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Math::BigInt

Section: User Contributed Perl Documentation (3)
Updated: 2018-11-08
Index

NAME

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

SYNOPSIS

  use Math::BigInt;  # or make it faster with huge numbers: install (optional)  # Math::BigInt::GMP and always use (it falls back to  # pure Perl if the GMP library is not installed):  # (See also the L<MATH LIBRARY> section!)  # warns if Math::BigInt::GMP cannot be found  use Math::BigInt lib => 'GMP';  # to suppress the warning use this:  # use Math::BigInt try => 'GMP';  # dies if GMP cannot be loaded:  # use Math::BigInt only => 'GMP';  my $str = '1234567890';  my @values = (64, 74, 18);  my $n = 1; my $sign = '-';  # Configuration methods (may be used as class methods and instance methods)  Math::BigInt->accuracy();     # get class accuracy  Math::BigInt->accuracy($n);   # set class accuracy  Math::BigInt->precision();    # get class precision  Math::BigInt->precision($n);  # set class precision  Math::BigInt->round_mode();   # get class rounding mode  Math::BigInt->round_mode($m); # set global round mode, must be one of                                # 'even', 'odd', '+inf', '-inf', 'zero',                                # 'trunc', or 'common'  Math::BigInt->config();       # return hash with configuration  # Constructor methods (when the class methods below are used as instance  # methods, the value is assigned the invocand)  $x = Math::BigInt->new($str);             # defaults to 0  $x = Math::BigInt->new('0x123');          # from hexadecimal  $x = Math::BigInt->new('0b101');          # from binary  $x = Math::BigInt->from_hex('cafe');      # from hexadecimal  $x = Math::BigInt->from_oct('377');       # from octal  $x = Math::BigInt->from_bin('1101');      # from binary  $x = Math::BigInt->from_base('why', 36);  # 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->bsin();             # sine  $x->bcos();             # cosine  $x->batan();            # inverse tangent  $x->batan2($y);         # two-argument inverse tangent  $x->bsqrt();            # calculate square root  $x->broot($y);          # $y'th root of $x (e.g. $y == 3 => cubic root)  $x->bfac();             # factorial of $x (1*2*3*4*..$x)  $x->blsft($n);          # left shift $n places in base 2  $x->blsft($n,$b);       # left shift $n places in base $b                          # returns (quo,rem) or quo (scalar context)  $x->brsft($n);          # right shift $n places in base 2  $x->brsft($n,$b);       # right shift $n places in base $b                          # returns (quo,rem) or quo (scalar context)  # Bitwise methods  $x->band($y);           # bitwise and  $x->bior($y);           # bitwise inclusive or  $x->bxor($y);           # bitwise exclusive or  $x->bnot();             # bitwise not (two's complement)  # Rounding methods  $x->round($A,$P,$mode); # round to accuracy or precision using                          # rounding mode $mode  $x->bround($n);         # accuracy: preserve $n digits  $x->bfround($n);        # $n > 0: round to $nth digit left of dec. point                          # $n < 0: round to $nth digit right of dec. point  $x->bfloor();           # round towards minus infinity  $x->bceil();            # round towards plus infinity  $x->bint();             # round towards zero  # Other mathematical methods  $x->bgcd($y);            # greatest common divisor  $x->blcm($y);            # least common multiple  # Object property methods (do not modify the invocand)  $x->sign();              # the sign, either +, - or NaN  $x->digit($n);           # the nth digit, counting from the right  $x->digit(-$n);          # the nth digit, counting from the left  $x->length();            # return number of digits in number  ($xl,$f) = $x->length(); # length of number and length of fraction                           # part, latter is always 0 digits long                           # for Math::BigInt objects  $x->mantissa();          # return (signed) mantissa as a Math::BigInt  $x->exponent();          # return exponent as a Math::BigInt  $x->parts();             # return (mantissa,exponent) as a Math::BigInt  $x->sparts();            # mantissa and exponent (as integers)  $x->nparts();            # mantissa and exponent (normalised)  $x->eparts();            # mantissa and exponent (engineering notation)  $x->dparts();            # integer and fraction part  # Conversion methods (do not modify the invocand)  $x->bstr();         # decimal notation, possibly zero padded  $x->bsstr();        # string in scientific notation with integers  $x->bnstr();        # string in normalized notation  $x->bestr();        # string in engineering notation  $x->bdstr();        # string in decimal notation  $x->to_hex();       # as signed hexadecimal string  $x->to_bin();       # as signed binary string  $x->to_oct();       # as signed octal string  $x->to_bytes();     # as byte string  $x->to_base($b);    # as string 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)

DESCRIPTION

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

Input

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

*: Leading and trailing whitespace is ignored.
*: Leading and trailing zeros are ignored.
*: If the string has a ``0x'' prefix, it is interpreted as a hexadecimal number.
*: If the string has a ``0b'' prefix, it is interpreted as a binary number.
*: One underline is allowed between any two digits.
*: If the string can not be interpreted, NaN is returned.

Octal numbers are typically prefixed by ``0'', but since leading zeros arestripped, these methods can not automatically recognize octal numbers, so usethe constructor from_oct() to interpret octal strings.

Some examples of valid string input

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

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

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

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

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

Output

Output values are usually Math::BigInt objects.

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

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

METHODS

Configuration methods

Each of the methods below (except config(), accuracy() and precision()) acceptsthree 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 createdafterwards.

accuracy()

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

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

Alternatively, one can round the results explicitly using one of ``round()'',``bround()'' or ``bfround()'' or by passing the desired accuracy to the methodas 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 ispossible to subclass Math::BigInt and make the globals of the subclass aliasesto the ones from Math::BigInt.

precision()

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

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

You might want to use ``accuracy()'' instead. With ``accuracy()'' you set thenumber of digits each result should have, with ``precision()'' you set theplace 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 ispossible to subclass Math::BigInt and make the globals of the subclass aliasesto the ones from Math::BigInt.

div_scale()

Set/get the fallback accuracy. This is the accuracy used when neither accuracynor precision is set explicitly. It is used when a computation might otherwiseattempt to return an infinite number of digits.

round_mode()

Set/get the rounding mode.

upgrade()

Set/get the class for upgrading. When a computation might result in anon-integer, the operands are upgraded to this class. This is used for instanceby bignum. The default is "undef", thus the following operation createsa Math::BigInt, not a Math::BigFloat:

    my $i = Math::BigInt->new(123);    my $f = Math::BigFloat->new('123.1');    print $i + $f, "\n";                # prints 246

downgrade()

Set/get the class for downgrading. The default is "undef". Downgrading is notdone by Math::BigInt.

modify()

    $x->modify('bpowd');

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

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

config()

    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-writeparameters 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

Constructor methods

new()

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

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

See ``Input'' for more info on accepted input formats.

from_hex()

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

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

from_oct()

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

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

from_bin()

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

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

from_bytes()

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

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

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

    $b = "\x4e";                             # one char byte string    $x = Math::BigInt->from_bytes($b);       # = 78    $y = unpack "C", $b;                     # ditto, but scalar    $b = "\xf3\x6b";                         # two char byte string    $x = Math::BigInt->from_bytes($b);       # = 62315    $y = unpack "S>", $b;                    # ditto, but scalar    $b = "\x2d\xe0\x49\xad";                 # four char byte string    $x = Math::BigInt->from_bytes($b);       # = 769673645    $y = unpack "L>", $b;                    # ditto, but scalar    $b = "\x2d\xe0\x49\xad\x2d\xe0\x49\xad"; # eight char byte string    $x = Math::BigInt->from_bytes($b);       # = 3305723134637787565    $y = unpack "Q>", $b;                    # ditto, but scalar

from_base()

Given a string, a base, and an optional collation sequence, interpret thestring as a number in the given base. The collation sequence describes thevalue of each character in the string.

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

These examples show standard binary, octal, and hexadecimal conversion. Allcases 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 defaultcollation sequence contains both uppercase and lowercase letters, sothe 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 casesare equivalent

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

bzero()

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

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

bone()

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

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

binf()

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

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

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

bnan()

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

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

    $x->bnan();

bpi()

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

Creates a new Math::BigInt object representing PI. If used as an instancemethod, assigns the value to the invocand. With Math::BigInt this alwaysreturns 3.

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

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

copy()

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

as_int()

as_number()

These methods are called when Math::BigInt encounters an object it doesn't knowhow 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 byfirst checking if $y is something that $x might be upgraded to. If that is thecase, no further attempts are made. The next is to see if $y supports themethod "as_int()". If it does, "as_int()" is called, but if it doesn't, thenext 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()") isexpected to return either an object that has the same class as $x, a subclassthereof, or a string that "ref($x)->new()" can parse to create an object.

"as_number()" is an alias to "as_int()". "as_number" was introduced inv1.22, while "as_int()" was introduced in v1.68.

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

Boolean methods

None of these methods modify the invocand object.

is_zero()

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

Returns true if the invocand is zero and false otherwise.

is_one( [ SIGN ])

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

Returns true if the invocand is one and false otherwise.

is_finite()

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

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

is_inf( [ SIGN ] )

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

Returns true if the invocand is infinite and false otherwise.

is_nan()

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

is_positive()

is_pos()

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

Returns true if the invocand is positive and false otherwise. A "NaN" isneither positive nor negative.

is_negative()

is_neg()

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

Returns true if the invocand is negative and false otherwise. A "NaN" isneither positive nor negative.

is_odd()

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

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

is_even()

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

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

is_int()

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

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

Comparison methods

None of these methods modify the invocand object. Note that a "NaN" is neitherless than, greater than, or equal to anything else, even a "NaN".

bcmp()

    $x->bcmp($y);

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

bacmp()

    $x->bacmp($y);

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

beq()

    $x -> beq($y);

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

bne()

    $x -> bne($y);

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

blt()

    $x -> blt($y);

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

ble()

    $x -> ble($y);

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

bgt()

    $x -> bgt($y);

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

bge()

    $x -> bge($y);

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

Arithmetic methods

These methods modify the invocand object and returns it.

bneg()

    $x->bneg();

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

babs()

    $x->babs();

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

bsgn()

    $x->bsgn();

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

bnorm()

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

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

binc()

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

bdec()

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

badd()

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

bsub()

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

bmul()

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

bmuladd()

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

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

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

bdiv()

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

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

The quotient is always the greatest integer less than or equal to thereal-valued quotient of the two operands, and the remainder (when it isnon-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'sbuilt-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 isbecause under 'use integer' Perl does what the underlying C library thinks isright, and this varies. However, ``use integer'' does not change the way thingsare done with Math::BigInt objects.

btdiv()

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

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

bmod()

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

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

btmod()

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

Returns the remainer after truncated division (T-division). See ``btdiv()''.

bmodinv()

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

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

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

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

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

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

bmodpow()

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

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

    $num ** $exp % $mod

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

"bmodpow" also supports negative exponents.

    bmodpow($num, -1, $mod)

is exactly equivalent to

    bmodinv($num, $mod)

bpow()

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

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

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

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

blog()

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

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

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

bexp()

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

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

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

Bitwise methods

band()

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

bior()

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

bxor()

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

bnot()

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

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

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

Rounding methods

round()

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

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

bround()

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

Rounds $x to an accuracy of $N digits.

bfround()

    $x->bfround($N);

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

    Input            N          Result    123456.123456    3          123500    123456.123456    2          123450    123456.123456   -2          123456.12    123456.123456   -3          123456.123

bfloor()

    $x->bfloor();

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

bceil()

    $x->bceil();

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

bint()

    $x->bint();

Round $x towards zero.

Other mathematical methods

bgcd()

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

Returns the greatest common divisor (GCD).

blcm()

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

Returns the least common multiple (LCM).

Object property methods

sign()

    $x->sign();

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

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

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

digit()

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

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

length()

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

Returns the number of digits in the decimal representation of the number. Inlist context, returns the length of the integer and fraction part. ForMath::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 partsince print calls "length()" in list context. Use something like:

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

mantissa()

    $x->mantissa();

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

exponent()

    $x->exponent();

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

parts()

    $x->parts();

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

sparts()

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

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

nparts()

Returns the significand (mantissa) and exponent corresponding to normalizednotation. In scalar context, only the significand is returned. For finitenon-zero numbers, the significand's absolute value is greater than or equal to1 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 aninteger, upgrading is performed or NaN is returned.

eparts()

Returns the significand (mantissa) and exponent corresponding to engineeringnotation. In scalar context, only the significand is returned. For finitenon-zero numbers, the significand's absolute value is greater than or equal to1 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 thesignificand can not be represented as an integer, upgrading is performed or NaNis returned.

dparts()

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

String conversion methods

bstr()

Returns a string representing the number using decimal notation. InMath::BigFloat, the output is zero padded according to the current accuracy orprecision, if any of those are defined.

bsstr()

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

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

bnstr()

Returns a string representing the number using normalized notation, the mostcommon variant of scientific notation. For finite non-zero numbers, theabsolute value of the significand is greater than or equal to 1 and less than10. The output corresponds to the output from "nparts()".

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

bestr()

Returns a string representing the number using engineering notation. For finitenon-zero numbers, the absolute value of the significand is greater than orequal to 1 and less than 1000, and the exponent is a multiple of 3. The outputcorresponds to the output from "eparts()".

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

bdstr()

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

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

to_hex()

    $x->to_hex();

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

to_bin()

    $x->to_bin();

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

to_oct()

    $x->to_oct();

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

to_bytes()

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

Returns a byte string representation of the number using big endian byteorder. The invocand must be a non-negative, finite integer. See also from_bytes().

to_base()

    $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 collationsequence is given, the collation sequence determines which characters are usedin 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 from_base() for information and examples.

as_hex()

    $x->as_hex();

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

as_bin()

    $x->as_bin();

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

as_oct()

    $x->as_oct();

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

as_bytes()

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

Other conversion methods

numify()

    print $x->numify();

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

ACCURACY and PRECISION

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

This section describes the accuracy/precision handling in Math::BigInt andMath::BigFloat as it used to be and as it is now, complete with an explanationof 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 (becausethese may differ from terms used by others people or documentation).

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

Precision P

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

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

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

For Math::BigInt objects, no padding occurs.

Accuracy A

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

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

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

For Math::BigInt objects, no padding occurs.

Fallback F

When both A and P are undefined, this is used as a fallback accuracy whendividing numbers.

Rounding mode R

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

Directed rounding

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, androunded to the fourth significant digit becomes 987.6 (A=4). 123.456 rounded tothe second place after the decimal point (P=-2) becomes 123.46. Thiscorresponds to the IEEE 754 rounding mode 'roundTowardZero'.

Rounding to nearest

These rounding modes round to the nearest digit. They differ in how theydetermine 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 theIEEE 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 theIEEE 754 rounding mode 'roundTiesToOdd'.
'+inf': Round towards plus infinity, i.e., always round up. E.g., when rounding to thenearest integer, 4.5 becomes 5, -5.5 becomes -5, and 4.501 also becomes 5. Thiscorresponds to the IEEE 754 rounding mode 'roundTiesToPositive'.
'-inf': Round towards minus infinity, i.e., always round down. E.g., when rounding tothe nearest integer, 4.5 becomes 4, -5.5 becomes -6, but 4.501 becomes 5. Thiscorresponds 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, but4.501 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 absolutevalue. E.g.,