<s>
In	O
computing	O
,	O
minifloats	B-Algorithm
are	O
floating-point	B-Algorithm
values	O
represented	O
with	O
very	O
few	O
bits	O
.	O
</s>
<s>
Additionally	O
,	O
they	O
are	O
frequently	O
encountered	O
as	O
a	O
pedagogical	O
tool	O
in	O
computer-science	O
courses	O
to	O
demonstrate	O
the	O
properties	O
and	O
structures	O
of	O
floating-point	B-Algorithm
arithmetic	I-Algorithm
and	O
IEEE	O
754	O
numbers	O
.	O
</s>
<s>
Minifloats	B-Algorithm
with	O
16	O
bits	O
are	O
half-precision	O
numbers	O
(	O
opposed	O
to	O
single	O
and	O
double	O
precision	O
)	O
.	O
</s>
<s>
There	O
are	O
also	O
minifloats	B-Algorithm
with	O
8	O
bits	O
or	O
even	O
fewer	O
.	O
</s>
<s>
Minifloats	B-Algorithm
can	O
be	O
designed	O
following	O
the	O
principles	O
of	O
the	O
IEEE	O
754	O
standard	O
.	O
</s>
<s>
In	O
this	O
case	O
they	O
must	O
obey	O
the	O
(	O
not	O
explicitly	O
written	O
)	O
rules	O
for	O
the	O
frontier	O
between	O
subnormal	B-Algorithm
and	I-Algorithm
normal	I-Algorithm
numbers	I-Algorithm
and	O
must	O
have	O
special	O
patterns	O
for	O
infinity	B-Application
and	O
NaN	O
.	O
</s>
<s>
Normalized	O
numbers	O
are	O
stored	O
with	O
a	O
biased	B-Algorithm
exponent	I-Algorithm
.	O
</s>
<s>
The	O
new	O
revision	O
of	O
the	O
standard	O
,	O
IEEE	O
754-2008	O
,	O
has	O
16-bit	O
binary	O
minifloats	B-Algorithm
.	O
</s>
<s>
The	O
Radeon	B-Device
R300	I-Device
and	O
R420	O
GPUs	O
used	O
an	O
"	O
fp24	O
"	O
floating-point	B-Algorithm
format	I-Algorithm
with	O
7	O
bits	O
of	O
exponent	O
and	O
16	O
bits	O
( +1	O
implicit	O
)	O
of	O
mantissa	O
.	O
</s>
<s>
"	O
Full	O
Precision	O
"	O
in	O
Direct3D	O
9.0	O
is	O
a	O
proprietary	O
24-bit	O
floating-point	B-Algorithm
format	I-Algorithm
.	O
</s>
<s>
Microsoft	O
's	O
D3D9	O
(	O
Shader	O
Model	O
2.0	O
)	O
graphics	O
API	B-General_Concept
initially	O
supported	O
both	O
FP24	O
(	O
as	O
in	O
ATI	O
's	O
R300	B-Device
chip	O
)	O
and	O
FP32	O
(	O
as	O
in	O
Nvidia	O
's	O
NV30	O
chip	O
)	O
as	O
"	O
Full	O
Precision	O
"	O
,	O
as	O
well	O
as	O
FP16	O
as	O
"	O
Partial	O
Precision	O
"	O
for	O
vertex	O
and	O
pixel	O
shader	O
calculations	O
performed	O
by	O
the	O
graphics	O
hardware	O
.	O
</s>
<s>
A	O
minifloat	B-Algorithm
is	O
usually	O
described	O
using	O
a	O
tuple	O
of	O
four	O
numbers	O
,	O
(	O
S	O
,	O
E	O
,	O
M	O
,	O
B	O
)	O
:	O
</s>
<s>
B	O
is	O
the	O
exponent	B-Algorithm
bias	I-Algorithm
.	O
</s>
<s>
A	O
minifloat	B-Algorithm
format	O
denoted	O
by	O
(	O
S	O
,	O
E	O
,	O
M	O
,	O
B	O
)	O
is	O
,	O
therefore	O
,	O
bits	O
long	O
.	O
</s>
<s>
In	O
computer	O
graphics	O
minifloats	B-Algorithm
are	O
sometimes	O
used	O
to	O
represent	O
only	O
integral	O
values	O
.	O
</s>
<s>
If	O
at	O
the	O
same	O
time	O
subnormal	O
values	O
should	O
exist	O
,	O
the	O
least	O
subnormal	B-Algorithm
number	I-Algorithm
has	O
to	O
be	O
1	O
.	O
</s>
<s>
In	O
this	O
example	O
,	O
a	O
minifloat	B-Algorithm
in	O
1byte	O
(	O
8	O
bit	O
)	O
with	O
1	O
sign	O
bit	O
,	O
4	O
exponent	O
bits	O
and	O
3	O
significand	O
bits	O
(	O
in	O
short	O
,	O
a	O
1.4.3.	O
−2	O
minifloat	B-Algorithm
)	O
is	O
used	O
to	O
represent	O
integral	O
values	O
.	O
</s>
<s>
The	O
only	O
free	O
value	O
is	O
the	O
exponent	B-Algorithm
bias	I-Algorithm
,	O
which	O
we	O
define	O
as	O
-2	O
for	O
integers	O
.	O
</s>
<s>
This	O
is	O
a	O
chart	O
of	O
all	O
possible	O
values	O
with	O
bias	O
1	O
when	O
treating	O
the	O
float	B-Algorithm
similarly	O
to	O
an	O
IEEE	O
float	B-Algorithm
.	O
</s>
<s>
Integral	O
minifloats	B-Algorithm
in	O
1byte	O
have	O
a	O
greater	O
range	O
of	O
±122	O
880	O
than	O
two's-complement	B-General_Concept
integer	O
with	O
a	O
range	O
−128	O
to	O
+127	O
.	O
</s>
<s>
They	O
also	O
have	O
greater	O
range	O
than	O
half-precision	O
minifloats	B-Algorithm
with	O
range	O
±65	O
504	O
,	O
also	O
compensated	O
by	O
lack	O
of	O
fractions	O
and	O
poor	O
precision	O
.	O
</s>
<s>
There	O
are	O
only	O
242	O
different	O
values	O
(	O
if	O
+0	O
and	O
−0	O
are	O
regarded	O
as	O
different	O
)	O
,	O
because	O
14	O
of	O
the	O
bit	O
patterns	O
represent	O
NaNs	O
.	O
</s>
<s>
The	O
values	O
between	O
0	O
and	O
16	O
have	O
the	O
same	O
bit	O
pattern	O
as	O
minifloat	B-Algorithm
or	O
two's-complement	B-General_Concept
integer	O
.	O
</s>
<s>
The	O
first	O
pattern	O
with	O
a	O
different	O
value	O
is	O
00010001	O
,	O
which	O
is	O
18	O
as	O
a	O
minifloat	B-Algorithm
and	O
17	O
as	O
a	O
two's-complement	B-General_Concept
integer	O
.	O
</s>
<s>
This	O
coincidence	O
does	O
not	O
occur	O
at	O
all	O
with	O
negative	O
values	O
,	O
because	O
this	O
minifloat	B-Algorithm
is	O
a	O
signed-magnitude	O
format	O
.	O
</s>
<s>
The	O
(	O
vertical	O
)	O
real	O
line	O
on	O
the	O
right	O
shows	O
clearly	O
the	O
varying	O
density	O
of	O
the	O
floating-point	B-Algorithm
values	O
–	O
a	O
property	O
which	O
is	O
common	O
to	O
any	O
floating-point	B-Algorithm
system	O
.	O
</s>
<s>
The	O
graphic	O
demonstrates	O
the	O
addition	O
of	O
even	O
smaller	O
(	O
1.3.2.3	O
)	O
-minifloats	O
with	O
6	O
bits	O
.	O
</s>
<s>
This	O
floating-point	B-Algorithm
system	O
follows	O
the	O
rules	O
of	O
IEEE	O
754	O
exactly	O
.	O
</s>
<s>
NaN	O
as	O
operand	O
produces	O
always	O
NaN	O
results	O
.	O
</s>
<s>
Inf−Inf	O
and	O
( −Inf	O
)	O
+Inf	O
results	O
in	O
NaN	O
too	O
(	O
green	O
area	O
)	O
.	O
</s>
<s>
The	O
range	O
of	O
the	O
finite	O
operands	O
is	O
filled	O
with	O
the	O
curves	O
x	O
+	O
y	O
=	O
c	O
,	O
where	O
c	O
is	O
always	O
one	O
of	O
the	O
representable	O
float	B-Algorithm
values	O
(	O
blue	O
and	O
red	O
for	O
positive	O
and	O
negative	O
results	O
respectively	O
)	O
.	O
</s>
<s>
Minifloats	B-Algorithm
are	O
also	O
commonly	O
used	O
in	O
embedded	O
devices	O
,	O
especially	O
on	O
microcontrollers	B-Architecture
where	O
floating-point	B-Algorithm
will	O
need	O
to	O
be	O
emulated	O
in	O
software	O
.	O
</s>
