<s>
The	O
GNU	B-Application
Multiple	I-Application
Precision	I-Application
Floating-Point	I-Application
Reliable	I-Application
Library	I-Application
(	O
GNU	B-Application
MPFR	I-Application
)	O
is	O
a	O
GNU	B-Operating_System
portable	O
C	B-Language
library	B-Library
for	O
arbitrary-precision	B-Algorithm
binary	B-Algorithm
floating-point	I-Algorithm
computation	O
with	O
correct	O
rounding	O
,	O
based	O
on	O
GNU	B-Application
Multi-Precision	I-Application
Library	I-Application
.	O
</s>
<s>
MPFR	B-Application
's	O
computation	O
is	O
both	O
efficient	O
and	O
has	O
a	O
well-defined	O
semantics	O
:	O
the	O
functions	O
are	O
completely	O
specified	O
on	O
all	O
the	O
possible	O
operands	O
and	O
the	O
results	O
do	O
not	O
depend	O
on	O
the	O
platform	O
.	O
</s>
<s>
This	O
is	O
done	O
by	O
copying	O
the	O
ideas	O
from	O
the	O
ANSI/IEEE	O
-754	O
standard	O
for	O
fixed-precision	O
floating-point	B-Algorithm
arithmetic	I-Algorithm
(	O
correct	O
rounding	O
and	O
exceptions	O
,	O
in	O
particular	O
)	O
.	O
</s>
<s>
Support	O
for	O
special	O
numbers	O
:	O
signed	B-Algorithm
zeros	I-Algorithm
( +0	O
and	O
−0	B-Algorithm
)	O
,	O
infinities	O
and	O
not-a-number	O
(	O
a	O
single	O
NaN	O
is	O
supported	O
:	O
MPFR	B-Application
does	O
not	O
differentiate	O
between	O
quiet	O
NaNs	O
and	O
signaling	O
NaNs	O
)	O
.	O
</s>
<s>
Each	O
number	O
has	O
its	O
own	O
precision	B-General_Concept
(	O
in	O
bits	O
since	O
MPFR	B-Application
uses	O
radix	O
2	O
)	O
.	O
</s>
<s>
The	O
floating-point	B-Algorithm
results	O
are	O
correctly	O
rounded	O
to	O
the	O
precision	B-General_Concept
of	O
the	O
target	O
variable	O
,	O
in	O
one	O
of	O
the	O
five	O
supported	O
rounding	O
modes	O
(	O
including	O
the	O
four	O
from	O
IEEE	O
754-1985	O
)	O
.	O
</s>
<s>
Supported	O
functions	O
:	O
MPFR	B-Application
implements	O
all	O
mathematical	O
functions	O
from	O
C99	B-Language
and	O
other	O
usual	O
mathematical	O
functions	O
:	O
the	O
logarithm	O
and	O
exponential	O
in	O
natural	O
base	O
,	O
base	O
2	O
and	O
base	O
10	O
,	O
the	O
log( 1+x	O
)	O
and	O
exp(x )	O
−1	O
functions	O
(	O
log1p	O
and	O
expm1	O
)	O
,	O
the	O
six	O
trigonometric	O
and	O
hyperbolic	O
functions	O
and	O
their	O
inverses	O
,	O
the	O
gamma	O
,	O
zeta	O
and	O
error	O
functions	O
,	O
the	O
arithmetic	O
–	O
geometric	O
mean	O
,	O
the	O
power	O
(	O
xy	O
)	O
function	O
.	O
</s>
<s>
Subnormal	B-Algorithm
numbers	I-Algorithm
are	O
not	O
supported	O
,	O
but	O
can	O
be	O
emulated	O
with	O
the	O
mpfr_subnormalize	O
function	O
.	O
</s>
<s>
MPFR	B-Application
is	O
not	O
able	O
to	O
track	O
the	O
accuracy	O
of	O
numbers	O
in	O
a	O
whole	O
program	O
or	O
expression	O
;	O
this	O
is	O
not	O
its	O
goal	O
.	O
</s>
<s>
Interval	B-Algorithm
arithmetic	I-Algorithm
packages	O
like	O
Arb	O
,	O
MPFI	O
,	O
or	O
Real	B-General_Concept
RAM	I-General_Concept
implementations	O
like	O
iRRAM	O
,	O
which	O
may	O
be	O
based	O
on	O
MPFR	B-Application
,	O
can	O
do	O
that	O
for	O
the	O
user	O
.	O
</s>
<s>
MPFR	B-Application
is	O
dependent	O
upon	O
the	O
GNU	B-Application
Multiple	I-Application
Precision	I-Application
Arithmetic	I-Application
Library	I-Application
(	O
GMP	B-Application
)	O
.	O
</s>
<s>
MPFR	B-Application
is	O
needed	O
to	O
build	O
the	O
GNU	B-Application
Compiler	I-Application
Collection	I-Application
(	O
GCC	B-Application
)	O
.	O
</s>
<s>
Other	O
software	O
uses	O
MPFR	B-Application
,	O
such	O
as	O
ALGLIB	B-Library
,	O
CGAL	B-Language
,	O
FLINT	B-Language
,	O
GNOME	B-Language
Calculator	I-Language
,	O
the	O
Julia	B-Application
language	I-Application
implementation	O
,	O
the	O
Magma	B-General_Concept
computer	I-General_Concept
algebra	I-General_Concept
system	I-General_Concept
,	O
Maple	B-Language
,	O
GNU	B-Operating_System
MPC	O
,	O
and	O
GNU	B-Language
Octave	I-Language
.	O
</s>
