Using accumulate and reduce with any type of execution policy produces large error for huge number of data

  accumulate, c++, execution, reduce, rounding

The result of the following code is very surprising to me. Why does using accumulate produce such a large error? I know that because it is single precisions, it will have a round-off error but when using reduce without any policy (single precision), this round-off error is not so significant! In fact, round-off errors is reasonable. But again, when I use reduce with policy (either of par seq unseq), its result will be the same with such a huge error compared to exact sum. Can anyone explain this?

#include <fstream>
#include <iomanip>
#include <execution>
#include <random>
#include <iostream>
#include <chrono>
#include <cfloat>
using namespace std;

int main() {

    const int N=100*1e6;

    default_random_engine g(time(0));
    uniform_real_distribution<float> d(0.0f,nextafter(1.0f, DBL_MAX));

    vector<float>  a;
    vector<double> b;
    double exact{0.0};
    float sum;


    for(auto i=0; i<N ;i++){
        a.push_back(d(g));

        b.push_back(static_cast<double>(a[i]));
    }

    exact=accumulate(b.begin(),b.end(),0.0);
    cout<<" exact sum is:  "<<exact<<endl;
    sum=accumulate(a.begin(),a.end(),0.0f);
    cout<<" using accumulate for float : "<<sum<<endl;
    sum=reduce(a.begin(),a.end());
    cout<<" using reduce for float : "<<sum<<endl;
    sum=reduce(execution::unseq,a.begin(),a.end());
    cout<<" using reduce with ploicy : "<<sum<<endl;

}

it’s result is :

  exact sum is:  4.99979e+07
 using accumulate for float : 1.67772e+07
 using reduce for float : 5.00006e+07
 using reduce with ploicy : 1.67772e+07

Source: Windows Questions C++

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