And if you want to choose a subset of math for programming to be a subset of, you'd be better off choosing set theory[1].
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[1] Programming can be thought of as an application of the lambda calculus, but that is calculus in that it calculates, not in that it involves integrals and so on.
I don't have a degree in compsci, but I have enough of a degree in pure mathematics to suggest a background in abstract algebra as well/instead (though recursion theory is even more applicable, if generally taught at a level far too elementary to be really useful; and furthermore, the Church-Turing Thesis is mad hax in terms of teaching one to rigorously define functions, and really promotes laziness at introductory levels).
The reason why I say this is that, much as analysis is all about finding clever upper and lower bounds for things, algebra is all about finding clever functions. One does almost nothing else.
Set theory is good, too, but in general one doesn't get into the definition of clever functions very quickly... the earliest theorems I can think of to really exploit functions (having rigorously defined them, which takes a while) are the Schroder-Bernstein Theorem, proof of equivalence of existence of a right inverse for all surjective functions and AoC, and maybe, in a very elementary way, the set-theoretical definition of natural numbers/principle of recursive definition.
With algebra, on the other hand, one tends to get the concept of a morphism almost right away, having defined a couple of binary structures, and then you are on the road to crying about kernels and well-defined formulae. By the second week. It is very wonderful, and very, very exhausting.
And, furthermore, number theory is usually taught (at least at the beginning) through algebra, which will drill the importance of a quick algorithm into your head very, very quickly (I am looking straight at you, Extended Euclidean Algorithm. Straight at you).
But if you want to work with rapid convergence, what you really need is a strong background in real analysis... I was first introduced to big O notation in a mathematical text, after all.
And then linear algebra is of absolutely vital importance for a lot of applications, and also tends to cover programming concepts extensively.
Of course, there's also discrete mathematics, which is very interested in much of the combinatorial things one might find useful... and, again, fast algorithms (at least as I learned it, both before college and during).
*sigh*
Well, as one of my professors has informed me, every great computer scientist began with his or her feet solidly drenched in pure mathematics. Of course, that person is somewhat biased, but you may take his words or leave them as you like.