sum vs add up

sum is a noun but add up is not a noun.

Word	Noun	Adjective	Verb	Adverb
sum	Yes	No	Yes	No
add up	No	No	Yes	No

As verbs, sum and add up are synonyms defined as:

Other synonyms of sum include add, add together, sum up, summate, tally, tot, tot up, total, tote up.

sum (noun)	add up (noun)
the whole amount
the choicest or most essential or most vital part of some idea or experience
a quantity obtained by the addition of a group of numbers
the final aggregate
a set containing all and only the members of two or more given sets
a quantity of money

sum (verb)	add up (verb)
determine the sum of	determine the sum of
be a summary of	be reasonable or logical or comprehensible
	to measure up to in kind or quality
	add up in number or quantity