Re: math function accuracy -- was Re: text2interval again /
On 2013-03-12 10:30:12 -0500, Ralph Baker Kearfott wrote:
> Is this argument meant to help us come to a consensus on
> what accuracy (if any) we are to demand of the end points
> of enclosing intervals for the elementary functions in
> P-1788, i.e. what should be in the second column of
> Table 8, p. 43 of the draft John just passed around.
> Are we saying the "accurate" requirement for "sin" might
> be too strict?
Perhaps, in particular because of the fact that the rounding direction
is not necessarily honored mainly the math functions. But it may also
not be available, e.g. in case of static rounding modes, in which case
the arithmetic operations (+, -, *, /, sqrt) are affected too.
So, the only requirements could be containment ("valid" accuracy mode)
and other accuracy modes could be recommendations.
But for 754-conforming P1788 implementations, you can add the
associated "tightest" requirements for each supported function
(this includes the required functions +, -, *, /, etc.).
As I've said, this is less a problem than with floating point, because
if the user gets a tight interval, he knows that the error is small.
He doesn't have to assume anything about the quality of the
implementation.
> Of course, error bounds for most of these functions
> are easy, e.g. sin(10^22) \in [-1,1],
For most of these, this is R (Entire). So, in practice, the P1788
implementer would have to talk with the developers of the numerical
library, unless he wants to rewrite everything for efficiency reasons
(e.g. to do a single range reduction for f([a,b]) when a and b are
close to each other).
> but the tightness (or in floating point parlance the "accuracy") of
> these bounds might be another matter.
--
Vincent Lefèvre <vincent@xxxxxxxxxx> - Web: <http://www.vinc17.net/>
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