The raison d'être of relational calculus is the formalization of query optimization, which is finding more efficient manners to execute the same query in a database.
A relational algebra expression might prescribe the following steps to retrieve the phone numbers and names of book stores that supply Some Sample Book: A relational calculus expression would formulate this query in the following descriptive or declarative manner: The relational algebra and the domain-independent relational calculus are logically equivalent: for any algebraic expression, there is an equivalent expression in the calculus, and vice versa.
The raison d'être of the relational calculus is the formalization of query optimization.
Query optimization consists in determining from a query the most efficient manner (or manners) to execute it.
Query optimization can be formalized as translating a relational calculus expression delivering an answer A into efficient relational algebraic expressions delivering the same answer A.