Postgresql – How to get postgres to log the stored procedures it is running

It is true that you cannot grant EXEC permissions on a function that returns a table. This type of function is effectively more of a view than a function. You need to grant SELECT instead, e.g.:

GRANT SELECT ON dbo.Table_Valued_Function TO [testuser];

So your script would look more like this (sorry, but I absolutely loathe INFORMATION_SCHEMA and much prefer to use the catalog views, which also don't need functions like OBJECTPROPERTY):

DECLARE 
    @sql      NVARCHAR(MAX) = N'',
    @username VARCHAR(255)  = 'testuser';

SELECT @sql += CHAR(13) + CHAR(10) + N'GRANT ' + CASE 
    WHEN type_desc LIKE 'SQL_%TABLE_VALUED_FUNCTION'
      OR type_desc = 'VIEW'
    THEN ' SELECT ' ELSE ' EXEC ' END 
    + ' ON ' + QUOTENAME(SCHEMA_NAME([schema_id])) 
    + '.' + QUOTENAME(name) 
    + ' TO ' + @username + ';'
 FROM sys.all_objects
WHERE is_ms_shipped = 0 AND
( 
    type_desc LIKE '%PROCEDURE' 
    OR type_desc LIKE '%FUNCTION'
    OR type_desc = 'VIEW'
);

PRINT @sql;
-- EXEC sp_executesql @sql;

Now you can grant EXEC on a schema, and always create these procedures in that schema (actually one of the purposes of schemas!), which @jgardner04 already suggested, however in order for this solution to apply to table-valued functions as well, you'd also have to grant SELECT. Which is okay if you're not storing any data in tables in that schema (or at least that you want to hide from them), but it will apply to any tables and views as well, which might not be your intention.

Another idea (e.g. if you can't, or don't want to, use schemas) is to write a DDL Trigger that captures the CREATE_PROCEDURE, CREATE_FUNCTION and CREATE_VIEW events, and grants permissions to a user (or a set of users, if you want to store them in a table):

CREATE TRIGGER ApplyPermissionsToAllProceduressAndFunctions -- be more creative!
    ON DATABASE FOR CREATE_PROCEDURE, CREATE_FUNCTION, CREATE_VIEW
AS
BEGIN
    SET NOCOUNT ON;

    DECLARE
        @sql       NVARCHAR(MAX),
        @EventData XML = EVENTDATA();

    ;WITH x ( sch, obj ) 
    AS
    (
        SELECT
          @EventData.value('(/EVENT_INSTANCE/SchemaName)[1]',  'NVARCHAR(255)'), 
          @EventData.value('(/EVENT_INSTANCE/ObjectName)[1]',  'NVARCHAR(255)')
    )
    SELECT @sql = N'GRANT ' + CASE 
      WHEN o.type_desc LIKE 'SQL_%TABLE_VALUED_FUNCTION' 
        OR o.type_desc = 'VIEW'
      THEN ' SELECT ' 
      ELSE ' EXEC ' END 
        + ' ON ' + QUOTENAME(x.sch) 
        + '.' + QUOTENAME(x.obj) 
        + ' TO testuser;' -- hard-code this, use a variable, or store in a table
    FROM x
    INNER JOIN sys.objects AS o
    ON o.[object_id] = OBJECT_ID(QUOTENAME(x.sch) + '.' + QUOTENAME(x.obj));

    EXEC sp_executesql @sql;
END
GO

The drawback I find with DDL Triggers is that you can quickly forget that they're there. So a year down the road when you decide to stop granting these permissions to all new objects, it might take a while to troubleshoot why it's still happening. At my last job we logged all actions invoked by DDL triggers to a central "event log" of sorts, and that was our go-to place for tracking down any actions that happened on the server that nobody seems to remember (and it was a DDL trigger about half the time). So you may consider adding some additional logic that will help with that.

EDIT

Adding code for schema-based, and I'll mention again that this will grant permissions on any procedures, functions and tables created in the foo schema.

CREATE SCHEMA foo;
GRANT EXEC, SELECT ON SCHEMA::foo TO testuser;

Now if you create the following procedure, testuser will be able to execute:

CREATE PROCEDURE foo.proc1
AS 
BEGIN
    SET NOCOUNT ON;
    SELECT 1;
END
GO

To be more precise, NOT FENCED routines run in the same process space as the database manager itself. The engine is written in C, so calling a not fenced routine is just like calling another C function from main(). This is where all the memory corruption and performance aspects come from: a not fenced routine has access to all the same resources -- memory, files, etc. -- as the database manager process itself.

For a FENCED routine the database manager launches a separate process (db2fmp), which in turn executes the routine code. As a result the operating system protection prevents a fenced routine to access any memory area or resource that belongs to the database manager.

SQL routines cannot be fenced, strictly speaking, because they don't "run", but they are even better than not fenced -- they are bytecode that the DB2 runtime engine itself executes, so there's no separate thread or process.

C and C++ routines can be fenced, in which case they execute in separate processes, or not fenced, in which case they are loaded into the database manager process space and called as functions.

Java routines can only be fenced by the fact that they need a separate process, Java virtual machine, to execute. If you declare them as NOT FENCED, the option is quietly ignored.

Having said all that, your only choice between fenced and not fenced is with C/C++ routines. Typically you would run the fenced for safety, changing to the not fenced mode only when you are pretty sure they can't harm the database manager and they need higher performance¹.

1 - Performance difference between fenced and not fenced routines comes from the overhead associated with allocating a fenced process, as well as with interprocess communications between the database manager and the not fenced routine. Even then, the fenced process is not created every time a fenced routine is called; a pool of them will be created and reused for such calls. All this means that you might see the benefit of declaring a routine as fenced only if that routine is called very frequently (as in, tens of times per second or more).