30 Sep
2001
30 Sep
'01
8:19 a.m.
On Sep 30, 12:06, Jeffrey S. Sharp wrote:
On Sun, 30 Sep 2001, Pete Turnbull wrote:
Did you notice that in a few cases, your carry wasn't the opposite of the
borrow, in the original table? But it sorted itself out so that it was
consistent once you used "A + inv(B) + 1" on the right hand side.
you're assuming (on the left) that
"carry" is the same as "borrow".
This works itself out exactly like that below.
However, you'll get the right answer if you
say that "A - B" is the
same as "A + inv(B) + 1" where "inv(B)" is the ones-complement of B.
Bingo! That fixed it:
* I get correct overflow flag values.
* The correct borrow flag value is the opposite of the carry out of the
addition. Now what's disturbing is that the same wrong ideas
that I was taught were
taught to an entire roomful of college students a few years ago, by a
professor in the School of Electrical and Computer Engineering.
Maybe part of the confusion arises because many processors (including the
8080) complement the carry flag at the end of a subtraction, so that it can
be used directly as a "borrow" flag in multibyte subtractions. Others
(like the 6502) don't do that.
The confusion over the overflow usually arises because people are told that
adding the two's complement of a number is equivalent to subtraction.
That's not so much "wrong" as "incomplete". As you've seen,
it's
arithmetically but not logically equivalent. If you combine the
carry-into-MSB from the compliment op with the carry-into-MSB from the
subsequent addition, they would be logically equivalent as well, but you'd
also need to do this for the carry flag otherwise subtracting 0 tends to
give the wrong result :-)
Since you're thinking about 2's complements, and I've mentioned you could
combine the carry-in from the complement operation with the carry-in in the
addition, you might like to think about different ways of performing the
2's complement. The way usually explained is "invert the bits and add a
1". There's another common way: find the *rightmost* '1' in the
number.
Invert every bit to its left. Done. Now where's the carry in that? Oh,
and you need to use a particular search order to find the "rightmost 1".
Thanks for the help.
You're welcome. It took me a few minutes to realise what was wrong, so it
was an interesting exercise :-)
--
Pete Peter Turnbull
Network Manager
University of York