How to optimize COUNT(*) performance on InnoDB by using index

Question

I have a largish but narrow InnoDB table with ~9m records. Doing count(*) or count(id) on the table is extremely slow (6+ seconds):

Answer 1

select max(id) - min(id) from xxx_table where ....

This will use "Select tables optimized away", and is very fast!!!

Note max(id) - min(id) is actually bigger than count(1).