Performance for drawing numbers from Poisson distribution with low mean

To draw random number from a Poisson distribution in C++, it is generally advised to use

RNG_type rng;
std::poisson_distribution<size_t> d(1e-6);
auto r = d(rng);

At each call of the std::poisson_distribution object, an entire sequence of random bits is consumed (e.g. 32 bits with std::mt19937, 64 bits for std::mt19937_64). It strikes me that with such low mean (mean = 1e-6), the vast majority of times, only a few bits are enough to determine that the value to return is 0. The other bits could then be cached for later use.

Assuming that a sequence of bits set to true is associated to a high returned value from the Poisson distribution, when using a mean of 1e-6, any sequence not starting with 19 trues necessarily returns a zero! Indeed,

1 - 1/2^19 < P(0, 1e-6) < 1 - 1/2^20

, where P(n, r) denotes the probability of drawing n from a Poisson distribution with mean r. An algorithm that does not waste bits would use one bit half of the time, two bits a quarter of the times, three bits an eighth of the times, ....

Is there an algorithm out there that can improve performance by consuming as few bits as possible when drawing Poisson numbers? Is there another way to improve performance compared to std::poisson_distribution when we consider a low mean?