🚀 go-pugleaf

jbs@ibm.com says:

> I believe IEEE chopped will be easier to implement and perform
> better than IEEE rounded. 

This is true, but only marginally.  The most important cost in IEEE (or VAX)
floating-point arithmetic is getting the correct unrounded result,
or at least enough of it to permit correct rounding.  This requires
provision of guard and (for chopping or unbiased rounding) sticky bits, 
postnormalization, 
development of full doubled-precision products,
careful division and sqrt, etc.

Once you've gotten the unrounded answer, rounding (as opposed to chopping) 
costs an extra add time and dealing with the possibility of a carry out
of the most significant bit.  Although these are real design costs, I doubt
they've affected performance of modern pipelined IEEE designs compared to
the cost of getting the unrounded result.

I suspect that the reason IBM 370 arithmetic isn't correctly chopped is 
that the cost of doing it correctly - maintaining a sticky bit for
subtraction - was deemed too high, and the computed result is more accurate
in those cases where rounding is done instead of chopping.  More accurate,
but not in a useful way, however - the size of the
error bounds you're likely to derive
will have to reflect chopping, but any algorithms that might like to depend
on correct chopping principles ( (x-z) < x if z is positive) will have to
do without.

In general, any process that involves repeated inexact conversions,
particularly binary->decimal->binary->decimal, or addition of many small
inexact quantities, will benefit from the better statistics of rounding
over chopping - drifting tendencies will be reduced, and error bounds
will be smaller.

For instance, during the 1970's one of the minor regional stock exchanges
noticed that its market stock average was declining in a bull market in which
most of the component stocks were rising in value.   There were a number
of factors at work in this phenomenon; using chopping arithmetic was
one of them.

Using enough precision can sometimes overcome deficiencies of chopping or
dirty arithmetic, otherwise Seymour Cray would have done things differently.
But it isn't always enough, as most Cray customers find out occasionally. 
-- 

David Hough

dgh@validgh.com		uunet!validgh!dgh	na.hough@na-net.ornl.gov