Re: converting to a wider data type
Van, Baker and P1788
On 11 Nov 2008, at 12:43, R. Baker Kearfott wrote:
> Van et al,
Is it really that complicated? This goes back to a basic difference
between us in point of view, reflected in both our view of floating
point conversions to intervals and interval-to-wider-interval
conversions. ... Looking at the underlying
mathematics, that means that, if [a,b] --> [aa,bb], what is
absolutely necessary is that aa <=a and bb <=b, when a, aa, b, and bb
are interpreted mathematically. I think IEEE 754-2008 directed rounding
between data types also specifies an "accuracy" of this conversion
in the sense that aa should be the largest number in the target
precision <= a and bb should be the smallest number in the target
precision >=a. a and b are completely defined mathematical
numbers, and there is no need to imagine any bits that aren't
actually there in a and b themselves. This statement is on the
"underlying model" level of discussion, but translates in a mechanical
way to bit patterns.
-----
Well stated, Baker! I agree 100%.
The great Jim Wilkinson used, and probably invented, the function
fl(x) = "real number x rounded to the nearest floating point value y",
with "nearest" in the sense of a chosen rounding mode.
It's clear from how *he* used it that y is just another real number like x, not something (a half-open
interval?) with "random bits". That's how everyone I know of has always done roundoff error analysis, let's
stick with it.
Best wishes
John Pryce
j.d.pryce@xxxxxxxxxxxx
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