Cancellation and Swamping
The IEEE standard 754 requires that the FAFA holds. That is: any arithmetic
operation on two ﬂoats returns the ﬂoat nearest the true value. Here we discuss two
principle ways information can be lost in this setting.
Consider what happens when adding a

large

number 0
.d
1
d
2
···
d
t
×
2
m
and a

small

number 0
.e
1
e
2
···
e
t
×
2
p
. Since only
t
digits can remain in the result, the
digits of the smaller number are eﬀectively shifted to the right about
m

p
places.
What were once signiﬁcant digits of the smaller number are now not as signiﬁcant,
and about
m

p
of them are lost completely:
0
. d
1
d
2
···
d
m

p
d
m

p
+1
···
d
t

1
d
t
+ 0
.
0
0
···
0
e
1
e
2
...
e
t

(
m

p
)
0
. d
1
d
2
···
d
m

p
f
m

p
+1
···
f
t

1
f
t
The result
is
the closest ﬂoat to the true answer; it is working the way it is
supposed to work. It is simply a fact of ﬂoating point arithmetic that in the
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 Fall '10
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 Addition, Subtraction, digits, FAFA, exact arithmetic z1

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