Math::BigFloat - Arbitrary size floating point math package |
Math::BigFloat - Arbitrary size floating point math package
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
All operators (including basic math operations) are overloaded if you declare your big floating point numbers as
$i = new Math::BigFloat '12_3.456_789_123_456_789E-2';
Operations with overloaded operators preserve the arguments, which is exactly what you expect.
Input to these routines are either BigFloat objects, or strings of the following four forms:
/^[+-]\d+$/
/^[+-]\d+\.\d*$/
/^[+-]\d+E[+-]?\d+$/
/^[+-]\d*\.\d+E[+-]?\d+$/
all with optional leading and trailing zeros and/or spaces. Additionally, numbers are allowed to have an underscore between any two digits.
Empty strings as well as other illegal numbers results in 'NaN'.
bnorm()
on a BigFloat object is now effectively a no-op, since the numbers
are always stored in normalized form. On a string, it creates a BigFloat
object.
Output values are BigFloat objects (normalized), except for bstr()
and bsstr().
The string output will always have leading and trailing zeros stripped and drop
a plus sign. bstr()
will give you always the form with a decimal point,
while bsstr()
(s for scientific) gives you the scientific notation.
Input bstr() bsstr() '-0' '0' '0E1' ' -123 123 123' '-123123123' '-123123123E0' '00.0123' '0.0123' '123E-4' '123.45E-2' '1.2345' '12345E-4' '10E+3' '10000' '1E4'
Some routines (is_odd()
, is_even()
, is_zero()
, is_one()
,
is_nan()
) return true or false, while others (bcmp()
, bacmp()
)
return either undef, <0, 0 or >0 and are suited for sort.
Actual math is done by using the class defined with with =
Class;> (which
defaults to BigInts) to represent the mantissa and exponent.
The sign /^[+-]$/
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.
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";
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
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:
$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).
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 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: General purpose packages should not be explicit about the library to use; let the script author decide which is best.
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';
If your script works with huge numbers and Calc is too slow for them, you can also for the loading of one of these libraries and if none of them can be used, the code will die:
use Math::BigFloat only => 'GMP,Pari';
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.
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.
Math::BigFloat
exports nothing by default, but can export the bpi()
method:
use Math::BigFloat qw/bpi/;
print bpi(10), "\n";
Please see the file BUGS in the CPAN distribution Math::BigInt for known bugs.
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!
bstr()
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.
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.
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.
$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()
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()
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.
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.
This program is free software; you may redistribute it and/or modify it under the same terms as Perl itself.
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 |