<s>
The	O
term	O
arithmetic	B-Algorithm
underflow	I-Algorithm
(	O
also	O
floating	B-Algorithm
point	I-Algorithm
underflow	B-Algorithm
,	O
or	O
just	O
underflow	B-Algorithm
)	O
is	O
a	O
condition	O
in	O
a	O
computer	B-Application
program	I-Application
where	O
the	O
result	O
of	O
a	O
calculation	O
is	O
a	O
number	O
of	O
more	O
precise	O
absolute	O
value	O
than	O
the	O
computer	O
can	O
actually	O
represent	O
in	O
memory	B-General_Concept
on	O
its	O
central	B-General_Concept
processing	I-General_Concept
unit	I-General_Concept
(	O
CPU	O
)	O
.	O
</s>
<s>
Arithmetic	B-Algorithm
underflow	I-Algorithm
can	O
occur	O
when	O
the	O
true	O
result	O
of	O
a	O
floating	B-Algorithm
point	I-Algorithm
operation	I-Algorithm
is	O
smaller	O
in	O
magnitude	O
(	O
that	O
is	O
,	O
closer	O
to	O
zero	O
)	O
than	O
the	O
smallest	O
value	O
representable	O
as	O
a	O
normal	B-Algorithm
floating	B-Algorithm
point	I-Algorithm
number	I-Algorithm
in	O
the	O
target	O
datatype	O
.	O
</s>
<s>
Underflow	B-Algorithm
can	O
in	O
part	O
be	O
regarded	O
as	O
negative	O
overflow	B-Algorithm
of	O
the	O
exponent	O
of	O
the	O
floating	B-Algorithm
point	I-Algorithm
value	I-Algorithm
.	O
</s>
<s>
For	O
example	O
,	O
if	O
the	O
exponent	O
part	O
can	O
represent	O
values	O
from	O
128	O
to	O
127	O
,	O
then	O
a	O
result	O
with	O
a	O
value	O
less	O
than	O
128	O
may	O
cause	O
underflow	B-Algorithm
.	O
</s>
<s>
Storing	O
values	O
that	O
are	O
too	O
low	O
in	O
an	O
integer	O
variable	O
(	O
e.g.	O
,	O
attempting	O
to	O
store	O
1	O
in	O
an	O
unsigned	B-Algorithm
integer	O
)	O
is	O
properly	O
referred	O
to	O
as	O
integer	O
,	O
or	O
more	O
broadly	O
,	O
integer	O
wraparound	O
.	O
</s>
<s>
The	O
term	O
underflow	B-Algorithm
normally	O
refers	O
to	O
floating	B-Algorithm
point	I-Algorithm
numbers	I-Algorithm
only	O
,	O
which	O
is	O
a	O
separate	O
issue	O
.	O
</s>
<s>
It	O
is	O
not	O
possible	O
in	O
most	O
floating-point	B-Algorithm
designs	O
to	O
store	O
a	O
too-low	O
value	O
,	O
as	O
usually	O
they	O
are	O
signed	O
and	O
have	O
a	O
negative	O
infinity	B-Application
value	O
.	O
</s>
<s>
The	O
interval	O
between	O
fminN	O
and	O
fminN	O
,	O
where	O
fminN	O
is	O
the	O
smallest	O
positive	O
normal	B-Algorithm
floating	B-Algorithm
point	I-Algorithm
value	I-Algorithm
,	O
is	O
called	O
the	O
underflow	B-Algorithm
gap	O
.	O
</s>
<s>
This	O
is	O
because	O
the	O
size	O
of	O
this	O
interval	O
is	O
many	O
orders	O
of	O
magnitude	O
larger	O
than	O
the	O
distance	O
between	O
adjacent	O
normal	B-Algorithm
floating	B-Algorithm
point	I-Algorithm
values	I-Algorithm
just	O
outside	O
the	O
gap	O
.	O
</s>
<s>
For	O
instance	O
,	O
if	O
the	O
floating	B-Algorithm
point	I-Algorithm
datatype	O
can	O
represent	O
20	O
bits	O
,	O
the	O
underflow	B-Algorithm
gap	O
is	O
221	O
times	O
larger	O
than	O
the	O
absolute	O
distance	O
between	O
adjacent	O
floating	B-Algorithm
point	I-Algorithm
values	I-Algorithm
just	O
outside	O
the	O
gap	O
.	O
</s>
<s>
In	O
older	O
designs	O
,	O
the	O
underflow	B-Algorithm
gap	O
had	O
just	O
one	O
usable	O
value	O
,	O
zero	O
.	O
</s>
<s>
When	O
an	O
underflow	B-Algorithm
occurred	O
,	O
the	O
true	O
result	O
was	O
replaced	O
by	O
zero	O
(	O
either	O
directly	O
by	O
the	O
hardware	O
,	O
or	O
by	O
system	O
software	O
handling	O
the	O
primary	O
underflow	B-Algorithm
condition	O
)	O
.	O
</s>
<s>
The	O
1984	O
edition	O
of	O
IEEE	O
754	O
introduced	O
subnormal	B-Algorithm
numbers	I-Algorithm
.	O
</s>
<s>
The	O
subnormal	B-Algorithm
numbers	I-Algorithm
(	O
including	O
zero	O
)	O
fill	O
the	O
underflow	B-Algorithm
gap	O
with	O
values	O
where	O
the	O
absolute	O
distance	O
between	O
adjacent	O
values	O
is	O
the	O
same	O
as	O
for	O
adjacent	O
values	O
just	O
outside	O
the	O
underflow	B-Algorithm
gap	O
.	O
</s>
<s>
This	O
enables	O
"	O
gradual	B-Algorithm
underflow	I-Algorithm
"	O
,	O
where	O
a	O
nearest	O
subnormal	O
value	O
is	O
used	O
,	O
just	O
as	O
a	O
nearest	O
normal	B-Algorithm
value	O
is	O
used	O
when	O
possible	O
.	O
</s>
<s>
Even	O
when	O
using	O
gradual	B-Algorithm
underflow	I-Algorithm
,	O
the	O
nearest	O
value	O
may	O
be	O
zero	O
.	O
</s>
<s>
The	O
absolute	O
distance	O
between	O
adjacent	O
floating	B-Algorithm
point	I-Algorithm
values	I-Algorithm
just	O
outside	O
the	O
gap	O
is	O
called	O
the	O
machine	B-Algorithm
epsilon	I-Algorithm
,	O
typically	O
characterized	O
by	O
the	O
largest	O
value	O
whose	O
sum	O
with	O
the	O
value	O
1	O
will	O
result	O
in	O
the	O
answer	O
with	O
value	O
1	O
in	O
that	O
floating	B-Algorithm
point	I-Algorithm
scheme	O
.	O
</s>
<s>
This	O
can	O
be	O
written	O
as	O
,	O
where	O
is	O
a	O
function	O
which	O
converts	O
the	O
real	O
value	O
into	O
the	O
floating	B-Algorithm
point	I-Algorithm
representation	I-Algorithm
.	O
</s>
<s>
While	O
the	O
machine	B-Algorithm
epsilon	I-Algorithm
is	O
not	O
to	O
be	O
confused	O
with	O
the	O
underflow	B-Algorithm
level	O
(	O
assuming	O
subnormal	B-Algorithm
numbers	I-Algorithm
)	O
,	O
it	O
is	O
closely	O
related	O
.	O
</s>
<s>
The	O
machine	B-Algorithm
epsilon	I-Algorithm
is	O
dependent	O
on	O
the	O
number	O
of	O
bits	O
which	O
make	O
up	O
the	O
significand	B-Algorithm
,	O
whereas	O
the	O
underflow	B-Algorithm
level	O
depends	O
on	O
the	O
number	O
of	O
digits	O
which	O
make	O
up	O
the	O
exponent	O
field	O
.	O
</s>
<s>
In	O
most	O
floating	B-Algorithm
point	I-Algorithm
systems	O
,	O
the	O
underflow	B-Algorithm
level	O
is	O
smaller	O
than	O
the	O
machine	B-Algorithm
epsilon	I-Algorithm
.	O
</s>
<s>
The	O
occurrence	O
of	O
an	O
underflow	B-Algorithm
may	O
set	O
a	O
(	O
"	O
sticky	O
"	O
)	O
status	O
bit	O
,	O
raise	O
an	O
exception	O
,	O
at	O
the	O
hardware	O
level	O
generate	O
an	O
interrupt	B-Application
,	O
or	O
may	O
cause	O
some	O
combination	O
of	O
these	O
effects	O
.	O
</s>
<s>
As	O
specified	O
in	O
IEEE	O
754	O
,	O
the	O
underflow	B-Algorithm
condition	O
is	O
only	O
signaled	O
if	O
there	O
is	O
also	O
a	O
loss	O
of	O
precision	O
.	O
</s>
<s>
However	O
,	O
if	O
the	O
user	O
is	O
trapping	B-Application
on	O
underflow	B-Algorithm
,	O
this	O
may	O
happen	O
regardless	O
of	O
consideration	O
for	O
loss	O
of	O
precision	O
.	O
</s>
<s>
The	O
default	O
handling	O
in	O
IEEE	O
754	O
for	O
underflow	B-Algorithm
(	O
as	O
well	O
as	O
other	O
exceptions	O
)	O
is	O
to	O
record	O
as	O
a	O
floating	B-Algorithm
point	I-Algorithm
status	O
that	O
underflow	B-Algorithm
has	O
occurred	O
.	O
</s>
