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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 the
GMP library first as it is a pre-requisite for MPFR.)

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

Documentation for current mpfr functions can be found at:
www.mpfr.org/mpfr-current/mpfr.html#Function-and-Type-Index``````

# 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 you're using '@' as
# the separator. If you must use double quotes, you'll 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
# it's 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 the "GMP_*" or the "MPFR_*" renditions.

Also available are:
mpfr-3.0.0 and later:
MPFR_RNDA (numeric value = 4): Round away from zero.
mpfr-4.0.0 and later:
MPFR_RNDF (numeric value = 5): Faithful rounding.

These last two rounding modes will cause a fatal error
if the mpfr library against which Math::MPFR has been
built is not sufficiently recent.

MPFR_RNDF is experimental - the computed value is either
that corresponding to MPFR_RNDD or that corresponding to
MPFR_RNDU. In particular when those values are identical,
i.e., when the result of the corresponding operation is
exactly representable, that exact result is returned.
Thus, the computed result can take at most two possible
values, and in absence of underflow/overflow, the
corresponding error is strictly less than one ulp (unit in
the last place) of that result and of the exact result.
For MPFR_RNDF, the returned value and the inexact flag are
unspecified, the divide-by-zero flag is as with other
roundings, and the underflow and overflow flags match what
would be obtained in the case the computed value is the
same as with MPFR_RNDD or MPFR_RNDU. The results may not
be reproducible, and it may not work with all functions.

MPFR_RNDN (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.

ROUNDING MODE FUNCTIONS section for the mode
"round to nearest, ties away from zero".``````

# 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.
Because these objects have not been blessed into the Math::MPFR
package, the overloaded operators will not work with them.``````

# 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 I've stopped
recommending it - and don't support the practice.``````

# PASSING __float128 VALUES

``````   There are 3 ways to pass __float128 values to/from
Math::MPFR. Option 1) is the preferred option:

1) Build perl (5.21.4 or later) with -Dusequadmath; build the
mpfr-4.0.0 (or later) library with the configure option
--enable-float128, and then build Math::MPFR by providing the
"F128=1" arg to the Makefile.PL (which will define the symbol
MPFR_WANT_FLOAT128):

perl Makefile.PL F128=1

Then you can pass your perl's __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.

NOTE:
If you have built perl with -Dusequadmath, but the mpfr
library was not built with --enable-float128 (or the F128=1
arg was not given to the Makefile.PL) , then both
Rmpfr_set_NV() and Rmpfr_get_NV() will still set/get
__float128 values.
evaluate that (__float128) NV to its full precision.
It's only Rmpfr_set_float128() and Rmpfr_get_float128() that
will not be available.

2) Install Math::Float128, build the mpfr-4.0.0 (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()

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

Rmpfr_set_str() and Rmpfr_get_str()``````

# PASSING _Decimal64 & _Decimal128 VALUES

``````   Install Math::Decimal64 and/or Math::Decimal128 and build the mpfr
library (version 3.1.0 or later for _Decimal64, version 4.1.0 or
later for _Decimal128) with the --enable-decimal-float option. The
Math::MPFR build process should detect the _Decimal64/_Decimal128
availability and include or exclude the support accordingly.

If the auto-detection fails to detect the availability, you can
override it by providing the "D64=1" and/or "D128=1" arg to the
the Makefile.PL (either of which will define
MPFR_WANT_DECIMAL_FLOATS):

perl Makefile.PL D64=1 D128=1

You can then pass _Decimal64/_Decimal128 values between Math::MPFR
and Math::Decimal64/Math::Decimal128 using:

Rmpfr_set_DECIMAL64() and Rmpfr_get_DECIMAL64()
Rmpfr_set_DECIMAL128() and Rmpfr_get_DECIMAL128()

NOTE: It is allowable to pass any or all of "D64=1", "D128=1" and
"F128=1" args to the Makefile.PL.

To force the removal of _Decimal64/Decimal128 support in
Math::MPFR, simply provide "D64=0" and/or "D128=0" as arguments to
the Makefile.PL:

perl Makefile.PL D64=0 D128=0``````

# FUNCTIONS

``````   These next 3 functions are demonstrated above (in SYNOPSIS):
\$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 generally 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 (or even as \$r *= 2).
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'.

"\$uj" means any integer that will fit into a C 'uintmax_t'. Don't
use any of these functions unless your perl's UV is at least as big
as a 'uintmax_t'.

"\$sj" means any integer that will fit into a C 'intmax_t'. Don't
use any of these functions unless your perl's IV is at least as big
as an 'intmax_t'.

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

"\$ld" means a long double. Don't use these functions if the precision
of your Perl's NV is less than the precision of a 'long double'.

"\$f128" means a __float128. Don't use these functions unless your
Perl's NV is a __float128 && mpfr has been configured with
'--enable-float128'. (Note that versions of mpfr prior to 4.0.0
cannot be configured with '--enable-float128'.)

"\$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 '1234567890987654321234567E7' or 'zsa34760sdfgq123r5@11'.
Valid bases for MPFR numbers are 2 to 62.

"\$rnd" is simply one of the 5 rounding mode values (discussed above).

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

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

ROUNDING MODE FUNCTIONS

Rmpfr_set_default_rounding_mode(\$rnd);
Sets the default rounding mode to \$rnd (where \$rnd can be one of
MPFR_RNDN, MPFR_RNDU, MPFR_RNDZ, MPFR_RNDD and MPFR_RNDA.
Note that MPFR_RNDA is available only if Math::MPFR has been built
against mpfr-3.0.0 or later.
The default rounding mode is to nearest initially (MPFR_RNDN).
The default rounding mode is the rounding mode that is used in

\$si = Rmpfr_get_default_rounding_mode();
Returns the numeric value (0, 1, 2, 3 or 4) 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.

\$si = Rmpfr_round_nearest_away(\&function, \$rop, @input_args);
This is a perl implementation (and not a wrapping) of
the mpfr_round_nearest_away macro introduced in
mpfr-4.0.0. You can use this function so long as
Math::MPFR has been built against mpfr-3.0.0 or later.
This rounding is defined in the same way as MPFR_RNDN,
except in case of tie, where the value away from zero is
returned.
The first arg is a reference to the perl subroutine you
wish to call, and the remaining args are the args that
the subroutine usually takes (minus the rounding arg).
For example:

\$op1, \$op2);

\$ret = Rmpfr_round_nearest_away(\&Rmpfr_strtofr, \$rop,
'1e-200', 10);

\$ret = Rmpfr_round_nearest_away(\&Rmpfr_prec_round,
\$rop, \$prec);

The "function" being called must be one that returns the
ternary int (-ve for "less than", 0 for "exact", and +ve for
"greater than").

Unlike the more efficient rndna(), this function requires that
Rmpfr_get_emin() is greater than Rmpfr_get_emin_min().
If you're not concerned about correct rounding (nearest away) of
the value +/- 0.25 * (2 ** Rmpfr_get_emin()) then you can instead
use the rndna function (immediately below).

\$si = rndna(\&function, \$rop, @input_args);
A more efficient version of the (above) Rmpfr_round_nearest_away
function. However, it will not correctly deal with the value
+/- 0.25 * (2 ** Rmpfr_get_emin()).
Unlike Rmpfr_round_nearest_away(), this function allows that
Rmpfr_get_emin() == Rmpfr_get_emin_min().

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

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);
\$si = Rmpfr_set_uj(\$rop, \$uj, \$rnd);
\$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,
# depending upon perl's nvtype.
\$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);# nvtype is __float128
# && mpfr-4.0.0 or later
# && mpfr lib built with
# --enable-float128 &&
# F128=1 cmd line arg
# given to Makefile.PL
\$si = Rmpfr_set_DECIMAL64(\$rop, \$D64, \$rnd) # mpfr-3.1.0 or later &&
# mpfr lib built with
# --enable-decimal-float
# && D64=1 arg given to
# Makefile.PL. \$D64 is a
# Math::Decimal64 object
\$si = Rmpfr_set_DECIMAL128(\$rop, \$D128, \$rnd) # mpfr-3.1.0 or later &&
# mpfr lib built with
# --enable-decimal-float
# && D128=1 arg given to
# Makefile.PL. \$D128 is a
# Math::Decimal128 object
\$si = Rmpfr_set_FLOAT128(\$rop, \$F128, \$rnd);# mpfr-4.0.0 and later
# && mpfr lib built with
# --enable-float128 &&
# F128=1 cmd line arg
# given to Makefile.PL.
# \$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.

\$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);
\$si = Rmpfr_set_sj_2exp(\$rop, \$sj, \$exp, \$rnd);
\$si = Rmpfr_set_z_2exp(\$rop, \$z, \$exp, \$rnd); # mpfr-3.0.0 & 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 initialised 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 won't 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 doesn't 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 62.

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 62. If \$arg is a string and no additional argument is
supplied, the base will be deduced.
See 'Rmpfr_set_str' above for an explanation of how that
deduction is done.

NOTE: If \$arg is *both* an NV (floating point value) and PV
(string), then the value specified by the PV (string) will be
used. This is probably what you want (less likely so with
perl-5.18.4 and earlier).
However, there's no guaranteed way for the new() function to
correctly tell and it's best to avoid passing such values, or
to explicitly use the value you want by doing an Rmpfr_init()
followed by the appropriate 'Rmpfr_set_*' function documented in
the previous section. Or, if such exists, you could instead call
the appropriate 'Rmpfr_init_set_*' function documented
immediately below.
Note that these functions (below) 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 initialised 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 will be set to 'Nan', '-Inf' or 'Inf' whenever \$op is
(respectively) a NaN, a negative infinity or a positive infinity.

(\$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.
There's 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.
Unlike Rmpfr_get_str() and Rmpfr_integer_string(), \$str will be set
to '@NaN@', '-@Inf@' or '@Inf@' whenever \$op is (respectively) a
NaN, a negative infinity or a positive infinity, as those are the
strings that the mpfr library assigns.

\$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".
\$str will be set to 'Nan', '-Inf' or 'Inf' whenever \$op is
(respectively) a NaN, a negative infinity or a positive infinity.
(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);
\$uj = Rmpfr_get_uj(\$op, \$rnd);
\$uv = Rmpfr_get_UV(\$op, \$rnd);
\$iv = Rmpfr_get_IV(\$op, \$rnd);
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 is __float128
# && mpfr-4.0.0 or later
# && mpfr lib built with
# --enable-float128 &&
# F128=1 cmd line arg
# given to Makefile.PL
\$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-3.1.0 or later &&
# mpfr lib built with
# --enable-decimal-float
# && D64=1 arg given to
# Makefile.PL. \$D64 is a
# Math::Decimal64 object
Rmpfr_get_FLOAT128(\$F128, \$op, \$rnd);# mpfr-4.0.0 and later &&
# mpfrlib built with
# --enable-float128 && F128=1
# arg given to Makefile.PL.
# \$F128 is a Math::Float128
# object.
Convert \$op to a 'double', a 'long double', a __float128, an 'NV',
a float, 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 if 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).

\$ui1 = Rmpfr_get_str_ndigits(\$ui2, \$ui3); # new with mpfr-4.1.0
Note: \$ui2 must be in the range 2..62 (inclusive).
Return the minimal integer \$ui1 such that any number of precision
\$ui3 bits, when output with \$ui1 digits in radix \$ui2 with rounding
to nearest, can be recovered exactly when read again, still with
rounding to nearest.
More precisely, we have \$ui1 = 1 + ceil(\$ui3 * log(2) / log(\$ui2)),
with \$ui3 decremented by 1 if \$ui2 is a power of 2.
This function wraps mpfr_get_str_ndigits if mpfr version is 4.1.0
or later. Otherwise it calls Rmpfr_get_str_ndigits_alt, which does
not require mpfr-4.1.0.

\$ui1 = Rmpfr_get_str_ndigits_alt(\$ui2, \$ui3);
Provided as a fallback (though it can also be called separately)
for Rmpfr_get_str_ndigits when mpfr version is older than 4.1.0.
Unlike Rmpfr_get_str_ndigits, this function does not require that
\$ui2 is in the range 2..62.

##########

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_z_div(\$rop, \$z, \$op, \$rnd);   # \$z is a mpz object.
\$si = Rmpfr_div_q(\$rop, \$op, \$q, \$rnd);   # \$q is a mpq object.
\$si = Rmpfr_q_div(\$rop, \$q, \$op, \$rnd);   # \$q is a mpq object.
NOTE: The mpfr library does not provide mpfr_z_div and
mpfr_q_div functions.
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 \$op ** (-1 / 2) 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 less than zero.

\$si = Rmpfr_rec_root(\$rop, \$op, \$ui, \$rnd);
NOTE: There is no such mpfr function as mpfr_rec_root.
This function originally provided as a perl implementation by
Vincent Lefevre - and rewritten by sisyphus as an XSub.
(See https://sympa.inria.fr/sympa/arc/mpfr/2016-12/msg00032.html)

Set \$rop to \$op ** (-1 / \$ui) rounded in the direction \$rnd.

May not correctly handle overflow or underflow.

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

\$si = Rmpfr_root(\$rop, \$op, \$ui \$rnd);
Deprecated in mpfr-4.0.0 - use Rmpfr_rootn_ui with mpfr-4.0.0.
Set \$rop to the \$ui'th root of \$op, rounded in the direction
\$rnd.  Return 0 if the operation is exact, a non-zero value
otherwise.

\$si = Rmpfr_rootn_ui(\$rop, \$op, \$ui \$rnd); # mpfr-4.0.0 and later
Same as Rmpfr_root except that \$rop is set to +0 (instead of -0)
when \$op is -0 and \$ui is even.
This function (and Rmpfr_cbrt) agree with the rootn function of
the IEEE 754-2008 standard (Section 9.2).

\$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 if 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_cmpabs_ui(\$op, \$ui);  # requires mpfr-4.1.0 and later
\$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
\$si = Rmpfr_cmp_NV(\$op, \$nv);
Compare 1st and 2nd args. In the case of 'Rmpfr_cmpabs()' and
'Rmpfr_cmpabs_ui' 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_total_order_p(\$op1, \$op2); # new with mpfr-4.1.0
This function implements the totalOrder predicate from IEEE
754-2008, where -NaN < -Inf < negative finite numbers < -0 < +0
< positive finite numbers < +Inf < +NaN. It returns a non-zero
value (true) when X is smaller than or equal to Y for this order
relation, and zero (false) otherwise.
Contrary to 'Rmpfr_cmp(\$x, \$y)', which returns a ternary value,
'Rmpfr_total_order_p' returns a binary value (zero or non-zero).
In particular, 'Rmpfr_total_order_p(\$x, \$x)' returns true,
'Rmpfr_total_order_p(-0, +0)' returns true and
'Rmpfr_total_order_p(+0, -0)' returns false.
The sign bit of NaN also matters.
This function falls back to our own implementation if the mpfr
version is older than 4.1.0 (because, in that case,
mpfr_total_order_p is unavailable).

\$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_log_ui(\$rop, \$ui, \$rnd); # mpfr-4.0.0 & later only
\$si = Rmpfr_log2(\$rop, \$op, \$rnd);
\$si = Rmpfr_log10(\$rop, \$op, \$rnd);
Set \$rop to the natural logarithm of \$op, the natural
logarithm  of \$ui, 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 if 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 if the result is less
than the actual value. Return a positive result if 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 if 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 if 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 if the result is exact. Return a negative value if
the result is less than the actual value. Return a positive
result if 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 if the result is exact. Return a negative value if
the result is less than the actual value. Return a positive
result if 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 if the result is exact
(this occurs in fact only when \$op is 0 i.e. the result is 1).
Return a negative value if the result is less than the actual
value. Return a positive result if 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 if 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
if the result is exact. Return a negative value if the
result is less than the actual value. Return a positive
result if 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
if the result is exact. Return a negative value if the
result is less than the actual value. Return a positive
result if 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 if 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 if
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 if 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_fmma(\$rop, \$op1, \$op2, \$op3, \$op4, \$rnd);
NOTE: Needs mpfr-4.0.0 or later
Set \$rop to \$op1 * \$op2 + \$op3 * \$op4, 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_fmms(\$rop, \$op1, \$op2, \$op3, \$op4, \$rnd);
NOTE: Needs mpfr-4.0.0 or later
Set \$rop to \$op1 * \$op2 - \$op3 * \$op4, 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 Euler's 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 Catalan's 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').

Rmpfr_free_cache2(\$ui);        # mpfr-4.0.0 and later only
Free various caches and pools used by MPFR internally, as
specified by \$ui, which is a set of flags:
a) those local to the current thread if flag
MPFR_FREE_LOCAL_CACHE is set;
b) those shared by all threads if flag
MPFR_FREE_GLOBAL_CACHE is set.

The other bits of \$ui are currently ignored and are reserved for
future use; they should be zero.
Note:
Rmpfr_free_cache2(MPFR_FREE_LOCAL_CACHE|MPFR_FREE_GLOBAL_CACHE)
is currently equivalent to mpfr_free_cache().

Rmpfr_free_pool()             # mpfr-4.0.0 and later only
Free the pools used by mpfr internally.
Note:
This function is automatically called after the thread-local
caches are freed (with mpfr_free_cache or mpfr_free_cache2).

\$si = Rmpfr_beta(\$rop, \$op1, \$op2, \$rnd); # mpfr-4.0.0 &
# later only
Set \$rop to the beta function at \$op1, \$op2, rounded
according to \$rnd.

\$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.
'Rmpfr_gamma' sets \$rop to NaN when \$op is negative.

\$si = Rmpfr_gamma_inc(\$rop, \$op1, \$op2, \$rnd); # mpfr-4.0.0 &
# later only
Set \$rop to the value of the incomplete Gamma function on \$op1
and \$op2, rounded in the direction \$rnd.
When \$op2 is zero and \$op1 is a negative value, \$rop is set to
NaN.
Note: the current implementation is slow for large values of
\$rop and \$op, in which case some internal overflow might
also occur.

(\$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.
As of mpfr-4.0.0 Rmpfr_eint() returns the value of the
E1/eint1 function for negative input. (With previous versions
of mpfr NaN was returned for negative argument.)

\$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.

\$si = Rmpfr_dot (\$rop, \@op1, \@op2, \$ui, \$rnd); # mpfr-4.1.0 &
# later only
Set \$rop to the dot product of elements of @op1 by those of
@op2, whose common size is \$ui (== scalar @op1), correctly
rounded in the direction \$rnd.
This function is experimental, and does not yet handle
intermediate overflows and underflows

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

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.  \$base may be in the range 2 to 62
(or -36..-2, 2 .. 62 if Math::MPFR has been built against
mpfr-4.1.0 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 there's 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 (let's 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 62.
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 initialised 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 there's the capability to read
from somewhere other than STDIN.
TRmpfr_inp_str(\$rop, *stdin, \$base, \$round);
To read from an open filehandle (let's call it FH):
TRmpfr_inp_str(\$rop, \*FH, \$base,  \$round);

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

\$si = Rmpfr_fpif_export (\$stream, \$op); # Needs mpfr-4.0.0
\$si = Rmpfr_fpif_export (\$op, \$stream); # Needs mpfr-4.0.0
Note: These function are experimental and their interface might
change in future versions of mpfr.
Export/import the number \$op to/from the stream \$stream in a
floating-point interchange format. In particular one can export
on a 32-bit computer and import on a 64-bit computer, or export
on a little-endian computer and import on a big-endian computer.
The precision of OP is stored too. The import function fails if
the precision (which is read from the stream) is greater than
MPFR_PREC_MAX.
Return 0 if the export/import was successful.

##########

EXCEPTIONS

\$si = Rmpfr_subnormalize (\$op, \$si, \$rnd);
This function rounds \$op emulating subnormal number arithmetic.
See the MPFR documentation for mpfr_subnormalize at:
www.mpfr.org/mpfr-current/mpfr.html#Function-and-Type-Index

\$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 assumes that \$op is the correctly rounded value of
some real value X in the direction \$rnd and some extended exponent
range, and that \$si1 is the corresponding ternary value. Thus \$si1
is negative if \$op is smaller than X, positive if \$op is larger than
X, and zero if \$op equals X.
This function modifies \$op if needed to be in the current range of
acceptable values. 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 new value of \$op equals the exact one X, a positive
value if that new value is larger than X, and a negative value if
it is smaller than X. Note that unlike most functions, the new
result \$op is compared to the (unknown) exact one X, 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 if the flag is set.

Rmpfr_flags_clear(\$mask);            # needs mpfr-4.0.0 or later
Rmpfr_flags_set(\$mask);              # needs mpfr-4.0.0 or later
\$mask  = Rmpfr_flags_save();         # needs mpfr-4.0.0 or later
\$mask is an integer value resulting from OR'ing (or adding) any of
the following constants together:
MPFR_FLAGS_UNDERFLOW
MPFR_FLAGS_OVERFLOW
MPFR_FLAGS_NAN
MPFR_FLAGS_INEXACT
MPFR_FLAGS_ERANGE
MPFR_FLAGS_DIVBY0
MPFR_FLAGS_ALL
MPFR_FLAGS_ALL is the same as all of those constants OR'ed (added)
together. Examples:

Clear the divby0 and nan flags, without affecting the status of
any other flags:
Rmpfr_flags_clear(MPFR_FLAGS_DIVBY0|MPFR_FLAGS_NAN);

Set the underflow and erange flags, without affecting the status of
any other flags:
Rmpfr_flags_set(MPFR_FLAGS_UNDERFLOW|MPFR_FLAGS_ERANGE);

Test the status of specific flags, ignoring all other flags. This
particular example will set \$mask to the value of MPFR_FLAGS_NAN
if the nan flag is set - else \$mask will be set to zero:

Return all flags:
Equivalently, this can be done with:

For this example, the overflow flag will be set && the nan flag
will be cleared. The underflow flag will remain untouched.

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

MISCELLANEOUS

\$str = doubletoa(\$nv [, \$x]); # Available only when
# \$Config{nvsize} == 8.
# Else use nvtoa().
Sets \$str to the value of the double \$nv.
\$str will appear in standard decimal format such that the
condition (\$str == \$nv) is true.
The grisu3 algorithm used by this function, whilst very quick,
will fail for about 0.5% of values. When this happens doubletoa
will (by default) fall back to using nvtoa() - which is slower,
but reliable.
Alternatively, providing a second argument to doubletoa (any
scalar will do) will ensure that sprintf("%.16e", \$nv) is
returned. This will be an accurate return, and it will be
quicker than the nvtoa() fallback, but it often provides more
digits than are necessary.
Whenever doubletoa reverts to a fallback routine,
\$Math::MPFR::doubletoa_fallback (which is initially set to 0)
will be incremented. This feature can be disabled by providing
the FB=0 argument to the 'perl Makefile.PL' step of the
Math::MPFR build.

\$nv = atodouble(\$str); # Needs mpfr-3.1.6 or later
Sets the \$nv to the value of the double that's represented by
the string \$str.
If \$Config{nvtype} is double, it therefore returns the same
value as atonv.
See atonv() below.

\$str = nvtoa(\$nv); # Available irrespective of \$Config{nvsize}
# and \$Config{nvtype}
Sets \$str to the value of \$nv.
\$str will appear in standard decimal format, containing the
*minimum* number of digits such that the condition (\$str == \$nv)
is true.
NOTE 1: Except for perls whose nvtype is __float128:
1) perl versions prior to 5.30.0 are buggy in their
assignemt of values to the NV;
2) perl versions from 5.30.0 on that don't define
\$Config{d_strtod} (or respectively \$Config{d_strtold}
for -Duselongdouble builds) are also buggy in their
assignmentof values to the NV.
The nvtoa() function looks only at the value that was
passed to it - which, with such buggy perls, may differ
from the intended value by a few ULPs.
On these buggy perls, the bug can be avoided by assigning
with atonv() - which requires mpfr-3.1.6 or later.
NOTE 2: Be wary of values assigned by perl to double-double NVs,
as such assignments may also be buggy, irrespective of
the perl version.
Again, with these perls, assign using atonv() - which
requires mpfr-3.1.6 or later.
With double-double NVs, nvtoa() currently requires
mpfr-4.0.0 or later.

\$nv = atonv(\$str); # Needs mpfr-3.1.6 or later
Sets \$nv to the NV value represented by the string \$str.
If \$str expresses a base 2 value, begin it with '0b' or '0B'.
If \$str expresses a base 16 value, begin it with '0x' or '0X.
For example:
atonv('0x0.89p5'), atonv('0x0.112@2'), atonv('17.125') and
atonv('0b0.10001001e5') all return 17.125.
NOTE: ORIGINALLY TOOK 2 ARGUMENTS - NOW TAKES ONLY 1, as of
Math-MPFR-4.07.

\$str = numtoa(\$sv);
As for nvtoa() but prints out UVs and IVs in integer format.
(If UV-precision is greater than NV-precision then nvtoa()
output can lose precision as it prints out UVs and IVs in floating
point format.)

\$sv = atonum(\$str); # Needs mpfr-3.1.6 or later
If \$str numifies to an IV, return that IV.
Else return the NV that is returned by atonv(\$str).

\$MPFR_version = Rmpfr_get_version();
Returns the version of the MPFR library (eg 4.0.2) 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 that's 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 for "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_float128_p(); # mpfr-4.0.0 and later only
Return a non-zero value if mpfr was compiled with __float128
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.

\$bool = Rmpfr_buildopt_sharedcache_p(); # mpfr-4.0.0 and later only
Return a non-zero value if MPFR was compiled so that all threads
share the same cache for the one MPFR constant, like `mpfr_const_pi'
or `mpfr_const_log2' (that is, MPFR was built with the
`--enable-shared-cache' 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_roundeven(\$rop, \$op); # mpfr-4.0.0 and later only
\$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, 'Rmpfr_roundeven' to the nearest
representable integer, rounding halfway cases with the even-
rounding rule 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_roundeven(\$rop, \$op, \$rnd);  # mpfr-4.0.0 & later
\$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, 'Rmpfr_rint_roundeven' to
the nearest integer,rounding halfway cases to the nearest even
integer 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 if 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);
(\$si2, \$si) = Rmpfr_fmodquo (\$rop, \$op1, \$op2, \$rnd); # mpfr-4.0.0 &
# later only
Set \$rop to the value of \$op - N*\$op2, rounded according to the
direction \$rnd, where N is the integer quotient of \$op1 divided by
\$op2, defined as follows: N is rounded toward zero for 'Rmpfr_fmod'
and 'Rmpfr_fmodquo', and to the nearest integer (ties rounded to
even) for 'mpfr_remainder' and 'mpfr_remquo'.
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' and 'Rmpfr_fmodquo store 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 if \$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().

\$iv = Math::MPFR::nok_pokflag(); # not exported
Returns the value of the nok_pok flag. This flag is
initialized to zero, but incemented by 1 whenever a
scalar that is both a float (NOK) and string (POK) is passed
to new() or to an overloaded operator. The value of the flag
therefore tells us how many times such events occurred . The
flag can be reset to 0 by running clear_nok_pok().

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

Math::MPFR::set_nok_pok(\$iv); # not exported
Resets the nok_pok 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).)

Math::MPFR::clear_nok_pok(); # not exported
Resets the nok_pok flag to 0.(Essentially the same
as running set_nok_pok(0).)

\$bytes = Math::MPFR::bytes(\$val, \$bits);
(The semantics for calling this function have changed as of
Math-MPFR-4.13.)
\$val must either be a string (eg '1.613e-45', '2.3', '0x17.8')
or a Math::MPFR object.
\$bits must be one of 53, 64, 2098 or 113.
\$bytes will be set to the hex representation of \$val.
If \$bits is 53, \$bytes will display the byte structure (in
hex) of the standard 8-byte double, \$val.
If \$bits is 64, the structure returned will be that of \$val
as an extended precison 10-byte long double.
If \$bits is 2098, the structure returned will be that of \$val
as the 16-byte IBM doubledouble.
If \$bits is 113, the structure returned will be that of \$val
as the quad precison 16-byte IEEE long double (or __float128,
which is identical in byte structure).

eg:
bytes('1.3', 53)   returns 3ff4cccccccccccd
bytes('1.3', 64)   returns 3fffa666666666666666
bytes('1.3', 2098) returns 3ff4cccccccccccdbc8999999999999a
bytes('1.3', 113)  returns 3fff4ccccccccccccccccccccccccccd

If \$val is a Math::MPFR object, its precision must be \$bits.

For many architectures and mpfr configurations, the structures
for all 4 allowed values of \$bits will be accessible.
In some cases, however, \$bits values of 64 and/or 113 might
throw a fatal error (with diagnostic message stating that the
requested byte structure is not currently available).

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

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.
Deprecated in mpfr-4.0.0. Use Rmpfr_nrandom with mpfr-4.0.0.
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.

\$si = Rmpfr_nrandom (\$rop, \$state, \$rnd);  # mpfr-4.0.0 and
# later only
Generate a random floating-point number according to a standard
normal Gaussian distribution (with mean zero and variance one).
The floating-point number \$rop 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.
This function is more efficient than Rmpfr_grandom.

\$si = Rmpfr_erandom(\$rop, \$state, \$rnd);  # mpfr-4.0.0 and
# later only
Generate one random floating-point number according to an
exponential distribution, with mean one.  Other characteristics
are identical to 'Rmpfr_nrandom'.

\$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 if \$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 if 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.)
""

All overloaded comparisons ( < <= > >= == != <=> ) between
Math::MPFR and Math::GMPz objects or between Math::MPFR and
Math::GMPq objects are now also permitted.
. allowed

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.

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 && the POK flag is unset.
Note therefore, that if the variable is both an NV (NOK flag
set) and PV (POK flag also set) then the string value in the
PV slot will be used. This is probably, but not necessarily,
what you want, and it's recommended not to pass such values

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, it's 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 didn't 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 sub's 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 didn't 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 there's 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 there's 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 there's 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 there's 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 doesn't define
# DBL_DIG.

\$LDBL_DIG = MPFR_LDBL_DIG; # Will be undef if float.h doesn't 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_prec, \$to_base);

NOTE: \$min_prec can be (very rarely) off by one if \$orig_prec is in
the millions, or if either \$orig_base or \$to_base are
outside of the range 2..64.

Example 1:
Let's say we have some base 10 integers comprising 16 base 10
digits, and we want to represent those numbers in base 2 (binary).
What is the minimum required number of bits, such that it can be
guaranteed that converting the base 2 representations back to base
10 will result in the original 16 digit representations ?

We can calculate that minimum required precision with:
\$min_prec = mpfr_min_inter_prec(\$orig_base, \$orig_prec, \$to_base);

In this 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 our base 2 representations provide at least 55
bits, we can pass 16-digit, base 10, integer values to them,
and be assured of retrieving the original base 10 representation when
we convert the base 2 representations back to base 10.
Sure ... not all 16-digit values require 55 bits, but there are some
that do ... and there are none that require more than  55 bits.

Example 2:
\$min_prec = mpfr_min_inter_prec(2, 53, 10);
\$min_prec is set to 17.
This tells us that a base 10 representation of a 53-bit integer needs
to comprise at least 17 digits if we are to be assured that assigning
that base 10 representation to a 53-bit integer will result in a
53-bit integer that is identical to the first.
Otherwise, there is no such assurance.

\$max_prec = mpfr_max_orig_prec (\$orig_base, \$to_base, \$to_prec);

NOTE: \$max_prec can be (very rarely) off by one if \$to_prec is in
the millions, or if either \$orig_base or \$to_base are
outside of the range 2..64.

For example:
To determine the maximum significant number of base 10 digits that
can be specified, when assigning to a 53-bit double. We have:

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

For this example that becomes:

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

which will set \$max_len to 15.
That is, so long as our base 10 integer consists of no more than
15 significant digits, we can assign it to a 53-bit double and be
assured of retrieving the original value upon converting that
double back to a 15-digit base 10 representation.
Otherwise, there is no such assurance.

It is to be expected that
mpfr_max_orig_len(10, 2, 53) == DBL_DIG
and
mpfr_max_orig_len(10, 2, 113) == FLT128_DIG
and
mpfr_max_orig_len(10, 2, \$long_double_prec) == LDBL_DIG

(where \$long_double_prec is the precision, in bits, of the
the C 'long double' type - usually either 53 or 64 or 113.)

#####################``````

# TODO

``    Look at wrapping mpfr_mp_memory_cleanup, mp_set_memory_functions``

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