Hardware Reference
In-Depth Information
3
Negative
underflow
5
Positive
underflow
4
Zero
1
Negative
overflow
2
Expressible
negative numbers
6
Expressible
positive numbers
7
Positive
overflow
-10
99
-10
-100
0
10
-100
10
99
Figure B-1.
The real number line can be divided into seven regions.
underflow error
. Underflow error is less serious than overflow error, because 0 is
often a satisfactory approximation to numbers in regions 3 and 5. A bank balance
of 10
−102
dollars is hardly better than a bank balance of 0.
Another important difference between floating-point numbers and real num-
bers is their density. Between any two real numbers,
x
and
y
, is another real num-
ber, no matter how close
x
is to
y
. This property comes from the fact that for any
distinct real numbers,
x
and
y
,
z
=
(
x
+
y
)/2 is a real number between them. The
real numbers form a continuum.
Floating-point numbers, in contrast, do not form a continuum. Exactly
179,100 positive numbers can be expressed in the five-digit, two-sign system used
above, 179,100 negative numbers, and 0 (which can be expressed in many ways),
for a total of 358,201 numbers. Of the infinite number of real numbers between
−
10
99
, only 358,201 of them can be specified by this notation.
They are symbolized by the dots in Fig. B-1. It is quite possible for the result of a
calculation to be one of the other numbers, even though it is in region 2 or 6. For
example,
10
+100
and
+
0. 999
×
10
3
divided by 3 cannot be expressed
exactly
in our system of
representation. If the result of a calculation cannot be expressed in the number
representation being used, the obvious thing to do is to use the nearest number that
can be expressed. This process is called
rounding
.
The spacing between adjacent expressible numbers is not constant throughout
region 2 or 6. The separation between
+
0. 100
×
10
99
and
10
99
+
0. 998
×
+
0. 999
×
is vastly
10
0
. However, when
the separation between a number and its successor is expressed as a percentage of
that number, there is no systematic variation throughout region 2 or 6. In other
words, the
relative error
introduced by rounding is approximately the same for
small numbers as large numbers.
Although the preceding discussion was in terms of a representation system
with a three-digit fraction and a two-digit exponent, the conclusions drawn are
valid for other representation systems as well. Changing the number of digits in
the fraction or exponent merely shifts the boundaries of regions 2 and 6 and
changes the number of expressible points in them. Increasing the number of digits
in the fraction increases the density of points and therefore improves the accuracy
10
0
and
more than the separation between
+
0. 998
×
+
0. 999
×
