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// <numeric> -*- C++ -*- // Copyright (C) 2001-2021 Free Software Foundation, Inc. // // This file is part of the GNU ISO C++ Library. This library 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; either version 3, or (at your option) // any later version. // This library 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. // Under Section 7 of GPL version 3, you are granted additional // permissions described in the GCC Runtime Library Exception, version // 3.1, as published by the Free Software Foundation. // You should have received a copy of the GNU General Public License and // a copy of the GCC Runtime Library Exception along with this program; // see the files COPYING3 and COPYING.RUNTIME respectively. If not, see // <http://www.gnu.org/licenses/>. /* * * Copyright (c) 1994 * Hewlett-Packard Company * * Permission to use, copy, modify, distribute and sell this software * and its documentation for any purpose is hereby granted without fee, * provided that the above copyright notice appear in all copies and * that both that copyright notice and this permission notice appear * in supporting documentation. Hewlett-Packard Company makes no * representations about the suitability of this software for any * purpose. It is provided "as is" without express or implied warranty. * * * Copyright (c) 1996,1997 * Silicon Graphics Computer Systems, Inc. * * Permission to use, copy, modify, distribute and sell this software * and its documentation for any purpose is hereby granted without fee, * provided that the above copyright notice appear in all copies and * that both that copyright notice and this permission notice appear * in supporting documentation. Silicon Graphics makes no * representations about the suitability of this software for any * purpose. It is provided "as is" without express or implied warranty. */ /** @file include/numeric * This is a Standard C++ Library header. */ #ifndef _GLIBCXX_NUMERIC #define _GLIBCXX_NUMERIC 1 #pragma GCC system_header #include <bits/c++config.h> #include <bits/stl_iterator_base_types.h> #include <bits/stl_numeric.h> #ifdef _GLIBCXX_PARALLEL # include <parallel/numeric> #endif #if __cplusplus >= 201402L # include <type_traits> # include <bit> # include <ext/numeric_traits.h> #endif #if __cplusplus >= 201703L # include <bits/stl_function.h> #endif #if __cplusplus > 201703L # include <limits> #endif /** * @defgroup numerics Numerics * * Components for performing numeric operations. Includes support for * complex number types, random number generation, numeric (n-at-a-time) * arrays, generalized numeric algorithms, and mathematical special functions. */ namespace std _GLIBCXX_VISIBILITY(default) { _GLIBCXX_BEGIN_NAMESPACE_VERSION #if __cplusplus >= 201402L namespace __detail { // Like std::abs, but supports unsigned types and returns the specified type, // so |std::numeric_limits<_Tp>::min()| is OK if representable in _Res. template<typename _Res, typename _Tp> constexpr _Res __abs_r(_Tp __val) { static_assert(sizeof(_Res) >= sizeof(_Tp), "result type must be at least as wide as the input type"); if (__val >= 0) return __val; #if defined _GLIBCXX_ASSERTIONS && defined _GLIBCXX_HAVE_BUILTIN_IS_CONSTANT_EVALUATED if (!__builtin_is_constant_evaluated()) // overflow already detected in constexpr __glibcxx_assert(__val != __gnu_cxx::__int_traits<_Res>::__min); #endif return -static_cast<_Res>(__val); } template<typename> void __abs_r(bool) = delete; // GCD implementation, using Stein's algorithm template<typename _Tp> constexpr _Tp __gcd(_Tp __m, _Tp __n) { static_assert(is_unsigned<_Tp>::value, "type must be unsigned"); if (__m == 0) return __n; if (__n == 0) return __m; const int __i = std::__countr_zero(__m); __m >>= __i; const int __j = std::__countr_zero(__n); __n >>= __j; const int __k = __i < __j ? __i : __j; // min(i, j) while (true) { if (__m > __n) { _Tp __tmp = __m; __m = __n; __n = __tmp; } __n -= __m; if (__n == 0) return __m << __k; __n >>= std::__countr_zero(__n); } } } // namespace __detail #if __cplusplus >= 201703L #define __cpp_lib_gcd_lcm 201606 // These were used in drafts of SD-6: #define __cpp_lib_gcd 201606 #define __cpp_lib_lcm 201606 /// Greatest common divisor template<typename _Mn, typename _Nn> constexpr common_type_t<_Mn, _Nn> gcd(_Mn __m, _Nn __n) noexcept { static_assert(is_integral_v<_Mn> && is_integral_v<_Nn>, "std::gcd arguments must be integers"); static_assert(_Mn(2) == 2 && _Nn(2) == 2, "std::gcd arguments must not be bool"); using _Ct = common_type_t<_Mn, _Nn>; const _Ct __m2 = __detail::__abs_r<_Ct>(__m); const _Ct __n2 = __detail::__abs_r<_Ct>(__n); return __detail::__gcd<make_unsigned_t<_Ct>>(__m2, __n2); } /// Least common multiple template<typename _Mn, typename _Nn> constexpr common_type_t<_Mn, _Nn> lcm(_Mn __m, _Nn __n) noexcept { static_assert(is_integral_v<_Mn> && is_integral_v<_Nn>, "std::lcm arguments must be integers"); static_assert(_Mn(2) == 2 && _Nn(2) == 2, "std::lcm arguments must not be bool"); using _Ct = common_type_t<_Mn, _Nn>; const _Ct __m2 = __detail::__abs_r<_Ct>(__m); const _Ct __n2 = __detail::__abs_r<_Ct>(__n); if (__m2 == 0 || __n2 == 0) return 0; _Ct __r = __m2 / __detail::__gcd<make_unsigned_t<_Ct>>(__m2, __n2); #if defined _GLIBCXX_HAVE_BUILTIN_IS_CONSTANT_EVALUATED if constexpr (is_signed_v<_Ct>) if (__builtin_is_constant_evaluated()) return __r * __n2; // constant evaluation can detect overflow here. #endif bool __overflow = __builtin_mul_overflow(__r, __n2, &__r); __glibcxx_assert(!__overflow); return __r; } #endif // C++17 #endif // C++14 #if __cplusplus > 201703L // midpoint # define __cpp_lib_interpolate 201902L template<typename _Tp> constexpr enable_if_t<__and_v<is_arithmetic<_Tp>, is_same<remove_cv_t<_Tp>, _Tp>, __not_<is_same<_Tp, bool>>>, _Tp> midpoint(_Tp __a, _Tp __b) noexcept { if constexpr (is_integral_v<_Tp>) { using _Up = make_unsigned_t<_Tp>; int __k = 1; _Up __m = __a; _Up __M = __b; if (__a > __b) { __k = -1; __m = __b; __M = __a; } return __a + __k * _Tp(_Up(__M - __m) / 2); } else // is_floating { constexpr _Tp __lo = numeric_limits<_Tp>::min() * 2; constexpr _Tp __hi = numeric_limits<_Tp>::max() / 2; const _Tp __abs_a = __a < 0 ? -__a : __a; const _Tp __abs_b = __b < 0 ? -__b : __b; if (__abs_a <= __hi && __abs_b <= __hi) [[likely]] return (__a + __b) / 2; // always correctly rounded if (__abs_a < __lo) // not safe to halve __a return __a + __b/2; if (__abs_b < __lo) // not safe to halve __b return __a/2 + __b; return __a/2 + __b/2; // otherwise correctly rounded } } template<typename _Tp> constexpr enable_if_t<is_object_v<_Tp>, _Tp*> midpoint(_Tp* __a, _Tp* __b) noexcept { static_assert( sizeof(_Tp) != 0, "type must be complete" ); return __a + (__b - __a) / 2; } #endif // C++20 #if __cplusplus >= 201703L #if __cplusplus > 201703L #define __cpp_lib_constexpr_numeric 201911L #endif /// @addtogroup numeric_ops /// @{ /** * @brief Calculate reduction of values in a range. * * @param __first Start of range. * @param __last End of range. * @param __init Starting value to add other values to. * @param __binary_op A binary function object. * @return The final sum. * * Reduce the values in the range `[first,last)` using a binary operation. * The initial value is `init`. The values are not necessarily processed * in order. * * This algorithm is similar to `std::accumulate` but is not required to * perform the operations in order from first to last. For operations * that are commutative and associative the result will be the same as * for `std::accumulate`, but for other operations (such as floating point * arithmetic) the result can be different. */ template<typename _InputIterator, typename _Tp, typename _BinaryOperation> _GLIBCXX20_CONSTEXPR _Tp reduce(_InputIterator __first, _InputIterator __last, _Tp __init, _BinaryOperation __binary_op) { using __ref = typename iterator_traits<_InputIterator>::reference; static_assert(is_invocable_r_v<_Tp, _BinaryOperation&, _Tp&, __ref>); static_assert(is_invocable_r_v<_Tp, _BinaryOperation&, __ref, _Tp&>); static_assert(is_invocable_r_v<_Tp, _BinaryOperation&, _Tp&, _Tp&>); static_assert(is_invocable_r_v<_Tp, _BinaryOperation&, __ref, __ref>); if constexpr (__is_random_access_iter<_InputIterator>::value) { while ((__last - __first) >= 4) { _Tp __v1 = __binary_op(__first[0], __first[1]); _Tp __v2 = __binary_op(__first[2], __first[3]); _Tp __v3 = __binary_op(__v1, __v2); __init = __binary_op(__init, __v3); __first += 4; } } for (; __first != __last; ++__first) __init = __binary_op(__init, *__first); return __init; } /** * @brief Calculate reduction of values in a range. * * @param __first Start of range. * @param __last End of range. * @param __init Starting value to add other values to. * @return The final sum. * * Reduce the values in the range `[first,last)` using addition. * Equivalent to calling `std::reduce(first, last, init, std::plus<>())`. */ template<typename _InputIterator, typename _Tp> _GLIBCXX20_CONSTEXPR inline _Tp reduce(_InputIterator __first, _InputIterator __last, _Tp __init) { return std::reduce(__first, __last, std::move(__init), plus<>()); } /** * @brief Calculate reduction of values in a range. * * @param __first Start of range. * @param __last End of range. * @return The final sum. * * Reduce the values in the range `[first,last)` using addition, with * an initial value of `T{}`, where `T` is the iterator's value type. * Equivalent to calling `std::reduce(first, last, T{}, std::plus<>())`. */ template<typename _InputIterator> _GLIBCXX20_CONSTEXPR inline typename iterator_traits<_InputIterator>::value_type reduce(_InputIterator __first, _InputIterator __last) { using value_type = typename iterator_traits<_InputIterator>::value_type; return std::reduce(__first, __last, value_type{}, plus<>()); } /** * @brief Combine elements from two ranges and reduce * * @param __first1 Start of first range. * @param __last1 End of first range. * @param __first2 Start of second range. * @param __init Starting value to add other values to. * @param __binary_op1 The function used to perform reduction. * @param __binary_op2 The function used to combine values from the ranges. * @return The final sum. * * Call `binary_op2(first1[n],first2[n])` for each `n` in `[0,last1-first1)` * and then use `binary_op1` to reduce the values returned by `binary_op2` * to a single value of type `T`. * * The range beginning at `first2` must contain at least `last1-first1` * elements. */ template<typename _InputIterator1, typename _InputIterator2, typename _Tp, typename _BinaryOperation1, typename _BinaryOperation2> _GLIBCXX20_CONSTEXPR _Tp transform_reduce(_InputIterator1 __first1, _InputIterator1 __last1, _InputIterator2 __first2, _Tp __init, _BinaryOperation1 __binary_op1, _BinaryOperation2 __binary_op2) { if constexpr (__and_v<__is_random_access_iter<_InputIterator1>, __is_random_access_iter<_InputIterator2>>) { while ((__last1 - __first1) >= 4) { _Tp __v1 = __binary_op1(__binary_op2(__first1[0], __first2[0]), __binary_op2(__first1[1], __first2[1])); _Tp __v2 = __binary_op1(__binary_op2(__first1[2], __first2[2]), __binary_op2(__first1[3], __first2[3])); _Tp __v3 = __binary_op1(__v1, __v2); __init = __binary_op1(__init, __v3); __first1 += 4; __first2 += 4; } } for (; __first1 != __last1; ++__first1, (void) ++__first2) __init = __binary_op1(__init, __binary_op2(*__first1, *__first2)); return __init; } /** * @brief Combine elements from two ranges and reduce * * @param __first1 Start of first range. * @param __last1 End of first range. * @param __first2 Start of second range. * @param __init Starting value to add other values to. * @return The final sum. * * Call `first1[n]*first2[n]` for each `n` in `[0,last1-first1)` and then * use addition to sum those products to a single value of type `T`. * * The range beginning at `first2` must contain at least `last1-first1` * elements. */ template<typename _InputIterator1, typename _InputIterator2, typename _Tp> _GLIBCXX20_CONSTEXPR inline _Tp transform_reduce(_InputIterator1 __first1, _InputIterator1 __last1, _InputIterator2 __first2, _Tp __init) { return std::transform_reduce(__first1, __last1, __first2, std::move(__init), plus<>(), multiplies<>()); } /** * @brief Transform the elements of a range and reduce * * @param __first Start of range. * @param __last End of range. * @param __init Starting value to add other values to. * @param __binary_op The function used to perform reduction. * @param __unary_op The function used to transform values from the range. * @return The final sum. * * Call `unary_op(first[n])` for each `n` in `[0,last-first)` and then * use `binary_op` to reduce the values returned by `unary_op` * to a single value of type `T`. */ template<typename _InputIterator, typename _Tp, typename _BinaryOperation, typename _UnaryOperation> _GLIBCXX20_CONSTEXPR _Tp transform_reduce(_InputIterator __first, _InputIterator __last, _Tp __init, _BinaryOperation __binary_op, _UnaryOperation __unary_op) { if constexpr (__is_random_access_iter<_InputIterator>::value) { while ((__last - __first) >= 4) { _Tp __v1 = __binary_op(__unary_op(__first[0]), __unary_op(__first[1])); _Tp __v2 = __binary_op(__unary_op(__first[2]), __unary_op(__first[3])); _Tp __v3 = __binary_op(__v1, __v2); __init = __binary_op(__init, __v3); __first += 4; } } for (; __first != __last; ++__first) __init = __binary_op(__init, __unary_op(*__first)); return __init; } /** @brief Output the cumulative sum of one range to a second range * * @param __first Start of input range. * @param __last End of input range. * @param __result Start of output range. * @param __init Initial value. * @param __binary_op Function to perform summation. * @return The end of the output range. * * Write the cumulative sum (aka prefix sum, aka scan) of the input range * to the output range. Each element of the output range contains the * running total of all earlier elements (and the initial value), * using `binary_op` for summation. * * This function generates an "exclusive" scan, meaning the Nth element * of the output range is the sum of the first N-1 input elements, * so the Nth input element is not included. */ template<typename _InputIterator, typename _OutputIterator, typename _Tp, typename _BinaryOperation> _GLIBCXX20_CONSTEXPR _OutputIterator exclusive_scan(_InputIterator __first, _InputIterator __last, _OutputIterator __result, _Tp __init, _BinaryOperation __binary_op) { while (__first != __last) { auto __v = __init; __init = __binary_op(__init, *__first); ++__first; *__result++ = std::move(__v); } return __result; } /** @brief Output the cumulative sum of one range to a second range * * @param __first Start of input range. * @param __last End of input range. * @param __result Start of output range. * @param __init Initial value. * @return The end of the output range. * * Write the cumulative sum (aka prefix sum, aka scan) of the input range * to the output range. Each element of the output range contains the * running total of all earlier elements (and the initial value), * using `std::plus<>` for summation. * * This function generates an "exclusive" scan, meaning the Nth element * of the output range is the sum of the first N-1 input elements, * so the Nth input element is not included. */ template<typename _InputIterator, typename _OutputIterator, typename _Tp> _GLIBCXX20_CONSTEXPR inline _OutputIterator exclusive_scan(_InputIterator __first, _InputIterator __last, _OutputIterator __result, _Tp __init) { return std::exclusive_scan(__first, __last, __result, std::move(__init), plus<>()); } /** @brief Output the cumulative sum of one range to a second range * * @param __first Start of input range. * @param __last End of input range. * @param __result Start of output range. * @param __binary_op Function to perform summation. * @param __init Initial value. * @return The end of the output range. * * Write the cumulative sum (aka prefix sum, aka scan) of the input range * to the output range. Each element of the output range contains the * running total of all earlier elements (and the initial value), * using `binary_op` for summation. * * This function generates an "inclusive" scan, meaning the Nth element * of the output range is the sum of the first N input elements, * so the Nth input element is included. */ template<typename _InputIterator, typename _OutputIterator, typename _BinaryOperation, typename _Tp> _GLIBCXX20_CONSTEXPR _OutputIterator inclusive_scan(_InputIterator __first, _InputIterator __last, _OutputIterator __result, _BinaryOperation __binary_op, _Tp __init) { for (; __first != __last; ++__first) *__result++ = __init = __binary_op(__init, *__first); return __result; } /** @brief Output the cumulative sum of one range to a second range * * @param __first Start of input range. * @param __last End of input range. * @param __result Start of output range. * @param __binary_op Function to perform summation. * @return The end of the output range. * * Write the cumulative sum (aka prefix sum, aka scan) of the input range * to the output range. Each element of the output range contains the * running total of all earlier elements, using `binary_op` for summation. * * This function generates an "inclusive" scan, meaning the Nth element * of the output range is the sum of the first N input elements, * so the Nth input element is included. */ template<typename _InputIterator, typename _OutputIterator, typename _BinaryOperation> _GLIBCXX20_CONSTEXPR _OutputIterator inclusive_scan(_InputIterator __first, _InputIterator __last, _OutputIterator __result, _BinaryOperation __binary_op) { if (__first != __last) { auto __init = *__first; *__result++ = __init; ++__first; if (__first != __last) __result = std::inclusive_scan(__first, __last, __result, __binary_op, std::move(__init)); } return __result; } /** @brief Output the cumulative sum of one range to a second range * * @param __first Start of input range. * @param __last End of input range. * @param __result Start of output range. * @return The end of the output range. * * Write the cumulative sum (aka prefix sum, aka scan) of the input range * to the output range. Each element of the output range contains the * running total of all earlier elements, using `std::plus<>` for summation. * * This function generates an "inclusive" scan, meaning the Nth element * of the output range is the sum of the first N input elements, * so the Nth input element is included. */ template<typename _InputIterator, typename _OutputIterator> _GLIBCXX20_CONSTEXPR inline _OutputIterator inclusive_scan(_InputIterator __first, _InputIterator __last, _OutputIterator __result) { return std::inclusive_scan(__first, __last, __result, plus<>()); } /** @brief Output the cumulative sum of one range to a second range * * @param __first Start of input range. * @param __last End of input range. * @param __result Start of output range. * @param __init Initial value. * @param __binary_op Function to perform summation. * @param __unary_op Function to transform elements of the input range. * @return The end of the output range. * * Write the cumulative sum (aka prefix sum, aka scan) of the input range * to the output range. Each element of the output range contains the * running total of all earlier elements (and the initial value), * using `__unary_op` to transform the input elements * and using `__binary_op` for summation. * * This function generates an "exclusive" scan, meaning the Nth element * of the output range is the sum of the first N-1 input elements, * so the Nth input element is not included. */ template<typename _InputIterator, typename _OutputIterator, typename _Tp, typename _BinaryOperation, typename _UnaryOperation> _GLIBCXX20_CONSTEXPR _OutputIterator transform_exclusive_scan(_InputIterator __first, _InputIterator __last, _OutputIterator __result, _Tp __init, _BinaryOperation __binary_op, _UnaryOperation __unary_op) { while (__first != __last) { auto __v = __init; __init = __binary_op(__init, __unary_op(*__first)); ++__first; *__result++ = std::move(__v); } return __result; } /** @brief Output the cumulative sum of one range to a second range * * @param __first Start of input range. * @param __last End of input range. * @param __result Start of output range. * @param __binary_op Function to perform summation. * @param __unary_op Function to transform elements of the input range. * @param __init Initial value. * @return The end of the output range. * * Write the cumulative sum (aka prefix sum, aka scan) of the input range * to the output range. Each element of the output range contains the * running total of all earlier elements (and the initial value), * using `__unary_op` to transform the input elements * and using `__binary_op` for summation. * * This function generates an "inclusive" scan, meaning the Nth element * of the output range is the sum of the first N input elements, * so the Nth input element is included. */ template<typename _InputIterator, typename _OutputIterator, typename _BinaryOperation, typename _UnaryOperation, typename _Tp> _GLIBCXX20_CONSTEXPR _OutputIterator transform_inclusive_scan(_InputIterator __first, _InputIterator __last, _OutputIterator __result, _BinaryOperation __binary_op, _UnaryOperation __unary_op, _Tp __init) { for (; __first != __last; ++__first) *__result++ = __init = __binary_op(__init, __unary_op(*__first)); return __result; } /** @brief Output the cumulative sum of one range to a second range * * @param __first Start of input range. * @param __last End of input range. * @param __result Start of output range. * @param __binary_op Function to perform summation. * @param __unary_op Function to transform elements of the input range. * @return The end of the output range. * * Write the cumulative sum (aka prefix sum, aka scan) of the input range * to the output range. Each element of the output range contains the * running total of all earlier elements, * using `__unary_op` to transform the input elements * and using `__binary_op` for summation. * * This function generates an "inclusive" scan, meaning the Nth element * of the output range is the sum of the first N input elements, * so the Nth input element is included. */ template<typename _InputIterator, typename _OutputIterator, typename _BinaryOperation, typename _UnaryOperation> _GLIBCXX20_CONSTEXPR _OutputIterator transform_inclusive_scan(_InputIterator __first, _InputIterator __last, _OutputIterator __result, _BinaryOperation __binary_op, _UnaryOperation __unary_op) { if (__first != __last) { auto __init = __unary_op(*__first); *__result++ = __init; ++__first; if (__first != __last) __result = std::transform_inclusive_scan(__first, __last, __result, __binary_op, __unary_op, std::move(__init)); } return __result; } /// @} group numeric_ops #endif // C++17 _GLIBCXX_END_NAMESPACE_VERSION } // namespace std #if __cplusplus >= 201703L // Parallel STL algorithms # if _PSTL_EXECUTION_POLICIES_DEFINED // If <execution> has already been included, pull in implementations # include <pstl/glue_numeric_impl.h> # else // Otherwise just pull in forward declarations # include <pstl/glue_numeric_defs.h> # define _PSTL_NUMERIC_FORWARD_DECLARED 1 # endif // Feature test macro for parallel algorithms # define __cpp_lib_parallel_algorithm 201603L #endif // C++17 #endif /* _GLIBCXX_NUMERIC */