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### Backend Requirements

The requirements on the `Backend` template argument to `number` are split up into sections: compulsory and optional.

Compulsory requirements have no default implementation in the library, therefore if the feature they implement is to be supported at all, then they must be implemented by the backend.

Optional requirements have default implementations that are called if the backend doesn't provide its own. Typically the backend will implement these to improve performance.

In the following tables, type B is the `Backend` template argument to `number`, `b` and `b2` are a variables of type B, `pb` is a variable of type B*, `cb`, `cb2` and `cb3` are constant variables of type `const B`, `rb` is a variable of type `B&&`, `a` and `a2` are variables of Arithmetic type, `s` is a variable of type `const char*`, `ui` is a variable of type `unsigned`, `bb` is a variable of type `bool`, `pa` is a variable of type pointer-to-arithmetic-type, `exp` is a variable of type `B::exp_type`, `pexp` is a variable of type `B::exp_type*`, `i` is a variable of type `int`, `pi` pointer to a variable of type `int`, B2 is another type that meets these requirements, b2 is a variable of type B2, `ss` is variable of type `std::streamsize` and `ff` is a variable of type `std::ios_base::fmtflags`.

Table 1.8. Compulsory Requirements on the Backend type.

Expression | Return Type | Comments | Throws
--- | --- | --- | ---
`B::signed_types` | `std::tuple<type-list>` | A list of signed integral types that can be assigned to type B. The types shall be listed in order of size, smallest first, and shall terminate in the type that is `std::intmax_t`. |
`B::unsigned_types` | `std::tuple<type-list>` | A list of unsigned integral types that can be assigned to type B. The types shall be listed in order of size, smallest first, and shall terminate in the type that is `std::uintmax_t`. |
`B::float_types` | `std::tuple<type-list>` | A list of floating-point types that can be assigned to type B.The types shall be listed in order of size, smallest first, and shall terminate in type `long double`. |
`B::exponent_type` | A signed integral type. | The type of the exponent of type B. This type is required only for floating-point types. |
`B()` |  | Default constructor. |
`B(cb)` |  | Copy Constructor. |
`b = b` | `B&` | Assignment operator. |
`b = a` | `B&` | Assignment from an Arithmetic type. The type of `a` shall be listed in one of the type lists `B::signed_types`, `B::unsigned_types` or `B::float_types`. |
`b = s` | `B&` | Assignment from a string. | Throws a `std::runtime_error` if the string could not be interpreted as a valid number.
`b.swap(b)` | `void` | Swaps the contents of its arguments. | `noexcept`
`cb.str(ss, ff)` | `std::string` | Returns the string representation of `b` with `ss` digits and formatted according to the flags set in `ff`. If `ss` is zero, then returns as many digits as are required to reconstruct the original value. |
`b.negate()` | `void` | Negates `b`. |
`cb.compare(cb2)` | `int` | Compares `cb` and `cb2`, returns a value less than zero if `cb < cb2`, a value greater than zero if `cb > cb2` and zero if `cb == cb2`. | `noexcept`
`cb.compare(a)` | `int` | Compares `cb` and `a`, returns a value less than zero if `cb < a`, a value greater than zero if `cb > a` and zero if `cb == a`. The type of `a` shall be listed in one of the type lists `B::signed_types`, `B::unsigned_types` or `B::float_types`. |
`eval_add(b, cb)` | `void` | Adds `cb` to `b`. |
`eval_subtract(b, cb)` | `void` | Subtracts `cb` from `b`. |
`eval_multiply(b, cb)` | `void` | Multiplies `b` by `cb`. |
`eval_divide(b, cb)` | `void` | Divides `b` by `cb`. | `std::overflow_error` if cb has the value zero, and `std::numeric_limits<number<B> >::has_infinity == false`
`eval_modulus(b, cb)` | `void` | Computes `b %= cb`, only required when `B` is an integer type. | `std::overflow_error` if cb has the value zero.
`eval_bitwise_and(b, cb)` | `void` | Computes `b &= cb`, only required when `B` is an integer type. |
`eval_bitwise_or(b, cb)` | `void` | Computes `b |= cb`, only required when `B` is an integer type. |
`eval_bitwise_xor(b, cb)` | `void` | Computes `b ^= cb`, only required when `B` is an integer type. |
`eval_complement(b, cb)` | `void` | Computes the ones-complement of `cb` and stores the result in `b`, only required when `B` is an integer type. |
`eval_left_shift(b, ui)` | `void` | Computes `b <<= ui`, only required when `B` is an integer type. |
`eval_right_shift(b, ui)` | `void` | Computes `b >>= ui`, only required when `B` is an integer type. |
`eval_convert_to(pa, cb)` | `void` | Converts `cb` to the type of `*pa` and store the result in `*pa`. Type `B` shall support conversion to at least types `std::intmax_t`, `std::uintmax_t` and `long long`. Conversion to other arithmetic types can then be synthesised using other operations. Conversions to other types are entirely optional. |
`eval_frexp(b, cb, pexp)` | `void` | Stores values in `b` and `*pexp` such that the value of `cb` is b * 2*pexp, only required when `B` is a floating-point type. |
`eval_ldexp(b, cb, exp)` | `void` | Stores a value in `b` that is cb * 2exp, only required when `B` is a floating-point type. |
`eval_frexp(b, cb, pi)` | `void` | Stores values in `b` and `*pi` such that the value of `cb` is b * 2*pi, only required when `B` is a floating-point type. | `std::runtime_error` if the exponent of cb is too large to be stored in an `int`.
`eval_ldexp(b, cb, i)` | `void` | Stores a value in `b` that is cb * 2i, only required when `B` is a floating-point type. |
`eval_floor(b, cb)` | `void` | Stores the floor of `cb` in `b`, only required when `B` is a floating-point type. |
`eval_ceil(b, cb)` | `void` | Stores the ceiling of `cb` in `b`, only required when `B` is a floating-point type. |
`eval_sqrt(b, cb)` | `void` | Stores the square root of `cb` in `b`, only required when `B` is a floating-point type. |
`boost::multiprecision::number_category<B>::type` | `std::integral_constant<int, N>` | `N` is one of the values `number_kind_integer`, `number_kind_floating_point`, `number_kind_complex`, `number_kind_rational` or `number_kind_fixed_point`. Defaults to `number_kind_floating_point`. |
`eval_conj(b, cb)` | `void` | Sets `b` to the complex conjugate of `cb`. Required for complex types only - other types have a sensible default. |
`eval_proj(b, cb)` | `void` | Sets `b` to the Riemann projection of `cb`. Required for complex types only - other types have a sensible default. |
`eval_real(b, cb)` | `void` | Sets `b` to the real part of `cb`. Required for complex types only - other types have a sensible default. |
`eval_imag(b, cb)` | `void` | Sets `b` to the imaginary of `cb`. Required for complex types only - other types have a sensible default. |
`eval_set_real(b, a)` | `void` | Sets the real part of `b` to `cb`. Required for complex types only - other types have a sensible default. |
`eval_set_imag(b, a)` | `void` | Sets the imaginary part of `b` to `cb`. Required for complex types only - other types have a sensible default. |

Table 1.9. Optional Requirements on the Backend Type

Expression | Returns | Comments | Throws
--- | --- | --- | ---
Construct and assign: |  |
`B(rb)` | `B` | Move constructor. Afterwards variable `rb` shall be in sane state, albeit with unspecified value. Only destruction and assignment to the moved-from variable `rb` need be supported after the operation. | `noexcept`
`b = rb` | `B&` | Move-assign. Afterwards variable `rb` shall be in sane state, albeit with unspecified value. Only destruction and assignment to the moved-from variable `rb` need be supported after the operation. | `noexcept`
`B(a)` | `B` | Direct construction from an arithmetic type. The type of `a` shall be listed in one of the type lists `B::signed_types`, `B::unsigned_types` or `B::float_types`. When not provided, this operation is simulated using default-construction followed by assignment. |
`B(b2)` | `B` | Copy constructor from a different back-end type. When not provided, a generic interconversion routine is used. This constructor may be `explicit` if the corresponding frontend constructor should also be `explicit`. |
`b = b2` | `b&` | Assignment operator from a different back-end type. When not provided, a generic interconversion routine is used. |
`assign_components(b, a, a)` | `void` | Assigns to `b` the two components in the following arguments. Only applies to rational and complex number types. When not provided, arithmetic operations are used to synthesise the result from the two values. |
`assign_components(b, b2, b2)` | `void` | Assigns to `b` the two components in the following arguments. Only applies to rational and complex number types. When not provided, arithmetic operations are used to synthesise the result from the two values. |
Comparisons: |  |
`eval_eq(cb, cb2)` | `bool` | Returns `true` if `cb` and `cb2` are equal in value. When not provided, the default implementation returns `cb.compare(cb2) == 0`. | `noexcept`
`eval_eq(cb, a)` | `bool` | Returns `true` if `cb` and `a` are equal in value. The type of `a` shall be listed in one of the type lists `B::signed_types`, `B::unsigned_types` or `B::float_types`. When not provided, return the equivalent of `eval_eq(cb, B(a))`. |
`eval_eq(a, cb)` | `bool` | Returns `true` if `cb` and `a` are equal in value. The type of `a` shall be listed in one of the type lists `B::signed_types`, `B::unsigned_types` or `B::float_types`. When not provided, the default version returns `eval_eq(cb, a)`. |
`eval_lt(cb, cb2)` | `bool` | Returns `true` if `cb` is less than `cb2` in value. When not provided, the default implementation returns `cb.compare(cb2) < 0`. | `noexcept`
`eval_lt(cb, a)` | `bool` | Returns `true` if `cb` is less than `a` in value. The type of `a` shall be listed in one of the type lists `B::signed_types`, `B::unsigned_types` or `B::float_types`. When not provided, the default implementation returns `eval_lt(cb, B(a))`. |
`eval_lt(a, cb)` | `bool` | Returns `true` if `a` is less than `cb` in value. The type of `a` shall be listed in one of the type lists `B::signed_types`, `B::unsigned_types` or `B::float_types`. When not provided, the default implementation returns `eval_gt(cb, a)`. |
`eval_gt(cb, cb2)` | `bool` | Returns `true` if `cb` is greater than `cb2` in value. When not provided, the default implementation returns `cb.compare(cb2) > 0`. | `noexcept`
`eval_gt(cb, a)` | `bool` | Returns `true` if `cb` is greater than `a` in value. The type of `a` shall be listed in one of the type lists `B::signed_types`, `B::unsigned_types` or `B::float_types`. When not provided, the default implementation returns `eval_gt(cb, B(a))`. |
`eval_gt(a, cb)` | `bool` | Returns `true` if `a` is greater than `cb` in value. The type of `a` shall be listed in one of the type lists `B::signed_types`, `B::unsigned_types` or `B::float_types`. When not provided, the default implementation returns `eval_lt(cb, a)`. |
`eval_is_zero(cb)` | `bool` | Returns `true` if `cb` is zero, otherwise `false`. The default version of this function returns `cb.compare(ui_type(0)) == 0`, where `ui_type` is `ui_type` is `typename std::tuple_element<0, typename B::unsigned_types>::type`. |
`eval_get_sign(cb)` | `int` | Returns a value < zero if `cb` is negative, a value > zero if `cb` is positive, and zero if `cb` is zero. The default version of this function returns `cb.compare(ui_type(0))`, where `ui_type` is `ui_type` is `typename std::tuple_element<0, typename B::unsigned_types>::type`. |
Basic arithmetic: |  |
`eval_add(b, a)` | `void` | Adds `a` to `b`. The type of `a` shall be listed in one of the type lists `B::signed_types`, `B::unsigned_types` or `B::float_types`. When not provided, the default version calls `eval_add(b, B(a))` |
`eval_add(b, cb, cb2)` | `void` | Add `cb` to `cb2` and stores the result in `b`. When not provided, does the equivalent of `b = cb; eval_add(b, cb2)`. |
`eval_add(b, cb, a)` | `void` | Add `cb` to `a` and stores the result in `b`. The type of `a` shall be listed in one of the type lists `B::signed_types`, `B::unsigned_types` or `B::float_types`. When not provided, does the equivalent of `eval_add(b, cb, B(a))`. |
`eval_add(b, a, cb)` | `void` | Add `a` to `cb` and stores the result in `b`. The type of `a` shall be listed in one of the type lists `B::signed_types`, `B::unsigned_types` or `B::float_types`. When not provided, does the equivalent of `eval_add(b, cb, a)`. |
`eval_subtract(b, a)` | `void` | Subtracts `a` from `b`. The type of `a` shall be listed in one of the type lists `B::signed_types`, `B::unsigned_types` or `B::float_types`. When not provided, the default version calls `eval_subtract(b, B(a))` |
`eval_subtract(b, cb, cb2)` | `void` | Subtracts `cb2` from `cb` and stores the result in `b`. When not provided, does the equivalent of `b = cb; eval_subtract(b, cb2)`. |
`eval_subtract(b, cb, a)` | `void` | Subtracts `a` from `cb` and stores the result in `b`. The type of `a` shall be listed in one of the type lists `B::signed_types`, `B::unsigned_types` or `B::float_types`. When not provided, does the equivalent of `eval_subtract(b, cb, B(a))`. |
`eval_subtract(b, a, cb)` | `void` | Subtracts `cb` from `a` and stores the result in `b`. The type of `a` shall be listed in one of the type lists `B::signed_types`, `B::unsigned_types` or `B::float_types`. When not provided, does the equivalent of `eval_subtract(b, cb, a); b.negate();`. |
`eval_multiply(b, a)` | `void` | Multiplies `b` by `a`. The type of `a` shall be listed in one of the type lists `B::signed_types`, `B::unsigned_types` or `B::float_types`. When not provided, the default version calls `eval_multiply(b, B(a))` |
`eval_multiply(b, cb, cb2)` | `void` | Multiplies `cb` by `cb2` and stores the result in `b`. When not provided, does the equivalent of `b = cb; eval_multiply(b, cb2)`. |
`eval_multiply(b, cb, a)` | `void` | Multiplies `cb` by `a` and stores the result in `b`. The type of `a` shall be listed in one of the type lists `B::signed_types`, `B::unsigned_types` or `B::float_types`. When not provided, does the equivalent of `eval_multiply(b, cb, B(a))`. |
`eval_multiply(b, a, cb)` | `void` | Multiplies `a` by `cb` and stores the result in `b`. The type of `a` shall be listed in one of the type lists `B::signed_types`, `B::unsigned_types` or `B::float_types`. When not provided, does the equivalent of `eval_multiply(b, cb, a)`. |
`eval_multiply_add(b, cb, cb2)` | `void` | Multiplies `cb` by `cb2` and adds the result to `b`. When not provided does the equivalent of creating a temporary `B t` and `eval_multiply(t, cb, cb2)` followed by `eval_add(b, t)`. |
`eval_multiply_add(b, cb, a)` | `void` | Multiplies `a` by `cb` and adds the result to `b`. The type of `a` shall be listed in one of the type lists `B::signed_types`, `B::unsigned_types` or `B::float_types`. When not provided does the equivalent of creating a temporary `B t` and `eval_multiply(t, cb, a)` followed by `eval_add(b, t)`. |
`eval_multiply_add(b, a, cb)` | `void` | Multiplies `a` by `cb` and adds the result to `b`. The type of `a` shall be listed in one of the type lists `B::signed_types`, `B::unsigned_types` or `B::float_types`. When not provided does the equivalent of `eval_multiply_add(b, cb, a)`. |
`eval_multiply_subtract(b, cb, cb2)` | `void` | Multiplies `cb` by `cb2` and subtracts the result from `b`. When not provided does the equivalent of creating a temporary `B t` and `eval_multiply(t, cb, cb2)` followed by `eval_subtract(b, t)`. |
`eval_multiply_subtract(b, cb, a)` | `void` | Multiplies `a` by `cb` and subtracts the result from `b`. The type of `a` shall be listed in one of the type lists `B::signed_types`, `B::unsigned_types` or `B::float_types`. When not provided does the equivalent of creating a temporary `B t` and `eval_multiply(t, cb, a)` followed by `eval_subtract(b, t)`. |
`eval_multiply_subtract(b, a, cb)` | `void` | Multiplies `a` by `cb` and subtracts the result from `b`. The type of `a` shall be listed in one of the type lists `B::signed_types`, `B::unsigned_types` or `B::float_types`. When not provided does the equivalent of `eval_multiply_subtract(b, cb, a)`. |
`eval_multiply_add(b, cb, cb2, cb3)` | `void` | Multiplies `cb` by `cb2` and adds the result to `cb3` storing the result in `b`. When not provided does the equivalent of `eval_multiply(b, cb, cb2)` followed by `eval_add(b, cb3)`. For brevity, only a version showing all arguments of type `B` is shown here, but you can replace up to any 2 of `cb`, `cb2` and `cb3` with any type listed in one of the type lists `B::signed_types`, `B::unsigned_types` or `B::float_types`. |
`eval_multiply_subtract(b, cb, cb2, cb3)` | `void` | Multiplies `cb` by `cb2` and subtracts from the result `cb3` storing the result in `b`. When not provided does the equivalent of `eval_multiply(b, cb, cb2)` followed by `eval_subtract(b, cb3)`. For brevity, only a version showing all arguments of type `B` is shown here, but you can replace up to any 2 of `cb`, `cb2` and `cb3` with any type listed in one of the type lists `B::signed_types`, `B::unsigned_types` or `B::float_types`. |
`eval_divide(b, a)` | `void` | Divides `b` by `a`. The type of `a` shall be listed in one of the type lists `B::signed_types`, `B::unsigned_types` or `B::float_types`. When not provided, the default version calls `eval_divide(b, B(a))` | `std::overflow_error` if `a` has the value zero, and `std::numeric_limits<number<B> >::has_infinity == false`
`eval_divide(b, cb, cb2)` | `void` | Divides `cb` by `cb2` and stores the result in `b`. When not provided, does the equivalent of `b = cb; eval_divide(b, cb2)`. | `std::overflow_error` if `cb2` has the value zero, and `std::numeric_limits<number<B> >::has_infinity == false`
`eval_divide(b, cb, a)` | `void` | Divides `cb` by `a` and stores the result in `b`. The type of `a` shall be listed in one of the type lists `B::signed_types`, `B::unsigned_types` or `B::float_types`. When not provided, does the equivalent of `eval_divide(b, cb, B(a))`. | `std::overflow_error` if `a` has the value zero, and `std::numeric_limits<number<B> >::has_infinity == false`
`eval_divide(b, a, cb)` | `void` | Divides `a` by `cb` and stores the result in `b`. The type of `a` shall be listed in one of the type lists `B::signed_types`, `B::unsigned_types` or `B::float_types`. When not provided, does the equivalent of `eval_divide(b, B(a), cb)`. | `std::overflow_error` if cb has the value zero, and `std::numeric_limits<number<B> >::has_infinity == false`
`eval_increment(b)` | void | Increments the value of `b` by one. When not provided, does the equivalent of `eval_add(b, static_cast<ui_type>(1u))`. Where `ui_type` is `typename std::tuple_element<0, typename B::unsigned_types>::type`. |
`eval_decrement(b)` | void | Decrements the value of `b` by one. When not provided, does the equivalent of `eval_subtract(b, static_cast<ui_type>(1u))`. Where `ui_type` is `typename std::tuple_element<0, typename B::unsigned_types>::type`. |
Integer specific operations: |  |
`eval_modulus(b, a)` | `void` | Computes `b %= cb`, only required when `B` is an integer type. The type of `a` shall be listed in one of the type lists `B::signed_types`, `B::unsigned_types` or `B::float_types`. When not provided, the default version calls `eval_modulus(b, B(a))` | `std::overflow_error` if `a` has the value zero.
`eval_modulus(b, cb, cb2)` | `void` | Computes `cb % cb2` and stores the result in `b`, only required when `B` is an integer type. When not provided, does the equivalent of `b = cb; eval_modulus(b, cb2)`. | `std::overflow_error` if `a` has the value zero.
`eval_modulus(b, cb, a)` | `void` | Computes `cb % a` and stores the result in `b`, only required when `B` is an integer type. The type of `a` shall be listed in one of the type lists `B::signed_types`, `B::unsigned_types` or `B::float_types`. When not provided, does the equivalent of `eval_modulus(b, cb, B(a))`. | `std::overflow_error` if `a` has the value zero.
`eval_modulus(b, a, cb)` | `void` | Computes `cb % a` and stores the result in `b`, only required when `B` is an integer type. The type of `a` shall be listed in one of the type lists `B::signed_types`, `B::unsigned_types` or `B::float_types`. When not provided, does the equivalent of `eval_modulus(b, B(a), cb)`. | `std::overflow_error` if `a` has the value zero.
`eval_bitwise_and(b, a)` | `void` | Computes `b &= cb`, only required when `B` is an integer type. The type of `a` shall be listed in one of the type lists `B::signed_types`, `B::unsigned_types` or `B::float_types`. When not provided, the default version calls `eval_bitwise_and(b, B(a))` |
`eval_bitwise_and(b, cb, cb2)` | `void` | Computes `cb & cb2` and stores the result in `b`, only required when `B` is an integer type. When not provided, does the equivalent of `b = cb; eval_bitwise_and(b, cb2)`. |
`eval_bitwise_and(b, cb, a)` | `void` | Computes `cb & a` and stores the result in `b`, only required when `B` is an integer type. The type of `a` shall be listed in one of the type lists `B::signed_types`, `B::unsigned_types` or `B::float_types`. When not provided, does the equivalent of `eval_bitwise_and(b, cb, B(a))`. |
`eval_bitwise_and(b, a, cb)` | `void` | Computes `cb & a` and stores the result in `b`, only required when `B` is an integer type. The type of `a` shall be listed in one of the type lists `B::signed_types`, `B::unsigned_types` or `B::float_types`. When not provided, does the equivalent of `eval_bitwise_and(b, cb, a)`. |
`eval_bitwise_or(b, a)` | `void` | Computes `b |= cb`, only required when `B` is an integer type. The type of `a` shall be listed in one of the type lists `B::signed_types`, `B::unsigned_types` or `B::float_types`. When not provided, the default version calls `eval_bitwise_or(b, B(a))` |
`eval_bitwise_or(b, cb, cb2)` | `void` | Computes `cb | cb2` and stores the result in `b`, only required when `B` is an integer type. When not provided, does the equivalent of `b = cb; eval_bitwise_or(b, cb2)`. |
`eval_bitwise_or(b, cb, a)` | `void` | Computes `cb | a` and stores the result in `b`, only required when `B` is an integer type. The type of `a` shall be listed in one of the type lists `B::signed_types`, `B::unsigned_types` or `B::float_types`. When not provided, does the equivalent of `eval_bitwise_or(b, cb, B(a))`. |
`eval_bitwise_or(b, a, cb)` | `void` | Computes `cb | a` and stores the result in `b`, only required when `B` is an integer type. The type of `a` shall be listed in one of the type lists `B::signed_types`, `B::unsigned_types` or `B::float_types`. When not provided, does the equivalent of `eval_bitwise_or(b, cb, a)`. |
`eval_bitwise_xor(b, a)` | `void` | Computes `b ^= cb`, only required when `B` is an integer type. The type of `a` shall be listed in one of the type lists `B::signed_types`, `B::unsigned_types` or `B::float_types`. When not provided, the default version calls `eval_bitwise_xor(b, B(a))` |
`eval_bitwise_xor(b, cb, cb2)` | `void` | Computes `cb ^ cb2` and stores the result in `b`, only required when `B` is an integer type. When not provided, does the equivalent of `b = cb; eval_bitwise_xor(b, cb2)`. |
`eval_bitwise_xor(b, cb, a)` | `void` | Computes `cb ^ a` and stores the result in `b`, only required when `B` is an integer type. The type of `a` shall be listed in one of the type lists `B::signed_types`, `B::unsigned_types` or `B::float_types`. When not provided, does the equivalent of `eval_bitwise_xor(b, cb, B(a))`. |
`eval_bitwise_xor(b, a, cb)` | `void` | Computes `a ^ cb` and stores the result in `b`, only required when `B` is an integer type. The type of `a` shall be listed in one of the type lists `B::signed_types`, `B::unsigned_types` or `B::float_types`. When not provided, does the equivalent of `eval_bitwise_xor(b, cb, a)`. |
`eval_left_shift(b, cb, ui)` | `void` | Computes `cb << ui` and stores the result in `b`, only required when `B` is an integer type. When not provided, does the equivalent of `b = cb; eval_left_shift(b, a);`. |
`eval_right_shift(b, cb, ui)` | `void` | Computes `cb >> ui` and stores the result in `b`, only required when `B` is an integer type. When not provided, does the equivalent of `b = cb; eval_right_shift(b, a);`. |
`eval_qr(cb, cb2, b, b2)` | `void` | Sets `b` to the result of `cb / cb2` and `b2` to the result of `cb % cb2`. Only required when `B` is an integer type. The default version of this function is synthesised from other operations above. | `std::overflow_error` if `a` has the value zero.
`eval_integer_modulus(cb, ui)` | `unsigned` | Returns the result of `cb % ui`. Only required when `B` is an integer type. The default version of this function is synthesised from other operations above. | `std::overflow_error` if `a` has the value zero.
`eval_lsb(cb)` | `unsigned` | Returns the index of the least significant bit that is set. Only required when `B` is an integer type. The default version of this function is synthesised from other operations above. |
`eval_msb(cb)` | `unsigned` | Returns the index of the most significant bit that is set. Only required when `B` is an integer type. The default version of this function is synthesised from other operations above. |
`eval_bit_test(cb, ui)` | `bool` | Returns true if `cb` has bit `ui` set. Only required when `B` is an integer type. The default version of this function is synthesised from other operations above. |
`eval_bit_set(b, ui)` | `void` | Sets the bit at index `ui` in `b`. Only required when `B` is an integer type. The default version of this function is synthesised from other operations above. |
`eval_bit_unset(b, ui)` | `void` | Unsets the bit at index `ui` in `b`. Only required when `B` is an integer type. The default version of this function is synthesised from other operations above. |
`eval_bit_flip(b, ui)` | `void` | Flips the bit at index `ui` in `b`. Only required when `B` is an integer type. The default version of this function is synthesised from other operations above. |
`eval_gcd(b, cb, cb2)` | `void` | Sets `b` to the greatest common divisor of `cb` and `cb2`. Only required when `B` is an integer type. The default version of this function is synthesised from other operations above. |
`eval_lcm(b, cb, cb2)` | `void` | Sets `b` to the least common multiple of `cb` and `cb2`. Only required when `B` is an integer type. The default version of this function is synthesised from other operations above. |
`eval_gcd(b, cb, a)` | `void` | Sets `b` to the greatest common divisor of `cb` and `cb2`. Only required when `B` is an integer type. The type of `a` shall be listed in one of the type lists `B::signed_types`, `B::unsigned_types` or `B::float_types`. The default version of this function calls `eval_gcd(b, cb, B(a))`. |
`eval_lcm(b, cb, a)` | `void` | Sets `b` to the least common multiple of `cb` and `cb2`. Only required when `B` is an integer type. The type of `a` shall be listed in one of the type lists `B::signed_types`, `B::unsigned_types` or `B::float_types`. The default version of this function calls `eval_lcm(b, cb, B(a))`. |
`eval_gcd(b, a, cb)` | `void` | Sets `b` to the greatest common divisor of `cb` and `a`. Only required when `B` is an integer type. The type of `a` shall be listed in one of the type lists `B::signed_types`, `B::unsigned_types` or `B::float_types`. The default version of this function calls `eval_gcd(b, cb, a)`. |
`eval_lcm(b, a, cb)` | `void` | Sets `b` to the least common multiple of `cb` and `a`. Only required when `B` is an integer type. The type of `a` shall be listed in one of the type lists `B::signed_types`, `B::unsigned_types` or `B::float_types`. The default version of this function calls `eval_lcm(b, cb, a)`. |
`eval_powm(b, cb, cb2, cb3)` | `void` | Sets `b` to the result of (cb^cb2)%cb3. The default version of this function is synthesised from other operations above. |
`eval_powm(b, cb, cb2, a)` | `void` | Sets `b` to the result of (cb^cb2)%a. The type of `a` shall be listed in one of the type lists `B::signed_types`, `B::unsigned_types`. The default version of this function is synthesised from other operations above. |
`eval_powm(b, cb, a, cb2)` | `void` | Sets `b` to the result of (cb^a)%cb2. The type of `a` shall be listed in one of the type lists `B::signed_types`, `B::unsigned_types`. The default version of this function is synthesised from other operations above. |
`eval_powm(b, cb, a, a2)` | `void` | Sets `b` to the result of (cb^a)%a2. The type of `a` shall be listed in one of the type lists `B::signed_types`, `B::unsigned_types`. The default version of this function is synthesised from other operations above. |
`eval_integer_sqrt(b, cb, b2)` | `void` | Sets `b` to the largest integer which when squared is less than `cb`, also sets `b2` to the remainder, ie to cb - b2. The default version of this function is synthesised from other operations above. |
Sign manipulation: |  |
`eval_abs(b, cb)` | `void` | Set `b` to the absolute value of `cb`. The default version of this functions assigns `cb` to `b`, and then calls `b.negate()` if `eval_get_sign(cb) < 0`. |
`eval_fabs(b, cb)` | `void` | Set `b` to the absolute value of `cb`. The default version of this functions assigns `cb` to `b`, and then calls `b.negate()` if `eval_get_sign(cb) < 0`. |
floating-point functions: |  |
`eval_fpclassify(cb)` | `int` | Returns one of the same values returned by `std::fpclassify`. Only required when `B` is an floating-point type. The default version of this function will only test for zero `cb`. |
`eval_trunc(b, cb)` | `void` | Performs the equivalent operation to `std::trunc` on argument `cb` and stores the result in `b`. Only required when `B` is an floating-point type. The default version of this function is synthesised from other operations above. |
`eval_round(b, cb)` | `void` | Performs the equivalent operation to `std::round` on argument `cb` and stores the result in `b`. Only required when `B` is an floating-point type. The default version of this function is synthesised from other operations above. |
`eval_exp(b, cb)` | `void` | Performs the equivalent operation to `std::exp` on argument `cb` and stores the result in `b`. Only required when `B` is an floating-point type. The default version of this function is synthesised from other operations above. |
`eval_exp2(b, cb)` | `void` | Performs the equivalent operation to `std::exp2` on argument `cb` and stores the result in `b`. Only required when `B` is an floating-point type. The default version of this function is implemented in terms of `eval_pow`. |
`eval_log(b, cb)` | `void` | Performs the equivalent operation to `std::log` on argument `cb` and stores the result in `b`. Only required when `B` is an floating-point type. The default version of this function is synthesised from other operations above. |
`eval_log10(b, cb)` | `void` | Performs the equivalent operation to `std::log10` on argument `cb` and stores the result in `b`. Only required when `B` is an floating-point type. The default version of this function is synthesised from other operations above. |
`eval_sin(b, cb)` | `void` | Performs the equivalent operation to `std::sin` on argument `cb` and stores the result in `b`. Only required when `B` is an floating-point type. The default version of this function is synthesised from other operations above. |
`eval_cos(b, cb)` | `void` | Performs the equivalent operation to `std::cos` on argument `cb` and stores the result in `b`. Only required when `B` is an floating-point type. The default version of this function is synthesised from other operations above. |
`eval_tan(b, cb)` | `void` | Performs the equivalent operation to `std::exp` on argument `cb` and stores the result in `b`. Only required when `B` is an floating-point type. The default version of this function is synthesised from other operations above. |
`eval_asin(b, cb)` | `void` | Performs the equivalent operation to `std::asin` on argument `cb` and stores the result in `b`. Only required when `B` is an floating-point type. The default version of this function is synthesised from other operations above. |
`eval_acos(b, cb)` | `void` | Performs the equivalent operation to `std::acos` on argument `cb` and stores the result in `b`. Only required when `B` is an floating-point type. The default version of this function is synthesised from other operations above. |
`eval_atan(b, cb)` | `void` | Performs the equivalent operation to `std::atan` on argument `cb` and stores the result in `b`. Only required when `B` is an floating-point type. The default version of this function is synthesised from other operations above. |
`eval_sinh(b, cb)` | `void` | Performs the equivalent operation to `std::sinh` on argument `cb` and stores the result in `b`. Only required when `B` is an floating-point type. The default version of this function is synthesised from other operations above. |
`eval_cosh(b, cb)` | `void` | Performs the equivalent operation to `std::cosh` on argument `cb` and stores the result in `b`. Only required when `B` is an floating-point type. The default version of this function is synthesised from other operations above. |
`eval_tanh(b, cb)` | `void` | Performs the equivalent operation to `std::tanh` on argument `cb` and stores the result in `b`. Only required when `B` is an floating-point type. The default version of this function is synthesised from other operations above. |
`eval_fmod(b, cb, cb2)` | `void` | Performs the equivalent operation to `std::fmod` on arguments `cb` and `cb2`, and store the result in `b`. Only required when `B` is an floating-point type. The default version of this function is synthesised from other operations above. |
`eval_modf(b, cb, pb)` | `void` | Performs the equivalent operation to `std::modf` on argument `cb`, and store the integer result in `*pb` and the fractional part in `b`. Only required when `B` is an floating-point type. The default version of this function is synthesised from other operations above. |
`eval_pow(b, cb, cb2)` | `void` | Performs the equivalent operation to `std::pow` on arguments `cb` and `cb2`, and store the result in `b`. Only required when `B` is an floating-point type. The default version of this function is synthesised from other operations above. |
`eval_atan2(b, cb, cb2)` | `void` | Performs the equivalent operation to `std::atan` on arguments `cb` and `cb2`, and store the result in `b`. Only required when `B` is an floating-point type. The default version of this function is synthesised from other operations above. |
`eval_scalbn(b, cb, e)` | `void` | Scales value `cb` by re, where r is the radix of the type. The default version of this function is implemented in terms of eval_ldexp, consequently this function must be provided for types with a radix other than 2.
`eval_scalbln(b, cb, e)` | `void` | Calls `eval_scalbn(b, cb, e)`.
`eval_ilogb(cb)` | `B::exponent_type` | Returns the exponent e of value `cb` such that 1 <= cb*r-e < r, where r is the radix of type B. The default version of this function is implemented in terms of eval_frexp, consequently this function must be provided for types with a radix other than 2.
`eval_remquo(b, cb, cb2, pi)` | `void` | Sets `b = cb - n * cb2` and stores `n` in `*pi`.
`eval_remquo(b, cb, a, pi)` | `void` | Default version converts a to type B and calls the overload above.
`eval_remquo(b, a, cb, pi)` | `void` | Default version converts a to type B and calls the overload above.
`eval_remainder(b, cb, cb2)` | `void` | Default version calls eval_remquo with a dummy final argument.
`eval_remainder(b, cb, a)` | `void` | Default version calls eval_remquo with a dummy final argument.
`eval_remainder(b, a, cb)` | `void` | Default version calls eval_remquo with a dummy final argument.
`eval_fdim(b, cb, cb2)` | `void` | Default version sets `b = cb - cb2` if `cb > cb2` and zero otherwise. Special cases are handled as in the C99 annex.
`eval_fdim(b, cb, a)` | `void` | Default version sets `b = cb - cb2` if `cb > cb2` and zero otherwise. Special cases are handled as in the C99 annex.
`eval_fdim(b, a, cb)` | `void` | Default version sets `b = cb - cb2` if `cb > cb2` and zero otherwise. Special cases are handled as in the C99 annex.
`eval_fmax(b, cb, cb2)` | `void` | Sets `b` to the larger of `cb` and `cb2`.
`eval_fmax(b, cb, a)` | `void` | Sets `b` to the larger of `cb` and `a`.
`eval_fmax(b, a, cb)` | `void` | Sets `b` to the larger of `cb` and `a`.
`eval_fmin(b, cb, cb2)` | `void` | Sets `b` to the smaller of `cb` and `cb2`.
`eval_fmin(b, cb, a)` | `void` | Sets `b` to the smaller of `cb` and `a`.
`eval_fmin(b, a, cb)` | `void` | Sets `b` to the smaller of `cb` and `a`.
`eval_hypot(b, cb, cb2)` | `void` | Sets `b` to the square root of the sum of the squares of `cb` and `cb2` without undue over or under flow.
`eval_hypot(b, cb, a)` | `void` | As above.
`eval_hypot(b, a, cb)` | `void` | As above.
`eval_logb(b, cb)` | `B::exponent_type` | Sets `b` to the exponent e of value `cb` such that 1 <= cb*r-b < r, where r is the radix of type B. The default version of this function is implemented in terms of `eval_ilogb`.
`eval_nearbyint(b, cb)` | `void` | Calls `eval_round(b, cb)`.
`eval_rint(b, cb)` | `void` | Calls `eval_nearbyint(b, cb)`.
`eval_log2(b, cb)` | `void` | Sets `b` to the logarithm base 2 of `cb`.
hashing: |  |
`hash_value(cb)` | `std::size_t` | Returns a hash value for the argument that is suitable for use with `std::hash` etc. If not provided then no automatic hashing support will be available for the number type.

When the tables above place no throws requirements on an operation, then it is up to each type modelling this concept to decide when or whether throwing an exception is desirable. However, thrown exceptions should always either be the type, or inherit from the type `std::runtime_error`. For example, a floating-point type might choose to throw `std::overflow_error` whenever the result of an operation would be infinite, and `std::underflow_error` whenever it would round to zero.

| Note
The non-member functions are all named with an "eval_" prefix to avoid conflicts with template classes of the same name - in point of fact this naming convention shouldn't be necessary, but rather works around some compiler bugs.

##### Overloadable Functions

Some of the C99 math functions do not have `eval_` functions but must be overloaded directly: these functions are either trivial or are forwarded to the Boost.Math implementations by default. The full list of these functions is:

    int           sign       (const number-or-expression-template-type&);
    int           signbit    (const number-or-expression-template-type&);
    number        changesign (const number-or-expression-template-type&);
    number        copysign   (const number-or-expression-template-type&, const number-or-expression-template-type&);
    number        asinh      (const number-or-expression-template-type&);
    number        acosh      (const number-or-expression-template-type&);
    number        atanh      (const number-or-expression-template-type&);
    number        cbrt       (const number-or-expression-template-type&);
    number        erf        (const number-or-expression-template-type&);
    number        erfc       (const number-or-expression-template-type&);
    number        expm1      (const number-or-expression-template-type&);
    number        log1p      (const number-or-expression-template-type&);
    number        tgamma     (const number-or-expression-template-type&);
    number        lgamma     (const number-or-expression-template-type&);
    long          lrint      (const number-or-expression-template-type&);
    long long     llrint     (const number-or-expression-template-type&);
    number        nextafter  (const number-or-expression-template-type&, const number-or-expression-template-type&);
    number        nexttoward (const number-or-expression-template-type&, const number-or-expression-template-type&);