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From: Terje Mathisen <terje.mathisen@tmsw.no>
Newsgroups: comp.lang.c,comp.arch
Subject: Re: Radians Or Degrees?
Date: Fri, 15 Mar 2024 12:16:27 +0100
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MitchAlsup1 wrote:
> And thirdly::
> 
> Let us postulate that the reduced argument r is indeed calculated with 
> 0.5ULP
> error.
> 
> What makes you think you can calculate the polynomial without introducing
> any more error ??

It should be obvious that any argument reduction step must return a 
value with significantly higher precision than the input(s), and that 
this higher precision value is then used in any polynomial evaluation.

With careful setup, it is very often possible to reduce the amount of 
extended-procision work needed to just one or two steps, i.e. for the 
classic Taylor sin(x) series, with x fairly small, the x^3 and higher 
terms can make do with double precision, so that the final step is to 
add the two parts of the leading x term: First the trailing part and 
then when adding the upper 53 bits of x you get a single rounding at 
this stage.

This is easier and better when done with 64-bit fixed-point values, 
augemented with a few 128-bit operations.

Terje

-- 
- <Terje.Mathisen at tmsw.no>
"almost all programming can be viewed as an exercise in caching"