quantity.hpp

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/*
 * Unless otherwise specified, the definitions of all units in this
 * file are from NIST Special Publication 811, found online at
 * http://physics.nist.gov/Document/sp811.pdf
 * Other sources: OED = Oxford English Dictionary
 */

#ifndef PHYS_UNITS_QUANTITY_HPP_INCLUDED
#define PHYS_UNITS_QUANTITY_HPP_INCLUDED

#include <cmath>
#include <cstdlib>
#include <utility> // std::declval
#include <type_traits>


namespace phys {


  namespace units {

#ifdef PHYS_UNITS_REP_TYPE
    using Rep = PHYS_UNITS_REP_TYPE;
#else
    using Rep = double;
#endif

    /*
     * declare now, define later.
     */
    template <typename Dims, typename T = Rep>
    class quantity;

    template <int D1, int D2, int D3, int D4 = 0, int D5 = 0, int D6 = 0, int D7 = 0,
              int D8 = 0>
    struct dimensions {
      enum {
        dim1 = D1,
        dim2 = D2,
        dim3 = D3,
        dim4 = D4,
        dim5 = D5,
        dim6 = D6,
        dim7 = D7,
        dim8 = D8,

        is_all_zero = D1 == 0 && D2 == 0 && D3 == 0 && D4 == 0 && D5 == 0 && D6 == 0 &&
                      D7 == 0 && D8 == 0,

        is_base = 1 == (D1 != 0) + (D2 != 0) + (D3 != 0) + (D4 != 0) + (D5 != 0) +
                           (D6 != 0) + (D7 != 0) + (D8 != 0) &&
                  1 == D1 + D2 + D3 + D4 + D5 + D6 + D7 + D8,
      };

      template <int R1, int R2, int R3, int R4, int R5, int R6, int R7, int R8>
      constexpr bool operator==(dimensions<R1, R2, R3, R4, R5, R6, R7, R8> const&) const {
        return D1 == R1 && D2 == R2 && D3 == R3 && D4 == R4 && D5 == R5 && D6 == R6 &&
               D7 == R7 && D8 == R8;
      }

      template <int R1, int R2, int R3, int R4, int R5, int R6, int R7, int R8>
      constexpr bool operator!=(
          dimensions<R1, R2, R3, R4, R5, R6, R7, R8> const& rhs) const {
        return !(*this == rhs);
      }
    };


    typedef dimensions<0, 0, 0> dimensionless_d;


    namespace detail {

      template <typename D, typename T>
      struct collapse {
        typedef quantity<D, T> type;
      };

      template <typename T>
      struct collapse<dimensionless_d, T> {
        typedef T type;
      };

      template <typename D, typename T>
      using Collapse = typename collapse<D, T>::type;

      // promote types of expression to result type.

      template <typename X, typename Y>
      using PromoteAdd = decltype(std::declval<X>() + std::declval<Y>());

      template <typename X, typename Y>
      using PromoteMul = decltype(std::declval<X>() * std::declval<Y>());

      /*
       * The following batch of structs are type generators to calculate
       * the correct type of the result of various operations.
       */

      template <typename DX, typename DY, typename T>
      struct product {
        enum {
          d1 = DX::dim1 + DY::dim1,
          d2 = DX::dim2 + DY::dim2,
          d3 = DX::dim3 + DY::dim3,
          d4 = DX::dim4 + DY::dim4,
          d5 = DX::dim5 + DY::dim5,
          d6 = DX::dim6 + DY::dim6,
          d7 = DX::dim7 + DY::dim7,
          d8 = DX::dim8 + DY::dim8
        };

        typedef dimensions<d1, d2, d3, d4, d5, d6, d7, d8> dim;

        typedef Collapse<dim, T> type;
      };

      template <typename DX, typename DY>
      using product_d = typename product<DX, DY, Rep>::dim;

      template <typename DX, typename DY, typename X, typename Y>
      using Product = typename product<DX, DY, PromoteMul<X, Y>>::type;

      template <typename DX, typename DY, typename T>
      struct quotient {
        enum {
          d1 = DX::dim1 - DY::dim1,
          d2 = DX::dim2 - DY::dim2,
          d3 = DX::dim3 - DY::dim3,
          d4 = DX::dim4 - DY::dim4,
          d5 = DX::dim5 - DY::dim5,
          d6 = DX::dim6 - DY::dim6,
          d7 = DX::dim7 - DY::dim7,
          d8 = DX::dim8 - DY::dim8
        };

        typedef dimensions<d1, d2, d3, d4, d5, d6, d7, d8> dim;

        typedef Collapse<dim, T> type;
      };

      template <typename DX, typename DY>
      using quotient_d = typename quotient<DX, DY, Rep>::dim;

      template <typename DX, typename DY, typename X, typename Y>
      using Quotient = typename quotient<DX, DY, PromoteMul<X, Y>>::type;

      template <typename D, typename T>
      struct reciprocal {
        enum {
          d1 = -D::dim1,
          d2 = -D::dim2,
          d3 = -D::dim3,
          d4 = -D::dim4,
          d5 = -D::dim5,
          d6 = -D::dim6,
          d7 = -D::dim7,
          d8 = -D::dim8
        };

        typedef dimensions<d1, d2, d3, d4, d5, d6, d7, d8> dim;

        typedef Collapse<dim, T> type;
      };

      template <typename DX, typename DY>
      using reciprocal_d = typename reciprocal<DX, Rep>::dim;

      template <typename D, typename X, typename Y>
      using Reciprocal = typename reciprocal<D, PromoteMul<X, Y>>::type;

      template <typename D, int N, typename T>
      struct power {
        enum {
          d1 = N * D::dim1,
          d2 = N * D::dim2,
          d3 = N * D::dim3,
          d4 = N * D::dim4,
          d5 = N * D::dim5,
          d6 = N * D::dim6,
          d7 = N * D::dim7,
          d8 = N * D::dim8
        };

        typedef dimensions<d1, d2, d3, d4, d5, d6, d7, d8> dim;

        typedef Collapse<dim, T> type;
      };

      template <typename DX, int N>
      using power_d = typename power<DX, N, Rep>::dim;

      template <typename D, int N, typename T>
      using Power = typename power<D, N, T>::type;

      template <typename D, int N, typename T>
      struct root {
        enum {
          all_even_multiples = D::dim1 % N == 0 && D::dim2 % N == 0 && D::dim3 % N == 0 &&
                               D::dim4 % N == 0 && D::dim5 % N == 0 && D::dim6 % N == 0 &&
                               D::dim7 % N == 0 && D::dim8 % N == 0
        };

        enum {
          d1 = D::dim1 / N,
          d2 = D::dim2 / N,
          d3 = D::dim3 / N,
          d4 = D::dim4 / N,
          d5 = D::dim5 / N,
          d6 = D::dim6 / N,
          d7 = D::dim7 / N,
          d8 = D::dim8 / N
        };

        typedef dimensions<d1, d2, d3, d4, d5, d6, d7, d8> dim;

        typedef Collapse<dim, T> type;
      };

      template <typename D, int N>
      using root_d = typename root<D, N, Rep>::dim;

      template <typename D, int N, typename T>
      using Root = typename detail::root<D, N, T>::type;

      constexpr struct magnitude_tag_t { } magnitude_tag{}; } // namespace detail

    template <typename Dims, typename T /*= Rep */>
    class quantity {
    public:
      typedef Dims dimension_type;

      typedef T value_type;

      typedef quantity<Dims, T> this_type;

      constexpr quantity()
          : m_value{} {}

      template <typename X>
      constexpr explicit quantity(detail::magnitude_tag_t, X x)
          : m_value(x) {}

      template <typename X>
      constexpr quantity(quantity<Dims, X> const& x)
          : m_value(x.magnitude()) {}

      //    /**
      //     * convert to compatible unit, for example: (3._dm).to(meter) gives 0.3;
      //     */
      //    constexpr value_type to( quantity const & x ) const {  return *this / x; }

      template <typename DX, typename X>
      constexpr auto to(quantity<DX, X> const& x) const
          -> detail::Quotient<Dims, DX, T, X> {
        return *this / x;
      }

      constexpr value_type magnitude() const { return m_value; }

      constexpr dimension_type dimension() const { return dimension_type{}; }

      static constexpr quantity zero() { return quantity{value_type(0.0)}; }
      //    static constexpr quantity zero = quantity{ value_type( 0.0 ) };

      static constexpr quantity infinity() {
        return quantity{value_type(std::numeric_limits<value_type>::infinity())};
      }

      template <typename _dim = dimension_type,
                std::enable_if_t<std::is_same_v<_dim, dimensionless_d>, bool> = true>
      operator double() const {
        return m_value;
      }

    private:
      constexpr explicit quantity(value_type x)
          : m_value{x} {}

    private:
      value_type m_value;

      enum { has_dimension = !Dims::is_all_zero };

      // static_assert( has_dimension, "quantity dimensions must not all be zero" ); //
      // MR: removed

    private:
      // friends:

      // arithmetic

      template <typename D, typename X, typename Y>
      friend constexpr quantity<D, X>& operator+=(quantity<D, X>& x,
                                                  quantity<D, Y> const& y);

      template <typename D, typename X>
      friend constexpr quantity<D, X> operator+(quantity<D, X> const& x);

      template <typename D, typename X, typename Y>
      friend constexpr quantity<D, detail::PromoteAdd<X, Y>> operator+(
          quantity<D, X> const& x, quantity<D, Y> const& y);

      template <typename D, typename X, typename Y>
      friend constexpr quantity<D, X>& operator-=(quantity<D, X>& x,
                                                  quantity<D, Y> const& y);

      template <typename D, typename X>
      friend constexpr quantity<D, X> operator-(quantity<D, X> const& x);

      template <typename D, typename X, typename Y>
      friend constexpr quantity<D, detail::PromoteAdd<X, Y>> operator-(
          quantity<D, X> const& x, quantity<D, Y> const& y);

      template <typename D, typename X, typename Y>
      friend constexpr quantity<D, X>& operator*=(quantity<D, X>& x, const Y& y);

      template <typename D, typename X, typename Y>
      friend constexpr quantity<D, detail::PromoteMul<X, Y>> operator*(
          quantity<D, X> const& x, const Y& y);

      template <typename D, typename X, typename Y>
      friend constexpr quantity<D, detail::PromoteMul<X, Y>> operator*(
          const X& x, quantity<D, Y> const& y);

      template <typename DX, typename DY, typename X, typename Y>
      friend constexpr detail::Product<DX, DY, X, Y> operator*(
          quantity<DX, X> const& lhs, quantity<DY, Y> const& rhs);

      template <typename D, typename X, typename Y>
      friend constexpr quantity<D, X>& operator/=(quantity<D, X>& x, const Y& y);

      template <typename D, typename X, typename Y>
      friend constexpr quantity<D, detail::PromoteMul<X, Y>> operator/(
          quantity<D, X> const& x, const Y& y);

      template <typename D, typename X, typename Y>
      friend constexpr detail::Reciprocal<D, X, Y> operator/(const X& x,
                                                             quantity<D, Y> const& y);

      template <typename DX, typename DY, typename X, typename Y>
      friend constexpr detail::Quotient<DX, DY, X, Y> operator/(quantity<DX, X> const& x,
                                                                quantity<DY, Y> const& y);

      // absolute value.

      template <typename D, typename X>
      friend constexpr quantity<D, X> abs(quantity<D, X> const& x);

      // powers and roots

      template <int N, typename D, typename X>
      friend constexpr detail::Power<D, N, X> nth_power(quantity<D, X> const& x);

      template <typename D, typename X>
      friend constexpr detail::Power<D, 2, X> square(quantity<D, X> const& x);

      template <typename D, typename X>
      friend constexpr detail::Power<D, 3, X> cube(quantity<D, X> const& x);

      template <int N, typename D, typename X>
      friend detail::Root<D, N, X> constexpr nth_root(quantity<D, X> const& x);

      template <typename D, typename X>
      friend detail::Root<D, 2, X> constexpr sqrt(quantity<D, X> const& x);

      template <typename D, typename X>
      friend detail::Root<D, 3, X> constexpr cbrt(quantity<D, X> const& x);

      // comparison

      template <typename D, typename X, typename Y>
      friend constexpr bool operator==(quantity<D, X> const& x, quantity<D, Y> const& y);

      template <typename D, typename X, typename Y>
      friend constexpr bool operator!=(quantity<D, X> const& x, quantity<D, Y> const& y);

      template <typename D, typename X, typename Y>
      friend constexpr bool operator<(quantity<D, X> const& x, quantity<D, Y> const& y);

      template <typename D, typename X, typename Y>
      friend constexpr bool operator<=(quantity<D, X> const& x, quantity<D, Y> const& y);

      template <typename D, typename X, typename Y>
      friend constexpr bool operator>(quantity<D, X> const& x, quantity<D, Y> const& y);

      template <typename D, typename X, typename Y>
      friend constexpr bool operator>=(quantity<D, X> const& x, quantity<D, Y> const& y);
    }; // namespace units

    // helper to check whether some type is a quantity
    template <typename TNonQuantityType>
    struct is_quantity : std::false_type {};

    template <typename Dims, typename T>
    struct is_quantity<quantity<Dims, T>> : std::true_type {};

    template <typename T>
    constexpr bool is_quantity_v = is_quantity<T>::value;

    // Give names to the seven fundamental dimensions of physical reality.

    typedef dimensions<1, 0, 0, 0, 0, 0, 0, 0> length_d;
    typedef dimensions<0, 1, 0, 0, 0, 0, 0, 0> mass_d;
    typedef dimensions<0, 0, 1, 0, 0, 0, 0, 0> time_interval_d;
    typedef dimensions<0, 0, 0, 1, 0, 0, 0, 0> electric_current_d;
    typedef dimensions<0, 0, 0, 0, 1, 0, 0, 0> thermodynamic_temperature_d;
    typedef dimensions<0, 0, 0, 0, 0, 1, 0, 0> amount_of_substance_d;
    typedef dimensions<0, 0, 0, 0, 0, 0, 1, 0> luminous_intensity_d;
    typedef dimensions<0, 0, 0, 0, 0, 0, 0, 1> hepenergy_d; // this is not an SI unit !

    // Addition operators


    template <typename D, typename X, typename Y>
    constexpr quantity<D, X>& operator+=(quantity<D, X>& x, quantity<D, Y> const& y) {
      return x.m_value += y.m_value, x;
    }


    template <typename D, typename X>
    constexpr quantity<D, X> operator+(quantity<D, X> const& x) {
      return quantity<D, X>(+x.m_value);
    }


    template <typename D, typename X, typename Y>
    constexpr quantity<D, detail::PromoteAdd<X, Y>> operator+(quantity<D, X> const& x,
                                                              quantity<D, Y> const& y) {
      return quantity<D, detail::PromoteAdd<X, Y>>(x.m_value + y.m_value);
    }

    // Subtraction operators


    template <typename D, typename X, typename Y>
    constexpr quantity<D, X>& operator-=(quantity<D, X>& x, quantity<D, Y> const& y) {
      return x.m_value -= y.m_value, x;
    }


    template <typename D, typename X>
    constexpr quantity<D, X> operator-(quantity<D, X> const& x) {
      return quantity<D, X>(-x.m_value);
    }


    template <typename D, typename X, typename Y>
    constexpr quantity<D, detail::PromoteAdd<X, Y>> operator-(quantity<D, X> const& x,
                                                              quantity<D, Y> const& y) {
      return quantity<D, detail::PromoteAdd<X, Y>>(x.m_value - y.m_value);
    }

    // Multiplication operators


    template <typename D, typename X, typename Y>
    constexpr quantity<D, X>& operator*=(quantity<D, X>& x, const Y& y) {
      return x.m_value *= y, x;
    }


    template <typename D, typename X, typename Y>
    constexpr quantity<D, detail::PromoteMul<X, Y>> operator*(quantity<D, X> const& x,
                                                              const Y& y) {
      return quantity<D, detail::PromoteMul<X, Y>>(x.m_value * y);
    }


    template <typename D, typename X, typename Y>
    constexpr quantity<D, detail::PromoteMul<X, Y>> operator*(const X& x,
                                                              quantity<D, Y> const& y) {
      return quantity<D, detail::PromoteMul<X, Y>>(x * y.m_value);
    }


    template <typename DX, typename DY, typename X, typename Y>
    constexpr detail::Product<DX, DY, X, Y> operator*(quantity<DX, X> const& lhs,
                                                      quantity<DY, Y> const& rhs) {
      return detail::Product<DX, DY, X, Y>(lhs.m_value * rhs.m_value);
    }

    // Division operators


    template <typename D, typename X, typename Y>
    constexpr quantity<D, X>& operator/=(quantity<D, X>& x, const Y& y) {
      return x.m_value /= y, x;
    }


    template <typename D, typename X, typename Y>
    constexpr quantity<D, detail::PromoteMul<X, Y>> operator/(quantity<D, X> const& x,
                                                              const Y& y) {
      return quantity<D, detail::PromoteMul<X, Y>>(x.m_value / y);
    }


    template <typename D, typename X, typename Y>
    constexpr detail::Reciprocal<D, X, Y> operator/(const X& x, quantity<D, Y> const& y) {
      return detail::Reciprocal<D, X, Y>(x / y.m_value);
    }


    template <typename DX, typename DY, typename X, typename Y>
    constexpr detail::Quotient<DX, DY, X, Y> operator/(quantity<DX, X> const& x,
                                                       quantity<DY, Y> const& y) {
      return detail::Quotient<DX, DY, X, Y>(x.m_value / y.m_value);
    }


    template <typename D, typename X>
    quantity<D, X> constexpr abs(quantity<D, X> const& x) {
      return quantity<D, X>(std::abs(x.m_value));
    }

    // General powers


    template <int N, typename D, typename X>
    detail::Power<D, N, X> constexpr nth_power(quantity<D, X> const& x) {
      return detail::Power<D, N, X>(std::pow(x.m_value, X(N)));
    }

    // Low powers defined separately for efficiency.


    template <typename D, typename X>
    constexpr detail::Power<D, 2, X> square(quantity<D, X> const& x) {
      return x * x;
    }


    template <typename D, typename X>
    constexpr detail::Power<D, 3, X> cube(quantity<D, X> const& x) {
      return x * x * x;
    }

    // General root


    template <int N, typename D, typename X>
    detail::Root<D, N, X> constexpr nth_root(quantity<D, X> const& x) {
      static_assert(detail::root<D, N, X>::all_even_multiples,
                    "root result dimensions must be integral");

      return detail::Root<D, N, X>(std::pow(x.m_value, X(1.0) / N));
    }

    // Low roots defined separately for convenience.


    template <typename D, typename X>
    detail::Root<D, 2, X> constexpr sqrt(quantity<D, X> const& x) {
      static_assert(detail::root<D, 2, X>::all_even_multiples,
                    "root result dimensions must be integral");

      return detail::Root<D, 2, X>(std::sqrt(x.m_value));
    }


    template <typename D, typename X>
    detail::Root<D, 3, X> constexpr cbrt(quantity<D, X> const& x) {
      static_assert(detail::root<D, 3, X>::all_even_multiples,
                    "root result dimensions must be integral");

      return detail::Root<D, 3, X>(std::cbrt(x.m_value));
    }

    // Comparison operators


    template <typename D, typename X, typename Y>
    constexpr bool operator==(quantity<D, X> const& x, quantity<D, Y> const& y) {
      return x.m_value == y.m_value;
    }


    template <typename D, typename X, typename Y>
    constexpr bool operator!=(quantity<D, X> const& x, quantity<D, Y> const& y) {
      return x.m_value != y.m_value;
    }


    template <typename D, typename X, typename Y>
    constexpr bool operator<(quantity<D, X> const& x, quantity<D, Y> const& y) {
      return x.m_value < y.m_value;
    }


    template <typename D, typename X, typename Y>
    constexpr bool operator<=(quantity<D, X> const& x, quantity<D, Y> const& y) {
      return x.m_value <= y.m_value;
    }


    template <typename D, typename X, typename Y>
    constexpr bool operator>(quantity<D, X> const& x, quantity<D, Y> const& y) {
      return x.m_value > y.m_value;
    }


    template <typename D, typename X, typename Y>
    constexpr bool operator>=(quantity<D, X> const& x, quantity<D, Y> const& y) {
      return x.m_value >= y.m_value;
    }


    template <typename DX, typename X>
    inline constexpr DX dimension(quantity<DX, X> const& q) {
      return q.dimension();
    }


    template <typename DX, typename X>
    inline constexpr X magnitude(quantity<DX, X> const& q) {
      return q.magnitude();
    }

    // The seven SI base units.  These tie our numbers to the real world.

    constexpr quantity<length_d> meter{detail::magnitude_tag, 1.0};
    constexpr quantity<mass_d> kilogram{detail::magnitude_tag, 1.0};
    constexpr quantity<time_interval_d> second{detail::magnitude_tag, 1.0};
    constexpr quantity<electric_current_d> ampere{detail::magnitude_tag, 1.0};
    constexpr quantity<thermodynamic_temperature_d> kelvin{detail::magnitude_tag, 1.0};
    constexpr quantity<amount_of_substance_d> mole{detail::magnitude_tag, 1.0};
    constexpr quantity<luminous_intensity_d> candela{detail::magnitude_tag, 1.0};
    constexpr quantity<hepenergy_d> electronvolt{detail::magnitude_tag, 1.0};

    // The standard SI prefixes.

    constexpr long double quetta = 1e+30L;
    constexpr long double ronna = 1e+27L;
    constexpr long double yotta = 1e+24L;
    constexpr long double zetta = 1e+21L;
    constexpr long double exa = 1e+18L;
    constexpr long double peta = 1e+15L;
    constexpr long double tera = 1e+12L;
    constexpr long double giga = 1e+9L;
    constexpr long double mega = 1e+6L;
    constexpr long double kilo = 1e+3L;
    constexpr long double hecto = 1e+2L;
    constexpr long double deka = 1e+1L;
    constexpr long double deci = 1e-1L;
    constexpr long double centi = 1e-2L;
    constexpr long double milli = 1e-3L;
    constexpr long double micro = 1e-6L;
    constexpr long double nano = 1e-9L;
    constexpr long double pico = 1e-12L;
    constexpr long double femto = 1e-15L;
    constexpr long double atto = 1e-18L;
    constexpr long double zepto = 1e-21L;
    constexpr long double yocto = 1e-24L;
    constexpr long double ronto = 1e-27L;
    constexpr long double quecto = 1e-30L;

    // Binary prefixes, pending adoption.

    constexpr long double kibi = 1024;
    constexpr long double mebi = 1024 * kibi;
    constexpr long double gibi = 1024 * mebi;
    constexpr long double tebi = 1024 * gibi;
    constexpr long double pebi = 1024 * tebi;
    constexpr long double exbi = 1024 * pebi;
    constexpr long double zebi = 1024 * exbi;
    constexpr long double yobi = 1024 * zebi;

    // The rest of the standard dimensional types, as specified in SP811.

    using absorbed_dose_d = dimensions<2, 0, -2>;
    using absorbed_dose_rate_d = dimensions<2, 0, -3>;
    using acceleration_d = dimensions<1, 0, -2>;
    using activity_of_a_nuclide_d = dimensions<0, 0, -1>;
    using angular_velocity_d = dimensions<0, 0, -1>;
    using angular_acceleration_d = dimensions<0, 0, -2>;
    using area_d = dimensions<2, 0, 0>;
    using capacitance_d = dimensions<-2, -1, 4, 2>;
    using concentration_d = dimensions<-3, 0, 0, 0, 0, 1>;
    using current_density_d = dimensions<-2, 0, 0, 1>;
    using dose_equivalent_d = dimensions<2, 0, -2>;
    using dynamic_viscosity_d = dimensions<-1, 1, -1>;
    using electric_charge_d = dimensions<0, 0, 1, 1>;
    using electric_charge_density_d = dimensions<-3, 0, 1, 1>;
    using electric_conductance_d = dimensions<-2, -1, 3, 2>;
    using electric_field_strenth_d = dimensions<1, 1, -3, -1>;
    using electric_flux_density_d = dimensions<-2, 0, 1, 1>;
    using electric_potential_d = dimensions<2, 1, -3, -1>;
    using electric_resistance_d = dimensions<2, 1, -3, -2>;
    using energy_d = dimensions<2, 1, -2>;
    using energy_density_d = dimensions<-1, 1, -2>;
    using exposure_d = dimensions<0, -1, 1, 1>;
    using force_d = dimensions<1, 1, -2>;
    using frequency_d = dimensions<0, 0, -1>;
    using heat_capacity_d = dimensions<2, 1, -2, 0, -1>;
    using heat_density_d = dimensions<0, 1, -2>;
    using heat_density_flow_rate_d = dimensions<0, 1, -3>;
    using heat_flow_rate_d = dimensions<2, 1, -3>;
    using heat_flux_density_d = dimensions<0, 1, -3>;
    using heat_transfer_coefficient_d = dimensions<0, 1, -3, 0, -1>;
    using illuminance_d = dimensions<-2, 0, 0, 0, 0, 0, 1>;
    using inductance_d = dimensions<2, 1, -2, -2>;
    using irradiance_d = dimensions<0, 1, -3>;
    using kinematic_viscosity_d = dimensions<2, 0, -1>;
    using luminance_d = dimensions<-2, 0, 0, 0, 0, 0, 1>;
    using luminous_flux_d = dimensions<0, 0, 0, 0, 0, 0, 1>;
    using magnetic_field_strength_d = dimensions<-1, 0, 0, 1>;
    using magnetic_flux_d = dimensions<2, 1, -2, -1>;
    using magnetic_flux_density_d = dimensions<0, 1, -2, -1>;
    using magnetic_permeability_d = dimensions<1, 1, -2, -2>;
    using mass_density_d = dimensions<-3, 1, 0>;
    using mass_flow_rate_d = dimensions<0, 1, -1>;
    using molar_energy_d = dimensions<2, 1, -2, 0, 0, -1>;
    using molar_entropy_d = dimensions<2, 1, -2, -1, 0, -1>;
    using moment_of_force_d = dimensions<2, 1, -2>;
    using permittivity_d = dimensions<-3, -1, 4, 2>;
    using power_d = dimensions<2, 1, -3>;
    using pressure_d = dimensions<-1, 1, -2>;
    using radiance_d = dimensions<0, 1, -3>;
    using radiant_intensity_d = dimensions<2, 1, -3>;
    using speed_d = dimensions<1, 0, -1>;
    using specific_energy_d = dimensions<2, 0, -2>;
    using specific_heat_capacity_d = dimensions<2, 0, -2, 0, -1>;
    using specific_volume_d = dimensions<3, -1, 0>;
    using substance_permeability_d = dimensions<-1, 0, 1>;
    using surface_tension_d = dimensions<0, 1, -2>;
    using thermal_conductivity_d = dimensions<1, 1, -3, 0, -1>;
    using thermal_diffusivity_d = dimensions<2, 0, -1>;
    using thermal_insulance_d = dimensions<0, -1, 3, 0, 1>;
    using thermal_resistance_d = dimensions<-2, -1, 3, 0, 1>;
    using thermal_resistivity_d = dimensions<-1, -1, 3, 0, 1>;
    using torque_d = dimensions<2, 1, -2>;
    using volume_d = dimensions<3, 0, 0>;
    using volume_flow_rate_d = dimensions<3, 0, -1>;
    using wave_number_d = dimensions<-1, 0, 0>;

    // Handy values.

    constexpr Rep pi{Rep(3.141592653589793238462L)};
    constexpr Rep percent{Rep(1) / 100};

    constexpr quantity<mass_d> gram{kilogram / 1000};

    // The derived SI units, as specified in SP811.

    constexpr Rep radian{Rep(1)};
    constexpr Rep steradian{Rep(1)};
    constexpr quantity<force_d> newton{meter * kilogram / square(second)};
    constexpr quantity<pressure_d> pascal{newton / square(meter)};
    constexpr quantity<energy_d> joule{newton * meter};
    constexpr quantity<power_d> watt{joule / second};
    constexpr quantity<electric_charge_d> coulomb{second * ampere};
    constexpr quantity<electric_potential_d> volt{watt / ampere};
    constexpr quantity<capacitance_d> farad{coulomb / volt};
    constexpr quantity<electric_resistance_d> ohm{volt / ampere};
    constexpr quantity<electric_conductance_d> siemens{ampere / volt};
    constexpr quantity<magnetic_flux_d> weber{volt * second};
    constexpr quantity<magnetic_flux_density_d> tesla{weber / square(meter)};
    constexpr quantity<inductance_d> henry{weber / ampere};
    constexpr quantity<thermodynamic_temperature_d> degree_celsius{kelvin};
    constexpr quantity<luminous_flux_d> lumen{candela * steradian};
    constexpr quantity<illuminance_d> lux{lumen / meter / meter};
    constexpr quantity<activity_of_a_nuclide_d> becquerel{1 / second};
    constexpr quantity<absorbed_dose_d> gray{joule / kilogram};
    constexpr quantity<dose_equivalent_d> sievert{joule / kilogram};
    constexpr quantity<frequency_d> hertz{1 / second};

    // The rest of the units approved for use with SI, as specified in SP811.
    // (However, use of these units is generally discouraged.)

    constexpr quantity<length_d> angstrom{Rep(1e-10L) * meter};
    constexpr quantity<area_d> are{Rep(1e+2L) * square(meter)};
    constexpr quantity<pressure_d> bar{Rep(1e+5L) * pascal};
    constexpr quantity<area_d> barn{Rep(1e-28L) * square(meter)};
    constexpr quantity<activity_of_a_nuclide_d> curie{Rep(3.7e+10L) * becquerel};
    constexpr quantity<time_interval_d> day{Rep(86400L) * second};
    constexpr Rep degree_angle{pi / 180};
    constexpr quantity<acceleration_d> gal{Rep(1e-2L) * meter / square(second)};
    constexpr quantity<area_d> hectare{Rep(1e+4L) * square(meter)};
    constexpr quantity<time_interval_d> hour{Rep(3600) * second};
    constexpr quantity<speed_d> knot{Rep(1852) / 3600 * meter / second};
    constexpr quantity<volume_d> liter{Rep(1e-3L) * cube(meter)};
    constexpr quantity<time_interval_d> minute{Rep(60) * second};
    constexpr Rep minute_angle{pi / 10800};
    constexpr quantity<length_d> mile_nautical{Rep(1852) * meter};
    constexpr quantity<absorbed_dose_d> rad{Rep(1e-2L) * gray};
    constexpr quantity<dose_equivalent_d> rem{Rep(1e-2L) * sievert};
    constexpr quantity<exposure_d> roentgen{Rep(2.58e-4L) * coulomb / kilogram};
    constexpr Rep second_angle{pi / 648000L};
    constexpr quantity<mass_d> ton_metric{Rep(1e+3L) * kilogram};

    // Alternate (non-US) spellings:

    constexpr quantity<length_d> metre{meter};
    constexpr quantity<volume_d> litre{liter};
    constexpr Rep deca{deka};
    constexpr quantity<mass_d> tonne{ton_metric};

    // cooked literals for base units;
    // these could also have been created with a script.

#define QUANTITY_DEFINE_SCALING_LITERAL(sfx, dim, factor)                   \
  constexpr quantity<dim, double> operator"" _##sfx(unsigned long long x) { \
    return quantity<dim, double>(detail::magnitude_tag, factor * x);        \
  }                                                                         \
  constexpr quantity<dim, double> operator"" _##sfx(long double x) {        \
    return quantity<dim, double>(detail::magnitude_tag, factor * x);        \
  }

#define QUANTITY_DEFINE_SCALING_LITERALS(pfx, dim, fact)    \
  QUANTITY_DEFINE_SCALING_LITERAL(Q##pfx, dim, fact* quetta) \
  QUANTITY_DEFINE_SCALING_LITERAL(R##pfx, dim, fact* ronna) \
  QUANTITY_DEFINE_SCALING_LITERAL(Y##pfx, dim, fact* yotta) \
  QUANTITY_DEFINE_SCALING_LITERAL(Z##pfx, dim, fact* zetta) \
  QUANTITY_DEFINE_SCALING_LITERAL(E##pfx, dim, fact* exa)   \
  QUANTITY_DEFINE_SCALING_LITERAL(P##pfx, dim, fact* peta)  \
  QUANTITY_DEFINE_SCALING_LITERAL(T##pfx, dim, fact* tera)  \
  QUANTITY_DEFINE_SCALING_LITERAL(G##pfx, dim, fact* giga)  \
  QUANTITY_DEFINE_SCALING_LITERAL(M##pfx, dim, fact* mega)  \
  QUANTITY_DEFINE_SCALING_LITERAL(k##pfx, dim, fact* kilo)  \
  QUANTITY_DEFINE_SCALING_LITERAL(h##pfx, dim, fact* hecto) \
  QUANTITY_DEFINE_SCALING_LITERAL(da##pfx, dim, fact* deka) \
  QUANTITY_DEFINE_SCALING_LITERAL(pfx, dim, fact * 1)       \
  QUANTITY_DEFINE_SCALING_LITERAL(d##pfx, dim, fact* deci)  \
  QUANTITY_DEFINE_SCALING_LITERAL(c##pfx, dim, fact* centi) \
  QUANTITY_DEFINE_SCALING_LITERAL(m##pfx, dim, fact* milli) \
  QUANTITY_DEFINE_SCALING_LITERAL(u##pfx, dim, fact* micro) \
  QUANTITY_DEFINE_SCALING_LITERAL(n##pfx, dim, fact* nano)  \
  QUANTITY_DEFINE_SCALING_LITERAL(p##pfx, dim, fact* pico)  \
  QUANTITY_DEFINE_SCALING_LITERAL(f##pfx, dim, fact* femto) \
  QUANTITY_DEFINE_SCALING_LITERAL(a##pfx, dim, fact* atto)  \
  QUANTITY_DEFINE_SCALING_LITERAL(z##pfx, dim, fact* zepto) \
  QUANTITY_DEFINE_SCALING_LITERAL(y##pfx, dim, fact* yocto) \
  QUANTITY_DEFINE_SCALING_LITERAL(r##pfx, dim, fact* ronto) \
  QUANTITY_DEFINE_SCALING_LITERAL(q##pfx, dim, fact* quecto)

#define QUANTITY_DEFINE_LITERALS(pfx, dim) QUANTITY_DEFINE_SCALING_LITERALS(pfx, dim, 1)


    namespace literals {

      QUANTITY_DEFINE_SCALING_LITERALS(g, mass_d, 1e-3)

      QUANTITY_DEFINE_LITERALS(m, length_d)
      QUANTITY_DEFINE_LITERALS(s, time_interval_d)
      QUANTITY_DEFINE_LITERALS(A, electric_current_d)
      QUANTITY_DEFINE_LITERALS(K, thermodynamic_temperature_d)
      QUANTITY_DEFINE_LITERALS(mol, amount_of_substance_d)
      QUANTITY_DEFINE_LITERALS(cd, luminous_intensity_d)

    } // namespace literals

  } // namespace units
} // namespace phys

#endif // PHYS_UNITS_QUANTITY_HPP_INCLUDED

/*
 * end of file
 */