NAME

Math::BigFloat - Arbitrary size floating point math package

SYNOPSIS

  use Math::BigFloat;

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

  my $pi = Math::BigFloat->bpi(100);    # PI to 100 digits

  # the following examples compute their result to 100 digits accuracy:
  my $cos  = Math::BigFloat->new(1)->bcos(100);         # cosinus(1)
  my $sin  = Math::BigFloat->new(1)->bsin(100);         # sinus(1)
  my $atan = Math::BigFloat->new(1)->batan(100);        # arcus tangens(1)

  my $atan2 = Math::BigFloat->new(  1 )->batan2( 1 ,100); # batan(1)
  my $atan2 = Math::BigFloat->new(  1 )->batan2( 8 ,100); # batan(1/8)
  my $atan2 = Math::BigFloat->new( -2 )->batan2( 1 ,100); # batan(-2)

  # Testing
  $x->is_zero();                # true if arg is +0
  $x->is_nan();                 # true if arg is NaN
  $x->is_one();                 # true if arg is +1
  $x->is_one('-');              # true if arg is -1
  $x->is_odd();                 # true if odd, false for even
  $x->is_even();                # true if even, false for odd
  $x->is_pos();                 # true if >= 0
  $x->is_neg();                 # true if <  0
  $x->is_inf(sign);             # true if +inf, or -inf (default is '+')

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

  # The following all modify their first argument. If you want to preserve
  # $x, use $z = $x->copy()->bXXX($y); See under L<CAVEATS> for why this is
  # necessary when mixing $a = $b assignments with non-overloaded math.
   
  # set 
  $x->bzero();                  # set $i to 0
  $x->bnan();                   # set $i to NaN
  $x->bone();                   # set $x to +1
  $x->bone('-');                # set $x to -1
  $x->binf();                   # set $x to inf
  $x->binf('-');                # set $x to -inf

  $x->bneg();                   # negation
  $x->babs();                   # absolute value
  $x->bnorm();                  # normalize (no-op)
  $x->bnot();                   # two's complement (bit wise not)
  $x->binc();                   # increment x by 1
  $x->bdec();                   # decrement x by 1
  
  $x->badd($y);                 # addition (add $y to $x)
  $x->bsub($y);                 # subtraction (subtract $y from $x)
  $x->bmul($y);                 # multiplication (multiply $x by $y)
  $x->bdiv($y);                 # divide, set $x to quotient
                                # return (quo,rem) or quo if scalar

  $x->bmod($y);                 # modulus ($x % $y)
  $x->bpow($y);                 # power of arguments ($x ** $y)
  $x->bmodpow($exp,$mod);       # modular exponentation (($num**$exp) % $mod))
  $x->blsft($y, $n);            # left shift by $y places in base $n
  $x->brsft($y, $n);            # right shift by $y places in base $n
                                # returns (quo,rem) or quo if in scalar context
  
  $x->blog();                   # logarithm of $x to base e (Euler's number)
  $x->blog($base);              # logarithm of $x to base $base (f.i. 2)
  $x->bexp();                   # calculate e ** $x where e is Euler's number
  
  $x->band($y);                 # bit-wise and
  $x->bior($y);                 # bit-wise inclusive or
  $x->bxor($y);                 # bit-wise exclusive or
  $x->bnot();                   # bit-wise not (two's complement)
   
  $x->bsqrt();                  # calculate square-root
  $x->broot($y);                # $y'th root of $x (e.g. $y == 3 => cubic root)
  $x->bfac();                   # factorial of $x (1*2*3*4*..$x)
   
  $x->bround($N);               # accuracy: preserve $N digits
  $x->bfround($N);              # precision: round to the $Nth digit

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

  # The following do not modify their arguments:

  bgcd(@values);                # greatest common divisor
  blcm(@values);                # lowest common multiplicator
  
  $x->bstr();                   # return string
  $x->bsstr();                  # return string in scientific notation

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

  $x->length();                 # number of digits (w/o sign and '.')
  ($l,$f) = $x->length();       # number of digits, and length of fraction

  $x->precision();              # return P of $x (or global, if P of $x undef)
  $x->precision($n);            # set P of $x to $n
  $x->accuracy();               # return A of $x (or global, if A of $x undef)
  $x->accuracy($n);             # set A $x to $n

  # these get/set the appropriate global value for all BigFloat objects
  Math::BigFloat->precision();  # Precision
  Math::BigFloat->accuracy();   # Accuracy
  Math::BigFloat->round_mode(); # rounding mode

DESCRIPTION

All operators (including basic math operations) are overloaded if you declare your big floating point numbers as

  $i = new Math::BigFloat '12_3.456_789_123_456_789E-2';

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

Canonical notation

Input to these routines are either BigFloat objects, or strings of the following four forms:

• /^[+-]\d+$/

• /^[+-]\d+\.\d*$/

• /^[+-]\d+E[+-]?\d+$/

• /^[+-]\d*\.\d+E[+-]?\d+$/

all with optional leading and trailing zeros and/or spaces. Additionally, numbers are allowed to have an underscore between any two digits.

Empty strings as well as other illegal numbers results in 'NaN'.

bnorm() on a BigFloat object is now effectively a no-op, since the numbers are always stored in normalized form. On a string, it creates a BigFloat object.

Output

Output values are BigFloat objects (normalized), except for bstr() and bsstr().

The string output will always have leading and trailing zeros stripped and drop a plus sign. bstr() will give you always the form with a decimal point, while bsstr() (s for scientific) gives you the scientific notation.

        Input                   bstr()          bsstr()
        '-0'                    '0'             '0E1'
        '  -123 123 123'        '-123123123'    '-123123123E0'
        '00.0123'               '0.0123'        '123E-4'
        '123.45E-2'             '1.2345'        '12345E-4'
        '10E+3'                 '10000'         '1E4'

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

Actual math is done by using the class defined with with = Class;> (which defaults to BigInts) to represent the mantissa and exponent.

The sign /^[+-]$/ is stored separately. The string 'NaN' is used to represent the result when input arguments are not numbers, as well as the result of dividing by zero.

mantissa(), exponent() and parts()

mantissa() and exponent() return the said parts of the BigFloat as BigInts such that:

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

($m,$e) = $x->parts(); is just a shortcut giving you both of them.

A zero is represented and returned as 0E1, not 0E0 (after Knuth).

Currently the mantissa is reduced as much as possible, favouring higher exponents over lower ones (e.g. returning 1e7 instead of 10e6 or 10000000e0). This might change in the future, so do not depend on it.

Accuracy vs. Precision

See also: Rounding.

Math::BigFloat supports both precision (rounding to a certain place before or after the dot) and accuracy (rounding to a certain number of digits). For a full documentation, examples and tips on these topics please see the large section about rounding in the Math::BigInt manpage.

Since things like sqrt(2) or 1 / 3 must presented with a limited accuracy lest a operation consumes all resources, each operation produces no more than the requested number of digits.

If there is no gloabl precision or accuracy set, and the operation in question was not called with a requested precision or accuracy, and the input $x has no accuracy or precision set, then a fallback parameter will be used. For historical reasons, it is called div_scale and can be accessed via:

        $d = Math::BigFloat->div_scale();               # query
        Math::BigFloat->div_scale($n);                  # set to $n digits

The default value for div_scale is 40.

In case the result of one operation has more digits than specified, it is rounded. The rounding mode taken is either the default mode, or the one supplied to the operation after the scale:

        $x = Math::BigFloat->new(2);
        Math::BigFloat->accuracy(5);            # 5 digits max
        $y = $x->copy()->bdiv(3);               # will give 0.66667
        $y = $x->copy()->bdiv(3,6);             # will give 0.666667
        $y = $x->copy()->bdiv(3,6,undef,'odd'); # will give 0.666667
        Math::BigFloat->round_mode('zero');
        $y = $x->copy()->bdiv(3,6);             # will also give 0.666667

Note that Math::BigFloat->accuracy() and Math::BigFloat->precision() set the global variables, and thus any newly created number will be subject to the global rounding immediately. This means that in the examples above, the 3 as argument to bdiv() will also get an accuracy of 5.

It is less confusing to either calculate the result fully, and afterwards round it explicitly, or use the additional parameters to the math functions like so:

        use Math::BigFloat;     
        $x = Math::BigFloat->new(2);
        $y = $x->copy()->bdiv(3);
        print $y->bround(5),"\n";               # will give 0.66667

        or

        use Math::BigFloat;     
        $x = Math::BigFloat->new(2);
        $y = $x->copy()->bdiv(3,5);             # will give 0.66667
        print "$y\n";

Rounding

ffround ( +$scale )

Rounds to the $scale'th place left from the '.', counting from the dot. The first digit is numbered 1.

ffround ( -$scale )

Rounds to the $scale'th place right from the '.', counting from the dot.

ffround ( 0 )

Rounds to an integer.

fround ( +$scale )

Preserves accuracy to $scale digits from the left (aka significant digits) and pads the rest with zeros. If the number is between 1 and -1, the significant digits count from the first non-zero after the '.'

fround ( -$scale ) and fround ( 0 )

These are effectively no-ops.

All rounding functions take as a second parameter a rounding mode from one of the following: 'even', 'odd', '+inf', '-inf', 'zero', 'trunc' or 'common'.

The default rounding mode is 'even'. By using Math::BigFloat->round_mode($round_mode); you can get and set the default mode for subsequent rounding. The usage of $Math::BigFloat::$round_mode is no longer supported. The second parameter to the round functions then overrides the default temporarily.

The as_number() function returns a BigInt from a Math::BigFloat. It uses 'trunc' as rounding mode to make it equivalent to:

        $x = 2.5;
        $y = int($x) + 2;

You can override this by passing the desired rounding mode as parameter to as_number():

        $x = Math::BigFloat->new(2.5);
        $y = $x->as_number('odd');      # $y = 3

METHODS

Math::BigFloat supports all methods that Math::BigInt supports, except it calculates non-integer results when possible. Please see the Math::BigInt manpage for a full description of each method. Below are just the most important differences:

accuracy

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

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

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

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

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

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

precision()

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

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

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

Note: You probably want to use accuracy() instead. With the accuracy manpage you set the number of digits each result should have, with precision you set the place where to round!

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).

bnok()

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