NetNews Usenet Archive 1993 #1/NN_1993_1.iso/spool/sci/math/17731

Internet Message Format | 1993-01-06 | 3.2 KB

Path: sparky!uunet!mcsun!sun4nl!cwi.nl!dik
From: [e-mail] (Dik T. Winter)
Newsgroups: sci.math
Subject: Re: Rounding Rules
Message-ID: <[e-mail]>
Date: 5 Jan 93 23:54:15 GMT
References: <[e-mail]> <[e-mail]> <a_rubin.726185517@dn66>
Sender: [e-mail]
Organization: CWI, Amsterdam
Lines: 51

(Thrown away all attributions to not confuse :-).)

 > >> The problem with many of the "rounding rules" is that they are biased.
 > >> In the IEEE round of rules, an attempt is made to make rounding as unbiased
 > >> as possible. Round to even is specified as the rule of choice, since it
 > >> avoids many of the problems of "round to infinity" or "round towards zero".
 >
 > >My personal favorite rounding rule is to round anything that has {0,1,2,3,4} in
 > >the "rounding place" "down" and anything that has {5,6,7,8,9} "up".  So as in
 > >the previous example, 1.13500 would be rounded to 1.14 and 1.14500 would be
 > >rounded to 1.15.
 >
 > >To this my teachers always gasped that this method was biased, and no amount of
 > >pursuasion using the above-mentioned sets could pursuade them.
 >
 > Well, it IS biased.  Rounding n places is biased by +1/2 10^-n.
 >
It depends on what context you are using.  It is true when rounding real
numbers (as in mathematics).  It is not true when rounding floating point
numbers.  When considering floating point arithmetic (with finite precision)
you should not consider all infinite precision results before rounding with
equal probability.  Rather you ought to consider the range of possible
inputs and the operation involved and conclude from that point whether the
rounding is biased or not.

However, this leads to a more interesting case: is unbiased rounding
in floating point arithmetic superior to biased rounding?  The answer
is *no*, there are unbiased roundings that are worse than simply
truncating to zero.  While the mean is correct, an actual result can
be worse.  An example provides the multiplication on the Cray.  (Now
bear with me, this is from memory so some numbers may be wrong, but
the way it works is like this.)  When multiplying two numbers (with
48 bit mantissa) together, the Cray does not calculate all 48*48 bits
of intermediate results to add them together.  Rather in the 48*96
array to hold the intermediate results it skips the last 16(?)
positions and adds 9 to the 80(?)th position.  The number 9 is indeed
the mean value of the omitted bits considering all inputs equally
probable.  When all is added together and additional 3 is added to
positions 50 and 51.  Next, if the leading result bit is zero the
number is shifted one position (adjusting the exponent).  Finally
the result is truncated to 48 bits.  It is not easy to see, but the
algorithm delivers an unbiased result, but it can be wrong by more
than 1 bit in the least significant position.

What considering the IEEE standard, it is easy to see that with regards
to biasedness they could have chosen either round to even or round to
odd.  The reason round to even was chosen is that it actually diminishes
the need to round on subsequent operations.
--
dik t. winter, cwi, kruislaan 413, 1098 sj  amsterdam, nederland
home: bovenover 215, 1025 jn  amsterdam, nederland; e-mail: [e-mail]