<s>
Extended	B-Algorithm
precision	I-Algorithm
refers	O
to	O
floating-point	B-Algorithm
number	I-Algorithm
formats	O
that	O
provide	O
greater	O
precision	B-General_Concept
than	O
the	O
basic	O
floating-point	B-Algorithm
formats	I-Algorithm
.	O
</s>
<s>
Extended	B-Algorithm
precision	I-Algorithm
formats	I-Algorithm
support	O
a	O
basic	O
format	O
by	O
minimizing	O
roundoff	O
and	O
overflow	O
errors	O
in	O
intermediate	O
values	O
of	O
expressions	O
on	O
the	O
base	O
format	O
.	O
</s>
<s>
In	O
contrast	O
to	O
extended	B-Algorithm
precision	I-Algorithm
,	O
arbitrary-precision	B-Algorithm
arithmetic	I-Algorithm
refers	O
to	O
implementations	O
of	O
much	O
larger	O
numeric	B-Algorithm
types	O
(	O
with	O
a	O
storage	O
count	O
that	O
usually	O
is	O
not	O
a	O
power	O
of	O
two	O
)	O
using	O
special	O
software	O
(	O
or	O
,	O
rarely	O
,	O
hardware	O
)	O
.	O
</s>
<s>
There	O
is	O
a	O
long	O
history	O
of	O
extended	O
floating-point	B-Algorithm
formats	I-Algorithm
reaching	O
back	O
nearly	O
to	O
the	O
middle	O
of	O
the	O
last	O
century	O
.	O
</s>
<s>
Various	O
manufacturers	O
have	O
used	O
different	O
formats	O
for	O
extended	B-Algorithm
precision	I-Algorithm
for	O
different	O
machines	O
.	O
</s>
<s>
In	O
many	O
cases	O
the	O
format	O
of	O
the	O
extended	B-Algorithm
precision	I-Algorithm
is	O
not	O
quite	O
the	O
same	O
as	O
a	O
scale-up	O
of	O
the	O
ordinary	O
single	O
-	O
and	O
double-precision	O
formats	O
it	O
is	O
meant	O
to	O
extend	O
.	O
</s>
<s>
In	O
a	O
few	O
cases	O
the	O
implementation	O
was	O
merely	O
a	O
software-based	O
change	O
in	O
the	O
floating-point	B-Algorithm
data	O
format	O
,	O
but	O
in	O
most	O
cases	O
extended	B-Algorithm
precision	I-Algorithm
was	O
implemented	O
in	O
hardware	O
,	O
either	O
built	O
into	O
the	O
central	B-General_Concept
processor	I-General_Concept
itself	O
,	O
or	O
more	O
often	O
,	O
built	O
into	O
the	O
hardware	O
of	O
an	O
optional	O
,	O
attached	O
processor	O
called	O
a	O
"	O
floating-point	B-General_Concept
unit	I-General_Concept
"	O
(	O
FPU	O
)	O
or	O
"	O
floating-point	B-General_Concept
processor	I-General_Concept
"	O
(	O
FPP	B-General_Concept
)	O
,	O
accessible	O
to	O
the	O
CPU	B-Device
as	O
a	O
fast	O
input	O
/	O
output	O
device	O
.	O
</s>
<s>
The	O
IBM	B-Device
1130	I-Device
,	O
sold	O
in	O
1965	O
,	O
offered	O
two	O
floating-point	B-Algorithm
formats	I-Algorithm
:	O
A	O
32-bit	O
"	O
standard	O
precision	B-General_Concept
"	O
format	O
and	O
a	O
40-bit	O
"	O
extended	B-Algorithm
precision	I-Algorithm
"	O
format	O
.	O
</s>
<s>
Standard	O
precision	B-General_Concept
format	O
contains	O
a	O
24-bit	O
two	B-General_Concept
's	I-General_Concept
complement	I-General_Concept
significand	B-Algorithm
while	O
extended	B-Algorithm
precision	I-Algorithm
utilizes	O
a	O
32-bit	O
two	B-General_Concept
's	I-General_Concept
complement	I-General_Concept
significand	B-Algorithm
.	O
</s>
<s>
The	O
latter	O
format	O
makes	O
full	O
use	O
of	O
the	O
CPU	B-Device
's	O
32-bit	O
integer	O
operations	O
.	O
</s>
<s>
The	O
characteristic	O
in	O
both	O
formats	O
is	O
an	O
8-bit	O
field	O
containing	O
the	O
power	O
of	O
two	O
biased	B-Algorithm
by	O
128	O
.	O
</s>
<s>
Floating-point	B-Algorithm
arithmetic	I-Algorithm
operations	O
are	O
performed	O
by	O
software	O
,	O
and	O
double	O
precision	B-General_Concept
is	O
not	O
supported	O
at	O
all	O
.	O
</s>
<s>
The	O
IBM	B-Application
System/360	I-Application
supports	O
a	O
32-bit	O
"	O
short	O
"	O
floating-point	B-Algorithm
format	I-Algorithm
and	O
a	O
64-bit	O
"	O
long	O
"	O
floating-point	B-Algorithm
format	I-Algorithm
.	O
</s>
<s>
The	O
360/85	O
and	O
follow-on	O
System/370	B-Device
add	O
support	O
for	O
a	O
128-bit	O
"	O
extended	O
"	O
format	O
.	O
</s>
<s>
These	O
formats	O
are	O
still	O
supported	O
in	O
the	O
current	O
design	B-Device
,	O
where	O
they	O
are	O
now	O
called	O
the	O
"	O
hexadecimal	B-Device
floating-point	I-Device
"	O
(	O
HFP	O
)	O
formats	O
.	O
</s>
<s>
The	O
Microsoft	B-Language
BASIC	I-Language
port	O
for	O
the	O
6502	B-General_Concept
CPU	B-Device
,	O
such	O
as	O
in	O
adaptations	O
like	O
Commodore	B-Language
BASIC	I-Language
,	O
AppleSoft	B-Language
BASIC	I-Language
,	O
KIM-1	O
BASIC	O
or	O
MicroTAN	O
BASIC	O
,	O
supports	O
an	O
extended	O
40-bit	O
variant	O
of	O
the	O
floating-point	B-Algorithm
format	I-Algorithm
Microsoft	O
Binary	O
Format	O
(	O
MBF	O
)	O
since	O
1977	O
.	O
</s>
<s>
The	O
IEEE	O
754	O
floating-point	B-Algorithm
standard	O
recommends	O
that	O
implementations	O
provide	O
extended	B-Algorithm
precision	I-Algorithm
formats	I-Algorithm
.	O
</s>
<s>
The	O
IA32	B-Device
,	O
x86-64	B-Device
,	O
and	O
Itanium	B-General_Concept
processors	O
support	O
an	O
80-bit	O
"	O
double	O
extended	O
"	O
extended	B-Algorithm
precision	I-Algorithm
format	I-Algorithm
with	O
a	O
64-bit	O
significand	B-Algorithm
.	O
</s>
<s>
The	O
Intel	B-Device
8087	I-Device
math	B-General_Concept
coprocessor	I-General_Concept
was	O
the	O
first	O
x86	B-Operating_System
device	O
which	O
supported	O
floating-point	B-Algorithm
arithmetic	I-Algorithm
in	O
hardware	O
.	O
</s>
<s>
It	O
was	O
designed	O
to	O
support	O
a	O
32-bit	O
"	O
single	O
precision	B-General_Concept
"	O
format	O
and	O
a	O
64-bit	O
"	O
double-precision	O
"	O
format	O
for	O
encoding	O
and	O
interchanging	O
floating-point	B-Algorithm
numbers	I-Algorithm
.	O
</s>
<s>
The	O
temporary	O
real	O
(	O
extended	O
)	O
format	O
was	O
designed	O
not	O
to	O
store	O
data	O
at	O
higher	O
precision	B-General_Concept
as	O
such	O
,	O
but	O
rather	O
primarily	O
to	O
allow	O
for	O
the	O
computation	O
of	O
double	O
results	O
more	O
reliably	O
and	O
accurately	O
by	O
minimising	O
overflow	O
and	O
roundoff-errors	O
in	O
intermediate	O
calculations	O
.	O
</s>
<s>
For	O
example	O
,	O
many	O
floating-point	B-Algorithm
algorithms	O
(	O
e.g.	O
</s>
<s>
exponentiation	O
)	O
suffer	O
from	O
significant	O
precision	B-General_Concept
loss	O
when	O
computed	O
using	O
the	O
most	O
direct	O
implementations	O
.	O
</s>
<s>
To	O
mitigate	O
such	O
issues	O
the	O
internal	O
registers	O
in	O
the	O
8087	B-Device
were	O
designed	O
to	O
hold	O
intermediate	O
results	O
in	O
an	O
80-bit	O
"	O
extended	B-Algorithm
precision	I-Algorithm
"	O
format	O
.	O
</s>
<s>
The	O
8087	B-Device
automatically	O
converts	O
numbers	O
to	O
this	O
format	O
when	O
loading	O
floating-point	B-Algorithm
registers	O
from	O
memory	B-General_Concept
and	O
also	O
converts	O
results	O
back	O
to	O
the	O
more	O
conventional	O
formats	O
when	O
storing	O
the	O
registers	O
back	O
into	O
memory	B-General_Concept
.	O
</s>
<s>
To	O
enable	O
intermediate	O
subexpression	O
results	O
to	O
be	O
saved	O
in	O
extended	B-Algorithm
precision	I-Algorithm
scratch	O
variables	O
and	O
continued	O
across	O
programming	O
language	O
statements	O
,	O
and	O
otherwise	O
interrupted	O
calculations	O
to	O
resume	O
where	O
they	O
were	O
interrupted	O
,	O
it	O
provides	O
instructions	O
which	O
transfer	O
values	O
between	O
these	O
internal	O
registers	O
and	O
memory	B-General_Concept
without	O
performing	O
any	O
conversion	O
,	O
which	O
therefore	O
enables	O
access	O
to	O
the	O
extended	O
format	O
for	O
calculations	O
–	O
also	O
reviving	O
the	O
issue	O
of	O
the	O
accuracy	O
of	O
functions	O
of	O
such	O
numbers	O
,	O
but	O
at	O
a	O
higher	O
precision	B-General_Concept
.	O
</s>
<s>
The	O
floating-point	B-General_Concept
units	I-General_Concept
(	O
FPU	O
)	O
on	O
all	O
subsequent	O
x86	B-Operating_System
processors	O
have	O
supported	O
this	O
format	O
.	O
</s>
<s>
As	O
a	O
result	O
,	O
software	O
can	O
be	O
developed	O
which	O
takes	O
advantage	O
of	O
the	O
higher	O
precision	B-General_Concept
provided	O
by	O
this	O
format	O
.	O
</s>
<s>
William	O
Kahan	O
,	O
a	O
primary	O
designer	O
of	O
the	O
x87	B-Application
arithmetic	O
and	O
initial	O
IEEE754	O
standard	O
proposal	O
notes	O
on	O
the	O
development	O
of	O
the	O
x87	B-Application
floating	B-Algorithm
point	I-Algorithm
:	O
"	O
An	O
extended	O
format	O
as	O
wide	O
as	O
we	O
dared	O
(	O
80bits	O
)	O
was	O
included	O
to	O
serve	O
the	O
same	O
support	O
role	O
as	O
the	O
13decimal	O
internal	O
format	O
serves	O
in	O
Hewlett-Packard	O
'	O
s	O
10decimal	O
calculators.	O
"	O
</s>
<s>
Moreover	O
,	O
Kahan	O
notes	O
that	O
64bits	O
was	O
the	O
widest	O
significand	B-Algorithm
across	O
which	O
carry	O
propagation	O
could	O
be	O
done	O
without	O
increasing	O
the	O
cycle	O
time	O
on	O
the	O
8087	B-Device
,	O
and	O
that	O
the	O
x87	B-Application
extended	B-Algorithm
precision	I-Algorithm
was	O
designed	O
to	O
be	O
extensible	O
to	O
higher	O
precision	B-General_Concept
in	O
future	O
processors	O
:	O
"	O
For	O
now	O
the	O
10-byte	O
Extended	O
format	O
is	O
a	O
tolerable	O
compromise	O
between	O
the	O
value	O
of	O
extra-precise	O
arithmetic	O
and	O
the	O
price	O
of	O
implementing	O
it	O
to	O
run	O
fast	O
;	O
very	O
soon	O
two	O
more	O
bytes	O
of	O
precision	B-General_Concept
will	O
become	O
tolerable	O
,	O
and	O
ultimately	O
a	O
16	O
byte	O
format	O
.	O
</s>
<s>
...	O
That	O
kind	O
of	O
gradual	O
evolution	O
towards	O
wider	O
precision	B-General_Concept
was	O
already	O
in	O
view	O
when	O
IEEE	O
Standard	O
754	O
for	O
Floating-Point	B-Algorithm
Arithmetic	I-Algorithm
was	O
framed.	O
"	O
</s>
<s>
The	O
Motorola	B-General_Concept
6888x	I-General_Concept
math	B-General_Concept
coprocessors	I-General_Concept
and	O
the	O
Motorola	B-Device
68040	I-Device
and	O
68060	B-General_Concept
processors	O
support	O
this	O
same	O
64-bit	O
significand	B-Algorithm
extended	B-Algorithm
precision	I-Algorithm
type	O
(	O
similar	O
to	O
the	O
Intel	O
format	O
although	O
padded	O
to	O
a	O
96-bit	O
format	O
with	O
16unused	O
bits	O
inserted	O
between	O
the	O
exponent	O
and	O
significand	B-Algorithm
fields	O
)	O
.	O
</s>
<s>
The	O
follow-on	O
Coldfire	B-Device
processors	O
do	O
not	O
support	O
this	O
96-bit	O
extended	B-Algorithm
precision	I-Algorithm
format	I-Algorithm
.	O
</s>
<s>
The	O
FPA10	O
math	B-General_Concept
coprocessor	I-General_Concept
for	O
early	O
ARM	B-Architecture
processors	I-Architecture
also	O
supports	O
this	O
extended	B-Algorithm
precision	I-Algorithm
type	O
(	O
similar	O
to	O
the	O
Intel	O
format	O
although	O
padded	O
to	O
a	O
96-bit	O
format	O
with	O
16zero	O
bits	O
inserted	O
between	O
the	O
sign	O
and	O
the	O
exponent	O
fields	O
)	O
,	O
but	O
without	O
correct	O
rounding	O
.	O
</s>
<s>
The	O
x87	B-Application
and	O
Motorola68881	O
80-bit	O
formats	O
meet	O
the	O
requirements	O
of	O
the	O
IEEE	O
754	O
double	O
extended	O
format	O
,	O
as	O
does	O
the	O
IEEE754	O
128-bit	O
format	O
.	O
</s>
<s>
The	O
x86	B-Operating_System
extended	B-Algorithm
precision	I-Algorithm
format	I-Algorithm
is	O
an	O
80-bit	O
format	O
first	O
implemented	O
in	O
the	O
Intel	B-Device
8087	I-Device
math	B-General_Concept
coprocessor	I-General_Concept
and	O
is	O
supported	O
by	O
all	O
processors	O
that	O
are	O
based	O
on	O
the	O
x86	B-Operating_System
design	I-Operating_System
that	O
incorporate	O
a	O
floating-point	B-General_Concept
unit	I-General_Concept
(	O
FPU	O
)	O
.	O
</s>
<s>
This	O
80-bit	O
format	O
uses	O
one	O
bit	O
for	O
the	O
sign	O
of	O
the	O
significand	B-Algorithm
,	O
15	O
bits	O
for	O
the	O
exponent	O
field	O
(	O
i.e.	O
</s>
<s>
the	O
same	O
range	O
as	O
the	O
128-bit	O
quadruple	O
precision	B-General_Concept
IEEE	O
754	O
format	O
)	O
and	O
64	O
bits	O
for	O
the	O
significand	B-Algorithm
.	O
</s>
<s>
The	O
exponent	O
field	O
is	O
biased	B-Algorithm
by	O
16383	O
,	O
meaning	O
that	O
16383	O
has	O
to	O
be	O
subtracted	O
from	O
the	O
value	O
in	O
the	O
exponent	O
field	O
to	O
compute	O
the	O
actual	O
power	O
of	O
2	O
.	O
</s>
<s>
An	O
exponent	O
field	O
value	O
of	O
32767	O
(	O
all	O
fifteen	O
bits	O
1	O
)	O
is	O
reserved	O
so	O
as	O
to	O
enable	O
the	O
representation	O
of	O
special	O
states	O
such	O
as	O
infinity	B-Application
and	O
Not	O
a	O
Number	O
.	O
</s>
<s>
If	O
the	O
exponent	O
field	O
is	O
zero	O
,	O
the	O
value	O
is	O
a	O
denormal	B-Algorithm
number	I-Algorithm
and	O
the	O
exponent	O
of	O
2	O
is	O
−16382	O
.	O
</s>
<s>
In	O
the	O
following	O
table	O
,	O
"	O
s	O
"	O
is	O
the	O
value	O
of	O
the	O
sign	O
bit	O
(	O
0	O
means	O
positive	O
,	O
1	O
means	O
negative	O
)	O
,	O
"	O
e	O
"	O
is	O
the	O
value	O
of	O
the	O
exponent	O
field	O
interpreted	O
as	O
a	O
positive	O
integer	O
,	O
and	O
"	O
m	O
"	O
is	O
the	O
significand	B-Algorithm
interpreted	O
as	O
a	O
positive	O
binary	O
number	O
where	O
the	O
binary	O
point	O
is	O
located	O
between	O
bits	O
63	O
and	O
62	O
.	O
</s>
<s>
+	O
Interpretation	O
of	O
the	O
fields	O
of	O
an	O
x86	B-Operating_System
Extended	B-Algorithm
Precision	I-Algorithm
value	O
Exponent	O
Significand	B-Algorithm
Meaning	O
All	O
Zeros	O
Bit	O
63	O
Bits	O
62-0	O
Zero	O
Zero	O
Zero	O
.	O
</s>
<s>
Non-zero	O
Denormal	B-Algorithm
.	O
</s>
<s>
The	O
value	O
is	O
One	O
Anything	O
Pseudo	O
Denormal	B-Algorithm
.	O
</s>
<s>
The	O
value	O
is	O
All	O
Ones	O
Bits	O
63	O
,	O
62	O
Bits	O
61-0	O
00	O
Zero	O
Pseudo-Infinity	O
.	O
</s>
<s>
The	O
sign	O
bit	O
gives	O
the	O
sign	O
of	O
the	O
infinity	B-Application
.	O
</s>
<s>
The	O
8087	B-Device
and	O
80287	O
treat	O
this	O
as	O
Infinity	B-Application
.	O
</s>
<s>
Non-zero	O
Pseudo	O
Not	O
a	O
Number	O
.	O
</s>
<s>
The	O
8087	B-Device
and	O
80287	O
treat	O
this	O
as	O
a	O
Signaling	O
Not	O
a	O
Number	O
.	O
</s>
<s>
The	O
8087	B-Device
and	O
80287	O
treat	O
this	O
as	O
a	O
Signaling	O
Not	O
a	O
Number	O
.	O
</s>
<s>
10	O
Zero	O
Infinity	B-Application
.	O
</s>
<s>
The	O
sign	O
bit	O
gives	O
the	O
sign	O
of	O
the	O
infinity	B-Application
.	O
</s>
<s>
The	O
8087	B-Device
and	O
80287	O
treat	O
this	O
as	O
a	O
Signaling	O
Not	O
a	O
Number	O
.	O
</s>
<s>
The	O
8087	B-Device
and	O
80287	O
coprocessors	B-General_Concept
used	O
the	O
pseudo-infinity	O
representation	O
for	O
infinities	O
.	O
</s>
<s>
Non-zero	O
Signalling	O
Not	O
a	O
Number	O
,	O
the	O
sign	O
bit	O
is	O
meaningless	O
.	O
</s>
<s>
11	O
Zero	O
Floating-point	B-Algorithm
Indefinite	O
,	O
the	O
result	O
of	O
invalid	O
calculations	O
such	O
as	O
square	O
root	O
of	O
a	O
negative	O
number	O
,	O
logarithm	O
of	O
a	O
negative	O
number	O
,	O
0/0	O
,	O
infinity	B-Application
/	O
infinity	B-Application
,	O
infinity	B-Application
times	O
0	O
,	O
and	O
others	O
when	O
the	O
processor	O
has	O
been	O
configured	O
to	O
not	O
generate	O
exceptions	O
for	O
invalid	O
operands	O
.	O
</s>
<s>
This	O
is	O
a	O
special	O
case	O
of	O
a	O
Quiet	O
Not	O
a	O
Number	O
.	O
</s>
<s>
Non-zero	O
Quiet	O
Not	O
a	O
Number	O
,	O
the	O
sign	O
bit	O
is	O
meaningless	O
.	O
</s>
<s>
The	O
8087	B-Device
and	O
80287	O
treat	O
this	O
as	O
a	O
Signaling	O
Not	O
a	O
Number	O
.	O
</s>
<s>
Only	O
generated	O
on	O
the	O
8087	B-Device
and	O
80287	O
.	O
</s>
<s>
In	O
contrast	O
to	O
the	O
single	O
and	O
double-precision	O
formats	O
,	O
this	O
format	O
does	O
not	O
utilize	O
an	O
implicit/hidden	O
bit	O
.	O
</s>
<s>
Rather	O
,	O
bit	O
63	O
contains	O
the	O
integer	O
part	O
of	O
the	O
significand	B-Algorithm
and	O
bits	O
62-0	O
hold	O
the	O
fractional	O
part	O
.	O
</s>
<s>
There	O
were	O
several	O
advantages	O
to	O
this	O
design	B-Device
when	O
the	O
8087	B-Device
was	O
being	O
developed	O
:	O
</s>
<s>
Calculations	O
can	O
be	O
completed	O
a	O
little	O
faster	O
if	O
all	O
bits	O
of	O
the	O
significand	B-Algorithm
are	O
present	O
in	O
the	O
register	O
.	O
</s>
<s>
A	O
64-bit	O
significand	B-Algorithm
provides	O
sufficient	O
precision	B-General_Concept
to	O
avoid	O
loss	O
of	O
precision	B-General_Concept
when	O
the	O
results	O
are	O
converted	O
back	O
to	O
double-precision	O
format	O
in	O
the	O
vast	O
number	O
of	O
cases	O
.	O
</s>
<s>
This	O
format	O
provides	O
a	O
mechanism	O
for	O
indicating	O
precision	B-General_Concept
loss	O
due	O
to	O
underflow	O
which	O
can	O
be	O
carried	O
through	O
further	O
operations	O
.	O
</s>
<s>
For	O
example	O
,	O
the	O
calculation	O
generates	O
the	O
intermediate	O
result	O
which	O
is	O
a	O
denormal	B-Algorithm
and	O
also	O
involves	O
precision	B-General_Concept
loss	O
.	O
</s>
<s>
The	O
80287	O
could	O
complete	O
this	O
calculation	O
and	O
indicate	O
the	O
loss	O
of	O
precision	B-General_Concept
by	O
returning	O
an	O
"	O
unnormal	O
"	O
result	O
(	O
exponent	O
not	O
0	O
,	O
bit	O
63	O
=	O
0	O
)	O
.	O
</s>
<s>
They	O
will	O
generate	O
a	O
denormal	B-Algorithm
if	O
an	O
underflow	O
occurs	O
but	O
will	O
generate	O
a	O
normalized	O
result	O
if	O
subsequent	O
operations	O
on	O
the	O
denormal	B-Algorithm
can	O
be	O
normalized	O
.	O
</s>
<s>
The	O
80-bit	B-Algorithm
floating-point	I-Algorithm
format	I-Algorithm
was	O
widely	O
available	O
by	O
1984	O
,	O
after	O
the	O
development	O
of	O
C	B-Language
,	O
Fortran	B-Application
and	O
similar	O
computer	O
languages	O
,	O
which	O
initially	O
offered	O
only	O
the	O
common	O
32	O
-	O
and	O
64-bit	O
floating-point	B-Algorithm
sizes	O
.	O
</s>
<s>
On	O
the	O
x86	B-Operating_System
design	I-Operating_System
most	O
C	B-Language
compilers	O
now	O
support	O
80-bit	O
extended	B-Algorithm
precision	I-Algorithm
via	O
the	O
long	B-Language
double	I-Language
type	O
,	O
and	O
this	O
was	O
specified	O
in	O
the	O
C99	B-Language
/	O
C11	O
standards	O
(	O
IEC	O
60559	O
floating-point	B-Algorithm
arithmetic	I-Algorithm
(	O
Annex	O
F	O
)	O
)	O
.	O
</s>
<s>
Compilers	O
on	O
x86	B-Operating_System
for	O
other	O
languages	O
often	O
support	O
extended	B-Algorithm
precision	I-Algorithm
as	O
well	O
,	O
sometimes	O
via	O
nonstandard	O
extensions	O
:	O
for	O
example	O
,	O
Turbo	B-Application
Pascal	I-Application
offers	O
an	O
extended	O
type	O
,	O
and	O
several	O
Fortran	B-Application
compilers	O
have	O
a	O
REAL*10	O
type	O
(	O
analogous	O
to	O
REAL*4	O
and	O
REAL*8	O
)	O
.	O
</s>
<s>
Such	O
compilers	O
also	O
typically	O
include	O
extended-precision	O
mathematical	O
subroutines	O
,	O
such	O
as	O
square	O
root	O
and	O
trigonometric	O
functions	O
,	O
in	O
their	O
standard	O
libraries	B-Library
.	O
</s>
<s>
The	O
80-bit	B-Algorithm
floating-point	I-Algorithm
format	I-Algorithm
has	O
a	O
range	O
(	O
including	O
subnormals	B-Algorithm
)	O
from	O
approximately	O
3.65	O
×10−4951	O
to	O
1.18	O
×104932	O
.	O
</s>
<s>
Although	O
log10(264 )	O
≅	O
19.266	O
,	O
this	O
format	O
is	O
usually	O
described	O
as	O
giving	O
approximately	O
eighteen	O
significant	O
digits	O
of	O
precision	B-General_Concept
(	O
the	O
floor	O
of	O
log10(263 )	O
,	O
the	O
minimum	O
guaranteed	O
precision	B-General_Concept
)	O
.	O
</s>
<s>
Bounds	O
on	O
conversion	O
between	O
decimal	O
and	O
binary	O
for	O
the	O
80-bit	O
format	O
can	O
be	O
given	O
as	O
follows	O
:	O
if	O
a	O
decimal	O
string	O
with	O
at	O
most	O
18	O
significant	O
digits	O
is	O
correctly	O
rounded	O
to	O
an	O
80-bit	O
IEEE	O
754	O
binary	B-Algorithm
floating-point	I-Algorithm
value	O
(	O
as	O
on	O
input	O
)	O
then	O
converted	O
back	O
to	O
the	O
same	O
number	O
of	O
significant	O
decimal	O
digits	O
(	O
as	O
for	O
output	O
)	O
,	O
then	O
the	O
final	O
string	O
will	O
exactly	O
match	O
the	O
original	O
;	O
while	O
,	O
conversely	O
,	O
if	O
an	O
80-bit	O
IEEE	O
754	O
binary	B-Algorithm
floating-point	I-Algorithm
value	O
is	O
correctly	O
converted	O
and	O
(	O
nearest	O
)	O
rounded	O
to	O
a	O
decimal	O
string	O
with	O
at	O
least	O
21	O
significant	O
decimal	O
digits	O
then	O
converted	O
back	O
to	O
binary	O
format	O
it	O
will	O
exactly	O
match	O
the	O
original	O
.	O
</s>
<s>
These	O
approximations	O
are	O
particularly	O
troublesome	O
when	O
specifying	O
the	O
best	O
value	O
for	O
constants	O
in	O
formulae	O
to	O
high	O
precision	B-General_Concept
,	O
as	O
might	O
be	O
calculated	O
via	O
arbitrary-precision	B-Algorithm
arithmetic	I-Algorithm
.	O
</s>
<s>
A	O
notable	O
example	O
of	O
the	O
need	O
for	O
a	O
minimum	O
of	O
64	O
bits	O
of	O
precision	B-General_Concept
in	O
the	O
significand	B-Algorithm
of	O
the	O
extended	B-Algorithm
precision	I-Algorithm
format	I-Algorithm
is	O
the	O
need	O
to	O
avoid	O
precision	B-General_Concept
loss	O
when	O
performing	O
exponentiation	O
on	O
double-precision	O
values	O
.	O
</s>
<s>
The	O
x86	B-Operating_System
floating-point	B-General_Concept
units	I-General_Concept
do	O
not	O
provide	O
an	O
instruction	O
that	O
directly	O
performs	O
exponentiation	O
.	O
</s>
<s>
In	O
order	O
to	O
avoid	O
precision	B-General_Concept
loss	O
,	O
the	O
intermediate	O
results	O
""	O
and	O
""	O
must	O
be	O
computed	O
with	O
much	O
higher	O
precision	B-General_Concept
,	O
because	O
effectively	O
both	O
the	O
exponent	O
and	O
the	O
significand	B-Algorithm
fields	O
of	O
must	O
fit	O
into	O
the	O
significand	B-Algorithm
field	O
of	O
the	O
intermediate	O
result	O
.	O
</s>
<s>
Subsequently	O
,	O
the	O
significand	B-Algorithm
field	O
of	O
the	O
intermediate	O
result	O
is	O
split	O
between	O
the	O
exponent	O
and	O
significand	B-Algorithm
fields	O
of	O
the	O
final	O
result	O
when	O
is	O
calculated	O
.	O
</s>
<s>
With	O
a	O
little	O
unpacking	O
,	O
an	O
IEEE	O
754	O
double-precision	O
value	O
can	O
be	O
represented	O
as	O
:	O
</s>
<s>
where	O
is	O
the	O
sign	O
of	O
the	O
exponent	O
(	O
either	O
0	O
or	O
1	O
)	O
,	O
is	O
the	O
unbiased	O
exponent	O
,	O
which	O
is	O
an	O
integer	O
that	O
ranges	O
from	O
0	O
to	O
1023	O
,	O
and	O
is	O
the	O
significand	B-Algorithm
which	O
is	O
a	O
53-bit	O
value	O
that	O
falls	O
in	O
the	O
range	O
.	O
</s>
<s>
For	O
purposes	O
of	O
this	O
discussion	O
does	O
not	O
have	O
53bits	O
of	O
precision	B-General_Concept
because	O
it	O
is	O
constrained	O
to	O
be	O
greater	O
than	O
or	O
equal	O
to	O
one	O
i.e.	O
</s>
<s>
the	O
hidden	O
bit	O
does	O
not	O
count	O
towards	O
the	O
precision	B-General_Concept
(	O
Note	O
that	O
in	O
situations	O
where	O
is	O
less	O
than	O
1	O
,	O
the	O
value	O
is	O
actually	O
a	O
de-normal	O
and	O
therefore	O
may	O
have	O
already	O
suffered	O
precision	B-General_Concept
loss	O
.	O
</s>
<s>
Taking	O
the	O
log	O
of	O
this	O
representation	O
of	O
a	O
double-precision	O
number	O
and	O
simplifying	O
results	O
in	O
the	O
following	O
:	O
</s>
<s>
This	O
result	O
demonstrates	O
that	O
when	O
taking	O
base2	O
logarithm	O
of	O
a	O
number	O
,	O
the	O
sign	O
of	O
the	O
exponent	O
of	O
the	O
original	O
value	O
becomes	O
the	O
sign	O
of	O
the	O
logarithm	O
,	O
the	O
exponent	O
of	O
the	O
original	O
value	O
becomes	O
the	O
integer	O
part	O
of	O
the	O
significand	B-Algorithm
of	O
the	O
logarithm	O
,	O
and	O
the	O
significand	B-Algorithm
of	O
the	O
original	O
value	O
is	O
transformed	O
into	O
the	O
fractional	O
part	O
of	O
the	O
significand	B-Algorithm
of	O
the	O
logarithm	O
.	O
</s>
<s>
Combining	O
10bits	O
to	O
the	O
left	O
of	O
the	O
radix	O
point	O
with	O
52bits	O
to	O
the	O
right	O
of	O
the	O
radix	O
point	O
means	O
that	O
the	O
significand	B-Algorithm
part	O
of	O
the	O
logarithm	O
must	O
be	O
computed	O
to	O
at	O
least	O
62bits	O
of	O
precision	B-General_Concept
.	O
</s>
<s>
In	O
practice	O
values	O
of	O
less	O
than	O
require	O
53bits	O
to	O
the	O
right	O
of	O
the	O
radix	O
point	O
and	O
values	O
of	O
less	O
than	O
require	O
54bits	O
to	O
the	O
right	O
of	O
the	O
radix	O
point	O
to	O
avoid	O
precision	B-General_Concept
loss	O
.	O
</s>
<s>
Balancing	O
this	O
requirement	O
for	O
added	O
precision	B-General_Concept
to	O
the	O
right	O
of	O
the	O
radix	O
point	O
,	O
exponents	O
less	O
than	O
512	O
only	O
require	O
9bits	O
to	O
the	O
left	O
of	O
the	O
radix	O
point	O
and	O
exponents	O
less	O
than	O
256	O
require	O
only	O
8bits	O
to	O
the	O
left	O
of	O
the	O
radix	O
point	O
.	O
</s>
<s>
Thus	O
the	O
integer	O
part	O
of	O
the	O
intermediate	O
result	O
(""	O
or	O
""	O
)	O
plus	O
a	O
bias	O
becomes	O
the	O
exponent	O
of	O
the	O
final	O
result	O
and	O
transformed	O
positive	O
fractional	O
part	O
of	O
the	O
intermediate	O
result	O
:	O
or	O
becomes	O
the	O
significand	B-Algorithm
of	O
the	O
final	O
result	O
.	O
</s>
<s>
In	O
order	O
to	O
supply	O
52bits	O
of	O
precision	B-General_Concept
to	O
the	O
final	O
result	O
,	O
the	O
positive	O
fractional	O
part	O
must	O
be	O
maintained	O
to	O
at	O
least	O
52bits	O
.	O
</s>
<s>
In	O
conclusion	O
,	O
the	O
exact	O
number	O
of	O
bits	O
of	O
precision	B-General_Concept
needed	O
in	O
the	O
significand	B-Algorithm
of	O
the	O
intermediate	O
result	O
is	O
somewhat	O
data	O
dependent	O
but	O
64bits	O
is	O
sufficient	O
to	O
avoid	O
precision	B-General_Concept
loss	O
in	O
the	O
vast	O
majority	O
of	O
exponentiation	O
computations	O
involving	O
double-precision	O
numbers	O
.	O
</s>
<s>
The	O
number	O
of	O
bits	O
needed	O
for	O
the	O
exponent	O
of	O
the	O
extended	B-Algorithm
precision	I-Algorithm
format	I-Algorithm
follows	O
from	O
the	O
requirement	O
that	O
the	O
product	O
of	O
two	O
double-precision	O
numbers	O
should	O
not	O
overflow	O
when	O
computed	O
using	O
the	O
extended	O
format	O
.	O
</s>
<s>
The	O
largest	O
possible	O
exponent	O
of	O
a	O
double-precision	O
value	O
is	O
1023	O
so	O
the	O
exponent	O
of	O
the	O
largest	O
possible	O
product	O
of	O
two	O
double-precision	O
numbers	O
is	O
2047	O
(	O
an	O
11-bit	O
value	O
)	O
.	O
</s>
<s>
Combining	O
these	O
requirements	O
:	O
1bit	O
for	O
the	O
sign	O
,	O
12bits	O
for	O
the	O
biased	B-Algorithm
exponent	I-Algorithm
,	O
and	O
64bits	O
for	O
the	O
significand	B-Algorithm
means	O
that	O
the	O
extended	B-Algorithm
precision	I-Algorithm
format	I-Algorithm
would	O
need	O
at	O
least	O
77bits	O
.	O
</s>
<s>
Engineering	O
considerations	O
resulted	O
in	O
the	O
final	O
definition	O
of	O
the	O
80-bit	O
format	O
(	O
in	O
particular	O
the	O
IEEE754	O
standard	O
requires	O
the	O
exponent	O
range	O
of	O
an	O
extended	B-Algorithm
precision	I-Algorithm
format	I-Algorithm
to	O
match	O
that	O
of	O
the	O
next	O
largest	O
,	O
quad	O
,	O
precision	B-General_Concept
format	O
which	O
is	O
15bits	O
)	O
.	O
</s>
<s>
Another	O
example	O
of	O
calculations	O
that	O
benefit	O
from	O
extended	B-Algorithm
precision	I-Algorithm
arithmetic	O
are	O
iterative	O
refinement	O
schemes	O
,	O
used	O
to	O
indirectly	O
clean	O
out	O
errors	O
accumulated	O
in	O
the	O
direct	O
solution	O
during	O
the	O
typically	O
very	O
large	O
number	O
of	O
calculations	O
made	O
for	O
numerical	O
linear	O
algebra	O
.	O
</s>
<s>
Some	O
C/C	O
++	O
implementations	O
(	O
e.g.	O
,	O
GNU	B-Application
Compiler	I-Application
Collection	I-Application
(	O
GCC	B-Application
)	O
,	O
Clang	B-Application
,	O
Intel	B-Language
C++	I-Language
)	O
implement	O
long	B-Language
double	I-Language
using	O
80-bit	O
floating-point	B-Algorithm
numbers	I-Algorithm
on	O
x86	B-Operating_System
systems	O
.	O
</s>
<s>
However	O
,	O
this	O
is	O
implementation-defined	O
behavior	O
and	O
is	O
not	O
required	O
,	O
but	O
allowed	O
by	O
the	O
standard	O
,	O
as	O
specified	O
for	O
IEEE	O
754	O
hardware	O
in	O
the	O
C99	B-Language
standard	I-Language
"	O
Annex	O
F	O
IEC	O
60559	O
floating-point	B-Algorithm
arithmetic	I-Algorithm
"	O
.	O
</s>
<s>
GCC	B-Application
also	O
provides	O
__float80	O
and	O
__float128	O
types	O
.	O
</s>
<s>
Some	O
Common	B-Language
Lisp	I-Language
implementations	O
(	O
e.g.	O
</s>
<s>
CMU	B-Language
Common	I-Language
Lisp	I-Language
,	O
Embeddable	B-Language
Common	I-Language
Lisp	I-Language
)	O
implement	O
long-float	O
using	O
80-bit	O
floating-point	B-Algorithm
numbers	I-Algorithm
on	O
x86	B-Operating_System
systems	O
.	O
</s>
<s>
D	B-Application
programming	I-Application
language	I-Application
implements	O
real	O
using	O
largest	O
floating-point	B-Algorithm
size	O
implemented	O
in	O
hardware	O
,	O
80	O
bits	O
for	O
x86	B-Operating_System
CPUs	B-Device
or	O
double	O
precision	B-General_Concept
,	O
whichever	O
is	O
larger	O
.	O
</s>
<s>
Object	B-Language
Pascal	I-Language
(	O
Delphi	B-Language
)	O
has	O
in	O
addition	O
to	O
Single	O
(	O
32-bit	O
)	O
and	O
Double	O
(	O
64-bit	O
)	O
,	O
an	O
Extended	O
(	O
80-bit	O
for	O
traditional	O
32-bit	O
targets	O
,	O
though	O
)	O
type	O
.	O
</s>
<s>
The	O
Racket	B-Operating_System
run-time	O
system	O
provides	O
the	O
80-bit	O
extflonum	O
datatype	O
on	O
x86	B-Operating_System
systems	O
.	O
</s>
<s>
The	O
Swift	B-Application
standard	O
library	O
provides	O
the	O
Float80	O
datatype	O
.	O
</s>
<s>
The	O
PowerBASIC	O
BASIC	O
compiler	O
provides	O
EXT	O
or	O
EXTENDED	O
10	O
byte	O
Extended-precision	B-Algorithm
floating-point	I-Algorithm
data	O
type	O
.	O
</s>
