Math::BigInt(3pm) Perl Programmers Reference Guide Math::BigInt(3pm)
NAME
Math::BigInt - Arbitrary size integer/float math package
SYNOPSIS
use Math::BigInt;
# or make it faster with huge numbers: install (optional)
# Math::BigInt::GMP and always use (it will fall back to
# pure Perl if the GMP library is not installed):
# (See also the L<MATH LIBRARY> section!)
# will warn if Math::BigInt::GMP cannot be found
use Math::BigInt lib => 'GMP';
# to suppress the warning use this:
# use Math::BigInt try => 'GMP';
# dies if GMP cannot be loaded:
# use Math::BigInt only => 'GMP';
my $str = '1234567890';
my @values = (64,74,18);
my $n = 1; my $sign = '-';
# Number creation
my $x = Math::BigInt->new($str); # defaults to 0
my $y = $x->copy(); # make a true copy
my $nan = Math::BigInt->bnan(); # create a NotANumber
my $zero = Math::BigInt->bzero(); # create a +0
my $inf = Math::BigInt->binf(); # create a +inf
my $inf = Math::BigInt->binf('-'); # create a -inf
my $one = Math::BigInt->bone(); # create a +1
my $mone = Math::BigInt->bone('-'); # create a -1
my $pi = Math::BigInt->bpi(); # returns '3'
# see Math::BigFloat::bpi()
$h = Math::BigInt->new('0x123'); # from hexadecimal
$b = Math::BigInt->new('0b101'); # from binary
$o = Math::BigInt->from_oct('0101'); # from octal
# Testing (don't modify their arguments)
# (return true if the condition is met, otherwise false)
$x->is_zero(); # if $x is +0
$x->is_nan(); # if $x is NaN
$x->is_one(); # if $x is +1
$x->is_one('-'); # if $x is -1
$x->is_odd(); # if $x is odd
$x->is_even(); # if $x is even
$x->is_pos(); # if $x > 0
$x->is_neg(); # if $x < 0
$x->is_inf($sign); # if $x is +inf, or -inf (sign is default '+')
$x->is_int(); # if $x is an integer (not a float)
# comparing and digit/sign extraction
$x->bcmp($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 pre-
# serve $x, use $z = $x->copy()->bXXX($y); See under L<CAVEATS> for
# why this is necessary when mixing $a = $b assignments with non-over-
# loaded math.
$x->bzero(); # set $x to 0
$x->bnan(); # set $x 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->bsgn(); # sign function (-1, 0, 1, or NaN)
$x->bnorm(); # normalize (no-op in BigInt)
$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
$x->bmuladd($y,$z); # $x = $x * $y + $z
$x->bmod($y); # modulus (x % y)
$x->bmodpow($y,$mod); # modular exponentiation (($x ** $y) % $mod)
$x->bmodinv($mod); # modular multiplicative inverse
$x->bpow($y); # power of arguments (x ** y)
$x->blsft($y); # left shift in base 2
$x->brsft($y); # right shift in base 2
# returns (quo,rem) or quo if in sca-
# lar context
$x->blsft($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 sca-
# lar context
$x->band($y); # bitwise and
$x->bior($y); # bitwise inclusive or
$x->bxor($y); # bitwise exclusive or
$x->bnot(); # bitwise 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)
$x->bnok($y); # x over y (binomial coefficient n over k)
$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
$x->round($A,$P,$mode); # round to accuracy or precision using
# mode $mode
$x->bround($n); # accuracy: preserve $n digits
$x->bfround($n); # $n > 0: round $nth digits,
# $n < 0: round to the $nth digit after the
# dot, no-op for BigInts
# The following do not modify their arguments in BigInt (are no-ops),
# but do so in BigFloat:
$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:
# greatest common divisor (no OO style)
my $gcd = Math::BigInt::bgcd(@values);
# lowest common multiple (no OO style)
my $lcm = Math::BigInt::blcm(@values);
$x->length(); # return number of digits in number
($xl,$f) = $x->length(); # length of number and length of fraction
# part, latter is always 0 digits long
# for BigInts
$x->exponent(); # return exponent as BigInt
$x->mantissa(); # return (signed) mantissa as BigInt
$x->parts(); # return (mantissa,exponent) as BigInt
$x->copy(); # make a true copy of $x (unlike $y = $x;)
$x->as_int(); # return as BigInt (in BigInt: same as copy())
$x->numify(); # return as scalar (might overflow!)
# conversion to string (do not modify their argument)
$x->bstr(); # normalized string (e.g. '3')
$x->bsstr(); # norm. string in scientific notation (e.g. '3E0')
$x->as_hex(); # as signed hexadecimal string with prefixed 0x
$x->as_bin(); # as signed binary string with prefixed 0b
$x->as_oct(); # as signed octal string with prefixed 0
# precision and accuracy (see section about rounding for more)
$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
# Global methods
Math::BigInt->precision(); # get/set global P for all BigInt objects
Math::BigInt->accuracy(); # get/set global A for all BigInt objects
Math::BigInt->round_mode(); # get/set global round mode, one of
# 'even', 'odd', '+inf', '-inf', 'zero',
# 'trunc' or 'common'
Math::BigInt->config(); # return hash containing configuration
DESCRIPTION
All operators (including basic math operations) are overloaded if you
declare your big integers as
$i = new Math::BigInt '123_456_789_123_456_789';
Operations with overloaded operators preserve the arguments which is
exactly what you expect.
Input
Input values to these routines may be any string, that looks like a
number and results in an integer, including hexadecimal and binary
numbers.
Scalars holding numbers may also be passed, but note that non-integer
numbers may already have lost precision due to the conversion to
float. Quote your input if you want BigInt to see all the digits:
$x = Math::BigInt->new(12345678890123456789); # bad
$x = Math::BigInt->new('12345678901234567890'); # good
You can include one underscore between any two digits.
This means integer values like 1.01E2 or even 1000E-2 are also
accepted. Non-integer values result in NaN.
Hexadecimal (prefixed with "0x") and binary numbers (prefixed with
"0b") are accepted, too. Please note that octal numbers are not
recognized by new(), so the following will print "123":
perl -MMath::BigInt -le 'print Math::BigInt->new("0123")'
To convert an octal number, use from_oct();
perl -MMath::BigInt -le 'print Math::BigInt->from_oct("0123")'
Currently, Math::BigInt::new() defaults to 0, while
Math::BigInt::new('') results in 'NaN'. This might change in the
future, so use always the following explicit forms to get a zero or
NaN:
$zero = Math::BigInt->bzero();
$nan = Math::BigInt->bnan();
"bnorm()" on a BigInt object is now effectively a no-op, since the
numbers are always stored in normalized form. If passed a string,
creates a BigInt object from the input.
Output
Output values are BigInt objects (normalized), except for the methods
which return a string (see "SYNOPSIS").
Some routines ("is_odd()", "is_even()", "is_zero()", "is_one()",
"is_nan()", etc.) return true or false, while others ("bcmp()",
"bacmp()") return either undef (if NaN is involved), <0, 0 or >0 and
are suited for sort.
METHODS
Each of the methods below (except config(), accuracy() and precision())
accepts three additional parameters. These arguments $A, $P and $R are
"accuracy", "precision" and "round_mode". Please see the section about
"ACCURACY and PRECISION" for more information.
config()
use Data::Dumper;
print Dumper ( Math::BigInt->config() );
print Math::BigInt->config()->{lib},"\n";
Returns a hash containing the configuration, e.g. the version number,
lib loaded etc. The following hash keys are currently filled in with
the appropriate information.
key Description
Example
============================================================
lib Name of the low-level math library
Math::BigInt::Calc
lib_version Version of low-level math library (see 'lib')
0.30
class The class name of config() you just called
Math::BigInt
upgrade To which class math operations might be upgraded
Math::BigFloat
downgrade To which class math operations might be downgraded
undef
precision Global precision
undef
accuracy Global accuracy
undef
round_mode Global round mode
even
version version number of the class you used
1.61
div_scale Fallback accuracy for div
40
trap_nan If true, traps creation of NaN via croak()
1
trap_inf If true, traps creation of +inf/-inf via croak()
1
The following values can be set by passing "config()" a reference to a
hash:
trap_inf trap_nan
upgrade downgrade precision accuracy round_mode div_scale
Example:
$new_cfg = Math::BigInt->config(
{ trap_inf => 1, precision => 5 }
);
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
Please see the section about "ACCURACY and PRECISION" for further
details.
Value must be greater than zero. Pass an undef value to disable it:
$x->accuracy(undef);
Math::BigInt->accuracy(undef);
Returns the current accuracy. For "$x->accuracy()" it will return
either the local accuracy, or if not defined, the global. This means
the return value represents the accuracy that will be in effect for $x:
$y = Math::BigInt->new(1234567); # unrounded
print Math::BigInt->accuracy(4),"\n"; # set 4, print 4
$x = Math::BigInt->new(123456); # $x will be automatic-
# ally rounded!
print "$x $y\n"; # '123500 1234567'
print $x->accuracy(),"\n"; # will be 4
print $y->accuracy(),"\n"; # also 4, since global is 4
print Math::BigInt->accuracy(5),"\n"; # set to 5, print 5
print $x->accuracy(),"\n"; # still 4
print $y->accuracy(),"\n"; # 5, since global is 5
Note: Works also for subclasses like Math::BigFloat. Each class has
it's own globals separated from Math::BigInt, but it is possible to
subclass Math::BigInt and make the globals of the subclass aliases to
the ones from Math::BigInt.
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 "accuracy()"
you set the number of digits each result should have, with
"precision()" you set the place where to round!
"precision()" sets or gets the global or local precision, aka at which
digit before or after the dot to round all results. A set global
precision also applies to all newly created numbers!
In Math::BigInt, passing a negative number precision has no effect
since no numbers have digits after the dot. In Math::BigFloat, it will
round all results to P digits after the dot.
Please see the section about "ACCURACY and PRECISION" for further
details.
Pass an undef value to disable it:
$x->precision(undef);
Math::BigInt->precision(undef);
Returns the current precision. For "$x->precision()" it will return
either the local precision of $x, or if not defined, the global. This
means the return value represents the prevision that will be in effect
for $x:
$y = Math::BigInt->new(1234567); # unrounded
print Math::BigInt->precision(4),"\n"; # set 4, print 4
$x = Math::BigInt->new(123456); # will be automatically rounded
print $x; # print "120000"!
Note: Works also for subclasses like Math::BigFloat. Each class has its
own globals separated from Math::BigInt, but it is possible to subclass
Math::BigInt and make the globals of the subclass aliases to the ones
from Math::BigInt.
brsft()
$x->brsft($y,$n);
Shifts $x right by $y in base $n. Default is base 2, used are usually
10 and 2, but others work, too.
Right shifting usually amounts to dividing $x by $n ** $y and
truncating the result:
$x = Math::BigInt->new(10);
$x->brsft(1); # same as $x >> 1: 5
$x = Math::BigInt->new(1234);
$x->brsft(2,10); # result 12
There is one exception, and that is base 2 with negative $x:
$x = Math::BigInt->new(-5);
print $x->brsft(1);
This will print -3, not -2 (as it would if you divide -5 by 2 and
truncate the result).
new()
$x = Math::BigInt->new($str,$A,$P,$R);
Creates a new BigInt object from a scalar or another BigInt object. The
input is accepted as decimal, hex (with leading '0x') or binary (with
leading '0b').
See "Input" for more info on accepted input formats.
from_oct()
$x = Math::BigInt->from_oct("0775"); # input is octal
Interpret the input as an octal string and return the corresponding
value. A "0" (zero) prefix is optional. A single underscore character
may be placed right after the prefix, if present, or between any two
digits. If the input is invalid, a NaN is returned.
from_hex()
$x = Math::BigInt->from_hex("0xcafe"); # input is hexadecimal
Interpret input as a hexadecimal string. A "0x" or "x" prefix is
optional. A single underscore character may be placed right after the
prefix, if present, or between any two digits. If the input is invalid,
a NaN is returned.
from_bin()
$x = Math::BigInt->from_bin("0b10011"); # input is binary
Interpret the input as a binary string. A "0b" or "b" prefix is
optional. A single underscore character may be placed right after the
prefix, if present, or between any two digits. If the input is invalid,
a NaN is returned.
bnan()
$x = Math::BigInt->bnan();
Creates a new BigInt object representing NaN (Not A Number). If used
on an object, it will set it to NaN:
$x->bnan();
bzero()
$x = Math::BigInt->bzero();
Creates a new BigInt object representing zero. If used on an object,
it will set it to zero:
$x->bzero();
binf()
$x = Math::BigInt->binf($sign);
Creates a new BigInt object representing infinity. The optional
argument is either '-' or '+', indicating whether you want infinity or
minus infinity. If used on an object, it will set it to infinity:
$x->binf();
$x->binf('-');
bone()
$x = Math::BigInt->binf($sign);
Creates a new BigInt object representing one. The optional argument is
either '-' or '+', indicating whether you want one or minus one. If
used on an object, it will set it to one:
$x->bone(); # +1
$x->bone('-'); # -1
is_one()/is_zero()/is_nan()/is_inf()
$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_inf(); # true if +inf
$x->is_inf('-'); # true if -inf (sign is default '+')
These methods all test the BigInt for being one specific value and
return true or false depending on the input. These are faster than
doing something like:
if ($x == 0)
is_pos()/is_neg()/is_positive()/is_negative()
$x->is_pos(); # true if > 0
$x->is_neg(); # true if < 0
The methods return true if the argument is positive or negative,
respectively. "NaN" is neither positive nor negative, while "+inf"
counts as positive, and "-inf" is negative. A "zero" is neither
positive nor negative.
These methods are only testing the sign, and not the value.
"is_positive()" and "is_negative()" are aliases to "is_pos()" and
"is_neg()", respectively. "is_positive()" and "is_negative()" were
introduced in v1.36, while "is_pos()" and "is_neg()" were only
introduced in v1.68.
is_odd()/is_even()/is_int()
$x->is_odd(); # true if odd, false for even
$x->is_even(); # true if even, false for odd
$x->is_int(); # true if $x is an integer
The return true when the argument satisfies the condition. "NaN",
"+inf", "-inf" are not integers and are neither odd nor even.
In BigInt, all numbers except "NaN", "+inf" and "-inf" are integers.
bcmp()
$x->bcmp($y);
Compares $x with $y and takes the sign into account. Returns -1, 0, 1
or undef.
bacmp()
$x->bacmp($y);
Compares $x with $y while ignoring their sign. Returns -1, 0, 1 or
undef.
sign()
$x->sign();
Return the sign, of $x, meaning either "+", "-", "-inf", "+inf" or NaN.
If you want $x to have a certain sign, use one of the following
methods:
$x->babs(); # '+'
$x->babs()->bneg(); # '-'
$x->bnan(); # 'NaN'
$x->binf(); # '+inf'
$x->binf('-'); # '-inf'
digit()
$x->digit($n); # return the nth digit, counting from right
If $n is negative, returns the digit counting from left.
bneg()
$x->bneg();
Negate the number, e.g. change the sign between '+' and '-', or between
'+inf' and '-inf', respectively. Does nothing for NaN or zero.
babs()
$x->babs();
Set the number to its absolute value, e.g. change the sign from '-' to
'+' and from '-inf' to '+inf', respectively. Does nothing for NaN or
positive numbers.
bsgn()
$x->bsgn();
Signum function. Set the number to -1, 0, or 1, depending on whether
the number is negative, zero, or positive, respectivly. Does not modify
NaNs.
bnorm()
$x->bnorm(); # normalize (no-op)
bnot()
$x->bnot();
Two's complement (bitwise not). This is equivalent to
$x->binc()->bneg();
but faster.
binc()
$x->binc(); # increment x by 1
bdec()
$x->bdec(); # decrement x by 1
badd()
$x->badd($y); # addition (add $y to $x)
bsub()
$x->bsub($y); # subtraction (subtract $y from $x)
bmul()
$x->bmul($y); # multiplication (multiply $x by $y)
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).
bdiv()
$x->bdiv($y); # divide, set $x to quotient
# return (quo,rem) or quo if scalar
bmod()
$x->bmod($y); # modulus (x % y)
bmodinv()
$x->bmodinv($mod); # modular multiplicative inverse
Returns the multiplicative inverse of $x modulo $mod. If
$y = $x -> copy() -> bmodinv($mod)
then $y is the number closest to zero, and with the same sign as $mod,
satisfying
($x * $y) % $mod = 1 % $mod
If $x and $y are non-zero, they must be relative primes, i.e.,
"bgcd($y, $mod)==1". '"NaN"' is returned when no modular multiplicative
inverse exists.
bmodpow()
$num->bmodpow($exp,$mod); # modular exponentiation
# ($num**$exp % $mod)
Returns the value of $num taken to the power $exp in the modulus $mod
using binary exponentiation. "bmodpow" is far superior to writing
$num ** $exp % $mod
because it is much faster - it reduces internal variables into the
modulus whenever possible, so it operates on smaller numbers.
"bmodpow" also supports negative exponents.
bmodpow($num, -1, $mod)
is exactly equivalent to
bmodinv($num, $mod)
bpow()
$x->bpow($y); # power of arguments (x ** y)
blog()
$x->blog($base, $accuracy); # logarithm of x to the base $base
If $base is not defined, Euler's number (e) is used:
print $x->blog(undef, 100); # log(x) to 100 digits
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).
See also "blog()".
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::BigInt->bpi(100), "\n"; # 3
Returns PI truncated to an integer, with the argument being ignored.
This means under BigInt this always returns 3.
If upgrading is in effect, returns PI, rounded to N digits with the
current rounding mode:
use Math::BigFloat;
use Math::BigInt upgrade => Math::BigFloat;
print Math::BigInt->bpi(3), "\n"; # 3.14
print Math::BigInt->bpi(100), "\n"; # 3.1415....
This method was added in v1.87 of Math::BigInt (June 2007).
bcos()
my $x = Math::BigInt->new(1);
print $x->bcos(100), "\n";
Calculate the cosinus of $x, modifying $x in place.
In BigInt, unless upgrading is in effect, the result is truncated to an
integer.
This method was added in v1.87 of Math::BigInt (June 2007).
bsin()
my $x = Math::BigInt->new(1);
print $x->bsin(100), "\n";
Calculate the sinus of $x, modifying $x in place.
In BigInt, unless upgrading is in effect, the result is truncated to an
integer.
This method was added in v1.87 of Math::BigInt (June 2007).
batan2()
my $x = Math::BigInt->new(1);
my $y = Math::BigInt->new(1);
print $y->batan2($x), "\n";
Calculate the arcus tangens of $y divided by $x, modifying $y in place.
In BigInt, unless upgrading is in effect, the result is truncated to an
integer.
This method was added in v1.87 of Math::BigInt (June 2007).
batan()
my $x = Math::BigFloat->new(0.5);
print $x->batan(100), "\n";
Calculate the arcus tangens of $x, modifying $x in place.
In BigInt, unless upgrading is in effect, the result is truncated to an
integer.
This method was added in v1.87 of Math::BigInt (June 2007).
blsft()
$x->blsft($y); # left shift in base 2
$x->blsft($y,$n); # left shift, in base $n (like 10)
brsft()
$x->brsft($y); # right shift in base 2
$x->brsft($y,$n); # right shift, in base $n (like 10)
band()
$x->band($y); # bitwise and
bior()
$x->bior($y); # bitwise inclusive or
bxor()
$x->bxor($y); # bitwise exclusive or
bnot()
$x->bnot(); # bitwise not (two's complement)
bsqrt()
$x->bsqrt(); # calculate square-root
broot()
$x->broot($N);
Calculates the N'th root of $x.
bfac()
$x->bfac(); # factorial of $x (1*2*3*4*..$x)
round()
$x->round($A,$P,$round_mode);
Round $x to accuracy $A or precision $P using the round mode
$round_mode.
bround()
$x->bround($N); # accuracy: preserve $N digits
bfround()
$x->bfround($N);
If N is > 0, rounds to the Nth digit from the left. If N < 0, rounds to
the Nth digit after the dot. Since BigInts are integers, the case N < 0
is a no-op for them.
Examples:
Input N Result
===================================================
123456.123456 3 123500
123456.123456 2 123450
123456.123456 -2 123456.12
123456.123456 -3 123456.123
bfloor()
$x->bfloor();
Set $x to the integer less or equal than $x. This is a no-op in BigInt,
but does change $x in BigFloat.
bceil()
$x->bceil();
Set $x to the integer greater or equal than $x. This is a no-op in
BigInt, but does change $x in BigFloat.
bgcd()
bgcd(@values); # greatest common divisor (no OO style)
blcm()
blcm(@values); # lowest common multiple (no OO style)
head2 length()
$x->length();
($xl,$fl) = $x->length();
Returns the number of digits in the decimal representation of the
number. In list context, returns the length of the integer and
fraction part. For BigInt's, the length of the fraction part will
always be 0.
exponent()
$x->exponent();
Return the exponent of $x as BigInt.
mantissa()
$x->mantissa();
Return the signed mantissa of $x as BigInt.
parts()
$x->parts(); # return (mantissa,exponent) as BigInt
copy()
$x->copy(); # make a true copy of $x (unlike $y = $x;)
as_int()/as_number()
$x->as_int();
Returns $x as a BigInt (truncated towards zero). In BigInt this is the
same as "copy()".
"as_number()" is an alias to this method. "as_number" was introduced in
v1.22, while "as_int()" was only introduced in v1.68.
bstr()
$x->bstr();
Returns a normalized string representation of $x.
bsstr()
$x->bsstr(); # normalized string in scientific notation
as_hex()
$x->as_hex(); # as signed hexadecimal string with prefixed 0x
as_bin()
$x->as_bin(); # as signed binary string with prefixed 0b
as_oct()
$x->as_oct(); # as signed octal string with prefixed 0
numify()
print $x->numify();
This returns a normal Perl scalar from $x. It is used automatically
whenever a scalar is needed, for instance in array index operations.
This loses precision, to avoid this use as_int() instead.
modify()
$x->modify('bpowd');
This method returns 0 if the object can be modified with the given
operation, or 1 if not.
This is used for instance by Math::BigInt::Constant.
upgrade()/downgrade()
Set/get the class for downgrade/upgrade operations. Thuis is used for
instance by bignum. The defaults are '', thus the following operation
will create a BigInt, not a BigFloat:
my $i = Math::BigInt->new(123);
my $f = Math::BigFloat->new('123.1');
print $i + $f,"\n"; # print 246
div_scale()
Set/get the number of digits for the default precision in divide
operations.
round_mode()
Set/get the current round mode.
ACCURACY and PRECISION
Since version v1.33, Math::BigInt and Math::BigFloat have full support
for accuracy and precision based rounding, both automatically after
every operation, as well as manually.
This section describes the accuracy/precision handling in Math::Big* as
it used to be and as it is now, complete with an explanation of all
terms and abbreviations.
Not yet implemented things (but with correct description) are marked
with '!', things that need to be answered are marked with '?'.
In the next paragraph follows a short description of terms used here
(because these may differ from terms used by others people or
documentation).
During the rest of this document, the shortcuts A (for accuracy), P
(for precision), F (fallback) and R (rounding mode) will be used.
Precision P
A fixed number of digits before (positive) or after (negative) the
decimal point. For example, 123.45 has a precision of -2. 0 means an
integer like 123 (or 120). A precision of 2 means two digits to the
left of the decimal point are zero, so 123 with P = 1 becomes 120. Note
that numbers with zeros before the decimal point may have different
precisions, because 1200 can have p = 0, 1 or 2 (depending on what the
initial value was). It could also have p < 0, when the digits after the
decimal point are zero.
The string output (of floating point numbers) will be padded with
zeros:
Initial value P A Result String
------------------------------------------------------------
1234.01 -3 1000 1000
1234 -2 1200 1200
1234.5 -1 1230 1230
1234.001 1 1234 1234.0
1234.01 0 1234 1234
1234.01 2 1234.01 1234.01
1234.01 5 1234.01 1234.01000
For BigInts, no padding occurs.
Accuracy A
Number of significant digits. Leading zeros are not counted. A number
may have an accuracy greater than the non-zero digits when there are
zeros in it or trailing zeros. For example, 123.456 has A of 6, 10203
has 5, 123.0506 has 7, 123.450000 has 8 and 0.000123 has 3.
The string output (of floating point numbers) will be padded with
zeros:
Initial value P A Result String
------------------------------------------------------------
1234.01 3 1230 1230
1234.01 6 1234.01 1234.01
1234.1 8 1234.1 1234.1000
For BigInts, no padding occurs.
Fallback F
When both A and P are undefined, this is used as a fallback accuracy
when dividing numbers.
Rounding mode R
When rounding a number, different 'styles' or 'kinds' of rounding are
possible. (Note that random rounding, as in Math::Round, is not
implemented.)
'trunc'
truncation invariably removes all digits following the rounding
place, replacing them with zeros. Thus, 987.65 rounded to tens (P=1)
becomes 980, and rounded to the fourth sigdig becomes 987.6 (A=4).
123.456 rounded to the second place after the decimal point (P=-2)
becomes 123.46.
All other implemented styles of rounding attempt to round to the
"nearest digit." If the digit D immediately to the right of the
rounding place (skipping the decimal point) is greater than 5, the
number is incremented at the rounding place (possibly causing a
cascade of incrementation): e.g. when rounding to units, 0.9 rounds
to 1, and -19.9 rounds to -20. If D < 5, the number is similarly
truncated at the rounding place: e.g. when rounding to units, 0.4
rounds to 0, and -19.4 rounds to -19.
However the results of other styles of rounding differ if the digit
immediately to the right of the rounding place (skipping the decimal
point) is 5 and if there are no digits, or no digits other than 0,
after that 5. In such cases:
'even'
rounds the digit at the rounding place to 0, 2, 4, 6, or 8 if it is
not already. E.g., when rounding to the first sigdig, 0.45 becomes
0.4, -0.55 becomes -0.6, but 0.4501 becomes 0.5.
'odd'
rounds the digit at the rounding place to 1, 3, 5, 7, or 9 if it is
not already. E.g., when rounding to the first sigdig, 0.45 becomes
0.5, -0.55 becomes -0.5, but 0.5501 becomes 0.6.
'+inf'
round to plus infinity, i.e. always round up. E.g., when rounding to
the first sigdig, 0.45 becomes 0.5, -0.55 becomes -0.5, and 0.4501
also becomes 0.5.
'-inf'
round to minus infinity, i.e. always round down. E.g., when rounding
to the first sigdig, 0.45 becomes 0.4, -0.55 becomes -0.6, but 0.4501
becomes 0.5.
'zero'
round to zero, i.e. positive numbers down, negative ones up. E.g.,
when rounding to the first sigdig, 0.45 becomes 0.4, -0.55 becomes
-0.5, but 0.4501 becomes 0.5.
'common'
round up if the digit immediately to the right of the rounding place
is 5 or greater, otherwise round down. E.g., 0.15 becomes 0.2 and
0.149 becomes 0.1.
The handling of A & P in MBI/MBF (the old core code shipped with Perl
versions <= 5.7.2) is like this:
Precision
* ffround($p) is able to round to $p number of digits after the decimal
point
* otherwise P is unused
Accuracy (significant digits)
* fround($a) rounds to $a significant digits
* only fdiv() and fsqrt() take A as (optional) parameter
+ other operations simply create the same number (fneg etc), or more (fmul)
of digits
+ rounding/truncating is only done when explicitly calling one of fround
or ffround, and never for BigInt (not implemented)
* fsqrt() simply hands its accuracy argument over to fdiv.
* the documentation and the comment in the code indicate two different ways
on how fdiv() determines the maximum number of digits it should calculate,
and the actual code does yet another thing
POD:
max($Math::BigFloat::div_scale,length(dividend)+length(divisor))
Comment:
result has at most max(scale, length(dividend), length(divisor)) digits
Actual code:
scale = max(scale, length(dividend)-1,length(divisor)-1);
scale += length(divisor) - length(dividend);
So for lx = 3, ly = 9, scale = 10, scale will actually be 16 (10+9-3).
Actually, the 'difference' added to the scale is calculated from the
number of "significant digits" in dividend and divisor, which is derived
by looking at the length of the mantissa. Which is wrong, since it includes
the + sign (oops) and actually gets 2 for '+100' and 4 for '+101'. Oops
again. Thus 124/3 with div_scale=1 will get you '41.3' based on the strange
assumption that 124 has 3 significant digits, while 120/7 will get you
'17', not '17.1' since 120 is thought to have 2 significant digits.
The rounding after the division then uses the remainder and $y to determine
whether it must round up or down.
? I have no idea which is the right way. That's why I used a slightly more
? simple scheme and tweaked the few failing testcases to match it.
This is how it works now:
Setting/Accessing
* You can set the A global via Math::BigInt->accuracy() or
Math::BigFloat->accuracy() or whatever class you are using.
* You can also set P globally by using Math::SomeClass->precision()
likewise.
* Globals are classwide, and not inherited by subclasses.
* to undefine A, use Math::SomeCLass->accuracy(undef);
* to undefine P, use Math::SomeClass->precision(undef);
* Setting Math::SomeClass->accuracy() clears automatically
Math::SomeClass->precision(), and vice versa.
* To be valid, A must be > 0, P can have any value.
* If P is negative, this means round to the P'th place to the right of the
decimal point; positive values mean to the left of the decimal point.
P of 0 means round to integer.
* to find out the current global A, use Math::SomeClass->accuracy()
* to find out the current global P, use Math::SomeClass->precision()
* use $x->accuracy() respective $x->precision() for the local
setting of $x.
* Please note that $x->accuracy() respective $x->precision()
return eventually defined global A or P, when $x's A or P is not
set.
Creating numbers
* When you create a number, you can give the desired A or P via:
$x = Math::BigInt->new($number,$A,$P);
* Only one of A or P can be defined, otherwise the result is NaN
* If no A or P is give ($x = Math::BigInt->new($number) form), then the
globals (if set) will be used. Thus changing the global defaults later on
will not change the A or P of previously created numbers (i.e., A and P of
$x will be what was in effect when $x was created)
* If given undef for A and P, NO rounding will occur, and the globals will
NOT be used. This is used by subclasses to create numbers without
suffering rounding in the parent. Thus a subclass is able to have its own
globals enforced upon creation of a number by using
$x = Math::BigInt->new($number,undef,undef):
use Math::BigInt::SomeSubclass;
use Math::BigInt;
Math::BigInt->accuracy(2);
Math::BigInt::SomeSubClass->accuracy(3);
$x = Math::BigInt::SomeSubClass->new(1234);
$x is now 1230, and not 1200. A subclass might choose to implement
this otherwise, e.g. falling back to the parent's A and P.
Usage
* If A or P are enabled/defined, they are used to round the result of each
operation according to the rules below
* Negative P is ignored in Math::BigInt, since BigInts never have digits
after the decimal point
* Math::BigFloat uses Math::BigInt internally, but setting A or P inside
Math::BigInt as globals does not tamper with the parts of a BigFloat.
A flag is used to mark all Math::BigFloat numbers as 'never round'.
Precedence
* It only makes sense that a number has only one of A or P at a time.
If you set either A or P on one object, or globally, the other one will
be automatically cleared.
* If two objects are involved in an operation, and one of them has A in
effect, and the other P, this results in an error (NaN).
* A takes precedence over P (Hint: A comes before P).
If neither of them is defined, nothing is used, i.e. the result will have
as many digits as it can (with an exception for fdiv/fsqrt) and will not
be rounded.
* There is another setting for fdiv() (and thus for fsqrt()). If neither of
A or P is defined, fdiv() will use a fallback (F) of $div_scale digits.
If either the dividend's or the divisor's mantissa has more digits than
the value of F, the higher value will be used instead of F.
This is to limit the digits (A) of the result (just consider what would
happen with unlimited A and P in the case of 1/3 :-)
* fdiv will calculate (at least) 4 more digits than required (determined by
A, P or F), and, if F is not used, round the result
(this will still fail in the case of a result like 0.12345000000001 with A
or P of 5, but this can not be helped - or can it?)
* Thus you can have the math done by on Math::Big* class in two modi:
+ never round (this is the default):
This is done by setting A and P to undef. No math operation
will round the result, with fdiv() and fsqrt() as exceptions to guard
against overflows. You must explicitly call bround(), bfround() or
round() (the latter with parameters).
Note: Once you have rounded a number, the settings will 'stick' on it
and 'infect' all other numbers engaged in math operations with it, since
local settings have the highest precedence. So, to get SaferRound[tm],
use a copy() before rounding like this:
$x = Math::BigFloat->new(12.34);
$y = Math::BigFloat->new(98.76);
$z = $x * $y; # 1218.6984
print $x->copy()->fround(3); # 12.3 (but A is now 3!)
$z = $x * $y; # still 1218.6984, without
# copy would have been 1210!
+ round after each op:
After each single operation (except for testing like is_zero()), the
method round() is called and the result is rounded appropriately. By
setting proper values for A and P, you can have all-the-same-A or
all-the-same-P modes. For example, Math::Currency might set A to undef,
and P to -2, globally.
?Maybe an extra option that forbids local A & P settings would be in order,
?so that intermediate rounding does not 'poison' further math?
Overriding globals
* you will be able to give A, P and R as an argument to all the calculation
routines; the second parameter is A, the third one is P, and the fourth is
R (shift right by one for binary operations like badd). P is used only if
the first parameter (A) is undefined. These three parameters override the
globals in the order detailed as follows, i.e. the first defined value
wins:
(local: per object, global: global default, parameter: argument to sub)
+ parameter A
+ parameter P
+ local A (if defined on both of the operands: smaller one is taken)
+ local P (if defined on both of the operands: bigger one is taken)
+ global A
+ global P
+ global F
* fsqrt() will hand its arguments to fdiv(), as it used to, only now for two
arguments (A and P) instead of one
Local settings
* You can set A or P locally by using $x->accuracy() or
$x->precision()
and thus force different A and P for different objects/numbers.
* Setting A or P this way immediately rounds $x to the new value.
* $x->accuracy() clears $x->precision(), and vice versa.
Rounding
* the rounding routines will use the respective global or local settings.
fround()/bround() is for accuracy rounding, while ffround()/bfround()
is for precision
* the two rounding functions take as the second parameter one of the
following rounding modes (R):
'even', 'odd', '+inf', '-inf', 'zero', 'trunc', 'common'
* you can set/get the global R by using Math::SomeClass->round_mode()
or by setting $Math::SomeClass::round_mode
* after each operation, $result->round() is called, and the result may
eventually be rounded (that is, if A or P were set either locally,
globally or as parameter to the operation)
* to manually round a number, call $x->round($A,$P,$round_mode);
this will round the number by using the appropriate rounding function
and then normalize it.
* rounding modifies the local settings of the number:
$x = Math::BigFloat->new(123.456);
$x->accuracy(5);
$x->bround(4);
Here 4 takes precedence over 5, so 123.5 is the result and $x->accuracy()
will be 4 from now on.
Default values
* R: 'even'
* F: 40
* A: undef
* P: undef
Remarks
* The defaults are set up so that the new code gives the same results as
the old code (except in a few cases on fdiv):
+ Both A and P are undefined and thus will not be used for rounding
after each operation.
+ round() is thus a no-op, unless given extra parameters A and P
Infinity and Not a Number
While BigInt has extensive handling of inf and NaN, certain quirks
remain.
oct()/hex()
These perl routines currently (as of Perl v.5.8.6) cannot handle
passed inf.
te@linux:~> perl -wle 'print 2 ** 3333'
inf
te@linux:~> perl -wle 'print 2 ** 3333 == 2 ** 3333'
1
te@linux:~> perl -wle 'print oct(2 ** 3333)'
0
te@linux:~> perl -wle 'print hex(2 ** 3333)'
Illegal hexadecimal digit 'i' ignored at -e line 1.
0
The same problems occur if you pass them Math::BigInt->binf()
objects. Since overloading these routines is not possible, this
cannot be fixed from BigInt.
==, !=, <, >, <=, >= with NaNs
BigInt's bcmp() routine currently returns undef to signal that a NaN
was involved in a comparison. However, the overload code turns that
into either 1 or '' and thus operations like "NaN != NaN" might
return wrong values.
log(-inf)
"log(-inf)" is highly weird. Since log(-x)=pi*i+log(x), then
log(-inf)=pi*i+inf. However, since the imaginary part is finite, the
real infinity "overshadows" it, so the number might as well just be
infinity. However, the result is a complex number, and since
BigInt/BigFloat can only have real numbers as results, the result is
NaN.
exp(), cos(), sin(), atan2()
These all might have problems handling infinity right.
INTERNALS
The actual numbers are stored as unsigned big integers (with separate
sign).
You should neither care about nor depend on the internal
representation; it might change without notice. Use ONLY method calls
like "$x->sign();" instead relying on the internal representation.
MATH LIBRARY
Math with the numbers is done (by default) by a module called
"Math::BigInt::Calc". This is equivalent to saying:
use Math::BigInt try => 'Calc';
You can change this backend library by using:
use Math::BigInt try => 'GMP';
Note: General purpose packages should not be explicit about the library
to use; let the script author decide which is best.
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::BigInt 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::BigInt try => 'Foo,Math::BigInt::Bar';
The library that is loaded last will be used. Note that this can be
overwritten at any time by loading a different library, and numbers
constructed with different libraries cannot be used in math operations
together.
What library to use?
Note: General purpose packages should not be explicit about the library
to use; let the script author decide which is best.
Math::BigInt::GMP and Math::BigInt::Pari are in cases involving big
numbers much faster than Calc, however it is slower when dealing with
very small numbers (less than about 20 digits) and when converting very
large numbers to decimal (for instance for printing, rounding,
calculating their length in decimal etc).
So please select carefully what library you want to use.
Different low-level libraries use different formats to store the
numbers. However, you should NOT depend on the number having a
specific format internally.
See the respective math library module documentation for further
details.
SIGN
The sign is either '+', '-', 'NaN', '+inf' or '-inf'.
A sign of 'NaN' is used to represent the result when input arguments
are not numbers or as a result of 0/0. '+inf' and '-inf' represent plus
respectively minus infinity. You will get '+inf' when dividing a
positive number by 0, and '-inf' when dividing any negative number by
0.
mantissa(), exponent() and parts()
"mantissa()" and "exponent()" return the said parts of the BigInt 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 that gives you both of them
in one go. Both the returned mantissa and exponent have a sign.
Currently, for BigInts $e is always 0, except +inf and -inf, where it
is "+inf"; and for NaN, where it is "NaN"; and for "$x == 0", where it
is 1 (to be compatible with Math::BigFloat's internal representation of
a zero as 0E1).
$m is currently just a copy of the original number. The relation
between $e and $m will stay always the same, though their real values
might change.
EXAMPLES
use Math::BigInt;
sub bint { Math::BigInt->new(shift); }
$x = Math::BigInt->bstr("1234") # string "1234"
$x = "$x"; # same as bstr()
$x = Math::BigInt->bneg("1234"); # BigInt "-1234"
$x = Math::BigInt->babs("-12345"); # BigInt "12345"
$x = Math::BigInt->bnorm("-0.00"); # BigInt "0"
$x = bint(1) + bint(2); # BigInt "3"
$x = bint(1) + "2"; # ditto (auto-BigIntify of "2")
$x = bint(1); # BigInt "1"
$x = $x + 5 / 2; # BigInt "3"
$x = $x ** 3; # BigInt "27"
$x *= 2; # BigInt "54"
$x = Math::BigInt->new(0); # BigInt "0"
$x--; # BigInt "-1"
$x = Math::BigInt->badd(4,5) # BigInt "9"
print $x->bsstr(); # 9e+0
Examples for rounding:
use Math::BigFloat;
use Test;
$x = Math::BigFloat->new(123.4567);
$y = Math::BigFloat->new(123.456789);
Math::BigFloat->accuracy(4); # no more A than 4
ok ($x->copy()->fround(),123.4); # even rounding
print $x->copy()->fround(),"\n"; # 123.4
Math::BigFloat->round_mode('odd'); # round to odd
print $x->copy()->fround(),"\n"; # 123.5
Math::BigFloat->accuracy(5); # no more A than 5
Math::BigFloat->round_mode('odd'); # round to odd
print $x->copy()->fround(),"\n"; # 123.46
$y = $x->copy()->fround(4),"\n"; # A = 4: 123.4
print "$y, ",$y->accuracy(),"\n"; # 123.4, 4
Math::BigFloat->accuracy(undef); # A not important now
Math::BigFloat->precision(2); # P important
print $x->copy()->bnorm(),"\n"; # 123.46
print $x->copy()->fround(),"\n"; # 123.46
Examples for converting:
my $x = Math::BigInt->new('0b1'.'01' x 123);
print "bin: ",$x->as_bin()," hex:",$x->as_hex()," dec: ",$x,"\n";
Autocreating constants
After "use Math::BigInt ':constant'" all the integer decimal,
hexadecimal and binary constants in the given scope are converted to
"Math::BigInt". This conversion happens at compile time.
In particular,
perl -MMath::BigInt=:constant -e 'print 2**100,"\n"'
prints the integer value of "2**100". Note that without conversion of
constants the expression 2**100 will be calculated as perl scalar.
Please note that strings and floating point constants are not affected,
so that
use Math::BigInt qw/:constant/;
$x = 1234567890123456789012345678901234567890
+ 123456789123456789;
$y = '1234567890123456789012345678901234567890'
+ '123456789123456789';
do not work. You need an explicit Math::BigInt->new() around one of the
operands. You should also quote large constants to protect loss of
precision:
use Math::BigInt;
$x = Math::BigInt->new('1234567889123456789123456789123456789');
Without the quotes Perl would convert the large number to a floating
point constant at compile time and then hand the result to BigInt,
which results in an truncated result or a NaN.
This also applies to integers that look like floating point constants:
use Math::BigInt ':constant';
print ref(123e2),"\n";
print ref(123.2e2),"\n";
will print nothing but newlines. Use either bignum or Math::BigFloat to
get this to work.
PERFORMANCE
Using the form $x += $y; etc over $x = $x + $y is faster, since a copy
of $x must be made in the second case. For long numbers, the copy can
eat up to 20% of the work (in the case of addition/subtraction, less
for multiplication/division). If $y is very small compared to $x, the
form $x += $y is MUCH faster than $x = $x + $y since making the copy of
$x takes more time then the actual addition.
With a technique called copy-on-write, the cost of copying with
overload could be minimized or even completely avoided. A test
implementation of COW did show performance gains for overloaded math,
but introduced a performance loss due to a constant overhead for all
other operations. So Math::BigInt does currently not COW.
The rewritten version of this module (vs. v0.01) is slower on certain
operations, like "new()", "bstr()" and "numify()". The reason are that
it does now more work and handles much more cases. The time spent in
these operations is usually gained in the other math operations so that
code on the average should get (much) faster. If they don't, please
contact the author.
Some operations may be slower for small numbers, but are significantly
faster for big numbers. Other operations are now constant (O(1), like
"bneg()", "babs()" etc), instead of O(N) and thus nearly always take
much less time. These optimizations were done on purpose.
If you find the Calc module to slow, try to install any of the
replacement modules and see if they help you.
Alternative math libraries
You can use an alternative library to drive Math::BigInt. See the
section "MATH LIBRARY" for more information.
For more benchmark results see
<http://bloodgate.com/perl/benchmarks.html>;.
SUBCLASSING
Subclassing Math::BigInt
The basic design of Math::BigInt allows simple subclasses with very
little work, as long as a few simple rules are followed:
o The public API must remain consistent, i.e. if a sub-class is
overloading addition, the sub-class must use the same name, in this
case badd(). The reason for this is that Math::BigInt is optimized to
call the object methods directly.
o The private object hash keys like "$x->{sign}" may not be changed,
but additional keys can be added, like "$x->{_custom}".
o Accessor functions are available for all existing object hash keys
and should be used instead of directly accessing the internal hash
keys. The reason for this is that Math::BigInt itself has a pluggable
interface which permits it to support different storage methods.
More complex sub-classes may have to replicate more of the logic
internal of Math::BigInt if they need to change more basic behaviors. A
subclass that needs to merely change the output only needs to overload
"bstr()".
All other object methods and overloaded functions can be directly
inherited from the parent class.
At the very minimum, any subclass will need to provide its own "new()"
and can store additional hash keys in the object. There are also some
package globals that must be defined, e.g.:
# Globals
$accuracy = undef;
$precision = -2; # round to 2 decimal places
$round_mode = 'even';
$div_scale = 40;
Additionally, you might want to provide the following two globals to
allow auto-upgrading and auto-downgrading to work correctly:
$upgrade = undef;
$downgrade = undef;
This allows Math::BigInt to correctly retrieve package globals from the
subclass, like $SubClass::precision. See t/Math/BigInt/Subclass.pm or
t/Math/BigFloat/SubClass.pm completely functional subclass examples.
Don't forget to
use overload;
in your subclass to automatically inherit the overloading from the
parent. If you like, you can change part of the overloading, look at
Math::String for an example.
UPGRADING
When used like this:
use Math::BigInt upgrade => 'Foo::Bar';
certain operations will 'upgrade' their calculation and thus the result
to the class Foo::Bar. Usually this is used in conjunction with
Math::BigFloat:
use Math::BigInt upgrade => 'Math::BigFloat';
As a shortcut, you can use the module "bignum":
use bignum;
Also good for one-liners:
perl -Mbignum -le 'print 2 ** 255'
This makes it possible to mix arguments of different classes (as in 2.5
+ 2) as well es preserve accuracy (as in sqrt(3)).
Beware: This feature is not fully implemented yet.
Auto-upgrade
The following methods upgrade themselves unconditionally; that is if
upgrade is in effect, they will always hand up their work:
bsqrt()
div()
blog()
bexp()
Beware: This list is not complete.
All other methods upgrade themselves only when one (or all) of their
arguments are of the class mentioned in $upgrade (This might change in
later versions to a more sophisticated scheme):
EXPORTS
"Math::BigInt" exports nothing by default, but can export the following
methods:
bgcd
blcm
CAVEATS
Some things might not work as you expect them. Below is documented what
is known to be troublesome:
bstr(), bsstr() and 'cmp'
Both "bstr()" and "bsstr()" as well as automated stringify via
overload now drop the leading '+'. The old code would return '+3', the
new returns '3'. This is to be consistent with Perl and to make "cmp"
(especially with overloading) to work as you expect. It also solves
problems with "Test.pm", because its "ok()" uses 'eq' internally.
Mark Biggar said, when asked about to drop the '+' altogether, or make
only "cmp" work:
I agree (with the first alternative), don't add the '+' on positive
numbers. It's not as important anymore with the new internal
form for numbers. It made doing things like abs and neg easier,
but those have to be done differently now anyway.
So, the following examples will now work all as expected:
use Test;
BEGIN { plan tests => 1 }
use Math::BigInt;
my $x = new Math::BigInt 3*3;
my $y = new Math::BigInt 3*3;
ok ($x,3*3);
print "$x eq 9" if $x eq $y;
print "$x eq 9" if $x eq '9';
print "$x eq 9" if $x eq 3*3;
Additionally, the following still works:
print "$x == 9" if $x == $y;
print "$x == 9" if $x == 9;
print "$x == 9" if $x == 3*3;
There is now a "bsstr()" method to get the string in scientific
notation aka 1e+2 instead of 100. Be advised that overloaded 'eq'
always uses bstr() for comparison, but Perl will represent some
numbers as 100 and others as 1e+308. If in doubt, convert both
arguments to Math::BigInt before comparing them as strings:
use Test;
BEGIN { plan tests => 3 }
use Math::BigInt;
$x = Math::BigInt->new('1e56'); $y = 1e56;
ok ($x,$y); # will fail
ok ($x->bsstr(),$y); # okay
$y = Math::BigInt->new($y);
ok ($x,$y); # okay
Alternatively, simple use "<=>" for comparisons, this will get it
always right. There is not yet a way to get a number automatically
represented as a string that matches exactly the way Perl represents
it.
See also the section about "Infinity and Not a Number" for problems in
comparing NaNs.
int()
"int()" will return (at least for Perl v5.7.1 and up) another BigInt,
not a Perl scalar:
$x = Math::BigInt->new(123);
$y = int($x); # BigInt 123
$x = Math::BigFloat->new(123.45);
$y = int($x); # BigInt 123
In all Perl versions you can use "as_number()" or "as_int" for the
same effect:
$x = Math::BigFloat->new(123.45);
$y = $x->as_number(); # BigInt 123
$y = $x->as_int(); # ditto
This also works for other subclasses, like Math::String.
If you want a real Perl scalar, use "numify()":
$y = $x->numify(); # 123 as scalar
This is seldom necessary, though, because this is done automatically,
like when you access an array:
$z = $array[$x]; # does work automatically
length
The following will probably not do what you expect:
$c = Math::BigInt->new(123);
print $c->length(),"\n"; # prints 30
It prints both the number of digits in the number and in the fraction
part since print calls "length()" in list context. Use something like:
print scalar $c->length(),"\n"; # prints 3
bdiv
The following will probably not do what you expect:
print $c->bdiv(10000),"\n";
It prints both quotient and remainder since print calls "bdiv()" in
list context. Also, "bdiv()" will modify $c, so be careful. You
probably want to use
print $c / 10000,"\n";
print scalar $c->bdiv(10000),"\n"; # or if you want to modify $c
instead.
The quotient is always the greatest integer less than or equal to the
real-valued quotient of the two operands, and the remainder (when it
is non-zero) always has the same sign as the second operand; so, for
example,
1 / 4 => ( 0, 1)
1 / -4 => (-1,-3)
-3 / 4 => (-1, 1)
-3 / -4 => ( 0,-3)
-11 / 2 => (-5,1)
11 /-2 => (-5,-1)
As a consequence, the behavior of the operator % agrees with the
behavior of Perl's built-in % operator (as documented in the perlop
manpage), and the equation
$x == ($x / $y) * $y + ($x % $y)
holds true for any $x and $y, which justifies calling the two return
values of bdiv() the quotient and remainder. The only exception to
this rule are when $y == 0 and $x is negative, then the remainder will
also be negative. See below under "infinity handling" for the
reasoning behind this.
Perl's 'use integer;' changes the behaviour of % and / for scalars,
but will not change BigInt's way to do things. This is because under
'use integer' Perl will do what the underlying C thinks is right and
this is different for each system. If you need BigInt's behaving
exactly like Perl's 'use integer', bug the author to implement it ;)
infinity handling
Here are some examples that explain the reasons why certain results
occur while handling infinity:
The following table shows the result of the division and the
remainder, so that the equation above holds true. Some "ordinary"
cases are strewn in to show more clearly the reasoning:
A / B = C, R so that C * B + R = A
=========================================================
5 / 8 = 0, 5 0 * 8 + 5 = 5
0 / 8 = 0, 0 0 * 8 + 0 = 0
0 / inf = 0, 0 0 * inf + 0 = 0
0 /-inf = 0, 0 0 * -inf + 0 = 0
5 / inf = 0, 5 0 * inf + 5 = 5
5 /-inf = 0, 5 0 * -inf + 5 = 5
-5/ inf = 0, -5 0 * inf + -5 = -5
-5/-inf = 0, -5 0 * -inf + -5 = -5
inf/ 5 = inf, 0 inf * 5 + 0 = inf
-inf/ 5 = -inf, 0 -inf * 5 + 0 = -inf
inf/ -5 = -inf, 0 -inf * -5 + 0 = inf
-inf/ -5 = inf, 0 inf * -5 + 0 = -inf
5/ 5 = 1, 0 1 * 5 + 0 = 5
-5/ -5 = 1, 0 1 * -5 + 0 = -5
inf/ inf = 1, 0 1 * inf + 0 = inf
-inf/-inf = 1, 0 1 * -inf + 0 = -inf
inf/-inf = -1, 0 -1 * -inf + 0 = inf
-inf/ inf = -1, 0 1 * -inf + 0 = -inf
8/ 0 = inf, 8 inf * 0 + 8 = 8
inf/ 0 = inf, inf inf * 0 + inf = inf
0/ 0 = NaN
These cases below violate the "remainder has the sign of the second of
the two arguments", since they wouldn't match up otherwise.
A / B = C, R so that C * B + R = A
========================================================
-inf/ 0 = -inf, -inf -inf * 0 + inf = -inf
-8/ 0 = -inf, -8 -inf * 0 + 8 = -8
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 (except overloaded operators) will
modify $y, and vice versa. Or in other words, "=" is only safe if you
modify your BigInts only via overloaded math. As soon as you use a
method call it breaks:
$x->bmul(2);
print "$x, $y\n"; # prints '10, 10'
If you want a true copy of $x, use:
$y = $x->copy();
You can also chain the calls like this, this will make first a copy
and then multiply it by 2:
$y = $x->copy()->bmul(2);
See also the documentation for overload.pm regarding "=".
bpow
"bpow()" (and the rounding functions) now modifies the first argument
and returns it, unlike the old code which left it alone and only
returned the result. This is to be consistent with "badd()" etc. The
first three will modify $x, the last one won't:
print bpow($x,$i),"\n"; # modify $x
print $x->bpow($i),"\n"; # ditto
print $x **= $i,"\n"; # the same
print $x ** $i,"\n"; # leave $x alone
The form "$x **= $y" is faster than "$x = $x ** $y;", though.
Overloading -$x
The following:
$x = -$x;
is slower than
$x->bneg();
since overload calls "sub($x,0,1);" instead of "neg($x)". The first
variant needs to preserve $x since it does not know that it later will
get overwritten. This makes a copy of $x and takes O(N), but
$x->bneg() is O(1).
Mixing different object types
In Perl you will get a floating point value if you do one of the
following:
$float = 5.0 + 2;
$float = 2 + 5.0;
$float = 5 / 2;
With overloaded math, only the first two variants will result in a
BigFloat:
use Math::BigInt;
use Math::BigFloat;
$mbf = Math::BigFloat->new(5);
$mbi2 = Math::BigInteger->new(5);
$mbi = Math::BigInteger->new(2);
# what actually gets called:
$float = $mbf + $mbi; # $mbf->badd()
$float = $mbf / $mbi; # $mbf->bdiv()
$integer = $mbi + $mbf; # $mbi->badd()
$integer = $mbi2 / $mbi; # $mbi2->bdiv()
$integer = $mbi2 / $mbf; # $mbi2->bdiv()
This is because math with overloaded operators follows the first
(dominating) operand, and the operation of that is called and returns
thus the result. So, Math::BigInt::bdiv() will always return a
Math::BigInt, regardless whether the result should be a Math::BigFloat
or the second operant is one.
To get a Math::BigFloat you either need to call the operation
manually, make sure the operands are already of the proper type or
casted to that type via Math::BigFloat->new():
$float = Math::BigFloat->new($mbi2) / $mbi; # = 2.5
Beware of simple "casting" the entire expression, this would only
convert the already computed result:
$float = Math::BigFloat->new($mbi2 / $mbi); # = 2.0 thus wrong!
Beware also of the order of more complicated expressions like:
$integer = ($mbi2 + $mbi) / $mbf; # int / float => int
$integer = $mbi2 / Math::BigFloat->new($mbi); # ditto
If in doubt, break the expression into simpler terms, or cast all
operands to the desired resulting type.
Scalar values are a bit different, since:
$float = 2 + $mbf;
$float = $mbf + 2;
will both result in the proper type due to the way the overloaded math
works.
This section also applies to other overloaded math packages, like
Math::String.
One solution to you problem might be autoupgrading|upgrading. See the
pragmas bignum, bigint and bigrat for an easy way to do this.
bsqrt()
"bsqrt()" works only good if the result is a big integer, e.g. the
square root of 144 is 12, but from 12 the square root is 3, regardless
of rounding mode. The reason is that the result is always truncated to
an integer.
If you want a better approximation of the square root, then use:
$x = Math::BigFloat->new(12);
Math::BigFloat->precision(0);
Math::BigFloat->round_mode('even');
print $x->copy->bsqrt(),"\n"; # 4
Math::BigFloat->precision(2);
print $x->bsqrt(),"\n"; # 3.46
print $x->bsqrt(3),"\n"; # 3.464
brsft()
For negative numbers in base see also brsft.
LICENSE
This program is free software; you may redistribute it and/or modify it
under the same terms as Perl itself.
SEE ALSO
Math::BigFloat, Math::BigRat and Math::Big as well as
Math::BigInt::BitVect, Math::BigInt::Pari and Math::BigInt::GMP.
The pragmas bignum, bigint and bigrat also might be of interest because
they solve the autoupgrading/downgrading issue, at least partly.
The package at
<http://search.cpan.org/search?mode=module&query=Math%3A%3ABigInt>
contains more documentation including a full version history,
testcases, empty subclass files and benchmarks.
AUTHORS
Original code by Mark Biggar, overloaded interface by Ilya Zakharevich.
Completely rewritten by Tels http://bloodgate.com in late 2000, 2001 -
2006 and still at it in 2007.
Many people contributed in one or more ways to the final beast, see the
file CREDITS for an (incomplete) list. If you miss your name, please
drop me a mail. Thank you!
perl v5.16.3 2023-10-25 Math::BigInt(3pm)
Linux-Distributionen
Huschi als Linux-Admin