select ...  
from user
where user.reg_date::date between '2016-02-01' and '2016-03-01'
and exists(select * from purchase where purchase.user_id = player.user_id)

CREATE TABLE public.user_test (
  user_id VARCHAR(32) NOT NULL,
  reg_date TIMESTAMP WITHOUT TIME ZONE,
  name VARCHAR(100),
  CONSTRAINT user_test_pkey PRIMARY KEY(user_id)
) 
WITH (oids = false);

CREATE INDEX user_test_day_idx ON public.user_test
  USING btree ((date_trunc('day'::text, reg_date)));

CREATE TABLE public.purchase_test (
  id SERIAL,
  user_id VARCHAR(32),
  date TIMESTAMP(6) WITHOUT TIME ZONE,
  sum INTEGER,
  CONSTRAINT purchase_test_pkey PRIMARY KEY(id)
) 
WITH (oids = false);

CREATE INDEX purchase_test_user_idx ON public.purchase_test
  USING btree (user_id COLLATE pg_catalog."default");

explain analyze
select user_test.* 
from user_test
where user_test.reg_date::date between '2016-02-01' and '2016-03-01'
and exists(select * from purchase_test where purchase_test.user_id = user_test.user_id)