How to understand that the standard error of redis hyperloglog is 0.81%？

[wangmm0220]

I am confused with hyperloglog standard error 0.81%, so I change rand() to $n+$j in https://github.com/redis/redis/blob/unstable/tests/unit/hyperloglog.tcl#L48

and change 5%->0.81% in https://github.com/redis/redis/blob/unstable/tests/unit/hyperloglog.tcl#L53

but error happens.

[zhugezy]

Interestring.
Here the "standard error" 0.81% means the "estimating error" has a great probability falling within (α-σ, α+σ). Here α is the average(or so-called Mathematical Expectation I'd prefer) of "estimating error", which is 100%. σ is the "standard error", which is 0.81%.

Paper:
Let σ ≈ 1.04/sqrt(m) represent the standard error; the estimates provided by HYPERLOGLOG
are expected to be within σ, 2σ, 3σ of the exact count in respectively 65%, 95%, 99% of all
the cases.

Notice the test run "assert" for $n/$j = 1000 times. Assuming that these 1000 checking are all independent to each other, we can get the test case passed by a probability of 65%^1000, which is nearly impossible.
However they aren't independent events, and it has a greater probability to get passed.