How can I use this parallel sum function to use vector instructions?

Question

As a kind of side project, I am working on a multi-threaded sum algorihm, which will be superior std::accumulatewhen working with a fairly large array. First, I will talk about my thinking process leading to this, but if you want to go straight to the problem, feel free to scroll to this part.

I found many parallel summary algorithms online, most of which take the following approach:

template <typename T, typename IT>
T parallel_sum(IT _begin, IT _end, T _init) {
    const auto size = distance(_begin, _end);
    static const auto n = thread::hardware_concurrency();
    if (size < 10000 || n == 1) return accumulate(_begin, _end, _init);
    vector<future<T>> partials;
    partials.reserve(n);
    auto chunkSize = size / n;
    for (unsigned i{ 0 }; i < n; i++) {
        partials.push_back(async(launch::async, [](IT _b, IT _e){
            return accumulate(_b, _e, T{0});
        }, next(_begin, i*chunkSize), (i==n-1)?_end:next(_begin, (i+1)*chunkSize)));
    }
    for (auto& f : partials) _init += f.get();
    return _init;
}

Assuming two threads (as reported thread::hardware_concurrency()), this function will access the elements in memory as follows:

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template <typename T, typename IT>
T parallel_sum2(IT _begin, IT _end, T _init) {
    const auto size = distance(_begin, _end);
    static const auto n = thread::hardware_concurrency();
    if (size < 10000 || n == 1) return accumulate(_begin, _end, _init);
    vector<future<T>> partials;
    partials.reserve(n);
    for (unsigned i{ 0 }; i < n; i++) {
        partials.push_back(async(launch::async, [](IT _b, IT _e, unsigned _s){
            T _ret{ 0 };
            for (; _b < _e; advance(_b, _s)) _ret += *_b;
            return _ret;
        }, next(_begin, i), _end, n));
    }
    for (auto& f : partials) _init += f.get();
    return _init;
}

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Answer 1

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