trig functions

[Dik T. Winter]

(I did already respond to the original article, but follow-up now.)

In article <9011131959.AA11389@> lmo at lsr-vax.UUCP ("Lance M. Optican - LMO") writes:
 > ----- Begin Included Message -----
...[lots of included message omitted]...
 > Is it reasonable to expect the cosine of (exactly) 0 to be (exactly) 1?
 > ----- End Included Message -----
 > It is reasonable to expect the maht library to perform well!
Yes, but that does not imply that cos(0)=1.
 >                                                  What standards
 > of compatibility are enforced with other machine architectures?
What do you mean?  Which machine architecture?  Honeywell?  (where cos(0)>1).

The problem is more complex than you think.  When implementing cos and sin
you have two algorithmic choices (if you use additional precision you've
got a few more choices):
1.  Promise that the functions are monotonic at pi/4, but do not promise that
    cos(0)=1.  This is what you get if you implement Cody-Waite's functions.
    You might even end up with cos(0)>1.  When you reduce the range of the
    argument to (-pi/2,pi/2) and do an approximation there you end up with
    this behaviour (not exactly true, but nearly).
2.  Promise that cos(0)=1, but do not promise that the functions are monotonic
    at pi/4 (i.e. in the neighbourhood of pi/4 it can happen that x>y but
    sin(x)<sin(y)).  This is the way I prefer it and implement them.  You get
    this behaviour if you reduce the range to (-pi/4,pi/4) and choose whether
    you need a cosine or a sine approximation.  This will also make the
    numerical derivative at x=0 correct; but that is a moot point as it will
    be very bad at x=pi/4!
In case 1 you can still make a choice:
3.  Special case x=0.  This may create non-monotonic behaviour in the
    neighbourhood of x=0 (it might even be that some results in the
    neighbourhood are larger than 1).

Still, even if cos(0)!=1 it may very well be within the accuracy constraints!
You should be much more apprehensive of implementations where sin(pi)=0.  Throw
them away, they are wrong!
--
dik t. winter, cwi, amsterdam, nederland
dik at cwi.nl