C++ exponential_distribution occasionally returns inf where 0 is expected

In a project, I am generating millions of expo(lambda) random variables, where lambda is potentially very large. When using std::exponential_distribution<float> I occasionally get a return value of inf. I would understand this if lambda were close to 0, but when lambda is large a value very close to zero is expected. For example, the following program usually terminates after several million draws:

#include<iostream>
#include<random>

int main(void)
{
    std::random_device rd;
    std::mt19937 generator(rd());

    float lambda = 1000000.0;
    float val = 0;

    for(int i = 0; i < 1000000000; ++i)
    {
        std::exponential_distribution<float> dist(lambda);
        val = dist(generator);

        if(isinf(val))
        {
            std::cout << i << " " << val << " failure" << std::endl;
            exit(0);
        }
    }

    return 0;
}

If there is some error (due to precision) in the function, why does it return inf instead of the more convenient 0.0? Is there any way to fix this, besides for manually checking that the output is finite? Thanks.