<s>
Hexadecimal	B-Device
floating	I-Device
point	I-Device
(	O
now	O
called	O
HFP	O
by	O
IBM	O
)	O
is	O
a	O
format	O
for	O
encoding	O
floating-point	B-Algorithm
numbers	I-Algorithm
first	O
introduced	O
on	O
the	O
IBM	B-Application
System/360	I-Application
computers	O
,	O
and	O
supported	O
on	O
subsequent	O
machines	O
based	O
on	O
that	O
architecture	O
,	O
as	O
well	O
as	O
machines	O
which	O
were	O
intended	O
to	O
be	O
application-compatible	O
with	O
System/360	B-Application
.	O
</s>
<s>
In	O
comparison	O
to	O
IEEE	O
754	O
floating	B-Algorithm
point	I-Algorithm
,	O
the	O
HFP	O
format	O
has	O
a	O
longer	O
significand	B-Algorithm
,	O
and	O
a	O
shorter	O
exponent	O
.	O
</s>
<s>
All	O
HFP	O
formats	O
have	O
7	O
bits	O
of	O
exponent	O
with	O
a	O
bias	B-Algorithm
of	O
64	O
.	O
</s>
<s>
The	O
number	O
is	O
represented	O
as	O
the	O
following	O
formula	O
:	O
( −1	O
)	O
sign	O
×	O
0.significand	O
×	O
16exponent−64	O
.	O
</s>
<s>
radix	O
(	O
hexadecimal	O
)	O
point	O
is	O
set	O
to	O
the	O
left	O
of	O
the	O
significand	B-Algorithm
(	O
fraction	O
in	O
IBM	O
documentation	O
and	O
the	O
figures	O
)	O
.	O
</s>
<s>
Consider	O
encoding	O
the	O
value	O
−	O
118.625	O
as	O
an	O
HFP	O
single-precision	O
floating-point	B-Algorithm
value	O
.	O
</s>
<s>
A	O
bias	B-Algorithm
of	O
+64	O
is	O
added	O
to	O
the	O
exponent	O
( +2	O
)	O
,	O
yielding	O
+66	O
,	O
which	O
is	O
10000102	O
.	O
</s>
<s>
Combining	O
the	O
sign	O
,	O
exponent	O
plus	O
bias	B-Algorithm
,	O
and	O
normalized	O
fraction	O
produces	O
this	O
encoding	O
:	O
</s>
<s>
Given	O
a	O
fraction	O
of	O
all-bits	O
zero	O
,	O
any	O
combination	O
of	O
positive	O
or	O
negative	O
sign	O
bit	O
and	O
a	O
non-zero	O
biased	B-Algorithm
exponent	I-Algorithm
will	O
yield	O
a	O
value	O
arithmetically	O
equal	O
to	O
zero	O
.	O
</s>
<s>
This	O
is	O
true	O
for	O
all	O
three	O
floating-point	B-Algorithm
precision	O
formats	O
.	O
</s>
<s>
Since	O
the	O
base	O
is	O
16	O
,	O
there	O
can	O
be	O
up	O
to	O
three	O
leading	O
zero	O
bits	O
in	O
the	O
binary	O
significand	B-Algorithm
.	O
</s>
<s>
A	O
conversion	O
of	O
single	O
precision	O
hexadecimal	O
float	B-Algorithm
to	O
decimal	O
string	O
would	O
require	O
at	O
least	O
9	O
significant	O
digits	O
(	O
i.e.	O
</s>
<s>
6	O
log10(16 )	O
+	O
1	O
≈	O
8.22	O
)	O
in	O
order	O
to	O
convert	O
back	O
to	O
the	O
same	O
hexadecimal	O
float	B-Algorithm
value	O
.	O
</s>
<s>
A	O
conversion	O
of	O
double	O
precision	O
hexadecimal	O
float	B-Algorithm
to	O
decimal	O
string	O
would	O
require	O
at	O
least	O
18	O
significant	O
digits	O
in	O
order	O
to	O
convert	O
back	O
to	O
the	O
same	O
hexadecimal	O
float	B-Algorithm
value	O
.	O
</s>
<s>
Called	O
extended-precision	O
by	O
IBM	O
,	O
a	O
quadruple-precision	O
HFP	O
format	O
was	O
added	O
to	O
the	O
System/370	O
series	O
and	O
was	O
available	O
on	O
some	O
S/360	B-Application
models	O
(	O
S/360	B-Application
-85	O
,	O
-195	O
,	O
and	O
others	O
by	O
special	O
request	O
or	O
simulated	O
by	O
OS	O
software	O
)	O
.	O
</s>
<s>
Starting	O
with	O
the	O
S/390	B-Device
G5	O
in	O
1998	O
,	O
IBM	O
mainframes	O
have	O
also	O
included	O
IEEE	O
binary	B-Algorithm
floating-point	I-Algorithm
units	O
which	O
conform	O
to	O
the	O
IEEE	O
754	O
Standard	O
for	O
Floating-Point	B-Algorithm
Arithmetic	I-Algorithm
.	O
</s>
<s>
IEEE	O
decimal	O
floating-point	B-Algorithm
was	O
added	O
to	O
IBM	B-Device
System	I-Device
z9	I-Device
GA2	O
in	O
2007	O
using	O
millicode	B-General_Concept
and	O
in	O
2008	O
to	O
the	O
IBM	B-Device
System	I-Device
z10	I-Device
in	O
hardware	O
.	O
</s>
<s>
Modern	O
IBM	O
mainframes	O
support	O
three	O
floating-point	B-Algorithm
radices	O
with	O
3	O
hexadecimal	O
(	O
HFP	O
)	O
formats	O
,	O
3	O
binary	O
(	O
BFP	O
)	O
formats	O
,	O
and	O
3	O
decimal	O
(	O
DFP	O
)	O
formats	O
.	O
</s>
<s>
There	O
are	O
two	O
floating-point	B-Algorithm
units	O
per	O
core	O
;	O
one	O
supporting	O
HFP	O
and	O
BFP	O
,	O
and	O
one	O
supporting	O
DFP	O
;	O
there	O
is	O
one	O
register	O
file	O
,	O
FPRs	O
,	O
which	O
holds	O
all	O
3	O
formats	O
.	O
</s>
<s>
Starting	O
with	O
the	O
z13	O
in	O
2015	O
,	O
processors	O
have	O
added	O
a	O
vector	O
facility	O
that	O
includes	O
32	O
vector	O
registers	O
,	O
each	O
128	O
bits	O
wide	O
;	O
a	O
vector	O
register	O
can	O
contain	O
two	O
64-bit	O
or	O
four	O
32-bit	O
floating-point	B-Algorithm
numbers	I-Algorithm
.	O
</s>
<s>
The	O
traditional	O
16	O
floating-point	B-Algorithm
registers	O
are	O
overlaid	O
on	O
the	O
new	O
vector	O
registers	O
so	O
some	O
data	O
can	O
be	O
manipulated	O
with	O
traditional	O
floating-point	B-Algorithm
instructions	O
or	O
with	O
the	O
newer	O
vector	O
instructions	O
.	O
</s>
<s>
GRIB	B-Application
(	O
GRIdded	O
Binary	O
)	O
data	O
files	O
to	O
exchange	O
the	O
output	O
of	O
weather	O
prediction	O
models	O
(	O
IEEE	O
single-precision	O
floating-point	B-Algorithm
format	I-Algorithm
in	O
current	O
version	O
)	O
,	O
</s>
<s>
SEG	O
Y	O
(	O
Society	O
of	O
Exploration	O
Geophysicists	O
Y	O
)	O
format	O
files	O
(	O
IEEE	O
single-precision	O
floating-point	B-Algorithm
was	O
added	O
to	O
the	O
format	O
in	O
2002	O
)	O
.	O
</s>
<s>
One	O
exception	O
is	O
the	O
SAS	O
5	O
Transport	O
file	O
format	O
,	O
which	O
the	O
FDA	O
requires	O
;	O
in	O
that	O
format	O
,	O
"	O
All	O
floating-point	B-Algorithm
numbers	I-Algorithm
in	O
the	O
file	O
are	O
stored	O
using	O
the	O
IBM	O
mainframe	O
representation	O
.	O
</s>
<s>
 [ ... ] 	O
Most	O
platforms	O
use	O
the	O
IEEE	O
representation	O
for	O
floating-point	B-Algorithm
numbers	I-Algorithm
.	O
</s>
<s>
Code	O
for	O
IBM	O
's	O
format	O
is	O
also	O
available	O
under	O
LGPLv2.1	B-Application
.	O
</s>
