I have written a constexpr sin function using c++14 and would like to know what I can do to improve it. I am trying to balance code clarity/maintainability with performance. My goal was to have every part of the algorithm that does not depend on a run-time parameter be evaluated at compile time.

#include <type_traits>
#include <limits>
#include <array>
#include <cmath>
#include <iostream>

template<std::size_t... Is> struct seq{};

template<std::size_t N, std::size_t... Is>
struct gen_seq: gen_seq<N-1, N, Is...> {};

template<std::size_t... Is>
struct gen_seq<0, Is...> : seq<Is...>{};

namespace math{
    template<typename T> constexpr T pi = 3.14159265358979323846264338327;
    template<typename T> constexpr T two_pi = 6.28318530717958647692528676656;
    template<typename T> constexpr T half_pi = pi<T> * 0.5;

    constexpr static double pi_v = pi<double>;
    constexpr static double two_pi_v = two_pi<double>;
    constexpr static double half_pi_v = half_pi<double>;

    template<class T,class dcy = std::decay_t<T>>
    constexpr inline std::enable_if_t<std::is_floating_point<T>::value,dcy> inverse(T value){
        return (value == 0) ? 0.0 : 1.0 / value;
    constexpr inline long double factorial(std::intmax_t const& n){
        if(n==0){return 1;}
        long double result = n;
        for(std::intmax_t i=n-1;i>0;--i){
            result *=i;
        return result;
    constexpr inline std::size_t max_factorial(){
        std::size_t i=0;
        long double d=0;
        while ((d= factorial(i))<std::numeric_limits<long double>::max()){++i;}
        return i;

    template<class base,std::size_t N>
    class trig_coeffs {
        using T = typename base::value_type;
        using array_type = std::array<T,N>;

        template<std::size_t ... NS>
        constexpr static inline array_type _coeffs(seq<NS ...>){
            return {{base::coeff(NS) ...}};
        constexpr static array_type coeffs=_coeffs(gen_seq<N>{});
    template<class base,std::size_t N>
    constexpr typename trig_coeffs<base,N>::array_type trig_coeffs<base,N>::coeffs;

    template<class base,std::size_t N, class dcy = std::decay_t<typename base::value_type>>
    constexpr std::enable_if_t<std::is_floating_point<dcy>::value,dcy>
    _sincos(typename base::value_type x) noexcept{
        using c = trig_coeffs<base,N>;

        if(std::isnan(x) && std::numeric_limits<dcy>::has_quiet_NaN){
            return static_cast<dcy>(std::numeric_limits<dcy>::quiet_NaN());
        else if(std::isinf(x) && std::numeric_limits<dcy>::has_infinity){
            return static_cast<dcy>(std::numeric_limits<dcy>::infinity());
            dcy result = 0.0;//result accumulator
            //do input range mapping
            dcy _x =base::range_reduce(x);
            //taylor series
                const dcy x_2 = _x*_x; //store x^2
                dcy pow = base::initial_condition(_x);
                for(auto&& cf: c::coeffs){
                    result +=  cf * pow;
            return result;
    namespace detail{
        template<class T>
        struct _sin{
            using value_type = T;
            constexpr static inline T coeff(std::size_t n)noexcept  {
                return (n % 2 ? 1 : -1) * inverse(factorial((2 * n)-1));
            constexpr static inline T range_reduce(T x)noexcept{
                T _x = x;
                _x += math::pi<T>;
                _x -= static_cast<std::size_t>(_x / math::two_pi<T>) * math::two_pi<T>;
                _x -= math::pi<T>;
                return _x;
            constexpr static inline T initial_condition(T x)noexcept{
                return x;
            constexpr static inline std::size_t default_N()noexcept{
                return 16;
    template<class T,std::size_t N = detail::_sin<T>::default_N()>
    constexpr inline std::decay_t<T> sin(T x)noexcept{
        return _sincos<detail::_sin<T>,N>(x);


int main(int argc,char** argv){
    double phs =0;
    double stp = math::two_pi_v/100.0;
    for(int i = 0;i<100;++i){
    return 0;

I have posted all of the code needed to compile the example. Live Example

  • 2
    \$\begingroup\$ for sequences generator, STL has std::index_sequence and std::make_index_sequence \$\endgroup\$ – MORTAL Jul 2 '16 at 5:34
  • \$\begingroup\$ For constexpr T pi = 3.14159265358979323846264338327, pi will lose precision whenever T is longer than double. At least you should use L suffix on the literal, so that all built-in types are supported well enough. Similarly in other such places. \$\endgroup\$ – Ruslan Mar 24 '18 at 7:25
  • \$\begingroup\$ It's not really constexpr, see coliru.stacked-crooked.com/a/2ac2f90cd81bc80e \$\endgroup\$ – Ruslan Mar 20 '19 at 14:55
  • \$\begingroup\$ @Ruslan: It's constexpr with C++17, I think, with std::array being constexpr'ified. \$\endgroup\$ – einpoklum Jul 28 '20 at 14:40
  • \$\begingroup\$ @einpoklum good to know, but note the title and tags of the OP. \$\endgroup\$ – Ruslan Jul 28 '20 at 14:47

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