mpfi_float

#include <boost/multiprecision/mpfi.hpp>

namespace boost{ namespace multiprecision{

template <unsigned Digits10>
class mpfi_float_backend;

typedef number<mpfi_float_backend<50> >    mpfi_float_50;
typedef number<mpfi_float_backend<100> >   mpfifloat_100;
typedef number<mpfi_float_backend<500> >   mpfifloat_500;
typedef number<mpfi_float_backend<1000> >  mpfi_float_1000;
typedef number<mpfi_float_backend<0> >     mpfi_float;

}} // namespaces

The mpfi_float_backend type is used in conjunction with number: It acts as a thin wrapper around the MPFI mpfi_t to provide an real-number type that is a drop-in replacement for the native C++ floating-point types, but with much greater precision and implementing interval arithmetic.

Type mpfi_float_backend can be used at fixed precision by specifying a non-zero Digits10 template parameter, or at variable precision by setting the template argument to zero. The typedefs mpfi_float_50, mpfi_float_100, mpfi_float_500, mpfi_float_1000 provide arithmetic types at 50, 100, 500 and 1000 decimal digits precision respectively. The typedef mpfi_float provides a variable precision type whose precision can be controlled via the numbers member functions.

	Note
	This type only provides `numeric_limits` support when the precision is fixed at compile time.

As well as the usual conversions from arithmetic and string types, instances of number<mpfi_float_backend<N> > are copy constructible and assignable from:

The MPFI native type mpfi_t.
The number wrappers around MPFI or MPFR: number<mpfi_float_backend<M> > and number<mpfr_float<M> >.
There is a two argument constructor taking two number<mpfr_float<M> > arguments specifying the interval.

It's also possible to access the underlying mpfi_t via the data() member function of mpfi_float_backend.

Things you should know when using this type:

A default constructed mpfi_float_backend is set to zero (Note that this is not the default MPFI behavior).
No changes are made to GMP or MPFR global settings, so this type can coexist with existing MPFR or GMP code.
The code can equally use MPIR in place of GMP - indeed that is the preferred option on Win32.
This backend supports rvalue-references and is move-aware, making instantiations of number on this backend move aware.
Conversion from a string results in a std::runtime_error being thrown if the string can not be interpreted as a valid floating-point number.
Division by zero results in an infinity.

There are some additional non member functions for working on intervals:

template <unsigned Digits10, expression_template_option ExpressionTemplates>
number<mpfr_float_backend<Digits10>, ExpressionTemplates> lower(const number<mpfi_float_backend<Digits10>, ExpressionTemplates>& val);

Returns the lower end of the interval.

template <unsigned Digits10, expression_template_option ExpressionTemplates>
number<mpfr_float_backend<Digits10>, ExpressionTemplates> upper(const number<mpfi_float_backend<Digits10>, ExpressionTemplates>& val);

Returns the upper end of the interval.

template <unsigned Digits10, expression_template_option ExpressionTemplates>
number<mpfr_float_backend<Digits10>, ExpressionTemplates> median(const number<mpfi_float_backend<Digits10>, ExpressionTemplates>& val);

Returns the mid point of the interval.

template <unsigned Digits10, expression_template_option ExpressionTemplates>
number<mpfr_float_backend<Digits10>, ExpressionTemplates> width(const number<mpfi_float_backend<Digits10>, ExpressionTemplates>& val);

Returns the absolute width of the interval.

template <unsigned Digits10, expression_template_option ExpressionTemplates>
number<mpfi_float_backend<Digits10>, ExpressionTemplates> intersect(
  const number<mpfi_float_backend<Digits10>, ExpressionTemplates>& a,
  const number<mpfi_float_backend<Digits10>, ExpressionTemplates>&  b);

Returns the interval which is the intersection of the a and b. Returns an unspecified empty interval if there is no such intersection.

template <unsigned Digits10, expression_template_option ExpressionTemplates>
number<mpfi_float_backend<Digits10>, ExpressionTemplates> hull(
  const number<mpfi_float_backend<Digits10>, ExpressionTemplates>& a,
  const number<mpfi_float_backend<Digits10>, ExpressionTemplates>&  b);

Returns the interval which is the union of a and b.

template <unsigned Digits10, expression_template_option ExpressionTemplates>
bool overlap(const number<mpfi_float_backend<Digits10>, ExpressionTemplates>& a,
             const number<mpfi_float_backend<Digits10>, ExpressionTemplates>&  b);

Returns true only if the intervals a and b overlap.

template <unsigned Digits10, expression_template_option ExpressionTemplates1, expression_template_option ExpressionTemplates2>
bool in(const number<mpfr_float_backend<Digits10>, ExpressionTemplates1>& a,
        const number<mpfi_float_backend<Digits10>, ExpressionTemplates2>&  b);

Returns true only if point a is contained within the interval b.

template <unsigned Digits10, expression_template_option ExpressionTemplates>
bool zero_in(const number<mpfi_float_backend<Digits10>, ExpressionTemplates>& a);

Returns true only if the interval a contains the value zero.

template <unsigned Digits10, expression_template_option ExpressionTemplates>
bool subset(const number<mpfi_float_backend<Digits10>, ExpressionTemplates>& a,
            const number<mpfi_float_backend<Digits10>, ExpressionTemplates>& b);

Returns true only if a is a subset of b.

template <unsigned Digits10, expression_template_option ExpressionTemplates>
bool proper_subset(const number<mpfi_float_backend<Digits10>, ExpressionTemplates>& a,
                   const number<mpfi_float_backend<Digits10>, ExpressionTemplates>& b);

Returns true only if a is a proper subset of b.

template <unsigned Digits10, expression_template_option ExpressionTemplates>
bool empty(const number<mpfi_float_backend<Digits10>, ExpressionTemplates>& a);

Returns true only if a is an empty interval, equivalent to upper(a) < lower(a).

template <unsigned Digits10, expression_template_option ExpressionTemplates>
bool singleton(const number<mpfi_float_backend<Digits10>, ExpressionTemplates>& a);

Returns true if lower(a) == upper(a).

MPFI example:

#include <boost/multiprecision/mpfi.hpp>
#include <boost/math/special_functions/gamma.hpp>
#include <iostream>

int main()
{
   using namespace boost::multiprecision;

   // Operations at variable precision and no numeric_limits support:
   mpfi_float a = 2;
   mpfi_float::default_precision(1000);
   std::cout << mpfi_float::default_precision() << std::endl;
   std::cout << sqrt(a) << std::endl; // print root-2

   // Operations at fixed precision and full numeric_limits support:
   mpfi_float_100 b = 2;
   std::cout << std::numeric_limits<mpfi_float_100>::digits << std::endl;
   // We can use any C++ std lib function:
   std::cout << log(b) << std::endl; // print log(2)

   // Access the underlying data:
   mpfi_t r;
   mpfi_init(r);
   mpfi_set(r, b.backend().data());

   // Construct some explicit intervals and perform set operations:
   mpfi_float_50 i1(1, 2), i2(1.5, 2.5);
   std::cout << intersect(i1, i2) << std::endl;
   std::cout << hull(i1, i2) << std::endl;
   std::cout << overlap(i1, i2) << std::endl;
   std::cout << subset(i1, i2) << std::endl;
   mpfi_clear(r);
   return 0;
}