There are two kinds of constexpr support in this library:

Declaring numeric literals

There are two backend types which are literals:

For example:

using namespace boost::multiprecision;

constexpr float128            f = 0.1Q   // OK, float128's are always literals in C++11

constexpr int128_t            i = 0;     // OK, fixed precision int128_t has no allocator.
constexpr uint1024_t          j = 0xFFFFFFFF00000000uLL;  // OK, fixed precision uint1024_t has no allocator.

constexpr checked_uint128_t   k = 1; // OK from C++14 and later, not supported for C++11.
constexpr checked_uint128_t   k = -1; // Error, as this would normally lead to a runtime failure (exception).
constexpr cpp_int             l = 2;  // Error, type is not a literal as it performs memory management.

There is also support for user defined-literals with cpp_int - these are limited to unchecked, fixed precision cpp_int's which are specified in hexadecimal notation. The suffixes supported are:

In each case, use of these suffixes with hexadecimal values produces a constexpr result.

Examples:

// Any use of user defined literals requires that we import the literal-operators into current scope first:
using namespace boost::multiprecision::literals;
//
// To keep things simple in the example, we'll make our types used visible to this scope as well:
using namespace boost::multiprecision;
//
// The value zero as a number<cpp_int_backend<4,4,signed_magnitude,unchecked,void> >:
constexpr auto a = 0x0_cppi;
// The type of each constant has 4 bits per hexadecimal digit,
// so this is of type uint256_t (ie number<cpp_int_backend<256,256,unsigned_magnitude,unchecked,void> >):
constexpr auto b = 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF_cppui;
//
// Smaller values can be assigned to larger values:
int256_t c = 0x1234_cppi; // OK
//
// However, this only works in constexpr contexts from C++14 onwards:
constexpr int256_t d = 0x1_cppi; // Compiler error in C++11, requires C++14
//
// Constants can be padded out with leading zeros to generate wider types:
constexpr uint256_t e = 0x0000000000000000000000000000000000000000000FFFFFFFFFFFFFFFFFFFFF_cppui; // OK
//
// However, specific-width types are best produced with specific-width suffixes,
// ones supported by default are `_cpp[u]i128`, `_cpp[u]i256`, `_cpp[u]i512`, `_cpp[u]i1024`.
//
constexpr int128_t f = 0x1234_cppi128; // OK, always produces an int128_t as the result.
constexpr uint1024_t g = 0xaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaabbbbbbbbbbbbbbbbbbbbbbbbbbccccccccccccccccccccc_cppui1024; // OK,
//  always produces an uint1024_t as the result.
//
// If other specific-width types are required, then there is a macro for generating the operators for these.
// The macro can be used at namespace scope only:
//
BOOST_MP_DEFINE_SIZED_CPP_INT_LITERAL(2048);
//
// Now we can create 2048-bit literals as well:
constexpr auto h = 0xff_cppi2048; // h is of type number<cpp_int_backend<2048,2048,signed_magnitude,unchecked,void> >
//
// Finally, negative values are handled via the unary minus operator:
//
constexpr int1024_t i = -0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF_cppui1024;
//
// Which means this also works:
constexpr int1024_t j = -g;   // OK: unary minus operator is constexpr.

constexpr arithmetic

The front end of the library is all constexpr from C++14 and later. Currently there are only two backend types that are constexpr aware: __float128 and cpp_int. More backends will follow at a later date.

Provided the compiler is GCC, type float128 support constexpr operations on all arithmetic operations from C++14, comparisons, abs, fabs, fpclassify, isnan, isinf, isfinite and isnormal are also fully supported, but the transcendental functions are not.

The cpp_int types support constexpr arithmetic, provided it is a fixed precision type with no allocator. It may also be a checked integer: in which case a compiler error will be generated on overflow or undefined behaviour. In addition the free functions abs, swap, multiply, add, subtract, divide_qr, integer_modulus, powm, lsb, msb, bit_test, bit_set, bit_unset, bit_flip, sqrt, gcd, lcm are all supported. Use of cpp_int in this way requires either a C++2a compiler (one which supports std::is_constant_evaluated() - currently only gcc-9 or clang-9 or later), or GCC-6 or later in C++14 mode. Compilers other than GCC and without std::is_constant_evaluated() will support a very limited set of operations: expect to hit roadblocks rather easily.

See compiler support for std::is_constant_evaluated;

For example given:

#include <boost/math/constants/constants.hpp> // For constant pi with full precision of type T.
// using  boost::math::constants::pi;

template <class T>
inline constexpr T circumference(T radius)
{
   return 2 * boost::math::constants::pi<T>() * radius;
}

template <class T>
inline constexpr T area(T radius)
{
   return boost::math::constants::pi<T>() * radius * radius;
}

We can now calculate areas and circumferences, using all compile-time constexpr arithmetic:

using boost::multiprecision::float128;

constexpr float128 radius = 2.25;
constexpr float128 c      = circumference(radius);
constexpr float128 a      = area(radius);

std::cout << "Circumference = " << c << std::endl;
std::cout << "Area = " << a << std::endl;

Note that these make use of the numeric constants from the Boost.Math constants library, which also happen to be constexpr. These usually have the full precision of the floating-point type, here 128-bit, about 36 decimal digits.

Calculating Hermite Polynomial coefficients at compile time

For a more interesting example, in constexpr_float_arithmetic_examples.cpp we define a simple class for constexpr polynomial arithmetic:

template <class T, unsigned Order>
struct const_polynomial;

Given this, we can use recurrence relations to calculate the coefficients for various orthogonal polynomials - in the example we use the Hermite polynomials. Only the constructor does any work - it uses the recurrence relations to calculate the coefficient array:

template <class T, unsigned Order>
class hermite_polynomial
{
   const_polynomial<T, Order> m_data;

 public:
   constexpr hermite_polynomial() : m_data(hermite_polynomial<T, Order - 1>().data() * const_polynomial<T, 1>{0, 2} - hermite_polynomial<T, Order - 1>().data().derivative())
   {
   }
   constexpr const const_polynomial<T, Order>& data() const
   {
      return m_data;
   }
   constexpr const T& operator[](std::size_t N)const
   {
      return m_data[N];
   }
   template <class U>
   constexpr T operator()(U val)const
   {
      return m_data(val);
   }
};

Now we just need to define H₀ and H₁ as termination conditions for the recurrence:

template <class T>
class hermite_polynomial<T, 0>
{
   const_polynomial<T, 0> m_data;

 public:
   constexpr hermite_polynomial() : m_data{1} {}
   constexpr const const_polynomial<T, 0>& data() const
   {
      return m_data;
   }
   constexpr const T& operator[](std::size_t N) const
   {
      return m_data[N];
   }
   template <class U>
   constexpr T operator()(U val)
   {
      return m_data(val);
   }
};

template <class T>
class hermite_polynomial<T, 1>
{
   const_polynomial<T, 1> m_data;

 public:
   constexpr hermite_polynomial() : m_data{0, 2} {}
   constexpr const const_polynomial<T, 1>& data() const
   {
      return m_data;
   }
   constexpr const T& operator[](std::size_t N) const
   {
      return m_data[N];
   }
   template <class U>
   constexpr T operator()(U val)
   {
      return m_data(val);
   }
};

We can now declare H₉ as a constexpr object, access the coefficients, and evaluate at an abscissa value, all at compile-time using constexpr arithmetic:

constexpr hermite_polynomial<float128, 9> h9;
//
// Verify that the polynomial's coefficients match the known values:
//
static_assert(h9[0] == 0);
static_assert(h9[1] == 30240);
static_assert(h9[2] == 0);
static_assert(h9[3] == -80640);
static_assert(h9[4] == 0);
static_assert(h9[5] == 48384);
static_assert(h9[6] == 0);
static_assert(h9[7] == -9216);
static_assert(h9[8] == 0);
static_assert(h9[9] == 512);
//
// Define an abscissa value to evaluate at:
constexpr float128 abscissa(0.5);
//
// Evaluate H_9(0.5) using all constexpr arithmetic, and check that it has the expected result:
static_assert(h9(abscissa) == 6481);

See constexpr_float_arithmetic_examples.cpp for working code.

Also since the coefficients to the Hermite polynomials are integers, we can also generate the Hermite coefficients using (fixed precision) cpp_ints: see constexpr_test_cpp_int_6.cpp.

constexpr Factorials

We can also generate integer factorials in constexpr_test_cpp_int_5.cpp like so:

template <class T>
constexpr T factorial(const T& a)
{
   return a ? a * factorial(a - 1) : 1;
}

and validate the result:

constexpr uint1024_t f1 = factorial(uint1024_t(31)); // Factorial 31!
static_assert(f1 == 0x1956ad0aae33a4560c5cd2c000000_cppi); // Expected result as an Boost.Multiprecision integer literal. 

Random constexpr values

Another example in constexpr_test_cpp_int_7.cpp generates a fresh multiprecision random number each time the file is compiled. It includes an C++ template implementation of the KISS random number algorithm by George Marsaglia for cpp_int integers.

constexpr uint1024_t rand = nth_random_value<uint1024_t>(1000);
std::cout << std::hex << rand << std::endl;

See also the random number generation section.