<s>
The	O
IEEE	O
754-2008	O
standard	O
includes	O
decimal	O
floating-point	O
number	O
formats	O
in	O
which	O
the	O
significand	B-Algorithm
and	O
the	O
exponent	O
(	O
and	O
the	O
payloads	O
of	O
NaNs	O
)	O
can	O
be	O
encoded	O
in	O
two	O
ways	O
,	O
referred	O
to	O
as	O
binary	B-Algorithm
encoding	I-Algorithm
and	O
decimal	O
encoding	O
.	O
</s>
<s>
Both	O
formats	O
break	O
a	O
number	O
down	O
into	O
a	O
sign	O
bit	O
s	O
,	O
an	O
exponent	O
q	O
(	O
between	O
qmin	O
and	O
qmax	O
)	O
,	O
and	O
a	O
p-digit	O
significand	B-Algorithm
c	O
(	O
between	O
0	O
and	O
10p−1	O
)	O
.	O
</s>
<s>
In	O
both	O
formats	O
the	O
range	O
of	O
possible	O
values	O
is	O
identical	O
,	O
but	O
they	O
differ	O
in	O
how	O
the	O
significand	B-Algorithm
c	O
is	O
represented	O
.	O
</s>
<s>
This	O
makes	O
conversion	O
to	O
decimal	O
form	O
efficient	O
,	O
but	O
requires	O
a	O
specialized	O
decimal	O
ALU	B-General_Concept
to	O
process	O
.	O
</s>
<s>
In	O
the	O
binary	B-Algorithm
integer	I-Algorithm
decimal	I-Algorithm
(	O
BID	O
)	O
encoding	O
,	O
it	O
is	O
encoded	O
as	O
a	O
binary	O
number	O
.	O
</s>
<s>
However	O
,	O
the	O
IEEE	O
formats	O
have	O
significands	B-Algorithm
of	O
3n+1	O
digits	O
,	O
which	O
would	O
generally	O
require	O
10n+4	O
binary	O
bits	O
to	O
represent	O
.	O
</s>
<s>
Using	O
the	O
Decimal32	O
encoding	O
(	O
with	O
a	O
significand	B-Algorithm
of	O
3*2+1	O
decimal	O
digits	O
)	O
as	O
an	O
example	O
(	O
e	O
stands	O
for	O
exponent	O
,	O
m	O
for	O
mantissa	B-Algorithm
,	O
i.e.	O
</s>
<s>
significand	B-Algorithm
)	O
:	O
</s>
<s>
If	O
the	O
significand	B-Algorithm
starts	O
with	O
0mmm	O
,	O
omitting	O
the	O
leading	O
0	O
bit	O
lets	O
the	O
significand	B-Algorithm
fit	O
into	O
23	O
bits	O
:	O
</s>
<s>
If	O
the	O
significand	B-Algorithm
starts	O
with	O
100m	O
,	O
omitting	O
the	O
leading	O
100	O
bits	O
lets	O
the	O
significand	B-Algorithm
fit	O
into	O
21	O
bits	O
.	O
</s>
<s>
Infinity	O
,	O
quiet	O
NaN	O
and	O
signaling	O
NaN	O
use	O
encodings	O
beginning	O
with	O
s	O
1111	O
:	O
</s>
<s>
The	O
Decimal64	O
and	O
Decimal128	O
encodings	O
have	O
larger	O
exponent	O
and	O
significand	B-Algorithm
fields	O
,	O
but	O
operate	O
in	O
a	O
similar	O
fashion	O
.	O
</s>
<s>
For	O
the	O
Decimal128	O
encoding	O
,	O
113	O
bits	O
of	O
significand	B-Algorithm
is	O
actually	O
enough	O
to	O
encode	O
34	O
decimal	O
digits	O
,	O
and	O
the	O
second	O
form	O
is	O
never	O
actually	O
required	O
.	O
</s>
<s>
If	O
the	O
result	O
of	O
a	O
calculation	O
is	O
inexact	O
the	O
largest	O
amount	O
of	O
significant	O
data	O
is	O
preserved	O
by	O
selecting	O
the	O
cohort	O
member	O
with	O
the	O
largest	O
integer	O
that	O
can	O
be	O
stored	O
in	O
the	O
significand	B-Algorithm
along	O
with	O
the	O
required	O
exponent	O
.	O
</s>
<s>
The	O
proposed	O
IEEE	O
754r	O
standard	O
limits	O
the	O
range	O
of	O
numbers	O
to	O
a	O
significand	B-Algorithm
of	O
the	O
form	O
10n1	O
,	O
where	O
n	O
is	O
the	O
number	O
of	O
whole	O
decimal	O
digits	O
that	O
can	O
be	O
stored	O
in	O
the	O
bits	O
available	O
so	O
that	O
decimal	O
rounding	O
is	O
effected	O
correctly	O
.	O
</s>
<s>
A	O
binary	B-Algorithm
encoding	I-Algorithm
is	O
inherently	O
less	O
efficient	O
for	O
conversions	O
to	O
or	O
from	O
decimal-encoded	O
data	O
,	O
such	O
as	O
strings	O
(	O
ASCII	B-Protocol
,	O
Unicode	O
,	O
etc	O
.	O
)	O
</s>
<s>
A	O
binary	B-Algorithm
encoding	I-Algorithm
is	O
therefore	O
best	O
chosen	O
only	O
when	O
the	O
data	O
are	O
binary	O
rather	O
than	O
decimal	O
.	O
</s>
