Background: I am entering college next year. I really enjoy making stuff and solving logical problems, so I'm thinking of majoring in compsci and working in software development. I hope to have the kind of job where I can work with implementing / improving algorithms and data structures on a regular basis.. as opposed to, say, a job that's purely concerned with mashing different libraries together, or 'finding the right APIs for the job'. (Hence the word 'low-level' in the title. No, I don't wish to write assembly all day.)
Thing is, I've never been the most logically-sharp person. Thus far I have only worked on hobby projects, but I find that I make the silliest of errors ever so often, and it can take me ages to find it. Like anywhere between three hours to a day to locate a simple segfault, off-by-one error, or other logical mistake. (Of course, I do other things in the meantime, like browsing SO, reddit, and the like..) It's not like I'm 'new' to programming either; I first tried C++ maybe five years ago.
My question is: is this normal? Should a programmer with any talent solve it in less time? Having read Spolsky's Smart and gets things done, where he talks about the large variance in programming speed, am I near the bottom of the curve, and therefore destined to work in companies that cannot afford to hire quality programmers?
I'd like to think that conceptually I'm okay -- I can grasp algorithms and concepts pretty well, I do fine in math and science, although I probably drop signs in my equations more often than the next guy. Still, grokking concepts makes me happy, and is the reason why I want to work with algorithms.
I'm hoping to hear from those of you with real-world programming experience.
TL;DR: I make many careless mistakes, should I not consider programming as a career?