The header <boost/multiprecision/fwd.hpp> contains forward declarations for class number plus all of the available backends in the this library:

namespace boost {
namespace multiprecision {

   enum expression_template_option
   {
      et_off = 0,
      et_on = 1
   };

   template <class Backend>
   struct expression_template_default { /*details*/ };

   template <class Backend, expression_template_option ExpressionTemplates = expression_template_default<Backend>::value>
   class number;

   template <class T>
   struct is_number;

   enum cpp_integer_type
   {
      signed_magnitude = 1,
      unsigned_magnitude = 0,
      signed_packed = 3,
      unsigned_packed = 2
   };

   enum cpp_int_check_type
   {
      checked = 1,
      unchecked = 0
   };

   enum mpfr_allocation_type
   {
      allocate_stack,
      allocate_dynamic
   };
   //
   // Logging functions for lagged_adaptor:
   //
   template <class Backend>
   void log_postfix_event(const Backend&, const char* /*event_description*/);
   template <class Backend, class T>
   void log_postfix_event(const Backend&, const T&, const char* /*event_description*/);
   template <class Backend>
   void log_prefix_event(const Backend&, const char* /*event_description*/);
   template <class Backend, class T>
   void log_prefix_event(const Backend&, const T&, const char* /*event_description*/);
   template <class Backend, class T, class U>
   void log_prefix_event(const Backend&, const T&, const U&, const char* /*event_description*/);
   template <class Backend, class T, class U, class V>
   void log_prefix_event(const Backend&, const T&, const U&, const V&, const char* /*event_description*/);

   namespace backends {

      template <class Backend>
      struct debug_adaptor;

      template <class Backend>
      struct logged_adaptor;

      template <class Backend>
      struct complex_adaptor;

      enum digit_base_type
      {
         digit_base_2 = 2,
         digit_base_10 = 10
      };

      template <unsigned Digits, digit_base_type DigitBase = digit_base_10, class Allocator = void, class Exponent = int, Exponent MinExponent = 0, Exponent MaxExponent = 0>
      class cpp_bin_float;

      template <unsigned Digits10, class ExponentType = std::int32_t, class Allocator = void>
      class cpp_dec_float;

      template <std::size_t MinBits = 0, std::size_t MaxBits = 0, boost::multiprecision::cpp_integer_type SignType = signed_magnitude, cpp_int_check_type Checked = unchecked, class Allocator = typename std::conditional<MinBits && (MinBits == MaxBits), void, std::allocator<limb_type> >::type>
      struct cpp_int_backend;

      struct float128_backend;

      struct gmp_int;
      struct gmp_rational;

      template <unsigned digits10>
      struct gmp_float;

      template <unsigned digits10>
      struct mpc_complex_backend;

      template <unsigned digits10>
      struct mpfi_float_backend;

      template <unsigned digits10, mpfr_allocation_type AllocationType = allocate_dynamic>
      struct mpfr_float_backend;

      template <>
      struct mpfr_float_backend<0, allocate_stack>;

      template <class Backend>
      struct rational_adaptor;

      struct tommath_int;
   }
   //
   // All the backends are imported into boost::multiprecision:
   //
   using boost::multiprecision::backends::complex_adaptor;
   using boost::multiprecision::backends::debug_adaptor;
   using boost::multiprecision::backends::logged_adaptor;
   using backends::cpp_bin_float;
   using backends::digit_base_10;
   using backends::digit_base_2;
   using boost::multiprecision::backends::cpp_dec_float;
   using boost::multiprecision::backends::cpp_int_backend;
   using boost::multiprecision::backends::float128_backend;
   using boost::multiprecision::backends::gmp_float;
   using boost::multiprecision::backends::gmp_int;
   using boost::multiprecision::backends::gmp_rational;
   using boost::multiprecision::backends::mpc_complex_backend;
   using boost::multiprecision::backends::mpfi_float_backend;
   using boost::multiprecision::backends::mpfr_float_backend;
   using boost::multiprecision::backends::rational_adaptor;
   using boost::multiprecision::backends::tommath_int;

   //
   // convenience typedefs:
   //

   using cpp_bin_float_50 = number<backends::cpp_bin_float<50> >;
   using cpp_bin_float_100 = number<backends::cpp_bin_float<100> >;

   using cpp_bin_float_single = number<backends::cpp_bin_float<24, backends::digit_base_2, void, std::int16_t, -126, 127>, et_off>;
   using cpp_bin_float_double = number<backends::cpp_bin_float<53, backends::digit_base_2, void, std::int16_t, -1022, 1023>, et_off>;
   using cpp_bin_float_double_extended = number<backends::cpp_bin_float<64, backends::digit_base_2, void, std::int16_t, -16382, 16383>, et_off>;
   using cpp_bin_float_quad = number<backends::cpp_bin_float<113, backends::digit_base_2, void, std::int16_t, -16382, 16383>, et_off>;
   using cpp_bin_float_oct = number<backends::cpp_bin_float<237, backends::digit_base_2, void, std::int32_t, -262142, 262143>, et_off>;

   template <unsigned Digits, backends::digit_base_type DigitBase = backends::digit_base_10, class Allocator = void, class Exponent = int, Exponent MinExponent = 0, Exponent MaxExponent = 0>
   using cpp_complex_backend = complex_adaptor<cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinExponent, MaxExponent> >;

   template <unsigned Digits, backends::digit_base_type DigitBase = digit_base_10, class Allocator = void, class Exponent = int, Exponent MinExponent = 0, Exponent MaxExponent = 0, expression_template_option ExpressionTemplates = et_off>
   using cpp_complex = number<complex_adaptor<cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinExponent, MaxExponent> >, ExpressionTemplates>;

   using cpp_complex_50 = cpp_complex<50>;
   using cpp_complex_100 = cpp_complex<100>;

   using cpp_complex_single = cpp_complex<24, backends::digit_base_2, void, std::int16_t, -126, 127>;
   using cpp_complex_double = cpp_complex<53, backends::digit_base_2, void, std::int16_t, -1022, 1023>;
   using cpp_complex_extended = cpp_complex<64, backends::digit_base_2, void, std::int16_t, -16382, 16383>;
   using cpp_complex_quad = cpp_complex<113, backends::digit_base_2, void, std::int16_t, -16382, 16383>;
   using cpp_complex_oct = cpp_complex<237, backends::digit_base_2, void, std::int32_t, -262142, 262143>;

   using cpp_dec_float_50 = number<cpp_dec_float<50> >;
   using cpp_dec_float_100 = number<cpp_dec_float<100> >;

   using cpp_int = number<cpp_int_backend<> >;
   using cpp_rational_backend = rational_adaptor<cpp_int_backend<> >;
   using cpp_rational = number<cpp_rational_backend>;

   // Fixed precision unsigned types:
   using uint128_t = number<cpp_int_backend<128, 128, unsigned_magnitude, unchecked, void> >;
   using uint256_t = number<cpp_int_backend<256, 256, unsigned_magnitude, unchecked, void> >;
   using uint512_t = number<cpp_int_backend<512, 512, unsigned_magnitude, unchecked, void> >;
   using uint1024_t = number<cpp_int_backend<1024, 1024, unsigned_magnitude, unchecked, void> >;

   // Fixed precision signed types:
   using int128_t = number<cpp_int_backend<128, 128, signed_magnitude, unchecked, void> >;
   using int256_t = number<cpp_int_backend<256, 256, signed_magnitude, unchecked, void> >;
   using int512_t = number<cpp_int_backend<512, 512, signed_magnitude, unchecked, void> >;
   using int1024_t = number<cpp_int_backend<1024, 1024, signed_magnitude, unchecked, void> >;

   // Over again, but with checking enabled this time:
   using checked_cpp_int = number<cpp_int_backend<0, 0, signed_magnitude, checked> >;
   using checked_cpp_rational_backend = rational_adaptor<cpp_int_backend<0, 0, signed_magnitude, checked> >;
   using checked_cpp_rational = number<checked_cpp_rational_backend>;
   // Fixed precision unsigned types:
   using checked_uint128_t = number<cpp_int_backend<128, 128, unsigned_magnitude, checked, void> >;
   using checked_uint256_t = number<cpp_int_backend<256, 256, unsigned_magnitude, checked, void> >;
   using checked_uint512_t = number<cpp_int_backend<512, 512, unsigned_magnitude, checked, void> >;
   using checked_uint1024_t = number<cpp_int_backend<1024, 1024, unsigned_magnitude, checked, void> >;

   // Fixed precision signed types:
   using checked_int128_t = number<cpp_int_backend<128, 128, signed_magnitude, checked, void> >;
   using checked_int256_t = number<cpp_int_backend<256, 256, signed_magnitude, checked, void> >;
   using checked_int512_t = number<cpp_int_backend<512, 512, signed_magnitude, checked, void> >;
   using checked_int1024_t = number<cpp_int_backend<1024, 1024, signed_magnitude, checked, void> >;

   template <class Number>
   using debug_adaptor_t = number<debug_adaptor<typename Number::backend_type>, Number::et>;
   template <class Number>
   using logged_adaptor_t = number<logged_adaptor<typename Number::backend_type>, Number::et>;

   using float128 = number<float128_backend, et_off>;
   using complex128 = number<complex_adaptor<float128_backend>, et_off>;

   using mpf_float_50 = number<gmp_float<50> >;
   using mpf_float_100 = number<gmp_float<100> >;
   using mpf_float_500 = number<gmp_float<500> >;
   using mpf_float_1000 = number<gmp_float<1000> >;
   using mpf_float = number<gmp_float<0> >;
   using mpz_int = number<gmp_int>;
   using mpq_rational = number<gmp_rational>;

   using mpc_complex_50 = number<mpc_complex_backend<50> >;
   using mpc_complex_100 = number<mpc_complex_backend<100> >;
   using mpc_complex_500 = number<mpc_complex_backend<500> >;
   using mpc_complex_1000 = number<mpc_complex_backend<1000> >;
   using mpc_complex = number<mpc_complex_backend<0> >;

   using mpfi_float_50 = number<mpfi_float_backend<50> >;
   using mpfi_float_100 = number<mpfi_float_backend<100> >;
   using mpfi_float_500 = number<mpfi_float_backend<500> >;
   using mpfi_float_1000 = number<mpfi_float_backend<1000> >;
   using mpfi_float = number<mpfi_float_backend<0> >;

   using mpfr_float_50 = number<mpfr_float_backend<50> >;
   using mpfr_float_100 = number<mpfr_float_backend<100> >;
   using mpfr_float_500 = number<mpfr_float_backend<500> >;
   using mpfr_float_1000 = number<mpfr_float_backend<1000> >;
   using mpfr_float = number<mpfr_float_backend<0> >;

   using static_mpfr_float_50 = number<mpfr_float_backend<50, allocate_stack> >;
   using static_mpfr_float_100 = number<mpfr_float_backend<100, allocate_stack> >;

   using tom_int = number<tommath_int>;
   using tommath_rational = rational_adaptor<tommath_int>;
   using tom_rational = number<tommath_rational>;

 } }