<s>
In	O
floating-point	B-Algorithm
arithmetic	I-Algorithm
,	O
the	O
Sterbenz	B-Algorithm
lemma	I-Algorithm
or	O
Sterbenz	O
's	O
lemma	O
is	O
a	O
theorem	O
giving	O
conditions	O
under	O
which	O
floating-point	B-Algorithm
differences	O
are	O
computed	O
exactly	O
.	O
</s>
<s>
The	O
Sterbenz	B-Algorithm
lemma	I-Algorithm
applies	O
to	O
IEEE	O
754	O
,	O
the	O
most	O
widely	O
used	O
floating-point	B-Algorithm
number	I-Algorithm
system	O
in	O
computers	O
.	O
</s>
<s>
Let	O
be	O
the	O
radix	O
of	O
the	O
floating-point	B-Algorithm
system	O
and	O
the	O
precision	O
.	O
</s>
<s>
If	O
is	O
zero	O
then	O
,	O
and	O
if	O
is	O
zero	O
then	O
,	O
so	O
the	O
result	O
is	O
trivial	O
because	O
floating-point	B-Algorithm
negation	O
is	O
always	O
exact	O
.	O
</s>
<s>
so	O
is	O
a	O
floating-point	B-Algorithm
number	I-Algorithm
.	O
</s>
<s>
For	O
example	O
,	O
the	O
difference	O
of	O
the	O
two	O
smallest	O
positive	O
normal	O
floating-point	B-Algorithm
numbers	I-Algorithm
and	O
is	O
which	O
is	O
necessarily	O
subnormal	O
.	O
</s>
<s>
In	O
floating-point	B-Algorithm
number	I-Algorithm
systems	O
without	O
subnormal	B-Algorithm
numbers	I-Algorithm
,	O
such	O
as	O
CPUs	O
in	O
nonstandard	O
flush-to-zero	O
mode	O
instead	O
of	O
the	O
standard	O
gradual	B-Algorithm
underflow	I-Algorithm
,	O
the	O
Sterbenz	B-Algorithm
lemma	I-Algorithm
does	O
not	O
apply	O
.	O
</s>
<s>
The	O
Sterbenz	B-Algorithm
lemma	I-Algorithm
may	O
be	O
contrasted	O
with	O
the	O
phenomenon	O
of	O
catastrophic	B-Algorithm
cancellation	I-Algorithm
:	O
</s>
<s>
The	O
Sterbenz	B-Algorithm
lemma	I-Algorithm
asserts	O
that	O
if	O
and	O
are	O
sufficiently	O
close	O
floating-point	B-Algorithm
numbers	I-Algorithm
then	O
their	O
difference	O
is	O
computed	O
exactly	O
by	O
floating-point	B-Algorithm
arithmetic	I-Algorithm
,	O
with	O
no	O
rounding	O
needed	O
.	O
</s>
<s>
The	O
phenomenon	O
of	O
catastrophic	B-Algorithm
cancellation	I-Algorithm
is	O
that	O
if	O
and	O
are	O
approximations	O
to	O
true	O
numbers	O
and	O
whether	O
the	O
approximations	O
arise	O
from	O
prior	O
rounding	O
error	O
or	O
from	O
series	O
truncation	O
or	O
from	O
physical	O
uncertainty	O
or	O
anything	O
elsethe	O
error	O
of	O
the	O
difference	O
from	O
the	O
desired	O
difference	O
is	O
inversely	O
proportional	O
to	O
.	O
</s>
<s>
In	O
other	O
words	O
,	O
the	O
Sterbenz	B-Algorithm
lemma	I-Algorithm
shows	O
that	O
subtracting	O
nearby	O
floating-point	B-Algorithm
numbers	I-Algorithm
is	O
exact	O
,	O
but	O
if	O
the	O
numbers	O
you	O
have	O
are	O
approximations	O
then	O
even	O
their	O
exact	O
difference	O
may	O
be	O
far	O
off	O
from	O
the	O
difference	O
of	O
numbers	O
you	O
wanted	O
to	O
subtract	O
.	O
</s>
<s>
The	O
Sterbenz	B-Algorithm
lemma	I-Algorithm
is	O
instrumental	O
in	O
proving	O
theorems	O
on	O
error	O
bounds	O
in	O
numerical	O
analysis	O
of	O
floating-point	B-Algorithm
algorithms	O
.	O
</s>
<s>
for	O
the	O
area	O
of	O
triangle	O
with	O
side	O
lengths	O
,	O
,	O
and	O
,	O
where	O
is	O
the	O
semi-perimeter	O
,	O
may	O
give	O
poor	O
accuracy	O
for	O
long	O
narrow	O
triangles	O
if	O
evaluated	O
directly	O
in	O
floating-point	B-Algorithm
arithmetic	I-Algorithm
.	O
</s>
<s>
can	O
be	O
proven	O
,	O
with	O
the	O
help	O
of	O
the	O
Sterbenz	B-Algorithm
lemma	I-Algorithm
,	O
to	O
have	O
low	O
forward	B-Algorithm
error	I-Algorithm
for	O
all	O
inputs	O
.	O
</s>
