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Re: about emp, mult by zero, exact results, exact endpoints.
Widest-Need expression processing provides a consistent and elegant solution to the problem of how to evaluate mixed-mode expressions.
It exists as an option in Sun's implementation of interval arithmetic in its Fortran compiler. See: 

<http://www.google.com/webhp?sourceid=chrome-instant&ion=1&ie=UTF-8#q=oracle%20widest-need&nfpr=1&ei=8YUaUaG8AszFiwKx3IDIBw&sqi=2&start=0&ion=1&bav=on.2,or.r_gc.r_pw.r_qf.&bvm=bv.42261806,d.cGE&fp=d5e4bc415c21e72b&biw=1024&bih=895>

and the two attached papers.



On 2/12/13 8:12 AM, Richard Fateman wrote:

what would sin(  [NaN,NaN; ill]) be?   [-1,1; what decoration???]
what would sin([NaN,NaN, emp) be?  [NaN,NaN; emp] ???
From a programming standpoint, at least one way of implementing multiplication would not work the way you suggest. If one does not have directed rounding 
per 754 but has to bump up and down, the results of rounding to nearest,
0*gg rounded up and 0*gg rounded down will differ and the result will not be [0,0], but [-eps,eps] for eps=smallest non-zero value in the chosen type (e.g. 2,2..*10^-308 for double). 

On the other hand, multiplication by a SCALAR INTEGER 0
or by a SCALAR float 0.0  could arguably result in something else.
For example, 0* [a,b] could result in a SCALAR INTEGER 0, 0.0d0*[a,b] could result in a scalar double-float 0.0d0. These would presumably not be decorated.
I know that some computer programming languages do not allow the type of returned result to vary in this way, so perhaps the interval concept should be strengthened 
to allow for intervals of the form [a,a;EXACT].

This would allow the representation of any "small enough" integer
as well as any rational number of the form r/2^s for "small enough r and s.
In those languages that allow exact rationals of arbitrary size, an operation that results in rational endpoints is not widened by bumping up or down; they are EXACT. I am personally unconvinced of the utility of having rational numbers of arbitrary precision as interval endpoints, but it makes 
sense mathematically.
This could be extended for those systems that provide access to the IEEE 754 INEXACT flag. That is, if there is no INEXACT operation (i.e. no rounding) in an interval arithmetic 
operation, the results are again EXACT.
I don't know if [a,b,EXACT] where a and b differ is especially more useful than [a,b,;DAC]. Again, I don't know if this has been discussed previously. apologies for possibly making a 
complicated system more complicated.

Richard Fateman


On 2/12/2013 12:13 AM, John Pryce wrote:

Guillaume

On 8 Feb 2013, at 06:05, Guillaume Melquiond wrote:

Let me quote you the relevant parts of 8.8.6:

"w_dw = phi(v_dv)" where v_dv is a box "v_dv_1, ..., v_dv_k"
"w \subseteq Rge(phi | v)" (23)
"p_dv_0(phi,v)" (24)
"dw = min(dv_0,dv_1,...,dv_k)" (25)
"where the minimum is taken with respect to the propagation order dac >
def > trv > emp > ill" (26)

As you can see from (25) above, the decoration dw of the output can
never be improved so that it becomes higher than the decorations
dv_0, ..., dv_k of the inputs.
Yes, that is true according to the current text. I have held to that principle for some time, partly for conceptual simplicity, partly because Nate strongly supported it.
But I think we should allow it to be varied. The simplest example is multiplying by a known zero, somewhere in an expression, so 
     f(x) = c*g(x)   where c=0
Interval evaluation will do
     ff = [0,0]*gg
and if gg has a better decoration than trv, surely ff should be given the best decoration, so com if supported? So def or dac can -> com.
Another IMPORTANT situation is the case(c,g,h) function where Dmitry has pointed out the 8.8.6 rules are incompatible with the intention that one can "streamline" case(cc,gg,hh) by not evaluating hh [resp. gg] if cc<0 [resp. cc>=0]. 

John Pryce

Attachment: widest-need.pdf
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Attachment: WidestNeed.PDF
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