Input

• 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-"new()> (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();

Output

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

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

Configuration methods

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

accuracy()

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

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

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

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

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

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

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

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

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

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

precision()

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

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

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

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

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

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

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

div_scale()

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

round_mode()

Set/get the rounding mode.

upgrade()

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

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

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

downgrade()

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

modify()

    $x->modify('bpowd');
    

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

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

config()

    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
    

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_dec()

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

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

from_hex()

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

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

from_oct()

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

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

from_bin()

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

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

from_bytes()

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

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

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

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

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

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

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

from_base()

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
    

from_base_num()

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
    

bzero()

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

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

bone()

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

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

binf()

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

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

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

bnan()

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

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

    $x->bnan();
    

bpi()

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

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

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

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

copy()

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

as_int()

as_number()

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

    $x -> badd($y);
    

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

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

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

Boolean methods

is_zero()

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

Returns true if the invocand is zero and false otherwise.

is_one( [ SIGN ])

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

Returns true if the invocand is one and false otherwise.

is_finite()

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

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

is_inf( [ SIGN ] )

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

Returns true if the invocand is infinite and false otherwise.

is_nan()

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

is_positive()

is_pos()

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

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

is_negative()

is_neg()

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

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

is_non_positive()

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

Returns true if the invocand is negative or zero.

is_non_negative()

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

Returns true if the invocand is positive or zero.

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

bcmp()

    $x->bcmp($y);
    

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

bacmp()

    $x->bacmp($y);
    

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

beq()

    $x -> beq($y);
    

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

bne()

    $x -> bne($y);
    

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

blt()

    $x -> blt($y);
    

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

ble()

    $x -> ble($y);
    

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

bgt()

    $x -> bgt($y);
    

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

bge()

    $x -> bge($y);
    

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

Arithmetic methods

bneg()

    $x->bneg();
    

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

babs()

    $x->babs();
    

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

bsgn()

    $x->bsgn();
    

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

bnorm()

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

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

binc()

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

bdec()

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

badd()

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

bsub()

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

bmul()

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

bmuladd()

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

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

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

bdiv()

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

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

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

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

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

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

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

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

btdiv()

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

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

bmod()

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

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

btmod()

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

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

bmodinv()

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

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

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

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

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

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

bmodpow()

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

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

    $num ** $exp % $mod
    

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

"bmodpow" also supports negative exponents.

    bmodpow($num, -1, $mod)
    

is exactly equivalent to

    bmodinv($num, $mod)
    

bpow()

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

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

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

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

blog()

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

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

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

bexp()

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

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

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

See also "blog()".

bnok()

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

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

buparrow()

uparrow()

    $a -> buparrow($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>.

backermann()

ackermann()

    $m -> backermann($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

bsin()

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

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

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

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

bcos()

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

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

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

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

batan()

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

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

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

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

batan2()

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

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

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

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

bsqrt()

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

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

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

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

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

broot()

    $x->broot($N);
    

Calculates the N'th root of $x.

bfac()

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

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.

bdfac()

    $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.

btfac()

    $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.

bmfac()

    $x->bmfac($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.

bfib()

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

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

The Fibonacci sequence is defined by

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

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

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

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

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

giving the bidirectional sequence

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

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

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

blucas()

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

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

The Lucas sequence is defined by

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

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

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

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

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

giving the bidirectional sequence

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

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

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

brsft()

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

The latter is equivalent to

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

blsft()

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

The latter is equivalent to

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

Bitwise methods

band()

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

bior()

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

bxor()

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

bnot()

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

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

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

Rounding methods

round()

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

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

bround()

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

Rounds $x to an accuracy of $N digits.

bfround()

    $x->bfround($N);
    

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

    Input            N          Result

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

bfloor()

    $x->bfloor();
    

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

bceil()

    $x->bceil();
    

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

bint()

    $x->bint();
    

Round $x towards zero.

Other mathematical methods

bgcd()

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

Returns the greatest common divisor (GCD).

blcm()

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

Returns the least common multiple (LCM).

Object property methods

sign()

    $x->sign();
    

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

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

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

digit()

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

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

digitsum()

    $x->digitsum();
    

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

bdigitsum()

    $x->bdigitsum();
    

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

length()

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

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

The following probably doesn't do what you expect:

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

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

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

mantissa()

    $x->mantissa();
    

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

exponent()

    $x->exponent();
    

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

parts()

    $x->parts();
    

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

sparts()

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

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

nparts()

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

eparts()

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

dparts()

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

String conversion methods

bstr()

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

bsstr()

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

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

bnstr()

Returns a string representing the number using normalized notation, the most common variant of scientific notation. For finite non-zero numbers, the absolute value of the significand is 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"
    

bestr()

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"
    

bdstr()

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

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

to_hex()

    $x->to_hex();
    

Returns a hexadecimal string representation of the number. 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 byte order. 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 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 from_base() for information and examples.

to_base_num()

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]
    

as_hex()

    $x->as_hex();
    

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

as_bin()

    $x->as_bin();
    

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

as_oct()

    $x->as_oct();
    

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

as_bytes()

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

Other conversion methods

numify()

    print $x->numify();
    

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

Utility methods

dec_str_to_dec_flt_str()

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
    

hex_str_to_dec_flt_str()

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 "dec_str_to_dec_flt_str()".

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

Some examples

    Input           Output
    0xff            +255e+0
    

Some examples

oct_str_to_dec_flt_str()

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 "dec_str_to_dec_flt_str()".

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

bin_str_to_dec_flt_str()

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 "dec_str_to_dec_flt_str()".

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

dec_str_to_dec_str()

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.

hex_str_to_dec_str()

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 "dec_str_to_dec_str()".

oct_str_to_dec_str()

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 "dec_str_to_dec_str()".

bin_str_to_dec_str()

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 "dec_str_to_dec_str()".

Precision P

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

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

Rounding mode R

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, 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'.

Rounding to nearest

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
    

MATH LIBRARY

The default library

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

Specifying a library

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';

Which library to use?

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

Math::BigInt::GMP, 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.

Loading multiple libraries

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"

SIGN

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.

Hexadecimal, octal, and binary floating point literals

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

Alternative math libraries

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

Subclassing Math::BigInt

• The public API must remain consistent, i.e. if a sub-class is overloading addition, the sub-class must use the same name, in this case badd(). The reason for this is that Math::BigInt is optimized to call the object methods directly.
• The private object hash keys like "$x->{sign}" may not be changed, but additional keys can be added, like "$x->{_custom}".
• Accessor functions are available for all existing object hash keys and should be used instead of directly accessing the internal hash keys. The reason for this is that Math::BigInt itself has a pluggable interface which permits it to support different storage methods.

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

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

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

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

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

    $upgrade = undef;
    $downgrade = undef;

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

Don't forget to

    use overload;

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

Auto-upgrade

    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.