There's no way to avoid the product join because it's not an equi-join.
What are you trying to achieve?
Table A is probably a calendar and you'll get multiple rows for each row in Table B.
If you really need that you better use EXPAND ON.
What described above requires the product join, every row of b compared to every row of a. This is going to be expensive for a medium table joined to a large table. Are you sure there is no other join condition available - account, customerid,...?
One way to improve it some is to do a separate qualification of a.TheDay against the MIN of b.SDay since it appears that there rows in a.TheDay that won't qualify based solely on that criteria. That requires two selects unioned together, one to get the rows that are less than the max and one to join the rows greater than or equal to the max to table b.
What is the result of select count(*) from tbla where TheDay < '2014-08-01'? Trying to exclude the rows in tbla that cannot match from the product join.
Is the desired result really the product join result? Or is it the list of rows from tblb that don't have a date in tbla? The product join result of all the combinations seems to not be very useful IMHO.
the result is 5041, the exclusion is irrelevant.
for the outer questions, i don't know the reply, i have to talk with the development enviroment.
"The product join result of all the combinations seems to not be very useful "... it's true.
The optimization have a limit called logistic! ;)