MirOS Manual: Math::BigInt(3p)


Math::BigInt(3p)Perl Programmers Reference Guide Math::BigInt(3p)

NAME

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

SYNOPSIS

       use Math::BigInt;

       # or make it faster: install (optional) Math::BigInt::GMP
       # and always use (it will fall back to pure Perl if the
       # GMP library is not installed):

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

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

       # Number creation
       $x = Math::BigInt->new($str);         # defaults to 0
       $y = $x->copy();                      # make a true copy
       $nan  = Math::BigInt->bnan();         # create a NotANumber
       $zero = Math::BigInt->bzero();        # create a +0
       $inf = Math::BigInt->binf();          # create a +inf
       $inf = Math::BigInt->binf('-');       # create a -inf
       $one = Math::BigInt->bone();          # create a +1
       $one = Math::BigInt->bone('-');       # create a -1

       # Testing (don't modify their arguments)
       # (return true if the condition is met, otherwise false)

       $x->is_zero();        # if $x is +0
       $x->is_nan();         # if $x is NaN
       $x->is_one();         # if $x is +1
       $x->is_one('-');      # if $x is -1
       $x->is_odd();         # if $x is odd
       $x->is_even();        # if $x is even
       $x->is_pos();         # if $x >= 0
       $x->is_neg();         # if $x <  0
       $x->is_inf($sign);    # if $x is +inf, or -inf (sign is default '+')
       $x->is_int();         # if $x is an integer (not a float)

       # comparing and digit/sign extration
       $x->bcmp($y);         # compare numbers (undef,<0,=0,>0)
       $x->bacmp($y);        # compare absolutely (undef,<0,=0,>0)
       $x->sign();           # return the sign, either +,- or NaN
       $x->digit($n);        # return the nth digit, counting from right
       $x->digit(-$n);       # return the nth digit, counting from left

       # The following all modify their first argument. If you want to preserve
       # $x, use $z = $x->copy()->bXXX($y); See under L<CAVEATS> for why this is
       # neccessary when mixing $a = $b assigments with non-overloaded math.

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       $x->bzero();          # set $x to 0
       $x->bnan();           # set $x to NaN
       $x->bone();           # set $x to +1
       $x->bone('-');        # set $x to -1
       $x->binf();           # set $x to inf
       $x->binf('-');        # set $x to -inf

       $x->bneg();           # negation
       $x->babs();           # absolute value
       $x->bnorm();          # normalize (no-op in BigInt)
       $x->bnot();           # two's complement (bit wise not)
       $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)
       $x->bdiv($y);         # divide, set $x to quotient
                             # return (quo,rem) or quo if scalar

       $x->bmod($y);            # modulus (x % y)
       $x->bmodpow($exp,$mod);  # modular exponentation (($num**$exp) % $mod))
       $x->bmodinv($mod);       # the inverse of $x in the given modulus $mod

       $x->bpow($y);            # power of arguments (x ** y)
       $x->blsft($y);           # left shift
       $x->brsft($y);           # right shift
       $x->blsft($y,$n);        # left shift, by base $n (like 10)
       $x->brsft($y,$n);        # right shift, by base $n (like 10)

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

       $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)

       $x->round($A,$P,$mode);  # round to accuracy or precision using mode $mode
       $x->bround($n);          # accuracy: preserve $n digits
       $x->bfround($n);         # round to $nth digit, no-op for BigInts

       # The following do not modify their arguments in BigInt (are no-ops),
       # but do so in BigFloat:

       $x->bfloor();            # return integer less or equal than $x
       $x->bceil();             # return integer greater or equal than $x

       # The following do not modify their arguments:

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       # greatest common divisor (no OO style)
       my $gcd = Math::BigInt::bgcd(@values);
       # lowest common multiplicator (no OO style)
       my $lcm = Math::BigInt::blcm(@values);

       $x->length();            # return number of digits in number
       ($xl,$f) = $x->length(); # length of number and length of fraction part,
                                # latter is always 0 digits long for BigInts

       $x->exponent();          # return exponent as BigInt
       $x->mantissa();          # return (signed) mantissa as BigInt
       $x->parts();             # return (mantissa,exponent) as BigInt
       $x->copy();              # make a true copy of $x (unlike $y = $x;)
       $x->as_int();            # return as BigInt (in BigInt: same as copy())
       $x->numify();            # return as scalar (might overflow!)

       # conversation to string (do not modify their argument)
       $x->bstr();              # normalized string (e.g. '3')
       $x->bsstr();             # norm. string in scientific notation (e.g. '3E0')
       $x->as_hex();            # as signed hexadecimal string with prefixed 0x
       $x->as_bin();            # as signed binary string with prefixed 0b

       # precision and accuracy (see section about rounding for more)
       $x->precision();         # return P of $x (or global, if P of $x undef)
       $x->precision($n);       # set P of $x to $n
       $x->accuracy();          # return A of $x (or global, if A of $x undef)
       $x->accuracy($n);        # set A $x to $n

       # Global methods
       Math::BigInt->precision();    # get/set global P for all BigInt objects
       Math::BigInt->accuracy();     # get/set global A for all BigInt objects
       Math::BigInt->round_mode();   # get/set global round mode, one of
                                     # 'even', 'odd', '+inf', '-inf', 'zero' or 'trunc'
       Math::BigInt->config();       # return hash containing configuration

DESCRIPTION

     All operators (inlcuding basic math operations) are over-
     loaded if you declare your big integers as

       $i = new Math::BigInt '123_456_789_123_456_789';

     Operations with overloaded operators preserve the arguments
     which is exactly what you expect.

     Input
       Input values to these routines may be any string, that
       looks like a number and results in an integer, including
       hexadecimal and binary numbers.

       Scalars holding numbers may also be passed, but note that
       non-integer numbers may already have lost precision due to
       the conversation to float. Quote your input if you want

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       BigInt to see all the digits:

               $x = Math::BigInt->new(12345678890123456789);   # bad
               $x = Math::BigInt->new('12345678901234567890'); # good

       You can include one underscore between any two digits.

       This means integer values like 1.01E2 or even 1000E-2 are
       also accepted. Non-integer values result in NaN.

       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();

       "bnorm()" on a BigInt object is now effectively a no-op,
       since the numbers are always stored in normalized form. If
       passed a string, creates a BigInt object from the input.

     Output
       Output values are BigInt objects (normalized), except for
       the methods which return a string (see SYNOPSIS).

       Some routines ("is_odd()", "is_even()", "is_zero()",
       "is_one()", "is_nan()", etc.) return true or false, while
       others ("bcmp()", "bacmp()") return either undef (if NaN
       is involved), <0, 0 or >0 and are suited for sort.

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.

     config

             use Data::Dumper;

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

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

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             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 acccuracy 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:

             trap_inf trap_nan
             upgrade downgrade precision accuracy round_mode div_scale

     Example:

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

     accuracy

             $x->accuracy(5);                # local for $x
             CLASS->accuracy(5);             # global for all members of CLASS
                                             # Note: This also applies to new()!

             $A = $x->accuracy();            # read out accuracy that affects $x
             $A = CLASS->accuracy();         # read out global accuracy

     Set or get the global or local accuracy, aka how many signi-
     ficant digits the results have. If you set a global accu-
     racy, then this also applies to new()!

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     Warning! The accuracy sticks, e.g. once you created a number
     under the influence of "CLASS->accuracy($A)", all results
     from math operations with that number will also be rounded.

     In most cases, you should probably round the results expli-
     citely using one of round(), bround() or bfround() or by
     passing the desired accuracy to the math operation as addi-
     tional parameter:

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

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

     Value must be greater than zero. Pass an undef value to dis-
     able it:

             $x->accuracy(undef);
             Math::BigInt->accuracy(undef);

     Returns the current accuracy. For "$x-"accuracy()> it will
     return either the local accuracy, or if not defined, the
     global. This means the return value represents the accuracy
     that will be in effect for $x:

             $y = Math::BigInt->new(1234567);        # unrounded
             print Math::BigInt->accuracy(4),"\n";   # set 4, print 4
             $x = Math::BigInt->new(123456);         # $x will be automatically rounded!
             print "$x $y\n";                        # '123500 1234567'
             print $x->accuracy(),"\n";              # will be 4
             print $y->accuracy(),"\n";              # also 4, since global is 4
             print Math::BigInt->accuracy(5),"\n";   # set to 5, print 5
             print $x->accuracy(),"\n";              # still 4
             print $y->accuracy(),"\n";              # 5, since global is 5

     Note: Works also for subclasses like Math::BigFloat. 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

             $x->precision(-2);      # local for $x, round at the second digit right of the dot
             $x->precision(2);       # ditto, round at the second digit left of the dot

             CLASS->precision(5);    # Global for all members of CLASS
                                     # This also applies to new()!
             CLASS->precision(-5);   # ditto

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             $P = CLASS->precision();        # read out global precision
             $P = $x->precision();           # read out precision that affects $x

     Note: You probably 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!

     "precision()" sets or gets the global or local precision,
     aka at which digit before or after the dot to round all
     results. A set global precision also applies to all newly
     created numbers!

     In Math::BigInt, passing a negative number precision has no
     effect since no numbers have digits after the dot. In
     Math::BigFloat, it will round all results to P digits after
     the dot.

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

     Pass an undef value to disable it:

             $x->precision(undef);
             Math::BigInt->precision(undef);

     Returns the current precision. For "$x-"precision()> it will
     return either the local precision of $x, or if not defined,
     the global. This means the return value represents the prev-
     ision that will be in effect for $x:

             $y = Math::BigInt->new(1234567);        # unrounded
             print Math::BigInt->precision(4),"\n";  # set 4, print 4
             $x = Math::BigInt->new(123456);         # will be automatically rounded
             print $x;                               # print "120000"!

     Note: Works also for subclasses like Math::BigFloat. 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.

     brsft

             $x->brsft($y,$n);

     Shifts $x right by $y in base $n. Default is base 2, used
     are usually 10 and 2, but others work, too.

     Right shifting usually amounts to dividing $x by $n ** $y
     and truncating the result:

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             $x = Math::BigInt->new(10);
             $x->brsft(1);                   # same as $x >> 1: 5
             $x = Math::BigInt->new(1234);
             $x->brsft(2,10);                # result 12

     There is one exception, and that is base 2 with negative $x:

             $x = Math::BigInt->new(-5);
             print $x->brsft(1);

     This will print -3, not -2 (as it would if you divide -5 by
     2 and truncate the result).

     new

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

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

     See Input for more info on accepted input formats.

     bnan

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

     Creates a new BigInt object representing NaN (Not A Number).
     If used on an object, it will set it to NaN:

             $x->bnan();

     bzero

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

     Creates a new BigInt object representing zero. If used on an
     object, it will set it to zero:

             $x->bzero();

     binf

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

     Creates a new BigInt object representing infinity. The
     optional argument is either '-' or '+', indicating whether
     you want infinity or minus infinity. If used on an object,
     it will set it to infinity:

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

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     bone

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

     Creates a new BigInt object representing one. The optional
     argument is either '-' or '+', indicating whether you want
     one or minus one. If used on an object, it will set it to
     one:

             $x->bone();             # +1
             $x->bone('-');          # -1

     is_one()/is_zero()/is_nan()/is_inf()

             $x->is_zero();                  # true if arg is +0
             $x->is_nan();                   # true if arg is NaN
             $x->is_one();                   # true if arg is +1
             $x->is_one('-');                # true if arg is -1
             $x->is_inf();                   # true if +inf
             $x->is_inf('-');                # true if -inf (sign is default '+')

     These methods all test the BigInt for beeing one specific
     value and return true or false depending on the input. These
     are faster than doing something like:

             if ($x == 0)

     is_pos()/is_neg()

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

     The methods return true if the argument is positive or nega-
     tive, respectively. "NaN" is neither positive nor negative,
     while "+inf" counts as positive, and "-inf" is negative. A
     "zero" is neither positive nor negative.

     These methods are only testing the sign, and not the value.

     "is_positive()" and "is_negative()" are aliase to "is_pos()"
     and "is_neg()", respectively. "is_positive()" and
     "is_negative()" were introduced in v1.36, while "is_pos()"
     and "is_neg()" were only introduced in v1.68.

     is_odd()/is_even()/is_int()

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

     The return true when the argument satisfies the condition.
     "NaN", "+inf", "-inf" are not integers and are neither odd

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     nor even.

     In BigInt, all numbers except "NaN", "+inf" and "-inf" are
     integers.

     bcmp

             $x->bcmp($y);

     Compares $x with $y and takes the sign into account. Returns
     -1, 0, 1 or undef.

     bacmp

             $x->bacmp($y);

     Compares $x with $y while ignoring their. Returns -1, 0, 1
     or undef.

     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 fol-
     lowing 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.

     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.

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     babs

             $x->babs();

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

     bnorm

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

     bnot

             $x->bnot();

     Two's complement (bit wise not). This is equivalent to

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

     but faster.

     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)

     bdiv

             $x->bdiv($y);                   # divide, set $x to quotient
                                             # return (quo,rem) or quo if scalar

     bmod

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

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     bmodinv

             num->bmodinv($mod);             # modular inverse

     Returns the inverse of $num in the given modulus $mod.
     '"NaN"' is returned unless $num is relatively prime to $mod,
     i.e. unless "bgcd($num, $mod)==1".

     bmodpow

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

     Returns the value of $num taken to the power $exp in the
     modulus $mod using binary exponentation.  "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)

     blsft

             $x->blsft($y);          # left shift
             $x->blsft($y,$n);       # left shift, in base $n (like 10)

     brsft

             $x->brsft($y);          # right shift
             $x->brsft($y,$n);       # right shift, in base $n (like 10)

     band

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

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     bior

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

     bxor

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

     bnot

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

     bsqrt

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

     bfac

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

     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

     bfround

             $x->bfround($N);              # round to $Nth digit, no-op for BigInts

     bfloor

             $x->bfloor();

     Set $x to the integer less or equal than $x. This is a no-op
     in BigInt, but does change $x in BigFloat.

     bceil

             $x->bceil();

     Set $x to the integer greater or equal than $x. This is a
     no-op in BigInt, but does change $x in BigFloat.

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     bgcd

             bgcd(@values);          # greatest common divisor (no OO style)

     blcm

             blcm(@values);          # lowest common multiplicator (no OO style)

     head2 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 BigInt's, the length of the
     fraction part will always be 0.

     exponent

             $x->exponent();

     Return the exponent of $x as BigInt.

     mantissa

             $x->mantissa();

     Return the signed mantissa of $x as BigInt.

     parts

             $x->parts();            # return (mantissa,exponent) as BigInt

     copy

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

     as_int

             $x->as_int();

     Returns $x as a BigInt (truncated towards zero). In BigInt
     this is the same as "copy()".

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

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     bstr

             $x->bstr();

     Returns a normalized string represantation of $x.

     bsstr

             $x->bsstr();            # normalized string in scientific notation

     as_hex

             $x->as_hex();           # as signed hexadecimal string with prefixed 0x

     as_bin

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

ACCURACY and PRECISION

     Since version v1.33, 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::Big* 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 oth-
     ers people or documentation).

     During the rest of this document, the shortcuts A (for accu-
     racy), P (for precision), F (fallback) and R (rounding mode)
     will be used.

     Precision P

     A fixed number of digits before (positive) or after (nega-
     tive) 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 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 preci-
     sions, because 1200 can have p = 0, 1 or 2 (depending on
     what the inital value was). It could also have p < 0, when
     the digits after the decimal point are zero.

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     The string output (of floating point numbers) will be 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 BigInts, 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,
     123.450000 has 8 and 0.000123 has 3.

     The string output (of floating point numbers) will be 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 BigInts, 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.

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       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 round-
       ing 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.

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

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

     Accuracy (significant digits)

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         * fround($a) rounds to $a significant digits
         * only fdiv() and fsqrt() take A as (optional) paramater
           + other operations simply create the same number (fneg etc), or more (fmul)
             of digits
           + rounding/truncating is only done when explicitly calling one of fround
             or ffround, and never for BigInt (not implemented)
         * fsqrt() simply hands its accuracy argument over to fdiv.
         * the documentation and the comment in the code indicate two different ways
           on how fdiv() determines the maximum number of digits it should calculate,
           and the actual code does yet another thing
           POD:
             max($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(divisior) - length(dividend);
           So for lx = 3, ly = 9, scale = 10, scale will actually be 16 (10+9-3).
           Actually, the 'difference' added to the scale is calculated from the
           number of "significant digits" in dividend and divisor, which is derived
           by looking at the length of the mantissa. 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 significant digits.
           The rounding after the division then uses the remainder and $y to determine
           wether 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

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         * You can set the A global via C<< Math::BigInt->accuracy() >> or
           C<< Math::BigFloat->accuracy() >> or whatever class you are using.
         * You can also set P globally by using C<< Math::SomeClass->precision() >>
           likewise.
         * Globals are classwide, and not inherited by subclasses.
         * to undefine A, use C<< Math::SomeCLass->accuracy(undef); >>
         * to undefine P, use C<< Math::SomeClass->precision(undef); >>
         * Setting C<< Math::SomeClass->accuracy() >> clears automatically
           C<< 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 C<< Math::SomeClass->accuracy() >>
         * to find out the current global P, use C<< Math::SomeClass->precision() >>
         * use C<< $x->accuracy() >> respective C<< $x->precision() >> for the local
           setting of C<< $x >>.
         * Please note that C<< $x->accuracy() >> respecive C<< $x->precision() >>
           return eventually defined global A or P, when C<< $x >>'s A or P is not
           set.

     Creating numbers
         * When you create a number, you can give it's 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, B<no> rounding will occur, and the globals will
           B<not> be used. This is used by subclasses to create numbers without
           suffering rounding in the parent. Thus a subclass is able to have it's own
           globals enforced upon creation of a number by using
           C<< $x = Math::BigInt->new($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

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         * 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 BigInts 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 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 precendence 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 fdiv/fsqrt) and will not
           be rounded.
         * There is another setting for fdiv() (and thus for fsqrt()). If neither of
           A or P is defined, fdiv() 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 :-)
         * fdiv 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 fdiv() and fsqrt() as exceptions to guard
             against overflows. You must explicitely 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()->fround(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.

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        ?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
         * fsqrt() will hand its arguments to fdiv(), 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 C<< $x->accuracy() >> or
           C<< $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.
         * C<< $x->accuracy() >> clears C<< $x->precision() >>, and vice versa.

     Rounding
         * the rounding routines will use the respective global or local settings.
           fround()/bround() is for accuracy rounding, while ffround()/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'
         * you can set/get the global R by using C<< Math::SomeClass->round_mode() >>
           or by setting C<< $Math::SomeClass::round_mode >>
         * after each operation, C<< $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 C<< $x->round($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.

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     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 fdiv):
           + 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 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 rou-
       tines is not possible, this cannot be fixed from BigInt.

     ==, !=, <, >, <=, >= with NaNs
       BigInt's bcmp() routine currently returns undef to signal
       that a NaN was involved in a comparisation. However, the
       overload code turns that into either 1 or '' and thus
       operations like "NaN != NaN" might return wrong values.

     log(-inf)
       "log(-inf)" is highly weird. Since log(-x)=pi*i+log(x),
       then log(-inf)=pi*i+inf. However, since the imaginary part
       is finite, the real infinity "overshadows" it, so the
       number might as well just be infinity. However, the result
       is a complex number, and since BigInt/BigFloat can only
       have real numbers as results, the result is NaN.

     exp(), cos(), sin(), atan2()
       These all might have problems handling infinity right.

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INTERNALS

     The actual numbers are stored as unsigned big integers (with
     seperate sign).

     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 lib => 'Calc';

     You can change this by using:

             use Math::BigInt lib => 'BitVect';

     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 lib => 'Foo,Math::BigInt::Bar';

     Since Math::BigInt::GMP is in almost all cases faster than
     Calc (especially in math involving really big numbers, where
     it is much faster), and there is no penalty if
     Math::BigInt::GMP is not installed, it is a good idea to
     always use the following:

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

     Different low-level libraries use different formats to store
     the numbers. 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 will
     get '+inf' when dividing a positive number by 0, and '-inf'
     when dividing any negative number by 0.

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     mantissa(), exponent() and parts()

     "mantissa()" and "exponent()" return the said parts of the
     BigInt such that:

             $m = $x->mantissa();
             $e = $x->exponent();
             $y = $m * ( 10 ** $e );
             print "ok\n" if $x == $y;

     "($m,$e) = $x->parts()" is just a shortcut that gives you
     both of them in one go. Both the returned mantissa and
     exponent have a sign.

     Currently, for BigInts $e is always 0, except for NaN, +inf
     and -inf, where it is "NaN"; and for "$x == 0", where it is
     1 (to be compatible with Math::BigFloat's internal represen-
     tation of a zero as 0E1).

     $m is currently just a copy of the original number. The
     relation between $e and $m will stay always the same, though
     their real values might change.

EXAMPLES

       use Math::BigInt;

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

       $x = Math::BigInt->bstr("1234")       # string "1234"
       $x = "$x";                            # same as bstr()
       $x = Math::BigInt->bneg("1234");      # BigInt "-1234"
       $x = Math::BigInt->babs("-12345");    # BigInt "12345"
       $x = Math::BigInt->bnorm("-0 00");    # BigInt "0"
       $x = bint(1) + bint(2);               # BigInt "3"
       $x = bint(1) + "2";                   # ditto (auto-BigIntify of "2")
       $x = bint(1);                         # BigInt "1"
       $x = $x + 5 / 2;                      # BigInt "3"
       $x = $x ** 3;                         # BigInt "27"
       $x *= 2;                              # BigInt "54"
       $x = Math::BigInt->new(0);            # BigInt "0"
       $x--;                                 # BigInt "-1"
       $x = Math::BigInt->badd(4,5)          # BigInt "9"
       print $x->bsstr();                    # 9e+0

     Examples for rounding:

       use Math::BigFloat;
       use Test;

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

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       ok ($x->copy()->fround(),123.4);      # even rounding
       print $x->copy()->fround(),"\n";      # 123.4
       Math::BigFloat->round_mode('odd');    # round to odd
       print $x->copy()->fround(),"\n";      # 123.5
       Math::BigFloat->accuracy(5);          # no more A than 5
       Math::BigFloat->round_mode('odd');    # round to odd
       print $x->copy()->fround(),"\n";      # 123.46
       $y = $x->copy()->fround(4),"\n";      # A = 4: 123.4
       print "$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()->fround(),"\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 will be calcu-
     lated as perl scalar.

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

             use Math::BigInt qw/:constant/;

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

     do not work. 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');

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     Without the quotes Perl would convert the large number to a
     floating point constant at compile time and then hand the
     result to 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";

     will print 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 operatons. 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 "num-
     ify()". The reason are that it does now more work and han-
     dles 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.

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     Alternative math libraries

     You can use an alternative library to drive Math::BigInt
     via:

             use Math::BigInt lib => 'Module';

     See "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 per-
       mits 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 will need to provide it's
     own "new()" and can store additional hash keys in the
     object. There are also some package globals that must be
     defined, e.g.:

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       # 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 glo-
     bals 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 will '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 oneliners:

             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.

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     Auto-upgrade

     The following methods upgrade themselves unconditionally;
     that is if upgrade is in effect, they will always hand up
     their work:

     bsqrt()
     div()
     blog()

     Beware: This list is not complete.

     All other methods upgrade themselves only when one (or all)
     of their arguments are of the class mentioned in $upgrade
     (This might change in later versions to a more sophisticated
     scheme):

BUGS

     broot() does not work
       The broot() function in BigInt may only work for small
       values. This will be fixed in a later version.

     Out of Memory!
       Under Perl prior to 5.6.0 having an "use Math::BigInt
       ':constant';" and "eval()" in your code will crash with
       "Out of memory". This is probably an overload/exporter
       bug. You can workaround by not having "eval()" and ':con-
       stant' at the same time or upgrade your Perl to a newer
       version.

     Fails to load Calc on Perl prior 5.6.0
       Since eval(' use ...') can not be used in conjunction with
       ':constant', BigInt will fall back to eval { require ... }
       when loading the math lib on Perls prior to 5.6.0. This
       simple replaces '::' with '/' and thus might fail on
       filesystems using a different seperator.

CAVEATS

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

     bstr(), bsstr() and 'cmp'
      Both "bstr()" and "bsstr()" as well as automated stringify
      via overload now drop the leading '+'. The old code would
      return '+3', the new returns '3'. 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", because it's "ok()" uses 'eq' internally.

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

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              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 will now work all as expected:

              use Test;
              BEGIN { plan tests => 1 }
              use Math::BigInt;

              my $x = new Math::BigInt 3*3;
              my $y = new Math::BigInt 3*3;

              ok ($x,3*3);
              print "$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 scien-
      tific notation aka 1e+2 instead of 100. Be advised that
      overloaded 'eq' always uses bstr() for comparisation, but
      Perl will represent 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;
              BEGIN { plan tests => 3 }
              use Math::BigInt;

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

      Alternatively, simple use "<=>" for comparisations, this
      will get it always 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()" will return (at least for Perl v5.7.1 and up)

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      another BigInt, not a Perl scalar:

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

      In all Perl versions you can use "as_number()" or "as_int"
      for the same effect:

              $x = Math::BigFloat->new(123.45);
              $y = $x->as_number();                   # BigInt 123
              $y = $x->as_int();                      # ditto

      This also works for other subclasses, like Math::String.

      It is yet unlcear whether overloaded int() should return a
      scalar or a BigInt.

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

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

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

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

     length
      The following will probably not 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 con-
      text. Use something like:

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

     bdiv
      The following will probably not do what you expect:

              print $c->bdiv(10000),"\n";

      It prints both quotient and remainder since print calls
      "bdiv()" in list context. Also, "bdiv()" will modify $c, so
      be carefull. You probably want to use

              print $c / 10000,"\n";
              print scalar $c->bdiv(10000),"\n";  # or if you want to modify $c

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      instead.

      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 nonzero) 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)

      As a consequence, the behavior of the operator % agrees
      with the behavior of Perl's built-in % operator (as docu-
      mented in the perlop manpage), and the equation

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

      holds true for any $x and $y, which justifies calling the
      two return values of bdiv() the quotient and remainder. The
      only exception to this rule are when $y == 0 and $x is
      negative, then the remainder will also be negative. See
      below under "infinity handling" for the reasoning behing
      this.

      Perl's 'use integer;' changes the behaviour of % and / for
      scalars, but will not change BigInt's way to do things.
      This is because under 'use integer' Perl will do what the
      underlying C thinks is right and this is different for each
      system. If you need BigInt's behaving exactly like Perl's
      'use integer', bug the author to implement it ;)

     infinity handling
      Here are some examples that explain the reasons why certain
      results occur while handling infinity:

      The following table shows the result of the division and
      the remainder, so that the equation above holds true. Some
      "ordinary" cases are strewn in to show more clearly the
      reasoning:

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              A /  B  =   C,     R so that C *    B +    R =    A
           =========================================================
              5 /   8 =   0,     5         0 *    8 +    5 =    5
              0 /   8 =   0,     0         0 *    8 +    0 =    0
              0 / inf =   0,     0         0 *  inf +    0 =    0
              0 /-inf =   0,     0         0 * -inf +    0 =    0
              5 / inf =   0,     5         0 *  inf +    5 =    5
              5 /-inf =   0,     5         0 * -inf +    5 =    5
              -5/ inf =   0,    -5         0 *  inf +   -5 =   -5
              -5/-inf =   0,    -5         0 * -inf +   -5 =   -5
             inf/   5 =  inf,    0       inf *    5 +    0 =  inf
            -inf/   5 = -inf,    0      -inf *    5 +    0 = -inf
             inf/  -5 = -inf,    0      -inf *   -5 +    0 =  inf
            -inf/  -5 =  inf,    0       inf *   -5 +    0 = -inf
               5/   5 =    1,    0         1 *    5 +    0 =    5
              -5/  -5 =    1,    0         1 *   -5 +    0 =   -5
             inf/ inf =    1,    0         1 *  inf +    0 =  inf
            -inf/-inf =    1,    0         1 * -inf +    0 = -inf
             inf/-inf =   -1,    0        -1 * -inf +    0 =  inf
            -inf/ inf =   -1,    0         1 * -inf +    0 = -inf
               8/   0 =  inf,    8       inf *    0 +    8 =    8
             inf/   0 =  inf,  inf       inf *    0 +  inf =  inf
               0/   0 =  NaN

      These cases below violate the "remainder has the sign of
      the second of the two arguments", since they wouldn't match
      up otherwise.

              A /  B  =   C,     R so that C *    B +    R =    A
           ========================================================
            -inf/   0 = -inf, -inf      -inf *    0 +  inf = -inf
              -8/   0 = -inf,   -8      -inf *    0 +    8 = -8

     Modifying and =
      Beware of:

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

      It will not do what you think, e.g. making a copy of $x.
      Instead it just makes a second reference to the same object
      and stores it in $y. Thus anything that modifies $x (except
      overloaded operators) will modify $y, and vice versa. Or in
      other words, "=" is only safe if you modify your BigInts
      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:

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              $y = $x->copy();

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

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

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

     bpow
      "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 will modify
      $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.

     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 will get overwritten. This makes a copy
      of $x and takes O(N), but $x->bneg() is O(1).

     Mixing different object types
      In Perl you will get a floating point value if you do one
      of the following:

              $float = 5.0 + 2;
              $float = 2 + 5.0;
              $float = 5 / 2;

      With overloaded math, only the first two variants will
      result in a BigFloat:

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

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              $mbf = Math::BigFloat->new(5);
              $mbi2 = Math::BigInteger->new(5);
              $mbi = Math::BigInteger->new(2);

                                              # what actually gets called:
              $float = $mbf + $mbi;           # $mbf->badd()
              $float = $mbf / $mbi;           # $mbf->bdiv()
              $integer = $mbi + $mbf;         # $mbi->badd()
              $integer = $mbi2 / $mbi;        # $mbi2->bdiv()
              $integer = $mbi2 / $mbf;        # $mbi2->bdiv()

      This is because math with overloaded operators follows the
      first (dominating) operand, and the operation of that is
      called and returns thus the result. So,
      Math::BigInt::bdiv() will always return a Math::BigInt,
      regardless whether the result should be a Math::BigFloat or
      the second operant is one.

      To get a Math::BigFloat you either need to call the opera-
      tion manually, make sure the operands are already of the
      proper type or casted to that type via
      Math::BigFloat->new():

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

      Beware of simple "casting" the entire expression, this
      would only convert the already computed result:

              $float = Math::BigFloat->new($mbi2 / $mbi);     # = 2.0 thus wrong!

      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;

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

      This section also applies to other overloaded math pack-
      ages, like Math::String.

      One solution to you problem might be
      autoupgrading|upgrading. See the pragmas bignum, bigint and

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      bigrat for an easy way to do this.

     bsqrt()
      "bsqrt()" works only good if the result is a big integer,
      e.g. the square root of 144 is 12, but from 12 the square
      root is 3, regardless of rounding mode. The reason is that
      the result is always 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

     brsft()
      For negative numbers in base see also brsft.

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, Math::BigRat and Math::Big as well as
     Math::BigInt::BitVect, Math::BigInt::Pari and
     Math::BigInt::GMP.

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

     The package at
     <http://search.cpan.org/search?mode=module&query=Math%3A%3ABigInt>
     contains more documentation including a full version his-
     tory, testcases, empty subclass files and benchmarks.

AUTHORS

     Original code by Mark Biggar, overloaded interface by Ilya
     Zakharevich. Completely rewritten by Tels
     http://bloodgate.com in late 2000, 2001 - 2004 and still at
     it in 2005.

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

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