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taylor_series_expansions.h
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#pragma once
#include <cmath>
#include <array>
#include <utility>
namespace taylor
{
namespace implementation
{
template<typename>
struct divide_by_n_p_1_impl;
template<std::size_t ...POS>
struct divide_by_n_p_1_impl<std::index_sequence<POS...> >
{ static constexpr const std::array<double, sizeof...(POS)> value = {1.0/double(POS + 1)...};
};
template<std::size_t ...POS>
constexpr const std::array<double, sizeof...(POS)> divide_by_n_p_1_impl<std::index_sequence<POS...> >::value;
#if 0
template<>
struct divide_by_n_p_1_impl<std::index_sequence<> >
{ static constexpr const std::array<double, 0> value = {};
};
#endif
template<std::size_t SIZE>
struct divide_by_n_p_1
{ typedef divide_by_n_p_1_impl<std::make_index_sequence<SIZE> > type;
};
template<std::size_t SIZE>
std::array<double, SIZE> exp(double _d)
{ std::array<double, SIZE> s;
double d = std::exp(_d);
#if 1
s[0] = d;
auto &r = divide_by_n_p_1<SIZE - 1>::type::value;
for (std::size_t i = 1; i < SIZE; ++i)
{ d *= r[i - 1];
s[i] = d;
}
#else
for (std::size_t i = 0; i < SIZE; ++i)
{ s[i] = d;
d /= i + 1;
}
#endif
return s;
}
template<typename>
struct n_p_1_divided_by_n_p_2_impl;
template<std::size_t ...POS>
struct n_p_1_divided_by_n_p_2_impl<std::index_sequence<POS...> >
{ static constexpr const std::array<double, sizeof...(POS)> value = {double(POS + 1)/double(POS + 2)...};
};
template<std::size_t ...POS>
constexpr const std::array<double, sizeof...(POS)> n_p_1_divided_by_n_p_2_impl<std::index_sequence<POS...> >::value;
template<std::size_t SIZE>
struct n_p_1_divided_by_n_p_2
{ typedef n_p_1_divided_by_n_p_2_impl<std::make_index_sequence<SIZE> > type;
};
template<std::size_t SIZE>
std::array<double, SIZE> log(const double _d)
{ std::array<double, SIZE> s;
const double d1 = 1.0/_d;
double d = d1;
#if 1
s[0] = std::log(_d);
s[1] = d;
auto &r = n_p_1_divided_by_n_p_2<SIZE - 2>::type::value;
for (std::size_t i = 2; i < SIZE; ++i)
{ d *= -d1*r[i - 2];
s[i] = d;
}
#else
s[0] = std::log(_d);
for (std::size_t i = 1; i < SIZE; ++i)
{ s[i] = d;
d *= -d1*i/(i + 1);
}
#endif
return s;
}
static const auto s_dLog10 = 1.0/std::log(10.0);
template<std::size_t SIZE>
std::array<double, SIZE> log10(const double _d)
{ std::array<double, SIZE> s;
const double d1 = 1.0/_d;
double d = d1*s_dLog10;
#if 1
s[0] = std::log10(_d);
s[1] = d;
auto &r = n_p_1_divided_by_n_p_2<SIZE - 2>::type::value;
for (std::size_t i = 2; i < SIZE; ++i)
{ d *= -d1*r[i - 2];
s[i] = d;
}
#else
s[0] = std::log10(_d);
for (std::size_t i = 1; i < SIZE; ++i)
{ s[i] = d;
d *= -d1*i/(i + 1);
}
#endif
return s;
}
template<std::size_t SIZE>
std::array<double, SIZE> sin(const double _d)
{ std::array<double, SIZE> s;
const double ds = std::sin(_d);
const double dc = std::cos(_d);
double d = 1.0;
//0=sin
//1=cos
//2=-sin
//3=-cos
//4=0
#if 1
s[0] = ds;
auto &r = divide_by_n_p_1<SIZE - 1>::type::value;
for (std::size_t i = 1; i < SIZE; ++i)
{ d *= r[i - 1];
s[i] = (i & 1 ? i & 2 ? -dc : dc : i & 2 ? -ds : ds)*d;
}
#else
for (std::size_t i = 0; i < SIZE; ++i)
{ s[i] = (i & 1 ? i & 2 ? -dc : dc : i & 2 ? -ds : ds)*d;
d /= i + 1;
}
#endif
return s;
}
template<std::size_t SIZE>
std::array<double, SIZE> cos(const double _d)
{ std::array<double, SIZE> s;
const double ds = std::sin(_d);
const double dc = std::cos(_d);
double d = 1.0;
//0=cos
//1=-sin
//2=-cos
//3=sin
//4=0
#if 1
s[0] = dc;
auto &r = divide_by_n_p_1<SIZE - 1>::type::value;
for (std::size_t i = 1; i < SIZE; ++i)
{ d *= r[i - 1];
s[i] = (i & 1 ? i & 2 ? ds : -ds : i & 2 ? -dc: dc)*d;
}
#else
for (std::size_t i = 0; i < SIZE; ++i)
{ s[i] = (i & 1 ? i & 2 ? ds : -ds : i & 2 ? -dc: dc)*d;
d /= i + 1;
}
#endif
return s;
}
template<std::size_t SIZE>
std::array<double, SIZE> sinh(const double _d)
{ std::array<double, SIZE> s;
const double ds = std::sinh(_d);
const double dc = std::cosh(_d);
double d = 1.0;
#if 1
s[0] = ds;
auto &r = divide_by_n_p_1<SIZE - 1>::type::value;
for (std::size_t i = 1; i < SIZE; ++i)
{ d *= r[i - 1];
s[i] = (i & 1 ? dc : ds)*d;
}
#else
for (std::size_t i = 0; i < SIZE; ++i)
{ s[i] = (i & 1 ? dc : ds)*d;
d /= i + 1;
}
#endif
return s;
}
template<std::size_t SIZE>
std::array<double, SIZE> cosh(const double _d)
{ std::array<double, SIZE> s;
const double ds = std::sinh(_d);
const double dc = std::cosh(_d);
double d = 1.0;
#if 1
s[0] = dc;
auto &r = divide_by_n_p_1<SIZE - 1>::type::value;
for (std::size_t i = 1; i < SIZE; ++i)
{ d *= r[i - 1];
s[i] = (i & 1 ? ds : dc)*d;
}
#else
for (std::size_t i = 0; i < SIZE; ++i)
{ s[i] = (i & 1 ? ds : dc)*d;
d /= i + 1;
}
#endif
return s;
}
template<std::size_t SIZE>
std::array<double, SIZE> sqrt(const double _d)
{ std::array<double, SIZE> s;
const double d0 = std::sqrt(_d);
const double d1 = 1.0/_d;
double dPow = 0.5;
double d = d0;
//0=x^0.5
//1=0.5*x^-0.5
//2=-0.25*x^-1.5
//3=
#if 1
s[0] = d;
auto &r = divide_by_n_p_1<SIZE - 1>::type::value;
for (std::size_t i = 1; i < SIZE; ++i)
{ d *= d1*dPow*r[i - 1]; // sqrt(_d)/_d*0.5, sqrt(_d)/_d*0.5/_d*-0.5
dPow -= 1.0; // -0.5
s[i] = d; // sqrt(_d),
}
#else
for (std::size_t i = 0; i < SIZE; ++i)
{ s[i] = d; // sqrt(_d),
d *= d1*dPow/(i + 1); // sqrt(_d)/_d*0.5, sqrt(_d)/_d*0.5/_d*-0.5
dPow -= 1.0; // -0.5
}
#endif
return s;
}
template<std::size_t SIZE>
std::array<double, SIZE> cbrt(const double _d)
{ std::array<double, SIZE> s;
const double d0 = std::cbrt(_d);
const double d1 = 1.0/_d;
double dPow = 1.0/3.0;
double d = d0;
//x^1.3
//pow*previous/x
//pow -= 1.0
#if 1
s[0] = d;
auto &r = divide_by_n_p_1<SIZE - 1>::type::value;
for (std::size_t i = 1; i < SIZE; ++i)
{ d *= d1*dPow*r[i - 1];
dPow -= 1.0;
s[i] = d;
}
#else
for (std::size_t i = 0; i < SIZE; ++i)
{ s[i] = d;
d *= d1*dPow/(i + 1);
dPow -= 1.0;
}
#endif
return s;
}
template<std::size_t SIZE>
std::array<double, SIZE> inverse(const double _d)
{ std::array<double, SIZE> s;
const double ds = 1.0/_d;
double d = ds;
//f0=x^-1/0!
//f1=-x^-2/1!
//f2=2x^-3/2!
//f3=-6x^-4/3!
#if 1
s[0] = d;
for (std::size_t i = 1; i < SIZE; ++i)
{ d *= ds;
s[i] = i & 1 ? -d : d;
}
#else
for (std::size_t i = 0; i < SIZE; ++i)
{ s[i] = i & 1 ? -d : d;
d *= ds;
}
#endif
return s;
}
}
}
#if 0
((n & 1) ? ((n & 2) ? -cosh_x0 : cosh_x0) : ((n & 2) ? -sinh_x0 : sinh_x0)) * term
#endif