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**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,$mod); # modular exponentiation (($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 $i = new Math::BigFloat '12_3.456_789_123_456_789E-2'; Operations with overloaded operators preserve the arguments, which is exactly what you expect.CanonicalnotationInput to these routines are either BigFloat objects, or strings of the following four forms: · "/^[+-]\d+$/" · "/^[+-]\d+\.\d*$/" · "/^[+-]\d+E[+-]?\d+$/" · "/^[+-]\d*\.\d+E[+-]?\d+$/" all with optional leading and trailing zeros and/or spaces. Additionally, numbers are allowed to have an underscore between any two digits. Empty strings as well as other illegal numbers results in 'NaN'.bnorm()on a BigFloat object is now effectively a no-op, since the numbers are always stored in normalized form. On a string, it creates a BigFloat object.OutputOutput values are BigFloat objects (normalized), except forbstr()andbsstr(). The string output will always have leading and trailing zeros stripped and drop a plus sign. "bstr()" will give you always the form with a decimal point, while "bsstr()" (s for scientific) gives you the scientific notation. Input bstr() bsstr() '-0' '0' '0E1' ' -123 123 123' '-123123123' '-123123123E0' '00.0123' '0.0123' '123E-4' '123.45E-2' '1.2345' '12345E-4' '10E+3' '10000' '1E4' Some routines ("is_odd()", "is_even()", "is_zero()", "is_one()", "is_nan()") return true or false, while others ("bcmp()", "bacmp()") return either undef, <0, 0 or >0 and are suited for sort. Actual math is done by using the class defined with "with => Class;" (which defaults to BigInts) to represent the mantissa and exponent. The sign "/^[+-]$/" 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,not0E0 (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.Accuracyvs.PrecisionSee 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 Math::BigInt. 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 global precision or accuracy set,andthe operation in question was not called with a requested precision or accuracy,andthe 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 thescale: $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 thusanynewly created number will be subject to the global roundingimmediately. This means that in the examples above, the 3 as argument to "bdiv()" will also get an accuracy of5. 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";Roundingffround ( +$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 Math::BigInt 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 $A = CLASS->accuracy(); # read out global accuracy Set or get the global or local accuracy, aka how many significant digits the results have. If you set a global accuracy, then this also applies tonew()! Warning! The accuracysticks, e.g. once you created a number under the influence of "CLASS->accuracy($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 ofround(),bround()orbfround()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 4300precision()$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 $x Note: You probably want to useaccuracy()instead. With accuracy you set the number of digits each result should have, with precision you set the place where to round!bexp()$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 alsobatan(). 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 $x in place. See alsobatan2(). This method was added in v1.87 of Math::BigInt (June 2007).bmuladd()$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 bignum or Math::BigInt to get this to work.MathlibraryMath 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 doesnotuse 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 Math::BigInt for more details than you ever wanted to know about using a different low-level library.UsingMath::BigInt::LiteFor 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 library 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, andVERYBADTHINGSwill happen when you use these together: my $flash_and_bang = $matter + $anti_matter; # Don't do this! stringify,bstr()Both stringify andbstr()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 print what you expect: print $c->bdiv(123.456),"\n"; It prints both quotient and remainder 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 thesameobject 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 aloneprecision()vs.accuracy()A common pitfall is to useprecision()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 accuracy 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 doesnot"taint" the result with an accuracy or precision setting, which would influence any further operation.

**SEE** **ALSO**

Math::BigInt, Math::BigRat and Math::Big as well as Math::BigInt::BitVect, Math::BigInt::Pari and Math::BigInt::GMP. The pragmas bignum, bigint and bigrat 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 <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.