Provided by: libmath-bigint-perl_1.999811-1_all bug

NAME

       Math::BigInt - Arbitrary size integer/float math package

SYNOPSIS

         use Math::BigInt;

         # or make it faster with huge numbers: install (optional)
         # Math::BigInt::GMP and always use (it falls back to
         # pure Perl if the GMP library is not installed):
         # (See also the L<MATH LIBRARY> section!)

         # warns if Math::BigInt::GMP cannot be found
         use Math::BigInt lib => 'GMP';

         # to suppress the warning use this:
         # use Math::BigInt try => 'GMP';

         # dies if GMP cannot be loaded:
         # use Math::BigInt only => 'GMP';

         my $str = '1234567890';
         my @values = (64, 74, 18);
         my $n = 1; my $sign = '-';

         # Configuration methods (may be used as class methods and instance methods)

         Math::BigInt->accuracy();     # get class accuracy
         Math::BigInt->accuracy($n);   # set class accuracy
         Math::BigInt->precision();    # get class precision
         Math::BigInt->precision($n);  # set class precision
         Math::BigInt->round_mode();   # get class rounding mode
         Math::BigInt->round_mode($m); # set global round mode, must be one of
                                       # 'even', 'odd', '+inf', '-inf', 'zero',
                                       # 'trunc', or 'common'
         Math::BigInt->config();       # return hash with configuration

         # Constructor methods (when the class methods below are used as instance
         # methods, the value is assigned the invocand)

         $x = Math::BigInt->new($str);         # defaults to 0
         $x = Math::BigInt->new('0x123');      # from hexadecimal
         $x = Math::BigInt->new('0b101');      # from binary
         $x = Math::BigInt->from_hex('cafe');  # from hexadecimal
         $x = Math::BigInt->from_oct('377');   # from octal
         $x = Math::BigInt->from_bin('1101');  # from binary
         $x = Math::BigInt->bzero();           # create a +0
         $x = Math::BigInt->bone();            # create a +1
         $x = Math::BigInt->bone('-');         # create a -1
         $x = Math::BigInt->binf();            # create a +inf
         $x = Math::BigInt->binf('-');         # create a -inf
         $x = Math::BigInt->bnan();            # create a Not-A-Number
         $x = Math::BigInt->bpi();             # returns pi

         $y = $x->copy();         # make a copy (unlike $y = $x)
         $y = $x->as_int();       # return as a Math::BigInt

         # Boolean methods (these don't modify the invocand)

         $x->is_zero();          # if $x is 0
         $x->is_one();           # if $x is +1
         $x->is_one("+");        # ditto
         $x->is_one("-");        # if $x is -1
         $x->is_inf();           # if $x is +inf or -inf
         $x->is_inf("+");        # if $x is +inf
         $x->is_inf("-");        # if $x is -inf
         $x->is_nan();           # if $x is NaN

         $x->is_positive();      # if $x > 0
         $x->is_pos();           # ditto
         $x->is_negative();      # if $x < 0
         $x->is_neg();           # ditto

         $x->is_odd();           # if $x is odd
         $x->is_even();          # if $x is even
         $x->is_int();           # if $x is an integer

         # Comparison methods

         $x->bcmp($y);           # compare numbers (undef, < 0, == 0, > 0)
         $x->bacmp($y);          # compare absolutely (undef, < 0, == 0, > 0)
         $x->beq($y);            # true if and only if $x == $y
         $x->bne($y);            # true if and only if $x != $y
         $x->blt($y);            # true if and only if $x < $y
         $x->ble($y);            # true if and only if $x <= $y
         $x->bgt($y);            # true if and only if $x > $y
         $x->bge($y);            # true if and only if $x >= $y

         # Arithmetic methods

         $x->bneg();             # negation
         $x->babs();             # absolute value
         $x->bsgn();             # sign function (-1, 0, 1, or NaN)
         $x->bnorm();            # normalize (no-op)
         $x->binc();             # increment $x by 1
         $x->bdec();             # decrement $x by 1
         $x->badd($y);           # addition (add $y to $x)
         $x->bsub($y);           # subtraction (subtract $y from $x)
         $x->bmul($y);           # multiplication (multiply $x by $y)
         $x->bmuladd($y,$z);     # $x = $x * $y + $z
         $x->bdiv($y);           # division (floored), set $x to quotient
                                 # return (quo,rem) or quo if scalar
         $x->btdiv($y);          # division (truncated), set $x to quotient
                                 # return (quo,rem) or quo if scalar
         $x->bmod($y);           # modulus (x % y)
         $x->btmod($y);          # modulus (truncated)
         $x->bmodinv($mod);      # modular multiplicative inverse
         $x->bmodpow($y,$mod);   # modular exponentiation (($x ** $y) % $mod)
         $x->bpow($y);           # power of arguments (x ** y)
         $x->blog();             # logarithm of $x to base e (Euler's number)
         $x->blog($base);        # logarithm of $x to base $base (e.g., base 2)
         $x->bexp();             # calculate e ** $x where e is Euler's number
         $x->bnok($y);           # x over y (binomial coefficient n over k)
         $x->bsin();             # sine
         $x->bcos();             # cosine
         $x->batan();            # inverse tangent
         $x->batan2($y);         # two-argument inverse tangent
         $x->bsqrt();            # calculate square-root
         $x->broot($y);          # $y'th root of $x (e.g. $y == 3 => cubic root)
         $x->bfac();             # factorial of $x (1*2*3*4*..$x)

         $x->blsft($n);          # left shift $n places in base 2
         $x->blsft($n,$b);       # left shift $n places in base $b
                                 # returns (quo,rem) or quo (scalar context)
         $x->brsft($n);          # right shift $n places in base 2
         $x->brsft($n,$b);       # right shift $n places in base $b
                                 # returns (quo,rem) or quo (scalar context)

         # Bitwise methods

         $x->band($y);           # bitwise and
         $x->bior($y);           # bitwise inclusive or
         $x->bxor($y);           # bitwise exclusive or
         $x->bnot();             # bitwise not (two's complement)

         # Rounding methods
         $x->round($A,$P,$mode); # round to accuracy or precision using
                                 # rounding mode $mode
         $x->bround($n);         # accuracy: preserve $n digits
         $x->bfround($n);        # $n > 0: round to $nth digit left of dec. point
                                 # $n < 0: round to $nth digit right of dec. point
         $x->bfloor();           # round towards minus infinity
         $x->bceil();            # round towards plus infinity
         $x->bint();             # round towards zero

         # Other mathematical methods

         $x->bgcd($y);            # greatest common divisor
         $x->blcm($y);            # least common multiple

         # Object property methods (do not modify the invocand)

         $x->sign();              # the sign, either +, - or NaN
         $x->digit($n);           # the nth digit, counting from the right
         $x->digit(-$n);          # the nth digit, counting from the left
         $x->length();            # return number of digits in number
         ($xl,$f) = $x->length(); # length of number and length of fraction
                                  # part, latter is always 0 digits long
                                  # for Math::BigInt objects
         $x->mantissa();          # return (signed) mantissa as a Math::BigInt
         $x->exponent();          # return exponent as a Math::BigInt
         $x->parts();             # return (mantissa,exponent) as a Math::BigInt
         $x->sparts();            # mantissa and exponent (as integers)
         $x->nparts();            # mantissa and exponent (normalised)
         $x->eparts();            # mantissa and exponent (engineering notation)
         $x->dparts();            # integer and fraction part

         # Conversion methods (do not modify the invocand)

         $x->bstr();         # decimal notation, possibly zero padded
         $x->bsstr();        # string in scientific notation with integers
         $x->bnstr();        # string in normalized notation
         $x->bestr();        # string in engineering notation
         $x->bdstr();        # string in decimal notation

         $x->to_hex();       # as signed hexadecimal string
         $x->to_bin();       # as signed binary string
         $x->to_oct();       # as signed octal string
         $x->to_bytes();     # as byte string

         $x->as_hex();       # as signed hexadecimal string with prefixed 0x
         $x->as_bin();       # as signed binary string with prefixed 0b
         $x->as_oct();       # as signed octal string with prefixed 0

         # Other conversion methods

         $x->numify();           # return as scalar (might overflow or underflow)

DESCRIPTION

       Math::BigInt provides support for arbitrary precision integers. Overloading is also
       provided for Perl operators.

   Input
       Input values to these routines may be any scalar number or string that looks like a number
       and represents an integer.

       •   Leading and trailing whitespace is ignored.

       •   Leading and trailing zeros are ignored.

       •   If the string has a "0x" prefix, it is interpreted as a hexadecimal number.

       •   If the string has a "0b" prefix, it is interpreted as a binary number.

       •   One underline is allowed between any two digits.

       •   If the string can not be interpreted, NaN is returned.

       Octal numbers are typically prefixed by "0", but since leading zeros are stripped, these
       methods can not automatically recognize octal numbers, so use the constructor from_oct()
       to interpret octal strings.

       Some examples of valid string input

           Input string                Resulting value
           123                         123
           1.23e2                      123
           12300e-2                    123
           0xcafe                      51966
           0b1101                      13
           67_538_754                  67538754
           -4_5_6.7_8_9e+0_1_0         -4567890000000

       Input given as scalar numbers might lose precision. Quote your input to ensure that no
       digits are lost:

           $x = Math::BigInt->new( 56789012345678901234 );   # bad
           $x = Math::BigInt->new('56789012345678901234');   # good

       Currently, Math::BigInt->new() defaults to 0, while Math::BigInt->new('') results in
       'NaN'. This might change in the future, so use always the following explicit forms to get
       a zero or NaN:

           $zero = Math::BigInt->bzero();
           $nan  = Math::BigInt->bnan();

   Output
       Output values are usually Math::BigInt objects.

       Boolean operators "is_zero()", "is_one()", "is_inf()", etc. return true or false.

       Comparison operators "bcmp()" and "bacmp()") return -1, 0, 1, or undef.

METHODS

   Configuration methods
       Each of the methods below (except config(), accuracy() and precision()) accepts three
       additional parameters. These arguments $A, $P and $R are "accuracy", "precision" and
       "round_mode". Please see the section about "ACCURACY and PRECISION" for more information.

       Setting a class variable effects all object instance that are created afterwards.

       accuracy()
               Math::BigInt->accuracy(5);      # set class accuracy
               $x->accuracy(5);                # set instance accuracy

               $A = Math::BigInt->accuracy();  # get class accuracy
               $A = $x->accuracy();            # get instance accuracy

           Set or get the accuracy, i.e., the number of significant digits. The accuracy must be
           an integer. If the accuracy is set to "undef", no rounding is done.

           Alternatively, one can round the results explicitly using one of "round()", "bround()"
           or "bfround()" or by passing the desired accuracy to the method as an additional
           parameter:

               my $x = Math::BigInt->new(30000);
               my $y = Math::BigInt->new(7);
               print scalar $x->copy()->bdiv($y, 2);               # prints 4300
               print scalar $x->copy()->bdiv($y)->bround(2);       # prints 4300

           Please see the section about "ACCURACY and PRECISION" for further details.

               $y = Math::BigInt->new(1234567);    # $y is not rounded
               Math::BigInt->accuracy(4);          # set class accuracy to 4
               $x = Math::BigInt->new(1234567);    # $x is rounded automatically
               print "$x $y";                      # prints "1235000 1234567"

               print $x->accuracy();       # prints "4"
               print $y->accuracy();       # also prints "4", since
                                           #   class accuracy is 4

               Math::BigInt->accuracy(5);  # set class accuracy to 5
               print $x->accuracy();       # prints "4", since instance
                                           #   accuracy is 4
               print $y->accuracy();       # prints "5", since no instance
                                           #   accuracy, and class accuracy is 5

           Note: Each class has it's own globals separated from Math::BigInt, but it is possible
           to subclass Math::BigInt and make the globals of the subclass aliases to the ones from
           Math::BigInt.

       precision()
               Math::BigInt->precision(-2);     # set class precision
               $x->precision(-2);               # set instance precision

               $P = Math::BigInt->precision();  # get class precision
               $P = $x->precision();            # get instance precision

           Set or get the precision, i.e., the place to round relative to the decimal point. The
           precision must be a integer. Setting the precision to $P means that each number is
           rounded up or down, depending on the rounding mode, to the nearest multiple of 10**$P.
           If the precision is set to "undef", no rounding is done.

           You might want to use "accuracy()" instead. With "accuracy()" you set the number of
           digits each result should have, with "precision()" you set the place where to round.

           Please see the section about "ACCURACY and PRECISION" for further details.

               $y = Math::BigInt->new(1234567);    # $y is not rounded
               Math::BigInt->precision(4);         # set class precision to 4
               $x = Math::BigInt->new(1234567);    # $x is rounded automatically
               print $x;                           # prints "1230000"

           Note: Each class has its own globals separated from Math::BigInt, but it is possible
           to subclass Math::BigInt and make the globals of the subclass aliases to the ones from
           Math::BigInt.

       div_scale()
           Set/get the fallback accuracy. This is the accuracy used when neither accuracy nor
           precision is set explicitly. It is used when a computation might otherwise attempt to
           return an infinite number of digits.

       round_mode()
           Set/get the rounding mode.

       upgrade()
           Set/get the class for upgrading. When a computation might result in a non-integer, the
           operands are upgraded to this class. This is used for instance by bignum. The default
           is "undef", thus the following operation creates a Math::BigInt, not a Math::BigFloat:

               my $i = Math::BigInt->new(123);
               my $f = Math::BigFloat->new('123.1');

               print $i + $f, "\n";                # prints 246

       downgrade()
           Set/get the class for downgrading. The default is "undef". Downgrading is not done by
           Math::BigInt.

       modify()
               $x->modify('bpowd');

           This method returns 0 if the object can be modified with the given operation, or 1 if
           not.

           This is used for instance by Math::BigInt::Constant.

       config()
               use Data::Dumper;

               print Dumper ( Math::BigInt->config() );
               print Math::BigInt->config()->{lib},"\n";
               print Math::BigInt->config('lib')},"\n";

           Returns a hash containing the configuration, e.g. the version number, lib loaded etc.
           The following hash keys are currently filled in with the appropriate information.

               key           Description
                             Example
               ============================================================
               lib           Name of the low-level math library
                             Math::BigInt::Calc
               lib_version   Version of low-level math library (see 'lib')
                             0.30
               class         The class name of config() you just called
                             Math::BigInt
               upgrade       To which class math operations might be
                             upgraded Math::BigFloat
               downgrade     To which class math operations might be
                             downgraded undef
               precision     Global precision
                             undef
               accuracy      Global accuracy
                             undef
               round_mode    Global round mode
                             even
               version       version number of the class you used
                             1.61
               div_scale     Fallback accuracy for div
                             40
               trap_nan      If true, traps creation of NaN via croak()
                             1
               trap_inf      If true, traps creation of +inf/-inf via croak()
                             1

           The following values can be set by passing "config()" a reference to a hash:

                   accuracy precision round_mode div_scale
                   upgrade downgrade trap_inf trap_nan

           Example:

               $new_cfg = Math::BigInt->config(
                   { trap_inf => 1, precision => 5 }
               );

   Constructor methods
       new()
               $x = Math::BigInt->new($str,$A,$P,$R);

           Creates a new Math::BigInt object from a scalar or another Math::BigInt object.  The
           input is accepted as decimal, hexadecimal (with leading '0x') or binary (with leading
           '0b').

           See "Input" for more info on accepted input formats.

       from_hex()
               $x = Math::BigInt->from_hex("0xcafe");    # input is hexadecimal

           Interpret input as a hexadecimal string. A "0x" or "x" prefix is optional. A single
           underscore character may be placed right after the prefix, if present, or between any
           two digits. If the input is invalid, a NaN is returned.

       from_oct()
               $x = Math::BigInt->from_oct("0775");      # input is octal

           Interpret the input as an octal string and return the corresponding value. A "0"
           (zero) prefix is optional. A single underscore character may be placed right after the
           prefix, if present, or between any two digits. If the input is invalid, a NaN is
           returned.

       from_bin()
               $x = Math::BigInt->from_bin("0b10011");   # input is binary

           Interpret the input as a binary string. A "0b" or "b" prefix is optional. A single
           underscore character may be placed right after the prefix, if present, or between any
           two digits. If the input is invalid, a NaN is returned.

       from_bytes()
               $x = Math::BigInt->from_bytes("\xf3\x6b");  # $x = 62315

           Interpret the input as a byte string, assuming big endian byte order. The output is
           always a non-negative, finite integer.

           In some special cases, from_bytes() matches the conversion done by unpack():

               $b = "\x4e";                             # one char byte string
               $x = Math::BigInt->from_bytes($b);       # = 78
               $y = unpack "C", $b;                     # ditto, but scalar

               $b = "\xf3\x6b";                         # two char byte string
               $x = Math::BigInt->from_bytes($b);       # = 62315
               $y = unpack "S>", $b;                    # ditto, but scalar

               $b = "\x2d\xe0\x49\xad";                 # four char byte string
               $x = Math::BigInt->from_bytes($b);       # = 769673645
               $y = unpack "L>", $b;                    # ditto, but scalar

               $b = "\x2d\xe0\x49\xad\x2d\xe0\x49\xad"; # eight char byte string
               $x = Math::BigInt->from_bytes($b);       # = 3305723134637787565
               $y = unpack "Q>", $b;                    # ditto, but scalar

       bzero()
               $x = Math::BigInt->bzero();
               $x->bzero();

           Returns a new Math::BigInt object representing zero. If used as an instance method,
           assigns the value to the invocand.

       bone()
               $x = Math::BigInt->bone();          # +1
               $x = Math::BigInt->bone("+");       # +1
               $x = Math::BigInt->bone("-");       # -1
               $x->bone();                         # +1
               $x->bone("+");                      # +1
               $x->bone('-');                      # -1

           Creates a new Math::BigInt object representing one. The optional argument is either
           '-' or '+', indicating whether you want plus one or minus one. If used as an instance
           method, assigns the value to the invocand.

       binf()
               $x = Math::BigInt->binf($sign);

           Creates a new Math::BigInt object representing infinity. The optional argument is
           either '-' or '+', indicating whether you want infinity or minus infinity.  If used as
           an instance method, assigns the value to the invocand.

               $x->binf();
               $x->binf('-');

       bnan()
               $x = Math::BigInt->bnan();

           Creates a new Math::BigInt object representing NaN (Not A Number). If used as an
           instance method, assigns the value to the invocand.

               $x->bnan();

       bpi()
               $x = Math::BigInt->bpi(100);        # 3
               $x->bpi(100);                       # 3

           Creates a new Math::BigInt object representing PI. If used as an instance method,
           assigns the value to the invocand. With Math::BigInt this always returns 3.

           If upgrading is in effect, returns PI, rounded to N digits with the current rounding
           mode:

               use Math::BigFloat;
               use Math::BigInt upgrade => "Math::BigFloat";
               print Math::BigInt->bpi(3), "\n";           # 3.14
               print Math::BigInt->bpi(100), "\n";         # 3.1415....

       copy()
               $x->copy();         # make a true copy of $x (unlike $y = $x)

       as_int()
       as_number()
           These methods are called when Math::BigInt encounters an object it doesn't know how to
           handle. For instance, assume $x is a Math::BigInt, or subclass thereof, and $y is
           defined, but not a Math::BigInt, or subclass thereof. If you do

               $x -> badd($y);

           $y needs to be converted into an object that $x can deal with. This is done by first
           checking if $y is something that $x might be upgraded to. If that is the case, no
           further attempts are made. The next is to see if $y supports the method "as_int()". If
           it does, "as_int()" is called, but if it doesn't, the next thing is to see if $y
           supports the method "as_number()". If it does, "as_number()" is called. The method
           "as_int()" (and "as_number()") is expected to return either an object that has the
           same class as $x, a subclass thereof, or a string that "ref($x)->new()" can parse to
           create an object.

           "as_number()" is an alias to "as_int()". "as_number" was introduced in v1.22, while
           "as_int()" was introduced in v1.68.

           In Math::BigInt, "as_int()" has the same effect as "copy()".

   Boolean methods
       None of these methods modify the invocand object.

       is_zero()
               $x->is_zero();              # true if $x is 0

           Returns true if the invocand is zero and false otherwise.

       is_one( [ SIGN ])
               $x->is_one();               # true if $x is +1
               $x->is_one("+");            # ditto
               $x->is_one("-");            # true if $x is -1

           Returns true if the invocand is one and false otherwise.

       is_finite()
               $x->is_finite();    # true if $x is not +inf, -inf or NaN

           Returns true if the invocand is a finite number, i.e., it is neither +inf, -inf, nor
           NaN.

       is_inf( [ SIGN ] )
               $x->is_inf();               # true if $x is +inf
               $x->is_inf("+");            # ditto
               $x->is_inf("-");            # true if $x is -inf

           Returns true if the invocand is infinite and false otherwise.

       is_nan()
               $x->is_nan();               # true if $x is NaN

       is_positive()
       is_pos()
               $x->is_positive();          # true if > 0
               $x->is_pos();               # ditto

           Returns true if the invocand is positive and false otherwise. A "NaN" is neither
           positive nor negative.

       is_negative()
       is_neg()
               $x->is_negative();          # true if < 0
               $x->is_neg();               # ditto

           Returns true if the invocand is negative and false otherwise. A "NaN" is neither
           positive nor negative.

       is_odd()
               $x->is_odd();               # true if odd, false for even

           Returns true if the invocand is odd and false otherwise. "NaN", "+inf", and "-inf" are
           neither odd nor even.

       is_even()
               $x->is_even();              # true if $x is even

           Returns true if the invocand is even and false otherwise. "NaN", "+inf", "-inf" are
           not integers and are neither odd nor even.

       is_int()
               $x->is_int();               # true if $x is an integer

           Returns true if the invocand is an integer and false otherwise. "NaN", "+inf", "-inf"
           are not integers.

   Comparison methods
       None of these methods modify the invocand object. Note that a "NaN" is neither less than,
       greater than, or equal to anything else, even a "NaN".

       bcmp()
               $x->bcmp($y);

           Returns -1, 0, 1 depending on whether $x is less than, equal to, or grater than $y.
           Returns undef if any operand is a NaN.

       bacmp()
               $x->bacmp($y);

           Returns -1, 0, 1 depending on whether the absolute value of $x is less than, equal to,
           or grater than the absolute value of $y. Returns undef if any operand is a NaN.

       beq()
               $x -> beq($y);

           Returns true if and only if $x is equal to $y, and false otherwise.

       bne()
               $x -> bne($y);

           Returns true if and only if $x is not equal to $y, and false otherwise.

       blt()
               $x -> blt($y);

           Returns true if and only if $x is equal to $y, and false otherwise.

       ble()
               $x -> ble($y);

           Returns true if and only if $x is less than or equal to $y, and false otherwise.

       bgt()
               $x -> bgt($y);

           Returns true if and only if $x is greater than $y, and false otherwise.

       bge()
               $x -> bge($y);

           Returns true if and only if $x is greater than or equal to $y, and false otherwise.

   Arithmetic methods
       These methods modify the invocand object and returns it.

       bneg()
               $x->bneg();

           Negate the number, e.g. change the sign between '+' and '-', or between '+inf' and
           '-inf', respectively. Does nothing for NaN or zero.

       babs()
               $x->babs();

           Set the number to its absolute value, e.g. change the sign from '-' to '+' and from
           '-inf' to '+inf', respectively. Does nothing for NaN or positive numbers.

       bsgn()
               $x->bsgn();

           Signum function. Set the number to -1, 0, or 1, depending on whether the number is
           negative, zero, or positive, respectively. Does not modify NaNs.

       bnorm()
               $x->bnorm();                        # normalize (no-op)

           Normalize the number. This is a no-op and is provided only for backwards
           compatibility.

       binc()
               $x->binc();                 # increment x by 1

       bdec()
               $x->bdec();                 # decrement x by 1

       badd()
               $x->badd($y);               # addition (add $y to $x)

       bsub()
               $x->bsub($y);               # subtraction (subtract $y from $x)

       bmul()
               $x->bmul($y);               # multiplication (multiply $x by $y)

       bmuladd()
               $x->bmuladd($y,$z);

           Multiply $x by $y, and then add $z to the result,

           This method was added in v1.87 of Math::BigInt (June 2007).

       bdiv()
               $x->bdiv($y);               # divide, set $x to quotient

           Divides $x by $y by doing floored division (F-division), where the quotient is the
           floored (rounded towards negative infinity) quotient of the two operands.  In list
           context, returns the quotient and the remainder. The remainder is either zero or has
           the same sign as the second operand. In scalar context, only the quotient is returned.

           The quotient is always the greatest integer less than or equal to the real-valued
           quotient of the two operands, and the remainder (when it is non-zero) always has the
           same sign as the second operand; so, for example,

                 1 /  4  => ( 0,  1)
                 1 / -4  => (-1, -3)
                -3 /  4  => (-1,  1)
                -3 / -4  => ( 0, -3)
               -11 /  2  => (-5,  1)
                11 / -2  => (-5, -1)

           The behavior of the overloaded operator % agrees with the behavior of Perl's built-in
           % operator (as documented in the perlop manpage), and the equation

               $x == ($x / $y) * $y + ($x % $y)

           holds true for any finite $x and finite, non-zero $y.

           Perl's "use integer" might change the behaviour of % and / for scalars. This is
           because under 'use integer' Perl does what the underlying C library thinks is right,
           and this varies. However, "use integer" does not change the way things are done with
           Math::BigInt objects.

       btdiv()
               $x->btdiv($y);              # divide, set $x to quotient

           Divides $x by $y by doing truncated division (T-division), where quotient is the
           truncated (rouneded towards zero) quotient of the two operands. In list context,
           returns the quotient and the remainder. The remainder is either zero or has the same
           sign as the first operand. In scalar context, only the quotient is returned.

       bmod()
               $x->bmod($y);               # modulus (x % y)

           Returns $x modulo $y, i.e., the remainder after floored division (F-division).  This
           method is like Perl's % operator. See "bdiv()".

       btmod()
               $x->btmod($y);              # modulus

           Returns the remainer after truncated division (T-division). See "btdiv()".

       bmodinv()
               $x->bmodinv($mod);          # modular multiplicative inverse

           Returns the multiplicative inverse of $x modulo $mod. If

               $y = $x -> copy() -> bmodinv($mod)

           then $y is the number closest to zero, and with the same sign as $mod, satisfying

               ($x * $y) % $mod = 1 % $mod

           If $x and $y are non-zero, they must be relative primes, i.e., "bgcd($y, $mod)==1".
           '"NaN"' is returned when no modular multiplicative inverse exists.

       bmodpow()
               $num->bmodpow($exp,$mod);           # modular exponentiation
                                                   # ($num**$exp % $mod)

           Returns the value of $num taken to the power $exp in the modulus $mod using binary
           exponentiation.  "bmodpow" is far superior to writing

               $num ** $exp % $mod

           because it is much faster - it reduces internal variables into the modulus whenever
           possible, so it operates on smaller numbers.

           "bmodpow" also supports negative exponents.

               bmodpow($num, -1, $mod)

           is exactly equivalent to

               bmodinv($num, $mod)

       bpow()
               $x->bpow($y);               # power of arguments (x ** y)

           "bpow()" (and the rounding functions) now modifies the first argument and returns it,
           unlike the old code which left it alone and only returned the result. This is to be
           consistent with "badd()" etc. The first three modifies $x, the last one won't:

               print bpow($x,$i),"\n";         # modify $x
               print $x->bpow($i),"\n";        # ditto
               print $x **= $i,"\n";           # the same
               print $x ** $i,"\n";            # leave $x alone

           The form "$x **= $y" is faster than "$x = $x ** $y;", though.

       blog()
               $x->blog($base, $accuracy);         # logarithm of x to the base $base

           If $base is not defined, Euler's number (e) is used:

               print $x->blog(undef, 100);         # log(x) to 100 digits

       bexp()
               $x->bexp($accuracy);                # calculate e ** X

           Calculates the expression "e ** $x" where "e" is Euler's number.

           This method was added in v1.82 of Math::BigInt (April 2007).

           See also "blog()".

       bnok()
               $x->bnok($y);               # x over y (binomial coefficient n over k)

           Calculates the binomial coefficient n over k, also called the "choose" function. The
           result is equivalent to:

               ( n )      n!
               | - |  = -------
               ( k )    k!(n-k)!

           This method was added in v1.84 of Math::BigInt (April 2007).

       bsin()
               my $x = Math::BigInt->new(1);
               print $x->bsin(100), "\n";

           Calculate the sine of $x, modifying $x in place.

           In Math::BigInt, unless upgrading is in effect, the result is truncated to an integer.

           This method was added in v1.87 of Math::BigInt (June 2007).

       bcos()
               my $x = Math::BigInt->new(1);
               print $x->bcos(100), "\n";

           Calculate the cosine of $x, modifying $x in place.

           In Math::BigInt, unless upgrading is in effect, the result is truncated to an integer.

           This method was added in v1.87 of Math::BigInt (June 2007).

       batan()
               my $x = Math::BigFloat->new(0.5);
               print $x->batan(100), "\n";

           Calculate the arcus tangens of $x, modifying $x in place.

           In Math::BigInt, unless upgrading is in effect, the result is truncated to an integer.

           This method was added in v1.87 of Math::BigInt (June 2007).

       batan2()
               my $x = Math::BigInt->new(1);
               my $y = Math::BigInt->new(1);
               print $y->batan2($x), "\n";

           Calculate the arcus tangens of $y divided by $x, modifying $y in place.

           In Math::BigInt, unless upgrading is in effect, the result is truncated to an integer.

           This method was added in v1.87 of Math::BigInt (June 2007).

       bsqrt()
               $x->bsqrt();                # calculate square-root

           "bsqrt()" returns the square root truncated to an integer.

           If you want a better approximation of the square root, then use:

               $x = Math::BigFloat->new(12);
               Math::BigFloat->precision(0);
               Math::BigFloat->round_mode('even');
               print $x->copy->bsqrt(),"\n";           # 4

               Math::BigFloat->precision(2);
               print $x->bsqrt(),"\n";                 # 3.46
               print $x->bsqrt(3),"\n";                # 3.464

       broot()
               $x->broot($N);

           Calculates the N'th root of $x.

       bfac()
               $x->bfac();                 # factorial of $x (1*2*3*4*..*$x)

           Returns the factorial of $x, i.e., the product of all positive integers up to and
           including $x.

       bdfac()
               $x->bdfac();                # double factorial of $x (1*2*3*4*..*$x)

           Returns the double factorial of $x. If $x is an even integer, returns the product of
           all positive, even integers up to and including $x, i.e., 2*4*6*...*$x. If $x is an
           odd integer, returns the product of all positive, odd integers, i.e., 1*3*5*...*$x.

       bfib()
               $F = $n->bfib();            # a single Fibonacci number
               @F = $n->bfib();            # a list of Fibonacci numbers

           In scalar context, returns a single Fibonacci number. In list context, returns a list
           of Fibonacci numbers. The invocand is the last element in the output.

           The Fibonacci sequence is defined by

               F(0) = 0
               F(1) = 1
               F(n) = F(n-1) + F(n-2)

           In list context, F(0) and F(n) is the first and last number in the output,
           respectively. For example, if $n is 12, then "@F = $n->bfib()" returns the following
           values, F(0) to F(12):

               0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144

           The sequence can also be extended to negative index n using the re-arranged recurrence
           relation

               F(n-2) = F(n) - F(n-1)

           giving the bidirectional sequence

                  n  -7  -6  -5  -4  -3  -2  -1   0   1   2   3   4   5   6   7
               F(n)  13  -8   5  -3   2  -1   1   0   1   1   2   3   5   8  13

           If $n is -12, the following values, F(0) to F(12), are returned:

               0, 1, -1, 2, -3, 5, -8, 13, -21, 34, -55, 89, -144

       blucas()
               $F = $n->blucas();          # a single Lucas number
               @F = $n->blucas();          # a list of Lucas numbers

           In scalar context, returns a single Lucas number. In list context, returns a list of
           Lucas numbers. The invocand is the last element in the output.

           The Lucas sequence is defined by

               L(0) = 2
               L(1) = 1
               L(n) = L(n-1) + L(n-2)

           In list context, L(0) and L(n) is the first and last number in the output,
           respectively. For example, if $n is 12, then "@L = $n->blucas()" returns the following
           values, L(0) to L(12):

               2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322

           The sequence can also be extended to negative index n using the re-arranged recurrence
           relation

               L(n-2) = L(n) - L(n-1)

           giving the bidirectional sequence

                  n  -7  -6  -5  -4  -3  -2  -1   0   1   2   3   4   5   6   7
               L(n)  29 -18  11  -7   4  -3   1   2   1   3   4   7  11  18  29

           If $n is -12, the following values, L(0) to L(-12), are returned:

               2, 1, -3, 4, -7, 11, -18, 29, -47, 76, -123, 199, -322

       brsft()
               $x->brsft($n);              # right shift $n places in base 2
               $x->brsft($n, $b);          # right shift $n places in base $b

           The latter is equivalent to

               $x -> bdiv($b -> copy() -> bpow($n))

       blsft()
               $x->blsft($n);              # left shift $n places in base 2
               $x->blsft($n, $b);          # left shift $n places in base $b

           The latter is equivalent to

               $x -> bmul($b -> copy() -> bpow($n))

   Bitwise methods
       band()
               $x->band($y);               # bitwise and

       bior()
               $x->bior($y);               # bitwise inclusive or

       bxor()
               $x->bxor($y);               # bitwise exclusive or

       bnot()
               $x->bnot();                 # bitwise not (two's complement)

           Two's complement (bitwise not). This is equivalent to, but faster than,

               $x->binc()->bneg();

   Rounding methods
       round()
               $x->round($A,$P,$round_mode);

           Round $x to accuracy $A or precision $P using the round mode $round_mode.

       bround()
               $x->bround($N);               # accuracy: preserve $N digits

           Rounds $x to an accuracy of $N digits.

       bfround()
               $x->bfround($N);

           Rounds to a multiple of 10**$N. Examples:

               Input            N          Result

               123456.123456    3          123500
               123456.123456    2          123450
               123456.123456   -2          123456.12
               123456.123456   -3          123456.123

       bfloor()
               $x->bfloor();

           Round $x towards minus infinity, i.e., set $x to the largest integer less than or
           equal to $x.

       bceil()
               $x->bceil();

           Round $x towards plus infinity, i.e., set $x to the smallest integer greater than or
           equal to $x).

       bint()
               $x->bint();

           Round $x towards zero.

   Other mathematical methods
       bgcd()
               $x -> bgcd($y);             # GCD of $x and $y
               $x -> bgcd($y, $z, ...);    # GCD of $x, $y, $z, ...

           Returns the greatest common divisor (GCD).

       blcm()
               $x -> blcm($y);             # LCM of $x and $y
               $x -> blcm($y, $z, ...);    # LCM of $x, $y, $z, ...

           Returns the least common multiple (LCM).

   Object property methods
       sign()
               $x->sign();

           Return the sign, of $x, meaning either "+", "-", "-inf", "+inf" or NaN.

           If you want $x to have a certain sign, use one of the following methods:

               $x->babs();                 # '+'
               $x->babs()->bneg();         # '-'
               $x->bnan();                 # 'NaN'
               $x->binf();                 # '+inf'
               $x->binf('-');              # '-inf'

       digit()
               $x->digit($n);       # return the nth digit, counting from right

           If $n is negative, returns the digit counting from left.

       length()
               $x->length();
               ($xl, $fl) = $x->length();

           Returns the number of digits in the decimal representation of the number. In list
           context, returns the length of the integer and fraction part. For Math::BigInt
           objects, the length of the fraction part is always 0.

           The following probably doesn't do what you expect:

               $c = Math::BigInt->new(123);
               print $c->length(),"\n";                # prints 30

           It prints both the number of digits in the number and in the fraction part since print
           calls "length()" in list context. Use something like:

               print scalar $c->length(),"\n";         # prints 3

       mantissa()
               $x->mantissa();

           Return the signed mantissa of $x as a Math::BigInt.

       exponent()
               $x->exponent();

           Return the exponent of $x as a Math::BigInt.

       parts()
               $x->parts();

           Returns the significand (mantissa) and the exponent as integers. In Math::BigFloat,
           both are returned as Math::BigInt objects.

       sparts()
           Returns the significand (mantissa) and the exponent as integers. In scalar context,
           only the significand is returned. The significand is the integer with the smallest
           absolute value. The output of "sparts()" corresponds to the output from "bsstr()".

           In Math::BigInt, this method is identical to "parts()".

       nparts()
           Returns the significand (mantissa) and exponent corresponding to normalized notation.
           In scalar context, only the significand is returned. For finite non-zero numbers, the
           significand's absolute value is greater than or equal to 1 and less than 10. The
           output of "nparts()" corresponds to the output from "bnstr()". In Math::BigInt, if the
           significand can not be represented as an integer, upgrading is performed or NaN is
           returned.

       eparts()
           Returns the significand (mantissa) and exponent corresponding to engineering notation.
           In scalar context, only the significand is returned. For finite non-zero numbers, the
           significand's absolute value is greater than or equal to 1 and less than 1000, and the
           exponent is a multiple of 3. The output of "eparts()" corresponds to the output from
           "bestr()". In Math::BigInt, if the significand can not be represented as an integer,
           upgrading is performed or NaN is returned.

       dparts()
           Returns the integer part and the fraction part. If the fraction part can not be
           represented as an integer, upgrading is performed or NaN is returned. The output of
           "dparts()" corresponds to the output from "bdstr()".

   String conversion methods
       bstr()
           Returns a string representing the number using decimal notation. In Math::BigFloat,
           the output is zero padded according to the current accuracy or precision, if any of
           those are defined.

       bsstr()
           Returns a string representing the number using scientific notation where both the
           significand (mantissa) and the exponent are integers. The output corresponds to the
           output from "sparts()".

                 123 is returned as "123e+0"
                1230 is returned as "123e+1"
               12300 is returned as "123e+2"
               12000 is returned as "12e+3"
               10000 is returned as "1e+4"

       bnstr()
           Returns a string representing the number using normalized notation, the most common
           variant of scientific notation. For finite non-zero numbers, the absolute value of the
           significand is less than or equal to 1 and less than 10.  The output corresponds to
           the output from "nparts()".

                 123 is returned as "1.23e+2"
                1230 is returned as "1.23e+3"
               12300 is returned as "1.23e+4"
               12000 is returned as "1.2e+4"
               10000 is returned as "1e+4"

       bestr()
           Returns a string representing the number using engineering notation. For finite non-
           zero numbers, the absolute value of the significand is less than or equal to 1 and
           less than 1000, and the exponent is a multiple of 3. The output corresponds to the
           output from "eparts()".

                 123 is returned as "123e+0"
                1230 is returned as "1.23e+3"
               12300 is returned as "12.3e+3"
               12000 is returned as "12e+3"
               10000 is returned as "10e+3"

       bdstr()
           Returns a string representing the number using decimal notation. The output
           corresponds to the output from "dparts()".

                 123 is returned as "123"
                1230 is returned as "1230"
               12300 is returned as "12300"
               12000 is returned as "12000"
               10000 is returned as "10000"

       to_hex()
               $x->to_hex();

           Returns a hexadecimal string representation of the number.

       to_bin()
               $x->to_bin();

           Returns a binary string representation of the number.

       to_oct()
               $x->to_oct();

           Returns an octal string representation of the number.

       to_bytes()
               $x = Math::BigInt->new("1667327589");
               $s = $x->to_bytes();                    # $s = "cafe"

           Returns a byte string representation of the number using big endian byte order. The
           invocand must be a non-negative, finite integer.

       as_hex()
               $x->as_hex();

           As, "to_hex()", but with a "0x" prefix.

       as_bin()
               $x->as_bin();

           As, "to_bin()", but with a "0b" prefix.

       as_oct()
               $x->as_oct();

           As, "to_oct()", but with a "0" prefix.

       as_bytes()
           This is just an alias for "to_bytes()".

   Other conversion methods
       numify()
               print $x->numify();

           Returns a Perl scalar from $x. It is used automatically whenever a scalar is needed,
           for instance in array index operations.

ACCURACY and PRECISION

       Math::BigInt and Math::BigFloat have full support for accuracy and precision based
       rounding, both automatically after every operation, as well as manually.

       This section describes the accuracy/precision handling in Math::BigInt and Math::BigFloat
       as it used to be and as it is now, complete with an explanation of all terms and
       abbreviations.

       Not yet implemented things (but with correct description) are marked with '!', things that
       need to be answered are marked with '?'.

       In the next paragraph follows a short description of terms used here (because these may
       differ from terms used by others people or documentation).

       During the rest of this document, the shortcuts A (for accuracy), P (for precision), F
       (fallback) and R (rounding mode) are be used.

   Precision P
       Precision is a fixed number of digits before (positive) or after (negative) the decimal
       point. For example, 123.45 has a precision of -2. 0 means an integer like 123 (or 120). A
       precision of 2 means at least two digits to the left of the decimal point are zero, so 123
       with P = 1 becomes 120. Note that numbers with zeros before the decimal point may have
       different precisions, because 1200 can have P = 0, 1 or 2 (depending on what the initial
       value was). It could also have p < 0, when the digits after the decimal point are zero.

       The string output (of floating point numbers) is padded with zeros:

           Initial value    P      A       Result          String
           ------------------------------------------------------------
           1234.01         -3              1000            1000
           1234            -2              1200            1200
           1234.5          -1              1230            1230
           1234.001         1              1234            1234.0
           1234.01          0              1234            1234
           1234.01          2              1234.01         1234.01
           1234.01          5              1234.01         1234.01000

       For Math::BigInt objects, no padding occurs.

   Accuracy A
       Number of significant digits. Leading zeros are not counted. A number may have an accuracy
       greater than the non-zero digits when there are zeros in it or trailing zeros. For
       example, 123.456 has A of 6, 10203 has 5, 123.0506 has 7, 123.45000 has 8 and 0.000123 has
       3.

       The string output (of floating point numbers) is padded with zeros:

           Initial value    P      A       Result          String
           ------------------------------------------------------------
           1234.01                 3       1230            1230
           1234.01                 6       1234.01         1234.01
           1234.1                  8       1234.1          1234.1000

       For Math::BigInt objects, no padding occurs.

   Fallback F
       When both A and P are undefined, this is used as a fallback accuracy when dividing
       numbers.

   Rounding mode R
       When rounding a number, different 'styles' or 'kinds' of rounding are possible.  (Note
       that random rounding, as in Math::Round, is not implemented.)

       'trunc'
           truncation invariably removes all digits following the rounding place, replacing them
           with zeros. Thus, 987.65 rounded to tens (P = 1) becomes 980, and rounded to the
           fourth sigdig becomes 987.6 (A = 4). 123.456 rounded to the second place after the
           decimal point (P = -2) becomes 123.46.

           All other implemented styles of rounding attempt to round to the "nearest digit." If
           the digit D immediately to the right of the rounding place (skipping the decimal
           point) is greater than 5, the number is incremented at the rounding place (possibly
           causing a cascade of incrementation): e.g. when rounding to units, 0.9 rounds to 1,
           and -19.9 rounds to -20. If D < 5, the number is similarly truncated at the rounding
           place: e.g. when rounding to units, 0.4 rounds to 0, and -19.4 rounds to -19.

           However the results of other styles of rounding differ if the digit immediately to the
           right of the rounding place (skipping the decimal point) is 5 and if there are no
           digits, or no digits other than 0, after that 5. In such cases:

       'even'
           rounds the digit at the rounding place to 0, 2, 4, 6, or 8 if it is not already. E.g.,
           when rounding to the first sigdig, 0.45 becomes 0.4, -0.55 becomes -0.6, but 0.4501
           becomes 0.5.

       'odd'
           rounds the digit at the rounding place to 1, 3, 5, 7, or 9 if it is not already. E.g.,
           when rounding to the first sigdig, 0.45 becomes 0.5, -0.55 becomes -0.5, but 0.5501
           becomes 0.6.

       '+inf'
           round to plus infinity, i.e. always round up. E.g., when rounding to the first sigdig,
           0.45 becomes 0.5, -0.55 becomes -0.5, and 0.4501 also becomes 0.5.

       '-inf'
           round to minus infinity, i.e. always round down. E.g., when rounding to the first
           sigdig, 0.45 becomes 0.4, -0.55 becomes -0.6, but 0.4501 becomes 0.5.

       'zero'
           round to zero, i.e. positive numbers down, negative ones up. E.g., when rounding to
           the first sigdig, 0.45 becomes 0.4, -0.55 becomes -0.5, but 0.4501 becomes 0.5.

       'common'
           round up if the digit immediately to the right of the rounding place is 5 or greater,
           otherwise round down. E.g., 0.15 becomes 0.2 and 0.149 becomes 0.1.

       The handling of A & P in MBI/MBF (the old core code shipped with Perl versions <= 5.7.2)
       is like this:

       Precision
             * bfround($p) is able to round to $p number of digits after the decimal
               point
             * otherwise P is unused

       Accuracy (significant digits)
             * bround($a) rounds to $a significant digits
             * only bdiv() and bsqrt() take A as (optional) parameter
               + other operations simply create the same number (bneg etc), or
                 more (bmul) of digits
               + rounding/truncating is only done when explicitly calling one
                 of bround or bfround, and never for Math::BigInt (not implemented)
             * bsqrt() simply hands its accuracy argument over to bdiv.
             * the documentation and the comment in the code indicate two
               different ways on how bdiv() determines the maximum number
               of digits it should calculate, and the actual code does yet
               another thing
               POD:
                 max($Math::BigFloat::div_scale,length(dividend)+length(divisor))
               Comment:
                 result has at most max(scale, length(dividend), length(divisor)) digits
               Actual code:
                 scale = max(scale, length(dividend)-1,length(divisor)-1);
                 scale += length(divisor) - length(dividend);
               So for lx = 3, ly = 9, scale = 10, scale will actually be 16 (10
               So for lx = 3, ly = 9, scale = 10, scale will actually be 16
               (10+9-3). Actually, the 'difference' added to the scale is cal-
               culated from the number of "significant digits" in dividend and
               divisor, which is derived by looking at the length of the man-
               tissa. Which is wrong, since it includes the + sign (oops) and
               actually gets 2 for '+100' and 4 for '+101'. Oops again. Thus
               124/3 with div_scale=1 will get you '41.3' based on the strange
               assumption that 124 has 3 significant digits, while 120/7 will
               get you '17', not '17.1' since 120 is thought to have 2 signif-
               icant digits. The rounding after the division then uses the
               remainder and $y to determine whether it must round up or down.
            ?  I have no idea which is the right way. That's why I used a slightly more
            ?  simple scheme and tweaked the few failing testcases to match it.

       This is how it works now:

       Setting/Accessing
             * You can set the A global via Math::BigInt->accuracy() or
               Math::BigFloat->accuracy() or whatever class you are using.
             * You can also set P globally by using Math::SomeClass->precision()
               likewise.
             * Globals are classwide, and not inherited by subclasses.
             * to undefine A, use Math::SomeCLass->accuracy(undef);
             * to undefine P, use Math::SomeClass->precision(undef);
             * Setting Math::SomeClass->accuracy() clears automatically
               Math::SomeClass->precision(), and vice versa.
             * To be valid, A must be > 0, P can have any value.
             * If P is negative, this means round to the P'th place to the right of the
               decimal point; positive values mean to the left of the decimal point.
               P of 0 means round to integer.
             * to find out the current global A, use Math::SomeClass->accuracy()
             * to find out the current global P, use Math::SomeClass->precision()
             * use $x->accuracy() respective $x->precision() for the local
               setting of $x.
             * Please note that $x->accuracy() respective $x->precision()
               return eventually defined global A or P, when $x's A or P is not
               set.

       Creating numbers
             * When you create a number, you can give the desired A or P via:
               $x = Math::BigInt->new($number,$A,$P);
             * Only one of A or P can be defined, otherwise the result is NaN
             * If no A or P is give ($x = Math::BigInt->new($number) form), then the
               globals (if set) will be used. Thus changing the global defaults later on
               will not change the A or P of previously created numbers (i.e., A and P of
               $x will be what was in effect when $x was created)
             * If given undef for A and P, NO rounding will occur, and the globals will
               NOT be used. This is used by subclasses to create numbers without
               suffering rounding in the parent. Thus a subclass is able to have its own
               globals enforced upon creation of a number by using
               $x = Math::BigInt->new($number,undef,undef):

                   use Math::BigInt::SomeSubclass;
                   use Math::BigInt;

                   Math::BigInt->accuracy(2);
                   Math::BigInt::SomeSubClass->accuracy(3);
                   $x = Math::BigInt::SomeSubClass->new(1234);

               $x is now 1230, and not 1200. A subclass might choose to implement
               this otherwise, e.g. falling back to the parent's A and P.

       Usage
             * If A or P are enabled/defined, they are used to round the result of each
               operation according to the rules below
             * Negative P is ignored in Math::BigInt, since Math::BigInt objects never
               have digits after the decimal point
             * Math::BigFloat uses Math::BigInt internally, but setting A or P inside
               Math::BigInt as globals does not tamper with the parts of a Math::BigFloat.
               A flag is used to mark all Math::BigFloat numbers as 'never round'.

       Precedence
             * It only makes sense that a number has only one of A or P at a time.
               If you set either A or P on one object, or globally, the other one will
               be automatically cleared.
             * If two objects are involved in an operation, and one of them has A in
               effect, and the other P, this results in an error (NaN).
             * A takes precedence over P (Hint: A comes before P).
               If neither of them is defined, nothing is used, i.e. the result will have
               as many digits as it can (with an exception for bdiv/bsqrt) and will not
               be rounded.
             * There is another setting for bdiv() (and thus for bsqrt()). If neither of
               A or P is defined, bdiv() will use a fallback (F) of $div_scale digits.
               If either the dividend's or the divisor's mantissa has more digits than
               the value of F, the higher value will be used instead of F.
               This is to limit the digits (A) of the result (just consider what would
               happen with unlimited A and P in the case of 1/3 :-)
             * bdiv will calculate (at least) 4 more digits than required (determined by
               A, P or F), and, if F is not used, round the result
               (this will still fail in the case of a result like 0.12345000000001 with A
               or P of 5, but this can not be helped - or can it?)
             * Thus you can have the math done by on Math::Big* class in two modi:
               + never round (this is the default):
                 This is done by setting A and P to undef. No math operation
                 will round the result, with bdiv() and bsqrt() as exceptions to guard
                 against overflows. You must explicitly call bround(), bfround() or
                 round() (the latter with parameters).
                 Note: Once you have rounded a number, the settings will 'stick' on it
                 and 'infect' all other numbers engaged in math operations with it, since
                 local settings have the highest precedence. So, to get SaferRound[tm],
                 use a copy() before rounding like this:

                   $x = Math::BigFloat->new(12.34);
                   $y = Math::BigFloat->new(98.76);
                   $z = $x * $y;                           # 1218.6984
                   print $x->copy()->bround(3);            # 12.3 (but A is now 3!)
                   $z = $x * $y;                           # still 1218.6984, without
                                                           # copy would have been 1210!

               + round after each op:
                 After each single operation (except for testing like is_zero()), the
                 method round() is called and the result is rounded appropriately. By
                 setting proper values for A and P, you can have all-the-same-A or
                 all-the-same-P modes. For example, Math::Currency might set A to undef,
                 and P to -2, globally.

            ?Maybe an extra option that forbids local A & P settings would be in order,
            ?so that intermediate rounding does not 'poison' further math?

       Overriding globals
             * you will be able to give A, P and R as an argument to all the calculation
               routines; the second parameter is A, the third one is P, and the fourth is
               R (shift right by one for binary operations like badd). P is used only if
               the first parameter (A) is undefined. These three parameters override the
               globals in the order detailed as follows, i.e. the first defined value
               wins:
               (local: per object, global: global default, parameter: argument to sub)
                 + parameter A
                 + parameter P
                 + local A (if defined on both of the operands: smaller one is taken)
                 + local P (if defined on both of the operands: bigger one is taken)
                 + global A
                 + global P
                 + global F
             * bsqrt() will hand its arguments to bdiv(), as it used to, only now for two
               arguments (A and P) instead of one

       Local settings
             * You can set A or P locally by using $x->accuracy() or
               $x->precision()
               and thus force different A and P for different objects/numbers.
             * Setting A or P this way immediately rounds $x to the new value.
             * $x->accuracy() clears $x->precision(), and vice versa.

       Rounding
             * the rounding routines will use the respective global or local settings.
               bround() is for accuracy rounding, while bfround() is for precision
             * the two rounding functions take as the second parameter one of the
               following rounding modes (R):
               'even', 'odd', '+inf', '-inf', 'zero', 'trunc', 'common'
             * you can set/get the global R by using Math::SomeClass->round_mode()
               or by setting $Math::SomeClass::round_mode
             * after each operation, $result->round() is called, and the result may
               eventually be rounded (that is, if A or P were set either locally,
               globally or as parameter to the operation)
             * to manually round a number, call $x->round($A,$P,$round_mode);
               this will round the number by using the appropriate rounding function
               and then normalize it.
             * rounding modifies the local settings of the number:

                   $x = Math::BigFloat->new(123.456);
                   $x->accuracy(5);
                   $x->bround(4);

               Here 4 takes precedence over 5, so 123.5 is the result and $x->accuracy()
               will be 4 from now on.

       Default values
             * R: 'even'
             * F: 40
             * A: undef
             * P: undef

       Remarks
             * The defaults are set up so that the new code gives the same results as
               the old code (except in a few cases on bdiv):
               + Both A and P are undefined and thus will not be used for rounding
                 after each operation.
               + round() is thus a no-op, unless given extra parameters A and P

Infinity and Not a Number

       While Math::BigInt has extensive handling of inf and NaN, certain quirks remain.

       oct()/hex()
           These perl routines currently (as of Perl v.5.8.6) cannot handle passed inf.

               te@linux:~> perl -wle 'print 2 ** 3333'
               Inf
               te@linux:~> perl -wle 'print 2 ** 3333 == 2 ** 3333'
               1
               te@linux:~> perl -wle 'print oct(2 ** 3333)'
               0
               te@linux:~> perl -wle 'print hex(2 ** 3333)'
               Illegal hexadecimal digit 'I' ignored at -e line 1.
               0

           The same problems occur if you pass them Math::BigInt->binf() objects. Since
           overloading these routines is not possible, this cannot be fixed from Math::BigInt.

INTERNALS

       You should neither care about nor depend on the internal representation; it might change
       without notice. Use ONLY method calls like "$x->sign();" instead relying on the internal
       representation.

   MATH LIBRARY
       Math with the numbers is done (by default) by a module called "Math::BigInt::Calc". This
       is equivalent to saying:

           use Math::BigInt try => 'Calc';

       You can change this backend library by using:

           use Math::BigInt try => 'GMP';

       Note: General purpose packages should not be explicit about the library to use; let the
       script author decide which is best.

       If your script works with huge numbers and Calc is too slow for them, you can also for the
       loading of one of these libraries and if none of them can be used, the code dies:

           use Math::BigInt only => 'GMP,Pari';

       The following would first try to find Math::BigInt::Foo, then Math::BigInt::Bar, and when
       this also fails, revert to Math::BigInt::Calc:

           use Math::BigInt try => 'Foo,Math::BigInt::Bar';

       The library that is loaded last is used. Note that this can be overwritten at any time by
       loading a different library, and numbers constructed with different libraries cannot be
       used in math operations together.

       What library to use?

       Note: General purpose packages should not be explicit about the library to use; let the
       script author decide which is best.

       Math::BigInt::GMP and Math::BigInt::Pari are in cases involving big numbers much faster
       than Calc, however it is slower when dealing with very small numbers (less than about 20
       digits) and when converting very large numbers to decimal (for instance for printing,
       rounding, calculating their length in decimal etc).

       So please select carefully what library you want to use.

       Different low-level libraries use different formats to store the numbers.  However, you
       should NOT depend on the number having a specific format internally.

       See the respective math library module documentation for further details.

   SIGN
       The sign is either '+', '-', 'NaN', '+inf' or '-inf'.

       A sign of 'NaN' is used to represent the result when input arguments are not numbers or as
       a result of 0/0. '+inf' and '-inf' represent plus respectively minus infinity. You get
       '+inf' when dividing a positive number by 0, and '-inf' when dividing any negative number
       by 0.

EXAMPLES

         use Math::BigInt;

         sub bigint { Math::BigInt->new(shift); }

         $x = Math::BigInt->bstr("1234")       # string "1234"
         $x = "$x";                            # same as bstr()
         $x = Math::BigInt->bneg("1234");      # Math::BigInt "-1234"
         $x = Math::BigInt->babs("-12345");    # Math::BigInt "12345"
         $x = Math::BigInt->bnorm("-0.00");    # Math::BigInt "0"
         $x = bigint(1) + bigint(2);           # Math::BigInt "3"
         $x = bigint(1) + "2";                 # ditto (auto-Math::BigIntify of "2")
         $x = bigint(1);                       # Math::BigInt "1"
         $x = $x + 5 / 2;                      # Math::BigInt "3"
         $x = $x ** 3;                         # Math::BigInt "27"
         $x *= 2;                              # Math::BigInt "54"
         $x = Math::BigInt->new(0);            # Math::BigInt "0"
         $x--;                                 # Math::BigInt "-1"
         $x = Math::BigInt->badd(4,5)          # Math::BigInt "9"
         print $x->bsstr();                    # 9e+0

       Examples for rounding:

         use Math::BigFloat;
         use Test::More;

         $x = Math::BigFloat->new(123.4567);
         $y = Math::BigFloat->new(123.456789);
         Math::BigFloat->accuracy(4);          # no more A than 4

         is ($x->copy()->bround(),123.4);      # even rounding
         print $x->copy()->bround(),"\n";      # 123.4
         Math::BigFloat->round_mode('odd');    # round to odd
         print $x->copy()->bround(),"\n";      # 123.5
         Math::BigFloat->accuracy(5);          # no more A than 5
         Math::BigFloat->round_mode('odd');    # round to odd
         print $x->copy()->bround(),"\n";      # 123.46
         $y = $x->copy()->bround(4),"\n";      # A = 4: 123.4
         print "$y, ",$y->accuracy(),"\n";     # 123.4, 4

         Math::BigFloat->accuracy(undef);      # A not important now
         Math::BigFloat->precision(2);         # P important
         print $x->copy()->bnorm(),"\n";       # 123.46
         print $x->copy()->bround(),"\n";      # 123.46

       Examples for converting:

         my $x = Math::BigInt->new('0b1'.'01' x 123);
         print "bin: ",$x->as_bin()," hex:",$x->as_hex()," dec: ",$x,"\n";

Autocreating constants

       After "use Math::BigInt ':constant'" all the integer decimal, hexadecimal and binary
       constants in the given scope are converted to "Math::BigInt". This conversion happens at
       compile time.

       In particular,

         perl -MMath::BigInt=:constant -e 'print 2**100,"\n"'

       prints the integer value of "2**100". Note that without conversion of constants the
       expression 2**100 is calculated using Perl scalars.

       Please note that strings and floating point constants are not affected, so that

           use Math::BigInt qw/:constant/;

           $x = 1234567890123456789012345678901234567890
                   + 123456789123456789;
           $y = '1234567890123456789012345678901234567890'
                   + '123456789123456789';

       does not give you what you expect. You need an explicit Math::BigInt->new() around one of
       the operands. You should also quote large constants to protect loss of precision:

           use Math::BigInt;

           $x = Math::BigInt->new('1234567889123456789123456789123456789');

       Without the quotes Perl would convert the large number to a floating point constant at
       compile time and then hand the result to Math::BigInt, which results in an truncated
       result or a NaN.

       This also applies to integers that look like floating point constants:

           use Math::BigInt ':constant';

           print ref(123e2),"\n";
           print ref(123.2e2),"\n";

       prints nothing but newlines. Use either bignum or Math::BigFloat to get this to work.

PERFORMANCE

       Using the form $x += $y; etc over $x = $x + $y is faster, since a copy of $x must be made
       in the second case. For long numbers, the copy can eat up to 20% of the work (in the case
       of addition/subtraction, less for multiplication/division). If $y is very small compared
       to $x, the form $x += $y is MUCH faster than $x = $x + $y since making the copy of $x
       takes more time then the actual addition.

       With a technique called copy-on-write, the cost of copying with overload could be
       minimized or even completely avoided. A test implementation of COW did show performance
       gains for overloaded math, but introduced a performance loss due to a constant overhead
       for all other operations. So Math::BigInt does currently not COW.

       The rewritten version of this module (vs. v0.01) is slower on certain operations, like
       "new()", "bstr()" and "numify()". The reason are that it does now more work and handles
       much more cases. The time spent in these operations is usually gained in the other math
       operations so that code on the average should get (much) faster. If they don't, please
       contact the author.

       Some operations may be slower for small numbers, but are significantly faster for big
       numbers. Other operations are now constant (O(1), like "bneg()", "babs()" etc), instead of
       O(N) and thus nearly always take much less time.  These optimizations were done on
       purpose.

       If you find the Calc module to slow, try to install any of the replacement modules and see
       if they help you.

   Alternative math libraries
       You can use an alternative library to drive Math::BigInt. See the section "MATH LIBRARY"
       for more information.

       For more benchmark results see <http://bloodgate.com/perl/benchmarks.html>.

SUBCLASSING

   Subclassing Math::BigInt
       The basic design of Math::BigInt allows simple subclasses with very little work, as long
       as a few simple rules are followed:

       •   The public API must remain consistent, i.e. if a sub-class is overloading addition,
           the sub-class must use the same name, in this case badd(). The reason for this is that
           Math::BigInt is optimized to call the object methods directly.

       •   The private object hash keys like "$x->{sign}" may not be changed, but additional keys
           can be added, like "$x->{_custom}".

       •   Accessor functions are available for all existing object hash keys and should be used
           instead of directly accessing the internal hash keys. The reason for this is that
           Math::BigInt itself has a pluggable interface which permits it to support different
           storage methods.

       More complex sub-classes may have to replicate more of the logic internal of Math::BigInt
       if they need to change more basic behaviors. A subclass that needs to merely change the
       output only needs to overload "bstr()".

       All other object methods and overloaded functions can be directly inherited from the
       parent class.

       At the very minimum, any subclass needs to provide its own "new()" and can store
       additional hash keys in the object. There are also some package globals that must be
       defined, e.g.:

           # Globals
           $accuracy = undef;
           $precision = -2;       # round to 2 decimal places
           $round_mode = 'even';
           $div_scale = 40;

       Additionally, you might want to provide the following two globals to allow auto-upgrading
       and auto-downgrading to work correctly:

           $upgrade = undef;
           $downgrade = undef;

       This allows Math::BigInt to correctly retrieve package globals from the subclass, like
       $SubClass::precision. See t/Math/BigInt/Subclass.pm or t/Math/BigFloat/SubClass.pm
       completely functional subclass examples.

       Don't forget to

           use overload;

       in your subclass to automatically inherit the overloading from the parent. If you like,
       you can change part of the overloading, look at Math::String for an example.

UPGRADING

       When used like this:

           use Math::BigInt upgrade => 'Foo::Bar';

       certain operations 'upgrade' their calculation and thus the result to the class Foo::Bar.
       Usually this is used in conjunction with Math::BigFloat:

           use Math::BigInt upgrade => 'Math::BigFloat';

       As a shortcut, you can use the module bignum:

           use bignum;

       Also good for one-liners:

           perl -Mbignum -le 'print 2 ** 255'

       This makes it possible to mix arguments of different classes (as in 2.5 + 2) as well es
       preserve accuracy (as in sqrt(3)).

       Beware: This feature is not fully implemented yet.

   Auto-upgrade
       The following methods upgrade themselves unconditionally; that is if upgrade is in effect,
       they always hands up their work:

           div bsqrt blog bexp bpi bsin bcos batan batan2

       All other methods upgrade themselves only when one (or all) of their arguments are of the
       class mentioned in $upgrade.

EXPORTS

       "Math::BigInt" exports nothing by default, but can export the following methods:

           bgcd
           blcm

CAVEATS

       Some things might not work as you expect them. Below is documented what is known to be
       troublesome:

       Comparing numbers as strings
           Both "bstr()" and "bsstr()" as well as stringify via overload drop the leading '+'.
           This is to be consistent with Perl and to make "cmp" (especially with overloading) to
           work as you expect. It also solves problems with "Test.pm" and Test::More, which
           stringify arguments before comparing them.

           Mark Biggar said, when asked about to drop the '+' altogether, or make only "cmp"
           work:

               I agree (with the first alternative), don't add the '+' on positive
               numbers.  It's not as important anymore with the new internal form
               for numbers.  It made doing things like abs and neg easier, but
               those have to be done differently now anyway.

           So, the following examples now works as expected:

               use Test::More tests => 1;
               use Math::BigInt;

               my $x = Math::BigInt -> new(3*3);
               my $y = Math::BigInt -> new(3*3);

               is($x,3*3, 'multiplication');
               print "$x eq 9" if $x eq $y;
               print "$x eq 9" if $x eq '9';
               print "$x eq 9" if $x eq 3*3;

           Additionally, the following still works:

               print "$x == 9" if $x == $y;
               print "$x == 9" if $x == 9;
               print "$x == 9" if $x == 3*3;

           There is now a "bsstr()" method to get the string in scientific notation aka 1e+2
           instead of 100. Be advised that overloaded 'eq' always uses bstr() for comparison, but
           Perl represents some numbers as 100 and others as 1e+308.  If in doubt, convert both
           arguments to Math::BigInt before comparing them as strings:

               use Test::More tests => 3;
               use Math::BigInt;

               $x = Math::BigInt->new('1e56'); $y = 1e56;
               is($x,$y);                     # fails
               is($x->bsstr(),$y);            # okay
               $y = Math::BigInt->new($y);
               is($x,$y);                     # okay

           Alternatively, simply use "<=>" for comparisons, this always gets it right. There is
           not yet a way to get a number automatically represented as a string that matches
           exactly the way Perl represents it.

           See also the section about "Infinity and Not a Number" for problems in comparing NaNs.

       int()
           "int()" returns (at least for Perl v5.7.1 and up) another Math::BigInt, not a Perl
           scalar:

               $x = Math::BigInt->new(123);
               $y = int($x);                           # 123 as a Math::BigInt
               $x = Math::BigFloat->new(123.45);
               $y = int($x);                           # 123 as a Math::BigFloat

           If you want a real Perl scalar, use "numify()":

               $y = $x->numify();                      # 123 as a scalar

           This is seldom necessary, though, because this is done automatically, like when you
           access an array:

               $z = $array[$x];                        # does work automatically

       Modifying and =
           Beware of:

               $x = Math::BigFloat->new(5);
               $y = $x;

           This makes a second reference to the same object and stores it in $y. Thus anything
           that modifies $x (except overloaded operators) also modifies $y, and vice versa. Or in
           other words, "=" is only safe if you modify your Math::BigInt objects only via
           overloaded math. As soon as you use a method call it breaks:

               $x->bmul(2);
               print "$x, $y\n";       # prints '10, 10'

           If you want a true copy of $x, use:

               $y = $x->copy();

           You can also chain the calls like this, this first makes a copy and then multiply it
           by 2:

               $y = $x->copy()->bmul(2);

           See also the documentation for overload.pm regarding "=".

       Overloading -$x
           The following:

               $x = -$x;

           is slower than

               $x->bneg();

           since overload calls "sub($x,0,1);" instead of "neg($x)". The first variant needs to
           preserve $x since it does not know that it later gets overwritten.  This makes a copy
           of $x and takes O(N), but $x->bneg() is O(1).

       Mixing different object types
           With overloaded operators, it is the first (dominating) operand that determines which
           method is called. Here are some examples showing what actually gets called in various
           cases.

               use Math::BigInt;
               use Math::BigFloat;

               $mbf  = Math::BigFloat->new(5);
               $mbi2 = Math::BigInt->new(5);
               $mbi  = Math::BigInt->new(2);
                                               # what actually gets called:
               $float = $mbf + $mbi;           # $mbf->badd($mbi)
               $float = $mbf / $mbi;           # $mbf->bdiv($mbi)
               $integer = $mbi + $mbf;         # $mbi->badd($mbf)
               $integer = $mbi2 / $mbi;        # $mbi2->bdiv($mbi)
               $integer = $mbi2 / $mbf;        # $mbi2->bdiv($mbf)

           For instance, Math::BigInt->bdiv() always returns a Math::BigInt, regardless of
           whether the second operant is a Math::BigFloat. To get a Math::BigFloat you either
           need to call the operation manually, make sure each operand already is a
           Math::BigFloat, or cast to that type via Math::BigFloat->new():

               $float = Math::BigFloat->new($mbi2) / $mbi;     # = 2.5

           Beware of casting the entire expression, as this would cast the result, at which point
           it is too late:

               $float = Math::BigFloat->new($mbi2 / $mbi);     # = 2

           Beware also of the order of more complicated expressions like:

               $integer = ($mbi2 + $mbi) / $mbf;               # int / float => int
               $integer = $mbi2 / Math::BigFloat->new($mbi);   # ditto

           If in doubt, break the expression into simpler terms, or cast all operands to the
           desired resulting type.

           Scalar values are a bit different, since:

               $float = 2 + $mbf;
               $float = $mbf + 2;

           both result in the proper type due to the way the overloaded math works.

           This section also applies to other overloaded math packages, like Math::String.

           One solution to you problem might be autoupgrading|upgrading. See the pragmas bignum,
           bigint and bigrat for an easy way to do this.

BUGS

       Please report any bugs or feature requests to "bug-math-bigint at rt.cpan.org", or through
       the web interface at <https://rt.cpan.org/Ticket/Create.html?Queue=Math-BigInt> (requires
       login).  We will be notified, and then you'll automatically be notified of progress on
       your bug as I make changes.

SUPPORT

       You can find documentation for this module with the perldoc command.

           perldoc Math::BigInt

       You can also look for information at:

       •   RT: CPAN's request tracker

           <https://rt.cpan.org/Public/Dist/Display.html?Name=Math-BigInt>

       •   AnnoCPAN: Annotated CPAN documentation

           <http://annocpan.org/dist/Math-BigInt>

       •   CPAN Ratings

           <http://cpanratings.perl.org/dist/Math-BigInt>

       •   Search CPAN

           <http://search.cpan.org/dist/Math-BigInt/>

       •   CPAN Testers Matrix

           <http://matrix.cpantesters.org/?dist=Math-BigInt>

       •   The Bignum mailing list

           •   Post to mailing list

               "bignum at lists.scsys.co.uk"

           •   View mailing list

               <http://lists.scsys.co.uk/pipermail/bignum/>

           •   Subscribe/Unsubscribe

               <http://lists.scsys.co.uk/cgi-bin/mailman/listinfo/bignum>

LICENSE

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

SEE ALSO

       Math::BigFloat and Math::BigRat as well as the backends Math::BigInt::FastCalc,
       Math::BigInt::GMP, and Math::BigInt::Pari.

       The pragmas bignum, bigint and bigrat also might be of interest because they solve the
       autoupgrading/downgrading issue, at least partly.

AUTHORS

       •   Mark Biggar, overloaded interface by Ilya Zakharevich, 1996-2001.

       •   Completely rewritten by Tels <http://bloodgate.com>, 2001-2008.

       •   Florian Ragwitz <flora@cpan.org>, 2010.

       •   Peter John Acklam <pjacklam@online.no>, 2011-.

       Many people contributed in one or more ways to the final beast, see the file CREDITS for
       an (incomplete) list. If you miss your name, please drop me a mail. Thank you!