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From: jlg@lanl-a.UUCP
Newsgroups: net.micro.68k,net.arch
Subject: Re: binary normalization
Message-ID: <13545@lanl-a.UUCP>
Date: Tue, 18-Sep-84 17:03:05 EDT
Article-I.D.: lanl-a.13545
Posted: Tue Sep 18 17:03:05 1984
Date-Received: Tue, 25-Sep-84 09:31:23 EDT
References: <1191@rti-sel.UUCP>, <4190@fortune.UUCP>, <420@watdcsu.UUCP>, <1422@wateng.UUCP>
Organization: Los Alamos National Laboratory
Lines: 32



It should be remembered that floating point numbers are a subset of the
rationals, not the reals.  Therefore, given the real line, only certain 
points are exactly representable in a give floating point number scheme.

If S(b,n,M) is the set of floating point numbers representable with n base
b digits between 1/M and M, then |S(b,n,M)| is the size of this set.  Note
that exponent range limits are excluded from this definition.  S can be 
regarded as a density function, i.e. the number of elements within a given
range.  A good definition of 'equivalent precision' for two floating point
systems would require that the two systems have the same density.  So 
define two systems as giving 'equivalent precision' if:

                         S(d,m,M)
                  lim   ---------- = 1
                 M->inf  S(b,m,M)

Matula [1] has shown that this is equivalent to:

         m = n log  b + log  [d(b-1)/b(d-1)] - log log  b
                  d        d                      d   d

Therefore, 24 bit binary is equivalent to about 6.27 hexadecimal digits.
Alternately, 24 bits = 7.49 decimal digits, and 6 digits hex = 7.16 decimal
digits.  IEEE clearly comes out superior to IBM (for example).  Restricting
the exponent range has no effect of this comparison, an application
which needs a wider range than provided by IEEE single would still be
better off with binary normalized arithmetic (use IEEE extended formats).

[1] A Formalization of Floating-Point Numeric Base Conversion, IEEE
    Transactions on Computers, vol. C-19, No. 8, August 1970, pp. 681-692.