<random>: Poisson approximation for binomial_distribution is not accurate

[MattStephanson]

For binomial_distribution(t,p) with t >= 25 and t*p < 1, it is approximated by a Poisson distribution. This is not very accurate, especially for smaller values of t.

Command-line test case

c:\temp>type test.cpp
#include <cmath>
#include <iostream>
#include <random>
#include <vector>

using namespace std;

int main() {
    constexpr int it_max  = 100'000;
    constexpr int n       = 25;
    constexpr double mean = 0.99;
    constexpr double p    = mean / n;
    constexpr double var  = n * p * (1.0 - p);

    mt19937 gen;
    binomial_distribution<> dist(n, p);

    vector<int> counts(n + 1);
    for (int i = 0; i < it_max; ++i) {
        ++counts[static_cast<size_t>(dist(gen))];
    }

    double sample_mean = 0.0;
    for (size_t i = 1; i < counts.size(); ++i) {
        sample_mean += static_cast<double>(i) * counts[i];
    }
    sample_mean /= it_max;

    double sample_var = 0.0;
    for (size_t i = 0; i < counts.size(); ++i) {
        sample_var += counts[i] * pow(i - sample_mean, 2);
    }
    sample_var /= it_max - 1;

    cout << "Expected variance: " << var << endl
         << "Observed variance: " << sample_var << endl;

    return 0;
}

c:\temp>cl /EHsc test.cpp
Microsoft (R) C/C++ Optimizing Compiler Version 19.28.29617 for x64
Copyright (C) Microsoft Corporation.  All rights reserved.

test.cpp
Microsoft (R) Incremental Linker Version 14.28.29617.0
Copyright (C) Microsoft Corporation.  All rights reserved.

/out:test.exe
test.obj

c:\temp>.\test.exe
Expected variance: 0.950796
Observed variance: 0.985299

Expected behavior
Observed variance should be within 0.01 (~95% CI) of expected.