<s>
In	O
numerical	B-General_Concept
analysis	I-General_Concept
,	O
one	O
or	O
more	O
guard	B-Algorithm
digits	I-Algorithm
can	O
be	O
used	O
to	O
reduce	O
the	O
amount	O
of	O
roundoff	B-Algorithm
error	O
.	O
</s>
<s>
That	O
is	O
to	O
say	O
,	O
the	O
roundoff	B-Algorithm
error	O
introduced	O
by	O
this	O
final	O
roundoff	B-Algorithm
makes	O
a	O
negligible	O
contribution	O
to	O
the	O
overall	O
uncertainty	O
.	O
</s>
<s>
Be	O
aware	O
that	O
roundoff	B-Algorithm
errors	O
can	O
accumulate	O
.	O
</s>
<s>
If	O
M	O
decimal	O
places	O
are	O
used	O
in	O
the	O
intermediate	O
calculation	O
,	O
we	O
say	O
there	O
are	O
M−N	O
guard	B-Algorithm
digits	I-Algorithm
.	O
</s>
<s>
Guard	B-Algorithm
digits	I-Algorithm
are	O
also	O
used	O
in	O
floating	O
point	O
operations	O
in	O
most	O
computer	O
systems	O
.	O
</s>
<s>
This	O
means	O
we	O
must	O
add	O
an	O
extra	O
digit	O
to	O
the	O
first	O
operand	O
—	O
a	O
guard	B-Algorithm
digit	I-Algorithm
.	O
</s>
<s>
Without	O
using	O
a	O
guard	B-Algorithm
digit	I-Algorithm
we	O
have	O
,	O
yielding	O
or	O
.	O
</s>
<s>
Therefore	O
,	O
we	O
can	O
see	O
how	O
important	O
guard	B-Algorithm
digits	I-Algorithm
can	O
be	O
.	O
</s>
<s>
An	O
example	O
of	O
the	O
error	O
caused	O
by	O
floating	O
point	O
roundoff	B-Algorithm
is	O
illustrated	O
in	O
the	O
following	O
C	B-Language
code	O
.	O
</s>
<s>
After	O
padding	O
the	O
second	O
number	O
(	O
i.e.	O
,	O
)	O
with	O
two	O
s	O
,	O
the	O
bit	O
after	O
is	O
the	O
guard	B-Algorithm
digit	I-Algorithm
,	O
and	O
the	O
bit	O
after	O
is	O
the	O
round	O
digit	O
.	O
</s>
<s>
The	O
result	O
after	O
rounding	B-Algorithm
is	O
as	O
opposed	O
to	O
,	O
without	O
the	O
extra	O
bits	O
(	O
guard	O
and	O
round	O
bits	O
)	O
,	O
i.e.	O
,	O
by	O
considering	O
only	O
.	O
</s>
