There is only one square root operator (prefix |/) defined in the standard catalog, and it takes an argument of type double precision. The scanner assigns an initial type of integer to the argument in this query expression:

SELECT |/ 40 AS "square root of 40";
 square root of 40
-------------------
 6.324555320336759
(1 row)

So the parser does a type conversion on the operand and the query is equivalent to:

SELECT |/ CAST(40 AS double precision) AS "square root of 40";

A string-like syntax is used for working with string types and for working with complex extension types. Strings with unspecified type are matched with likely operator candidates.

An example with one unspecified argument:

SELECT text 'abc' || 'def' AS "text and unknown";

 text and unknown
------------------
 abcdef
(1 row)

In this case the parser looks to see if there is an operator taking text for both arguments. Since there is, it assumes that the second argument should be interpreted as type text.

Here is a concatenation of two values of unspecified types:

SELECT 'abc' || 'def' AS "unspecified";

 unspecified
-------------
 abcdef
(1 row)

In this case there is no initial hint for which type to use, since no types are specified in the query. So, the parser looks for all candidate operators and finds that there are candidates accepting both string-category and bit-string-category inputs. Since string category is preferred when available, that category is selected, and then the preferred type for strings, text, is used as the specific type to resolve the unknown-type literals as.

The Postgres Pro operator catalog has several entries for the prefix operator @, all of which implement absolute-value operations for various numeric data types. One of these entries is for type float8, which is the preferred type in the numeric category. Therefore, Postgres Pro will use that entry when faced with an unknown input:

SELECT @ '-4.5' AS "abs";
 abs
-----
 4.5
(1 row)

Here the system has implicitly resolved the unknown-type literal as type float8 before applying the chosen operator. We can verify that float8 and not some other type was used:

SELECT @ '-4.5e500' AS "abs";

ERROR:  "-4.5e500" is out of range for type double precision

On the other hand, the prefix operator ~ (bitwise negation) is defined only for integer data types, not for float8. So, if we try a similar case with ~, we get:

SELECT ~ '20' AS "negation";

ERROR:  operator is not unique: ~ "unknown"
HINT:  Could not choose a best candidate operator. You might need to add
explicit type casts.

This happens because the system cannot decide which of the several possible ~ operators should be preferred. We can help it out with an explicit cast:

SELECT ~ CAST('20' AS int8) AS "negation";

 negation
----------
      -21
(1 row)

Here is another example of resolving an operator with one known and one unknown input:

SELECT array[1,2] <@ '{1,2,3}' as "is subset";

 is subset
-----------
 t
(1 row)

The Postgres Pro operator catalog has several entries for the infix operator <@, but the only two that could possibly accept an integer array on the left-hand side are array inclusion (anyarray <@ anyarray) and range inclusion (anyelement <@ anyrange). Since none of these polymorphic pseudo-types (see Section 8.21) are considered preferred, the parser cannot resolve the ambiguity on that basis. However, Step 3.f tells it to assume that the unknown-type literal is of the same type as the other input, that is, integer array. Now only one of the two operators can match, so array inclusion is selected. (Had range inclusion been selected, we would have gotten an error, because the string does not have the right format to be a range literal.)

Users sometimes try to declare operators applying just to a domain type. This is possible but is not nearly as useful as it might seem, because the operator resolution rules are designed to select operators applying to the domain's base type. As an example consider

CREATE DOMAIN mytext AS text CHECK(...);
CREATE FUNCTION mytext_eq_text (mytext, text) RETURNS boolean AS ...;
CREATE OPERATOR = (procedure=mytext_eq_text, leftarg=mytext, rightarg=text);
CREATE TABLE mytable (val mytext);

SELECT * FROM mytable WHERE val = 'foo';

This query will not use the custom operator. The parser will first see if there is a mytext = mytext operator (Step 2.a), which there is not; then it will consider the domain's base type text, and see if there is a text = text operator (Step 2.b), which there is; so it resolves the unknown-type literal as text and uses the text = text operator. The only way to get the custom operator to be used is to explicitly cast the literal:

SELECT * FROM mytable WHERE val = text 'foo';

so that the mytext = text operator is found immediately according to the exact-match rule. If the best-match rules are reached, they actively discriminate against operators on domain types. If they did not, such an operator would create too many ambiguous-operator failures, because the casting rules always consider a domain as castable to or from its base type, and so the domain operator would be considered usable in all the same cases as a similarly-named operator on the base type.