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# Related Modules

Module Version: 1.36   Source   Latest Release: perl-5.6.2

# 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"

# Testing
\$x->is_zero();                # return whether arg is zero or not
\$x->is_nan();                 # return whether arg is NaN or not
\$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->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->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->bsub(\$y);                 # subtraction (subtract \$y from \$x)
\$x->bmul(\$y);                 # multiplication (multiply \$x by \$y)
\$x->bdiv(\$y);                 # divide, set \$x to quotient
# return (quo,rem) or quo if scalar

\$x->bmod(\$y);                 # modulus (x % y)
\$x->bpow(\$y);                 # power of arguments (x ** y)
\$x->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->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
blcm(@values);                # lowest common multiplicator

\$x->bstr();                   # normalized string
\$x->bsstr();                  # normalized string in scientific notation
\$x->length();                 # return number of digits in number
(\$x,\$f) = \$x->length();       # length of number and length of fraction part

\$x->exponent();               # return exponent as BigInt
\$x->mantissa();               # return 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())```

# 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 `/^\s*[+-]?[\d]+\.?[\d]*E?[+-]?[\d]*\$/`.

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::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.

# 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 other 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 such a number should be padded with zeros:
!
!      Initial value   P       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.

## Fallback F

When both A and P are undefined, this is used as a fallback accuracy.

## 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 reminder 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
* 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
* use \$x->accuracy() for the local setting of \$x.
* to find out the current global P, take \$Math::SomeClass::precision
* use \$x->precision() for the local setting```
Creating numbers
``` !* When you create a number, there should be a way to define its A & P
* When a number without specific A or P is created, but the globals are
defined, these should be used to round the number immediately and also
stored locally with the number. 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) ```
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 should result in a warning or an error,
!  probably in 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 1 more digit 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 cannot 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 place by one for binary operations like add). 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: smaller 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.```
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::rnd_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,\$rnd_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, and math with them is done (by default) by a module called Math::BigInt::Calc. This is equivalent to:

`        use Math::BigInt lib => 'calc';`

You can change this by using:

`        use Math::BigInt lib => 'BitVect';`

('Math::BitInt::BitVect' works, too.)

Calc.pm uses as internal format an array of elements of base 100000 digits with the least significant digit first, BitVect.pm uses a bit vector of base 2, most significant bit first.

The sign `/^[+-]\$/` is stored separately. The string 'NaN' is used to represent the result when input arguments are not numbers. '+inf' and '-inf' represent infinity.

You should neither care about nor depend on the internal representation; it might change without notice. Use only method calls like `\$x->sign();` instead of relying on the internal hash keys like in `\$x->{sign};`.

## 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 where it will be NaN and for \$x == 0, then 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 qw(bstr bint);
\$x = bstr("1234")                     # string "1234"
\$x = "\$x";                            # same as bstr()
\$x = bneg("1234")                     # Bigint "-1234"
\$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 = new Math::BigInt;                # BigInt "0"
\$x--;                                 # BigInt "-1"
\$x = Math::BigInt->badd(4,5)          # BigInt "9"
\$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
\$Math::BigFloat::precision = 2;       # P important
print \$x->copy()->bnorm(),"\n";       # 123.46
print \$x->copy()->fround(),"\n";      # 123.46```

# Autocreating constants

After `use Math::BigInt ':constant'` all the integer decimal 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.

# 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. This is currently not implemented.

The new version of this module is slower on new(), bstr() and numify(). 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.

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

## Replacing the math library

You can use an alternative library to drive Math::BigInt via:

`        use Math::BigInt lib => 'Module';`

The default is called Math::BigInt::Calc and is a pure-perl base 100,000 math package that consists of the standard routine present in earlier versions of Math::BigInt.

There are also Math::BigInt::Scalar (primarily for testing) and Math::BigInt::BitVect; these and others can be found via http://search.cpan.org/:

```        use Math::BigInt lib => 'BitVect';

my \$x = Math::BigInt->new(2);
print \$x ** (1024*1024);```

# BUGS

:constant and eval()

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.

# 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```
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.

bdiv

The following will probably not do what you expect:

`        print \$c->bdiv(10000),"\n";`

It prints both quotient and reminder 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)```

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.

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 ;)

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 will modify \$y, and vice versa.

```        \$x->bmul(2);
print "\$x, \$y\n";       # prints '10, 10'```

If you want a true copy of \$x, use:

`        \$y = \$x->copy();`

See also the documentation for overload.pm regarding `=`.

bpow

`bpow()` (and the rounding functions) now modifies the first argument and return 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 will be gone. Stay tuned...

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, this one's operation 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.

bsqrt()

`bsqrt()` works only good if the result is an 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```