math::bigint(3)

NAME

Math::BigInt - Arbitrary size integer math package

SYNOPSIS

use Math::BigInt;
# Number creation
$x = Math::BigInt->new($str);         # defaults to 0
$nan  = Math::BigInt->bnan();          #  create  a  NotANumber
$zero = Math::BigInt->bzero();        # create a +0
$inf = Math::BigInt->binf();          # create a +inf
$inf = Math::BigInt->binf('-');       # create a -inf
$one = Math::BigInt->bone();          # create a +1
$one = Math::BigInt->bone('-');       # create a -1
# Testing
$x->is_zero();                # true if arg is +0
$x->is_nan();                 # true if arg is NaN
$x->is_one();                 # true if arg is +1
$x->is_one('-');              # true if arg is -1
$x->is_odd();                  #  true if odd, false for
even
$x->is_even();                # true if even, false  for
odd
$x->is_positive();            # true if >= 0
$x->is_negative();            # true if <  0
$x->is_inf(sign);              #  true  if +inf, or -inf
(sign is default '+')
$x->is_int();                 # true if $x is an integer
(not a float)
$x->bcmp($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:
# set
$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->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)
$x->bmodpow($exp,$mod);        #  modular  exponentation
(($num**$exp) % $mod))
$x->bmodinv($mod);            # the inverse of $x in the
given modulus $mod
$x->bpow($y);                 # power of arguments (x **
y)
$x->blsft($y);                # left shift
$x->brsft($y);                # right shift
$x->blsft($y,$n);             # left shift, by  base  $n
(like 10)
$x->brsft($y,$n);              # right shift, by base $n
(like 10)
$x->band($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->bfac();                      #   factorial   of   $x
(1*2*3*4*..$x)
$x->round($A,$P,$round_mode); #  round  to  accuracy  or
precision using mode $r
$x->bround($N);                #  accuracy:  preserve $N
digits
$x->bfround($N);              # round to $Nth digit, noop for BigInts
# The following do not modify their arguments in BigInt,
but do 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:
bgcd(@values);                # greatest common  divisor
(no OO style)
blcm(@values);                 # lowest common multiplicator (no OO style)
$x->length();                 # return number of  digits
in number
($x,$f)  =  $x->length();        #  length of number and
length of fraction part,
                              # latter is always 0  digits long for BigInt's
$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_number();               #  return  as  BigInt (in
BigInt: same as copy())
# conversation to string
$x->bstr();                   # normalized string
$x->bsstr();                   #  normalized  string  in
scientific notation
$x->as_hex();                  #  as  signed hexadecimal
string with prefixed 0x
$x->as_bin();                 # as signed binary  string
with prefixed 0b
Math::BigInt->config();        #  return hash containing
configuration/version
# 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 P $x to $n
Math::BigInt->precision();    # get/set global P for all
BigInt objects
Math::BigInt->accuracy();     # get/set global A for all
BigInt objects

DESCRIPTION

All operators (inlcuding basic math operations) are over loaded if you declare your big integers as: $i = new Math::BigInt '123_456_789_123_456_789';
Operations with overloaded operators preserve the argu ments which is exactly what you expect.
Canonical notation: Big integer values are strings of the form "/^[+-]+$/"
with leading zeros suppressed.

'-0' canonical value '-0',

normalized '0'
' -123_123_123' canonical value

'-123123123'
'1_23_456_7890' canonical value

'1234567890'; Input
Input values to these routines may be either Math::Big
Int objects or strings of the form
"/^[+-]?[]+.?[]*E?[+-]?[]*$/".

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.

Math::BigInt::new() defaults to 0, while Math::Big Int::new('') results in 'NaN'.

bnorm() on a BigInt object is now effectively a no-op, since the numbers are always stored in normalized form.
On a string, it creates a BigInt object.; Output
Output values are BigInt objects (normalized), except
for bstr(), which returns a string in normalized form.
Some routines ("is_odd()", "is_even()", "is_zero()",
"is_one()", "is_nan()") return true or false, while oth
ers ("bcmp()", "bacmp()") return either undef, <0, 0 or
>0 and are suited for sort.

METHODS

Each of the methods below accepts three additional parame
ters. These arguments $A, $P and $R are accuracy, preci
sion and round_mode. Please see more in the section about
ACCURACY and ROUNDIND.

config: use Data::Dumper;; print Dumper ( Math::BigInt->config() );
Returns a hash containing the configuration, e.g. the ver sion number, lib loaded etc.
accuracy: $x->accuracy(5); # local for $x
$class->accuracy(5); # global for all; members of $class
Set or get the global or local accuracy, aka how many sig nificant digits the results have. 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 accu racy that will be in effect for $x:: $y = Math::BigInt->new(1234567); # unround; ed
print Math::BigInt->accuracy(4),"0; # set 4,; print 4
$x = Math::BigInt->new(123456); # will be; automatically rounded
print "$x $y0; # '123500; 1234567'
print $x->accuracy(),"0; # will be 4
print $y->accuracy(),"0; # also 4,; since global is 4
print Math::BigInt->accuracy(5),"0; # set to 5,; print 5
print $x->accuracy(),"0; # still 4
print $y->accuracy(),"0; # 5, since; global is 5
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 string or another Big Int object. The input is accepted as decimal, hex (with leading '0x') or binary (with leading '0b').
bnan: $x = Math::BigInt->bnan();
Creates a new BigInt object representing NaN (Not A Num ber). 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
$x->is_inf('-'); # true if -inf; (sign is default '+')
These methods all test the BigInt for beeing one specific value and return true or false depending on the input. These are faster than doing something like:: if ($x == 0)
is_positive()/is_negative(): $x->is_positive(); # true if >= 0
$x->is_negative(); # 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 positive.
These methods are only testing the sign, and not the value.
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.
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. Returns -1, 0, 1 or undef.
sign: $x->sign();
Return the sign, of $x, meaning either "+", "-", "-inf", "+inf" or NaN.
bcmp: $x->digit($n); # return the nth digit,; counting from right
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 it's absolute value, e.g. change the sign from '-' to '+' and from '-inf' to '+inf', respec tively. Does nothing for NaN or positive numbers.
bnorm: $x->bnorm(); # normalize (no-op)
bnot: $x->bnot(); # two's complement (bit; wise not)
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)
bdiv: $x->bdiv($y); # divide, set $x to quo; tient
# return (quo,rem) or quo

if scalar
bmod: $x->bmod($y); # modulus (x % y)
bmodinv: $num->bmodinv($mod); # modular inverse
Returns the inverse of $num in the given modulus $mod. '"NaN"' is returned unless $num is relatively prime to $mod, i.e. unless "bgcd($num, $mod)==1".
bmodpow: $num->bmodpow($exp,$mod); # modular exponentation; ($num**$exp % $mod)
Returns the value of $num taken to the power $exp in the modulus $mod using binary exponentation. "bmodpow" is far superior to writing: $num ** $exp % $mod
because "bmodpow" is much faster--it reduces internal variables into the modulus whenever possible, so it oper ates on smaller numbers.
"bmodpow" also supports negative exponents.: bmodpow($num, -1, $mod)
is exactly equivalent to: bmodinv($num, $mod)
bpow: $x->bpow($y); # power of arguments (x **; y)
blsft: $x->blsft($y); # left shift
$x->blsft($y,$n); # left shift, by base $n; (like 10)
brsft: $x->brsft($y); # right shift
$x->brsft($y,$n); # right shift, by 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 com; plement)
bsqrt: $x->bsqrt(); # calculate square-root
bfac: $x->bfac(); # factorial of $x; (1*2*3*4*..$x)
round: $x->round($A,$P,$round_mode); # round to accuracy or; precision using mode $r
bround: $x->bround($N); # accuracy: preserve $N; digits
bfround: $x->bfround($N); # round to $Nth digit, no; op for BigInts
bfloor: $x->bfloor();
Set $x to the integer less or equal than $x. This is a noop 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 multipli; cator (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,expo; nent) as BigInt
copy: $x->copy(); # make a true copy of $x; (unlike $y = $x;)
as_number: $x->as_number(); # return as BigInt (in; BigInt: same as copy())
bsrt: $x->bstr(); # normalized string
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

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 manu
ally.

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 (nega
tive) the decimal point. For example, 123.45 has a preci
sion of -2. 0 means an integer like 123 (or 120). A preci
sion 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 dif
ferent precisions, because 1200 can have p = 0, 1 or 2
(depending on what the inital 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 fall back 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 incre
mentation): e.g. when rounding to units, 0.9 rounds to
1, and -19.9 rounds to -20. If D < 5, the number is sim
ilarly truncated at the rounding place: e.g. when round
ing 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.
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) para; mater
+ 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 indi

cate two different ways

on how fdiv() determines the maximum number of dig

its it should calculate,
and the actual code does yet another thing
POD:
max($Math::BigFloat::div_scale,length(divi

dend)+length(divisor))

Comment:

result has at most max(scale, length(dividend),

length(divisor)) digits

Actual code:

scale = max(scale, length(dividend)-1,length(divi

sor)-1);
scale += length(divisior) - length(dividend);

So for lx = 3, ly = 9, scale = 10, scale will actu

ally be 16 (10+9-3).
Actually, the 'difference' added to the scale is

calculated from the
number of "significant digits" in dividend and divi

sor, which is derived
by looking at the length of the mantissa. Which is

wrong, since it includes
the + sign (oups) and actually gets 2 for '+100' and

4 for '+101'. Oups
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 sig

nificant digits.
The rounding after the division then uses the re

mainder and $y to determine
wether 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->accura

cy() or
Math::BigFloat->accuracy() or whatever class you are

using.

* You can also set P globally by using Math::Some

Class->precision() likewise.
* Globals are classwide, and not inherited by sub

classes.
* to undefine A, use Math::SomeCLass->accuracy(undef);
* to undefine P, use Math::SomeClass->precision(un

def);
* Setting Math::SomeClass->accuracy() clears automati

cally

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, take Math::Some

Class->accuracy()
* to find out the current global P, take Math::Some

Class->precision()
* use $x->accuracy() respective $x->precision() for

the local setting of $x.
* Please note that $x->accuracy() respecive $x->preci

sion() fall back to the

defined globals, when $x's A or P is not set.

Creating numbers

* When you create a number, you can give it's desired

A or P via:
$x = Math::BigInt->new($number,$A,$P);

* Only one of A or P can be defined, otherwise the re

sult is NaN
* If no A or P is give ($x = Math::BigInt->new($num

ber) 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, B<no> rounding will oc

cur, and the globals will

B<not> be used. This is used by subclasses to create

numbers without
suffering rounding in the parent. Thus a subclass is

able to have it's 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::BigInts internally, but

setting A or P inside

Math::BigInt as globals should not tamper with the

parts of a BigFloat.
Thus a flag is used to mark all Math::BigFloat num

bers as 'never round'

Precedence

* It only makes sense that a number has only one of A

or P at a time.
Since you can set/get both A and P, there is a rule

that will practically
enforce only A or P to be in effect at a time, even

if both are set.
This is called precedence.

* 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 precendence over P (Hint: A comes before P).

If A is defined, it

is used, otherwise P is used. 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 in

stead 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 three modes:

+ 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 explicitely 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 op

erations 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 set
tings would be in order, ?so that intermediate rounding does not 'poison' fur
ther 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, parame

ter: 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 and P locally by using $x->accuracy()

and $x->precision()
and thus force different A and P for different ob

jects/numbers.

* Setting A or P this way immediately rounds $x to the

new value.
* $x->accuracy() clears $x->precision(), and vice ver

sa.

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 param

eter one of the

following rounding modes (R):
'even', 'odd', '+inf', '-inf', 'zero', 'trunc'

* you can set and 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 re

sult 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 param

eters A and P

INTERNALS

The actual numbers are stored as unsigned big integers
(with seperate sign). You should neither care about nor
depend on the internal representation; it might change
without notice. Use only method calls like "$x->sign();"
instead relying on the internal hash keys like in
"$x->{sign};".

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 lib => 'Calc';
You can change this by using:: use Math::BigInt lib => 'BitVect';
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 lib => 'Foo,Math::BigInt::Bar';
Calc.pm uses as internal format an array of elements of some decimal base (usually 1e5 or 1e7) with the least sig nificant digit first, while BitVect.pm uses a bit vector of base 2, most significant bit first. Other modules might use even different means of representing the numbers. See the respective module documentation for further details.
SIGN
The sign is either '+', '-', 'NaN', '+inf' or '-inf' and stored seperately.
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 "ok0 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 will be always 0, except for NaN, +inf and -inf, where it will be NaN; and for $x == 0, where it will be 1 (to be compatible with Math::BigFloat's internal representation of a zero as 0E1).
$m will always be a copy of the original number. The rela tion between $e and $m might change in the future, but will always be equivalent in a numerical sense, e.g. $m might get minimized.

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-Big
Intify 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(),"0; # 123.4
Math::BigFloat->round_mode('odd'); # round to odd
print $x->copy()->fround(),"0; # 123.5
Math::BigFloat->accuracy(5); # no more A than 5
Math::BigFloat->round_mode('odd'); # round to odd
print $x->copy()->fround(),"0; # 123.46
$y = $x->copy()->fround(4),"0; # A = 4: 123.4
print "$y, ",$y->accuracy(),"0; # 123.4, 4; Math::BigFloat->accuracy(undef); # A not important; now
Math::BigFloat->precision(2); # P important
print $x->copy()->bnorm(),"0; # 123.46
print $x->copy()->fround(),"0; # 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,"0;

Autocreating constants

After "use Math::BigInt ':constant'" all the integer deci mal, 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,"0'
prints the integer value of "2**100". Note that without
conversion of constants the expression 2**100 will be cal
culated 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::Big; Int->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),"0;
print ref(123.2e2),"0;
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 multiplica
tion/division). If $y is very small compared to $x, the
form $x += $y is MUCH faster than $x = $x + $y since mak
ing the copy of $x takes more time then the actual addi
tion.

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 operatons.

The rewritten version of this module is slower on certain
operations, like new(), bstr() and numify(). The reason are that it does now more work and handles more cases. The
time spent in these operations is usually gained in the
other operations so that programs on the average should
get faster. If they don't, please contect 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 via:: use Math::BigInt lib => 'Module';
See "MATH LIBRARY" for more information.
For more benchmark results see <http://blood gate.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:

� 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.
� The private object hash keys like "$x-"{sign}> may not: be changed, but additional keys can be added, like
"$x-"{_custom}>.
� Accessor functions are available for all existing object: hash keys and should be used instead of directly access
ing the internal hash keys. The reason for this is that
Math::BigInt itself has a pluggable interface which per
mits 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 it's 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 oneliners:: 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()
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):

BUGS

Out of Memory!: Under Perl prior to 5.6.0 having an "use Math::BigInt
':constant';" and "eval()" in your code will crash with
"Out of memory". This is probably an overload/exporter
bug. You can workaround by not having "eval()" and
':constant' at the same time or upgrade your Perl to a
newer version.
Fails to load Calc on Perl prior 5.6.0: Since eval(' use ...') can not be used in conjunction
with ':constant', BigInt will fall back to eval {
require ... } when loading the math lib on Perls prior
to 5.6.0. This simple replaces '::' with '/' and thus
might fail on filesystems using a different seperator.

CAVEATS

Some things might not work as you expect them. Below is
documented what is known to be troublesome:

stringify, bstr(), bsstr() and 'cmp': Both stringify and bstr() 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, it's ok() uses 'eq' internally.; Mark 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 sci entific notation aka 1e+2 instead of 100. Be advised that overloaded 'eq' always uses bstr() for comparisation, but Perl will represent some numbers as 100 and others as 1e+308. If in doubt, convert both arguments to Math::Big Int before doing eq:: 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 comparisations, that 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.
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()" for the same effect:: $x = Math::BigFloat->new(123.45);
$y = $x->as_number(); # BigInt; 123
This also works for other subclasses, like Math::String.
It is yet unlcear whether overloaded int() should return a scalar or a BigInt.
length: The following will probably not do what you expect:

$c = Math::BigInt->new(123);
print $c->length(),"0; # 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(),"0; # prints 3
bdiv: The following will probably not do what you expect:

print $c->bdiv(10000),"0;
It prints both quotient and remainder since print calls "bdiv()" in list context. Also, "bdiv()" will modify $c, so be carefull. You probably want to use: print $c / 10000,"0;
print scalar $c->bdiv(10000),"0; # 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 nonzero) 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 docu mented 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 behing 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 differ ent 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 cer
tain 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 / inf = 0, 0 0 * inf + 0 =

0 /-inf = 0, 0 0 * -inf + 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, $y0; # 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),"0; # modify $x
print $x->bpow($i),"0; # ditto
print $x **= $i,"0; # the same
print $x ** $i,"0; # 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).
With Copy-On-Write, this issue would be gone, but C-o-W is not implemented since it is slower for all other things.
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::Big_ Int::bdiv() will always return a Math::BigInt, regardless whether the result should be a Math::BigFloat or the sec ond operant is one.
To get a Math::BigFloat you either need to call the oper ation 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 pack ages, like Math::String.
One solution to you problem might be autoupgrading.
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.; 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(),"0; # 4

Math::BigFloat->precision(2);
print $x->bsqrt(),"0; # 3.46
print $x->bsqrt(3),"0; # 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.

AUTHORS

Original code by Mark Biggar, overloaded interface by Ilya
Zakharevich. Completely rewritten by Tels http://blood
gate.com in late 2000, 2001.

docs.sk

comprehensive documentation repository

In section

See also