Re: Midpoint paper
On 2012-02-08 16:47:08 -0600, Nate Hayes wrote:
> Attached is a new draft of the midpoint paper. Section 2.1 and the
> conclusion has been completely re-written to reflect recent discussions.
Page 6, with property (26) at Level 2, property (16) is not necessarily
satisfied for a number format that is not symmetrical. For instance,
consider:
* F = { -oo, -53, 51, 54, 100, 148, 153, 200, 204, +oo }
* X = [100,200]
* Y = [100,204]
Then
* midpoint(X) = 148, radius(X) = max(|RD(-48)|,|RU(52)|) = 54
* midpoint(Y) = 153, radius(Y) = max(|RD(-53)|,|RU(51)|) = 53
If the number format is symmetrical, property (16) is satisfied,
assuming that midpoint([a,b]) is defined by RN((a+b)/2). Indeed,
let X = [u,v] and Y = [u,v'] with v' > v, and let's try to build
a case where radius(Y) < radius(X). We necessarily have:
* midpoint(Y) > midpoint(X)
* midpoint(Y) - midpoint(X) > v' - v
* d = (u+v)/2 - midpoint(X) > 0 otherwise
radius(X) = RU(midpoint(X) - u) <= RU(midpoint(Y) - u) = radius(Y)
* midpoint(Y) - (u+v)/2 >= d since midpoint(X) = RN((u+v)/2)
* midpoint(X) + midpoint(Y) >= u + v
* midpoint(Y) - u >= v - midpoint(X)
* radius(X) = RU(v - midpoint(X)) <= RU(midpoint(Y) - u) = radius(Y)
which shows that one can't have radius(Y) < radius(X).
Concerning the proof of property (18), due to property (16), you can
consider only the largest bounded interval, [-Fmax,Fmax].
--
Vincent Lefèvre <vincent@xxxxxxxxxx> - Web: <http://www.vinc17.net/>
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Work: CR INRIA - computer arithmetic / AriC project (LIP, ENS-Lyon)