In his famous essay How to Become a Hacker, Eric Raymond says that you need basically zero mathematical training. Now this essay is talking about hacking as in writing open-source system-level software; for example if you wanted to write a disk backup utility or a network monitoring program.
The range of software can vary from fairly straightforward (web page scripting) to very complex (airline reservation systems), with a rich spectrum in between. For most of it, you don't need any math training at all. However a few points:
- I can think of no instance in which having some mathematical training hurts you. Math teaches us how to think carefully and critically, which is a central skill for good scientists, including computer programmers. We still require calculus, linear algebra, and discrete math classes for a B.S. in Computer Science for a reason.
- There are programming careers where math knowledge is essential. These include most HPC jobs, optimization, physical simulation, cryptography, finance, weather forecasting, traffic routing, etc
- If you want a job at a top-level place (eg, Google) you will be expected to know something about math. Or rather about complexity, data structures, and algorithms which really are math. For example, if you don't realize that your algorithm running in $T(n)=2T(n/2) + O(n)$ is considerably faster than another running in $O(n^2)$, then you will probably be shown the door.
Even working as a database developer years ago, I found my math training useful (I had to implement arbitrary-precision arithmetic in software for the DECIMAL datatype that is part of standard SQL... doing division correctly and efficiently is non-obvious).
Finally, a lifelong passion for math has helped me engage better with like-minded programmers. I don't find as much affinity with coders whose preoccupation is minutiae about syntax or language fads. The beautiful part of programming, at its core, is really just mathematics.