# Math::BigFloat

```Math::BigFloat(3)      Perl Programmers Reference Guide      Math::BigFloat(3)

NAME
Math::BigFloat - Arbitrary size floating point math package

SYNOPSIS
use Math::BigFloat;

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

# 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
# neccessary when mixing \$a = \$b assigments 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->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->blsft(\$y);                # left shift
\$x->brsft(\$y);                # right shift
# return (quo,rem) or quo if scalar

\$x->blog();                   # logarithm of \$x to base e (Euler's number)
\$x->blog(\$base);              # logarithm of \$x to base \$base (f.i. 2)

\$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 (inlcuding 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:

o "/^[+-]\d+\$/"

o "/^[+-]\d+\.\d*\$/"

o "/^[+-]\d+E[+-]?\d+\$/"

o "/^[+-]\d*\.\d+E[+-]?\d+\$/"

ally, 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 num-
bers 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 scien-
tific 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

Math::BigFloat supports both precision and accuracy. For a full docu-
mentation, examples and tips on these topics please see the large sec-
tion in Math::BigInt.

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

Please refer to BigInt's documentation for the precedence rules of
which accuracy/precision setting will be used.

If there is no gloabl precision set, and the operation inquestion 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 is 40 digits.

In case the result of one operation has more precision 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->precision(5);           # 5 digits max
\$y = \$x->copy()->bdiv(3);               # will give 0.66666
\$y = \$x->copy()->bdiv(3,6);             # will give 0.666666
\$y = \$x->copy()->bdiv(3,6,'odd');       # will give 0.666667
Math::BigFloat->round_mode('zero');
\$y = \$x->copy()->bdiv(3,6);             # will give 0.666666

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' or 'trunc'.

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 temporar-
ily.

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

EXAMPLES

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 num-
ber.

Please note that ':constant' does not affect integer constants, nor
binary nor hexadecimal constants. Use bignum or Math::BigInt 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 => 'BitVect';

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

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

Calc.pm uses as internal format an array of elements of some decimal
base (usually 1e7, but this might be differen for some systems) with
the least significant digit first, while BitVect.pm uses a bit vector
of base 2, most significant bit first. Other modules might use even
different means of representing the numbers. See the respective module
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 neccessary things (like BigInt) when they are
needed, and automatically.

Use the lib, Luke! And see "Using Math::BigInt::Lite" for more details

Using Math::BigInt::Lite

It is possible to use Math::BigInt::Lite with Math::BigFloat:

# 1
use Math::BigFloat with => 'Math::BigInt::Lite';

There is no need to "use Math::BigInt" or "use Math::BigInt::Lite", but
you can combine these if you want. For instance, you may want to use
Math::BigInt objects in your main script, too.

# 2
use Math::BigInt;
use Math::BigFloat with => 'Math::BigInt::Lite';

Of course, you can combine this with the "lib" parameter.

# 3
use Math::BigFloat with => 'Math::BigInt::Lite', lib => 'GMP,Pari';

There is no need for a "use Math::BigInt;" statement, even if you want
to use Math::BigInt's, since Math::BigFloat will needs Math::BigInt and

# 4
use Math::BigInt;
use Math::BigFloat with => 'Math::BigInt::Lite', lib => 'GMP,Pari';

Notice that the module with the last "lib" will "win" and thus it's lib
will be used if the lib is available:

# 5
use Math::BigInt lib => 'Bar,Baz';
use Math::BigFloat with => 'Math::BigInt::Lite', lib => 'Foo';

That would try to load Foo, Bar, Baz and Calc (in that order). Or in
other words, Math::BigFloat will try to retain previously loaded libs
when you don't specify it onem but if you specify one, it will try to

"Foo,Bar,Baz,Calc", but independend of which lib exists, the result is
the same as trying the latter load alone, except for the fact that one
of Bar or Baz might be loaded needlessly in an intermidiate step (and
thus hang around and waste memory). If neither Bar nor Baz exist (or
don't work/compile), they will still be tried to be loaded, but this is
this type of usage is not recommended due to these issues.

The old way (loading the lib only in BigInt) still works though:

# 6
use Math::BigInt lib => 'Bar,Baz';
use Math::BigFloat;

You can even load Math::BigInt afterwards:

# 7
use Math::BigFloat;
use Math::BigInt lib => 'Bar,Baz';

But this has the same problems like #5, it will first load Calc
(Math::BigFloat needs Math::BigInt and thus loads it) and then later
Bar or Baz, depending on which of them works and is usable/loadable.
Since this loads Calc unnecc., it is not recommended.

Since it also possible to just require Math::BigFloat, this poses the
question about what libary this will use:

require Math::BigFloat;
my \$x = Math::BigFloat->new(123); \$x += 123;

It will use Calc. Please note that the call to import() is still done,
but only when you use for the first time some Math::BigFloat math (it
is triggered via any constructor, so the first time you create a
Math::BigFloat, the load will happen in the background). This means:

require Math::BigFloat;
Math::BigFloat->import ( lib => 'Foo,Bar' );

would be the same as:

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

But don't try to be clever to insert some operations in between:

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

While this works, it loads Calc needlessly. But maybe you just wanted
that?

Examples #3 is highly recommended for daily usage.

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

CAVEATS
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
Math::BigInt for reasoning and details.

bdiv
The following will probably not do what you expect:

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

It prints both quotient and reminder since print works in list con-
text. Also, bdiv() will modify \$c, so be carefull. You probably want
to use

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

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 Math::BigInt 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

Math::BigInt, Math::BigRat and Math::Big as well as Math::Big-
Int::BitVect, Math::BigInt::Pari and  Math::BigInt::GMP.

The pragmas bignum, bigint and bigrat might also be of interest because

The package at <http://search.cpan.org/search?mode=mod-
ule&query=Math%3A%3ABigInt> contains more documentation including a
full version history, testcases, empty subclass files and benchmarks.