Based on the official doc:
'PostgreSQL provides two predefined ranking functions, which take into account lexical, proximity, and structural information; that is, they consider how often the query terms appear in the document, how close together the terms are in the document, and how important is the part of the document where they occur.'
The definition of ts_rank:
'Ranks vectors based on the frequency of their matching lexemes.'
Definition of ts_rank_cd:
'This function computes the cover density ranking for the given document vector and query, as described in Clarke, Cormack, and Tudhope's "Relevance Ranking for One to Three Term Queries" in the journal "Information Processing and Management", 1999. Cover density is similar to ts_rankranking except that the proximity of matching lexemes to each other is taken into consideration.'
Testing the ts_rank function with these examples:
SELECT ts_rank( to_tsvector('name lastname name lastname'), to_tsquery('name & lastname'));
ts_rank: 0,3400053
SELECT ts_rank( to_tsvector('name lastname zzzzzzzz tttttt name lastname'), to_tsquery('name & lastname'));
ts_rank: 0,3344279
SELECT ts_rank( to_tsvector('name lastname xxxxx yyyyyyy zzzzzzzz tttttt name lastname'), to_tsquery('name & lastname'));
ts_rank: 0,3187879
why the function ts_rank gives different results however I have the same number of searched items on the different examples? does the distance was token into consideration into the ts_rank. If yes what's is the difference between ts_rank and ts_rank_cd which is differ from the first by the proximity of matching lexemes?