1 // <numeric> -*- C++ -*-
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16 // Under Section 7 of GPL version 3, you are granted additional
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18 // 3.1, as published by the Free Software Foundation.
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22 // see the files COPYING3 and COPYING.RUNTIME respectively. If not, see
23 // <http://www.gnu.org/licenses/>.
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39 * Copyright (c) 1996,1997
40 * Silicon Graphics Computer Systems, Inc.
42 * Permission to use, copy, modify, distribute and sell this software
43 * and its documentation for any purpose is hereby granted without fee,
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51 /** @file include/numeric
52 * This is a Standard C++ Library header.
55 #ifndef _GLIBCXX_NUMERIC
56 #define _GLIBCXX_NUMERIC 1
58 #pragma GCC system_header
60 #include <bits/c++config.h>
61 #include <bits/stl_iterator_base_types.h>
62 #include <bits/stl_numeric.h>
64 #ifdef _GLIBCXX_PARALLEL
65 # include <parallel/numeric>
68 #if __cplusplus >= 201402L
69 # include <type_traits>
73 #if __cplusplus >= 201703L
74 # include <bits/stl_function.h>
77 #if __cplusplus > 201703L
82 * @defgroup numerics Numerics
84 * Components for performing numeric operations. Includes support for
85 * complex number types, random number generation, numeric (n-at-a-time)
86 * arrays, generalized numeric algorithms, and mathematical special functions.
89 namespace std _GLIBCXX_VISIBILITY(default)
91 _GLIBCXX_BEGIN_NAMESPACE_VERSION
93 #if __cplusplus >= 201402L
96 // std::abs is not constexpr, doesn't support unsigned integers,
97 // and std::abs(std::numeric_limits<T>::min()) is undefined.
98 template<typename _Up, typename _Tp>
102 static_assert(is_unsigned<_Up>::value, "result type must be unsigned");
103 static_assert(sizeof(_Up) >= sizeof(_Tp),
104 "result type must be at least as wide as the input type");
105 return __val < 0 ? -(_Up)__val : (_Up)__val;
108 template<typename _Up> void __absu(bool) = delete;
110 // GCD implementation, using Stein's algorithm
111 template<typename _Tp>
113 __gcd(_Tp __m, _Tp __n)
115 static_assert(is_unsigned<_Tp>::value, "type must be unsigned");
122 const int __i = std::__countr_zero(__m);
124 const int __j = std::__countr_zero(__n);
126 const int __k = __i < __j ? __i : __j; // min(i, j)
142 __n >>= std::__countr_zero(__n);
146 // LCM implementation
147 template<typename _Tp>
149 __lcm(_Tp __m, _Tp __n)
151 return (__m != 0 && __n != 0)
152 ? (__m / __detail::__gcd(__m, __n)) * __n
155 } // namespace __detail
157 #if __cplusplus >= 201703L
159 #define __cpp_lib_gcd_lcm 201606
160 // These were used in drafts of SD-6:
161 #define __cpp_lib_gcd 201606
162 #define __cpp_lib_lcm 201606
164 /// Greatest common divisor
165 template<typename _Mn, typename _Nn>
166 constexpr common_type_t<_Mn, _Nn>
167 gcd(_Mn __m, _Nn __n) noexcept
169 static_assert(is_integral_v<_Mn>, "std::gcd arguments must be integers");
170 static_assert(is_integral_v<_Nn>, "std::gcd arguments must be integers");
171 static_assert(_Mn(2) != _Mn(1), "std::gcd arguments must not be bool");
172 static_assert(_Nn(2) != _Nn(1), "std::gcd arguments must not be bool");
173 using _Up = make_unsigned_t<common_type_t<_Mn, _Nn>>;
174 return __detail::__gcd(__detail::__absu<_Up>(__m),
175 __detail::__absu<_Up>(__n));
178 /// Least common multiple
179 template<typename _Mn, typename _Nn>
180 constexpr common_type_t<_Mn, _Nn>
181 lcm(_Mn __m, _Nn __n) noexcept
183 static_assert(is_integral_v<_Mn>, "std::lcm arguments must be integers");
184 static_assert(is_integral_v<_Nn>, "std::lcm arguments must be integers");
185 static_assert(_Mn(2) == 2, "std::lcm arguments must not be bool");
186 static_assert(_Nn(2) == 2, "std::lcm arguments must not be bool");
187 using _Up = make_unsigned_t<common_type_t<_Mn, _Nn>>;
188 return __detail::__lcm(__detail::__absu<_Up>(__m),
189 __detail::__absu<_Up>(__n));
195 #if __cplusplus > 201703L
198 # define __cpp_lib_interpolate 201902L
200 template<typename _Tp>
202 enable_if_t<__and_v<is_arithmetic<_Tp>, is_same<remove_cv_t<_Tp>, _Tp>,
203 __not_<is_same<_Tp, bool>>>,
205 midpoint(_Tp __a, _Tp __b) noexcept
207 if constexpr (is_integral_v<_Tp>)
209 using _Up = make_unsigned_t<_Tp>;
220 return __a + __k * _Tp(_Up(__M - __m) / 2);
224 constexpr _Tp __lo = numeric_limits<_Tp>::min() * 2;
225 constexpr _Tp __hi = numeric_limits<_Tp>::max() / 2;
226 const _Tp __abs_a = __a < 0 ? -__a : __a;
227 const _Tp __abs_b = __b < 0 ? -__b : __b;
228 if (__abs_a <= __hi && __abs_b <= __hi) [[likely]]
229 return (__a + __b) / 2; // always correctly rounded
230 if (__abs_a < __lo) // not safe to halve __a
232 if (__abs_b < __lo) // not safe to halve __b
234 return __a/2 + __b/2; // otherwise correctly rounded
238 template<typename _Tp>
239 constexpr enable_if_t<is_object_v<_Tp>, _Tp*>
240 midpoint(_Tp* __a, _Tp* __b) noexcept
242 static_assert( sizeof(_Tp) != 0, "type must be complete" );
243 return __a + (__b - __a) / 2;
247 #if __cplusplus >= 201703L
249 #if __cplusplus > 201703L
250 #define __cpp_lib_constexpr_numeric 201911L
253 /// @addtogroup numeric_ops
257 * @brief Calculate reduction of values in a range.
259 * @param __first Start of range.
260 * @param __last End of range.
261 * @param __init Starting value to add other values to.
262 * @param __binary_op A binary function object.
263 * @return The final sum.
265 * Reduce the values in the range `[first,last)` using a binary operation.
266 * The initial value is `init`. The values are not necessarily processed
269 * This algorithm is similar to `std::accumulate` but is not required to
270 * perform the operations in order from first to last. For operations
271 * that are commutative and associative the result will be the same as
272 * for `std::accumulate`, but for other operations (such as floating point
273 * arithmetic) the result can be different.
275 template<typename _InputIterator, typename _Tp, typename _BinaryOperation>
278 reduce(_InputIterator __first, _InputIterator __last, _Tp __init,
279 _BinaryOperation __binary_op)
281 using __ref = typename iterator_traits<_InputIterator>::reference;
282 static_assert(is_invocable_r_v<_Tp, _BinaryOperation&, _Tp&, __ref>);
283 static_assert(is_invocable_r_v<_Tp, _BinaryOperation&, __ref, _Tp&>);
284 static_assert(is_invocable_r_v<_Tp, _BinaryOperation&, _Tp&, _Tp&>);
285 static_assert(is_invocable_r_v<_Tp, _BinaryOperation&, __ref, __ref>);
286 if constexpr (__is_random_access_iter<_InputIterator>::value)
288 while ((__last - __first) >= 4)
290 _Tp __v1 = __binary_op(__first[0], __first[1]);
291 _Tp __v2 = __binary_op(__first[2], __first[3]);
292 _Tp __v3 = __binary_op(__v1, __v2);
293 __init = __binary_op(__init, __v3);
297 for (; __first != __last; ++__first)
298 __init = __binary_op(__init, *__first);
303 * @brief Calculate reduction of values in a range.
305 * @param __first Start of range.
306 * @param __last End of range.
307 * @param __init Starting value to add other values to.
308 * @return The final sum.
310 * Reduce the values in the range `[first,last)` using addition.
311 * Equivalent to calling `std::reduce(first, last, init, std::plus<>())`.
313 template<typename _InputIterator, typename _Tp>
316 reduce(_InputIterator __first, _InputIterator __last, _Tp __init)
317 { return std::reduce(__first, __last, std::move(__init), plus<>()); }
320 * @brief Calculate reduction of values in a range.
322 * @param __first Start of range.
323 * @param __last End of range.
324 * @return The final sum.
326 * Reduce the values in the range `[first,last)` using addition, with
327 * an initial value of `T{}`, where `T` is the iterator's value type.
328 * Equivalent to calling `std::reduce(first, last, T{}, std::plus<>())`.
330 template<typename _InputIterator>
332 inline typename iterator_traits<_InputIterator>::value_type
333 reduce(_InputIterator __first, _InputIterator __last)
335 using value_type = typename iterator_traits<_InputIterator>::value_type;
336 return std::reduce(__first, __last, value_type{}, plus<>());
340 * @brief Combine elements from two ranges and reduce
342 * @param __first1 Start of first range.
343 * @param __last1 End of first range.
344 * @param __first2 Start of second range.
345 * @param __init Starting value to add other values to.
346 * @param __binary_op1 The function used to perform reduction.
347 * @param __binary_op2 The function used to combine values from the ranges.
348 * @return The final sum.
350 * Call `binary_op2(first1[n],first2[n])` for each `n` in `[0,last1-first1)`
351 * and then use `binary_op1` to reduce the values returned by `binary_op2`
352 * to a single value of type `T`.
354 * The range beginning at `first2` must contain at least `last1-first1`
357 template<typename _InputIterator1, typename _InputIterator2, typename _Tp,
358 typename _BinaryOperation1, typename _BinaryOperation2>
361 transform_reduce(_InputIterator1 __first1, _InputIterator1 __last1,
362 _InputIterator2 __first2, _Tp __init,
363 _BinaryOperation1 __binary_op1,
364 _BinaryOperation2 __binary_op2)
366 if constexpr (__and_v<__is_random_access_iter<_InputIterator1>,
367 __is_random_access_iter<_InputIterator2>>)
369 while ((__last1 - __first1) >= 4)
371 _Tp __v1 = __binary_op1(__binary_op2(__first1[0], __first2[0]),
372 __binary_op2(__first1[1], __first2[1]));
373 _Tp __v2 = __binary_op1(__binary_op2(__first1[2], __first2[2]),
374 __binary_op2(__first1[3], __first2[3]));
375 _Tp __v3 = __binary_op1(__v1, __v2);
376 __init = __binary_op1(__init, __v3);
381 for (; __first1 != __last1; ++__first1, (void) ++__first2)
382 __init = __binary_op1(__init, __binary_op2(*__first1, *__first2));
387 * @brief Combine elements from two ranges and reduce
389 * @param __first1 Start of first range.
390 * @param __last1 End of first range.
391 * @param __first2 Start of second range.
392 * @param __init Starting value to add other values to.
393 * @return The final sum.
395 * Call `first1[n]*first2[n]` for each `n` in `[0,last1-first1)` and then
396 * use addition to sum those products to a single value of type `T`.
398 * The range beginning at `first2` must contain at least `last1-first1`
401 template<typename _InputIterator1, typename _InputIterator2, typename _Tp>
404 transform_reduce(_InputIterator1 __first1, _InputIterator1 __last1,
405 _InputIterator2 __first2, _Tp __init)
407 return std::transform_reduce(__first1, __last1, __first2,
409 plus<>(), multiplies<>());
413 * @brief Transform the elements of a range and reduce
415 * @param __first Start of range.
416 * @param __last End of range.
417 * @param __init Starting value to add other values to.
418 * @param __binary_op The function used to perform reduction.
419 * @param __unary_op The function used to transform values from the range.
420 * @return The final sum.
422 * Call `unary_op(first[n])` for each `n` in `[0,last-first)` and then
423 * use `binary_op` to reduce the values returned by `unary_op`
424 * to a single value of type `T`.
426 template<typename _InputIterator, typename _Tp,
427 typename _BinaryOperation, typename _UnaryOperation>
430 transform_reduce(_InputIterator __first, _InputIterator __last, _Tp __init,
431 _BinaryOperation __binary_op, _UnaryOperation __unary_op)
433 if constexpr (__is_random_access_iter<_InputIterator>::value)
435 while ((__last - __first) >= 4)
437 _Tp __v1 = __binary_op(__unary_op(__first[0]),
438 __unary_op(__first[1]));
439 _Tp __v2 = __binary_op(__unary_op(__first[2]),
440 __unary_op(__first[3]));
441 _Tp __v3 = __binary_op(__v1, __v2);
442 __init = __binary_op(__init, __v3);
446 for (; __first != __last; ++__first)
447 __init = __binary_op(__init, __unary_op(*__first));
451 /** @brief Output the cumulative sum of one range to a second range
453 * @param __first Start of input range.
454 * @param __last End of input range.
455 * @param __result Start of output range.
456 * @param __init Initial value.
457 * @param __binary_op Function to perform summation.
458 * @return The end of the output range.
460 * Write the cumulative sum (aka prefix sum, aka scan) of the input range
461 * to the output range. Each element of the output range contains the
462 * running total of all earlier elements (and the initial value),
463 * using `binary_op` for summation.
465 * This function generates an "exclusive" scan, meaning the Nth element
466 * of the output range is the sum of the first N-1 input elements,
467 * so the Nth input element is not included.
469 template<typename _InputIterator, typename _OutputIterator, typename _Tp,
470 typename _BinaryOperation>
473 exclusive_scan(_InputIterator __first, _InputIterator __last,
474 _OutputIterator __result, _Tp __init,
475 _BinaryOperation __binary_op)
477 while (__first != __last)
480 __init = __binary_op(__init, *__first);
482 *__result++ = std::move(__v);
487 /** @brief Output the cumulative sum of one range to a second range
489 * @param __first Start of input range.
490 * @param __last End of input range.
491 * @param __result Start of output range.
492 * @param __init Initial value.
493 * @return The end of the output range.
495 * Write the cumulative sum (aka prefix sum, aka scan) of the input range
496 * to the output range. Each element of the output range contains the
497 * running total of all earlier elements (and the initial value),
498 * using `std::plus<>` for summation.
500 * This function generates an "exclusive" scan, meaning the Nth element
501 * of the output range is the sum of the first N-1 input elements,
502 * so the Nth input element is not included.
504 template<typename _InputIterator, typename _OutputIterator, typename _Tp>
506 inline _OutputIterator
507 exclusive_scan(_InputIterator __first, _InputIterator __last,
508 _OutputIterator __result, _Tp __init)
510 return std::exclusive_scan(__first, __last, __result, std::move(__init),
514 /** @brief Output the cumulative sum of one range to a second range
516 * @param __first Start of input range.
517 * @param __last End of input range.
518 * @param __result Start of output range.
519 * @param __binary_op Function to perform summation.
520 * @param __init Initial value.
521 * @return The end of the output range.
523 * Write the cumulative sum (aka prefix sum, aka scan) of the input range
524 * to the output range. Each element of the output range contains the
525 * running total of all earlier elements (and the initial value),
526 * using `binary_op` for summation.
528 * This function generates an "inclusive" scan, meaning the Nth element
529 * of the output range is the sum of the first N input elements,
530 * so the Nth input element is included.
532 template<typename _InputIterator, typename _OutputIterator,
533 typename _BinaryOperation, typename _Tp>
536 inclusive_scan(_InputIterator __first, _InputIterator __last,
537 _OutputIterator __result, _BinaryOperation __binary_op,
540 for (; __first != __last; ++__first)
541 *__result++ = __init = __binary_op(__init, *__first);
545 /** @brief Output the cumulative sum of one range to a second range
547 * @param __first Start of input range.
548 * @param __last End of input range.
549 * @param __result Start of output range.
550 * @param __binary_op Function to perform summation.
551 * @return The end of the output range.
553 * Write the cumulative sum (aka prefix sum, aka scan) of the input range
554 * to the output range. Each element of the output range contains the
555 * running total of all earlier elements, using `binary_op` for summation.
557 * This function generates an "inclusive" scan, meaning the Nth element
558 * of the output range is the sum of the first N input elements,
559 * so the Nth input element is included.
561 template<typename _InputIterator, typename _OutputIterator,
562 typename _BinaryOperation>
565 inclusive_scan(_InputIterator __first, _InputIterator __last,
566 _OutputIterator __result, _BinaryOperation __binary_op)
568 if (__first != __last)
570 auto __init = *__first;
571 *__result++ = __init;
573 if (__first != __last)
574 __result = std::inclusive_scan(__first, __last, __result,
575 __binary_op, std::move(__init));
580 /** @brief Output the cumulative sum of one range to a second range
582 * @param __first Start of input range.
583 * @param __last End of input range.
584 * @param __result Start of output range.
585 * @return The end of the output range.
587 * Write the cumulative sum (aka prefix sum, aka scan) of the input range
588 * to the output range. Each element of the output range contains the
589 * running total of all earlier elements, using `std::plus<>` for summation.
591 * This function generates an "inclusive" scan, meaning the Nth element
592 * of the output range is the sum of the first N input elements,
593 * so the Nth input element is included.
595 template<typename _InputIterator, typename _OutputIterator>
597 inline _OutputIterator
598 inclusive_scan(_InputIterator __first, _InputIterator __last,
599 _OutputIterator __result)
600 { return std::inclusive_scan(__first, __last, __result, plus<>()); }
602 /** @brief Output the cumulative sum of one range to a second range
604 * @param __first Start of input range.
605 * @param __last End of input range.
606 * @param __result Start of output range.
607 * @param __init Initial value.
608 * @param __binary_op Function to perform summation.
609 * @param __unary_op Function to transform elements of the input range.
610 * @return The end of the output range.
612 * Write the cumulative sum (aka prefix sum, aka scan) of the input range
613 * to the output range. Each element of the output range contains the
614 * running total of all earlier elements (and the initial value),
615 * using `__unary_op` to transform the input elements
616 * and using `__binary_op` for summation.
618 * This function generates an "exclusive" scan, meaning the Nth element
619 * of the output range is the sum of the first N-1 input elements,
620 * so the Nth input element is not included.
622 template<typename _InputIterator, typename _OutputIterator, typename _Tp,
623 typename _BinaryOperation, typename _UnaryOperation>
626 transform_exclusive_scan(_InputIterator __first, _InputIterator __last,
627 _OutputIterator __result, _Tp __init,
628 _BinaryOperation __binary_op,
629 _UnaryOperation __unary_op)
631 while (__first != __last)
634 __init = __binary_op(__init, __unary_op(*__first));
636 *__result++ = std::move(__v);
641 /** @brief Output the cumulative sum of one range to a second range
643 * @param __first Start of input range.
644 * @param __last End of input range.
645 * @param __result Start of output range.
646 * @param __binary_op Function to perform summation.
647 * @param __unary_op Function to transform elements of the input range.
648 * @param __init Initial value.
649 * @return The end of the output range.
651 * Write the cumulative sum (aka prefix sum, aka scan) of the input range
652 * to the output range. Each element of the output range contains the
653 * running total of all earlier elements (and the initial value),
654 * using `__unary_op` to transform the input elements
655 * and using `__binary_op` for summation.
657 * This function generates an "inclusive" scan, meaning the Nth element
658 * of the output range is the sum of the first N input elements,
659 * so the Nth input element is included.
661 template<typename _InputIterator, typename _OutputIterator,
662 typename _BinaryOperation, typename _UnaryOperation, typename _Tp>
665 transform_inclusive_scan(_InputIterator __first, _InputIterator __last,
666 _OutputIterator __result,
667 _BinaryOperation __binary_op,
668 _UnaryOperation __unary_op,
671 for (; __first != __last; ++__first)
672 *__result++ = __init = __binary_op(__init, __unary_op(*__first));
676 /** @brief Output the cumulative sum of one range to a second range
678 * @param __first Start of input range.
679 * @param __last End of input range.
680 * @param __result Start of output range.
681 * @param __binary_op Function to perform summation.
682 * @param __unary_op Function to transform elements of the input range.
683 * @return The end of the output range.
685 * Write the cumulative sum (aka prefix sum, aka scan) of the input range
686 * to the output range. Each element of the output range contains the
687 * running total of all earlier elements,
688 * using `__unary_op` to transform the input elements
689 * and using `__binary_op` for summation.
691 * This function generates an "inclusive" scan, meaning the Nth element
692 * of the output range is the sum of the first N input elements,
693 * so the Nth input element is included.
695 template<typename _InputIterator, typename _OutputIterator,
696 typename _BinaryOperation, typename _UnaryOperation>
699 transform_inclusive_scan(_InputIterator __first, _InputIterator __last,
700 _OutputIterator __result,
701 _BinaryOperation __binary_op,
702 _UnaryOperation __unary_op)
704 if (__first != __last)
706 auto __init = __unary_op(*__first);
707 *__result++ = __init;
709 if (__first != __last)
710 __result = std::transform_inclusive_scan(__first, __last, __result,
711 __binary_op, __unary_op,
717 /// @} group numeric_ops
720 _GLIBCXX_END_NAMESPACE_VERSION
723 #if __cplusplus >= 201703L
724 // Parallel STL algorithms
725 # if _PSTL_EXECUTION_POLICIES_DEFINED
726 // If <execution> has already been included, pull in implementations
727 # include <pstl/glue_numeric_impl.h>
729 // Otherwise just pull in forward declarations
730 # include <pstl/glue_numeric_defs.h>
731 # define _PSTL_NUMERIC_FORWARD_DECLARED 1
734 // Feature test macro for parallel algorithms
735 # define __cpp_lib_parallel_algorithm 201603L
738 #endif /* _GLIBCXX_NUMERIC */