| Date: Sun, 24 Jul 2005 22:38:24 -0400
 | From: Paul Schlie <xxxxxx@comcast.net>
 |
 | > From: Aubrey Jaffer <xxxxxx@alum.mit.edu>
 | >  | Date: Thu, 21 Jul 2005 20:50:05 -0400
 | >  | From: Paul Schlie <xxxxxx@comcast.net>
 |  ....
 | >  | - thereby the range of all numerical transforms map to a
 | >  | correspondingly representable domain (although may optionally
 | >  | signal a run-time exception as may be desired in certain
 | >  | circumstances).
 | >
 | > This would have been nice, but the smallest (unnormalized)
 | > numbers in IEEE-754 are not symmetrical with the largest
 | > magnitude numbers:
 |
 | - which would imply that unnormalized values would be invalid
 | inexact values. (as only normalized floats would satisfy the
 | symmetry specified)

If you eliminate unnormalized numbers, then the difference between all
pairs of positive flonums having the most negative exponent is 0,
which isn't right.  The way to achieve symmetry is to choose a
different bias for the exponent.  But you have work through the math;
possibly one of the exponent values would also need to be unused.