Perl 5 version 8.0 documentation

# Math::BigInt

## 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, no-op 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 A \$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 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.

• Canonical notation

Big integer values are strings of the form `/^[+-]\d+\$/` 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::BigInt objects or strings of the form `/^[+-]?[\d]+\.?[\d]*E?[+-]?[\d]*\$/` .

You can include one underscore between any two digits. The input string may have leading and trailing whitespace, which will be ignored. In later versions, a more strict (no whitespace at all) or more lax (whitespace allowed everywhere) input checking will also be possible.

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::BigInt::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 others (`bcmp()` , `bacmp()` ) return either undef, <0, 0 or >0 and are suited for sort.

## METHODS

Each of the methods below accepts three additional parameters. These arguments \$A, \$P and \$R are accuracy, precision 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 version 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 significant 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 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);		# will be automatically 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`

### 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 BigInt 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 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 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', respectively. 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`

`  \$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 quotient				# 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 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)`

### 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 complement)`

### 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 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 multiplicator (no OO style)`

`	\$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_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 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 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`

### 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`

### 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.

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) paramater    + 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(divisior) - 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 (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 significant digits.    The rounding after the division then uses the remainder 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->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, take Math::SomeClass->accuracy()  * to find out the current global P, take Math::SomeClass->precision()  * use \$x->accuracy() respective \$x->precision() for the local setting of \$x.  * Please note that \$x->accuracy() respecive \$x->precision() 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 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, B<no> rounding will occur, 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 numbers 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 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 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 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 and P locally by using \$x->accuracy() and \$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'  * 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 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`

## 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 significant digit first, while BitVect.pm uses a bit vector of base 2, most significant bit first. Other modules might use even different means of representing the numbers. See the respective module documentation for further details.

### 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 "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` 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 relation 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-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 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';`

For more benchmark results see http://bloodgate.com/perl/benchmarks.html.

## 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 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 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;`

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.

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;`

`	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 comparisation, but Perl will represent some numbers as 100 and others as 1e+308. If in doubt, convert both arguments to Math::BigInt 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(),"\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 carefull. You probably want to use

`	print \$c / 10000,"\n";	print scalar \$c->bdiv(10000),"\n";  # or if you want to modify \$c`

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 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 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 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.

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::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.

• 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(),"\n";		# 4	Math::BigFloat->precision(2);	print \$x->bsqrt(),"\n";			# 3.46	print \$x->bsqrt(3),"\n";		# 3.464`
• brsft()