I always thought that algebraic functions are not guaranteed to be defined given certain parameters. It's just that perfectly algebraic functions don't throw errors, they silently return +-infinity. Like the asymptotes in `tan x`.
Algebraic functions are just syntactic notation. You can sit down and convert from one notation to another, like how cos(5) is a number quite close to 0.28366218546322625, and deriving one representation from the other does take resources and time, because it's a physical process performed by a person or computer.
But sin, cos, tan, cotan, log and all their friends by themselves don't compute, they are just a different kind of notation for numbers.
Which is why I find the desire to make functions in programming like algebraic functions silly - by definition they are two completely different things. One is a specification for a process that produces binary-encoded numbers, the other is a syntactic notation for real numbers.
You seem to be implicitly defining "computation" as that which a physical computer does (machines, biological brains...). Something that literally consumes physical resources. By that logic, a Turing machine is not a computer and lambda calculus is not about computation. As I said, you could spin the semantics that way but is it useful? Does that give us useful insights?
Functions are not just syntactic notation. Functions are, by definition, mappings from set A to set B. They don't have anything to do with notation. "cos(x)" is merely a notation, yes, but not of a number but of a function. This is an important distinction. "cos(5)" evaluates to a certain number, yes, but it's not just syntactic sugar for that number. Not to mention that functions don't need to operate on sets of numbers.
It's precisely that sense of function that things like Haskell (or Erlang, or Prolog) try to introduce - a different syntax for writing the same value.
A pure function is just explaining that a particular value is functions applied to other particular (stand in) values (and a way to introduce names to values). These are only useful for explaining which computation you'd like to happen when the program actually goes to carry one out - I'd like the number which is the same as the cosine function applied to 5, which tells it precisely how to compute that number (assuming it can build an algorithm equivalent to cosine). Of course, we could use another pure definition (that cosine of x is the same as a series expansion about x) to allow the compiler to replace the desired cosine function with something that has intrinsics on the system - evaluation a polynomial at the point 5.
So in essence, we can use functions as a tool to describe what value we want from the program (or various structural properties), in such a way that the compiler can correctly infer how to build an algorithm to do so, and tie it correctly in to our (actually) algorithmic code.
Minor aside: functions (in the math world) apply in contexts besides the real numbers, including being syntactic notation for binary numbers (or fields of order 2^n).
http://math.stackexchange.com/questions/189621/is-tan-pi-2-u...
No, they are a notation for ideas. The fact that they have numerical values associated with them is true but misses the point that trigonometric functions are primarily defined with respect to each other and with respect to certain geometric concepts.
In other words, cos(0) equals 1, but 1 has many more meanings than just cos(0). I would much rather have a student tell me that tan(x) = sin(x) / cos(x) than to say tan(x) is just a number.
Most generally, a function is not a number, it is a relation between two sets. One can define such things in a number of ways, but one way is to describe the relation via a formal language such as the lambda calculus. The lambda calculus is not technically a "programming language", it is a formal system of meaningless symbols and a set of rules for forming, manipulating, and eliminating them.
Although there is no fixed relation between reduction rules in the lambda calculus and physical resources in a computer, one can still compare the number of reduction steps required to determine which form in the co-domain of a function defined in the lambda calculus corresponds to a specific form in its domain, and this will provide a reasonable stand-in for comparisons between resource usage of similar computer programs.
So, really, computation is not completely foreign to mathematics, and mathematical functions and "computing functions" are not completely different animals, just... distantly related? Some languages are more mathematical in their approach than others.