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Manual Reference Pages  -  MATH::MPFR (3)

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NAME

Math::MPFR - perl interface to the MPFR (floating point) library.

DEPENDENCIES

```

This module needs the MPFR and GMP C libraries. (Install GMP
first as it is a pre-requisite for MPFR.)

The GMP library is available from http://gmplib.org
The MPFR library is available from http://www.mpfr.org/

```

DESCRIPTION

```

A bigfloat module utilising the MPFR library. Basically
this module simply wraps the mpfr floating point functions
provided by that library.
The following documentation heavily plagiarises the mpfr
documentation.

```

SYNOPSIS

```

use Math::MPFR qw(:mpfr);

# @ can be used to separate mantissa from exponent. For bases
# that are <= 10, e or E can also be used.
# Use single quotes for string assignment if youre using @ as
# the separator. If you must use double quotes, youll have to
# escape the @.

my \$str = .123542@2; # mantissa = (.)123452
# exponent = 2
#Alternatively:
# my \$str = ".123542\@2";
# or:
# my \$str = 12.3542;
# or:
# my \$str = 1.23542e1;
# or:
# my \$str = 1.23542E1;

my \$base = 10;
my \$rnd = MPFR_RNDZ; # See ROUNDING MODE

# Create an Math::MPFR object that holds an initial
# value of \$str (in base \$base) and has the default
# precision. \$bn1 is the number. \$nok will either be 0
# indicating that the string was a valid number string, or
# -1, indicating that the string contained at least one
# invalid numeric character.
# See COMBINED INITIALISATION AND ASSIGNMENT, below.
my (\$bn1, \$nok) = Rmpfr_init_set_str(\$str, \$base, \$rnd);

# Or use the new() constructor - also documented below
# in COMBINED INITIALISATION AND ASSIGNMENT.
# my \$bn1 = Math::MPFR->new(\$str);

# Create another Math::MPFR object with precision
# of 100 bits and an initial value of NaN.
my \$bn2 = Rmpfr_init2(100);

# Assign the value -2314.451 to \$bn1.
Rmpfr_set_d(\$bn2, -2314.451, MPFR_RNDN);

# Create another Math::MPFR object that holds
# an initial value of NaN and has the default precision.
my \$bn3 = Rmpfr_init();

# Or using instead the new() constructor:
# my \$bn3 = Math::MPFR->new();

# Perform some operations ... see FUNCTIONS below.

.
.

# print out the value held by \$bn1 (in octal):
print Rmpfr_get_str(\$bn1, 8, 0, \$rnd), "\n";

# print out the value held by \$bn1 (in decimal):
print Rmpfr_get_str(\$bn1, 10, 0, \$rnd), "\n";
print \$bn1, "\n"; # is base 10, and uses e rather than @.

# print out the value held by \$bn1 (in base 16) using the
# TRmpfr_out_str function. (No newline is printed - unless
# its supplied as the optional fifth arg. See the
# TRmpfr_out_str documentation below.)
TRmpfr_out_str(*stdout, 16, 0, \$bn1, \$rnd);

```

ROUNDING MODE

```

One of 4 values:
GMP_RNDN (numeric value = 0): Round to nearest.
GMP_RNDZ (numeric value = 1): Round towards zero.
GMP_RNDU (numeric value = 2): Round towards +infinity.
GMP_RNDD (numeric value = 3): Round towards -infinity.

With the release of mpfr-3.0.0, the same rounding values
are renamed to:
MPFR_RNDN (numeric value = 0): Round to nearest.
MPFR_RNDZ (numeric value = 1): Round towards zero.
MPFR_RNDU (numeric value = 2): Round towards +infinity.
MPFR_RNDD (numeric value = 3): Round towards -infinity.

You can use either rendition with Math-MPFR-3.0 or later.

The mpfr-3.0.0 library also provides:
MPFR_RNDA (numeric value = 4): Round away from zero.

It, too, can be used with Math-MPFR-3.0 or later, but
will cause a fatal error iff the mpfr library against
which Math::MPFR is built is earlier than version 3.0.0.

The `round to nearest mode works as in the IEEE
P754 standard: in case the number to be rounded
lies exactly in the middle of two representable
numbers, it is rounded to the one with the least
significant bit set to zero.  For example, the
number 5, which is represented by (101) in binary,
is rounded to (100)=4 with a precision of two bits,
and not to (110)=6.  This rule avoids the "drift"
phenomenon mentioned by Knuth in volume 2 of
The Art of Computer Programming (section 4.2.2,
pages 221-222).

Most Math::MPFR functions take as first argument the
destination variable, as second and following arguments
the input variables, as last argument a rounding mode,
and have a return value of type `int. If this value
is zero, it usually means that the value stored in the
destination variable is the exact result of the
corresponding mathematical function. If the returned
value is positive (resp. negative), it usually means
the value stored in the destination variable is greater
(resp. lower) than the exact result.  For example with
the `GMP_RNDU rounding mode, the returned value is
usually positive, except when the result is exact, in
which case it is zero.  In the case of an infinite
result, it is considered as inexact when it was
obtained by overflow, and exact otherwise.  A
NaN result (Not-a-Number) always corresponds to an
inexact return value.

```

MEMORY MANAGEMENT

```

Objects are created with new() or with the Rmpfr_init*
functions. All of these functions return an object that has
been blessed into the package Math::MPFR.
They will therefore be automatically cleaned up by the
DESTROY() function whenever they go out of scope.

For each Rmpfr_init* function there is a corresponding function
called Rmpfr_init*_nobless which returns an unblessed object.
If you create Math::MPFR objects using the _nobless
versions, it will then be up to you to clean up the memory
associated with these objects by calling Rmpfr_clear(\$op)
for each object, or Rmpfr_clears(\$op1, \$op2, ....).
Alternatively such objects will be cleaned up when the script
ends. I dont know why you would want to create unblessed
objects. The point is that you can if you want to.
The test suite does no testing of unblessed objects ... beware
of bugs if you go down that path.

```

MIXING GMP OBJECTS WITH MPFR OBJECTS

```

Some of the Math::MPFR functions below take as arguments
one or more of the GMP types mpz (integer), mpq
(rational) and mpf (floating point). (Such functions are
marked as taking mpz/mpq/mpf arguments.)
For these functions to work you need to have loaded either:

objects only - NOT mpf and mpq objects.)

AND/OR

2) Math::GMPz (for mpz types), Math::GMPq (for mpq types)
and Math::GMPf (for mpf types).

You may also be able to use objects from the GMP module
that ships with the GMP sources. I get occasional
segfaults when I try to do that, so Ive stopped
recommending it - and dont support the practice.

```

PASSING __float128 VALUES

```

There are 3 ways to pass __float128 values to/from
Math::MPFR:

1) Install Math::Float128, build the mpfr-3.2.0 (or later)
library with the configure option --enable-float128, and build
Math::MPFR by providing the "F128=1" arg to the Makefile.pl:

perl Makefile.PL F128=1

Then you can pass the values of the Math::Float128 objects to
and from Math::MPFR objects using:

Rmpfr_set_FLOAT128() and Rmpfr_get_FLOAT128()

2) Build perl (5.21.4 or later) with -Dusequadmath; build the
mpfr-3.2.0 (or later) library with the configure option
--enable-float128, and then build Math::MPFR by providing the
"F128=1" arg to the Makefile.pl:

perl Makefile.PL F128=1

Then you can pass your perls __float128 NV values directly
to/from Math::MPFR using:

Rmpfr_set_float128() or Rmpfr_set_NV() and
Rmpfr_get_float128() or Rmpfr_get_NV()

NV will evaluate that (__float128) NV to its full precision.
And assigning the NV as Math::MPFR->new(\$nv) will also work as
intended.

3) Convert the __float128 values to a string and pass them to
and from Math::MPFR using:

Rmpfr_set_str() and Rmpfr_get_str()

```

FUNCTIONS

```

These next 3 functions are demonstrated above:
\$rop = Rmpfr_init();
\$rop = Rmpfr_init2(\$p);
\$str = Rmpfr_get_str(\$op, \$base, \$digits, \$rnd); # 1 < \$base < 37
The third argument to Rmpfr_get_str() specifies the number of digits
required to be output in the mantissa. (Trailing zeroes are removed.)
If \$digits is 0, the number of digits of the mantissa is chosen
large enough so that re-reading the printed value with the same
precision, assuming both output and input use rounding to nearest,
will recover the original value of \$op.

The following functions are simply wrappers around an mpfr
function of the same name. eg. Rmpfr_swap() is a wrapper around
mpfr_swap().

"\$rop", "\$op1", "\$op2", etc. are Math::MPFR objects - the
return value of one of the Rmpfr_init* functions. They are in fact
references to mpfr structures. The "\$op" variables are the operands
and "\$rop" is the variable that stores the result of the operation.
Generally, \$rop, \$op1, \$op2, etc. can be the same perl variable
referencing the same mpfr structure, though often they will be
distinct perl variables referencing distinct mpfr structures.
Eg something like Rmpfr_add(\$r1, \$r1, \$r1, \$rnd),
where \$r1 *is* the same reference to the same mpfr structure,
would add \$r1 to itself and store the result in \$r1. Alternatively,
as \$r1 += \$r1. Otoh, Rmpfr_add(\$r1, \$r2, \$r3, \$rnd), where each of the
arguments is a different reference to a different mpfr structure
would add \$r2 to \$r3 and store the result in \$r1. Alternatively
it could be coded as \$r1 = \$r2 + \$r3.

"\$ui" means any integer that will fit into a C unsigned long int,

"\$si" means any integer that will fit into a C signed long int.

"\$sj" means any integer that will fit into a C intmax_t. Dont
use any of these functions unless your perl was compiled with 64
bit support.

"\$double" is a C double and "\$float" is a C float ... but both will
be represented in Perl as an NV.

"\$bool" means a value (usually a signed long int) in which
the only interest is whether it evaluates as false or true.

"\$str" simply means a string of symbols that represent a number,
eg 1234567890987654321234567@7 which might be a base 10 number,
or zsa34760sdfgq123r5@11 which would have to represent at least
a base 36 number (because "z" is a valid digit only in bases 36
and above). Valid bases for MPFR numbers are 0 and 2 to 36 (2 to 62
if Math::MPFR has been built against mpfr-3.0.0 or later).

"\$rnd" is simply one of the 4 rounding mode values (discussed above).

"\$p" is the (signed int) value for precision.

##############

ROUNDING MODES

Rmpfr_set_default_rounding_mode(\$rnd);
Sets the default rounding mode to \$rnd.
The default rounding mode is to nearest initially (GMP_RNDN).
The default rounding mode is the rounding mode that

\$si = Rmpfr_get_default_rounding_mode();
Returns the numeric value (0, 1, 2 or 3) of the
current default rounding mode. This will initially be 0.

\$si = Rmpfr_prec_round(\$rop, \$p, \$rnd);
Rounds \$rop according to \$rnd with precision \$p, which may be
different from that of \$rop.  If \$p is greater or equal to the
precision of \$rop, then new space is allocated for the mantissa,
and it is filled with zeroes.  Otherwise, the mantissa is rounded
to precision \$p with the given direction. In both cases, the
precision of \$rop is changed to \$p.  The returned value is zero
when the result is exact, positive when it is greater than the
original value of \$rop, and negative when it is smaller.  The
precision \$p can be any integer between RMPFR_PREC_MIN and
RMPFR_PREC_MAX.

##########

EXCEPTIONS

\$si =  Rmpfr_get_emin();
\$si =  Rmpfr_get_emax();
Return the (current) smallest and largest exponents
allowed for a floating-point variable.

\$si = Rmpfr_get_emin_min();
\$si = Rmpfr_get_emin_max();
\$si = Rmpfr_get_emax_min();
\$si = Rmpfr_get_emax_max();
Return the minimum and maximum of the smallest and largest
exponents allowed for `mpfr_set_emin and `mpfr_set_emax. These
values are implementation dependent

\$bool =  Rmpfr_set_emin(\$si);
\$bool =  Rmpfr_set_emax(\$si);
Set the smallest and largest exponents allowed for a
floating-point variable.  Return a non-zero value when \$si is not
in the range of exponents accepted by the implementation (in that
case the smallest or largest exponent is not changed), and zero
otherwise. If the user changes the exponent range, it is her/his
responsibility to check that all current floating-point variables
are in the new allowed range (for example using `Rmpfr_check_range,
otherwise the subsequent behaviour will be undefined, in the sense
of the ISO C standard.

\$si2 = Rmpfr_check_range(\$op, \$si1, \$rnd);
This function has changed from earlier implementations.
It now forces \$op to be in the current range of acceptable
values, \$si1 the current ternary value: negative if \$op is
smaller than the exact value, positive if \$op is larger than the
exact value and zero if \$op is exact (before the call). It generates
an underflow or an overflow if the exponent of \$op is outside the
current allowed range; the value of \$si1 may be used to avoid a
double rounding. This function returns zero if the rounded result
is equal to the exact one, a positive value if the rounded result
is larger than the exact one, a negative value if the rounded
result is smaller than the exact one. Note that unlike most
functions, the result is compared to the exact one, not the input
value \$op, i.e. the ternary value is propagated.
Note: If \$op is an infinity and \$si1 is different from zero
(i.e., if the rounded result is an inexact infinity), then the
overflow flag is set.

Rmpfr_set_underflow();
Rmpfr_set_overflow();
Rmpfr_set_nanflag();
Rmpfr_set_inexflag();
Rmpfr_set_erangeflag();
Rmpfr_set_divby0();     # mpfr-3.1.0 and later only
Rmpfr_clear_underflow();
Rmpfr_clear_overflow();
Rmpfr_clear_nanflag();
Rmpfr_clear_inexflag();
Rmpfr_clear_erangeflag();
Rmpfr_clear_divby0();   # mpfr-3.1.0 and later only
Set/clear the underflow, overflow, invalid, inexact, erange and
divide-by-zero flags.

Rmpfr_clear_flags();
Clear all global flags (underflow, overflow, inexact, invalid,
erange and divide-by-zero).

\$bool = Rmpfr_underflow_p();
\$bool = Rmpfr_overflow_p();
\$bool = Rmpfr_nanflag_p();
\$bool = Rmpfr_inexflag_p();
\$bool = Rmpfr_erangeflag_p();
\$bool = Rmpfr_divby0_p();   # mpfr-3.1.0 and later only
Return the corresponding (underflow, overflow, invalid, inexact,
erange, divide-by-zero) flag, which is non-zero iff the flag is set.

\$si = Rmpfr_subnormalize (\$op, \$si, \$rnd);
See the MPFR documentation for mpfr_subnormalize().

##############

INITIALIZATION

A variable should be initialized once only.

First read the section MEMORY MANAGEMENT (above).

Rmpfr_set_default_prec(\$p);
Set the default precision to be *exactly* \$p bits.  The
precision of a variable means the number of bits used to store its
mantissa.  All subsequent calls to `mpfr_init will use this
precision, but previously initialized variables are unaffected.
This default precision is set to 53 bits initially.  The precision
can be any integer between RMPFR_PREC_MIN and RMPFR_PREC_MAX.

\$ui = Rmpfr_get_default_prec();
Returns the default MPFR precision in bits.

\$rop = Math::MPFR->new();
\$rop = Math::MPFR::new();
\$rop = new Math::MPFR();
\$rop = Rmpfr_init();
\$rop = Rmpfr_init_nobless();
Initialize \$rop, and set its value to NaN. The precision
of \$rop is the default precision, which can be changed
by a call to `Rmpfr_set_default_prec.

\$rop = Rmpfr_init2(\$p);
\$rop = Rmpfr_init2_nobless(\$p);
Initialize \$rop, set its precision to be *exactly* \$p bits,
and set its value to NaN.  To change the precision of a
variable which has already been initialized,
use `Rmpfr_set_prec instead.  The precision \$p can be
any integer between RMPFR_PREC_MIN and RMPFR_PREC_MAX.

@rops = Rmpfr_inits(\$how_many);
@rops = Rmpfr_inits_nobless(\$how_many);
Returns an array of \$how_many Math::MPFR objects - initialized,
with a value of NaN, and with default precision.
(These functions do not wrap mpfr_inits.)

@rops = Rmpfr_inits2(\$p, \$how_many);
@rops = Rmpfr_inits2_nobless(\$p, \$how_many);
Returns an array of \$how_many Math::MPFR objects - initialized,
with a value of NaN, and with precision of \$p.
(These functions do not wrap mpfr_inits2.)

Rmpfr_set_prec(\$op, \$p);
Reset the precision of \$op to be *exactly* \$p bits.
The previous value stored in \$op is lost.  The precision
\$p can be any integer between RMPFR_PREC_MIN and
RMPFR_PREC_MAX. If you want to keep the previous
value stored in \$op, use Rmpfr_prec_round instead.

\$si = Rmpfr_get_prec(\$op);
Return the precision actually used for assignments of \$op,
i.e. the number of bits used to store its mantissa.

Rmpfr_set_prec_raw(\$rop, \$p);
Reset the precision of \$rop to be *exactly* \$p bits.  The only
difference with `mpfr_set_prec is that \$p is assumed to be small
enough so that the mantissa fits into the current allocated
memory space for \$rop. Otherwise an error will occur.

\$min_prec = Rmpfr_min_prec(\$op);
(This function is implemented only when Math::MPFR is built
against mpfr-3.0.0 or later. The mpfr_min_prec function was
not present in earlier versions of mpfr.)
\$min_prec is set to the minimal number of bits required to store
the significand of \$op, and 0 for special values, including 0.
(Warning: the returned value can be less than RMPFR_PREC_MIN.)

\$minimum_precision = RMPFR_PREC_MIN;
\$maximum_precision = RMPFR_PREC_MAX;
Returns the minimum/maximum precision for Math::MPFR objects
allowed by the mpfr library being used.

##########

ASSIGNMENT

\$si = Rmpfr_set(\$rop, \$op, \$rnd);
\$si = Rmpfr_set_ui(\$rop, \$ui, \$rnd);
\$si = Rmpfr_set_si(\$rop, \$si, \$rnd);
\$si = Rmpfr_set_sj(\$rop, \$sj, \$rnd); # 64 bit
\$si = Rmpfr_set_uj(\$rop, \$uj, \$rnd); # 64 bit
\$si = Rmpfr_set_d(\$rop, \$double, \$rnd);
\$si = Rmpfr_set_ld(\$rop, \$ld, \$rnd); # long double
\$si = Rmpfr_set_NV(\$rop, \$nv, \$rnd); # double/long double/__float128
\$si = Rmpfr_set_LD(\$rop, \$LD, \$rnd); # \$LD is a Math::LongDouble object
\$si = Rmpfr_set_z(\$rop, \$z, \$rnd); # \$z is a mpz object.
\$si = Rmpfr_set_q(\$rop, \$q, \$rnd); # \$q is a mpq object.
\$si = Rmpfr_set_f(\$rop, \$f, \$rnd); # \$f is a mpf object.
\$si = Rmpfr_set_flt(\$rop, \$float, \$rnd); # mpfr-3.0.0 and later only
\$si = Rmpfr_set_float128(\$rop, \$f128, \$rnd); # mpfr-3.2.0 and later
\$si = Rmpfr_set_DECIMAL64(\$rop, \$D64, \$rnd) # mpfr-3.1.1 and later
# only. \$D64 is a
# Math::Decimal64 object
\$si = Rmpfr_set_FLOAT128(\$rop, \$F128, \$rnd) # mpfr-3.2.0 and later
# only. \$F128 is a
# Math::Float128 object
Set the value of \$rop from 2nd arg, rounded to the precision of
\$rop towards the given direction \$rnd.  Please note that even a
long int may have to be rounded if the destination precision
is less than the machine word width.  The return value is zero
when \$rop=2nd arg, positive when \$rop>2nd arg, and negative when
\$rop<2nd arg.  For `mpfr_set_d, be careful that the input
number \$double may not be exactly representable as a double-precision
number (this happens for 0.1 for instance), in which case it is
first rounded by the C compiler to a double-precision number,
and then only to a mpfr floating-point number.

NOTE: If your perls nvtype is long double use Rmpfr_set_ld() or
Rmpfr_set_NV(), but if your perls nvtype is double and you want
to set a value whose precision is that of long double, then
install Math::LongDouble and use Rmpfr_set_LD().
Rmpfr_set_NV simply calls either mpfr_set_ld, mpfr_set_ld, or
mpfr_set_float128 as appropriate for your Math::MPFR and perl
configuration.

\$si = Rmpfr_set_ui_2exp(\$rop, \$ui, \$exp, \$rnd);
\$si = Rmpfr_set_si_2exp(\$rop, \$si, \$exp, \$rnd);
\$si = Rmpfr_set_uj_2exp(\$rop, \$sj, \$exp, \$rnd); # 64 bit
\$si = Rmpfr_set_sj_2exp(\$rop, \$sj, \$exp, \$rnd); # 64 bit
\$si = Rmpfr_set_z_2exp(\$rop, \$z, \$exp, \$rnd); # mpfr-3.0.0 and later only
Set the value of \$rop from the 2nd arg multiplied by two to the
power \$exp, rounded towards the given direction \$rnd.  Note that
the input 0 is converted to +0. (\$z is a GMP mpz object.)

\$si = Rmpfr_set_str(\$rop, \$str, \$base, \$rnd);
Set \$rop to the value of \$str in base \$base (0,2..36 or, if
Math::MPFR has been built against mpfr-3.0.0 or later, (0,2..62),
rounded in direction \$rnd to the precision of \$rop.
The exponent is read in decimal.  This function returns 0 if
the entire string is a valid number in base \$base. otherwise
it returns -1.
If -1 is returned:
1) the non-numeric flag (which was initalised to 0) will be
incremented. You can query/clear/reset the value of the
flag with (resp.) nnumflag()/clear_nnum()/set_nnum() - all
of which are documented below (in "MISCELLANEOUS");
2) A warning will be emitted if \$Math::MPFR::NNW is set to 1
(default is 0).
If \$base is zero, the base is set according to the following
rules:
if the string starts with 0b or 0B the base is set to 2;
if the string starts with 0x or 0X the base is set to 16;
otherwise the base is set to 10.
The following exponent symbols can be used:
@ - can be used for any base;
e or E - can be used only with bases <= 10;
p or P - can be used to introduce binary exponents with
Rmpfr_inp_str (below).
Because of the special significance of the @ symbol in perl,
make sure you assign to strings using single quotes, not
double quotes, when using @ as the exponent marker. If you
must use double quotes (which is hard to believe) then you
need to escape the @. ie the following two assignments are
equivalent:
Rmpfr_set_str(\$rop, .1234@-5, 10, GMP_RNDN);
Rmpfr_set_str(\$rop, ".1234\@-5", 10, GMP_RNDN);
But the following assignment wont do what you want:
Rmpfr_set_str(\$rop, ".1234@-5", 10, GMP_RNDN);

Rmpfr_strtofr(\$rop, \$str, \$base, \$rnd);
Read a floating point number from a string \$str in base \$base,
rounded in the direction \$rnd. If successful, the result is
stored in \$rop. If \$str doesnt start with a valid number then
\$rop is set to zero.
Parsing follows the standard C `strtod function with some
extensions.  Case is ignored. After optional leading whitespace,
one has a subject sequence consisting of an optional sign (`+ or
`-), and either numeric data or special data. The subject
sequence is defined as the longest initial subsequence of the
input string, starting with the first non-whitespace character,
that is of the expected form.
The form of numeric data is a non-empty sequence of significand
digits with an optional decimal point, and an optional exponent
consisting of an exponent prefix followed by an optional sign and
a non-empty sequence of decimal digits. A significand digit is
either a decimal digit or a Latin letter (62 possible characters),
with `a = 10, `b = 11, ..., `z = 36; its value must be strictly
less than the base.  The decimal point can be either the one
defined by the current locale or the period (the first one is
accepted for consistency with the C standard and the practice, the
second one is accepted to allow the programmer to provide MPFR
numbers from strings in a way that does not depend on the current
locale).  The exponent prefix can be `e or `E for bases up to
10, or `@ in any base; it indicates a multiplication by a power
of the base. In bases 2 and 16, the exponent prefix can also be
`p or `P, in which case it introduces a binary exponent: it
indicates a multiplication by a power of 2 (there is a difference
only for base 16).  The value of an exponent is always written in
base 10.  In base 2, the significand can start with `0b or `0B,

If the argument \$base is 0, then the base is automatically detected
as follows. If the significand starts with `0b or `0B, base 2 is
assumed. If the significand starts with `0x or `0X, base 16 is
assumed. Otherwise base 10 is assumed. Other allowable values for
\$base are 2 to 36 (2 to 62 if Math::MPFR has been built against
mpfr-3.0.0 or later).

Note: The exponent must contain at least a digit. Otherwise the
possible exponent prefix and sign are not part of the number
(which ends with the significand). Similarly, if `0b, `0B, `0x
or `0X is not followed by a binary/hexadecimal digit, then the
subject sequence stops at the character `0.
Special data (for infinities and NaN) can be `@inf@ or
`@nan@(n-char-sequence), and if BASE <= 16, it can also be
`infinity, `inf, `nan or `nan(n-char-sequence), all case
insensitive.  A `n-char-sequence is a non-empty string containing
only digits, Latin letters and the underscore (0, 1, 2, ..., 9, a,
b, ..., z, A, B, ..., Z, _). Note: one has an optional sign for
all data, even NaN.
The function returns a usual ternary value.

Rmpfr_set_str_binary(\$rop, \$str);
Removed in Math-MPFR-3.30. Should have been removed long ago.
Set \$rop to the value of the binary number in \$str, which has to
be of the form +/-xxxx.xxxxxxEyy. The exponent is read in decimal,
but is interpreted as the power of two to be multiplied by the
mantissa.  The mantissa length of \$str has to be less or equal to
the precision of \$rop, otherwise an error occurs.  If \$str starts
with `N, it is interpreted as NaN (Not-a-Number); if it starts
with `I after the sign, it is interpreted as infinity, with the
corresponding sign.

Rmpfr_set_inf(\$rop, \$si);
Rmpfr_set_nan(\$rop);
Rmpfr_set_zero(\$rop, \$si); # mpfr-3.0.0 and later only.
Set the variable \$rop to infinity or NaN (Not-a-Number) or zero
respectively. In mpfr_set_inf and mpfr_set_zero, the sign of \$rop
is positive if 2nd arg >= 0. Else the sign is negative.

Rmpfr_swap(\$op1, \$op2);
Swap the values \$op1 and \$op2 efficiently. Warning: the precisions
are exchanged too; in case the precisions are different, `mpfr_swap
is thus not equivalent to three `mpfr_set calls using a third
auxiliary variable.

################################################

COMBINED INITIALIZATION AND ASSIGNMENT

NOTE: Do NOT use these functions if \$rop has already
been initialised. Use the Rmpfr_set* functions in the
section ASSIGNMENT (above).

First read the section MEMORY MANAGEMENT (above).

\$rop = Math::MPFR->new(\$arg);
\$rop = Math::MPFR::new(\$arg);
\$rop = new Math::MPFR(\$arg);
Returns a Math::MPFR object with the value of \$arg, rounded
in the default rounding direction, with default precision.
\$arg can be either a number (signed integer, unsigned integer,
signed fraction or unsigned fraction), a string that
represents a numeric value, or an object (of type Math::GMPf,
Math::GMPq, Math::GMPz, orMath::GMP) If \$arg is a string, an
optional additional argument that specifies the base of the
number can be supplied to new(). Legal values for base are 0
and 2 to 36 (2 to 62 if Math::MPFR has been built against
mpfr-3.0.0 or later). If \$arg is a string and no
base. See Rmpfr_set_str above for an explanation of how
that deduction is attempted. For finer grained control, use
one of the Rmpfr_init_set_* functions documented immediately
below.
Note that these functions return a list of 2 values.

(\$rop, \$si) = Rmpfr_init_set(\$op, \$rnd);
(\$rop, \$si) = Rmpfr_init_set_nobless(\$op, \$rnd);
(\$rop, \$si) = Rmpfr_init_set_ui(\$ui, \$rnd);
(\$rop, \$si) = Rmpfr_init_set_ui_nobless(\$ui, \$rnd);
(\$rop, \$si) = Rmpfr_init_set_si(\$si, \$rnd);
(\$rop, \$si) = Rmpfr_init_set_si_nobless(\$si, \$rnd);
(\$rop, \$si) = Rmpfr_init_set_d(\$double, \$rnd);
(\$rop, \$si) = Rmpfr_init_set_d_nobless(\$double, \$rnd);
(\$rop, \$si) = Rmpfr_init_set_ld(\$double, \$rnd);
(\$rop, \$si) = Rmpfr_init_set_ld_nobless(\$double, \$rnd);
(\$rop, \$si) = Rmpfr_init_set_f(\$f, \$rnd);# \$f is a mpf object
(\$rop, \$si) = Rmpfr_init_set_f_nobless(\$f, \$rnd);# \$f is a mpf object
(\$rop, \$si) = Rmpfr_init_set_z(\$z, \$rnd);# \$z is a mpz object
(\$rop, \$si) = Rmpfr_init_set_z_nobless(\$z, \$rnd);# \$z is a mpz object
(\$rop, \$si) = Rmpfr_init_set_q(\$q, \$rnd);# \$q is a mpq object
(\$rop, \$si) = Rmpfr_init_set_q_nobless(\$q, \$rnd);# \$q is a mpq object
Initialize \$rop and set its value from the 1st arg, rounded to
direction \$rnd. The precision of \$rop will be taken from the
active default precision, as set by `Rmpfr_set_default_prec.
If \$rop = 1st arg, \$si is zero. If \$rop > 1st arg, \$si is positive.
If \$rop < 1st arg, \$si is negative.

(\$rop, \$si) = Rmpfr_init_set_str(\$str, \$base, \$rnd);
(\$rop, \$si) = Rmpfr_init_set_str_nobless(\$str, \$base, \$rnd);
Initialize \$rop and set its value from \$str in base \$base,
rounded to direction \$rnd. If \$str was a valid number, then
\$si will be set to 0. Else it will be set to -1.
If \$si is -1 :
1) the non-numeric flag (which was initalised to 0) will be
incremented. You can query/clear/reset the value of the
flag with (resp.) nnumflag()/clear_nnum()/set_nnum() - all
of which are documented below (in "MISCELLANEOUS");
2) A warning will be emitted if \$Math::MPFR::NNW is set to 1
(default is 0).
See `Rmpfr_set_str (above) and Rmpfr_inp_str (below).

##########

CONVERSION

\$str = Rmpfr_get_str(\$op, \$base, \$digits, \$rnd);
Returns a string of the form, eg, 8.3456712@2
which means 834.56712.
The third argument to Rmpfr_get_str() specifies the number of digits
required to be output in the mantissa. (Trailing zeroes are removed.)
If \$digits is 0, the number of digits of the mantissa is chosen
large enough so that re-reading the printed value with the same
precision, assuming both output and input use rounding to nearest,
will recover the original value of \$op.

(\$str, \$si) = Rmpfr_deref2(\$op, \$base, \$digits, \$rnd);
Returns the mantissa to \$str (as a string of digits, prefixed with
a minus sign if \$op is negative), and returns the exponent to \$si.
Theres an implicit decimal point to the left of the first digit in
\$str. The third argument to Rmpfr_deref2() specifies the number of
digits required to be output in the mantissa.
If \$digits is 0, the number of digits of the mantissa is chosen
large enough so that re-reading the printed value with the same
precision, assuming both output and input use rounding to nearest,
will recover the original value of \$op.

\$str = Rmpfr_integer_string(\$op, \$base, \$rnd);
Returns the truncated integer value of \$op as a string. (No exponent
is returned). For example, if \$op contains the value 2.3145679e2,
\$str will be set to "231".
(This function is mainly to provide a simple means of getting sj
and uj values on a 64-bit perl where the MPFR library does not
support mpfr_get_uj and mpfr_get_sj functions - which may happen,
for example, with libraries built with Microsoft Compilers.)

\$bool = Rmpfr_fits_ushort_p(\$op, \$rnd); # fits in unsigned short
\$bool = Rmpfr_fits_sshort_p(\$op, \$rnd); # fits in signed short
\$bool = Rmpfr_fits_uint_p(\$op, \$rnd); # fits in unsigned int
\$bool = Rmpfr_fits_sint_p(\$op, \$rnd); # fits in signed int
\$bool = Rmpfr_fits_ulong_p(\$op, \$rnd); # fits in unsigned long
\$bool = Rmpfr_fits_slong_p(\$op, \$rnd); # fits in signed long
\$bool = Rmpfr_fits_uintmax_p(\$op, \$rnd); # fits in uintmax_t
\$bool = Rmpfr_fits_intmax_p(\$op, \$rnd); # fits in intmax_t
\$bool = Rmpfr_fits_IV_p(\$op, \$rnd); # fits in perl IV
\$bool = Rmpfr_fits_UV_p(\$op, \$rnd); # fits in perl UV
Return non-zero if \$op would fit in the respective data
type, when rounded to an integer in the direction \$rnd.

\$ui = Rmpfr_get_ui(\$op, \$rnd);
\$si = Rmpfr_get_si(\$op, \$rnd);
\$sj = Rmpfr_get_sj(\$op, \$rnd); # 64 bit builds only
\$uj = Rmpfr_get_uj(\$op, \$rnd); # 64 bit builds only
\$uv = Rmpfr_get_UV(\$op, \$rnd); # 32 and 64 bit
\$iv = Rmpfr_get_IV(\$op, \$rnd); # 32 and 64 bit
Convert \$op to an unsigned long long, a signed long, a
signed long long, an `unsigned long long, a UV, or an
IV - after rounding it with respect to \$rnd.
If \$op is NaN, the result is undefined. If \$op is too big
for the return type, it returns the maximum or the minimum
of the corresponding C type, depending on the direction of
the overflow. The flag erange is then also set.

\$double = Rmpfr_get_d(\$op, \$rnd);
\$ld     = Rmpfr_get_ld(\$op, \$rnd);
\$f128   = Rmpfr_get_float128(\$op, \$rnd); # nvtype must be __float128
# mpfr-3.2.0 or later
\$nv     = Rmpfr_get_NV(\$op, \$rnd);    # double/long double/__float128
\$float  = Rmpfr_get_flt(\$op, \$rnd);   # mpfr-3.0.0 and later.
Rmpfr_get_LD(\$LD, \$op, \$rnd); # \$LD is a Math::LongDouble object.
Rmpfr_get_DECIMAL64(\$d64, \$op, \$rnd); # mpfr-3.1.1 and later.
# \$d64 is a Math::Decimal64
# object.
Rmpfr_get_FLOAT128(\$F128, \$op, \$rnd); # mpfr-3.2.0 and later.
# \$F128 is a Math::Float128
# object.
Convert \$op to a double a long double an NV, a float, a
__float128, a Math::LongDouble object, a Math::Decimal64 object, or
a Math::Float128 object using the rounding mode \$rnd.

\$double = Rmpfr_get_d1(\$op);
Convert \$op to a double, using the default MPFR rounding mode
(see function `mpfr_set_default_rounding_mode).

\$si = Rmpfr_get_z_exp(\$z, \$op); # \$z is a mpz object
\$si = Rmpfr_get_z_2exp(\$z, \$op); # \$z is a mpz object
(Identical functions. Use either - get_z_exp might one day
be removed.)
Puts the mantissa of \$rop into \$z, and returns the exponent
\$si such that \$rop == \$z * (2 ** \$ui).

\$si = Rmpfr_get_z(\$z, \$op, \$rnd); # \$z is a mpz object.
Convert \$op to an mpz object (\$z), after rounding it with respect
to RND. If built against mpfr-3.0.0 or later, return the usual
ternary value. (The function returns undef when using mpfr-2.x.x.)
If \$op is NaN or Inf, the result is undefined.

\$si = Rmpfr_get_f (\$f, \$op, \$rnd); # \$f is a Math::GMPf object.
Convert \$op to a `mpf_t, after rounding it with respect to \$rnd.
When built against mpfr-3.0.0 or later, this function returns the
usual ternary value. (If \$op is NaN or Inf, then the erange flag
will be set.) When built against earlier versions of mpfr,
return zero iff no error occurred.In particular a non-zero value
is returned if \$op is NaN or Inf. which do not exist in `mpf.

Rmpfr_get_q (\$q, \$op); # \$q is a Math::GMPq object.
Convert \$op to a rational value. \$q will be set to the exact
value contained in \$op - hence no need for a rounding argument.
If \$op is NaN or Inf, then \$q is set to zero and the erange flag
will be set.

\$d = Rmpfr_get_d_2exp (\$exp, \$op, \$rnd); # \$d is NV (double)
\$d = Rmpfr_get_ld_2exp (\$exp, \$op, \$rnd); # \$d is NV (long double)
Set \$exp and \$d such that 0.5<=abs(\$d)<1 and \$d times 2 raised
to \$exp equals \$op rounded to double (resp. long double)
precision, using the given rounding mode.  If \$op is zero, then a
zero of the same sign (or an unsigned zero, if the implementation
does not have signed zeros) is returned, and \$exp is set to 0.
If \$op is NaN or an infinity, then the corresponding double
precision (resp. long-double precision) value is returned, and
\$exp is undefined.

\$si1 = Rmpfr_frexp(\$si2, \$rop, \$op, \$rnd); # mpfr-3.1.0 and later only
Set \$si and \$rop such that 0.5<=abs(\$rop)<1 and \$rop * (2 ** \$exp)
equals \$op rounded to the precision of \$rop, using the given
rounding mode. If \$op is zero, then \$rop is set to zero (of the same
sign) and \$exp is set to 0. If \$op is  NaN or an infinity, then \$rop
is set to the same value and the value of \$exp is meaningless (and
should be ignored).

##########

ARITHMETIC

\$si = Rmpfr_add(\$rop, \$op1, \$op2, \$rnd);
\$si = Rmpfr_add_ui(\$rop, \$op, \$ui, \$rnd);
\$si = Rmpfr_add_si(\$rop, \$op, \$si1, \$rnd);
\$si = Rmpfr_add_d(\$rop, \$op, \$double, \$rnd);
\$si = Rmpfr_add_z(\$rop, \$op, \$z, \$rnd); # \$z is a mpz object.
\$si = Rmpfr_add_q(\$rop, \$op, \$q, \$rnd); # \$q is a mpq object.
Set \$rop to 2nd arg + 3rd arg rounded in the direction \$rnd.
The return  value is zero if \$rop is exactly 2nd arg + 3rd arg,
positive if \$rop is larger than 2nd arg + 3rd arg, and negative
if \$rop is smaller than 2nd arg + 3rd arg.

\$si = Rmpfr_sum(\$rop, \@ops, scalar(@ops), \$rnd);
@ops is an array consisting entirely of Math::MPFR objects.
Set \$rop to the sum of all members of @ops, rounded in the direction
\$rnd. \$si is zero when the computed value is the exact value, and
non-zero when this cannot be guaranteed, without giving the direction
of the error as the other functions do.

\$si = Rmpfr_sub(\$rop, \$op1, \$op2, \$rnd);
\$si = Rmpfr_sub_ui(\$rop, \$op, \$ui, \$rnd);
\$si = Rmpfr_sub_z(\$rop, \$op, \$z, \$rnd); # \$z is a mpz object.
\$si = Rmpfr_z_sub(\$rop, \$z, \$op, \$rnd); # mpfr-3.1.0 and later only
\$si = Rmpfr_sub_q(\$rop, \$op, \$q, \$rnd); # \$q is a mpq object.
\$si = Rmpfr_ui_sub(\$rop, \$ui, \$op, \$rnd);
\$si = Rmpfr_si_sub(\$rop, \$si1, \$op, \$rnd);
\$si = Rmpfr_sub_si(\$rop, \$op, \$si1, \$rnd);
\$si = Rmpfr_sub_d(\$rop, \$op, \$double, \$rnd);
\$si = Rmpfr_d_sub(\$rop, \$double, \$op, \$rnd);
Set \$rop to 2nd arg - 3rd arg rounded in the direction \$rnd.
The return value is zero if \$rop is exactly 2nd arg - 3rd arg,
positive if \$rop is larger than 2nd arg - 3rd arg, and negative
if \$rop is smaller than 2nd arg - 3rd arg.

\$si = Rmpfr_mul(\$rop, \$op1, \$op2, \$rnd);
\$si = Rmpfr_mul_ui(\$rop, \$op, \$ui, \$rnd);
\$si = Rmpfr_mul_si(\$rop, \$op, \$si1, \$rnd);
\$si = Rmpfr_mul_d(\$rop, \$op, \$double, \$rnd);
\$si = Rmpfr_mul_z(\$rop, \$op, \$z, \$rnd); # \$z is a mpz object.
\$si = Rmpfr_mul_q(\$rop, \$op, \$q, \$rnd); # \$q is a mpq object.
Set \$rop to 2nd arg * 3rd arg rounded in the direction \$rnd.
Return 0 if the result is exact, a positive value if \$rop is
greater than 2nd arg times 3rd arg, a negative value otherwise.

\$si = Rmpfr_div(\$rop, \$op1, \$op2, \$rnd);
\$si = Rmpfr_div_ui(\$rop, \$op, \$ui, \$rnd);
\$si = Rmpfr_ui_div(\$rop, \$ui, \$op, \$rnd);
\$si = Rmpfr_div_si(\$rop, \$op, \$si1, \$rnd);
\$si = Rmpfr_si_div(\$rop, \$si1, \$op, \$rnd);
\$si = Rmpfr_div_d(\$rop, \$op, \$double, \$rnd);
\$si = Rmpfr_d_div(\$rop, \$double, \$op, \$rnd);
\$si = Rmpfr_div_z(\$rop, \$op, \$z, \$rnd); # \$z is a mpz object.
\$si = Rmpfr_div_q(\$rop, \$op, \$q, \$rnd); # \$q is a mpq object.
Set \$rop to 2nd arg / 3rd arg rounded in the direction \$rnd.
These functions return 0 if the division is exact, a positive
value when \$rop is larger than 2nd arg divided by 3rd arg,
and a negative value otherwise.

\$si = Rmpfr_sqr(\$rop, \$op, \$rnd);
Set \$rop to the square of \$op, rounded in direction \$rnd.

\$si = Rmpfr_sqrt(\$rop, \$op, \$rnd);
\$si = Rmpfr_sqrt_ui(\$rop, \$ui, \$rnd);
Set \$rop to the square root of the 2nd arg rounded in the
direction \$rnd. Set \$rop to NaN if 2nd arg is negative.
Return 0 if the operation is exact, a non-zero value otherwise.

\$si = Rmpfr_rec_sqrt(\$rop, \$op, \$rnd);
Set \$rop to the reciprocal square root of \$op rounded in the
direction \$rnd. Set \$rop to +Inf if \$op is 0, and 0 if \$op is
+Inf. Set \$rop to NaN if \$op is negative.

\$si = Rmpfr_cbrt(\$rop, \$op, \$rnd);
Set \$rop to the cubic root (defined over the real numbers)
of \$op, rounded in the direction \$rnd.

\$si = Rmpfr_root(\$rop, \$op, \$ui \$rnd);
Set \$rop to the \$uith root of \$op, rounded in the direction
\$rnd.  Return 0 if the operation is exact, a non-zero value
otherwise.

\$si = Rmpfr_pow_ui(\$rop, \$op, \$ui, \$rnd);
\$si = Rmpfr_pow_si(\$rop, \$op, \$si, \$rnd);
\$si = Rmpfr_ui_pow_ui(\$rop, \$ui, \$ui, \$rnd);
\$si = Rmpfr_ui_pow(\$rop, \$ui, \$op, \$rnd);
\$si = Rmpfr_pow(\$rop, \$op1, \$op2, \$rnd);
\$si = Rmpfr_pow_z(\$rop, \$op1, \$z, \$rnd); # \$z is a mpz object
Set \$rop to 2nd arg raised to 3rd arg, rounded to the directio
\$rnd with the precision of \$rop.  Return zero iff the result is
exact, a positive value when the result is greater than 2nd arg
to the power 3rd arg, and a negative value when it is smaller.
See the MPFR documentation for documentation regarding special
cases.

\$si = Rmpfr_neg(\$rop, \$op, \$rnd);
Set \$rop to -\$op rounded in the direction \$rnd. Just
changes the sign if \$rop and \$op are the same variable.

\$si = Rmpfr_abs(\$rop, \$op, \$rnd);
Set \$rop to the absolute value of \$op, rounded in the direction
\$rnd. Return 0 if the result is exact, a positive value if \$rop
is larger than the absolute value of \$op, and a negative value
otherwise.

\$si = Rmpfr_dim(\$rop, \$op1, \$op2, \$rnd);
Set \$rop to the positive difference of \$op1 and \$op2, i.e.,
\$op1 - \$op2 rounded in the direction \$rnd if \$op1 > \$op2, and
+0 otherwise. \$rop is set to NaN when \$op1 or \$op2 is NaN.

\$si = Rmpfr_mul_2exp(\$rop, \$op, \$ui, \$rnd);
\$si = Rmpfr_mul_2ui(\$rop, \$op, \$ui, \$rnd);
\$si = Rmpfr_mul_2si(\$rop, \$op, \$si, \$rnd);
Set \$rop to 2nd arg times 2 raised to 3rd arg rounded to the
direction \$rnd. Just increases the exponent by 3rd arg when
\$rop and 2nd arg are identical. Return zero when \$rop = 2nd
arg, a positive value when \$rop > 2nd arg, and a negative
value when \$rop < 2nd arg.  Note: The `Rmpfr_mul_2exp function
is defined for compatibility reasons; you should use

\$si = Rmpfr_div_2exp(\$rop, \$op, \$ui, \$rnd);
\$si = Rmpfr_div_2ui(\$rop, \$op, \$ui, \$rnd);
\$si = Rmpfr_div_2si(\$rop, \$op, \$si, \$rnd);
Set \$rop to 2nd arg divided by 2 raised to 3rd arg rounded to
the direction \$rnd. Just decreases the exponent by 3rd arg
when \$rop and 2nd arg are identical.  Return zero when
\$rop = 2nd arg, a positive value when \$rop > 2nd arg, and a
negative value when \$rop < 2nd arg.  Note: The `Rmpfr_div_2exp
function is defined for compatibility reasons; you should

##########

COMPARISON

\$si = Rmpfr_cmp(\$op1, \$op2);
\$si = Rmpfr_cmpabs(\$op1, \$op2);
\$si = Rmpfr_cmp_ui(\$op, \$ui);
\$si = Rmpfr_cmp_si(\$op, \$si);
\$si = Rmpfr_cmp_d(\$op, \$double);
\$si = Rmpfr_cmp_ld(\$op, \$ld); # long double
\$si = Rmpfr_cmp_z(\$op, \$z); # \$z is a mpz object
\$si = Rmpfr_cmp_q(\$op, \$q); # \$q is a mpq object
\$si = Rmpfr_cmp_f(\$op, \$f); # \$f is a mpf object
Compare 1st and 2nd args. In the case of Rmpfr_cmpabs()
compare the absolute values of the 2 args.  Return a positive
value if 1st arg > 2nd arg, zero if 1st arg = 2nd arg, and a
negative value if 1st arg < 2nd arg.  Both args are considered
to their full own precision, which may differ. In case 1st and
2nd args are of same sign but different, the absolute value
returned is one plus the absolute difference of their exponents.
If one of the operands is NaN (Not-a-Number), return zero
and set the erange flag.

\$si = Rmpfr_cmp_ui_2exp(\$op, \$ui, \$si);
\$si = Rmpfr_cmp_si_2exp(\$op, \$si, \$si);
Compare 1st arg and 2nd arg multiplied by two to the power
3rd arg.

\$bool = Rmpfr_eq(\$op1, \$op2, \$ui);
The mpfr library function mpfr_eq may change in future
releases of the mpfr library (post 2.4.0). If that happens,
the change will also be relected in Rmpfr_eq.
Return non-zero if the first \$ui bits of \$op1 and \$op2 are
equal, zero otherwise.  I.e., tests if \$op1 and \$op2 are
approximately equal.

\$bool = Rmpfr_nan_p(\$op);
Return non-zero if \$op is Not-a-Number (NaN), zero otherwise.

\$bool = Rmpfr_inf_p(\$op);
Return non-zero if \$op is plus or minus infinity, zero otherwise.

\$bool = Rmpfr_number_p(\$op);
Return non-zero if \$op is an ordinary number, i.e. neither
Not-a-Number nor plus or minus infinity.

\$bool = Rmpfr_zero_p(\$op);
Return non-zero if \$op is zero. Else return 0.

\$bool = Rmpfr_regular_p(\$op); # mpfr-3.0.0 and later only
Return non-zero if \$op is a regular number (i.e. neither NaN,
nor an infinity nor zero). Return zero otherwise.

Rmpfr_reldiff(\$rop, \$op1, \$op2, \$rnd);
Compute the relative difference between \$op1 and \$op2 and
store the result in \$rop.  This function does not guarantee
the exact rounding on the relative difference; it just
computes abs(\$op1-\$op2)/\$op1, using the rounding mode
\$rnd for all operations.

\$si = Rmpfr_sgn(\$op);
Return a positive value if op > 0, zero if \$op = 0, and a
negative value if \$op < 0.  Its result is not specified
when \$op is NaN (Not-a-Number).

\$bool = Rmpfr_greater_p(\$op1, \$op2);
Return non-zero if \$op1 > \$op2, zero otherwise.

\$bool = Rmpfr_greaterequal_p(\$op1, \$op2);
Return non-zero if \$op1 >= \$op2, zero otherwise.

\$bool = Rmpfr_less_p(\$op1, \$op2);
Return non-zero if \$op1 < \$op2, zero otherwise.

\$bool = Rmpfr_lessequal_p(\$op1, \$op2);
Return non-zero if \$op1 <= \$op2, zero otherwise.

\$bool = Rmpfr_lessgreater_p(\$op1, \$op2);
Return non-zero if \$op1 < \$op2 or \$op1 > \$op2 (i.e. neither
\$op1, nor \$op2 is NaN, and \$op1 <> \$op2), zero otherwise
(i.e. \$op1 and/or \$op2 are NaN, or \$op1 = \$op2).

\$bool = Rmpfr_equal_p(\$op1, \$op2);
Return non-zero if \$op1 = \$op2, zero otherwise
(i.e. \$op1 and/or \$op2 are NaN, or \$op1 <> \$op2).

\$bool = Rmpfr_unordered_p(\$op1, \$op2);
Return non-zero if \$op1 or \$op2 is a NaN
(i.e. they cannot be compared), zero otherwise.

#######

SPECIAL

\$si = Rmpfr_log(\$rop, \$op, \$rnd);
\$si = Rmpfr_log2(\$rop, \$op, \$rnd);
\$si = Rmpfr_log10(\$rop, \$op, \$rnd);
Set \$rop to the natural logarithm of \$op, log2(\$op) or
log10(\$op), respectively, rounded in the direction rnd.

\$si = Rmpfr_exp(\$rop, \$op, \$rnd);
\$si = Rmpfr_exp2(\$rop, \$op, \$rnd);
\$si = Rmpfr_exp10(\$rop, \$op, \$rnd);
Set rop to the exponential of op, to 2 power of op or to
10 power of op, respectively, rounded in the direction rnd.

\$si = Rmpfr_sin(\$rop \$op, \$rnd);
\$si = Rmpfr_cos(\$rop, \$op, \$rnd);
\$si = Rmpfr_tan(\$rop, \$op, \$rnd);
Set \$rop to the sine/cosine/tangent respectively of \$op,
rounded to the direction \$rnd with the precision of \$rop.
Return 0 iff the result is exact (this occurs in fact only
when \$op is 0 i.e. the sine is 0, the cosine is 1, and the
tangent is 0). Return a negative value iff the result is less
than the actual value. Return a positive result iff the
return is greater than the actual value.

\$si = Rmpfr_sin_cos(\$rop1, \$rop2, \$op, \$rnd);
Set simultaneously \$rop1 to the sine of \$op and
\$rop2 to the cosine of \$op, rounded to the direction \$rnd
with their corresponding precisions.  Return 0 iff both
results are exact.

\$si = Rmpfr_sinh_cosh(\$rop1, \$rop2, \$op, \$rnd);
Set simultaneously \$rop1 to the hyperbolic sine of \$op and
\$rop2 to the hyperbolic cosine of \$op, rounded in the direction
\$rnd with the corresponding precision of \$rop1 and \$rop2 which
must be different variables. Return 0 iff both results are
exact.

\$si = Rmpfr_acos(\$rop, \$op, \$rnd);
\$si = Rmpfr_asin(\$rop, \$op, \$rnd);
\$si = Rmpfr_atan(\$rop, \$op, \$rnd);
Set \$rop to the arc-cosine, arc-sine or arc-tangent of \$op,
rounded to the direction \$rnd with the precision of \$rop.
Return 0 iff the result is exact. Return a negative value iff
the result is less than the actual value. Return a positive
result iff the return is greater than the actual value.

\$si = Rmpfr_atan2(\$rop, \$op1, \$op2, \$rnd);
Set \$rop to the tangent of \$op1/\$op2, rounded to the
direction \$rnd with the precision of \$rop.
Return 0 iff the result is exact. Return a negative value iff
the result is less than the actual value. Return a positive
result iff the return is greater than the actual value.
See the MPFR documentation for details regarding special cases.

\$si = Rmpfr_cosh(\$rop, \$op, \$rnd);
\$si = Rmpfr_sinh(\$rop, \$op, \$rnd);
\$si = Rmpfr_tanh(\$rop, \$op, \$rnd);
Set \$rop to the hyperbolic cosine/hyperbolic sine/hyperbolic
tangent respectively of \$op, rounded to the direction \$rnd
with the precision of \$rop.  Return 0 iff the result is exact
(this occurs in fact only when \$op is 0 i.e. the result is 1).
Return a negative value iff the result is less than the actual
value. Return a positive result iff the return is greater than
the actual value.

\$si = Rmpfr_acosh(\$rop, \$op, \$rnd);
\$si = Rmpfr_asinh(\$rop, \$op, \$rnd);
\$si = Rmpfr_atanh(\$rop, \$op, \$rnd);
Set \$rop to the inverse hyperbolic cosine, sine or tangent
of \$op, rounded to the direction \$rnd with the precision of
\$rop.  Return 0 iff the result is exact.

\$si = Rmpfr_sec (\$rop, \$op, \$rnd);
\$si = Rmpfr_csc (\$rop, \$op, \$rnd);
\$si = Rmpfr_cot (\$rop, \$op, \$rnd);
Set \$rop to the secant of \$op, cosecant of \$op,
cotangent of \$op, rounded in the direction RND. Return 0
iff the result is exact. Return a negative value iff the
result is less than the actual value. Return a positive
result iff the return is greater than the actual value.

\$si = Rmpfr_sech (\$rop, \$op, \$rnd);
\$si = Rmpfr_csch (\$rop, \$op, \$rnd);
\$si = Rmpfr_coth (\$rop, \$op, \$rnd);
Set \$rop to the hyperbolic secant of \$op, cosecant of \$op,
cotangent of \$op, rounded in the direction RND. Return 0
iff the result is exact. Return a negative value iff the
result is less than the actual value. Return a positive
result iff the return is greater than the actual value.

\$bool = Rmpfr_fac_ui(\$rop, \$ui, \$rnd);
Set \$rop to the factorial of \$ui, rounded to the direction
\$rnd with the precision of \$rop.  Return 0 iff the
result is exact.

\$bool = Rmpfr_log1p(\$rop, \$op, \$rnd);
Set \$rop to the logarithm of one plus \$op, rounded to the
direction \$rnd with the precision of \$rop.  Return 0 iff
the result is exact (this occurs in fact only when \$op is 0
i.e. the result is 0).

\$bool = Rmpfr_expm1(\$rop, \$op, \$rnd);
Set \$rop to the exponential of \$op minus one, rounded to the
direction \$rnd with the precision of \$rop.  Return 0 iff the
result is exact (this occurs in fact only when \$op is 0 i.e
the result is 0).

\$si = Rmpfr_fma(\$rop, \$op1, \$op2, \$op3, \$rnd);
Set \$rop to \$op1 * \$op2 + \$op3, rounded to the direction
\$rnd.

\$si = Rmpfr_fms(\$rop, \$op1, \$op2, \$op3, \$rnd);
Set \$rop to \$op1 * \$op2 - \$op3, rounded to the direction
\$rnd.

\$si = Rmpfr_agm(\$rop, \$op1, \$op2, \$rnd);
Set \$rop to the arithmetic-geometric mean of \$op1 and \$op2,
rounded to the direction \$rnd with the precision of \$rop.
Return zero if \$rop is exact, a positive value if \$rop is
larger than the exact value, or a negative value if \$rop
is less than the exact value.

\$si = Rmpfr_hypot (\$rop, \$op1, \$op2, \$rnd);
Set \$rop to the Euclidean norm of \$op1 and \$op2, i.e. the
square root of the sum of the squares of \$op1 and \$op2,
rounded in the direction \$rnd. Special values are currently
handled as described in Section F.9.4.3 of the ISO C99
standard, for the hypot function (note this may change in
future versions): If \$op1 or \$op2 is an infinity, then plus
infinity is returned in \$rop, even if the other number is
NaN.

\$si = Rmpfr_ai(\$rop, \$op, \$rnd); # mpfr-3.0.0 and later only
Set \$rop to the value of the Airy function Ai on \$op,
rounded in the direction \$rnd.  When \$op is NaN, \$rop is
always set to NaN. When \$op is +Inf or -Inf, \$rop is +0.
The current implementation is not intended to be used with
large arguments.  It works with \$op typically smaller than
500. For larger arguments, other methods should be used and
will be implemented soon.

\$si = Rmpfr_const_log2(\$rop, \$rnd);
Set \$rop to the logarithm of 2 rounded to the direction
\$rnd with the precision of \$rop. This function stores the
computed value to avoid another calculation if a lower or
equal precision is requested.
Return zero if \$rop is exact, a positive value if \$rop is
larger than the exact value, or a negative value if \$rop
is less than the exact value.

\$si = Rmpfr_const_pi(\$rop, \$rnd);
Set \$rop to the value of Pi rounded to the direction \$rnd
with the precision of \$rop. This function uses the Borwein,
Borwein, Plouffe formula which directly gives the expansion
of Pi in base 16.
Return zero if \$rop is exact, a positive value if \$rop is
larger than the exact value, or a negative value if \$rop
is less than the exact value.

\$si = Rmpfr_const_euler(\$rop, \$rnd);
Set \$rop to the value of Eulers constant 0.577...  rounded
to the direction \$rnd with the precision of \$rop.
Return zero if \$rop is exact, a positive value if \$rop is
larger than the exact value, or a negative value if \$rop
is less than the exact value.

\$si = Rmpfr_const_catalan(\$rop, \$rnd);
Set \$rop to the value of Catalans constant 0.915...
rounded to the direction \$rnd with the precision of \$rop.
Return zero if \$rop is exact, a positive value if \$rop is
larger than the exact value, or a negative value if \$rop
is less than the exact value.

Rmpfr_free_cache();
Free the cache used by the functions computing constants if
needed (currently `mpfr_const_log2, `mpfr_const_pi and
`mpfr_const_euler).

\$si = Rmpfr_gamma(\$rop, \$op, \$rnd);
\$si = Rmpfr_lngamma(\$rop, \$op, \$rnd);
Set \$rop to the value of the Gamma function on \$op
(and, respectively, its natural logarithm) rounded
to the direction \$rnd. Return zero if \$rop is exact, a
positive value if \$rop is larger than the exact value, or a
negative value if \$rop is less than the exact value.

(\$signp, \$si) = Rmpfr_lgamma (\$rop, \$op, \$rnd);
Set \$rop to the value of the logarithm of the absolute value
of the Gamma function on \$op, rounded in the direction \$rnd.
The sign (1 or -1) of Gamma(\$op) is returned in \$signp.
When \$op is an infinity or a non-positive integer, +Inf is
returned. When \$op is NaN, -Inf or a negative integer, \$signp
is undefined, and when \$op is 0, \$signp is the sign of the zero.

\$si = Rmpfr_digamma (\$rop, \$op, \$rnd); # mpfr-3.0.0 and later only
Set \$rop to the value of the Digamma (sometimes also called Psi)
function on \$op, rounded in the direction \$rnd.  When \$op is a
negative integer, set \$rop to NaN.

\$si = Rmpfr_zeta(\$rop, \$op, \$rnd);
\$si = Rmpfr_zeta_ui(\$rop, \$ul, \$rnd);
Set \$rop to the value of the Riemann Zeta function on 2nd arg,
rounded to the direction \$rnd. Return zero if \$rop is exact,
a positive value if \$rop is larger than the exact value, or
a negative value if \$rop is less than the exact value.

\$si = Rmpfr_erf(\$rop, \$op, \$rnd);
Set \$rop to the value of the error function on \$op,
rounded to the direction \$rnd. Return zero if \$rop is exact,
a positive value if \$rop is larger than the exact value, or
a negative value if \$rop is less than the exact value.

\$si = Rmpfr_erfc(\$rop, \$op, \$rnd);
Set \$rop to the complementary error function on \$op,
rounded to the direction \$rnd. Return zero if \$rop is exact,
a positive value if \$rop is larger than the exact value, or
a negative value if \$rop is less than the exact value.

\$si = Rmpfr_j0 (\$rop, \$op, \$rnd);
\$si = Rmpfr_j1 (\$rop, \$op, \$rnd);
\$si = Rmpfr_jn (\$rop, \$si2, \$op, \$rnd);
Set \$rop to the value of the first order Bessel function of
order 0, 1 and \$si2 on \$op, rounded in the direction \$rnd.
When \$op is NaN, \$rop is always set to NaN. When \$op is plus
or minus Infinity, \$rop is set to +0. When \$op is zero, and
\$si2 is not zero, \$rop is +0 or -0 depending on the parity
and sign of \$si2, and the sign of \$op.

\$si = Rmpfr_y0 (\$rop, \$op, \$rnd);
\$si = Rmpfr_y1 (\$rop, \$op, \$rnd);
\$si = Rmpfr_yn (\$rop, \$si2, \$op, \$rnd);
Set \$rop to the value of the second order Bessel function of
order 0, 1 and \$si2 on \$op, rounded in the direction \$rnd.
When \$op is NaN or negative, \$rop is always set to NaN.
When \$op is +Inf, \$rop is +0. When \$op is zero, \$rop is +Inf
or -Inf depending on the parity and sign of \$si2.

\$si = Rmpfr_eint (\$rop, \$op, \$rnd)
Set \$rop to the exponential integral of \$op, rounded in the
direction \$rnd. See the MPFR documentation for details.

\$si = Rmpfr_li2 (\$rop, \$op, \$rnd);
Set \$rop to real part of the dilogarithm of \$op, rounded in the
direction \$rnd. The dilogarithm function is defined here as
the integral of -log(1-t)/t from 0 to x.

#############

I-O FUNCTIONS

\$ui = Rmpfr_out_str([\$prefix,] \$op, \$base, \$digits, \$round [, \$suffix]);
Output \$op to STDOUT, as a string of digits in base \$base,
rounded in direction \$round.  The base may vary from 2 to 36
(2 to 62 if Math::MPFR has been built against mpfr-3.0.0 or later).
Print \$digits significant digits exactly, or if \$digits is 0,
enough digits so that \$op can be read back exactly
(see Rmpfr_get_str). In addition to the significant
digits, a decimal point at the right of the first digit and a
trailing exponent in base 10, in the form `eNNN, are printed
If \$base is greater than 10, `@ will be used instead of `e
as exponent delimiter. The optional arguments, \$prefix and
\$suffix, are strings that will be prepended/appended to the
mpfr_out_str output. Return the number of bytes written (not
counting those contained in \$suffix and \$prefix), or if an error
occurred, return 0. (Note that none, one or both of \$prefix and
\$suffix can be supplied.)

\$ui = TRmpfr_out_str([\$prefix,] \$stream, \$base, \$digits, \$op, \$round [, \$suffix]);
As for Rmpfr_out_str, except that theres the capability to print
to somewhere other than STDOUT. Note that the order of the args
is different (to match the order of the mpfr_out_str args).
To print to STDERR:
TRmpfr_out_str(*stderr, \$base, \$digits, \$op, \$round);
To print to an open filehandle (lets call it FH):
TRmpfr_out_str(\*FH, \$base, \$digits, \$op, \$round);

\$ui = Rmpfr_inp_str(\$rop, \$base, \$round);
Input a string in base \$base from STDIN, rounded in
direction \$round, and put the read float in \$rop.  The string
is of the form `M@N or, if the base is 10 or less, alternatively
`MeN or `MEN, or, if the base is 16, alternatively `MpB or
`MPB. `M is the mantissa in the specified base, `N is the
exponent written in decimal for the specified base, and in base 16,
`B is the binary exponent written in decimal (i.e. it indicates
the power of 2 by which the mantissa is to be scaled).
The argument \$base may be in the range 2 to 36 (2 to 62 if Math::MPFR
has been built against mpfr-3.0.0 or later).
Special values can be read as follows (the case does not matter):
`@NaN@, `@Inf@, `+@Inf@ and `-@Inf@, possibly followed by
other characters; if the base is smaller or equal to 16, the
following strings are accepted too: `NaN, `Inf, `+Inf and
`-Inf.
Return the number of bytes read, or if non-numeric characters were
encountered in the input, return 0.
If 0 is returned:
1) the non-numeric flag (which was initalised to 0) will be
incremented. You can query/clear/reset the value of the
flag with (resp.) nnumflag()/clear_nnum()/set_nnum() - all
of which are documented below;
2) A warning will be emitted if \$Math::MPFR::NNW is set to 1
(default is 0).

\$ui = TRmpfr_inp_str(\$rop, \$stream, \$base, \$round);
As for Rmpfr_inp_str, except that theres the capability to read
from somewhere other than STDIN.
TRmpfr_inp_str(\$rop, *stdin, \$base, \$round);
To read from an open filehandle (lets call it FH):
TRmpfr_inp_str(\$rop, \*FH, \$base,  \$round);

Rmpfr_print_binary(\$op);
Removed in Math-MPFR-3.30. Should have been removed long ago.
Output \$op on stdout in raw binary format (the exponent is in
decimal, yet).

Rmpfr_dump(\$op);
Output "\$op\n" on stdout in base 2.
As with Rmpfr_print_binary the exponent is in base 10.

#############

MISCELLANEOUS

\$MPFR_version = Rmpfr_get_version();
Returns the version of the MPFR library (eg 2.1.0) being used by
Math::MPFR.

\$GMP_version = Math::MPFR::gmp_v();
Returns the version of the gmp library (eg. 4.1.3) being used by
the mpfr library thats being used by Math::MPFR.
The function is not exportable.

\$ui = MPFR_VERSION;
An integer whose value is dependent upon the major, minor and
patchlevel values of the MPFR library against which Math::MPFR
was built.
This value is from the mpfr.h that was in use when the compilation
of Math::MPFR took place.

\$ui = MPFR_VERSION_MAJOR;
The x in the x.y.z of the MPFR library version.
This value is from the mpfr.h that was in use when the compilation
of Math::MPFR took place.

\$ui = MPFR_VERSION_MINOR;
The y in the x.y.z of the MPFR library version.
This value is from the mpfr.h that was in use when the compilation
of Math::MPFR took place.

\$ui = MPFR_VERSION_PATCHLEVEL;
The z in the x.y.z of the MPFR library version.
This value is from the mpfr.h that was in use when the compilation
of Math::MPFR took place.

\$string = MPFR_VERSION_STRING;
\$string is set to the version of the MPFR library (eg 2.1.0)
against which Math::MPFR was built.
This value is from the mpfr.h that was in use when the compilation
of Math::MPFR took place.

\$ui = MPFR_VERSION_NUM(\$major, \$minor, \$patchlevel);
Returns the value for MPFR_VERSION on "MPFR-\$major.\$minor.\$patchlevel".

\$str = Rmpfr_get_patches();
Return a string containing the ids of the patches applied to the
MPFR library (contents of the `PATCHES file), separated by spaces.
Note: If the program has been compiled with an older MPFR version and
is dynamically linked with a new MPFR library version, the ids of the
patches applied to the old (compile-time) MPFR version are not
available (however this information should not have much interest
in general).

\$bool = Rmpfr_buildopt_tls_p(); # mpfr-3.0.0 and later only
Return a non-zero value if mpfr was compiled as thread safe using
compiler-level Thread Local Storage (that is mpfr was built with
the `--enable-thread-safe configure option), else return zero.

\$bool = Rmpfr_buildopt_decimal_p(); # mpfr-3.0.0 and later only
Return a non-zero value if mpfr was compiled with decimal float
support (that is mpfr was built with the `--enable-decimal-float
configure option), return zero otherwise.

\$bool = Rmpfr_buildopt_gmpinternals_p(); # mpfr-3.1.0 and later only
Return a non-zero value if mpfr was compiled with gmp internals
(that is, mpfr was built with either --with-gmp-build or
--enable-gmp-internals configure option), return zero otherwise.

\$str = Rmpfr_buildopt_tune_case(); # mpfr-3.1.0 and later only
Return a string saying which thresholds file has been used at
compile time.  This file is normally selected from the processor
type. If "make tune" has been used, then it will return
"src/mparam.h". Otherwise it will say which official mparam.h file
has been used.

\$si = Rmpfr_rint(\$rop, \$op, \$rnd);
\$si = Rmpfr_ceil(\$rop, \$op);
\$si = Rmpfr_floor(\$rop, \$op);
\$si = Rmpfr_round(\$rop, \$op);
\$si = Rmpfr_trunc(\$rop, \$op);
Set \$rop to \$op rounded to an integer. `Rmpfr_ceil rounds to the
next higher representable integer, `Rmpfr_floor to the next lower,
`Rmpfr_round to the nearest representable integer, rounding
halfway cases away from zero, and `Rmpfr_trunc to the
representable integer towards zero. `Rmpfr_rint behaves like one
of these four functions, depending on the rounding mode.  The
returned value is zero when the result is exact, positive when it
is greater than the original value of \$op, and negative when it is
smaller.  More precisely, the returned value is 0 when \$op is an
integer representable in \$rop, 1 or -1 when \$op is an integer that
is not representable in \$rop, 2 or -2 when \$op is not an integer.

\$si = Rmpfr_rint_ceil(\$rop, \$op, \$rnd);
\$si = Rmpfr_rint_floor(\$rop, \$op, \$rnd);
\$si = Rmpfr_rint_round(\$rop, \$op, \$rnd);
\$si = Rmpfr_rint_trunc(\$rop, \$op, \$rnd):
Set \$rop to \$op rounded to an integer. `Rmpfr_rint_ceil rounds to
the next higher or equal integer, `Rmpfr_rint_floor to the next
lower or equal integer, `Rmpfr_rint_round to the nearest integer,
rounding halfway cases away from zero, and `Rmpfr_rint_trunc to
the next integer towards zero.  If the result is not
representable, it is rounded in the direction \$rnd. The returned
value is the ternary value associated with the considered
round-to-integer function (regarded in the same way as any other
mathematical function).

\$si = Rmpfr_frac(\$rop, \$op, \$round);
Set \$rop to the fractional part of \$op, having the same sign as \$op,
rounded in the direction \$round (unlike in `mpfr_rint, \$round
affects only how the exact fractional part is rounded, not how
the fractional part is generated).

\$si = Rmpfr_modf (\$rop1, \$rop2, \$op, \$rnd);
Set simultaneously \$rop1 to the integral part of \$op and \$rop2
to the fractional part of \$op, rounded in the direction RND with
the corresponding precision of \$rop1 and \$rop2 (equivalent to
`Rmpfr_trunc(\$rop1, \$op, \$rnd) and `Rmpfr_frac(\$rop1, \$op, \$rnd)).
The variables \$rop1 and \$rop2 must be different. Return 0 iff both
results are exact.

\$si = Rmpfr_remainder(\$rop, \$op1, \$op2, \$rnd);
\$si = Rmpfr_fmod(\$rop, \$op1, \$op2, \$rnd);
(\$si2, \$si) = Rmpfr_remquo (\$rop, \$op1, \$op2, \$rnd);
Set \$rop to the remainder of the division of \$op1 by \$op2, with
quotient rounded toward zero for Rmpfr_fmod and to the nearest
integer (ties rounded to even) for Rmpfr_remainder and
Rmpfr_remquo, and \$rop rounded according to the direction \$rnd.
Special values are handled as described in Section F.9.7.1 of the
ISO C99 standard: If \$op1 is infinite or \$op2 is zero, \$rop is NaN.
If \$op2 is infinite and \$op1 is finite, \$rop is \$op1 rounded to
the precision of \$rop. If \$rop is zero, it has the sign of \$op1.
The return value is the ternary value corresponding to \$rop.
Additionally, `Rmpfr_remquo stores the low significant bits from
the quotient in \$si2 (more precisely the number of bits in a `long
minus one), with the sign of \$op1 divided by \$op2 (except if those
low bits are all zero, in which case zero is returned).  Note that
\$op1 may be so large in magnitude relative to \$op2 that an exact
representation of the quotient is not practical.  `Rmpfr_remainder
and `Rmpfr_remquo functions are useful for additive argument
reduction.

\$si = Rmpfr_integer_p(\$op);
Return non-zero iff \$op is an integer.

Rmpfr_nexttoward(\$op1, \$op2);
If \$op1 or \$op2 is NaN, set \$op1 to NaN. Otherwise, if \$op1 is
different from \$op2, replace \$op1 by the next floating-point number
(with the precision of \$op1 and the current exponent range) in the
direction of \$op2, if there is one (the infinite values are seen as
the smallest and largest floating-point numbers). If the result is
zero, it keeps the same sign. No underflow or overflow is generated.

Rmpfr_nextabove(\$op1);
Equivalent to `mpfr_nexttoward where \$op2 is plus infinity.

Rmpfr_nextbelow(\$op1);
Equivalent to `mpfr_nexttoward where \$op2 is minus infinity.

\$si = Rmpfr_min(\$rop, \$op1, \$op2, \$round);
Set \$rop to the minimum of \$op1 and \$op2. If \$op1 and \$op2
are both NaN, then \$rop is set to NaN. If \$op1 or \$op2 is
NaN, then \$rop is set to the numeric value. If \$op1 and
\$op2 are zeros of different signs, then \$rop is set to -0.

\$si = Rmpfr_max(\$rop, \$op1, \$op2, \$round);
Set \$rop to the maximum of \$op1 and \$op2. If \$op1 and \$op2
are both NaN, then \$rop is set to NaN. If \$op1 or \$op2 is
NaN, then \$rop is set to the numeric value. If \$op1 and
\$op2 are zeros of different signs, then \$rop is set to +0.

\$iv = Math::MPFR::nnumflag(); # not exported
Returns the value of the non-numeric flag. This flag is
initialized to zero, but incemented by 1 whenever the
a string containing non-numeric characters is passed to an
mpfr function. The value of the flag therefore tells us how
many times such strings were passed to mpfr functions . The
flag can be reset to 0 by running clear_nnum().

Math::MPFR::set_nnum(\$iv); # not exported
Resets the global non-numeric flag to the value specified by
\$iv.

Math::MPFR::clear_nnum(); # not exported
Resets the global non-numeric flag to 0.(Essentially the same
as running set_nnum(0).)

\$bytes = Math::MPFR::bytes(\$val, \$type);
\$type must be either double, long double, double-double,
or __float128, though both upper and lower cases of the
characters is acceptable.
\$val must either be a string (eg 1.6e+45, 2.3, 0x17.8)
or a Math::MPFR object.
For the given value expressed by the string (or encapsulated in
the object) the hex representation of that value for the given
(\$type) datatype is returned.
If \$val is a Math::MPFR object, its precision must be 53 if
\$type is double, 64 if \$type is long double, 106 if \$type is
double-double, or 113 if \$type is __float128.
NOTE: Setting \$type to __float128 causes a fatal error if
Math::MPFR::MPFR_WANT_FLOAT128() returns false.

##############

RANDOM NUMBERS

Rmpfr_urandomb(@r, \$state);
Each member of @r is a Math::MPFR object.
\$state is a reference to a gmp_randstate_t structure.
Set each member of @r to a uniformly distributed random
float in the interval 0 <= \$_ < 1.
Before using this function you must first create \$state
by calling one of the 4 Rmpfr_randinit functions, then
seed \$state by calling one of the 2 Rmpfr_randseed functions.
The memory associated with \$state will be freed automatically
when \$state goes out of scope.

Rmpfr_random2(\$rop, \$si, \$ui); # not implemented in
# mpfr-3.0.0 and later
Attempting to use this function when Math::MPFR has been
built against mpfr-3.0.0 (or later) will cause the program
to die, with an appropriate error message.
Generate a random float of at most abs(\$si) limbs, with long
strings of zeros and ones in the binary representation.
The exponent of the number is in the interval -\$ui to
\$ui.  This function is useful for testing functions and
algorithms, since this kind of random numbers have proven
to be more likely to trigger corner-case bugs.  Negative
random numbers are generated when \$si is negative.

\$si = Rmpfr_urandom (\$rop, \$state, \$rnd); # mpfr-3.0.0 and
# later only
Generate a uniformly distributed random float.  The
floating-point number \$rop can be seen as if a random real
number is generated according to the continuous uniform
distribution on the interval[0, 1] and then rounded in the
direction RND.
Before using this function you must first create \$state
by calling one of the Rmpfr_randinit functions (below), then
seed \$state by calling one of the Rmpfr_randseed functions.

\$si = Rmpfr_grandom(\$rop1, \$rop2, \$state, \$rnd);
Available only with mpfr-3.1.0 and later.
Generate two random floats according to a standard normal
gaussian distribution. The floating-point numbers \$rop1 and
\$rop2 can be seen as if a random real number were generated
according to the standard normal gaussian distribution and
then rounded in the direction \$rnd.
Before using this function you must first create \$state
by calling one of the Rmpfr_randinit functions (below), then
seed \$state by calling one of the Rmpfr_randseed functions.

\$state = Rmpfr_randinit_default();
Initialise \$state with a default algorithm. This will be
a compromise between speed and randomness, and is
recommended for applications with no special requirements.

\$state = Rmpfr_randinit_mt();
Initialize state for a Mersenne Twister algorithm. This
algorithm is fast and has good randomness properties.

\$state = Rmpfr_randinit_lc_2exp(\$a, \$c, \$m2exp);
This function is not tested in the test suite.
Use with caution - I often select values here that cause
Rmpf_urandomb() to behave non-randomly.
Initialise \$state with a linear congruential algorithm:
X = (\$a * X + \$c) % 2 ** \$m2exp
The low bits in X are not very random - for this reason
only the high half of each X is actually used.
\$c and \$m2exp sre both unsigned longs.
\$a can be any one of Math::GMP, or Math::GMPz objects.
Or it can be a string.
If it is a string of hex digits it must be prefixed with
either OX or Ox. If it is a string of octal digits it must
be prefixed with O. Else it is assumed to be a decimal
integer. No other bases are allowed.

\$state = Rmpfr_randinit_lc_2exp_size(\$ui);
Initialise state as per Rmpfr_randinit_lc_2exp. The values
for \$a, \$c. and \$m2exp are selected from a table, chosen
so that \$ui bits (or more) of each X will be used.

Rmpfr_randseed(\$state, \$seed);
\$state is a reference to a gmp_randstate_t strucure (the
return value of one of the Rmpfr_randinit functions).
\$seed is the seed. It can be any one of Math::GMP,
or Math::GMPz objects. Or it can be a string of digits.
If it is a string of hex digits it must be prefixed with
either OX or Ox. If it is a string of octal digits it must
be prefixed with O. Else it is assumed to be a decimal
integer. No other bases are allowed.

Rmpfr_randseed_ui(\$state, \$ui);
\$state is a reference to a gmp_randstate_t strucure (the
return value of one of the Rmpfr_randinit functions).
\$ui is the seed.

#########

INTERNALS

\$bool = Rmpfr_can_round(\$op, \$ui, \$rnd1, \$rnd2, \$p);
Assuming \$op is an approximation of an unknown number X in direction
\$rnd1 with error at most two to the power E(b)-\$ui where E(b) is
the exponent of \$op, returns 1 if one is able to round exactly X
to precision \$p with direction \$rnd2, and 0 otherwise. This
function *does not modify* its arguments.

\$str = Rmpfr_print_rnd_mode(\$rnd);
Return a string ("MPFR_RNDD", "MPFR_RNDU", "MPFR_RNDN", "MPFR_RNDZ",
"MPFR_RNDA") corresponding to the rounding mode \$rnd, or return
undef if \$rnd is an invalid rounding mode.

\$si = Rmpfr_get_exp(\$op);
Get the exponent of \$op, assuming that \$op is a non-zero
ordinary number.

\$si = Rmpfr_set_exp(\$op, \$si);
Set the exponent of \$op if \$si is in the current exponent
range, and return 0 (even if \$op is not a non-zero
ordinary number); otherwise, return a non-zero value.

\$si = Rmpfr_signbit (\$op);
Return a non-zero value iff \$op has its sign bit set (i.e. if it is
negative, -0, or a NaN whose representation has its sign bit set).

\$si2 = Rmpfr_setsign (\$rop, \$op, \$si, \$rnd);
Set the value of \$rop from \$op, rounded towards the given direction
\$rnd, then set/clear its sign bit if \$si is true/false (even when
\$op is a NaN).

\$si = Rmpfr_copysign (\$rop, \$op1, \$op2, \$rnd);
Set the value of \$rop from \$op1, rounded towards the given direction
\$rnd, then set its sign bit to that of \$op2 (even when \$op1 or \$op2
is a NaN). This function is equivalent to:
Rmpfr_setsign (\$rop, \$op1, Rmpfr_signbit (\$op2), \$rnd).

####################

only - see step 4. below) and Math::MPFR objects.
Overloaded operations are performed using the current
"default rounding mode" (which you can determine using the
Rmpfr_get_default_rounding_mode function, and change using
the Rmpfr_set_default_rounding_mode function).

the overload subroutine converts that string operand to a
Math::MPFR object with *current default precision*, and using
the *current default rounding mode*.

Note that any comparison using the spaceship operator ( <=> )
will return undef iff either/both of the operands is a NaN.
All comparisons ( < <= > >= == != <=> ) involving one or more
NaNs will set the erange flag.

For the purposes of the overloaded not, ! and bool
operators, a "false" Math::MPFR object is one whose value is
either 0 (including -0) or NaN.
(A "true" Math::MPFR object is, of course, simply one that
is not "false".)

+ - * / ** sqrt (Return object has default precision)
+= -= *= /= **= ++ -- (Precision remains unchanged)
< <= > >= == != <=>
! bool
abs atan2 cos sin log exp (Return object has default precision)
int (On perl 5.8 only, NA on perl 5.6. The return object
has default precision)
= (The copy has the same precision as the copied object.)
""

is allowed.
Let \$M be a Math::MPFR object, and \$G be any one of a Math::GMPz,
Math::GMPq or Math::GMPf object. Then it is now permissible to
do:

\$M + \$G;
\$M - \$G;
\$M * \$G;
\$M / \$G;
\$M ** \$G;

In each of the above, a Math::MPFR object containing the result
of the operation is returned.It is also now permissible to do:

\$M += \$G;
\$M -= \$G;
\$M *= \$G;
\$M /= \$G;
\$M **= \$G;

If you have version 0.35 (or later) of Math::GMPz, Math::GMPq
and Math::GMPf, it is also permissible to do:

\$G + \$M;
\$G - \$M;
\$G * \$M;
\$G / \$M;
\$G ** \$M;

Again, each of those operations returns a Math::MPFR object
containing the result of the operation.
Each operation is conducted using current default rounding mode.

NOTE: If \$G is a Math::GMPq object or a Math::GMPz object, then
the value of \$G/\$M is calculated by doing 1/(\$M/\$G). This
involves *2* roundings of the value that is returned - once when
\$M/\$G is calculated, and again when the inverse is calculated.

Math::GMPq object, it is necessary to convert the Math::GMPq
object to an mpfr_t (the type of value encapsulated in the
Math::MPFR object). This conversion is done using current default
precision and current default rounding mode.

The following is still NOT ALLOWED, and will cause a fatal error:

\$G += \$M;
\$G -= \$M;
\$G *= \$M;
\$G /= \$M;
\$G **= \$M;

In those situations where the overload subroutine operates on 2
perl variables, then obviously one of those perl variables is
a Math::MPFR object. To determine the value of the other variable
the subroutine works through the following steps (in order),
using the first value it finds, or croaking if it gets
to step 6:

1. If the variable is a UV (unsigned integer value) then that
value is used. The variable is considered to be a UV if
(perl 5.8) the UOK flag is set or if (perl 5.6) SvIsUV()
returns true.

2. If the variable is an IV (signed integer value) then that
value is used. The variable is considered to be an IV if the
IOK flag is set.

3. If the variable is an NV (floating point value) then that
value is used. The variable is considered to be an NV if the
NOK flag is set.

4. If the variable is a string (ie the POK flag is set) then the
value of that string is used. If the POK flag is set, but the
string is not a valid number, the subroutine croaks with an
appropriate error message. If the string starts with 0b or
0B it is regarded as a base 2 number. If it starts with 0x
or 0X it is regarded as a base 16 number. Otherwise it is
regarded as a base 10 number.

5. If the variable is a Math::MPFR, Math::GMPz, Math::GMPf, or
Math::GMPq object then the value of that object is used.

6. If none of the above is true, then the second variable is
deemed to be of an invalid type. The subroutine croaks with
an appropriate error message.

#####################

FORMATTED OUTPUT

NOTE: When using the P (precision) type specifier, instead of
providing \$prec to the P specifier, its now advisable
to provide prec_cast(\$prec). The P specifier expects an
mp_prec_t but, prior to 3.18, we could pass it only an IV.
This didnt work on at least some big-endian machines if
the size of the IV was greater than the size of the
mp_prec_t.
The Math::MPFR::Prec package (which is part of this
distribution) exists solely to provide the prec_cast sub.
And the prec_cast subs return value should be passed *only*
to the P type specifier. Nothing else will understand it.
Passing it to something other than the P specifier may
produce a garbage result - might even cause a segfault.

prec_cast(\$prec);

Ensures that the P type specifier will provide correct results.
In Math::MPFR versions prior to 3.18 we could do only (eg) :
Rmpfr_printf("%Pu\n", Rmpfr_get_prec(\$op));
But that didnt work correctly for all architectures. As of 3.18,
that can be rewritten as:
Rmpfr_printf("%Pu\n", prec_cast(Rmpfr_get_prec(\$op)));
which should work on all architectures.

Rmpfr_printf(\$format_string, [\$rnd,] \$var);

This function (unlike the MPFR counterpart) is limited to taking
2 or 3 arguments - the format string, optionally a rounding argument,
and the variable to be formatted.
That is, you can currently printf only one variable at a time.
If theres no variable to be formatted, just add a 0 as the final
argument. ie this will work fine:
Rmpfr_printf("hello world\n", 0);
NOTE: The rounding argument \$rnd can be provided *only* if \$var is a
Math::MPFR object. To do otherwise is a fatal error.
See the mpfr documentation for details re the formatting options:
http://www.mpfr.org/mpfr-current/mpfr.html#Formatted-Output-Functions

Rmpfr_fprintf(\$fh, \$format_string, [\$rnd,] \$var);

This function (unlike the MPFR counterpart) is limited to taking
3 or 4 arguments - the filehandle, the format string, optionally a
rounding argument, and the variable to be formatted. That is, you
can printf only one variable at a time.
If theres no variable to be formatted, just add a 0 as the final
argument. ie this will work fine:
Rmpfr_fprintf(\$fh, "hello world\n", 0);
NOTE: The rounding argument \$rnd can be provided *only* if \$var is a
Math::MPFR object. To do otherwise is a fatal error.
See the mpfr documentation for details re the formatting options:
http://www.mpfr.org/mpfr-current/mpfr.html#Formatted-Output-Functions

Rmpfr_sprintf(\$buffer, \$format_string, [\$rnd,] \$var, \$buflen);

This function (unlike the MPFR counterpart) is limited to taking
4 or 5 arguments - the buffer, the format string, optionally a
rounding argument, the variable to be formatted and the size of the
buffer (\$buflen) into which the result will be written. \$buflen
must specify a size (characters) that is at least large enough to
accommodate the formatted string (including the terminating NULL).
The formatted string will be placed in \$buffer.
If theres no variable to be formatted, just insert a 0 as the
value for \$var. ie this will work fine:
Rmpfr_sprintf(\$buffer, "hello world", 0, \$buflen);
NOTE: The rounding argument \$rnd can be provided *only* if \$var is a
Math::MPFR object. To do otherwise is a fatal error.
See the mpfr documentation for details re the formatting options:
http://www.mpfr.org/mpfr-current/mpfr.html#Formatted-Output-Functions

Rmpfr_snprintf(\$buffer, \$bytes, \$format_string, [\$rnd,] \$var, \$buflen);

This function (unlike the MPFR counterpart) is limited to taking
5 or 6 arguments - the buffer, the number of bytes to be written,
the format string, optionally a rounding argument, the variable
to be formatted and the size of the buffer (\$buflen).  \$buflen must
specify a size (characters) that is at least large enough to
accommodate the formatted string (including the terminating NULL).
The formatted string will be placed in \$buffer.
If theres no variable to be formatted, just insert a 0 as the
value for \$arg. ie this will work fine:
Rmpfr_snprintf(\$buffer, 12, "hello world", 0, \$buflen);
NOTE: The rounding argument \$rnd can be provided *only* if \$var is a
Math::MPFR object. To do otherwise is a fatal error.
See the mpfr documentation for further details:
http://www.mpfr.org/mpfr-current/mpfr.html#Formatted-Output-Functions

#####################

BASE CONVERSIONS

\$DBL_DIG  = MPFR_DBL_DIG;  # Will be undef if float.h doesnt define
# DBL_DIG.

\$LDBL_DIG = MPFR_LDBL_DIG; # Will be undef if float.h doesnt define
# LDBL_DIG.

\$FLT128_DIG = MPFR_FLT128_DIG; # Will be undef if quadmath.h has not
# define FLT128_DIG
.
\$min_prec = mpfr_min_inter_prec(\$orig_base, \$orig_length, \$to_base);
\$max_len  = mpfr_max_orig_len(\$orig_base, \$to_base, \$to_prec);
\$min_base = mpfr_min_inter_base(\$orig_base, \$orig_length, \$to_prec);
\$max_base = mpfr_max_orig_base(\$orig_length, \$to_base, \$to_prec);

The last 4 of the above functions establish the relationship between
\$orig_base, \$orig_length, \$to_base and \$to_prec.
Given any 3 of those 4, theres a function there to determine the
value of the 4th.

Lets say we have some base 10 floating point numbers comprising 16
significant digits, and we want to convert those numbers to a base 2
data type (say, long double).
If we then convert the value of that long double to a 16-digit base 10
float are we guaranteed of getting the original value back ?
It all depends upon the precision of the long double type, and the
min_inter_prec() subroutine will tell you what the minimum
required precision is (in order to be sure of getting the original
value back). We have:

\$min_prec = mpfr_min_inter_prec(\$orig_base, \$orig_length, \$to_base);

In our example case that becomes:

\$min_prec = mpfr_min_inter_prec(10, 16, 2);

which will set \$min_prec to 55.
That is, so long as the long double type has a precision of at least 55
bits, you can pass 16-digit, base 10, floating point values to it and
back again, and be assured of retrieving the original value.
(Naturally, this is assuming absence of buggy behaviour, and correct
rounding practice.)

Similarly, you might like to know the maximum significant number of
base 10 digits that can be specified, when assigning to (say) a
53-bit double. We have:

\$max_len = mpfr_max_orig_len(\$orig_base, \$to_base, \$to_prec);

For this second example that becomes:

\$max_len = mpfr_max_orig_len(10, 2, 53);

which will set \$max_len to 15.

That is, so long as your base 10 float consists of no more than 15
siginificant digits, you can pass it to a 53-bit double and back again,
and be assured of retrieving the original value.
(Again, we assume absence of bugs and correct rounding practice.)

It is to be expected that
mpfr_max_orig_len(10, 2, \$double_prec)
and
mpfr_max_orig_len(10, 2, \$long_double_prec)
will (resp.) return the same values as MPFR_DBL_DIG and MPFR_LDBL_DIG.
(\$double_prec is the precision, in bits, of the C double type,
and \$long_double_prec is the precision, in bits, of the C long double
type.)

The last 2 of the above subroutines (ie mpfr_min_inter_base and
mpfr_max_orig_base) are provided mainly for completeness.
Normally, there wouldnt be a need to use these last 2 forms ... but
who knows ...

The above examples demonstrate usage in relation to conversion between
bases 2 and 10. The functions apply just as well to conversions between
bases of any values.

The Math::LongDouble module provides 4 identical functions, prefixed
with ld_ instead of mpfr_ (to avoid name clashes).
Similarly, it provides constants (prefixed with LD_ instead of
MPFR_) that reflect the values of float.hs DBL_DIG and LDBL_DIG.

#####################

```

BUGS

```

You can get segfaults if you pass the wrong type of argument to the
functions - so if you get a segfault, the first thing to do is to
check that the argument types you have supplied are appropriate.

```

ACKNOWLEDGEMENTS

```

Thanks to Vincent Lefevre for providing corrections to errors
and omissions, and suggesting improvements (which were duly
put in place).

```

```

This program is free software; you may redistribute it and/or
modify it under the same terms as Perl itself.

```

AUTHOR

```

Sisyphus <sisyphus at(@) cpan dot (.) org>

```
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