float16_t.hpp

#ifndef FLOAT16_T_HPP_INCLUDED_OSDIJSALKJS8OU4LKJAFSOIUASFD98U3LJKASFOIJFFDDDDDF
#define FLOAT16_T_HPP_INCLUDED_OSDIJSALKJS8OU4LKJAFSOIUASFD98U3LJKASFOIJFFDDDDDF
//
// inspired by:
// https://github.com/acgessler/half_float
// https://github.com/x448/float16
//
#include <cstdint>
#include <limits>
#include <iostream>
#include <cmath>
#include <bitset>
#include <type_traits>

#ifdef _MSC_VER
#pragma warning( push )
#pragma warning( disable : 4146) // we use this as a bit trick, so no warning pz
#endif

namespace half
{
    // credit goes to David Lin <https://gist.github.com/davll/9679518>
    #ifdef __GNUC__
    constexpr inline int using_gnu_c = 1;
    #else
    constexpr inline int using_gnu_c = 0;
    #endif

    namespace half_private
    {
        constexpr inline std::uint32_t _uint32_sels( std::uint32_t test, std::uint32_t a, std::uint32_t b ) noexcept
        {
            const std::uint32_t mask = ( ( ( std::int32_t )test ) >> 31 );
            const std::uint32_t sel_a = ( a & mask );
            const std::uint32_t sel_b = ( b & ~mask );
            const std::uint32_t result = ( sel_a | sel_b );
            return ( result );
        }

        constexpr inline std::uint32_t _uint32_selb( std::uint32_t mask, std::uint32_t a, std::uint32_t b ) noexcept
        {
            const std::uint32_t sel_a = ( a & mask );
            const std::uint32_t sel_b = ( b & ~mask );
            const std::uint32_t result = ( sel_a | sel_b );
            return ( result );
        }

        constexpr inline std::uint16_t _uint16_sels( std::uint16_t test, std::uint16_t a, std::uint16_t b ) noexcept
        {
            const std::uint16_t mask = ( ( ( int16_t )test ) >> 15 );
            const std::uint16_t sel_a = ( a & mask );
            const std::uint16_t sel_b = ( b & ~mask );
            const std::uint16_t result = ( sel_a | sel_b );
            return ( result );
        }

        constexpr inline std::uint32_t _uint32_cntlz( std::uint32_t x ) noexcept
        {
#ifdef __GNUC__
            //if constexpr( using_gnu_c )
            {
                std::uint32_t is_x_nez_msb = ( -x );
                std::uint32_t nlz = __builtin_clz( x );
                std::uint32_t result = _uint32_sels( is_x_nez_msb, nlz, 0x00000020 );
                return ( result );
            }
#else
            //else
            {
                const std::uint32_t x0 = ( x >> 1 );
                const std::uint32_t x1 = ( x | x0 );
                const std::uint32_t x2 = ( x1 >> 2 );
                const std::uint32_t x3 = ( x1 | x2 );
                const std::uint32_t x4 = ( x3 >> 4 );
                const std::uint32_t x5 = ( x3 | x4 );
                const std::uint32_t x6 = ( x5 >> 8 );
                const std::uint32_t x7 = ( x5 | x6 );
                const std::uint32_t x8 = ( x7 >> 16 );
                const std::uint32_t x9 = ( x7 | x8 );
                const std::uint32_t xA = ( ~x9 );
                const std::uint32_t xB = ( xA >> 1 );
                const std::uint32_t xC = ( xB & 0x55555555 );
                const std::uint32_t xD = ( xA - xC );
                const std::uint32_t xE = ( xD & 0x33333333 );
                const std::uint32_t xF = ( xD >> 2 );
                const std::uint32_t x10 = ( xF & 0x33333333 );
                const std::uint32_t x11 = ( xE + x10 );
                const std::uint32_t x12 = ( x11 >> 4 );
                const std::uint32_t x13 = ( x11 + x12 );
                const std::uint32_t x14 = ( x13 & 0x0f0f0f0f );
                const std::uint32_t x15 = ( x14 >> 8 );
                const std::uint32_t x16 = ( x14 + x15 );
                const std::uint32_t x17 = ( x16 >> 16 );
                const std::uint32_t x18 = ( x16 + x17 );
                const std::uint32_t x19 = ( x18 & 0x0000003f );
                return ( x19 );
            }
#endif // NOT __GNUC__
        }


        constexpr inline std::uint16_t _uint16_cntlz( std::uint16_t x ) noexcept
        {
#ifdef __GNUC__
            #//if constexpr( using_gnu_c )
            {
                std::uint16_t nlz32 = ( std::uint16_t )_uint32_cntlz( ( std::uint32_t )x );
                std::uint32_t nlz = ( nlz32 - 16 );
                return ( nlz );
            }
#else
            //else
            {
                const std::uint16_t x0 = ( x >> 1 );
                const std::uint16_t x1 = ( x | x0 );
                const std::uint16_t x2 = ( x1 >> 2 );
                const std::uint16_t x3 = ( x1 | x2 );
                const std::uint16_t x4 = ( x3 >> 4 );
                const std::uint16_t x5 = ( x3 | x4 );
                const std::uint16_t x6 = ( x5 >> 8 );
                const std::uint16_t x7 = ( x5 | x6 );
                const std::uint16_t x8 = ( ~x7 );
                const std::uint16_t x9 = ( ( x8 >> 1 ) & 0x5555 );
                const std::uint16_t xA = ( x8 - x9 );
                const std::uint16_t xB = ( xA & 0x3333 );
                const std::uint16_t xC = ( ( xA >> 2 ) & 0x3333 );
                const std::uint16_t xD = ( xB + xC );
                const std::uint16_t xE = ( xD >> 4 );
                const std::uint16_t xF = ( ( xD + xE ) & 0x0f0f );
                const std::uint16_t x10 = ( xF >> 8 );
                const std::uint16_t x11 = ( ( xF + x10 ) & 0x001f );
                return ( x11 );
            }
#endif // NOT __GNUC__
        }

    }//namespace half_private


    constexpr inline std::uint16_t float_to_half( std::uint32_t f ) noexcept
    {
        const std::uint32_t one = ( 0x00000001 );
        const std::uint32_t f_s_mask = ( 0x80000000 );
        const std::uint32_t f_e_mask = ( 0x7f800000 );
        const std::uint32_t f_m_mask = ( 0x007fffff );
        const std::uint32_t f_m_hidden_bit = ( 0x00800000 );
        const std::uint32_t f_m_round_bit = ( 0x00001000 );
        const std::uint32_t f_snan_mask = ( 0x7fc00000 );
        const std::uint32_t f_e_pos = ( 0x00000017 );
        const std::uint32_t h_e_pos = ( 0x0000000a );
        const std::uint32_t h_e_mask = ( 0x00007c00 );
        const std::uint32_t h_snan_mask = ( 0x00007e00 );
        const std::uint32_t h_e_mask_value = ( 0x0000001f );
        const std::uint32_t f_h_s_pos_offset = ( 0x00000010 );
        const std::uint32_t f_h_bias_offset = ( 0x00000070 );
        const std::uint32_t f_h_m_pos_offset = ( 0x0000000d );
        const std::uint32_t h_nan_min = ( 0x00007c01 );
        const std::uint32_t f_h_e_biased_flag = ( 0x0000008f );
        const std::uint32_t f_s = ( f & f_s_mask );
        const std::uint32_t f_e = ( f & f_e_mask );
        const std::uint16_t h_s = ( f_s >> f_h_s_pos_offset );
        const std::uint32_t f_m = ( f & f_m_mask );
        const std::uint16_t f_e_amount = ( f_e >> f_e_pos );
        const std::uint32_t f_e_half_bias = ( f_e_amount - f_h_bias_offset );
        const std::uint32_t f_snan = ( f & f_snan_mask );
        const std::uint32_t f_m_round_mask = ( f_m & f_m_round_bit );
        const std::uint32_t f_m_round_offset = ( f_m_round_mask << one );
        const std::uint32_t f_m_rounded = ( f_m + f_m_round_offset );
        const std::uint32_t f_m_denorm_sa = ( one - f_e_half_bias );
        const std::uint32_t f_m_with_hidden = ( f_m_rounded | f_m_hidden_bit );
        const std::uint32_t f_m_denorm = ( f_m_with_hidden >> f_m_denorm_sa );
        const std::uint32_t h_m_denorm = ( f_m_denorm >> f_h_m_pos_offset );
        const std::uint32_t f_m_rounded_overflow = ( f_m_rounded & f_m_hidden_bit );
        const std::uint32_t m_nan = ( f_m >> f_h_m_pos_offset );
        const std::uint32_t h_em_nan = ( h_e_mask | m_nan );
        const std::uint32_t h_e_norm_overflow_offset = ( f_e_half_bias + 1 );
        const std::uint32_t h_e_norm_overflow = ( h_e_norm_overflow_offset << h_e_pos );
        const std::uint32_t h_e_norm = ( f_e_half_bias << h_e_pos );
        const std::uint32_t h_m_norm = ( f_m_rounded >> f_h_m_pos_offset );
        const std::uint32_t h_em_norm = ( h_e_norm | h_m_norm );
        const std::uint32_t is_h_ndenorm_msb = ( f_h_bias_offset - f_e_amount );
        const std::uint32_t is_f_e_flagged_msb = ( f_h_e_biased_flag - f_e_half_bias );
        const std::uint32_t is_h_denorm_msb = ( ~is_h_ndenorm_msb );
        const std::uint32_t is_f_m_eqz_msb = ( f_m - 1 );
        const std::uint32_t is_h_nan_eqz_msb = ( m_nan - 1 );
        const std::uint32_t is_f_inf_msb = ( is_f_e_flagged_msb & is_f_m_eqz_msb );
        const std::uint32_t is_f_nan_underflow_msb = ( is_f_e_flagged_msb & is_h_nan_eqz_msb );
        const std::uint32_t is_e_overflow_msb = ( h_e_mask_value - f_e_half_bias );
        const std::uint32_t is_h_inf_msb = ( is_e_overflow_msb | is_f_inf_msb );
        const std::uint32_t is_f_nsnan_msb = ( f_snan - f_snan_mask );
        const std::uint32_t is_m_norm_overflow_msb = ( -f_m_rounded_overflow );
        const std::uint32_t is_f_snan_msb = ( ~is_f_nsnan_msb );
        const std::uint32_t h_em_overflow_result = half_private::_uint32_sels( is_m_norm_overflow_msb, h_e_norm_overflow, h_em_norm );
        const std::uint32_t h_em_nan_result = half_private::_uint32_sels( is_f_e_flagged_msb, h_em_nan, h_em_overflow_result );
        const std::uint32_t h_em_nan_underflow_result = half_private::_uint32_sels( is_f_nan_underflow_msb, h_nan_min, h_em_nan_result );
        const std::uint32_t h_em_inf_result = half_private::_uint32_sels( is_h_inf_msb, h_e_mask, h_em_nan_underflow_result );
        const std::uint32_t h_em_denorm_result = half_private::_uint32_sels( is_h_denorm_msb, h_m_denorm, h_em_inf_result );
        const std::uint32_t h_em_snan_result = half_private::_uint32_sels( is_f_snan_msb, h_snan_mask, h_em_denorm_result );
        const std::uint32_t h_result = ( h_s | h_em_snan_result );
        return ( std::uint16_t )( h_result );
    }

    constexpr inline std::uint32_t half_to_float( std::uint16_t h ) noexcept
    {
        const std::uint32_t h_e_mask = ( 0x00007c00 );
        const std::uint32_t h_m_mask = ( 0x000003ff );
        const std::uint32_t h_s_mask = ( 0x00008000 );
        const std::uint32_t h_f_s_pos_offset = ( 0x00000010 );
        const std::uint32_t h_f_e_pos_offset = ( 0x0000000d );
        const std::uint32_t h_f_bias_offset = ( 0x0001c000 );
        const std::uint32_t f_e_mask = ( 0x7f800000 );
        const std::uint32_t f_m_mask = ( 0x007fffff );
        const std::uint32_t h_f_e_denorm_bias = ( 0x0000007e );
        const std::uint32_t h_f_m_denorm_sa_bias = ( 0x00000008 );
        const std::uint32_t f_e_pos = ( 0x00000017 );
        const std::uint32_t h_e_mask_minus_one = ( 0x00007bff );
        const std::uint32_t h_e = ( h & h_e_mask );
        const std::uint32_t h_m = ( h & h_m_mask );
        const std::uint32_t h_s = ( h & h_s_mask );
        const std::uint32_t h_e_f_bias = ( h_e + h_f_bias_offset );
        const std::uint32_t h_m_nlz = half_private::_uint32_cntlz( h_m );
        const std::uint32_t f_s = ( h_s << h_f_s_pos_offset );
        const std::uint32_t f_e = ( h_e_f_bias << h_f_e_pos_offset );
        const std::uint32_t f_m = ( h_m << h_f_e_pos_offset );
        const std::uint32_t f_em = ( f_e | f_m );
        const std::uint32_t h_f_m_sa = ( h_m_nlz - h_f_m_denorm_sa_bias );
        const std::uint32_t f_e_denorm_unpacked = ( h_f_e_denorm_bias - h_f_m_sa );
        const std::uint32_t h_f_m = ( h_m << h_f_m_sa );
        const std::uint32_t f_m_denorm = ( h_f_m & f_m_mask );
        const std::uint32_t f_e_denorm = ( f_e_denorm_unpacked << f_e_pos );
        const std::uint32_t f_em_denorm = ( f_e_denorm | f_m_denorm );
        const std::uint32_t f_em_nan = ( f_e_mask | f_m );
        const std::uint32_t is_e_eqz_msb = ( h_e - 1 );
        const std::uint32_t is_m_nez_msb = ( -h_m );
        const std::uint32_t is_e_flagged_msb = ( h_e_mask_minus_one - h_e );
        const std::uint32_t is_zero_msb = ( is_e_eqz_msb & ~is_m_nez_msb );
        const std::uint32_t is_inf_msb = ( is_e_flagged_msb & ~is_m_nez_msb );
        const std::uint32_t is_denorm_msb = ( is_m_nez_msb & is_e_eqz_msb );
        const std::uint32_t is_nan_msb = ( is_e_flagged_msb & is_m_nez_msb );
        const std::uint32_t is_zero = ( ( ( std::int32_t )is_zero_msb ) >> 31 );
        const std::uint32_t f_zero_result = ( f_em & ~is_zero );
        const std::uint32_t f_denorm_result = half_private::_uint32_sels( is_denorm_msb, f_em_denorm, f_zero_result );
        const std::uint32_t f_inf_result = half_private::_uint32_sels( is_inf_msb, f_e_mask, f_denorm_result );
        const std::uint32_t f_nan_result = half_private::_uint32_sels( is_nan_msb, f_em_nan, f_inf_result );
        const std::uint32_t f_result = ( f_s | f_nan_result );
        return ( f_result );
    }

    constexpr inline std::uint16_t half_add( std::uint16_t x, std::uint16_t y ) noexcept
    {
        const std::uint16_t one = ( 0x0001 );
        const std::uint16_t msb_to_lsb_sa = ( 0x000f );
        const std::uint16_t h_s_mask = ( 0x8000 );
        const std::uint16_t h_e_mask = ( 0x7c00 );
        const std::uint16_t h_m_mask = ( 0x03ff );
        const std::uint16_t h_m_msb_mask = ( 0x2000 );
        const std::uint16_t h_m_msb_sa = ( 0x000d );
        const std::uint16_t h_m_hidden = ( 0x0400 );
        const std::uint16_t h_e_pos = ( 0x000a );
        const std::uint16_t h_e_bias_minus_one = ( 0x000e );
        const std::uint16_t h_m_grs_carry = ( 0x4000 );
        const std::uint16_t h_m_grs_carry_pos = ( 0x000e );
        const std::uint16_t h_grs_size = ( 0x0003 );
        const std::uint16_t h_snan = ( 0xfe00 );
        const std::uint16_t h_e_mask_minus_one = ( 0x7bff );
        const std::uint16_t h_grs_round_carry = ( one << h_grs_size );
        const std::uint16_t h_grs_round_mask = ( h_grs_round_carry - one );
        const std::uint16_t x_e = ( x & h_e_mask );
        const std::uint16_t y_e = ( y & h_e_mask );
        const std::uint16_t is_y_e_larger_msb = ( x_e - y_e );
        const std::uint16_t a = half_private::_uint16_sels( is_y_e_larger_msb, y, x );
        const std::uint16_t a_s = ( a & h_s_mask );
        const std::uint16_t a_e = ( a & h_e_mask );
        const std::uint16_t a_m_no_hidden_bit = ( a & h_m_mask );
        const std::uint16_t a_em_no_hidden_bit = ( a_e | a_m_no_hidden_bit );
        const std::uint16_t b = half_private::_uint16_sels( is_y_e_larger_msb, x, y );
        const std::uint16_t b_s = ( b & h_s_mask );
        const std::uint16_t b_e = ( b & h_e_mask );
        const std::uint16_t b_m_no_hidden_bit = ( b & h_m_mask );
        const std::uint16_t b_em_no_hidden_bit = ( b_e | b_m_no_hidden_bit );
        const std::uint16_t is_diff_sign_msb = ( a_s ^ b_s );
        const std::uint16_t is_a_inf_msb = ( h_e_mask_minus_one - a_em_no_hidden_bit );
        const std::uint16_t is_b_inf_msb = ( h_e_mask_minus_one - b_em_no_hidden_bit );
        const std::uint16_t is_undenorm_msb = ( a_e - 1 );
        const std::uint16_t is_undenorm = ( ( ( int16_t )is_undenorm_msb ) >> 15 );
        const std::uint16_t is_both_inf_msb = ( is_a_inf_msb & is_b_inf_msb );
        const std::uint16_t is_invalid_inf_op_msb = ( is_both_inf_msb & b_s );
        const std::uint16_t is_a_e_nez_msb = ( -a_e );
        const std::uint16_t is_b_e_nez_msb = ( -b_e );
        const std::uint16_t is_a_e_nez = ( ( ( int16_t )is_a_e_nez_msb ) >> 15 );
        const std::uint16_t is_b_e_nez = ( ( ( int16_t )is_b_e_nez_msb ) >> 15 );
        const std::uint16_t a_m_hidden_bit = ( is_a_e_nez & h_m_hidden );
        const std::uint16_t b_m_hidden_bit = ( is_b_e_nez & h_m_hidden );
        const std::uint16_t a_m_no_grs = ( a_m_no_hidden_bit | a_m_hidden_bit );
        const std::uint16_t b_m_no_grs = ( b_m_no_hidden_bit | b_m_hidden_bit );
        const std::uint16_t diff_e = ( a_e - b_e );
        const std::uint16_t a_e_unbias = ( a_e - h_e_bias_minus_one );
        const std::uint16_t a_m = ( a_m_no_grs << h_grs_size );
        const std::uint16_t a_e_biased = ( a_e >> h_e_pos );
        const std::uint16_t m_sa_unbias = ( a_e_unbias >> h_e_pos );
        const std::uint16_t m_sa_default = ( diff_e >> h_e_pos );
        const std::uint16_t m_sa_unbias_mask = ( is_a_e_nez_msb & ~is_b_e_nez_msb );
        const std::uint16_t m_sa = half_private::_uint16_sels( m_sa_unbias_mask, m_sa_unbias, m_sa_default );
        const std::uint16_t b_m_no_sticky = ( b_m_no_grs << h_grs_size );
        const std::uint16_t sh_m = ( b_m_no_sticky >> m_sa );
        const std::uint16_t sticky_overflow = ( one << m_sa );
        const std::uint16_t sticky_mask = ( sticky_overflow - 1 );
        const std::uint16_t sticky_collect = ( b_m_no_sticky & sticky_mask );
        const std::uint16_t is_sticky_set_msb = ( -sticky_collect );
        const std::uint16_t sticky = ( is_sticky_set_msb >> msb_to_lsb_sa );
        const std::uint16_t b_m = ( sh_m | sticky );
        const std::uint16_t is_c_m_ab_pos_msb = ( b_m - a_m );
        const std::uint16_t c_inf = ( a_s | h_e_mask );
        const std::uint16_t c_m_sum = ( a_m + b_m );
        const std::uint16_t c_m_diff_ab = ( a_m - b_m );
        const std::uint16_t c_m_diff_ba = ( b_m - a_m );
        const std::uint16_t c_m_smag_diff = half_private::_uint16_sels( is_c_m_ab_pos_msb, c_m_diff_ab, c_m_diff_ba );
        const std::uint16_t c_s_diff = half_private::_uint16_sels( is_c_m_ab_pos_msb, a_s, b_s );
        const std::uint16_t c_s = half_private::_uint16_sels( is_diff_sign_msb, c_s_diff, a_s );
        const std::uint16_t c_m_smag_diff_nlz = half_private::_uint16_cntlz( c_m_smag_diff );
        const std::uint16_t diff_norm_sa = ( c_m_smag_diff_nlz - one );
        const std::uint16_t is_diff_denorm_msb = ( a_e_biased - diff_norm_sa );
        const std::uint16_t is_diff_denorm = ( ( ( int16_t )is_diff_denorm_msb ) >> 15 );
        const std::uint16_t is_a_or_b_norm_msb = ( -a_e_biased );
        const std::uint16_t diff_denorm_sa = ( a_e_biased - 1 );
        const std::uint16_t c_m_diff_denorm = ( c_m_smag_diff << diff_denorm_sa );
        const std::uint16_t c_m_diff_norm = ( c_m_smag_diff << diff_norm_sa );
        const std::uint16_t c_e_diff_norm = ( a_e_biased - diff_norm_sa );
        const std::uint16_t c_m_diff_ab_norm = half_private::_uint16_sels( is_diff_denorm_msb, c_m_diff_denorm, c_m_diff_norm );
        const std::uint16_t c_e_diff_ab_norm = ( c_e_diff_norm & ~is_diff_denorm );
        const std::uint16_t c_m_diff = half_private::_uint16_sels( is_a_or_b_norm_msb, c_m_diff_ab_norm, c_m_smag_diff );
        const std::uint16_t c_e_diff = half_private::_uint16_sels( is_a_or_b_norm_msb, c_e_diff_ab_norm, a_e_biased );
        const std::uint16_t is_diff_eqz_msb = ( c_m_diff - 1 );
        const std::uint16_t is_diff_exactly_zero_msb = ( is_diff_sign_msb & is_diff_eqz_msb );
        const std::uint16_t is_diff_exactly_zero = ( ( ( int16_t )is_diff_exactly_zero_msb ) >> 15 );
        const std::uint16_t c_m_added = half_private::_uint16_sels( is_diff_sign_msb, c_m_diff, c_m_sum );
        const std::uint16_t c_e_added = half_private::_uint16_sels( is_diff_sign_msb, c_e_diff, a_e_biased );
        const std::uint16_t c_m_carry = ( c_m_added & h_m_grs_carry );
        const std::uint16_t is_c_m_carry_msb = ( -c_m_carry );
        const std::uint16_t c_e_hidden_offset = ( ( c_m_added & h_m_grs_carry ) >> h_m_grs_carry_pos );
        const std::uint16_t c_m_sub_hidden = ( c_m_added >> one );
        const std::uint16_t c_m_no_hidden = half_private::_uint16_sels( is_c_m_carry_msb, c_m_sub_hidden, c_m_added );
        const std::uint16_t c_e_no_hidden = ( c_e_added + c_e_hidden_offset );
        const std::uint16_t c_m_no_hidden_msb = ( c_m_no_hidden & h_m_msb_mask );
        const std::uint16_t undenorm_m_msb_odd = ( c_m_no_hidden_msb >> h_m_msb_sa );
        const std::uint16_t undenorm_fix_e = ( is_undenorm & undenorm_m_msb_odd );
        const std::uint16_t c_e_fixed = ( c_e_no_hidden + undenorm_fix_e );
        const std::uint16_t c_m_round_amount = ( c_m_no_hidden & h_grs_round_mask );
        const std::uint16_t c_m_rounded = ( c_m_no_hidden + c_m_round_amount );
        const std::uint16_t c_m_round_overflow = ( ( c_m_rounded & h_m_grs_carry ) >> h_m_grs_carry_pos );
        const std::uint16_t c_e_rounded = ( c_e_fixed + c_m_round_overflow );
        const std::uint16_t c_m_no_grs = ( ( c_m_rounded >> h_grs_size ) & h_m_mask );
        const std::uint16_t c_e = ( c_e_rounded << h_e_pos );
        const std::uint16_t c_em = ( c_e | c_m_no_grs );
        const std::uint16_t c_normal = ( c_s | c_em );
        const std::uint16_t c_inf_result = half_private::_uint16_sels( is_a_inf_msb, c_inf, c_normal );
        const std::uint16_t c_zero_result = ( c_inf_result & ~is_diff_exactly_zero );
        const std::uint16_t c_result = half_private::_uint16_sels( is_invalid_inf_op_msb, h_snan, c_zero_result );
        return ( c_result );
    }

    constexpr inline std::uint16_t half_mul( std::uint16_t x, std::uint16_t y ) noexcept
    {
        const std::uint32_t one = ( 0x00000001 );
        const std::uint32_t h_s_mask = ( 0x00008000 );
        const std::uint32_t h_e_mask = ( 0x00007c00 );
        const std::uint32_t h_m_mask = ( 0x000003ff );
        const std::uint32_t h_m_hidden = ( 0x00000400 );
        const std::uint32_t h_e_pos = ( 0x0000000a );
        const std::uint32_t h_e_bias = ( 0x0000000f );
        const std::uint32_t h_m_bit_count = ( 0x0000000a );
        const std::uint32_t h_m_bit_half_count = ( 0x00000005 );
        const std::uint32_t h_nan_min = ( 0x00007c01 );
        const std::uint32_t h_e_mask_minus_one = ( 0x00007bff );
        const std::uint32_t h_snan = ( 0x0000fe00 );
        const std::uint32_t m_round_overflow_bit = ( 0x00000020 );
        const std::uint32_t m_hidden_bit = ( 0x00100000 );
        const std::uint32_t a_s = ( x & h_s_mask );
        const std::uint32_t b_s = ( y & h_s_mask );
        const std::uint32_t c_s = ( a_s ^ b_s );
        const std::uint32_t x_e = ( x & h_e_mask );
        const std::uint32_t x_e_eqz_msb = ( x_e - 1 );
        const std::uint32_t a = half_private::_uint32_sels( x_e_eqz_msb, y, x );
        const std::uint32_t b = half_private::_uint32_sels( x_e_eqz_msb, x, y );
        const std::uint32_t a_e = ( a & h_e_mask );
        const std::uint32_t b_e = ( b & h_e_mask );
        const std::uint32_t a_m = ( a & h_m_mask );
        const std::uint32_t b_m = ( b & h_m_mask );
        const std::uint32_t a_e_amount = ( a_e >> h_e_pos );
        const std::uint32_t b_e_amount = ( b_e >> h_e_pos );
        const std::uint32_t a_m_with_hidden = ( a_m | h_m_hidden );
        const std::uint32_t b_m_with_hidden = ( b_m | h_m_hidden );
        const std::uint32_t c_m_normal = ( a_m_with_hidden * b_m_with_hidden );
        const std::uint32_t c_m_denorm_biased = ( a_m_with_hidden * b_m );
        const std::uint32_t c_e_denorm_unbias_e = ( h_e_bias - a_e_amount );
        const std::uint32_t c_m_denorm_round_amount = ( c_m_denorm_biased & h_m_mask );
        const std::uint32_t c_m_denorm_rounded = ( c_m_denorm_biased + c_m_denorm_round_amount );
        const std::uint32_t c_m_denorm_inplace = ( c_m_denorm_rounded >> h_m_bit_count );
        const std::uint32_t c_m_denorm_unbiased = ( c_m_denorm_inplace >> c_e_denorm_unbias_e );
        const std::uint32_t c_m_denorm = ( c_m_denorm_unbiased & h_m_mask );
        const std::uint32_t c_e_amount_biased = ( a_e_amount + b_e_amount );
        const std::uint32_t c_e_amount_unbiased = ( c_e_amount_biased - h_e_bias );
        const std::uint32_t is_c_e_unbiased_underflow = ( ( ( std::int32_t )c_e_amount_unbiased ) >> 31 );
        const std::uint32_t c_e_underflow_half_sa = ( -c_e_amount_unbiased );
        const std::uint32_t c_e_underflow_sa = ( c_e_underflow_half_sa << one );
        const std::uint32_t c_m_underflow = ( c_m_normal >> c_e_underflow_sa );
        const std::uint32_t c_e_underflow_added = ( c_e_amount_unbiased & ~is_c_e_unbiased_underflow );
        const std::uint32_t c_m_underflow_added = half_private::_uint32_selb( is_c_e_unbiased_underflow, c_m_underflow, c_m_normal );
        const std::uint32_t is_mul_overflow_test = ( c_e_underflow_added & m_round_overflow_bit );
        const std::uint32_t is_mul_overflow_msb = ( -is_mul_overflow_test );
        const std::uint32_t c_e_norm_radix_corrected = ( c_e_underflow_added + 1 );
        const std::uint32_t c_m_norm_radix_corrected = ( c_m_underflow_added >> one );
        const std::uint32_t c_m_norm_hidden_bit = ( c_m_norm_radix_corrected & m_hidden_bit );
        const std::uint32_t is_c_m_norm_no_hidden_msb = ( c_m_norm_hidden_bit - 1 );
        const std::uint32_t c_m_norm_lo = ( c_m_norm_radix_corrected >> h_m_bit_half_count );
        const std::uint32_t c_m_norm_lo_nlz = half_private::_uint16_cntlz( static_cast<uint16_t>(c_m_norm_lo) );
        const std::uint32_t is_c_m_hidden_nunderflow_msb = ( c_m_norm_lo_nlz - c_e_norm_radix_corrected );
        const std::uint32_t is_c_m_hidden_underflow_msb = ( ~is_c_m_hidden_nunderflow_msb );
        const std::uint32_t is_c_m_hidden_underflow = ( ( ( std::int32_t )is_c_m_hidden_underflow_msb ) >> 31 );
        const std::uint32_t c_m_hidden_underflow_normalized_sa = ( c_m_norm_lo_nlz >> one );
        const std::uint32_t c_m_hidden_underflow_normalized = ( c_m_norm_radix_corrected << c_m_hidden_underflow_normalized_sa );
        const std::uint32_t c_m_hidden_normalized = ( c_m_norm_radix_corrected << c_m_norm_lo_nlz );
        const std::uint32_t c_e_hidden_normalized = ( c_e_norm_radix_corrected - c_m_norm_lo_nlz );
        const std::uint32_t c_e_hidden = ( c_e_hidden_normalized & ~is_c_m_hidden_underflow );
        const std::uint32_t c_m_hidden = half_private::_uint32_sels( is_c_m_hidden_underflow_msb, c_m_hidden_underflow_normalized, c_m_hidden_normalized );
        const std::uint32_t c_m_normalized = half_private::_uint32_sels( is_c_m_norm_no_hidden_msb, c_m_hidden, c_m_norm_radix_corrected );
        const std::uint32_t c_e_normalized = half_private::_uint32_sels( is_c_m_norm_no_hidden_msb, c_e_hidden, c_e_norm_radix_corrected );
        const std::uint32_t c_m_norm_round_amount = ( c_m_normalized & h_m_mask );
        const std::uint32_t c_m_norm_rounded = ( c_m_normalized + c_m_norm_round_amount );
        const std::uint32_t is_round_overflow_test = ( c_e_normalized & m_round_overflow_bit );
        const std::uint32_t is_round_overflow_msb = ( -is_round_overflow_test );
        const std::uint32_t c_m_norm_inplace = ( c_m_norm_rounded >> h_m_bit_count );
        const std::uint32_t c_m = ( c_m_norm_inplace & h_m_mask );
        const std::uint32_t c_e_norm_inplace = ( c_e_normalized << h_e_pos );
        const std::uint32_t c_e = ( c_e_norm_inplace & h_e_mask );
        const std::uint32_t c_em_nan = ( h_e_mask | a_m );
        const std::uint32_t c_nan = ( a_s | c_em_nan );
        const std::uint32_t c_denorm = ( c_s | c_m_denorm );
        const std::uint32_t c_inf = ( c_s | h_e_mask );
        const std::uint32_t c_em_norm = ( c_e | c_m );
        const std::uint32_t is_a_e_flagged_msb = ( h_e_mask_minus_one - a_e );
        const std::uint32_t is_b_e_flagged_msb = ( h_e_mask_minus_one - b_e );
        const std::uint32_t is_a_e_eqz_msb = ( a_e - 1 );
        const std::uint32_t is_a_m_eqz_msb = ( a_m - 1 );
        const std::uint32_t is_b_e_eqz_msb = ( b_e - 1 );
        const std::uint32_t is_b_m_eqz_msb = ( b_m - 1 );
        const std::uint32_t is_b_eqz_msb = ( is_b_e_eqz_msb & is_b_m_eqz_msb );
        const std::uint32_t is_a_eqz_msb = ( is_a_e_eqz_msb & is_a_m_eqz_msb );
        const std::uint32_t is_c_nan_via_a_msb = ( is_a_e_flagged_msb & ~is_b_e_flagged_msb );
        const std::uint32_t is_c_nan_via_b_msb = ( is_b_e_flagged_msb & ~is_b_m_eqz_msb );
        const std::uint32_t is_c_nan_msb = ( is_c_nan_via_a_msb | is_c_nan_via_b_msb );
        const std::uint32_t is_c_denorm_msb = ( is_b_e_eqz_msb & ~is_a_e_flagged_msb );
        const std::uint32_t is_a_inf_msb = ( is_a_e_flagged_msb & is_a_m_eqz_msb );
        const std::uint32_t is_c_snan_msb = ( is_a_inf_msb & is_b_eqz_msb );
        const std::uint32_t is_c_nan_min_via_a_msb = ( is_a_e_flagged_msb & is_b_eqz_msb );
        const std::uint32_t is_c_nan_min_via_b_msb = ( is_b_e_flagged_msb & is_a_eqz_msb );
        const std::uint32_t is_c_nan_min_msb = ( is_c_nan_min_via_a_msb | is_c_nan_min_via_b_msb );
        const std::uint32_t is_c_inf_msb = ( is_a_e_flagged_msb | is_b_e_flagged_msb );
        const std::uint32_t is_overflow_msb = ( is_round_overflow_msb | is_mul_overflow_msb );
        const std::uint32_t c_em_overflow_result = half_private::_uint32_sels( is_overflow_msb, h_e_mask, c_em_norm );
        const std::uint32_t c_common_result = ( c_s | c_em_overflow_result );
        const std::uint32_t c_zero_result = half_private::_uint32_sels( is_b_eqz_msb, c_s, c_common_result );
        const std::uint32_t c_nan_result = half_private::_uint32_sels( is_c_nan_msb, c_nan, c_zero_result );
        const std::uint32_t c_nan_min_result = half_private::_uint32_sels( is_c_nan_min_msb, h_nan_min, c_nan_result );
        const std::uint32_t c_inf_result = half_private::_uint32_sels( is_c_inf_msb, c_inf, c_nan_min_result );
        const std::uint32_t c_denorm_result = half_private::_uint32_sels( is_c_denorm_msb, c_denorm, c_inf_result );
        const std::uint32_t c_result = half_private::_uint32_sels( is_c_snan_msb, h_snan, c_denorm_result );
        return ( std::uint16_t )( c_result );
    }

    constexpr inline std::uint16_t half_neg( std::uint16_t h ) noexcept
    {
        return h ^ 0x8000;
    }

    constexpr inline std::uint16_t half_sub( std::uint16_t ha, std::uint16_t hb ) noexcept
    {
        return half_add( ha, half_neg( hb ) );
    }

}//namespace half


namespace numeric
{
    constexpr inline unsigned long const version = 20210121UL;
    #ifdef DEBUG
    constexpr inline unsigned long const float16_debug_mode = 1;
    #else
    constexpr inline unsigned long const float16_debug_mode = 0;
    #endif

    namespace float16_t_private
    {

        union float16
        {
            std::uint16_t bits_;
            struct
            {
                std::uint16_t frac_ : 10;
                std::uint16_t exp_  : 5;
                std::uint16_t sign_ : 1;
            } ieee_;
        };

        template< typename T >
        constexpr inline float16 to_float16( T val ) noexcept
        {
            static_assert(std::is_integral<T>::value, "Integral required.");
            return float16{ static_cast<std::uint16_t>( val ) };
        }

        inline std::ostream& operator<<( std::ostream& os, float16 const& f16 )
        {
            os << std::bitset<1>( f16.ieee_.sign_ ) << " "
               << std::bitset<5>( f16.ieee_.exp_ ) << " "
               << std::bitset<10>( f16.ieee_.frac_ );
            return os;
        }

        union float32
        {
            std::uint32_t bits_;
            struct
            {
                uint32_t frac_ : 23;
                uint32_t exp_  : 8;
                uint32_t sign_ : 1;
            } ieee_;
            float float_;
        };

        inline std::ostream& operator<<( std::ostream& os, float32 const& f32 )
        {
            os << std::bitset<1>( f32.ieee_.sign_ ) << " "
               << std::bitset<8>( f32.ieee_.exp_ ) << " "
               << std::bitset<23>( f32.ieee_.frac_ ) << " ("
               << f32.float_ << ")";
            return os;
        }

        inline constexpr float16 float32_to_float16( float input ) noexcept
        {
            float32 f32 = {};
            f32.float_ = input;
            float16 f16 = {};
            f16.bits_ = half::float_to_half( f32.bits_ );
            return f16;
        }

        inline constexpr float32 float16_to_float32( std::uint16_t input ) noexcept
        {
            float32 f32{};
            f32.bits_ = half::half_to_float( input );
            return f32;
        }

    }//namespace float16_private

    struct float16_t
    {
        float16_t_private::float16 data_;

#if __cpp_constexpr >= 201907L
        constexpr inline float16_t() noexcept = default;
#else
        inline float16_t() noexcept = default;
#endif
        constexpr inline float16_t( float16_t const& ) noexcept = default;
        constexpr inline float16_t( float16_t&& ) noexcept = default;
        constexpr inline float16_t( float other ) noexcept : data_ { float16_t_private::float32_to_float16( other ) } { }
        constexpr inline float16_t( double other ) noexcept : data_ { float16_t_private::float32_to_float16(static_cast<float>(other)) } { }
        constexpr inline float16_t( int other ) noexcept : data_{ float16_t_private::float32_to_float16(static_cast<float>(other)) } { }
        constexpr inline float16_t( std::uint16_t bits ) noexcept : data_{ bits } { }

        constexpr inline float16_t& operator = ( float16_t const& ) noexcept = default;
        constexpr inline float16_t& operator = ( float16_t&& ) noexcept = default;
        constexpr inline float16_t& operator = ( std::uint16_t bits ) noexcept
        {
            data_.bits_ = bits;
            return *this;
        }
        constexpr inline float16_t& operator = ( float other ) noexcept
        {
            data_ = float16_t_private::float32_to_float16( other );
            return *this;
        }

        constexpr inline operator float() const noexcept
        {
            auto f32 = float16_t_private::float16_to_float32( data_.bits_ );
            return f32.float_;
        }

        constexpr inline operator std::uint16_t() const noexcept
        {
            return data_.bits_;
        }

        constexpr inline float16_t& operator += ( float16_t v ) noexcept
        {
            data_.bits_ = half::half_add( data_.bits_, v.data_.bits_ );
            return *this;
        }

        constexpr inline float16_t& operator -= ( float16_t v ) noexcept
        {
            data_.bits_ = half::half_sub( data_.bits_, v.data_.bits_ );
            return *this;
        }

        constexpr inline float16_t& operator *= ( float16_t v ) noexcept
        {
            data_.bits_ = half::half_mul( data_.bits_, v.data_.bits_ );
            return *this;
        }

        constexpr inline float16_t& operator /= ( float16_t v ) noexcept
        {
            *this = float(*this) / float(v);
            return *this;
        }

        constexpr inline float16_t& operator += ( float v ) noexcept
        {
            auto f16 =  float16_t_private::float32_to_float16( v );
            data_.bits_ = half::half_add( data_.bits_, f16.bits_ );
            return *this;
        }

        constexpr inline float16_t& operator -= ( float v ) noexcept
        {
            auto f16 =  float16_t_private::float32_to_float16( v );
            data_.bits_ = half::half_sub( data_.bits_, f16.bits_ );
            return *this;
        }

        constexpr inline float16_t& operator *= ( float v ) noexcept
        {
            auto f16 =  float16_t_private::float32_to_float16( v );
            data_.bits_ = half::half_mul( data_.bits_, f16.bits_ );
            return *this;
        }

        constexpr inline float16_t& operator /= ( float v ) noexcept
        {
            *this = float(*this) / v;
            return *this;
        }

        constexpr inline float16_t operator -- () noexcept //--f
        {
            *this -= float16_t{static_cast<std::uint16_t>(0x3c00)};
            return *this;
        }

        constexpr inline float16_t operator -- (int) noexcept // f--
        {
            float16_t ans{*this};
            *this -= float16_t{static_cast<std::uint16_t>(0x3c00)};
            return ans;
        }

        constexpr inline float16_t operator ++ () noexcept //++f
        {
            *this += float16_t{static_cast<std::uint16_t>(0x3c00)};
            return *this;
        }

        constexpr inline float16_t operator ++ (int) noexcept // f++
        {
            float16_t ans{*this};
            *this += float16_t{static_cast<std::uint16_t>(0x3c00)};
            return ans;
        }

        constexpr inline float16_t operator - () const noexcept
        {
            return float16_t{ static_cast<std::uint16_t>((data_.bits_ & 0x7fff) | (data_.bits_ ^ 0x8000 )) };
        }

        constexpr inline float16_t operator + () const noexcept
        {
            return *this;
        }

    }; //struct float16_t

    constexpr inline float16_t         fp16_infinity{ static_cast<std::uint16_t>(0x7c00) };
    constexpr inline float16_t         fp16_max{ static_cast<std::uint16_t>(0x7bff) }; //65504
    constexpr inline float16_t         fp16_max_subnormal{ static_cast<std::uint16_t>(0x3ff) }; // 0.00006097
    constexpr inline float16_t         fp16_min{ static_cast<std::uint16_t>(0xfbff) };
    constexpr inline float16_t         fp16_min_positive{ static_cast<std::uint16_t>(0x400) };
    constexpr inline float16_t         fp16_min_positive_subnormal{ static_cast<std::uint16_t>(0x1) };
    constexpr inline float16_t         fp16_nan{ static_cast<std::uint16_t>(0x7e00) };
    constexpr inline float16_t         fp16_infinity_negative{ static_cast<std::uint16_t>(0xfc00) };
    constexpr inline float16_t         fp16_epsilon{ static_cast<std::uint16_t>(0x1400) };

    constexpr inline float16_t         fp16_one{ static_cast<std::uint16_t>(0x3c00) };
    constexpr inline float16_t         fp16_one_negative{ static_cast<std::uint16_t>(0x4000) };
    constexpr inline float16_t         fp16_two{ static_cast<std::uint16_t>(0x4000) };
    constexpr inline float16_t         fp16_two_negative{ static_cast<std::uint16_t>(0xc000) };
    constexpr inline float16_t         fp16_half{ static_cast<std::uint16_t>(0x3800) };
    constexpr inline float16_t         fp16_half_negative{ static_cast<std::uint16_t>(0x3b00) };
    constexpr inline float16_t         fp16_zero{ static_cast<std::uint16_t>(0x0) };
    constexpr inline float16_t         fp16_zero_negative{ static_cast<std::uint16_t>(0x8000) };
    constexpr inline float16_t         fp16_e{ static_cast<std::uint16_t>(0x4170) };
    constexpr inline float16_t         fp16_pi{ static_cast<std::uint16_t>(0x4248) };

    constexpr inline float16_t operator + ( float16_t lhs, float16_t rhs ) noexcept
    {
        float16_t ans{ lhs };
        ans += rhs;
        return ans;
    }

    constexpr inline float16_t operator - ( float16_t lhs, float16_t rhs ) noexcept
    {
        float16_t ans{ lhs };
        ans -= rhs;
        return ans;
    }

    constexpr inline float16_t operator * ( float16_t lhs, float16_t rhs ) noexcept
    {
        float16_t ans{ lhs };
        ans *= rhs;
        return ans;
    }

    constexpr inline float16_t operator / ( float16_t lhs, float16_t rhs ) noexcept
    {
        return float(lhs) / float(rhs);
    }

    constexpr inline bool operator < ( float16_t lhs, float16_t rhs ) noexcept
    {
        auto const& l_ieee = lhs.data_.ieee_;
        auto const& r_ieee = rhs.data_.ieee_;

        if ( l_ieee.sign_ == 1 )
        {
            if ( r_ieee.sign_ == 0 ) return true;
            if ( l_ieee.exp_ > r_ieee.exp_ ) return true;
            if ( l_ieee.exp_ < r_ieee.exp_ ) return false;
            if ( l_ieee.frac_ > r_ieee.frac_ ) return true;
            return false;
        }

        if ( r_ieee.sign_ == 1 ) return false;
        if ( l_ieee.exp_ > r_ieee.exp_ ) return false;
        if ( l_ieee.exp_ < r_ieee.exp_ ) return true;
        if ( l_ieee.frac_ >= r_ieee.frac_ ) return false;
        return true;
    }

    constexpr inline bool operator == ( float16_t lhs, float16_t rhs ) noexcept
    {
        return lhs.data_.bits_ == rhs.data_.bits_;
    }

    constexpr inline bool operator <= ( float16_t lhs, float16_t rhs ) noexcept
    {
        return (lhs < rhs) || (lhs == rhs);
    }

    constexpr inline bool operator > ( float16_t lhs, float16_t rhs ) noexcept
    {
        return !( lhs <= rhs );
    }

    constexpr inline bool operator >= ( float16_t lhs, float16_t rhs ) noexcept
    {
        return !( lhs < rhs );
    }

    constexpr inline bool operator != ( float16_t lhs, float16_t rhs ) noexcept
    {
        return !( lhs == rhs );
    }

    template<typename CharT, class Traits>
    std::basic_ostream<CharT, Traits>& operator << ( std::basic_ostream<CharT, Traits>& os, float16_t const& f )
    {
        std::basic_ostringstream<CharT, Traits> __s;
        __s.flags(os.flags());
        __s.imbue(os.getloc());
        __s.precision(os.precision());

        __s << std::scientific << float(f);
        if constexpr( float16_debug_mode )
        {
            __s << "[0x" << std::hex << f.data_.bits_ << "]";
            __s << "(";
            __s  << std::bitset<1>( f.data_.ieee_.sign_ ) << " ";
            __s  << std::bitset<5>( f.data_.ieee_.exp_ ) << " ";
            __s  << std::bitset<10>( f.data_.ieee_.frac_ ) << ")";
        }

        return os << __s.str();
    }

    template<typename CharT, class Traits>
    std::basic_istream<CharT, Traits>& operator >> ( std::basic_istream<CharT, Traits>& is, float16_t& f )
    {
        bool __fail = true;
        float __v;

        if ( is >> __v )
        {
            __fail = false;
            f = __v;
        }

        if (__fail) is.setstate(std::ios_base::failbit);

        return is;
    }


    //TODO: all functions in <cmath>
    constexpr inline float16_t abs( float16_t f ) noexcept
    {
        float16_t ans{f};
        ans.data_.bits_ &= 0x7fff;
        return ans;
    }

    namespace float16_t_private
    {
        auto constexpr inline make_unary_function = []( auto const& func ) noexcept
        {
            return [func]( float16_t f ) -> float16_t { return func( float(f) ); };
        };

        auto constexpr inline make_binary_function = []( auto const& func ) noexcept
        {
            return [func]( float16_t f1, float16_t f2 ) -> float16_t { return func( float(f1), float(f2) ); };
        };

        auto constexpr inline make_trinary_function = []( auto const& func ) noexcept
        {
            return [func]( float16_t f1, float16_t f2, float16_t f3 ) -> float16_t { return func( float(f1), float(f2), float(f3) ); };
        };

    }//float_t_private

    constexpr inline auto fmod = float16_t_private::make_binary_function( []( float f1, float f2 ) { return std::fmod( f1, f2 ); } );
    constexpr inline auto remainder = float16_t_private::make_binary_function( []( float f1, float f2 ) { return std::remainder( f1, f2 ); } );
    //remquo ??
    constexpr inline auto fma = float16_t_private::make_trinary_function( []( float f1, float f2, float f3 ) { return std::fma( f1, f2, f3 ); } );
    constexpr inline auto fmax = float16_t_private::make_binary_function( []( float f1, float f2 ) { return std::fmax( f1, f2 ); } );
    constexpr inline auto fmin = float16_t_private::make_binary_function( []( float f1, float f2 ) { return std::fmax( f1, f2 ); } );
    constexpr inline auto fdim = float16_t_private::make_binary_function( []( float f1, float f2 ) { return std::fdim( f1, f2 ); } );
    constexpr inline auto lerp = float16_t_private::make_trinary_function( []( float f1, float f2, float f3 ) { return f1 + f3 * (f2 - f1); } );
    constexpr inline auto exp = float16_t_private::make_unary_function( [](float f){ return std::exp(f); } );
    constexpr inline auto exp2 = float16_t_private::make_unary_function( [](float f){ return std::exp2(f); } );
    constexpr inline auto expm1 = float16_t_private::make_unary_function( [](float f){ return std::expm1(f); } );
    constexpr inline auto log = float16_t_private::make_unary_function( [](float f){ return std::log(f); } );
    constexpr inline auto log10 = float16_t_private::make_unary_function( [](float f){ return std::log10(f); } );
    constexpr inline auto log2 = float16_t_private::make_unary_function( [](float f){ return std::log2(f); } );
    constexpr inline auto log1p = float16_t_private::make_unary_function( [](float f){ return std::log1p(f); } );
    constexpr inline auto pow = float16_t_private::make_binary_function( []( float f1, float f2 ) { return std::pow( f1, f2 ); } );
    constexpr inline auto sqrt = float16_t_private::make_unary_function( [](float f){ return std::sqrt(f); } );
    constexpr inline auto cbrt = float16_t_private::make_unary_function( [](float f){ return std::cbrt(f); } );

    inline float16_t hypot( float16_t f1, float16_t f2 )
    {
        return std::hypot( float(f1), float(f2) );
    }

    /* -- Should be valid after c++17 for 3 parameter hypot
    inline float16_t hypot( float16_t f1, float16_t f2, float16_t )
    {
        return std::hypot( float(f1), float(f2), float(f3) );
    }
    */

    constexpr inline auto sin = float16_t_private::make_unary_function( [](float f){ return std::sin(f); } );
    constexpr inline auto sinh = float16_t_private::make_unary_function( [](float f){ return std::sinh(f); } );
    constexpr inline auto cos = float16_t_private::make_unary_function( [](float f){ return std::cos(f); } );
    constexpr inline auto cosh = float16_t_private::make_unary_function( [](float f){ return std::cosh(f); } );
    constexpr inline auto tan = float16_t_private::make_unary_function( [](float f){ return std::tan(f); } );
    constexpr inline auto tanh = float16_t_private::make_unary_function( [](float f){ return std::tanh(f); } );
    constexpr inline auto asin = float16_t_private::make_unary_function( [](float f){ return std::asin(f); } );
    constexpr inline auto asinh = float16_t_private::make_unary_function( [](float f){ return std::asinh(f); } );
    constexpr inline auto acos = float16_t_private::make_unary_function( [](float f){ return std::acos(f); } );
    constexpr inline auto acosh = float16_t_private::make_unary_function( [](float f){ return std::acosh(f); } );
    constexpr inline auto atan = float16_t_private::make_unary_function( [](float f){ return std::atan(f); } );
    constexpr inline auto atanh = float16_t_private::make_unary_function( [](float f){ return std::atanh(f); } );
    constexpr inline auto atan2 = float16_t_private::make_binary_function( [](float f1, float f2){ return std::atan2(f1, f2); } );

    constexpr inline auto erf = float16_t_private::make_unary_function( [](float f){ return std::erf(f); } );
    constexpr inline auto erfc = float16_t_private::make_unary_function( [](float f){ return std::erfc(f); } );
    constexpr inline auto tgamma = float16_t_private::make_unary_function( [](float f){ return std::tgamma(f); } );
    constexpr inline auto lgamma = float16_t_private::make_unary_function( [](float f){ return std::lgamma(f); } );
    constexpr inline auto ceil = float16_t_private::make_unary_function( [](float f){ return std::ceil(f); } );
    constexpr inline auto floor = float16_t_private::make_unary_function( [](float f){ return std::floor(f); } );
    constexpr inline auto trunc = float16_t_private::make_unary_function( [](float f){ return std::trunc(f); } );
    constexpr inline auto round = float16_t_private::make_unary_function( [](float f){ return std::round(f); } );
    constexpr inline auto nearbyint = float16_t_private::make_unary_function( [](float f){ return std::nearbyint(f); } );
    constexpr inline auto rint = float16_t_private::make_unary_function( [](float f){ return std::rint(f); } );

    //Floating point manipulation functions not defined
    //frexp ldexp, modf, scalbn, ilogb

    constexpr inline auto logb = float16_t_private::make_unary_function( [](float f){ return std::logb(f); } );
    constexpr inline auto nextafter = float16_t_private::make_binary_function( [](float f1, float f2){ return std::nextafter(f1, f2); } );
    constexpr inline auto copysign = float16_t_private::make_binary_function( [](float f1, float f2){ return std::copysign(f1, f2); } );

    constexpr inline bool is_nan( float16_t f16 ) noexcept
    {
        return (std::uint16_t(f16) & 0x7fff) > 0x7f80;
    }

    constexpr inline bool is_inf( float16_t f16 ) noexcept
    {
        return (std::uint16_t(f16) & 0x7fff) == 0x7f80;
    }

    constexpr inline bool is_finite( float16_t f16 ) noexcept
    {
        return (std::uint16_t(f16) & 0x7f80) != 0x7f80;
    }

    constexpr inline bool is_normal( float16_t f16 ) noexcept
    {
        auto const exponent = std::uint16_t(f16) & 0x7f80;
        return (exponent != 0x7f80) && (exponent != 0);
    }

    constexpr inline bool is_positive( float16_t f16 ) noexcept
    {
        return ((std::uint16_t(f16)) & 0x8000) == 0;
    }

    constexpr inline bool is_negative( float16_t f16 ) noexcept
    {
        return (std::uint16_t(f16)) & 0x8000;
    }

    //special functions not defined
    // assoc_laguerre, asso_legendre, hermite, legendre, laguerre, sph_bessel, sph_legendre, sph_neumann
    //

#if __STDCPP_MATH_SPEC_FUNCS__ >= 201003L
    constexpr inline auto beta = float16_t_private::make_binary_function( [](float f1, float f2){ return std::beta(f1, f2); } );
    constexpr inline auto comp_ellint_1 = float16_t_private::make_unary_function( [](float f){ return std::comp_ellint_1(f); } );
    constexpr inline auto comp_ellint_2 = float16_t_private::make_unary_function( [](float f){ return std::comp_ellint_2(f); } );
    constexpr inline auto comp_ellint_3 = float16_t_private::make_binary_function( [](float f1, float f2){ return std::comp_ellint_3(f1, f2); } );
    constexpr inline auto cyl_bessel_i = float16_t_private::make_binary_function( [](float f1, float f2){ return std::cyl_bessel_i(f1, f2); } );
    constexpr inline auto cyl_bessel_j = float16_t_private::make_binary_function( [](float f1, float f2){ return std::cyl_bessel_j(f1, f2); } );
    constexpr inline auto cyl_bessel_k = float16_t_private::make_binary_function( [](float f1, float f2){ return std::cyl_bessel_k(f1, f2); } );
    constexpr inline auto cyl_neumann = float16_t_private::make_binary_function( [](float f1, float f2){ return std::cyl_neumann(f1, f2); } );
    constexpr inline auto ellint_1 = float16_t_private::make_binary_function( [](float f1, float f2){ return std::ellint_1(f1, f2); } );
    constexpr inline auto ellint_2 = float16_t_private::make_binary_function( [](float f1, float f2){ return std::ellint_2(f1, f2); } );
    constexpr inline auto ellint_3 = float16_t_private::make_trinary_function( [](float f1, float f2, float f3){ return std::ellint_3(f1, f2, f3); } );
    constexpr inline auto expint = float16_t_private::make_unary_function( [](float f){ return std::expint(f); } );
    constexpr inline auto riemann_zeta = float16_t_private::make_unary_function( [](float f){ return std::riemann_zeta(f); } );
#endif

}//namespace numeric


#if 1
namespace std
{

    template<> struct numeric_limits<numeric::float16_t>
    {
        static constexpr bool is_specialized = true;
        static constexpr numeric::float16_t min() noexcept { return numeric::fp16_min_positive; }
        static constexpr numeric::float16_t max() noexcept { return numeric::fp16_max; }
        static constexpr numeric::float16_t lowest() noexcept { return numeric::fp16_min; }
        static constexpr int digits = 11;
        static constexpr int digits10 = 3;
        static constexpr int max_digits10 = 4;
        static constexpr bool is_signed = true;
        static constexpr bool is_integer = false;
        static constexpr bool is_exact = false;
        static constexpr int radix = std::numeric_limits<float>::radix;
        static constexpr numeric::float16_t epsilon() noexcept { return numeric::fp16_epsilon; }
        static constexpr numeric::float16_t round_error() noexcept { return numeric::fp16_half; }
        static constexpr int min_exponent = -13;
        static constexpr int min_exponent10 = -4;
        static constexpr int max_exponent = 14;
        static constexpr int max_exponent10 = 4;
        static constexpr bool has_infinity = true;
        static constexpr bool has_quiet_NaN = true;
        static constexpr bool has_signaling_NaN = has_quiet_NaN;
        static constexpr std::float_denorm_style has_denorm = denorm_present;
        static constexpr bool has_denorm_loss = false;
        static constexpr numeric::float16_t infinity() noexcept { return numeric::fp16_infinity; }
        static constexpr numeric::float16_t quiet_NaN() noexcept { return numeric::fp16_nan; }
        static constexpr numeric::float16_t signaling_NaN() noexcept { return numeric::fp16_nan; }//TODO: emit signal??
        static constexpr numeric::float16_t denorm_min() noexcept { return numeric::fp16_min_positive_subnormal; }
        static constexpr bool is_iec559 = has_infinity && has_quiet_NaN && has_denorm == denorm_present;
        static constexpr bool is_bounded = true;
        static constexpr bool is_modulo = false;
        static constexpr bool traps = false;
        static constexpr bool tinyness_before = true;
        static constexpr std::float_round_style round_style = round_to_nearest;
    };

    template<> inline constexpr bool is_floating_point_v<numeric::float16_t> = true;
    template<> inline constexpr bool is_arithmetic_v<numeric::float16_t> = true;
    template<> inline constexpr bool is_signed_v<numeric::float16_t> = true;

}

#endif

#endif

#ifdef _MSC_VER
#pragma warning( pop )
#endif