Theory cs could care less about coding abilities. It's never a bad thing to know how to code, but I can go though the entire PhD program without coding a single line of code. (Well not really, one of the professors in my PhD program committee recommended me to do one systems course. other than that, I just prove theorems. )
Yes, we've got a bunch of people like that in our PhD program as well - but 100% of them, despite not writing a single line of code for years, could still easily get a paying job in industry and be mostly okay (though less happy) there.
If you can prove theorems, then you can also code CRUD websites by following a tutorial, and get paid for that.
If you can't write code if your life depended on it but can write papers on human interface or methodology issues, then you can get a job as a system analyst in any large software development company and keep not writing code there.
You then have to store how many levels down the real, as opposed to index, data is. If you have looped enough to have reached an index that is small, the depth count will on average be so large that it takes about as much space as your original data.
*presuming that you went to an undergraduate institution where the variance in student intelligence is large enough to detect an appreciable difference between the avg intelligence in these two majors.
The phenomenon of the Computer Science graduate who can't write a FizzBuzz program, or even the post-graduate who can't write a simple recursive function, is well attested. But a Math student who can't handle a recursive definition is unlikely to make it through the first term.
Why do you care about the high demand? We can put that aside for now--if you are a competent coder, you are in high demand...
If you want to have more math, then some subfield in CS theory is the way to go.
CS theory have lot of elegant math. complexity, data structure, algorithms, combinatorial optimization, computational geometry. All of them have nice set of mathematical tools you can use. There are also unexpected ones that uses more traditional mathematics, like universal algebra for CSP, functional analysis in graph embedding with little distortion, and topology for computational topology(well that seems obvious, there are certain uses for computational topology, read up on persistent topology, which I guess is part of machine learning now).
Of course, the demands are low for pure theory students. However you can do some practical work. For example http://www.tokutek.com/ , founded by professors who specialize in cache oblivious data structures. Some more practical ones include cache oblivious data structures, sublinear time algorithms, string related algorithms. In Google, there are researchers working on how to optimize ads.
Also, I just don't see how you are going to write non-boilerplate code anywhere. everything eventually become repetitive(unless you use Haskell, anything new become a paper.)
No, because as I said, learning statistics has helped my critical thinking improve - and that's something everyone needs. There was even an example in my comment regarding reduced susceptibility to yellow journalism, which can only be a positive to both the individual and society in general.
I also use simple algebra a lot in my every day life. By solving simple equations you can with little effort derive formulas for calculations you have never done before.
Statistics are useful in the way the parent post mentioned.
Logarithms are useful for better understanding some statistics, but this has less every day use than the two above.
And while I do not use calculus in my every day life I still find it very useful to reason about ordinary things in the terms of derivates and integrals. It has given my new ways to think about problems.
This is a common technique of switching between two algorithms. iirc, I never saw this as an material in undergrad algorithms course.
I wrote on another problem that uses this technique. Mix ternary search and linear search to find the minima of an first decreasing then increasing array.
