<s>
The	O
Pentium	B-Device
FDIV	I-Device
bug	I-Device
is	O
a	O
hardware	B-Error_Name
bug	I-Error_Name
affecting	O
the	O
floating-point	B-General_Concept
unit	I-General_Concept
(	O
FPU	O
)	O
of	O
the	O
early	B-General_Concept
Intel	I-General_Concept
Pentium	I-General_Concept
processors	I-General_Concept
.	O
</s>
<s>
Because	O
of	O
the	O
bug	O
,	O
the	O
processor	O
would	O
return	O
incorrect	O
binary	B-Algorithm
floating	I-Algorithm
point	I-Algorithm
results	O
when	O
dividing	O
certain	O
pairs	O
of	O
high-precision	O
numbers	O
.	O
</s>
<s>
Missing	O
values	O
in	O
a	O
lookup	O
table	O
used	O
by	O
the	O
FPU	O
's	O
floating-point	B-Algorithm
division	O
algorithm	O
led	O
to	O
calculations	O
acquiring	O
small	O
errors	O
.	O
</s>
<s>
The	O
severity	O
of	O
the	O
FDIV	B-Device
bug	I-Device
is	O
debated	O
.	O
</s>
<s>
Though	O
rarely	O
encountered	O
by	O
most	O
users	O
(	O
Byte	B-Application
magazine	O
estimated	O
that	O
1	O
in	O
9	O
billion	O
floating	B-Algorithm
point	I-Algorithm
divides	O
with	O
random	O
parameters	O
would	O
produce	O
inaccurate	O
results	O
)	O
,	O
both	O
the	O
flaw	O
and	O
Intel	O
's	O
initial	O
handling	O
of	O
the	O
matter	O
were	O
heavily	O
criticized	O
by	O
the	O
tech	O
community	O
.	O
</s>
<s>
In	O
order	O
to	O
improve	O
the	O
speed	O
of	O
floating-point	B-Algorithm
division	O
calculations	O
on	O
the	O
Pentium	B-General_Concept
chip	O
over	O
the	O
486DX	B-General_Concept
,	O
Intel	O
opted	O
to	O
replace	O
the	O
shift-and-subtract	O
division	O
algorithm	O
with	O
the	O
Sweeney	O
,	O
Robertson	O
,	O
and	O
Tocher	O
(	O
SRT	O
)	O
algorithm	O
.	O
</s>
<s>
The	O
SRT	O
algorithm	O
can	O
generate	O
two	O
bits	O
of	O
the	O
division	O
result	O
per	O
clock	O
cycle	O
,	O
whereas	O
the	O
486	B-General_Concept
's	O
algorithm	O
could	O
only	O
generate	O
one	O
.	O
</s>
<s>
When	O
the	O
original	O
array	O
for	O
the	O
Pentium	B-General_Concept
was	O
compiled	O
,	O
five	O
values	O
were	O
not	O
correctly	O
downloaded	O
into	O
the	O
equipment	O
that	O
etches	O
the	O
arrays	O
into	O
the	O
chips	O
–	O
thus	O
five	O
of	O
the	O
array	O
cells	O
contained	O
zero	O
when	O
they	O
should	O
have	O
contained	O
+2	O
.	O
</s>
<s>
Performing	O
this	O
calculation	O
in	O
any	O
software	O
that	O
used	O
the	O
floating-point	B-Algorithm
coprocessor	O
,	O
such	O
as	O
Windows	B-Application
Calculator	I-Application
,	O
would	O
allow	O
users	O
to	O
discover	O
whether	O
their	O
Pentium	B-General_Concept
chip	O
was	O
affected	O
.	O
</s>
<s>
As	O
a	O
result	O
,	O
the	O
value	O
returned	O
by	O
a	O
flawed	O
Pentium	B-General_Concept
processor	O
is	O
incorrect	O
at	O
or	O
beyond	O
four	O
digits	O
:	O
</s>
<s>
Thomas	B-Device
Nicely	I-Device
,	O
a	O
professor	O
of	O
mathematics	O
at	O
Lynchburg	O
College	O
,	O
had	O
written	O
code	O
to	O
enumerate	O
primes	O
,	O
twin	O
primes	O
,	O
prime	O
triplets	O
,	O
and	O
prime	O
quadruplets	O
.	O
</s>
<s>
Nicely	O
noticed	O
some	O
inconsistencies	O
in	O
the	O
calculations	O
on	O
June	O
13	O
,	O
1994	O
,	O
shortly	O
after	O
adding	O
a	O
Pentium	B-General_Concept
system	O
to	O
his	O
group	O
of	O
computers	O
,	O
but	O
was	O
unable	O
to	O
eliminate	O
other	O
factors	O
(	O
such	O
as	O
programming	O
errors	O
,	O
motherboard	B-Device
chipsets	O
,	O
etc	O
.	O
)	O
</s>
<s>
On	O
October	O
30	O
,	O
1994	O
,	O
Nicely	O
sent	O
an	O
email	O
describing	O
the	O
bug	O
to	O
various	O
academic	O
contacts	O
,	O
requesting	O
reports	O
of	O
testing	O
for	O
the	O
flaw	O
on	O
486-DX4s	B-Device
,	O
Pentiums	B-General_Concept
and	O
Pentium	B-General_Concept
clones	O
.	O
</s>
<s>
The	O
bug	O
acquired	O
the	O
name	O
"	O
Pentium	B-Device
FDIV	I-Device
bug	I-Device
"	O
from	O
the	O
x86	B-Device
assembly	I-Device
language	I-Device
mnemonic	I-Device
for	O
floating-point	B-Algorithm
division	O
,	O
the	O
most	O
frequently	O
used	O
instruction	O
affected	O
.	O
</s>
<s>
The	O
story	O
first	O
appeared	O
in	O
the	O
press	O
on	O
November	O
7	O
,	O
1994	O
,	O
in	O
an	O
article	O
in	O
Electronic	O
Engineering	O
Times	O
,	O
"	O
Intel	O
fixes	O
a	O
Pentium	B-General_Concept
FPU	O
glitch	O
"	O
by	O
Alexander	O
Wolfe	O
,	O
and	O
was	O
subsequently	O
picked	O
up	O
by	O
CNN	B-Architecture
in	O
a	O
segment	O
aired	O
on	O
November	O
22	O
.	O
</s>
<s>
At	O
this	O
point	O
,	O
Intel	O
acknowledged	O
the	O
floating-point	B-Algorithm
flaw	O
,	O
but	O
claimed	O
that	O
it	O
was	O
not	O
serious	O
and	O
would	O
not	O
affect	O
most	O
users	O
.	O
</s>
<s>
The	O
motive	O
behind	O
IBM	O
's	O
decision	O
was	O
questioned	O
by	O
some	O
in	O
the	O
industry	O
;	O
IBM	O
produced	O
the	O
PowerPC	B-Architecture
CPUs	O
at	O
the	O
time	O
,	O
and	O
potentially	O
stood	O
to	O
benefit	O
from	O
any	O
reputational	O
damage	O
to	O
the	O
Pentium	B-General_Concept
or	O
Intel	O
as	O
a	O
company	O
.	O
</s>
<s>
However	O
,	O
the	O
decision	O
led	O
to	O
corporate	O
buyers	O
of	O
PC	O
equipment	O
demanding	O
replacements	O
of	O
existing	O
Pentium	B-General_Concept
CPUs	O
,	O
and	O
soon	O
afterwards	O
other	O
PC	O
manufacturers	O
began	O
offering	O
"	O
no	O
questions	O
asked	O
"	O
replacements	O
of	O
flawed	O
Pentium	B-General_Concept
chips	O
.	O
</s>
<s>
The	O
growing	O
dissatisfaction	O
with	O
Intel	O
's	O
response	O
led	O
to	O
the	O
company	O
offering	O
to	O
replace	O
all	O
flawed	O
Pentium	B-General_Concept
processors	O
on	O
request	O
on	O
December	O
20	O
.	O
</s>
<s>
Intel	O
's	O
response	O
to	O
the	O
FDIV	B-Device
bug	I-Device
has	O
been	O
cited	O
as	O
a	O
case	O
of	O
the	O
public	O
relations	O
impact	O
of	O
a	O
problem	O
eclipsing	O
the	O
practical	O
impact	O
of	O
said	O
problem	O
on	O
customers	O
.	O
</s>
<s>
In	O
the	O
aftermath	O
of	O
the	O
bug	O
and	O
subsequent	O
recall	O
,	O
there	O
was	O
a	O
marked	O
increase	O
in	O
the	O
use	O
of	O
formal	O
verification	O
of	O
hardware	O
floating	B-Algorithm
point	I-Algorithm
operations	O
across	O
the	O
semiconductor	O
industry	O
.	O
</s>
<s>
In	O
the	O
development	O
of	O
the	O
Pentium	B-General_Concept
4	I-General_Concept
,	O
symbolic	B-Application
trajectory	I-Application
evaluation	I-Application
and	O
theorem	O
proving	O
were	O
used	O
to	O
find	O
a	O
number	O
of	O
bugs	O
that	O
could	O
have	O
led	O
to	O
a	O
similar	O
recall	O
incident	O
had	O
they	O
gone	O
undetected	O
.	O
</s>
<s>
The	O
first	O
Intel	O
microarchitecture	O
to	O
use	O
formal	O
verification	O
as	O
the	O
primary	O
method	O
of	O
validation	O
was	O
Nehalem	B-Device
,	O
developed	O
in	O
2008	O
.	O
</s>
<s>
The	O
FDIV	B-Device
bug	I-Device
affects	O
the	O
60	O
and	O
66	O
MHz	O
Pentium	B-General_Concept
P5	B-General_Concept
800	O
in	O
stepping	B-General_Concept
levels	I-General_Concept
prior	O
to	O
D1	O
,	O
and	O
the	O
75	O
,	O
90	O
,	O
and	O
100	O
MHz	O
Pentium	B-General_Concept
P54C	O
600	O
in	O
steppings	B-General_Concept
prior	O
to	O
B5	O
.	O
</s>
<s>
This	O
fix	O
does	O
carry	O
a	O
measurable	O
speed	O
penalty	O
-	O
worst	O
case	O
for	O
a	O
program	O
doing	O
nothing	O
but	O
FDIV	B-Device
operations	O
with	O
bad	O
divisors	O
the	O
running	O
time	O
would	O
double	O
since	O
each	O
FDIV	B-Device
would	O
take	O
about	O
80	O
instead	O
of	O
40	O
clock	O
cycles	O
.	O
</s>
<s>
With	O
more	O
random	O
divisors	O
the	O
average	O
time	O
per	O
FDIV	B-Device
was	O
approximately	O
50	O
clock	O
cycles	O
,	O
i.e.	O
</s>
<s>
Since	O
FDIV	B-Device
is	O
a	O
rare	O
operation	O
in	O
most	O
programs	O
,	O
the	O
normal	O
slowdown	O
with	O
the	O
fix	O
installed	O
was	O
typically	O
a	O
percent	O
or	O
less	O
.	O
</s>
<s>
The	O
main	O
challenge	O
faced	O
by	O
software	O
companies	O
was	O
implementing	O
the	O
fix	O
in	O
pre-existing	O
software	O
,	O
much	O
of	O
which	O
relied	O
on	O
libraries	B-Library
outside	O
their	O
control	O
.	O
</s>
<s>
Some	O
companies	O
,	O
such	O
as	O
Wolfram	B-Application
Research	I-Application
,	O
opted	O
to	O
directly	O
patch	O
the	O
machine	B-Language
code	I-Language
of	O
existing	O
executables	O
to	O
replace	O
the	O
FDIV	B-Device
opcode	O
with	O
an	O
illegal	O
instruction	O
.	O
</s>
<s>
Microsoft	O
offered	O
operating	O
system	O
level	O
workarounds	O
in	O
versions	O
of	O
Windows	B-Application
up	O
to	O
Windows	B-Application
XP	O
.	O
</s>
