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

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

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

NNAAMMEE #

 Math::BigInt - arbitrary size integer math package

SSYYNNOOPPSSIISS #

   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!)

   # to warn if Math::BigInt::GMP cannot be found, use
   use Math::BigInt lib => 'GMP';

   # to suppress the warning if Math::BigInt::GMP cannot be found, use
   # use Math::BigInt try => 'GMP';

   # to die if Math::BigInt::GMP cannot be found, use
   # 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->from_base('why', 36);  # from any base
   $x = Math::BigInt->from_base_num([1, 0], 2);  # from any base
   $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)
   $x->buparrow($n, $y);   # Knuth's up-arrow notation
   $x->backermann($y);     # the Ackermann function
   $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->bdfac();            # double factorial of $x ($x*($x-2)*($x-4)*...)
   $x->btfac();            # triple factorial of $x ($x*($x-3)*($x-6)*...)
   $x->bmfac($k);          # $k'th multi-factorial of $x ($x*($x-$k)*...)

   $x->blsft($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
   $x->fparts();            # numerator and denominator
   $x->numerator();         # numerator
   $x->denominator();       # denominator

   # 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
   $x->to_base($b);    # as string in any base
   $x->to_base_num($b);   # as array of integers in any base

   $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)

DDEESSCCRRIIPPTTIIOONN #

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

IInnppuutt Input values to these routines may be any scalar number or string that looks like a number and represents an integer. Anything that is accepted by Perl as a literal numeric constant should be accepted by this module, except that finite non-integers return NaN.

 •   Leading and trailing whitespace is ignored.

 •   Leading zeros are ignored, except for floating point numbers with a
     binary exponent, in which case the number is interpreted as an octal
     floating point number. For example, "01.4p+0" gives 1.5, "00.4p+0"
     gives 0.5, but "0.4p+0" gives a NaN. And while "0377" gives 255,
     "0377p0" gives 255.

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

 •   If the string has a "0o" or "0O" prefix, it is interpreted as an
     octal number. A floating point literal with a "0" prefix is also
     interpreted as an octal number.

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

 •   Underline characters are allowed in the same way as they are allowed
     in literal numerical constants.

 •   If the string can not be interpreted, or does not represent a finite
     integer, NaN is returned.

 •   For hexadecimal, octal, and binary floating point numbers, the
     exponent must be separated from the significand (mantissa) by the
     letter "p" or "P", not "e" or "E" as with decimal numbers.

 Some examples of valid string input

     Input string                Resulting value

     123                         123
     1.23e2                      123
     12300e-2                    123

     67_538_754                  67538754
     -4_5_6.7_8_9e+0_1_0         -4567890000000

     0x13a                       314
     0x13ap0                     314
     0x1.3ap+8                   314
     0x0.00013ap+24              314
     0x13a000p-12                314

     0o472                       314
     0o1.164p+8                  314
     0o0.0001164p+20             314
     0o1164000p-10               314

     0472                        472     Note!
     01.164p+8                   314
     00.0001164p+20              314
     01164000p-10                314

     0b100111010                 314
     0b1.0011101p+8              314
     0b0.00010011101p+12         314
     0b100111010000p-3           314

 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-"nneeww(())> (no input argument) and
 "Math::BigInt-"new("")> return 0. This might change in the future, so
 always use the following explicit forms to get a zero:

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

OOuuttppuutt 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.

MMEETTHHOODDSS #

CCoonnffiigguurraattiioonn mmeetthhooddss Each of the methods below (except ccoonnffiigg(()), aaccccuurraaccyy(()) and pprreecciissiioonn(())) 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.

 aaccccuurraaccyy(())
         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
     "rroouunndd(())", "bbrroouunndd(())" or "bbffrroouunndd(())" 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.

 pprreecciissiioonn(())
         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 "aaccccuurraaccyy(())" instead. With "aaccccuurraaccyy(())" you set
     the number of digits each result should have, with "pprreecciissiioonn(())" 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.

 ddiivv__ssccaallee(())
     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.

 rroouunndd__mmooddee(())
     Set/get the rounding mode.

 uuppggrraaddee(())
     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", i.e., no upgrading.

         # with no upgrading
         $x = Math::BigInt->new(12);
         $y = Math::BigInt->new(5);
         print $x / $y, "\n";                # 2 as a Math::BigInt

         # with upgrading to Math::BigFloat
         Math::BigInt -> upgrade("Math::BigFloat");
         print $x / $y, "\n";                # 2.4 as a Math::BigFloat

         # with upgrading to Math::BigRat (after loading Math::BigRat)
         Math::BigInt -> upgrade("Math::BigRat");
         print $x / $y, "\n";                # 12/5 as a Math::BigRat

 ddoowwnnggrraaddee(())
     Set/get the class for downgrading. The default is "undef", i.e., no
     downgrading. Downgrading is not done by Math::BigInt.

 mmooddiiffyy(())
         $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.

 ccoonnffiigg(())
         Math::BigInt->config("trap_nan" => 1);      # set
         $accu = Math::BigInt->config("accuracy");   # get

     Set or get class variables. Read-only parameters are marked as RO.
     Read-write parameters are marked as RW. The following parameters are
     supported.

         Parameter       RO/RW   Description
                                 Example
         ============================================================
         lib             RO      Name of the math backend library
                                 Math::BigInt::Calc
         lib_version     RO      Version of the math backend library
                                 0.30
         class           RO      The class of config you just called
                                 Math::BigRat
         version         RO      version number of the class you used
                                 0.10
         upgrade         RW      To which class numbers are upgraded
                                 undef
         downgrade       RW      To which class numbers are downgraded
                                 undef
         precision       RW      Global precision
                                 undef
         accuracy        RW      Global accuracy
                                 undef
         round_mode      RW      Global round mode
                                 even
         div_scale       RW      Fallback accuracy for division etc.
                                 40
         trap_nan        RW      Trap NaNs
                                 undef
         trap_inf        RW      Trap +inf/-inf
                                 undef

CCoonnssttrruuccttoorr mmeetthhooddss nneeww(()) $x = Math::BigInt->new($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'), octal (with leading ('0o') or binary (with
     leading '0b').

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

 ffrroomm__ddeecc(())
         $x = Math::BigInt->from_dec("314159");    # input is decimal

     Interpret input as a decimal. It is equivalent to nneeww(()), but does not
     accept anything but strings representing finite, decimal numbers.

 ffrroomm__hheexx(())
         $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.

 ffrroomm__oocctt(())
         $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.

 ffrroomm__bbiinn(())
         $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.

 ffrroomm__bbyytteess(())
         $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, ffrroomm__bbyytteess(()) matches the conversion done by
     uunnppaacckk(()):

         $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

 ffrroomm__bbaassee(())
     Given a string, a base, and an optional collation sequence, interpret
     the string as a number in the given base. The collation sequence
     describes the value of each character in the string.

     If a collation sequence is not given, a default collation sequence is
     used. If the base is less than or equal to 36, the collation sequence
     is the string consisting of the 36 characters "0" to "9" and "A" to
     "Z". In this case, the letter case in the input is ignored. If the
     base is greater than 36, and smaller than or equal to 62, the
     collation sequence is the string consisting of the 62 characters "0"
     to "9", "A" to "Z", and "a" to "z". A base larger than 62 requires
     the collation sequence to be specified explicitly.

     These examples show standard binary, octal, and hexadecimal
     conversion. All cases return 250.

         $x = Math::BigInt->from_base("11111010", 2);
         $x = Math::BigInt->from_base("372", 8);
         $x = Math::BigInt->from_base("fa", 16);

     When the base is less than or equal to 36, and no collation sequence
     is given, the letter case is ignored, so both of these also return
     250:

         $x = Math::BigInt->from_base("6Y", 16);
         $x = Math::BigInt->from_base("6y", 16);

     When the base greater than 36, and no collation sequence is given,
     the default collation sequence contains both uppercase and lowercase
     letters, so the letter case in the input is not ignored:

         $x = Math::BigInt->from_base("6S", 37);         # $x is 250
         $x = Math::BigInt->from_base("6s", 37);         # $x is 276
         $x = Math::BigInt->from_base("121", 3);         # $x is 16
         $x = Math::BigInt->from_base("XYZ", 36);        # $x is 44027
         $x = Math::BigInt->from_base("Why", 42);        # $x is 58314

     The collation sequence can be any set of unique characters. These two
     cases are equivalent

         $x = Math::BigInt->from_base("100", 2, "01");   # $x is 4
         $x = Math::BigInt->from_base("|--", 2, "-|");   # $x is 4

 ffrroomm__bbaassee__nnuumm(())
     Returns a new Math::BigInt object given an array of values and a
     base. This method is equivalent to "from_base()", but works on
     numbers in an array rather than characters in a string. Unlike
     "from_base()", all input values may be arbitrarily large.

         $x = Math::BigInt->from_base_num([1, 1, 0, 1], 2)     # $x is 13
         $x = Math::BigInt->from_base_num([3, 125, 39], 128)   # $x is 65191

 bbzzeerroo(())
         $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.

 bboonnee(())
         $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.

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

 bbnnaann(())
         $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();

 bbppii(())
         $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....

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

 aass__iinntt(())
 aass__nnuummbbeerr(())
     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()".

BBoooolleeaann mmeetthhooddss None of these methods modify the invocand object.

 iiss__zzeerroo(())
         $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.

 iiss__ffiinniittee(())
         $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.

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

 iiss__ppoossiittiivvee(())
 iiss__ppooss(())
         $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.

 iiss__nneeggaattiivvee(())
 iiss__nneegg(())
         $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.

 iiss__nnoonn__ppoossiittiivvee(())
         $x->is_non_positive();      # true if <= 0

     Returns true if the invocand is negative or zero.

 iiss__nnoonn__nneeggaattiivvee(())
         $x->is_non_negative();      # true if >= 0

     Returns true if the invocand is positive or zero.

 iiss__oodddd(())
         $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.

 iiss__eevveenn(())
         $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.

 iiss__iinntt(())
         $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.

CCoommppaarriissoonn mmeetthhooddss 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”.

 bbccmmpp(())
         $x->bcmp($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.

 bbaaccmmpp(())
         $x->bacmp($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.

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

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

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

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

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

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

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

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

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

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

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

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

AArriitthhmmeettiicc mmeetthhooddss These methods modify the invocand object and returns it.

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

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

 bbaabbss(())
         $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.

 bbssggnn(())
         $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.

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

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

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

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

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

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

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

 bbmmuullaadddd(())
         $x->bmuladd($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).

 bbddiivv(())
         $x->bdiv($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.

 bbttddiivv(())
         $x->btdiv($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.

 bbmmoodd(())
         $x->bmod($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 "bbddiivv(())".

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

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

 bbmmooddiinnvv(())
         $x->bmodinv($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.

 bbmmooddppooww(())
         $num->bmodpow($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)

 bbppooww(())
         $x->bpow($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.

 bblloogg(())
         $x->blog($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

 bbeexxpp(())
         $x->bexp($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 "bblloogg(())".

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

     Calculates the binomial coefficient n over k, also called the
     "choose" function, which is

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

     when n and k are non-negative. This method implements the full
     Kronenburg extension (Kronenburg, M.J. "The Binomial Coefficient for
     Negative Arguments."  18 May 2011. http://arxiv.org/abs/1105.3689/)
     illustrated by the following pseudo-code:

         if n >= 0 and k >= 0:
             return binomial(n, k)
         if k >= 0:
             return (-1)^k*binomial(-n+k-1, k)
         if k <= n:
             return (-1)^(n-k)*binomial(-k-1, n-k)
         else
             return 0

     The behaviour is identical to the behaviour of the Maple and
     Mathematica function for negative integers n, k.

 bbuuppaarrrrooww(())
 uuppaarrrrooww(())
         $a -> buparrow($n, $b);         # modifies $a
         $x = $a -> uparrow($n, $b);     # does not modify $a

     This method implements Knuth's up-arrow notation, where $n is a non-
     negative integer representing the number of up-arrows. $n = 0 gives
     multiplication, $n = 1 gives exponentiation, $n = 2 gives tetration,
     $n = 3 gives hexation etc. The following illustrates the relation
     between the first values of $n.

     See <https://en.wikipedia.org/wiki/Knuth%27s_up-arrow_notation>.

 bbaacckkeerrmmaannnn(())
 aacckkeerrmmaannnn(())
         $m -> backermann($n);           # modifies $a
         $x = $m -> ackermann($n);       # does not modify $a

     This method implements the Ackermann function:

                  / n + 1              if m = 0
        A(m, n) = | A(m-1, 1)          if m > 0 and n = 0
                  \ A(m-1, A(m, n-1))  if m > 0 and n > 0

     Its value grows rapidly, even for small inputs. For example, A(4, 2)
     is an integer of 19729 decimal digits.

     See https://en.wikipedia.org/wiki/Ackermann_function

 bbssiinn(())
         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).

 bbccooss(())
         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).

 bbaattaann(())
         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).

 bbaattaann22(())
         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).

 bbssqqrrtt(())
         $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

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

     Calculates the N'th root of $x.

 bbffaacc(())
         $x->bfac();             # factorial of $x

     Returns the factorial of $x, i.e., $x*($x-1)*($x-2)*...*2*1, the
     product of all positive integers up to and including $x. $x must be >
     -1. The factorial of N is commonly written as N!, or N!1, when using
     the multifactorial notation.

 bbddffaacc(())
         $x->bdfac();                # double factorial of $x

     Returns the double factorial of $x, i.e., $x*($x-2)*($x-4)*... $x
     must be > -2. The double factorial of N is commonly written as N!!,
     or N!2, when using the multifactorial notation.

 bbttffaacc(())
         $x->btfac();            # triple factorial of $x

     Returns the triple factorial of $x, i.e., $x*($x-3)*($x-6)*... $x
     must be > -3. The triple factorial of N is commonly written as N!!!,
     or N!3, when using the multifactorial notation.

 bbmmffaacc(())
         $x->bmfac($k);          # $k'th multifactorial of $x

     Returns the multi-factorial of $x, i.e., $x*($x-$k)*($x-2*$k)*... $x
     must be > -$k. The multi-factorial of N is commonly written as N!K.

 bbffiibb(())
         $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

 bblluuccaass(())
         $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

 bbrrssfftt(())
         $x->brsft($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))

 bbllssfftt(())
         $x->blsft($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))

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

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

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

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

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

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

RRoouunnddiinngg mmeetthhooddss rroouunndd(()) $x->round($A,$P,$round_mode);

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

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

     Rounds $x to an accuracy of $N digits.

 bbffrroouunndd(())
         $x->bfround($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

 bbfflloooorr(())
         $x->bfloor();

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

 bbcceeiill(())
         $x->bceil();

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

 bbiinntt(())
         $x->bint();

     Round $x towards zero.

OOtthheerr mmaatthheemmaattiiccaall mmeetthhooddss bbggccdd(()) $x -> bgcd($y); # GCD of $x and $y $x -> bgcd($y, $z, …); # GCD of $x, $y, $z, …

     Returns the greatest common divisor (GCD).

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

     Returns the least common multiple (LCM).

OObbjjeecctt pprrooppeerrttyy mmeetthhooddss ssiiggnn(()) $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'

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

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

 ddiiggiittssuumm(())
         $x->digitsum();

     Computes the sum of the base 10 digits and returns it.

 bbddiiggiittssuumm(())
         $x->bdigitsum();

     Computes the sum of the base 10 digits and assigns the result to the
     invocand.

 lleennggtthh(())
         $x->length();
         ($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

 mmaannttiissssaa(())
         $x->mantissa();

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

 eexxppoonneenntt(())
         $x->exponent();

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

 ppaarrttss(())
         $x->parts();

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

 ssppaarrttss(())
     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()".

 nnppaarrttss(())
     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.

 eeppaarrttss(())
     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.

 ddppaarrttss(())
     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()".

 ffppaarrttss(())
     Returns the smallest possible numerator and denominator so that the
     numerator divided by the denominator gives back the original value.
     For finite numbers, both values are integers. Mnemonic: fraction.

 nnuummeerraattoorr(())
     Together with "ddeennoommiinnaattoorr(())", returns the smallest integers so that
     the numerator divided by the denominator reproduces the original
     value. With Math::BigInt, nnuummeerraattoorr(()) simply returns a copy of the
     invocand.

 ddeennoommiinnaattoorr(())
     Together with "nnuummeerraattoorr(())", returns the smallest integers so that
     the numerator divided by the denominator reproduces the original
     value. With Math::BigInt, ddeennoommiinnaattoorr(()) always returns either a 1 or
     a NaN.

SSttrriinngg ccoonnvveerrssiioonn mmeetthhooddss bbssttrr(()) 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.

 bbssssttrr(())
     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"

 bbnnssttrr(())
     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 greater 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"

 bbeessttrr(())
     Returns a string representing the number using engineering notation.
     For finite non-zero numbers, the absolute value of the significand is
     greater 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"

 bbddssttrr(())
     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"

 ttoo__hheexx(())
         $x->to_hex();

     Returns a hexadecimal string representation of the number. See also
     ffrroomm__hheexx(()).

 ttoo__bbiinn(())
         $x->to_bin();

     Returns a binary string representation of the number. See also
     ffrroomm__bbiinn(()).

 ttoo__oocctt(())
         $x->to_oct();

     Returns an octal string representation of the number. See also
     ffrroomm__oocctt(()).

 ttoo__bbyytteess(())
         $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. See
     also ffrroomm__bbyytteess(()).

 ttoo__bbaassee(())
         $x = Math::BigInt->new("250");
         $x->to_base(2);     # returns "11111010"
         $x->to_base(8);     # returns "372"
         $x->to_base(16);    # returns "fa"

     Returns a string representation of the number in the given base. If a
     collation sequence is given, the collation sequence determines which
     characters are used in the output.

     Here are some more examples

         $x = Math::BigInt->new("16")->to_base(3);       # returns "121"
         $x = Math::BigInt->new("44027")->to_base(36);   # returns "XYZ"
         $x = Math::BigInt->new("58314")->to_base(42);   # returns "Why"
         $x = Math::BigInt->new("4")->to_base(2, "-|");  # returns "|--"

     See ffrroomm__bbaassee(()) for information and examples.

 ttoo__bbaassee__nnuumm(())
     Converts the given number to the given base. This method is
     equivalent to "_to_base()", but returns numbers in an array rather
     than characters in a string. In the output, the first element is the
     most significant. Unlike "_to_base()", all input values may be
     arbitrarily large.

         $x = Math::BigInt->new(13);
         $x->to_base_num(2);                         # returns [1, 1, 0, 1]

         $x = Math::BigInt->new(65191);
         $x->to_base_num(128);                       # returns [3, 125, 39]

 aass__hheexx(())
         $x->as_hex();

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

 aass__bbiinn(())
         $x->as_bin();

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

 aass__oocctt(())
         $x->as_oct();

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

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

OOtthheerr ccoonnvveerrssiioonn mmeetthhooddss nnuummiiffyy(()) print $x->numify();

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

UUttiilliittyy mmeetthhooddss These utility methods are made public

 ddeecc__ssttrr__ttoo__ddeecc__fflltt__ssttrr(())
     Takes a string representing any valid number using decimal notation
     and converts it to a string representing the same number using
     decimal floating point notation. The output consists of five parts
     joined together: the sign of the significand, the absolute value of
     the significand as the smallest possible integer, the letter "e", the
     sign of the exponent, and the absolute value of the exponent. If the
     input is invalid, nothing is returned.

         $str2 = $class -> dec_str_to_dec_flt_str($str1);

     Some examples

         Input           Output
         31400.00e-4     +314e-2
         -0.00012300e8   -123e+2
         0               +0e+0

 hheexx__ssttrr__ttoo__ddeecc__fflltt__ssttrr(())
     Takes a string representing any valid number using hexadecimal
     notation and converts it to a string representing the same number
     using decimal floating point notation. The output has the same format
     as that of "ddeecc__ssttrr__ttoo__ddeecc__fflltt__ssttrr(())".

         $str2 = $class -> hex_str_to_dec_flt_str($str1);

     Some examples

         Input           Output
         0xff            +255e+0

     Some examples

 oocctt__ssttrr__ttoo__ddeecc__fflltt__ssttrr(())
     Takes a string representing any valid number using octal notation and
     converts it to a string representing the same number using decimal
     floating point notation. The output has the same format as that of
     "ddeecc__ssttrr__ttoo__ddeecc__fflltt__ssttrr(())".

         $str2 = $class -> oct_str_to_dec_flt_str($str1);

 bbiinn__ssttrr__ttoo__ddeecc__fflltt__ssttrr(())
     Takes a string representing any valid number using binary notation
     and converts it to a string representing the same number using
     decimal floating point notation. The output has the same format as
     that of "ddeecc__ssttrr__ttoo__ddeecc__fflltt__ssttrr(())".

         $str2 = $class -> bin_str_to_dec_flt_str($str1);

 ddeecc__ssttrr__ttoo__ddeecc__ssttrr(())
     Takes a string representing any valid number using decimal notation
     and converts it to a string representing the same number using
     decimal notation. If the number represents an integer, the output
     consists of a sign and the absolute value. If the number represents a
     non-integer, the output consists of a sign, the integer part of the
     number, the decimal point ".", and the fraction part of the number
     without any trailing zeros. If the input is invalid, nothing is
     returned.

 hheexx__ssttrr__ttoo__ddeecc__ssttrr(())
     Takes a string representing any valid number using hexadecimal
     notation and converts it to a string representing the same number
     using decimal notation. The output has the same format as that of
     "ddeecc__ssttrr__ttoo__ddeecc__ssttrr(())".

 oocctt__ssttrr__ttoo__ddeecc__ssttrr(())
     Takes a string representing any valid number using octal notation and
     converts it to a string representing the same number using decimal
     notation. The output has the same format as that of
     "ddeecc__ssttrr__ttoo__ddeecc__ssttrr(())".

 bbiinn__ssttrr__ttoo__ddeecc__ssttrr(())
     Takes a string representing any valid number using binary notation
     and converts it to a string representing the same number using
     decimal notation. The output has the same format as that of
     "ddeecc__ssttrr__ttoo__ddeecc__ssttrr(())".

AACCCCUURRAACCYY aanndd PPRREECCIISSIIOONN 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.

PPrreecciissiioonn PP 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.

AAccccuurraaccyy AA 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.

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

RRoouunnddiinngg mmooddee RR When rounding a number, different ‘styles’ or ‘kinds’ of rounding are possible. (Note that random rounding, as in Math::Round, is not implemented.)

 _D_i_r_e_c_t_e_d _r_o_u_n_d_i_n_g

 These round modes always round in the same direction.

 'trunc'
     Round towards zero. Remove all digits following the rounding place,
     i.e., replace them with zeros. Thus, 987.65 rounded to tens (P=1)
     becomes 980, and rounded to the fourth significant digit becomes
     987.6 (A=4). 123.456 rounded to the second place after the decimal
     point (P=-2) becomes 123.46. This corresponds to the IEEE 754
     rounding mode 'roundTowardZero'.

 _R_o_u_n_d_i_n_g _t_o _n_e_a_r_e_s_t

 These rounding modes round to the nearest digit. They differ in how they
 determine which way to round in the ambiguous case when there is a tie.

 'even'
     Round towards the nearest even digit, e.g., when rounding to nearest
     integer, -5.5 becomes -6, 4.5 becomes 4, but 4.501 becomes 5. This
     corresponds to the IEEE 754 rounding mode 'roundTiesToEven'.

 'odd'
     Round towards the nearest odd digit, e.g., when rounding to nearest
     integer, 4.5 becomes 5, -5.5 becomes -5, but 5.501 becomes 6. This
     corresponds to the IEEE 754 rounding mode 'roundTiesToOdd'.

 '+inf'
     Round towards plus infinity, i.e., always round up. E.g., when
     rounding to the nearest integer, 4.5 becomes 5, -5.5 becomes -5, and
     4.501 also becomes 5. This corresponds to the IEEE 754 rounding mode
     'roundTiesToPositive'.

 '-inf'
     Round towards minus infinity, i.e., always round down. E.g., when
     rounding to the nearest integer, 4.5 becomes 4, -5.5 becomes -6, but
     4.501 becomes 5. This corresponds to the IEEE 754 rounding mode
     'roundTiesToNegative'.

 'zero'
     Round towards zero, i.e., round positive numbers down and negative
     numbers up.  E.g., when rounding to the nearest integer, 4.5 becomes
     4, -5.5 becomes -5, but 4.501 becomes 5. This corresponds to the IEEE
     754 rounding mode 'roundTiesToZero'.

 'common'
     Round away from zero, i.e., round to the number with the largest
     absolute value. E.g., when rounding to the nearest integer, -1.5
     becomes -2, 1.5 becomes 2 and 1.49 becomes 1. This corresponds to the
     IEEE 754 rounding mode 'roundTiesToAway'.

 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

IInnffiinniittyy aanndd NNoott aa NNuummbbeerr While Math::BigInt has extensive handling of inf and NaN, certain quirks remain.

 oocctt(())/hheexx(())
     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->bbiinnff(())
     objects. Since overloading these routines is not possible, this
     cannot be fixed from Math::BigInt.

IINNTTEERRNNAALLSS #

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

MMAATTHH LLIIBBRRAARRYY #

 The mathematical computations are performed by a backend library. It is
 not required to specify which backend library to use, but some backend
 libraries are much faster than the default library.

 _T_h_e _d_e_f_a_u_l_t _l_i_b_r_a_r_y

 The default library is Math::BigInt::Calc, which is implemented in pure
 Perl and hence does not require a compiler.

 _S_p_e_c_i_f_y_i_n_g _a _l_i_b_r_a_r_y

 The simple case

     use Math::BigInt;

 is equivalent to saying

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

 You can use a different backend library with, e.g.,

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

 which attempts to load the Math::BigInt::GMP library, and falls back to
 the default library if the specified library can't be loaded.

 Multiple libraries can be specified by separating them by a comma, e.g.,

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

 If you request a specific set of libraries and do not allow fallback to
 the default library, specify them using "only",

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

 If you prefer a specific set of libraries, but want to see a warning if
 the fallback library is used, specify them using "lib",

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

 The following first tries to find Math::BigInt::Foo, then
 Math::BigInt::Bar, and if this also fails, reverts to Math::BigInt::Calc:

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

 _W_h_i_c_h _l_i_b_r_a_r_y _t_o _u_s_e_?

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

 Math::BigInt::GMP, Math::BigInt::Pari, and Math::BigInt::GMPz are in
 cases involving big numbers much faster than Math::BigInt::Calc. However
 these libraries are 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,
 so mixing them won't work. You should not depend on the number having a
 specific internal format.

 See the respective math library module documentation for further details.

 _L_o_a_d_i_n_g _m_u_l_t_i_p_l_e _l_i_b_r_a_r_i_e_s

 The first library that is successfully loaded is the one that will be
 used. Any further attempts at loading a different module will be ignored.
 This is to avoid the situation where module A requires math library X,
 and module B requires math library Y, causing modules A and B to be
 incompatible. For example,

     use Math::BigInt;                   # loads default "Calc"
     use Math::BigFloat only => "GMP";   # ignores "GMP"

SSIIGGNN #

 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.

EEXXAAMMPPLLEESS #

   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 ("2" becomes a Math::BigInt)
   $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";

NNUUMMEERRIICC LLIITTEERRAALLSS #

 After "use Math::BigInt ':constant'" all numeric literals in the given
 scope are converted to "Math::BigInt" objects. This conversion happens at
 compile time. Every non-integer is convert to a NaN.

 For example,

     perl -MMath::BigInt=:constant -le 'print 2**150'

 prints the exact value of "2**150". Note that without conversion of
 constants to objects the expression "2**150" is calculated using Perl
 scalars, which leads to an inaccurate result.

 Please note that strings are not affected, so that

     use Math::BigInt qw/:constant/;

     $x = "1234567890123456789012345678901234567890"
             + "123456789123456789";

 does give you what you expect. You need an explicit Math::BigInt->nneeww(())
 around at least one of the operands. You should also quote large
 constants to prevent loss of precision:

     use Math::BigInt;

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

 Without the quotes Perl first converts the large number to a floating
 point constant at compile time, and then converts the result to a
 Math::BigInt object at run time, which results in an inaccurate result.

HHeexxaaddeecciimmaall,, ooccttaall,, aanndd bbiinnaarryy ffllooaattiinngg ppooiinntt lliitteerraallss Perl (and this module) accepts hexadecimal, octal, and binary floating point literals, but use them with care with Perl versions before v5.32.0, because some versions of Perl silently give the wrong result. Below are some examples of different ways to write the number decimal 314.

 Hexadecimal floating point literals:

     0x1.3ap+8         0X1.3AP+8
     0x1.3ap8          0X1.3AP8
     0x13a0p-4         0X13A0P-4

 Octal floating point literals (with "0" prefix):

     01.164p+8         01.164P+8
     01.164p8          01.164P8
     011640p-4         011640P-4

 Octal floating point literals (with "0o" prefix) (requires v5.34.0):

     0o1.164p+8        0O1.164P+8
     0o1.164p8         0O1.164P8
     0o11640p-4        0O11640P-4

 Binary floating point literals:

     0b1.0011101p+8    0B1.0011101P+8
     0b1.0011101p8     0B1.0011101P8
     0b10011101000p-2  0B10011101000P-2

PPEERRFFOORRMMAANNCCEE #

 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.

AAlltteerrnnaattiivvee mmaatthh lliibbrraarriieess 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>.

SSUUBBCCLLAASSSSIINNGG #

SSuubbccllaassssiinngg MMaatthh::::BBiiggIInntt 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 bbaadddd(()). 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.

UUPPGGRRAADDIINNGG #

 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 ssqqrrtt(3)).

 Beware: This feature is not fully implemented yet.

AAuuttoo--uuppggrraaddee 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.

EEXXPPOORRTTSS #

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

     bgcd
     blcm

CCAAVVEEAATTSS #

 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 bbssttrr(()) 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.

 iinntt(())
     "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 ssaammee 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->bbnneegg(()) 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->bbddiivv(()) 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->nneeww(()):

         $float = Math::BigFloat->new($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.

BBUUGGSS #

 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.

SSUUPPPPOORRTT #

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

     perldoc Math::BigInt

 You can also look for information at:

 •   GitHub

     <https://github.com/pjacklam/p5-Math-BigInt>

 •   RT: CPAN's request tracker

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

 •   MetaCPAN

     <https://metacpan.org/release/Math-BigInt>

 •   CPAN Testers Matrix

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

 •   CPAN Ratings

     <https://cpanratings.perl.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>

LLIICCEENNSSEE #

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

SSEEEE AALLSSOO #

 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.

AAUUTTHHOORRSS #

 •   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@gmail.com>, 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!

perl v5.36.3 2023-02-15 Math::BigInt(3p)