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Showing content from https://en.cppreference.com/w/cpp/language/../algorithm/../numeric/fenv/../../algorithm/reduce.html below:

std::reduce - cppreference.com

template< class InputIt > typename std::iterator_traits<InputIt>::value_type reduce( InputIt first, InputIt last ); (1) (since C++17)
(constexpr since C++20) template< class ExecutionPolicy, class ForwardIt > typename std::iterator_traits<ForwardIt>::value_type reduce( ExecutionPolicy&& policy, ForwardIt first, ForwardIt last ); (2) (since C++17)

template< class InputIt, class T > T reduce( InputIt first, InputIt last, T init );

(3) (since C++17)
(constexpr since C++20) template< class ExecutionPolicy, class ForwardIt, class T > T reduce( ExecutionPolicy&& policy, ForwardIt first, ForwardIt last, T init ); (4) (since C++17)

template< class InputIt, class T, class BinaryOp > T reduce( InputIt first, InputIt last, T init, BinaryOp op );

(5) (since C++17)
(constexpr since C++20) template< class ExecutionPolicy, class ForwardIt, class T, class BinaryOp > T reduce( ExecutionPolicy&& policy, ForwardIt first, ForwardIt last, T init, BinaryOp op ); (6) (since C++17) 3)

Equivalent to

reduce(first, last, init, std::plus<>())

5) Reduces the range [first, last), possibly permuted and aggregated in unspecified manner, along with the initial value init over op.

2,4,6) Same as (1,3,5), but executed according to policy.

These overloads participate in overload resolution only if all following conditions are satisfied:

Given binary_op as the actual binary operation:

The result is non-deterministic if the binary_op is not associative or not commutative (such as floating-point addition).

If any of the following values is not convertible to T, the program is ill-formed:

binary_op(init, *first)
binary_op(*first, init)
binary_op(init, init)
binary_op(*first, *first)

If any of the following conditions is satisfied, the behavior is undefined:

T is not MoveConstructible.
binary_op modifies any element of [first, last).
binary_op invalidates any iterator or subrange of [first, last].

[edit] Parameters first, last - the pair of iterators defining the range of elements to apply the algorithm to init - the initial value of the generalized sum policy - the execution policy to use op - binary FunctionObject that will be applied in unspecified order to the result of dereferencing the input iterators, the results of other op and init. Type requirements -InputIt must meet the requirements of LegacyInputIterator. -ForwardIt must meet the requirements of LegacyForwardIterator. [edit] Return value 1-4)

The generalized sum of

init

and the elements of

[first, last)

over

std::plus<>()

5,6) The generalized sum of init and the elements of [first, last) over op.

The generalized sum of a group of elements over an binary operation binary_op is defined as follows:

If the group only has one element, the sum is the value of the element.
Otherwise, performs the following operations in order:

Takes any two elements elem1 and elem2 from the group.
Calculates binary_op(elem1, elem2) and puts the result back to the group.
Repeats steps 1 and 2 until there is only one element in the group.

[edit] Complexity

Given \(\scriptsize N\)N as std::distance(first, last):

1-4) \(\scriptsize O(N)\)O(N)

applications of

std::plus<>()

5,6) \(\scriptsize O(N)\)O(N) applications of op.

[edit] Exceptions

The overloads with a template parameter named ExecutionPolicy report errors as follows:

If execution of a function invoked as part of the algorithm throws an exception and ExecutionPolicy is one of the standard policies, std::terminate is called. For any other ExecutionPolicy, the behavior is implementation-defined.
If the algorithm fails to allocate memory, std::bad_alloc is thrown.

[edit] Notes

std::reduce behaves like std::accumulate except the elements of the range may be grouped and rearranged in arbitrary order.

[edit] Example

Side-by-side comparison between std::reduce and std::accumulate:

#if PARALLEL
#include <execution>
#define SEQ std::execution::seq,
#define PAR std::execution::par,
#else
#define SEQ
#define PAR
#endif
 
#include <chrono>
#include <iomanip>
#include <iostream>
#include <locale>
#include <numeric>
#include <utility>
#include <vector>
 
int main()
{
    std::cout.imbue(std::locale("en_US.UTF-8"));
    std::cout << std::fixed << std::setprecision(1);
 
    auto eval = [](auto fun)
    {
        const auto t1 = std::chrono::high_resolution_clock::now();
        const auto [name, result] = fun();
        const auto t2 = std::chrono::high_resolution_clock::now();
        const std::chrono::duration<double, std::milli> ms = t2 - t1;
        std::cout << std::setw(28) << std::left << name << "sum: "
                  << result << '\t' << "time: " << ms.count() << " ms\n";
    };
 
    {
        const std::vector<double> v(100'000'007, 0.1);
 
        eval([&v]{ return std::pair{"std::accumulate (double)",
            std::accumulate(v.cbegin(), v.cend(), 0.0)}; });
        eval([&v]{ return std::pair{"std::reduce (seq, double)",
            std::reduce(SEQ v.cbegin(), v.cend())}; });
        eval([&v]{ return std::pair{"std::reduce (par, double)",
            std::reduce(PAR v.cbegin(), v.cend())}; });
    }
 
    {
        const std::vector<long> v(100'000'007, 1);
 
        eval([&v]{ return std::pair{"std::accumulate (long)",
            std::accumulate(v.cbegin(), v.cend(), 0l)}; });
        eval([&v]{ return std::pair{"std::reduce (seq, long)",
            std::reduce(SEQ v.cbegin(), v.cend())}; });
        eval([&v]{ return std::pair{"std::reduce (par, long)",
            std::reduce(PAR v.cbegin(), v.cend())}; });
    }
}

Possible output:

// POSIX: g++ -std=c++23 ./example.cpp -ltbb -O3; ./a.out
std::accumulate (double)    sum: 10,000,000.7	time: 356.9 ms
std::reduce (seq, double)   sum: 10,000,000.7	time: 140.1 ms
std::reduce (par, double)   sum: 10,000,000.7	time: 140.1 ms
std::accumulate (long)      sum: 100,000,007	time: 46.0 ms
std::reduce (seq, long)     sum: 100,000,007	time: 67.3 ms
std::reduce (par, long)     sum: 100,000,007	time: 63.3 ms
 
// POSIX: g++ -std=c++23 ./example.cpp -ltbb -O3 -DPARALLEL; ./a.out
std::accumulate (double)    sum: 10,000,000.7	time: 353.4 ms
std::reduce (seq, double)   sum: 10,000,000.7	time: 140.7 ms
std::reduce (par, double)   sum: 10,000,000.7	time: 24.7 ms
std::accumulate (long)      sum: 100,000,007	time: 42.4 ms
std::reduce (seq, long)     sum: 100,000,007	time: 52.0 ms
std::reduce (par, long)     sum: 100,000,007	time: 23.1 ms

[edit] See also sums up or folds a range of elements
(function template) [edit] applies a function to a range of elements, storing results in a destination range
(function template) [edit] applies an invocable, then reduces out of order
(function template) [edit] left-folds a range of elements
(algorithm function object)[edit]

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