SELECT l2_distance('{}/2'::sparsevec, '{1:3,2:4}/2');
 l2_distance 
-------------
           5
(1 row)

SELECT l2_distance('{}/2'::sparsevec, '{2:1}/2');
 l2_distance 
-------------
           1
(1 row)

SELECT '{}/2'::sparsevec <-> '{1:3,2:4}/2';
 ?column? 
----------
        5
(1 row)

SELECT inner_product('{1:1,2:2}/2'::sparsevec, '{1:2,2:4}/2');
 inner_product 
---------------
            10
(1 row)

SELECT sparsevec_negative_inner_product('{1:1,2:2}/2', '{1:2,2:4}/2');
 sparsevec_negative_inner_product 
----------------------------------
                              -10
(1 row)

SELECT cosine_distance('{1:1,2:2}/2'::sparsevec, '{1:2,2:4}/2');
 cosine_distance 
-----------------
               0
(1 row)

SELECT cosine_distance('{1:1,2:2}/2'::sparsevec, '{}/2');
 cosine_distance 
-----------------
             NaN
(1 row)

SELECT cosine_distance('{1:1,2:1}/2'::sparsevec, '{1:-1,2:-1}/2');
 cosine_distance 
-----------------
               2
(1 row)

SELECT cosine_distance('{1:2}/2'::sparsevec, '{2:2}/2');
 cosine_distance 
-----------------
               1
(1 row)

SELECT cosine_distance('{}/1'::sparsevec, '{}/1');
 cosine_distance 
-----------------
             NaN
(1 row)

SELECT cosine_distance('{1:2}/2'::sparsevec, '{1:1}/3');
ERROR:  different sparsevec dimensions 2 and 3