] ]
[[cosh_one] [cosh(1)] [1.54308] [] ]

[[[*Phi]] [ Phidias golden ratio] [[@http://en.wikipedia.org/wiki/Golden_ratio Phidias golden ratio]] [] ]
[[phi] [(1 + [radic]5) /2] [1.61803] [finance] ]
[[ln_phi] [ln([phi])] [0.48121] [] ]
[[one_div_ln_phi] [1/ln([phi])] [2.07808] [] ]

[[[*Euler's Gamma]] [] [] [] ]
[[euler] [euler] [0.577215] [[@http://en.wikipedia.org/wiki/Euler%E2%80%93Mascheroni_constant Euler-Mascheroni gamma constant]] ]
[[one_div_euler] [1/euler] [1.73245] [] ]
[[euler_sqr] [euler[super 2]] [0.333177] [] ]

[[[*Misc]] [] [] [] ]
[[zeta_two] [[zeta](2)] [1.64493] [[@http://en.wikipedia.org/wiki/Riemann_zeta_function Riemann zeta function]] ]
[[zeta_three] [[zeta](3)] [1.20205] [[@http://en.wikipedia.org/wiki/Riemann_zeta_function Riemann zeta function]] ]
[[catalan] [['K]] [0.915965] [[@http://mathworld.wolfram.com/CatalansConstant.html Catalan (or Glaisher) combinatorial constant] ]]
[[glaisher] [['A]] [1.28242] [[@https://oeis.org/A074962/constant Decimal expansion of Glaisher-Kinkelin constant] ]]
[[khinchin] [['k]] [2.685452] [[@https://oeis.org/A002210/constant Decimal expansion of Khinchin constant] ]]

[[extreme_value_skewness] [12[radic]6 [zeta](3)/ [pi][super 3]] [1.139547] [Extreme value distribution] ]
[[rayleigh_skewness] [2[radic][pi]([pi]-3)/(4 - [pi])[super 3/2]] [0.631110] [Rayleigh distribution skewness] ]
[[rayleigh_kurtosis_excess] [-(6[pi][super 2]-24[pi]+16)/(4-[pi])[super 2]] [0.245089] [[@http://en.wikipedia.org/wiki/Rayleigh_distribution Rayleigh distribution kurtosis excess]] ]
[[rayleigh_kurtosis] [3+(6[pi][super 2]-24[pi]+16)/(4-[pi])[super 2]] [3.245089] [Rayleigh distribution kurtosis] ]
[[first_feigenbaum] [] [4.6692016] [[@https://en.wikipedia.org/wiki/Feigenbaum_constants First Feigenbaum constant]] ]
[[plastic] [Real solution of x[super 3] = x + 1] [1.324717957] [[@https://en.wikipedia.org/wiki/Plastic_number Plastic constant]] ]
[[gauss] [Reciprocal of agm(1, [radic]2)] [0.8346268] [[@https://en.wikipedia.org/wiki/Gauss%27s_constant Gauss's constant]] ]
[[dottie] [Solution of cos(x) = x] [0.739085] [[@https://en.wikipedia.org/wiki/Dottie_number Dottie's number]] ]
[[reciprocal_fibonacci] [Sum of reciprocals of Fibonacci numbers] [3.359885666] [[@https://en.wikipedia.org/wiki/Reciprocal_Fibonacci_constant Reciprocal Fibonacci constant]] ]
[[laplace_limit] [] [.6627434193] [[@https://en.wikipedia.org/wiki/Laplace_limit Laplace Limit]] ]
] [/table]


[note Integer values are [*not included] in this list of math constants, however interesting,
because they can be so easily and exactly constructed, even for UDT, for example: `static_cast<cpp_float>(42)`.]

[tip If you know the approximate value of the constant, you can search for the value to find Boost.Math chosen name in this table.]
[tip Bernoulli numbers are available at __bernoulli_numbers.]
[tip Factorials are available at __factorial.]

[endsect]  [/section:constants The constants]

[section:new_const Defining New Constants]

The library provides some helper code to assist in defining new constants;
the process for defining a constant called `my_constant` goes like this:

1. [*Define a function that calculates the value of the constant].
This should be a template function, and be placed in `boost/math/constants/calculate_constants.hpp`
if the constant is to be added to this library,
or else defined at the top of your source file if not.

The function should look like this:

  namespace boost{ namespace math{ namespace constants{ namespace detail{

  template <class Real>
  template <int N>
  Real constant_my_constant<Real>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC((std::integral_constant<int, N>)))
  {
    int required_precision = N ? N : tools::digits<Real>();
    Real result = /* value computed to required_precision bits */ ;
    return result;
  }

  }}}} // namespaces

Then define a placeholder for the constant itself:

   namespace boost{ namespace math{ namespace constants{

   BOOST_DEFINE_MATH_CONSTANT(my_constant, 0.0, "0");

   }}}


For example, to calculate [pi]/2, add to `boost/math/constants/calculate_constants.hpp`

  template <class T>
  template<int N>
  inline T constant_half_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC((std::integral_constant<int, N>)))
  {
     BOOST_MATH_STD_USING
     return pi<T, policies::policy<policies::digits2<N> > >() / static_cast<T>(2);
  }

Then to `boost/math/constants/constants.hpp` add:

   BOOST_DEFINE_MATH_CONSTANT(half_pi, 0.0, "0");  // Actual values are temporary, we'll replace them later.

[note Previously defined constants like pi and e can be used, but by *not simply calling* `pi<T>()`;
specifying the precision via the policy
`pi<T, policies::policy<policies::digits2<N> > >()`
is essential to ensure full accuracy.]

[warning Newly defined constants can only be used once they are included in
`boost/math/constants/constants.hpp`. So if you add
`template <class T, class N> T constant_my_constant{...}`,
then you cannot define `constant_my_constant`
until you add the temporary `BOOST_DEFINE_MATH_CONSTANT(my_constant, 0.0, "0")`.
Failing to do this will result in surprising compile errors:
``
  error C2143: syntax error : missing ';' before '<'
  error C2433: 'constant_root_two_div_pi' : 'inline' not permitted on data declarations
  error C2888: 'T constant_root_two_div_pi' : symbol cannot be defined within namespace 'detail'
  error C2988: unrecognizable template declaration/definition
``
]

2. [*You will need an arbitrary precision type to use to calculate the value].
Boost.Multiprecision has a large number of arbitrary precision floating point types to aid with this.

3. It is necessary to link to the Boost.Regex library,
and probably to your chosen arbitrary precision type library.

4. You need to add `libs\math\include_private` to your compiler's include path as the needed
header is not installed in the usual places by default (this avoids a cyclic dependency between
the Math and Multiprecision library's headers).

5. The complete program to generate the constant `half_pi` using function `calculate_half_pi` is then:

   #define USE_CPP_FLOAT // If required.
   #include <boost/math/constants/generate.hpp>

   int main()
   {
      BOOST_CONSTANTS_GENERATE(half_pi);
   }

The output from the program is a snippet of C++ code
(actually a macro call) that can be cut and pasted
into `boost/math/constants/constants.hpp` or else into your own code, for example:

[pre
  BOOST_DEFINE_MATH_CONSTANT(half_pi, 1.570796326794896619231321691639751442e+00, "1.57079632679489661923132169163975144209858469968755291048747229615390820314310449931401741267105853399107404326e+00");
]

This macro BOOST_DEFINE_MATH_CONSTANT  inserts a C++ struct code snippet that
declares the `float`, `double` and `long double` versions of the constant,
plus a decimal digit string representation correct to 100 decimal
digits, and all the meta-programming machinery needed to select between them.

The result of an expanded macro for Pi is shown below.

[import ./pp_pi.hpp]

[preprocessed_pi]


[endsect] [/section:new_const Defining New Constants]

[section:constants_faq Math Constants FAQs]

[h4 Why are ['these] Constants Chosen?]
It is, of course, impossible to please everyone with a list like this.

Some of the criteria we have used are:

* Used in Boost.Math.
* Commonly used.
* Expensive to compute.
* Requested by users.
* [@http://en.wikipedia.org/wiki/Mathematical_constant Used in science and mathematics.]
* No integer values (because so cheap to construct).

(You can easily define your own if found convenient, for example: `FPT one =static_cast<FPT>(42);`).

[h4 How are constants named?]
* Not macros, so no upper case.
* All lower case (following C++ standard names).
* No CamelCase.
* Underscore as _ delimiter between words.
* Numbers spelt as words rather than decimal digits (except following pow).
* Abbreviation conventions:
  * root for square root.
  * cbrt for cube root.
  * pow for pow function using decimal digits like pow23 for n[super 2/3].
  * div for divided by or operator /.
  * minus for operator -, plus for operator +.
  * sqr for squared.
  * cubed for cubed n[super 3].
  * words for greek, like [pi], [zeta] and [Gamma].
  * words like half, third, three_quarters, sixth for fractions.  (Digit(s) can get muddled).
  * log10 for log[sub 10]
  * ln for log[sub e]

[h4 How are the constants derived?]

The constants have all been calculated using high-precision software working
with up to 300-bit precision giving about 100 decimal digits.
(The precision can be arbitrarily chosen and is limited only by compute time).

[h4 How Accurate are the constants?]
The minimum accuracy chosen (100 decimal digits) exceeds the
accuracy of reasonably-foreseeable floating-point hardware (256-bit)
and should meet most high-precision computations.

[h4 How are the constants tested?]

# Comparison using Boost.Test BOOST_CHECK_CLOSE_FRACTION using long double literals,
with at least 35 decimal digits, enough to be accurate for all long double implementations.
The tolerance is usually twice `long double epsilon`.

# Comparison with calculation at long double precision.
This often requires a slightly higher tolerance than two epsilon
because of computational noise from round-off etc,
especially when trig and other functions are called.

# Comparison with independent published values,
for example, using [@http://oeis.org/ The On-Line Encyclopedia of Integer Sequences (OEIS)]
again using at least 35 decimal digits strings.

# Comparison with independently calculated values using arbitrary precision tools like
[@http://www.wolfram.com/mathematica/ Mathematica], again using at least 35 decimal digits literal strings.

[warning We have not yet been able to [*check] that
[*all] constants are accurate at the full arbitrary precision,
at present 100 decimal digits.
But certain key values like `e` and `pi` appear to be accurate
and internal consistencies suggest that others are this accurate too.
]

[h4 Why is Portability important?]

Code written using math constants is easily portable even when using different
floating-point types with differing precision.

It is a mistake to expect that results of computations will be [*identical], but
you can achieve the [*best accuracy possible for the floating-point type in use].

This has no extra cost to the user, but reduces irritating,
and often confusing and very hard-to-trace effects,
caused by the intrinsically limited precision of floating-point calculations.

A harmless symptom of this limit is a spurious least-significant digit;
at worst, slightly inaccurate constants sometimes cause iterating algorithms
to diverge wildly because internal comparisons just fail.

[h4 What is the Internal Format of the constants, and why?]

See [link math_toolkit.tutorial tutorial] above for normal use,
but this FAQ explains the internal details used for the constants.

Constants are stored as 100 decimal digit values.
However, some compilers do not accept decimal digits strings as long as this.
So the constant is split into two parts, with the first containing at least
128-bit long double precision (35 decimal digits),
and for consistency should be in scientific format with a signed exponent.

The second part is the value of the constant expressed as a string literal,
accurate to at least 100 decimal digits (in practice that means at least 102 digits).
Again for consistency use scientific format with a signed exponent.

For types with precision greater than a long double,
then if T is constructible `T `is constructible from a `const char*`
then it's directly constructed from the string,
otherwise we fall back on lexical_cast to convert to type `T`.
(Using a string is necessary because you can't use a numeric constant
since even a `long double` might not have enough digits).

So, for example, a constant like pi is internally defined as

  BOOST_DEFINE_MATH_CONSTANT(pi, 3.141592653589793238462643383279502884e+00, "3.14159265358979323846264338327950288419716939937510582097494459230781640628620899862803482534211706798214808651e+00");

In this case the significand is 109 decimal digits, ensuring 100 decimal digits are exact, and exponent is zero.

See [link math_toolkit.new_const defining new constants] to calculate new constants.

A macro definition like this can be pasted into user code where convenient,
or into `boost/math/constants.hpp` if it is to be added to the Boost.Math library.

[h4 What Floating-point Types could I use?]

Apart from the built-in floating-point types `float`, `double`, `long double`,
there are several arbitrary precision floating-point classes available,
but most are not licensed for commercial use.

[h5  Boost.Multiprecision by Christopher Kormanyos and John Maddock]

This is a well-maintained and fully featured library which provides numerous high-precision types, as well as wrappers to other popular libraries like MPFR.

[h5 NTL class quad_float]

[@http://shoup.net/ntl/ NTL] by Victor Shoup has fixed and arbitrary high precision fixed and floating-point types.
However none of these are licenced for commercial use.

  #include <NTL/quad_float.h> // quad precision 106-bit, about 32 decimal digits.
  using NTL::to_quad_float; // Less precise than arbitrary precision NTL::RR.

NTL class `quad_float`, which gives a form of quadruple precision,
106-bit significand (but without an extended exponent range.)
With an IEC559/IEEE 754 compatible processor,
for example Intel X86 family, with 64-bit double, and 53-bit significand,
using the significands of [*two] 64-bit doubles,
if `std::numeric_limits<double>::digits10` is 16,
then we get about twice the precision,
so `std::numeric_limits<quad_float>::digits10()` should be 32.
(the default `std::numeric_limits<RR>::digits10()` should be about 40).
(which seems to agree with experiments).
We output constants (including some noisy bits,
an approximation to `std::numeric_limits<RR>::max_digits10()`)
by adding 2 or 3 extra decimal digits, so using `quad_float::SetOutputPrecision(32 + 3);`

Apple Mac/Darwin uses a similar ['doubledouble] 106-bit for its built-in `long double` type.

[note The precision of all `doubledouble` floating-point types is rather odd and values given are only approximate.]

[*New projects should use __multiprecision.]

[h5 NTL class RR]

Arbitrary precision floating point with NTL class RR,
default is 150 bit (about 50 decimal digits)
used here with 300 bit to output 100 decimal digits,
enough for many practical non-'number-theoretic' C++ applications.

__NTL is [*not licenced for commercial use].

This class is used in Boost.Math and is an option when using big_number projects to calculate new math constants.

[*New projects should use __multiprecision.]

[h5 GMP and MPFR]

[@http://gmplib.org GMP] and [@http://www.mpfr.org/ MPFR] have also been used to compute constants,
but are licensed under the [@http://www.gnu.org/copyleft/lesser.html Lesser GPL license]
and are [*not licensed for commercial use].

[h4 What happened to a previous collection of constants proposed for Boost?]

A review concluded that the way in which the constants were presented did not meet many peoples needs.
None of the methods proposed met many users' essential requirement to allow writing simply `pi` rather than `pi()`.
Many science and engineering equations look difficult to read because function call brackets can be confused
with the many other brackets often needed.  All the methods then proposed of avoiding the brackets failed to meet all needs,
often on grounds of complexity and lack of applicability to various realistic scenarios.

So the simple namespace method, proposed on its own, but rejected at the first review,
has been added to allow users to have convenient access to `float`, `double` and `long double` values,
but combined with template struct and functions to allow simultaneous use
with other non-built-in floating-point types.


[h4 Why do the constants (internally) have a struct rather than a simple function?]

A function mechanism was provided by in previous versions of Boost.Math.

The new mechanism is to permit partial specialization. See Custom Specializing a constant above.
It should also allow use with other packages like [@http://www.ttmath.org/ ttmath Bignum C++ library.]

[h4 Where can I find other high precision constants?]

# Constants with very high precision and good accuracy (>40 decimal digits)
from Simon Plouffe's web based collection [@http://pi.lacim.uqam.ca/eng/].
# [@https://oeis.org/ The On-Line Encyclopedia of Integer Sequences (OEIS)]
# Checks using  printed text optically scanned values and converted from:
D. E. Knuth, Art of Computer Programming, Appendix A, Table 1, Vol 1, ISBN 0 201 89683 4 (1997)
# M. Abrahamovitz & I. E. Stegun, National Bureau of Standards, Handbook of Mathematical Functions,
a reference source for formulae now superseded by
# Frank W. Olver, Daniel W. Lozier, Ronald F. Boisvert, Charles W. Clark, NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0-521-14063-8, 2010.
# John F Hart, Computer Approximations, Kreiger (1978) ISBN 0 88275 642 7.
# Some values from Cephes Mathematical Library, Stephen L. Moshier
and CALC100 100 decimal digit Complex Variable Calculator Program, a DOS utility.
# Xavier Gourdon, Pascal Sebah, 50 decimal digits constants at [@http://numbers.computation.free.fr/Constants/constants.html Number, constants and computation].

[h4 Where are Physical Constants?]

Not here, because physical constants:

* Are measurements, not truly constants.
* Are not truly constant and keeping changing as metrology improves.
* Have intrinsic uncertainty.
* Mathematical constants are stored and represented at varying precision, but should never be inaccurate.

Some physical constants may be available in Boost.Units.

[endsect] [/section:FAQ FAQ]

[endmathpart] [/section:constants Mathematical Constants]

[/
  Copyright 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).
]


