computer algebra - Bounds on remainder term of power series of elementary functions

This is mainly a question about the remainder term of power series for elementary functions.

I'm very interested in aspects of calculating or computing elementary operations and functions, by which I mean:

• trigonometric: $sin$, $cos$, $tan$


• inverse trig.: $sin^{-1}$, $cos^{-1}$, $tan^{-1}$


• log and exponential: $ln$, $exp$


• hyperbolic: $sinh$, $cosh$, $tanh$


• inverse hyp.: $sinh^{-1}$, $cosh^{-1}$, $tanh^{-1}$


• powers, reciprocation, $sqrt{ }$

perhaps also:

• gamma function: $Gamma$


• and a few other important functions

There are many contexts (of calculation). For example:

• real versus complex arguments


• known, fixed precision versus variable precision


• numerical versus symbolic

There are many approaches and techniques available too. For example:

• power series expansions and polynomial approximations


• use of relationships between the functions


• use of periodic or similar properties to shrink the domain


• lookup tables and interpolation


• CORDIC (used within some hand calculators I believe)


• exact methods


• interval or other error-tracking methods

Some good references to certain aspects include:

The main gap in my knowledge is in finding bounds for the error or remainder term in partial power series expansions of certain of the above functions. Some are fairly simple to determine, whilst others seem to be awkward.

Any pointers on this matter would be much appreciated.

Likewise for any further references on any other aspects of or techniques for calculating elementary functions.