Re: Sine and Cosine Accuracy

[Andrew Haley <aph at redhat dot com>]

chris jefferson writes:
 > Scott Robert Ladd wrote:
 > 
 > >Marc Espie wrote:
 > >  
 > >
 > >>Heck, I can plot trajectories on a sphere that do not follow great circles,
 > >>and that extend over 360 degrees in longitude.  I don't see why I should be
 > >>restricted from doing that.
 > >>    
 > >>
 > >
 > >Can you show me a circumstance where sin(x - 2 * pi) and sin(x + 2 * pi)
 > >are not equal to sin(x)?
 > >
 > >Using an earlier example in these threads, do you deny that
 > >sin(pow(2.0,90.0)) == sin(5.15314063427653548) ==
 > >sin(-1.130044672903051) -- assuming no use of
 > >-funsafe-math-optimizations, of course?
 > >
 > >  
 > >
 > I would like to say yes, I disagree that this should be true. By your 
 > argument, why isn't sin(pow(2.0,90.0)+1) == sin(6.153104..)? Also, how 
 > the heck do you intend to actually calculate that value? You can't just 
 > keep subtracting multiples of 2*pi from pow(2.0, 90.0) else nothing will 
 > happen,

Actually you can, and this is how real floating-point packages work.  

Rether than speculate how things _might_ work, I invite you to have a
look at glibc sysdeps/ieee754/dbl-64/sincos32.c.  Accurate techniques
for range reduction are quite well-known, and this list is not an
appropriate place for tutorials on floating-point arithmetic.

Andrew.