Re: Sine and Cosine Accuracy

[Scott Robert Ladd <scott dot ladd at coyotegulch dot com>]

Paul Koning wrote:
>  Scott> Yes, but within the defined mathematical ranges for sine and
>  Scott> cosine -- [0, 2 * PI) -- the processor intrinsics are quite
>  Scott> accurate.
> 
> Huh?  Sine and consine are mathematically defined for all finite
> inputs. 

Defined, yes. However, I'm speaking as a mathematician in this case, not
a programmer. Pick up an trig book, and it will have a statement similar
to this one, taken from a text (Trigonometry Demystified, Gibilisco,
McGraw-Hill, 2003) randomly grabbed from the shelf next to me:

"These trigonometric identities apply to angles in the *standard range*
of 0 rad <= theta < 2 * PI rad. Angles outside the standard range are
converted to values within the standard range by adding or subtracting
the appropriate multiple of 2 * PI rad. You might hear of an angle with
negative measurement or with a measure more than 2 * PI rad, but this
can always be converted..."

I can assure you that other texts (of which I have several) make similar
statements.

> Yes, normally the first step is to reduce the arguments to a small
> range around zero and then do the series expansion after that, because
> the series expansion convergest fastest near zero.  But sin(100) is
> certainly a valid call, even if not a common one.

I *said* that such statements are outside the standard range of
trigonometric identities. Writing sin(100) is not a matter of necessity,
nor should people using "regular" math be penalized in speed or accuracy
for extreme cases.

..Scott