Re: Motion P1788.1/M004.01

[Walter Mascarenhas <walter.mascarenhas@xxxxxxxxx>]

PS, in the previous message, I meant to say that

there is a gap between the rational numbers with

a bounded denominator in the interval [-M,M],

otherwise my statement is obviously false.

On Wed, May 18, 2016 at 3:56 AM, Walter Mascarenhas <walter.mascarenhas@xxxxxxxxx> wrote:

> Richardson's theorem does not apply to IEEE 754 (my context), since
> only finite sets of rational numbers are involved -- no Pi, log 2, or
> (to mention the one reamining requirement after some of the original
> ones were found to be inessential) an exact sin() function.

The log and atan functions are part of IEEE 754. So, math expressions
as those involved in Richardson's theorem can be written with IEEE 754
constants and operations.

 The math expressions can be written in IEEE, but their meaning is different.

In other words, the _expression_ "sin(sin(x))" in IEEE means something different
from the _expression_ "sin(sin(x))" in the pure mathematical sense.

   This difference is essential for the practical implementation of trigonometric

functions. Argument reduction techniques for instance rely heavily on the

fact that there is a gap between the floating point numbers and the

multiples of pi.

   I am not familiar with the proofs of Richardson's theorem, but I would

not be surprised if they relied on the fact that the gap between multiples

of pi in the interval [-M,M] and the rational numbers on this interval

goes to zero as M tends to infinity.

   In the IEEE world there is no such things as "the limit as M

tends to infinity", and algorithms can and do take advantage

of this fact. This is why it is not clear to me either whether

Richardson's theorem applies in this context.