BUG #14011: select count(distinct column) does not utilizes indices to improve performance

The following bug has been logged on the website:

Bug reference: 14011
Logged by: Glen Takahashi
Email address: gtakahashi(at)palantir(dot)com
PostgreSQL version: 9.3.6
Operating system: Red Hat Enterprise Linux 6.7
Description:

Doing a select count(distinct column) does not utilize indices and takes a
very long time to compute.
For example

Create table test_table (number integer);
Create index on test_table (number);
Insert into test_table (number) select generate_series(1,100000); (I did
this ~500 times to replicate the scale of data we have)
Select count(distinct number) from test_table;
Count 100000 Time: 63419.600ms
Select count(1) from (select distinct number from test_table) a;
Count 100000 Time: 10743.186 ms