++ed by:
Sisyphus

NAME

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

DEPENDENCIES

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

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

DESCRIPTION

   A bigfloat module utilising the MPFR library. Basically
   this module simply wraps the 'mpfr' floating point functions
   provided by that library.
   Operator overloading is also available.
   The following documentation heavily plagiarises the mpfr
   documentation.
   See also the Math::MPFR test suite for some examples of usage.

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.
   # see 'OPERATOR OVERLOADING' below for docs re
   # operator overloading

   .
   .

   # 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";
   # or just make use of overloading :
   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 rendition with Math-MPFR-3.0 or later.

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

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

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

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

MEMORY MANAGEMENT

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

   For each Rmpfr_init* function there is a corresponding function
   called Rmpfr_init*_nobless which returns an unblessed object.
   If you create Math::MPFR objects using the '_nobless'
   versions, it will then be up to you to clean up the memory
   associated with these objects by calling Rmpfr_clear($op) 
   for each object, or Rmpfr_clears($op1, $op2, ....).
   Alternatively such objects will be cleaned up when the script
   ends. I don't know why you would want to create unblessed
   objects. The point is that you can if you want to.
   The test suite does no testing of unblessed objects ... beware
   of bugs if you go down that path.

MIXING GMP OBJECTS WITH MPFR OBJECTS

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

   1) Math::GMP from CPAN. (This module provides access to mpz
      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.     

FUNCTIONS

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

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

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

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

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

   "$sj" means any integer that will fit into a C 'intmax_t'. Don't
   use any of these functions unless your perl was compiled with 64
   bit support.

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

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

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

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

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

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

   ROUNDING MODES

   Rmpfr_set_default_rounding_mode($rnd);
    Sets the default rounding mode to $rnd.
    The default rounding mode is to nearest initially (GMP_RNDN).
    The default rounding mode is the rounding mode that
    is used in overloaded operations.

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

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

   ##########

   EXCEPTIONS

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

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

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

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

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

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

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

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

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

   INITIALIZATION

   A variable should be initialized once only.

   First read the section 'MEMORY MANAGEMENT' (above).

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

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

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

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

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

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

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

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

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

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

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

   ##########

   ASSIGNMENT

   $si = Rmpfr_set($rop, $op, $rnd);
   $si = Rmpfr_set_ui($rop, $ui, $rnd);
   $si = Rmpfr_set_si($rop, $si, $rnd);
   $si = Rmpfr_set_sj($rop, $sj, $rnd); # 64 bit
   $si = Rmpfr_set_uj($rop, $uj, $rnd); # 64 bit
   $si = Rmpfr_set_d($rop, $double, $rnd);
   $si = Rmpfr_set_ld($rop, $ld, $rnd); # long double
   $si = Rmpfr_set_LD($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
   $si = Rmpfr_set_decimal64($rop, $d64, $rnd) # mpfr-3.1.1 and later
                                               # only. $d64 is a
                                               # Math::Decimal64 object 
   $si = Rmpfr_set_float128($rop, $f128, $rnd) # mpfr-3.2.0 and later
                                               # only. $f128 is a
                                               # Math::Float128 object 
    Set the value of $rop from 2nd arg, rounded to the precision of
    $rop towards the given direction $rnd.  Please note that even a 
    'long int' may have to be rounded if the destination precision
    is less than the machine word width.  The return value is zero
    when $rop=2nd arg, positive when $rop>2nd arg, and negative when 
    $rop<2nd arg.  For `mpfr_set_d', be careful that the input
    number $double may not be exactly representable as a double-precision
    number (this happens for 0.1 for instance), in which case it is
    first rounded by the C compiler to a double-precision number,
    and then only to a mpfr floating-point number.

    NOTE: If your perl's nvtype is 'long double' use Rmpfr_set_ld(), but
    your perl's nvtype is 'double' and you want to set a value whose
    precision is that of 'long double', then install Math::LongDouble
    and use Rmpfr_set_LD().

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

   $si = Rmpfr_set_str($rop, $str, $base, $rnd);
    Set $rop to the value of $str in base $base (0,2..36 or, if
    Math::MPFR has been built against mpfr-3.0.0 or later, 0,2..62),
    rounded in direction $rnd to the precision of $rop. 
    The exponent is read in decimal.  This function returns 0 if
    the entire string is a valid number in base $base. otherwise
    it returns -1. If $base is zero, the base is set according to 
    the following rules:
     if the string starts with '0b' or '0B' the base is set to 2;
     if the string starts with '0x' or '0X' the base is set to 16;
     otherwise the base is set to 10.
    The following exponent symbols can be used:
     '@' - can be used for any base;
     'e' or 'E' - can be used only with bases <= 10;
     'p' or 'P' - can be used to introduce binary exponents with
                  hexadecimal or binary strings.
    See the MPFR library documentation for more details. See also
    'Rmpfr_inp_str' (below). 
    Because of the special significance of the '@' symbol in perl,
    make sure you assign to strings using single quotes, not
    double quotes, when using '@' as the exponent marker. If you 
    must use double quotes (which is hard to believe) then you
    need to escape the '@'. ie the following two assignments are
    equivalent:
     Rmpfr_set_str($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',
    and in base 16, it can start with `0x' or `0X'.

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

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

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

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

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

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

   COMBINED INITIALIZATION AND ASSIGNMENT

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

   First read the section 'MEMORY MANAGEMENT' (above).

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

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

   ($rop, $si) = Rmpfr_init_set_str($str, $base, $rnd);
   ($rop, $si) = Rmpfr_init_set_str_nobless($str, $base, $rnd);
     Initialize $rop and set its value from $str in base $base,
     rounded to direction $rnd. If $str was a valid number, then
     $si will be set to 0. Else it will be set to -1.
     See `Rmpfr_set_str' (above) and 'Rmpfr_inp_str' (below).

   ##########

   CONVERSION

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

   ($str, $si) = Rmpfr_deref2($op, $base, $digits, $rnd);
    Returns the mantissa to $str (as a string of digits, prefixed with
    a minus sign if $op is negative), and returns the exponent to $si.
    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.

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

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

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

   $double = Rmpfr_get_d($op, $rnd);
   $ld     = Rmpfr_get_ld($op, $rnd);
   $nv     = Rmpfr_get_NV($op, $rnd);
   $float  = Rmpfr_get_flt($op, $rnd);   # mpfr-3.0.0 and later.
   Rmpfr_get_LD($LD, $op, $rnd); # $LD is a Math::LongDouble object.
   Rmpfr_get_decimal64($d64, $op, $rnd); # mpfr-3.1.1 and later.
                                         # $d64 is a Math::Decimal64
                                         # object.
   Rmpfr_get_float128($f128, $op, $rnd); # mpfr-3.2.0 and later.
                                         # $f128 is a Math::Float128
                                         # object.
    Convert $op to a 'double' a 'long double' an 'NV', a float, a
    Math::LongDouble object, a Math::Decimal64 object, or a
    Math::Float128 object using the rounding mode $rnd.

    NOTE: If your perl's nvtype is 'long double' use Rmpfr_get_ld(), but
    if your perl's nvtype is 'double' and you want to get a value whose
    precision is that of 'long double', then install Math::LongDouble and
    use Rmpfr_get_LD(). 

   $double = Rmpfr_get_d1($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 an mpf object.
    Convert $op to a `mpf_t', after rounding it with respect to $rnd.
    When built against mpfr-3.0.0 or later, this function returns the
    usual ternary value. (If $op is NaN or Inf, then the erange flag
    will be set.) When built against earlier versions of mpfr,
    return zero iff no error occurred.In particular a non-zero value
    is returned if $op is NaN or Inf. which do not exist in `mpf'.

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

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

   ##########

   ARITHMETIC

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

   $si = Rmpfr_div_2exp($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
    use `Rmpfr_div_2ui' (or `Rmpfr_div_2si') instead.

   ##########
     
   COMPARISON

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


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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

   #######

   SPECIAL

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

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

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

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

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

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

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


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

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

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

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

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

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

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

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

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

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

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

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


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

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

   $si = Rmpfr_const_euler($rop, $rnd);
    Set $rop to the value of 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').

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

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

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

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

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

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

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

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

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

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

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

   I-O FUNCTIONS

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

   $ui = TRmpfr_out_str([$prefix,] $stream, $base, $digits, $op, $round [, $suffix]);
    As for Rmpfr_out_str, except that 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);
    BEST TO USE TRmpfr_inp_str INSTEAD.
    Input a string in base $base from STDIN, rounded in
    direction $round, and put the read float in $rop.  The string
    is of the form `M@N' or, if the base is 10 or less, alternatively
    `MeN' or `MEN', or, if the base is 16, alternatively `MpB' or
    `MPB'. `M' is the mantissa in the specified base, `N' is the 
    exponent written in decimal for the specified base, and in base 16,
    `B' is the binary exponent written in decimal (i.e. it indicates
    the power of 2 by which the mantissa is to be scaled).
    The argument $base may be in the range 2 to 36 (2 to 62 if Math::MPFR
    has been built against mpfr-3.0.0 or later).
    Special values can be read as follows (the case does not matter):
    `@NaN@', `@Inf@', `+@Inf@' and `-@Inf@', possibly followed by
    other characters; if the base is smaller or equal to 16, the
    following strings are accepted too: `NaN', `Inf', `+Inf' and
    `-Inf'.
    Return the number of bytes read, or if an error occurred, return 0.

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

   Rmpfr_print_binary($op);
    Output $op on stdout in raw binary format (the exponent is in
    decimal, yet).

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

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

   MISCELLANEOUS

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

   $GMP_version = Math::MPFR::gmp_v();
    Returns the version of the gmp library (eg. 4.1.3) being used by
    the mpfr library 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, $patchlevel);
    Returns the value for MPFR_VERSION on "MPFR-$major.$minor.$patchlevel".

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

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

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

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

   $str = Rmpfr_buildopt_tune_case(); # mpfr-3.1.0 and later only
    Return a string saying which thresholds file has been used at
    compile time.  This file is normally selected from the processor
    type.

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

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

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

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

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

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

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

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

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

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

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

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

   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 3 Rgmp_randinit functions, then
    seed $state by calling one of the 2 Rgmp_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 Rgmp_randinit functions (below), then
    seed $state by calling one of the Rgmp_randseed functions.

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

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

   $state = Rgmp_randinit_lc_2exp($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.

   $state = Rgmp_randinit_lc_2exp_size($ui);
    Initialise state as per Rgmp_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.

   Rgmp_randseed($state, $seed);
    $state is a reference to a gmp_randstate_t strucure (the
    return value of one of the Rgmp_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.

   Rgmp_randseed_ui($state, $ui);
    $state is a reference to a gmp_randstate_t strucure (the
    return value of one of the Rgmp_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.

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

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

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

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

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

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

   OPERATOR OVERLOADING

    Overloading works with numbers, strings (bases 2, 10, and 16
    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).

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

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

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

    The following operators are overloaded:
     + - * / ** 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.)
     ""

    As of version 3.13 of Math::MPFR, some cross-class overloading
    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;

    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
    and, if there's a need for the operation to create a Math::MPFR
    object, the created object will be given current default precision.

    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 an unsigned long then that value is used.
       The variable is considered to be an unsigned long if 
       (perl 5.8) the UOK flag is set or if (perl 5.6) SvIsUV() 
       returns true.(In the case of perls built with
       -Duse64bitint, the variable is treated as an unsigned long
       long int if the UOK flag is set.)

    2. If the variable is a signed long int, then that value is used.
       The variable is considered to be a signed long int if the
       IOK flag is set. (In the case of perls built with
       -Duse64bitint, the variable is treated as a signed long long
       int if the IOK flag is set.)

    3. If the variable is a double, then that value is used. The
       variable is considered to be a double if the NOK flag is set.
       (In the case of perls built with -Duselongdouble, the variable
       is treated as a long double if the NOK flag is set.)

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

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

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

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

   FORMATTED OUTPUT

   NOTE: When using the 'P' (precision) type specifier, instead of
         providing $prec to the 'P' specifier, it's now advisable
         to provide prec_cast($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).
    If you prefer to have the resultant string returned (rather than
    stored in $buffer), use Rmpfrf_sprintf_ret instead.
    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

   $string = Rmpfr_sprintf_ret($format_string, [$rnd,] $var, $buflen);

    As for Rmpfr_sprintf, but returns the formatted string, rather than
    storing it in $buffer.  $buflen must specify a size (characters)
    that is at least large enough to accommodate the formatted string
    (including the terminating NULL).
    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).
    If you prefer to have the resultant string returned (rather than
    stored in $buffer), use Rmpfrf_sprintf_ret instead. 
    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

   $string = Rmpfr_snprintf_ret($bytes, $format_string, [$rnd,] $var, $buflen);

    As for Rmpfr_snprintf, but returns the formatted string, rather than
    storing it in $buffer.  $buflen must specify a size (characters) that
    is at least large enough to accommodate the formatted string
    (including the terminating NULL).
    See the mpfr documentation for details re the formatting options:
    http://www.mpfr.org/mpfr-current/mpfr.html#Formatted-Output-Functions

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

   BASE CONVERSIONS

   $DBL_DIG  = MPFR_DBL_DIG;  # Will be 0 if float.h doesn't define
                              # DBL_DIG.

   $LDBL_DIG = MPFR_LDBL_DIG; # Will be 0 if float.h doesn't define
                              # LDBL_DIG.

   $min_prec = mpfr_min_inter_prec($orig_base, $orig_length, $to_base);
   $max_len  = mpfr_max_orig_len($orig_base, $to_base, $to_prec);
   $min_base = mpfr_min_inter_base($orig_base, $orig_length, $to_prec);
   $max_base = mpfr_max_orig_base($orig_length, $to_base, $to_prec);

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

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

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

   In our example case that becomes:

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

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

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

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

   For this second example that becomes:

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

   which will set $max_len to 15.

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

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

   The last 2 of the above subroutines (ie mpfr_min_inter_base and
   mpfr_max_orig_base) are provided mainly for completeness.
   Normally, there wouldn't be a need to use these last 2 forms ... but
   who knows ...

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

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

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

BUGS

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

ACKNOWLEDGEMENTS

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

LICENSE

    This program is free software; you may redistribute it and/or 
    modify it under the same terms as Perl itself.
    Copyright 2006-2013 Sisyphus

AUTHOR

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



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