Calculator algorithms

See CORDIC. See also this question: https://stackoverflow.com/questions/2169641/where-to-find-algorithms-for-standard-math-functions/2169666 and this review: http://mathdl.maa.org/mathDL/19/?pa=reviews&sa=viewBook&bookId=65790

I would recommend reading Gerald Rising's Inside your Calculator (which has a supplementary website); there is a nice discussion of the methods used by some calculators that is suitable at the undergraduate level.

Otherwise, to really figure out what methods they are using, it might help to search the technical notes of the manufacturer's websites. For instance, Texas Instruments has notes like this one on their "knowledge base" that discuss "what's under the hood", though not in detail of course. (Sometimes, hobbyist sites like this one also discuss calculator algorithms.)

The GNU Multiple Precision Arithmetic Library (GMP) has a very good documentation describing how they implemented all the arithmetic functions for their multiple precision library:

http://gmplib.org/manual/

Give it a try!

UPDATE: I know the GMP is used for big number computation, but arithmetic is arithmetic, and to make big number computations you must understand small number computations, so I am sure (or at least "almost" sure) the documentation mention what is enough to for calculator arithmetic.