Does there exist a good reference on the algorithms used by calculators, especially on the trigonometric and transcendental functions?
I would still like to know how Casio generates its random numbers. I still wonder if they are any good.
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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.)
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
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:
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.
The algorithms are typically common, but the implementation changes.
Early calculators like the HP-41 stored numbers in BCD format. Modern machines are more likely to use IEE-754 formats, in single or double precision.
For modern stuff, any math library should have what you want. For older formats, look for websites like the HP calculator Museum.