Math::BigFloat - Arbitrary size floating point math package

- NAME
- SYNOPSIS
- DESCRIPTION

- Canonical notation
- Output
- `mantissa()`, `exponent()` and `parts()`
- Accuracy vs. Precision
- Rounding

- METHODS

- accuracy
- `precision()`
- `bexp()`
- `bnok()`
- `bpi()`
- `bcos()`
- `bsin()`
- batan2()
- `batan()`
- `bmuladd()`

- Autocreating constants

- Math library
- Using Math::BigInt::Lite

- EXPORTS
- BUGS
- CAVEATS
- SEE ALSO
- LICENSE
- AUTHORS

# 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)

Calculates the binomial coefficient n over k, also called the "choose" function. The result is equivalent to:

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

This method was added in v1.84 of Math::BigInt (April 2007).

## `bpi()`

            print Math::BigFloat->bpi(100), "\n";

Calculate PI to N digits (including the 3 before the dot). The result is rounded according to the current rounding mode, which defaults to "even".

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

## `bcos()`

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

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

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

## `bsin()`

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

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

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

## batan2()

            my $y = Math::BigFloat->new(2);
            my $x = Math::BigFloat->new(3);
            print $y->batan2($x), "\n";

Calculate the arcus tanges of `$y` divided by `$x`, modifying $y in place. See also batan().

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

## `batan()`

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

Calculate the arcus tanges of $x, modifying $x in place. See also batan2().

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

## `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).

# Autocreating constants

After `use Math::BigFloat ':constant'` all the floating point constants in the given scope are converted to `Math::BigFloat`. This conversion happens at compile time.

In particular

      perl -MMath::BigFloat=:constant -e 'print 2E-100,"\n"'

prints the value of `2E-100`. Note that without conversion of constants the expression 2E-100 will be calculated as normal floating point number.

Please note that ':constant' does not affect integer constants, nor binary nor hexadecimal constants. Use the bignum manpage or the Math::BigInt manpage to get this to work.

## Math library

Math with the numbers is done (by default) by a module called Math::BigInt::Calc. This is equivalent to saying:

            use Math::BigFloat lib => 'Calc';

You can change this by using:

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

Note: The keyword 'lib' will warn when the requested library could not be loaded. To suppress the warning use 'try' instead:

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

To turn the warning into a `die()`, use 'only' instead:

            use Math::BigFloat only => 'GMP';

The following would first try to find Math::BigInt::Foo, then Math::BigInt::Bar, and when this also fails, revert to Math::BigInt::Calc:

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

See the respective low-level library documentation for further details.

Please note that Math::BigFloat does not use the denoted library itself, but it merely passes the lib argument to Math::BigInt. So, instead of the need to do:

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

you can roll it all into one line:

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

It is also possible to just require Math::BigFloat:

            require Math::BigFloat;

This will load the necessary things (like BigInt) when they are needed, and automatically.

See the Math::BigInt manpage for more details than you ever wanted to know about using a different low-level library.

## Using Math::BigInt::Lite

For backwards compatibility reasons it is still possible to request a different storage class for use with Math::BigFloat:

            use Math::BigFloat with => 'Math::BigInt::Lite';

However, this request is ignored, as the current code now uses the low-level math libary for directly storing the number parts.

# EXPORTS

`Math::BigFloat` exports nothing by default, but can export the `bpi()` method:

            use Math::BigFloat qw/bpi/;

            print bpi(10), "\n";

# BUGS

Please see the file BUGS in the CPAN distribution Math::BigInt for known bugs.

# CAVEATS

Do not try to be clever to insert some operations in between switching libraries:

            require Math::BigFloat;
            my $matter = Math::BigFloat->bone() + 4;        # load BigInt and Calc
            Math::BigFloat->import( lib => 'Pari' );        # load Pari, too
            my $anti_matter = Math::BigFloat->bone()+4;     # now use Pari

This will create objects with numbers stored in two different backend libraries, and VERY BAD THINGS will happen when you use these together:

            my $flash_and_bang = $matter + $anti_matter;    # Don't do this!

stringify, `bstr()`

  Both stringify and `bstr()` now drop the leading '+'. The old code would return '+1.23', the new returns '1.23'. See the documentation in the Math::BigInt manpage for reasoning and details.

bdiv

  The following will probably not print what you expect:

              print $c->bdiv(123.456),"\n";

  It prints both quotient and reminder since print works in list context. Also, `bdiv()` will modify $c, so be careful. You probably want to use

              print $c / 123.456,"\n";
              print scalar $c->bdiv(123.456),"\n";  # or if you want to modify $c

  instead.

brsft

  The following will probably not print what you expect:

              my $c = Math::BigFloat->new('3.14159');
              print $c->brsft(3,10),"\n";     # prints 0.00314153.1415

  It prints both quotient and remainder, since print calls `brsft()` in list context. Also, `$c->brsft()` will modify $c, so be careful. You probably want to use

              print scalar $c->copy()->brsft(3,10),"\n";
              # or if you really want to modify $c
              print scalar $c->brsft(3,10),"\n";

  instead.

Modifying and =

  Beware of:

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

  It will not do what you think, e.g. making a copy of $x. Instead it just makes a second reference to the same object and stores it in $y. Thus anything that modifies $x will modify $y (except overloaded math operators), and vice versa. See the Math::BigInt manpage for details and how to avoid that.

bpow

  `bpow()` now modifies the first argument, unlike the old code which left it alone and only returned the result. This is to be consistent with `badd()` etc. The first will modify $x, the second one won't:

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

`precision()` vs. `accuracy()`

  A common pitfall is to use precision() when you want to round a result to a certain number of digits:

              use Math::BigFloat;

              Math::BigFloat->precision(4);           # does not do what you think it does
              my $x = Math::BigFloat->new(12345);     # rounds $x to "12000"!
              print "$x\n";                           # print "12000"
              my $y = Math::BigFloat->new(3);         # rounds $y to "0"!
              print "$y\n";                           # print "0"
              $z = $x / $y;                           # 12000 / 0 => NaN!
              print "$z\n";
              print $z->precision(),"\n";             # 4

  Replacing precision with the accuracy manpage is probably not what you want, either:

              use Math::BigFloat;

              Math::BigFloat->accuracy(4);            # enables global rounding:
              my $x = Math::BigFloat->new(123456);    # rounded immediately to "12350"
              print "$x\n";                           # print "123500"
              my $y = Math::BigFloat->new(3);         # rounded to "3
              print "$y\n";                           # print "3"
              print $z = $x->copy()->bdiv($y),"\n";   # 41170
              print $z->accuracy(),"\n";              # 4

  What you want to use instead is:

              use Math::BigFloat;

              my $x = Math::BigFloat->new(123456);    # no rounding
              print "$x\n";                           # print "123456"
              my $y = Math::BigFloat->new(3);         # no rounding
              print "$y\n";                           # print "3"
              print $z = $x->copy()->bdiv($y,4),"\n"; # 41150
              print $z->accuracy(),"\n";              # undef

  In addition to computing what you expected, the last example also does not "taint" the result with an accuracy or precision setting, which would influence any further operation.

# SEE ALSO

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

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

The package at http://search.cpan.org/~tels/Math-BigInt contains more documentation including a full version history, testcases, empty subclass files and benchmarks.

# LICENSE

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

# AUTHORS

Mark Biggar, overloaded interface by Ilya Zakharevich. Completely rewritten by Tels http://bloodgate.com in 2001 - 2006, and still at it in 2007.

Math::BigFloat - Arbitrary size floating point math package