<random>: Improve binomial_distribution accuracy

[MattStephanson]

Fixes #1530.

• For small mean, replaces Poisson approximation with exact waiting time method based on cumulative sum of exponential variates.
• Uses cheap but rigorous approximation of exit condition for fast return on first iteration.

[MattStephanson]

BTW, for reference, the underlying algorithm is from "Non-Uniform Random Variate Generation", Luc Devroye, section X.4, "Second waiting time method".

[MattStephanson]

There is an inconsistency wrt Rand and _Rand

I think I'm going to restore the original version anyway, because 1.0 - _Par0._Mean makes it impossible to return anything other than 0 when _Par0._Mean < DBL_EPSILON.

[StephanTLavavej]

Looks good to me - the logic here is significantly beyond my experience so I really appreciate the PR. 😻 I will push changes for 2 trivial things I noticed.

[statementreply]

The proposed change doesn't correctly handle the case that n * (1-p) < 1.

When p > 0.5, except for the small n case, the original algorithm generates B(n, p) as n - B(n, 1-p). See binomial_distribution::param_type::_Init():

_Pp    = _Px < 0.5 ? _Px : (1.0 - _Px);
_Mean  = _Tx * _Pp;

and the return below:

return _Par0._Px == _Par0._Pp ? _Res : static_cast<_Ty>(_Par0._Tx - _Res);

Test case:

#include <cmath>
#include <iostream>
#include <random>
#include <vector>

using namespace std;

int main() {
    constexpr int it_max  = 10'000'000;
    constexpr int n       = 25;
    constexpr double mean = n - 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;

    cout << "Expected mean: " << mean << endl
         << "Observed mean: " << sample_mean << endl;

    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;
}

Pre-PR output:

Expected mean: 24.01
Observed mean: 24.0104
Expected variance: 0.950796
Observed variance: 0.988557

Post-PR output:

Expected mean: 24.01
Observed mean: 0.990254
Expected variance: 0.950796
Observed variance: 0.950525

(Sorry I missed this in my previous review)

[MattStephanson]

@StephanTLavavej I've pushed a bugfix based on @statementreply's feedback after your approval.

[CaseyCarter]

FYI: I've pushed a merge with main, and updated the list of libcxx test skips to indicate that my guess that #1530 caused std/numerics/rand/rand.dis/rand.dist.bern/rand.dist.bern.bin/eval.PR44847.pass.cpp to fail was wrong.

[StephanTLavavej]

Thanks for noticing and fixing this accuracy bug! 🎯 🎯 🎯