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

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

SELECT '{1:1,2:2,3:3}/3'::sparsevec <= '{1:1,2:2,3:3}/3';
 ?column? 
----------
 t
(1 row)

SELECT '{1:1,2:2,3:3}/3'::sparsevec <= '{1:1,2:2}/2';
 ?column? 
----------
 f
(1 row)

SELECT '{1:1,2:2,3:3}/3'::sparsevec = '{1:1,2:2,3:3}/3';
 ?column? 
----------
 t
(1 row)

SELECT '{1:1,2:2,3:3}/3'::sparsevec = '{1:1,2:2}/2';
 ?column? 
----------
 f
(1 row)

SELECT '{1:1,2:2,3:3}/3'::sparsevec != '{1:1,2:2,3:3}/3';
 ?column? 
----------
 f
(1 row)

SELECT '{1:1,2:2,3:3}/3'::sparsevec != '{1:1,2:2}/2';
 ?column? 
----------
 t
(1 row)

SELECT '{1:1,2:2,3:3}/3'::sparsevec >= '{1:1,2:2,3:3}/3';
 ?column? 
----------
 t
(1 row)

SELECT '{1:1,2:2,3:3}/3'::sparsevec >= '{1:1,2:2}/2';
 ?column? 
----------
 t
(1 row)

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

SELECT '{1:1,2:2,3:3}/3'::sparsevec > '{1:1,2:2}/2';
 ?column? 
----------
 t
(1 row)

SELECT sparsevec_cmp('{1:1,2:2,3:3}/3', '{1:1,2:2,3:3}/3');
 sparsevec_cmp 
---------------
             0
(1 row)

SELECT sparsevec_cmp('{1:1,2:2,3:3}/3', '{}/3');
 sparsevec_cmp 
---------------
             1
(1 row)

SELECT sparsevec_cmp('{}/3', '{1:1,2:2,3:3}/3');
 sparsevec_cmp 
---------------
            -1
(1 row)

SELECT sparsevec_cmp('{1:1,2:2}/2', '{1:1,2:2,3:3}/3');
 sparsevec_cmp 
---------------
            -1
(1 row)

SELECT sparsevec_cmp('{1:1,2:2,3:3}/3', '{1:1,2:2}/2');
 sparsevec_cmp 
---------------
             1
(1 row)

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

SELECT sparsevec_cmp('{1:2,2:3}/2', '{1:1,2:2,3:3}/3');
 sparsevec_cmp 
---------------
             1
(1 row)

SELECT round(l2_norm('{1:1,2:1}/2'::sparsevec)::numeric, 5);
  round  
---------
 1.41421
(1 row)

SELECT l2_norm('{1:3,2:4}/2'::sparsevec);
 l2_norm 
---------
       5
(1 row)

SELECT l2_norm('{2:1}/2'::sparsevec);
 l2_norm 
---------
       1
(1 row)

SELECT l2_norm('{1:3e37,2:4e37}/2'::sparsevec)::real;
 l2_norm 
---------
   5e+37
(1 row)

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

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

SELECT l2_distance('{2:4}/2'::sparsevec, '{1:3}/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 inner_product('{1:1,2:2}/2'::sparsevec, '{1:3}/1');
ERROR:  different sparsevec dimensions 2 and 1
SELECT inner_product('{1:3e38}/1'::sparsevec, '{1:3e38}/1');
 inner_product 
---------------
      Infinity
(1 row)

SELECT inner_product('{1:1,3:3,5:5}/5'::sparsevec, '{2:4,3:6,4:8}/5');
 inner_product 
---------------
            18
(1 row)

SELECT '{1:1,2:2}/2'::sparsevec <#> '{1:3,2:4}/2';
 ?column? 
----------
      -11
(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 
-----------------
               0
(1 row)

SELECT cosine_distance('{1:1}/2'::sparsevec, '{2:2}/2');
 cosine_distance 
-----------------
               1
(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('{1:1,2:2}/2'::sparsevec, '{1:3}/1');
ERROR:  different sparsevec dimensions 2 and 1
SELECT cosine_distance('{1:1,2:1}/2'::sparsevec, '{1:1.1,2:1.1}/2');
 cosine_distance 
-----------------
               0
(1 row)

SELECT cosine_distance('{1:1,2:1}/2'::sparsevec, '{1:-1.1,2:-1.1}/2');
 cosine_distance 
-----------------
               2
(1 row)

SELECT cosine_distance('{1:3e38}/1'::sparsevec, '{1:3e38}/1');
 cosine_distance 
-----------------
             NaN
(1 row)

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

SELECT '{1:1,2:2}/2'::sparsevec <=> '{1:2,2:4}/2';
 ?column? 
----------
        0
(1 row)

SELECT l1_distance('{}/2'::sparsevec, '{1:3,2:4}/2');
 l1_distance 
-------------
           7
(1 row)

SELECT l1_distance('{}/2'::sparsevec, '{2:1}/2');
 l1_distance 
-------------
           1
(1 row)

SELECT l1_distance('{1:1,2:2}/2'::sparsevec, '{1:3}/1');
ERROR:  different sparsevec dimensions 2 and 1
SELECT l1_distance('{1:3e38}/1'::sparsevec, '{1:-3e38}/1');
 l1_distance 
-------------
    Infinity
(1 row)

SELECT l1_distance('{1:1,3:3,5:5,7:7}/8'::sparsevec, '{2:2,4:4,6:6,8:8}/8');
 l1_distance 
-------------
          36
(1 row)

SELECT l1_distance('{1:1,3:3,5:5,7:7,9:9}/9'::sparsevec, '{2:2,4:4,6:6,8:8}/9');
 l1_distance 
-------------
          45
(1 row)

SELECT l2_normalize('{1:3,2:4}/2'::sparsevec);
  l2_normalize   
-----------------
 {1:0.6,2:0.8}/2
(1 row)

SELECT l2_normalize('{1:3}/2'::sparsevec);
 l2_normalize 
--------------
 {1:1}/2
(1 row)

SELECT l2_normalize('{2:0.1}/2'::sparsevec);
 l2_normalize 
--------------
 {2:1}/2
(1 row)

SELECT l2_normalize('{}/2'::sparsevec);
 l2_normalize 
--------------
 {}/2
(1 row)

SELECT l2_normalize('{1:3e38}/1'::sparsevec);
 l2_normalize 
--------------
 {1:1}/1
(1 row)

SELECT l2_normalize('{1:3e38,2:1e-37}/2'::sparsevec);
 l2_normalize 
--------------
 {1:1}/2
(1 row)

SELECT l2_normalize('{2:3e37,4:3e-37,6:4e37,8:4e-37}/9'::sparsevec);
  l2_normalize   
-----------------
 {2:0.6,6:0.8}/9
(1 row)