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OVERVIEW OF NUMERIC PROCESSING
in arithmetic and store operations when the format of the destination cannot exactly represent the
infinitely precise true result. For example, a real number may
be
rounded if it
is
stored in a shorter
real format, or in an integer format.
Or, the infinitely precise true result may be rounded when it
is
returned
to
a register.
The
NPX
has four rounding modes, selectable by the
RC
field in the control word
(see
figure
1-5).
Given a true result b that cannot be represented by the target data type, the 80287 determines the two
representable numbers
a and c that most closely bracket b in value (a < b <
c).
The processor then
rounds (changes)
b to a or
to
c according
to
the mode selected
by
the
RC
field
as
shown in table
1-6.
Round introduces an error in a result that
is
less than one unit in the last place
to
which the result
is
rounded. "Round
to
nearest"
is
the default mode and
is
suitable for most applications; it provides the
most accurate and statistically unbiased estimate of the true result. The chop mode
is
provided for
integer arithmetic applications.
"Round up" and "round down" are termed directed rounding and can
be
used to implement interval
arithmetic. Interval arithmetic generates a certifiable result independent of the occurrence of rounding
and other errors. The upper and lower bounds of an interval may
be
computed by executing
an
algorithm
twice, rounding up in one pass and down in the other.
Precision Control
The 80287 allows results to
be
calculated with either 64, 53, or
24
bits of precision in the significand
as
selected by the precision control (PC) field of the control word. The default setting, and the one
that
is
best suited for most applications,
is
the full
64
bits of significance provided by the temporary-
real format. The other settings are required by the proposed IEEE standard, and are provided to obtain
compatibility with the specifications of certain existing programming languages. Specifying less
preci-
sion nullifies the advantages of the temporary real format's extended fraction length, and does not
increase execution speed. When reduced precision
is
specified, the rounding of the fractional value
clears the unused bits
on
the right to zeros.
Infinity Control
The 80287's system of real numbers may be closed by either of two models of infinity. These two means
of closing the number system, projective and affine closure, are illustrated schematically
in
figure
1-10.
The setting of the IC field in the control word selects one model or the other. The default
Table 1·6. Rounding Modes
RC
Field Rounding Mode
Rounding Action
00
Round to nearest
Closer to b of a or
c;
if equally close,
select even number (the one whose
least significant bit
is
zero).
01
Round down (toward
-00)
a
10
Round
up
(toward
+00)
c
11
Chop (toward
0)
Smaller
in
magnitude of a or c
NOTE:
a < b <
c;
a and c are representable, b
is
not.
1-19