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From: anton@mips.complang.tuwien.ac.at (Anton Ertl)
Newsgroups: comp.arch
Subject: Re: Round to Nearest at Binade
Date: Tue, 22 Apr 2025 15:09:09 GMT
Organization: Institut fuer Computersprachen, Technische Universitaet Wien
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mitchalsup@aol.com (MitchAlsup1) writes:
>In a message over on Unum NG, there was a statement which got
>me thinking about whether Round to Nearest is properly defined
>in IEEE 754 (all the way back to 1985) in a theoretic sense
>not in a practical sense.
>
>Consider a calculation which produces in a result prior to
>rounding with a fraction:: 1.1111111...11111 and 1 as the
>½ ULP bit--properly ½ way between nearest representable
>points except for the binade switching points. (ignore sticky}
>
>Since rounding up increments the exponent, the closest number
>above the result is 2 units of ½ ULP above

No.

Let's consider a (before rounding) 4-bit mantissa (plus hidden bit),
and a 3-bit mantissa after rounding, and the full-length unrounded
number in binary is

x = 1.1111 e0

The closest number above is (before normalization)

y = 10.0000 e0

and after normalization and with the correct number of mantissa bits:

y = 1.000 e1

Nothing is lost in this normalization.  The closest representable
number below x is

z = 1.111 e0

Both y and z are representable in FP with a 3 bit mantissa (plus
hidden bit), and both are 0.0001 e0 away from x.  So when you round
from 1.1111 e0 to a 3-bit mantissa with round-to-nearest-or-even, the
effect you claim does not exist.  And thanks to the or-even rule, the
rounded number is y.

However, there is one case where the effect probably exists. Consider

1.1111 eM

where M is the maximum finite exponent for our FP numbers.  Do you
round up towards infinity, or down to the maximum finite FP number
with 3-bit mantissa?  I have not tried, but I expect rounding to
infinity in this case.

If you rounded to the maximum finite FP number, at what intermediate
result would you stop doing that and round to infinity?

>while the closest
>number below is only 1 unit of ½ ULP below. A span of 3 units
>of ½ ULP. The midpoint of a span of 3×½ is 3/4 not 1/2.

Unums may have this problem, but IEEE FP numbers don't.

>I know this ship sailed <gosh> 40 years ago, and I am not trying
>to alter it, I am trying to wrap my head around the numerics of
>it only.

Maybe hanging around in a Unum NG is not good for you:-)

- anton
-- 
'Anyone trying for "industrial quality" ISA should avoid undefined behavior.'
  Mitch Alsup, <c17fcd89-f024-40e7-a594-88a85ac10d20o@googlegroups.com>