So if you need 1000 random numbers, generate from 1 to 1 million.
If you don't check for clashes, the 50% chance of failure is too much. Probably even 0.1% is too much, so you'd need more elaborate approach.
If you do check for clashes, you can generate from 1 to 2000 with little overhead.
Then you can choose how many collisions to accept on average. (If the answer is zero, then it makes more sense to look at the probability of one or more collisions.)
The square root approximation works well for large numbers, but leaves out some factors that are relevant for small numbers.