```Subject: Re: Why is (round 2.5) = 2?
From: Erik Naggum <erik@naggum.no>
Date: 1995/04/22
Newsgroups: comp.lang.lisp
Message-ID: <19950422T110155Z.enag@naggum.no>

[Dan Britt]

|   Except that in the range 0 | 1 ... 9 | 10, zero equals ten (with
|   respect to rounding).  And, the real number line _does_ include
|   integers, so the range is 0 (exactly, not 0.00...01) through
|   0.49999....  There are exactly as many real numbers between 0 and
|   0.49999... inclusive as there are between 0.5 and 0.99999... inclusive,
|   so the habit of rounding .5 down half of the time means rounding down
|   more often than rounding up.  Finite precision arithmetic may either
|   mitigate or exacerbate that; I don't know.  But this subject has to do
|   with open vs closed intervals on the real number line. The interval [0,
|   .5) is the same size as the interval [.5, 1) (though it's been so long
|   I might have the symbols reversed), and that [.5 means include .5,
|   whereas .5) means don't include it.

I don't know whether this is a useful way to see it, but it does show that
round-up is a better solution under the stated premises.  the question is
whether those premises are optimal.  I don't think so.

instead of looking at the size of the ranges [0,.5) and [.5,1), we should
look at the range of numbers that round to a given integer.  e.g., the
numbers that round to zero are in the range [-.5,+.5).  this range is
slightly off balance, and it is this imbalance that accumulates errors.  in
a round-to-even scheme, the numbers that round to an even integer n are in
the range [n-.5,n+.5], which balances perfectly.  the numbers that round to
an odd integer n is in the range (n-.5,n+.5), which also balances.  this
balance yields correct results.

#<Erik>
--
sufficiently advanced political correctness is indistinguishable from irony

```