🚀 go-pugleaf

In article <81mmk2.tjm@netcom.com> tjm@netcom.com (Thomas J. Merritt) writes:
>I would like to be able to calculate sin/cos/... with ~60 digits of
>precision using a floating point format that has a 224 bit mantissa.
>
>I'm looking for references on how to calculate the coefficients that
>would be needed for the polynomial approximations.  Cody and Waite
>provide a greate discussion on how to write standard functions but only
>list a limited set of coefficients none of which are helpful.
>Computer Approximations by Hart, Et. Al., probably has enough theory in
>it to derive things.  The tables that they provide would be great but,
>agian, they do not provide the requiste precision for a 224 bit
>mantissa.  I would much prefer a more applied discussion rather than
>a theoretical one.

Hart et al describe algorithms (I believe it's "Remez' 2nd algorithm"
but I'm a little fuzzy on that) to *compute* from "first principles"
the coefficients for approximations of *any* desired accuracy.  To do
this you need (only) high-precision arithmetic and reference values,
both of which can probably be had most easily from your local symbolic
algebra system.  For example:


Script started on Wed Jul 29 17:02:10 1992
% maple
    |\^/|      MAPLE V
._|\|   |/|_.  Copyright (c) 1981-1990 by the University of Waterloo.
 \  MAPLE  /   All rights reserved.  MAPLE is a registered trademark of
 <____ ____>   Waterloo Maple Software.
      |        Type ? for help.
> Digits := 60;

                                  Digits := 60

> cos(1.0);

         .540302305868139717400936607442976603732310420617922227670097

> 4.0 * arctan(1.0);

         3.14159265358979323846264338327950288419716939937510582097494

> quit();
bytes used=161648, alloc=196572, time=.400
% exit
% 
script done on Wed Jul 29 17:04:11 1992


- Jonathan Thornburg
  <jonathan@einstein.ph.utexas.edu> or <jonathan@hermes.chpc.utexas.edu>
  University of Texas at Austin / Physics Dept / Center for Relativity
  and (for a few more months) U of British Columbia / {Astronomy,Physics}