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symbolic
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/*
Copyright (C) 2018-2024 Geoffrey Daniels. https://gpdaniels.com/
This program is free software: you can redistribute it and/or modify
it under the terms of the GNU General Public License as published by
the Free Software Foundation, version 3 of the License only.
This program is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
GNU General Public License for more details.
You should have received a copy of the GNU General Public License
along with this program. If not, see <https://www.gnu.org/licenses/>.
*/
#pragma once
#ifndef GTL_MATH_SYMBOLIC_HPP
#define GTL_MATH_SYMBOLIC_HPP
// Summary: Compile time symbolic differentiation using template metaprogramming. [wip]
#if defined(_MSC_VER)
# pragma warning(push, 0)
#endif
#include <algorithm>
#include <cmath>
#include <string>
#include <vector>
#if defined(_MSC_VER)
# pragma warning(pop)
#endif
namespace gtl {
namespace symbolic {
///////////////////////////////////////////////////////////////////////////
/// Parameters / Variables / Constants.
///////////////////////////////////////////////////////////////////////////
template <
unsigned long long int parameter_index
>
class parameter final {
public:
constexpr inline parameter() = default;
public:
template <
typename scalar_type,
typename value_type,
typename... value_types
>
constexpr static inline scalar_type solve(
value_type value,
value_types... values
) {
if constexpr(parameter_index != 0) {
static_cast<void>(value);
return parameter<parameter_index - 1>::template solve<scalar_type>(values...);
}
else {
(static_cast<void>(values), ...);
return static_cast<scalar_type>(value);
}
}
public:
static inline std::string to_string(
) {
return "param[" + std::to_string(parameter_index) + "]";
}
};
template <
unsigned long long int variable_index
>
class variable final {
public:
constexpr inline variable() = default;
public:
template <
typename scalar_type,
typename value_type,
typename... value_types
>
constexpr static inline scalar_type solve(
value_type value,
value_types... values
) {
if constexpr(variable_index != 0) {
static_cast<void>(value);
return parameter<variable_index - 1>::template solve<scalar_type>(values...);
}
else {
(static_cast<void>(values), ...);
return static_cast<scalar_type>(value);
}
}
public:
static inline std::string to_string(
) {
return "const[" + std::to_string(variable_index) + "]";
}
};
template <
typename constant_type
>
class constant final {
public:
using type = constant_type;
public:
constexpr inline constant() = default;
public:
template <
typename scalar_type,
typename... value_types
>
constexpr static inline scalar_type solve(
value_types... values
) {
(static_cast<void>(values), ...);
return static_cast<scalar_type>(constant_type::value);
}
public:
static inline std::string to_string(
) {
return std::to_string(constant_type::value);
}
};
///////////////////////////////////////////////////////////////////////////
/// Missing Operations.
///////////////////////////////////////////////////////////////////////////
namespace missing_operations {
template <
typename argument_type
>
constexpr inline auto plus(
argument_type value
) {
return +value;
}
template <
typename argument_type
>
constexpr inline auto minus(
argument_type value
) {
return -value;
}
template <
typename... argument_types
>
constexpr inline auto add(
argument_types... values
) {
return (values + ...);
}
template <
typename... argument_types
>
constexpr inline auto subtract(
argument_types... values
) {
return (values - ...);
}
template <
typename... argument_types
>
constexpr inline auto multiply(
argument_types... values
) {
return (values * ...);
}
template <
typename... argument_types
>
constexpr inline auto divide(
argument_types... values
) {
return (values / ...);
}
template <
typename argument_type
>
constexpr int sign(
argument_type value
) {
return (argument_type(0) < value) - (value < argument_type(0));
}
template <
typename argument_type
>
constexpr argument_type abs(
argument_type value
) {
//if (value == argument_type(0)) {
// return argument_type(0);
//}
if (value < argument_type(0)) {
return -value;
}
return value;
}
// https://github.com/elbeno/constexpr/blob/master/src/include/cx_math.h
// https://github.com/a2flo/floor/blob/master/constexpr/const_math.hpp
// https://github.com/kthohr/gcem
//TODO: This does not work for large or small numbers.
template <
typename argument_type
>
constexpr argument_type ceil(
argument_type value
) {
// Casting to fixed point truncates the value, which is ceil(value) for negative values.
// Have to add 1 for positive values unless value is already ceiled.
return static_cast<long long int>(value) + (value == static_cast<argument_type>(static_cast<long long int>(value))) * (value > 0);
}
//TODO: This does not work for large or small numbers.
template <
typename argument_type
>
constexpr argument_type floor(
argument_type value
) {
// Casting to fixed point truncates the value, which is floor(value) for positive values.
// Have to substract 1 for negative values unless value is already floored.
return static_cast<long long int>(value) - (value == static_cast<argument_type>(static_cast<long long int>(value))) * (value < 0);
}
template <
typename argument_type
>
constexpr argument_type min(
argument_type lhs_value,
argument_type rhs_value
) {
if (lhs_value < rhs_value) {
return lhs_value;
}
return rhs_value;
}
template <
typename argument_type
>
constexpr argument_type max(
argument_type lhs_value,
argument_type rhs_value
) {
if (lhs_value < rhs_value) {
return rhs_value;
}
return lhs_value;
}
}
///////////////////////////////////////////////////////////////////////////
/// Constant number types.
///////////////////////////////////////////////////////////////////////////
template <
int constant_value
>
class integer final {
public:
constexpr inline integer() = default;
public:
constexpr static inline int value = constant_value;
};
template <
int constant_value
>
inline constexpr auto operator+(
const integer<constant_value> value
) {
static_cast<void>(value);
return integer<constant_value>{};
}
template <
int constant_value
>
inline constexpr auto operator-(
const integer<constant_value> value
) {
static_cast<void>(value);
return integer<-constant_value>{};
}
template <
int constant_value,
int other_value
>
inline constexpr auto operator+(
const integer<constant_value> lhs,
const integer<other_value> rhs
) {
static_cast<void>(lhs);
static_cast<void>(rhs);
return integer<constant_value + other_value>{};
}
template <
int constant_value,
int other_value
>
inline constexpr auto operator-(
const integer<constant_value> lhs,
const integer<other_value> rhs
) {
static_cast<void>(lhs);
static_cast<void>(rhs);
return integer<constant_value - other_value>{};
}
template <
int constant_value,
int other_value
>
inline constexpr auto operator*(
const integer<constant_value> lhs,
const integer<other_value> rhs
) {
static_cast<void>(lhs);
static_cast<void>(rhs);
return integer<constant_value * other_value>{};
}
template <
int constant_value,
int other_value
>
inline constexpr auto operator/(
const integer<constant_value> lhs,
const integer<other_value> rhs
) {
static_cast<void>(lhs);
static_cast<void>(rhs);
return integer<constant_value / other_value>{};
}
template <
int constant_value
>
auto sign(
const integer<constant_value> value
) {
static_cast<void>(value);
return integer<missing_operations::sign(constant_value)>{};
}
template <
int constant_value
>
auto abs(
const integer<constant_value> value
) {
static_cast<void>(value);
return integer<missing_operations::abs(constant_value)>{};
}
template <
int constant_value,
int other_value
>
auto min(
const integer<constant_value> lhs,
const integer<other_value> rhs
) {
static_cast<void>(lhs);
static_cast<void>(rhs);
return integer<missing_operations::min(constant_value, other_value)>{};
}
template <
int constant_value,
int other_value
>
auto max(
const integer<constant_value> lhs,
const integer<other_value> rhs
) {
static_cast<void>(lhs);
static_cast<void>(rhs);
return integer<missing_operations::max(constant_value, other_value)>{};
}
using minus_three = constant<integer<-3>>;
using minus_two = constant<integer<-2>>;
using minus_one = constant<integer<-1>>;
using zero = constant<integer<0>>;
using one = constant<integer<1>>;
using two = constant<integer<2>>;
using three = constant<integer<3>>;
template<
signed long long int whole,
unsigned long long int decimal,
unsigned long long int exponent
>
class real final {
public:
constexpr static float pow10(int e) {
return e == 0 ? 1.0f : e > 0 ? 10.0f * pow10(e - 1) : 0.1f * pow10(e + 1);
}
constexpr static float log10(int e) {
// TODO: This is definitely not right...
static_cast<void>(e);
return 1;
}
constexpr static const float value = (whole + (decimal / log10(decimal))) * pow10(exponent);
};
template<
bool sign,
unsigned int mantissa,
unsigned int exponent
>
class real_float final {
public:
# if 0
constexpr float abs(float x) { return x<0 ? -x : x; }
constexpr int get_exponent(float x) {
return abs(x) >= 2 ? get_exponent(x / 2) + 1 : abs(x) < 1 ? get_exponent(x * 2) - 1 : 0;
}
// Multiplies a floating point value x by FLT_RADIX raised to power exp.
constexpr float get_scalbn(float value, int exp) {
return exp == 0 ? value : exp > 0 ? get_scalbn(value * 2, exp - 1) : get_scalbn(value / 2, exp + 1);
}
constexpr unsigned get_mantissa(float x) {
// remove hidden 1 and bias the exponent to get integer
return abs(x) < std::numeric_limits<float>::infinity() ? scalbn(scalbn(abs(x), -get_exponent(x)) - 1, 23) : 0;
}
#endif
constexpr static float pow2(int e) {
return e == 0 ? 1.0f : e > 0 ? 2.0f * pow2(e - 1) : 0.5f * pow2(e + 1);
}
constexpr static const float value = (sign ? -1.0f : 1.0f) * mantissa * pow2(exponent);
};
///////////////////////////////////////////////////////////////////////////
/// Template metaprograms.
///////////////////////////////////////////////////////////////////////////
template<
typename lhs_type,
typename rhs_type
>
class is_same_type_test final {
public:
constexpr static const bool value = false;
};
template<
typename type
>
class is_same_type_test<type, type> final {
public:
constexpr static const bool value = true;
};
template<
typename lhs_type,
typename rhs_type
>
constexpr static const bool is_same_type = is_same_type_test<lhs_type, rhs_type>::value;
///////////////////////////////////////////////////////////////////////////
template<
typename type
>
class is_parameter_type_test final {
public:
constexpr static const bool value = false;
};
template<
unsigned long long int parameter_index
>
class is_parameter_type_test<parameter<parameter_index>> final {
public:
constexpr static const bool value = true;
};
template<
typename type
>
constexpr static const bool is_parameter_type = is_parameter_type_test<type>::value;
///////////////////////////////////////////////////////////////////////////
template<
typename type
>
class is_variable_type_test final {
public:
constexpr static const bool value = false;
};
template<
unsigned long long int variable_index
>
class is_variable_type_test<variable<variable_index>> final {
public:
constexpr static const bool value = true;
};
template<
typename type
>
constexpr static const bool is_variable_type = is_variable_type_test<type>::value;
///////////////////////////////////////////////////////////////////////////
template<
typename type
>
class is_constant_type_test final {
public:
constexpr static const bool value = false;
};
template<
typename constant_type
>
class is_constant_type_test<constant<constant_type>> final {
public:
constexpr static const bool value = true;
};
template<
typename type
>
constexpr static const bool is_constant_type = is_constant_type_test<type>::value;
///////////////////////////////////////////////////////////////////////////
template<
typename type
>
constexpr static const bool is_input_type = is_parameter_type<type> || is_variable_type<type> || is_constant_type<type>;
///////////////////////////////////////////////////////////////////////////
template <
typename operation_handler,
typename... argument_types
>
class operation;
template<
typename type
>
class is_operation_type_test final {
public:
constexpr static const bool value = false;
};
template<
typename operation_type,
typename... argument_types
>
class is_operation_type_test<operation<operation_type, argument_types...>> final {
public:
constexpr static const bool value = true;
};
template<
typename operation_type
>
class is_operation_type_test<operation<operation_type>> final {
public:
constexpr static const bool value = false;
};
///////////////////////////////////////////////////////////////////////////
template<
typename type
>
constexpr static const bool is_operation_type = is_operation_type_test<type>::value;
///////////////////////////////////////////////////////////////////////////
template<
typename type
>
constexpr static const bool is_symbolic_type = is_input_type<type> || is_operation_type<type>;
///////////////////////////////////////////////////////////////////////////
/// Operations.
///////////////////////////////////////////////////////////////////////////
template <
typename operation_handler,
typename... argument_types
>
class operation final {
private:
static_assert((is_symbolic_type<argument_types> && ...), "All arguments must be symbolic types.");
public:
constexpr inline operation() = default;
public:
template <
typename scalar_type,
typename... value_types
>
constexpr static inline scalar_type solve(
value_types... values
) {
return operation_handler::template apply<scalar_type>(argument_types::template solve<scalar_type>(values...)...);
}
public:
static inline std::string to_string(
) {
std::string argument_strings[] = { argument_types::to_string()... };
std::string result = std::string(operation_handler::to_string()) + "(";
for (std::size_t argument_index = 0; argument_index < sizeof...(argument_types); ++argument_index) {
if (argument_index > 0) {
result += ",";
}
result += argument_strings[argument_index];
}
result += ")";
return result;
}
};
#define GTL_DEFINE_OPERATION_HANDLER( \
OPERATION, \
FUNCTION \
) \
class OPERATION ## _handler final { \
public: \
constexpr inline OPERATION ## _handler() = default; \
public: \
template < \
typename scalar_type, \
typename... value_types \
> \
constexpr static inline scalar_type apply( \
const value_types... values \
) { \
return static_cast<scalar_type>(FUNCTION); \
} \
public: \
static inline const char* to_string( \
) { \
return #OPERATION; \
} \
}; \
\
template < \
typename... argument_types \
> \
using OPERATION ## _op = operation< \
OPERATION ## _handler, \
argument_types... \
>; \
GTL_DEFINE_OPERATION_HANDLER(plus, missing_operations::plus(values...))
GTL_DEFINE_OPERATION_HANDLER(minus, missing_operations::minus(values...))
GTL_DEFINE_OPERATION_HANDLER(add, missing_operations::add(values...))
GTL_DEFINE_OPERATION_HANDLER(subtract, missing_operations::subtract(values...))
GTL_DEFINE_OPERATION_HANDLER(multiply, missing_operations::multiply(values...))
GTL_DEFINE_OPERATION_HANDLER(divide, missing_operations::divide(values...))
#define GTL_DEFINE_OPERATION_UNARY_OPERATOR( \
OPERATION, \
SYMBOL \
) \
template < \
typename argument_type, \
typename... argument_types \
> \
constexpr inline OPERATION ## _op< \
argument_type \
> \
operator SYMBOL( \
argument_type argument \
) { \
static_cast<void>(argument); \
return OPERATION ## _op< \
argument_type \
>{}; \
} \
#define GTL_DEFINE_OPERATION_NARY_OPERATOR( \
OPERATION, \
SYMBOL \
) \
template < \
typename argument_type, \
typename... argument_types \
> \
constexpr inline OPERATION ## _op< \
argument_type, \
argument_types... \
> \
operator SYMBOL( \
argument_type argument, \
argument_types... arguments \
) { \
static_cast<void>(argument); \
(static_cast<void>(arguments), ...); \
return OPERATION ## _op< \
argument_type, \
argument_types... \
>{}; \
} \
GTL_DEFINE_OPERATION_UNARY_OPERATOR(plus, +)
GTL_DEFINE_OPERATION_UNARY_OPERATOR(minus, -)
GTL_DEFINE_OPERATION_NARY_OPERATOR(add, +)
GTL_DEFINE_OPERATION_NARY_OPERATOR(subtract, -)
GTL_DEFINE_OPERATION_NARY_OPERATOR(multiply, *)
GTL_DEFINE_OPERATION_NARY_OPERATOR(divide, /)
#define GTL_DEFINE_OPERATION_OVERLOAD( \
OPERATION \
) \
template < \
typename... argument_types \
> \
constexpr inline operation< \
OPERATION ## _handler, \
argument_types... \
> OPERATION(const argument_types&... arguments) { \
(static_cast<void>(arguments), ...); \
return operation< \
OPERATION ## _handler, \
argument_types... \
>{}; \
} \
GTL_DEFINE_OPERATION_HANDLER(sign, missing_operations::sign(values...))
GTL_DEFINE_OPERATION_OVERLOAD(sign)
GTL_DEFINE_OPERATION_HANDLER(abs, missing_operations::abs(values...))
GTL_DEFINE_OPERATION_OVERLOAD(abs)
GTL_DEFINE_OPERATION_HANDLER(ceil, std::ceil(values...))
GTL_DEFINE_OPERATION_OVERLOAD(ceil)
GTL_DEFINE_OPERATION_HANDLER(floor, std::floor(values...))
GTL_DEFINE_OPERATION_OVERLOAD(floor)
GTL_DEFINE_OPERATION_HANDLER(exp, std::exp(values...))
GTL_DEFINE_OPERATION_OVERLOAD(exp)
GTL_DEFINE_OPERATION_HANDLER(log, std::log(values...))
GTL_DEFINE_OPERATION_OVERLOAD(log)
GTL_DEFINE_OPERATION_HANDLER(exp2, std::exp2(values...))
GTL_DEFINE_OPERATION_OVERLOAD(exp2)
GTL_DEFINE_OPERATION_HANDLER(log2, std::log2(values...))
GTL_DEFINE_OPERATION_OVERLOAD(log2)
GTL_DEFINE_OPERATION_HANDLER(sin, std::sin(values...))
GTL_DEFINE_OPERATION_OVERLOAD(sin)
GTL_DEFINE_OPERATION_HANDLER(asin, std::asin(values...))
GTL_DEFINE_OPERATION_OVERLOAD(asin)
GTL_DEFINE_OPERATION_HANDLER(cos, std::cos(values...))
GTL_DEFINE_OPERATION_OVERLOAD(cos)
GTL_DEFINE_OPERATION_HANDLER(acos, std::acos(values...))
GTL_DEFINE_OPERATION_OVERLOAD(acos)
GTL_DEFINE_OPERATION_HANDLER(tan, std::tan(values...))
GTL_DEFINE_OPERATION_OVERLOAD(tan)
GTL_DEFINE_OPERATION_HANDLER(atan, std::atan(values...))
GTL_DEFINE_OPERATION_OVERLOAD(atan)
GTL_DEFINE_OPERATION_HANDLER(sinh, std::sinh(values...))
GTL_DEFINE_OPERATION_OVERLOAD(sinh)
GTL_DEFINE_OPERATION_HANDLER(asinh, std::asinh(values...))
GTL_DEFINE_OPERATION_OVERLOAD(asinh)
GTL_DEFINE_OPERATION_HANDLER(cosh, std::cosh(values...))
GTL_DEFINE_OPERATION_OVERLOAD(cosh)
GTL_DEFINE_OPERATION_HANDLER(acosh, std::acosh(values...))
GTL_DEFINE_OPERATION_OVERLOAD(acosh)
GTL_DEFINE_OPERATION_HANDLER(tanh, std::tanh(values...))
GTL_DEFINE_OPERATION_OVERLOAD(tanh)
GTL_DEFINE_OPERATION_HANDLER(atanh, std::atanh(values...))
GTL_DEFINE_OPERATION_OVERLOAD(atanh)
GTL_DEFINE_OPERATION_HANDLER(sqrt, std::sqrt(values...))
GTL_DEFINE_OPERATION_OVERLOAD(sqrt)
GTL_DEFINE_OPERATION_HANDLER(cbrt, std::cbrt(values...))
GTL_DEFINE_OPERATION_OVERLOAD(cbrt)
GTL_DEFINE_OPERATION_HANDLER(min, missing_operations::min(values...))
GTL_DEFINE_OPERATION_OVERLOAD(min)
GTL_DEFINE_OPERATION_HANDLER(max, missing_operations::max(values...))
GTL_DEFINE_OPERATION_OVERLOAD(max)
GTL_DEFINE_OPERATION_HANDLER(hypot, std::hypot(values...))
GTL_DEFINE_OPERATION_OVERLOAD(hypot)
GTL_DEFINE_OPERATION_HANDLER(atan2, std::atan2(values...))
GTL_DEFINE_OPERATION_OVERLOAD(atan2)
GTL_DEFINE_OPERATION_HANDLER(pow, std::pow(values...))
GTL_DEFINE_OPERATION_OVERLOAD(pow)
///////////////////////////////////////////////////////////////////////////
/// Simplification.
///////////////////////////////////////////////////////////////////////////
template <
unsigned long long int parameter_index
>
constexpr auto simplify_expression(
parameter<parameter_index> value
) {
return value;
}
template <
unsigned long long int variable_index
>
constexpr auto simplify_expression(
variable<variable_index> value
) {
return value;
}
template <
typename constant_type
>
constexpr auto simplify_expression(
constant<constant_type> value
) {
return value;
}
///////////////////////////////////////////////////////////////////////////
template<
typename... argument_types
>
class argument_pack;
template<
typename argument_type,
typename... argument_types
>
class argument_pack<argument_type, argument_types...> final {
private:
static_assert(is_symbolic_type<argument_type> && (is_symbolic_type<argument_types> && ...), "All arguments of an argument pack must be symbolic types.");
public:
using first = argument_type;
using remaining = argument_pack<argument_types...>;
public:
static std::vector<std::string> to_strings(
) {
std::vector<std::string> result;
result.push_back(first::to_string());
std::vector<std::string> result_remaining = remaining::to_strings();
result.insert(std::end(result), std::begin(result_remaining), std::end(result_remaining));
return result;
}
};
template<
>
class argument_pack<> final {
public:
static std::vector<std::string> to_strings(
) {
return {};
}
};
template<
typename... argument_types
>
using argument_first = typename argument_pack<argument_types...>::first;
///////////////////////////////////////////////////////////////////////////
template<typename... argument_types>
constexpr inline auto simplifier_stages() {
static_assert((is_symbolic_type<argument_types> && ...), "All arguments must be symbolic types.");
return argument_pack<argument_types...>{};
}
template<typename... argument_types, typename function_stage_type, typename... function_stage_types>
constexpr inline auto simplifier_stages(function_stage_type function_stage, function_stage_types... function_stages) {
static_assert((is_symbolic_type<argument_types> && ...), "All arguments must be symbolic types.");
auto simplified = function_stage();
if constexpr(is_same_type<argument_pack<argument_types...>, decltype(simplified)>) {
return simplifier_stages<argument_types...>(function_stages...);
}
else {
return simplified;
}
}
///////////////////////////////////////////////////////////////////////////
template<typename... function_types>
class function_collection final
: function_types... {
public:
constexpr inline function_collection(
function_types... functions
) : function_types{functions}... {
}
public:
using function_types::operator()...;
};
// Template argument deduction guide.
template<typename... function_types>
function_collection(
function_types... functions
) -> function_collection<function_types...>;
///////////////////////////////////////////////////////////////////////////
template<
typename... argument_types_lhs,
typename... argument_types_rhs,
typename... argument_pack_types
>
constexpr auto merge_argument_packs(
argument_pack<argument_types_lhs...> pack_lhs,
argument_pack<argument_types_rhs...> pack_rhs,
argument_pack_types... packs
) {
static_assert((is_symbolic_type<argument_types_lhs> && ...), "All arguments must be symbolic types.");
static_assert((is_symbolic_type<argument_types_rhs> && ...), "All arguments must be symbolic types.");
static_cast<void>(pack_lhs);
static_cast<void>(pack_rhs);
(static_cast<void>(packs), ...);
return merge_argument_packs(argument_pack<argument_types_lhs..., argument_types_rhs...>{}, argument_pack_types{}...);
}
template<
typename... argument_types
>
constexpr auto merge_argument_packs(
argument_pack<argument_types...> pack
) {
static_assert((is_symbolic_type<argument_types> && ...), "All arguments must be symbolic types.");
static_cast<void>(pack);
return argument_pack<argument_types...>{};
}
constexpr auto merge_argument_packs(
) {
return argument_pack<>{};
}
///////////////////////////////////////////////////////////////////////////
template<
typename... argument_types,
typename... function_types
>
constexpr auto simplify_argument(
function_types... functions
) {
static_assert((is_symbolic_type<argument_types> && ...), "All arguments must be symbolic types.");
return merge_argument_packs(
function_collection{functions...}(
argument_types{}
) ...);
}
template<
typename... argument_types,
typename... function_types
>
constexpr auto simplify_argument_pair(
function_types... functions
) {
static_assert((is_symbolic_type<argument_types> && ...), "All arguments must be symbolic types.");