[OpenSPARC-T2-DV] / tools / perl-5.8.0 / man / man3 / Math::BigInt.3

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.\" ========================================================================
.\"
.IX Title "Math::BigInt 3"
.TH Math::BigInt 3 "2002-06-01" "perl v5.8.0" "Perl Programmers Reference Guide"
.SH "NAME"
Math::BigInt \- Arbitrary size integer math package
.SH "SYNOPSIS"
.IX Header "SYNOPSIS"
.Vb 1
\&  use Math::BigInt;
.Ve
.PP
.Vb 8
\&  # 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
.Ve
.PP
.Vb 11
\&  # 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)
.Ve
.PP
.Vb 5
\&  $x->bcmp($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
.Ve
.PP
.Vb 1
\&  # The following all modify their first argument:
.Ve
.PP
.Vb 7
\&  # 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
.Ve
.PP
.Vb 6
\&  $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
.Ve
.PP
.Vb 5
\&  $x->badd($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
.Ve
.PP
.Vb 3
\&  $x->bmod($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
.Ve
.PP
.Vb 5
\&  $x->bpow($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)
.Ve
.PP
.Vb 4
\&  $x->band($y);                 # bitwise and
\&  $x->bior($y);                 # bitwise inclusive or
\&  $x->bxor($y);                 # bitwise exclusive or
\&  $x->bnot();                   # bitwise not (two's complement)
.Ve
.PP
.Vb 2
\&  $x->bsqrt();                  # calculate square-root
\&  $x->bfac();                   # factorial of $x (1*2*3*4*..$x)
.Ve
.PP
.Vb 3
\&  $x->round($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
.Ve
.PP
.Vb 3
\&  # 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
.Ve
.PP
.Vb 1
\&  # The following do not modify their arguments:
.Ve
.PP
.Vb 2
\&  bgcd(@values);                # greatest common divisor (no OO style)
\&  blcm(@values);                # lowest common multiplicator (no OO style)
.Ve
.PP
.Vb 3
\&  $x->length();                 # return number of digits in number
\&  ($x,$f) = $x->length();       # length of number and length of fraction part,
\&                                # latter is always 0 digits long for BigInt's
.Ve
.PP
.Vb 5
\&  $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())
.Ve
.PP
.Vb 5
\&  # 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
.Ve
.PP
.Vb 1
\&  Math::BigInt->config();       # return hash containing configuration/version
.Ve
.PP
.Vb 5
\&  # 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
.Ve
.PP
.Vb 2
\&  Math::BigInt->precision();    # get/set global P for all BigInt objects
\&  Math::BigInt->accuracy();     # get/set global A for all BigInt objects
.Ve
.SH "DESCRIPTION"
.IX Header "DESCRIPTION"
All operators (inlcuding basic math operations) are overloaded if you
declare your big integers as
.PP
.Vb 1
\&  $i = new Math::BigInt '123_456_789_123_456_789';
.Ve
.PP
Operations with overloaded operators preserve the arguments which is
exactly what you expect.
.IP "Canonical notation" 2
.IX Item "Canonical notation"
Big integer values are strings of the form \f(CW\*(C`/^[+\-]\ed+$/\*(C'\fR with leading
zeros suppressed.
.Sp
.Vb 3
\&   '-0'                            canonical value '-0', normalized '0'
\&   '   -123_123_123'               canonical value '-123123123'
\&   '1_23_456_7890'                 canonical value '1234567890'
.Ve
.IP "Input" 2
.IX Item "Input"
Input values to these routines may be either Math::BigInt objects or
strings of the form \f(CW\*(C`/^[+\-]?[\ed]+\e.?[\ed]*E?[+\-]?[\ed]*$/\*(C'\fR.
.Sp
You can include one underscore between any two digits. The input string may
have leading and trailing whitespace, which will be ignored. In later
versions, a more strict (no whitespace at all) or more lax (whitespace
allowed everywhere) input checking will also be possible.
.Sp
This means integer values like 1.01E2 or even 1000E\-2 are also accepted.
Non integer values result in NaN.
.Sp
\&\fIMath::BigInt::new()\fR defaults to 0, while Math::BigInt::new('') results
in 'NaN'.
.Sp
\&\fIbnorm()\fR on a BigInt object is now effectively a no\-op, since the numbers 
are always stored in normalized form. On a string, it creates a BigInt 
object.
.IP "Output" 2
.IX Item "Output"
Output values are BigInt objects (normalized), except for \fIbstr()\fR, which
returns a string in normalized form.
Some routines (\f(CW\*(C`is_odd()\*(C'\fR, \f(CW\*(C`is_even()\*(C'\fR, \f(CW\*(C`is_zero()\*(C'\fR, \f(CW\*(C`is_one()\*(C'\fR,
\&\f(CW\*(C`is_nan()\*(C'\fR) return true or false, while others (\f(CW\*(C`bcmp()\*(C'\fR, \f(CW\*(C`bacmp()\*(C'\fR)
return either undef, <0, 0 or >0 and are suited for sort.
.SH "METHODS"
.IX Header "METHODS"
Each of the methods below accepts three additional parameters. These arguments
\&\f(CW$A\fR, \f(CW$P\fR and \f(CW$R\fR are accuracy, precision and round_mode. Please see more in the
section about \s-1ACCURACY\s0 and \s-1ROUNDIND\s0.
.Sh "config"
.IX Subsection "config"
.Vb 1
\&        use Data::Dumper;
.Ve
.PP
.Vb 1
\&        print Dumper ( Math::BigInt->config() );
.Ve
.PP
Returns a hash containing the configuration, e.g. the version number, lib
loaded etc.
.Sh "accuracy"
.IX Subsection "accuracy"
.Vb 2
\&        $x->accuracy(5);                # local for $x
\&        $class->accuracy(5);            # global for all members of $class
.Ve
.PP
Set or get the global or local accuracy, aka how many significant digits the
results have. Please see the section about \*(L"\s-1ACCURACY\s0 \s-1AND\s0 \s-1PRECISION\s0\*(R" for
further details.
.PP
Value must be greater than zero. Pass an undef value to disable it:
.PP
.Vb 2
\&        $x->accuracy(undef);
\&        Math::BigInt->accuracy(undef);
.Ve
.PP
Returns the current accuracy. For \f(CW\*(C`$x\-\*(C'\fR\fIaccuracy()\fR> 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 \f(CW$x:\fR
.PP
.Vb 9
\&        $y = Math::BigInt->new(1234567);        # unrounded
\&        print Math::BigInt->accuracy(4),"\en";   # set 4, print 4
\&        $x = Math::BigInt->new(123456);         # will be automatically rounded
\&        print "$x $y\en";                        # '123500 1234567'
\&        print $x->accuracy(),"\en";              # will be 4
\&        print $y->accuracy(),"\en";              # also 4, since global is 4
\&        print Math::BigInt->accuracy(5),"\en";   # set to 5, print 5
\&        print $x->accuracy(),"\en";              # still 4
\&        print $y->accuracy(),"\en";              # 5, since global is 5
.Ve
.Sh "brsft"
.IX Subsection "brsft"
.Vb 1
\&        $x->brsft($y,$n);
.Ve
.PP
Shifts \f(CW$x\fR right by \f(CW$y\fR in base \f(CW$n\fR. Default is base 2, used are usually 10 and
2, but others work, too.
.PP
Right shifting usually amounts to dividing \f(CW$x\fR by \f(CW$n\fR ** \f(CW$y\fR and truncating the
result:
.PP
.Vb 4
\&        $x = Math::BigInt->new(10);
\&        $x->brsft(1);                   # same as $x >> 1: 5
\&        $x = Math::BigInt->new(1234);
\&        $x->brsft(2,10);                # result 12
.Ve
.PP
There is one exception, and that is base 2 with negative \f(CW$x:\fR
.PP
.Vb 2
\&        $x = Math::BigInt->new(-5);
\&        print $x->brsft(1);
.Ve
.PP
This will print \-3, not \-2 (as it would if you divide \-5 by 2 and truncate the
result).
.Sh "new"
.IX Subsection "new"
.Vb 1
\&        $x = Math::BigInt->new($str,$A,$P,$R);
.Ve
.PP
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').
.Sh "bnan"
.IX Subsection "bnan"
.Vb 1
\&        $x = Math::BigInt->bnan();
.Ve
.PP
Creates a new BigInt object representing NaN (Not A Number).
If used on an object, it will set it to NaN:
.PP
.Vb 1
\&        $x->bnan();
.Ve
.Sh "bzero"
.IX Subsection "bzero"
.Vb 1
\&        $x = Math::BigInt->bzero();
.Ve
.PP
Creates a new BigInt object representing zero.
If used on an object, it will set it to zero:
.PP
.Vb 1
\&        $x->bzero();
.Ve
.Sh "binf"
.IX Subsection "binf"
.Vb 1
\&        $x = Math::BigInt->binf($sign);
.Ve
.PP
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:
.PP
.Vb 2
\&        $x->binf();
\&        $x->binf('-');
.Ve
.Sh "bone"
.IX Subsection "bone"
.Vb 1
\&        $x = Math::BigInt->binf($sign);
.Ve
.PP
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:
.PP
.Vb 2
\&        $x->bone();             # +1
\&        $x->bone('-');          # -1
.Ve
.Sh "\fIis_one()\fP/\fIis_zero()\fP/\fIis_nan()\fP/\fIis_inf()\fP"
.IX Subsection "is_one()/is_zero()/is_nan()/is_inf()"
.Vb 6
\&        $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
\&        $x->is_inf('-');                # true if -inf (sign is default '+')
.Ve
.PP
These methods all test the BigInt for beeing one specific value and return
true or false depending on the input. These are faster than doing something
like:
.PP
.Vb 1
\&        if ($x == 0)
.Ve
.Sh "\fIis_positive()\fP/\fIis_negative()\fP"
.IX Subsection "is_positive()/is_negative()"
.Vb 2
\&        $x->is_positive();              # true if >= 0
\&        $x->is_negative();              # true if <  0
.Ve
.PP
The methods return true if the argument is positive or negative, respectively.
\&\f(CW\*(C`NaN\*(C'\fR is neither positive nor negative, while \f(CW\*(C`+inf\*(C'\fR counts as positive, and
\&\f(CW\*(C`\-inf\*(C'\fR is negative. A \f(CW\*(C`zero\*(C'\fR is positive.
.PP
These methods are only testing the sign, and not the value.
.Sh "\fIis_odd()\fP/\fIis_even()\fP/\fIis_int()\fP"
.IX Subsection "is_odd()/is_even()/is_int()"
.Vb 3
\&        $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
.Ve
.PP
The return true when the argument satisfies the condition. \f(CW\*(C`NaN\*(C'\fR, \f(CW\*(C`+inf\*(C'\fR,
\&\f(CW\*(C`\-inf\*(C'\fR are not integers and are neither odd nor even.
.Sh "bcmp"
.IX Subsection "bcmp"
.Vb 1
\&        $x->bcmp($y);
.Ve
.PP
Compares \f(CW$x\fR with \f(CW$y\fR and takes the sign into account.
Returns \-1, 0, 1 or undef.
.Sh "bacmp"
.IX Subsection "bacmp"
.Vb 1
\&        $x->bacmp($y);
.Ve
.PP
Compares \f(CW$x\fR with \f(CW$y\fR while ignoring their. Returns \-1, 0, 1 or undef.
.Sh "sign"
.IX Subsection "sign"
.Vb 1
\&        $x->sign();
.Ve
.PP
Return the sign, of \f(CW$x\fR, meaning either \f(CW\*(C`+\*(C'\fR, \f(CW\*(C`\-\*(C'\fR, \f(CW\*(C`\-inf\*(C'\fR, \f(CW\*(C`+inf\*(C'\fR or NaN.
.Sh "bcmp"
.IX Subsection "bcmp"
.Vb 1
\&  $x->digit($n);                # return the nth digit, counting from right
.Ve
.Sh "bneg"
.IX Subsection "bneg"
.Vb 1
\&        $x->bneg();
.Ve
.PP
Negate the number, e.g. change the sign between '+' and '\-', or between '+inf'
and '\-inf', respectively. Does nothing for NaN or zero.
.Sh "babs"
.IX Subsection "babs"
.Vb 1
\&        $x->babs();
.Ve
.PP
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.
.Sh "bnorm"
.IX Subsection "bnorm"
.Vb 1
\&  $x->bnorm();                  # normalize (no-op)
.Ve
.Sh "bnot"
.IX Subsection "bnot"
.Vb 1
\&  $x->bnot();                   # two's complement (bit wise not)
.Ve
.Sh "binc"
.IX Subsection "binc"
.Vb 1
\&  $x->binc();                   # increment x by 1
.Ve
.Sh "bdec"
.IX Subsection "bdec"
.Vb 1
\&  $x->bdec();                   # decrement x by 1
.Ve
.Sh "badd"
.IX Subsection "badd"
.Vb 1
\&  $x->badd($y);                 # addition (add $y to $x)
.Ve
.Sh "bsub"
.IX Subsection "bsub"
.Vb 1
\&  $x->bsub($y);                 # subtraction (subtract $y from $x)
.Ve
.Sh "bmul"
.IX Subsection "bmul"
.Vb 1
\&  $x->bmul($y);                 # multiplication (multiply $x by $y)
.Ve
.Sh "bdiv"
.IX Subsection "bdiv"
.Vb 2
\&  $x->bdiv($y);                 # divide, set $x to quotient
\&                                # return (quo,rem) or quo if scalar
.Ve
.Sh "bmod"
.IX Subsection "bmod"
.Vb 1
\&  $x->bmod($y);                 # modulus (x % y)
.Ve
.Sh "bmodinv"
.IX Subsection "bmodinv"
.Vb 1
\&  $num->bmodinv($mod);          # modular inverse
.Ve
.PP
Returns the inverse of \f(CW$num\fR in the given modulus \f(CW$mod\fR.  '\f(CW\*(C`NaN\*(C'\fR' is
returned unless \f(CW$num\fR is relatively prime to \f(CW$mod\fR, i.e. unless
\&\f(CW\*(C`bgcd($num, $mod)==1\*(C'\fR.
.Sh "bmodpow"
.IX Subsection "bmodpow"
.Vb 1
\&  $num->bmodpow($exp,$mod);     # modular exponentation ($num**$exp % $mod)
.Ve
.PP
Returns the value of \f(CW$num\fR taken to the power \f(CW$exp\fR in the modulus
\&\f(CW$mod\fR using binary exponentation.  \f(CW\*(C`bmodpow\*(C'\fR is far superior to
writing
.PP
.Vb 1
\&  $num ** $exp % $mod
.Ve
.PP
because \f(CW\*(C`bmodpow\*(C'\fR is much faster\*(--it reduces internal variables into
the modulus whenever possible, so it operates on smaller numbers.
.PP
\&\f(CW\*(C`bmodpow\*(C'\fR also supports negative exponents.
.PP
.Vb 1
\&  bmodpow($num, -1, $mod)
.Ve
.PP
is exactly equivalent to
.PP
.Vb 1
\&  bmodinv($num, $mod)
.Ve
.Sh "bpow"
.IX Subsection "bpow"
.Vb 1
\&  $x->bpow($y);                 # power of arguments (x ** y)
.Ve
.Sh "blsft"
.IX Subsection "blsft"
.Vb 2
\&  $x->blsft($y);                # left shift
\&  $x->blsft($y,$n);             # left shift, by base $n (like 10)
.Ve
.Sh "brsft"
.IX Subsection "brsft"
.Vb 2
\&  $x->brsft($y);                # right shift 
\&  $x->brsft($y,$n);             # right shift, by base $n (like 10)
.Ve
.Sh "band"
.IX Subsection "band"
.Vb 1
\&  $x->band($y);                 # bitwise and
.Ve
.Sh "bior"
.IX Subsection "bior"
.Vb 1
\&  $x->bior($y);                 # bitwise inclusive or
.Ve
.Sh "bxor"
.IX Subsection "bxor"
.Vb 1
\&  $x->bxor($y);                 # bitwise exclusive or
.Ve
.Sh "bnot"
.IX Subsection "bnot"
.Vb 1
\&  $x->bnot();                   # bitwise not (two's complement)
.Ve
.Sh "bsqrt"
.IX Subsection "bsqrt"
.Vb 1
\&  $x->bsqrt();                  # calculate square-root
.Ve
.Sh "bfac"
.IX Subsection "bfac"
.Vb 1
\&  $x->bfac();                   # factorial of $x (1*2*3*4*..$x)
.Ve
.Sh "round"
.IX Subsection "round"
.Vb 1
\&  $x->round($A,$P,$round_mode); # round to accuracy or precision using mode $r
.Ve
.Sh "bround"
.IX Subsection "bround"
.Vb 1
\&  $x->bround($N);               # accuracy: preserve $N digits
.Ve
.Sh "bfround"
.IX Subsection "bfround"
.Vb 1
\&  $x->bfround($N);              # round to $Nth digit, no-op for BigInts
.Ve
.Sh "bfloor"
.IX Subsection "bfloor"
.Vb 1
\&        $x->bfloor();
.Ve
.PP
Set \f(CW$x\fR to the integer less or equal than \f(CW$x\fR. This is a no-op in BigInt, but
does change \f(CW$x\fR in BigFloat.
.Sh "bceil"
.IX Subsection "bceil"
.Vb 1
\&        $x->bceil();
.Ve
.PP
Set \f(CW$x\fR to the integer greater or equal than \f(CW$x\fR. This is a no-op in BigInt, but
does change \f(CW$x\fR in BigFloat.
.Sh "bgcd"
.IX Subsection "bgcd"
.Vb 1
\&  bgcd(@values);                # greatest common divisor (no OO style)
.Ve
.Sh "blcm"
.IX Subsection "blcm"
.Vb 1
\&  blcm(@values);                # lowest common multiplicator (no OO style)
.Ve
.PP
head2 length
.PP
.Vb 2
\&        $x->length();
\&        ($xl,$fl) = $x->length();
.Ve
.PP
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.
.Sh "exponent"
.IX Subsection "exponent"
.Vb 1
\&        $x->exponent();
.Ve
.PP
Return the exponent of \f(CW$x\fR as BigInt.
.Sh "mantissa"
.IX Subsection "mantissa"
.Vb 1
\&        $x->mantissa();
.Ve
.PP
Return the signed mantissa of \f(CW$x\fR as BigInt.
.Sh "parts"
.IX Subsection "parts"
.Vb 1
\&  $x->parts();                  # return (mantissa,exponent) as BigInt
.Ve
.Sh "copy"
.IX Subsection "copy"
.Vb 1
\&  $x->copy();                   # make a true copy of $x (unlike $y = $x;)
.Ve
.Sh "as_number"
.IX Subsection "as_number"
.Vb 1
\&  $x->as_number();              # return as BigInt (in BigInt: same as copy())
.Ve
.Sh "bsrt"
.IX Subsection "bsrt"
.Vb 1
\&  $x->bstr();                   # normalized string
.Ve
.Sh "bsstr"
.IX Subsection "bsstr"
.Vb 1
\&  $x->bsstr();                  # normalized string in scientific notation
.Ve
.Sh "as_hex"
.IX Subsection "as_hex"
.Vb 1
\&  $x->as_hex();                 # as signed hexadecimal string with prefixed 0x
.Ve
.Sh "as_bin"
.IX Subsection "as_bin"
.Vb 1
\&  $x->as_bin();                 # as signed binary string with prefixed 0b
.Ve
.SH "ACCURACY and PRECISION"
.IX Header "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.
.PP
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.
.PP
Not yet implemented things (but with correct description) are marked with '!',
things that need to be answered are marked with '?'.
.PP
In the next paragraph follows a short description of terms used here (because
these may differ from terms used by others people or documentation).
.PP
During the rest of this document, the shortcuts A (for accuracy), P (for
precision), F (fallback) and R (rounding mode) will be used.
.Sh "Precision P"
.IX Subsection "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.
.PP
The string output (of floating point numbers) will be padded with zeros:
.PP
.Vb 9
\&        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
.Ve
.PP
For BigInts, no padding occurs.
.Sh "Accuracy A"
.IX Subsection "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.
.PP
The string output (of floating point numbers) will be padded with zeros:
.PP
.Vb 5
\&        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
.Ve
.PP
For BigInts, no padding occurs.
.Sh "Fallback F"
.IX Subsection "Fallback F"
When both A and P are undefined, this is used as a fallback accuracy when
dividing numbers.
.Sh "Rounding mode R"
.IX Subsection "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.)
.IP "'trunc'" 2
.IX Item "'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.
.Sp
All other implemented styles of rounding attempt to round to the
\&\*(L"nearest digit.\*(R" 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.
.Sp
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:
.IP "'even'" 2
.IX Item "'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.
.IP "'odd'" 2
.IX Item "'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.
.IP "'+inf'" 2
.IX Item "'+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.
.IP "'\-inf'" 2
.IX Item "'-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.
.IP "'zero'" 2
.IX Item "'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.
.PP
The handling of A & P in \s-1MBI/MBF\s0 (the old core code shipped with Perl
versions <= 5.7.2) is like this:
.IP "Precision" 2
.IX Item "Precision"
.Vb 3
\&  * ffround($p) is able to round to $p number of digits after the decimal
\&    point
\&  * otherwise P is unused
.Ve
.IP "Accuracy (significant digits)" 2
.IX Item "Accuracy (significant digits)"
.Vb 29
\&  * fround($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.
.Ve
.PP
This is how it works now:
.IP "Setting/Accessing" 2
.IX Item "Setting/Accessing"
.Vb 17
\&  * 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.
.Ve
.IP "Creating numbers" 2
.IX Item "Creating numbers"
.Vb 12
\&  * 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):
.Ve
.Sp
.Vb 2
\&        use Math::Bigint::SomeSubclass;
\&        use Math::BigInt;
.Ve
.Sp
.Vb 3
\&        Math::BigInt->accuracy(2);
\&        Math::BigInt::SomeSubClass->accuracy(3);
\&        $x = Math::BigInt::SomeSubClass->new(1234);
.Ve
.Sp
.Vb 2
\&    $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.
.Ve
.IP "Usage" 2
.IX Item "Usage"
.Vb 7
\&  * 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'
.Ve
.IP "Precedence" 2
.IX Item "Precedence"
.Vb 30
\&  * 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:
.Ve
.Sp
.Vb 6
\&        $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!
.Ve
.Sp
.Vb 6
\&    + 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.
.Ve
.Sp
.Vb 2
\& ?Maybe an extra option that forbids local A & P settings would be in order,
\& ?so that intermediate rounding does not 'poison' further math?
.Ve
.IP "Overriding globals" 2
.IX Item "Overriding globals"
.Vb 16
\&  * 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
.Ve
.IP "Local settings" 2
.IX Item "Local settings"
.Vb 4
\&  * 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.
.Ve
.IP "Rounding" 2
.IX Item "Rounding"
.Vb 15
\&  * 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:
.Ve
.Sp
.Vb 3
\&        $x = Math::BigFloat->new(123.456);
\&        $x->accuracy(5);
\&        $x->bround(4);
.Ve
.Sp
.Vb 2
\&    Here 4 takes precedence over 5, so 123.5 is the result and $x->accuracy()
\&    will be 4 from now on.
.Ve
.IP "Default values" 2
.IX Item "Default values"
.Vb 4
\&  * R: 'even'
\&  * F: 40
\&  * A: undef
\&  * P: undef
.Ve
.IP "Remarks" 2
.IX Item "Remarks"
.Vb 5
\&  * The defaults are set up so that the new code gives the same results as
\&    the old code (except in a few cases on fdiv):
\&    + Both A and P are undefined and thus will not be used for rounding
\&      after each operation.
\&    + round() is thus a no-op, unless given extra parameters A and P
.Ve
.SH "INTERNALS"
.IX Header "INTERNALS"
The actual numbers are stored as unsigned big integers (with seperate sign).
You should neither care about nor depend on the internal representation; it
might change without notice. Use only method calls like \f(CW\*(C`$x\->sign();\*(C'\fR
instead relying on the internal hash keys like in \f(CW\*(C`$x\->{sign};\*(C'\fR. 
.Sh "\s-1MATH\s0 \s-1LIBRARY\s0"
.IX Subsection "MATH LIBRARY"
Math with the numbers is done (by default) by a module called
Math::BigInt::Calc. This is equivalent to saying:
.PP
.Vb 1
\&        use Math::BigInt lib => 'Calc';
.Ve
.PP
You can change this by using:
.PP
.Vb 1
\&        use Math::BigInt lib => 'BitVect';
.Ve
.PP
The following would first try to find Math::BigInt::Foo, then
Math::BigInt::Bar, and when this also fails, revert to Math::BigInt::Calc:
.PP
.Vb 1
\&        use Math::BigInt lib => 'Foo,Math::BigInt::Bar';
.Ve
.PP
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.
.Sh "\s-1SIGN\s0"
.IX Subsection "SIGN"
The sign is either '+', '\-', 'NaN', '+inf' or '\-inf' and stored seperately.
.PP
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.
.Sh "\fImantissa()\fP, \fIexponent()\fP and \fIparts()\fP"
.IX Subsection "mantissa(), exponent() and parts()"
\&\f(CW\*(C`mantissa()\*(C'\fR and \f(CW\*(C`exponent()\*(C'\fR return the said parts of the BigInt such
that:
.PP
.Vb 4
\&        $m = $x->mantissa();
\&        $e = $x->exponent();
\&        $y = $m * ( 10 ** $e );
\&        print "ok\en" if $x == $y;
.Ve
.PP
\&\f(CW\*(C`($m,$e) = $x\->parts()\*(C'\fR is just a shortcut that gives you both of them
in one go. Both the returned mantissa and exponent have a sign.
.PP
Currently, for BigInts \f(CW$e\fR will be always 0, except for NaN, +inf and \-inf,
where it will be NaN; and for \f(CW$x\fR == 0, where it will be 1
(to be compatible with Math::BigFloat's internal representation of a zero as
\&\f(CW0E1\fR).
.PP
\&\f(CW$m\fR will always be a copy of the original number. The relation between \f(CW$e\fR
and \f(CW$m\fR might change in the future, but will always be equivalent in a
numerical sense, e.g. \f(CW$m\fR might get minimized.
.SH "EXAMPLES"
.IX Header "EXAMPLES"
.Vb 1
\&  use Math::BigInt;
.Ve
.PP
.Vb 1
\&  sub bint { Math::BigInt->new(shift); }
.Ve
.PP
.Vb 15
\&  $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
.Ve
.PP
Examples for rounding:
.PP
.Vb 2
\&  use Math::BigFloat;
\&  use Test;
.Ve
.PP
.Vb 3
\&  $x = Math::BigFloat->new(123.4567);
\&  $y = Math::BigFloat->new(123.456789);
\&  Math::BigFloat->accuracy(4);          # no more A than 4
.Ve
.PP
.Vb 9
\&  ok ($x->copy()->fround(),123.4);      # even rounding
\&  print $x->copy()->fround(),"\en";      # 123.4
\&  Math::BigFloat->round_mode('odd');    # round to odd
\&  print $x->copy()->fround(),"\en";      # 123.5
\&  Math::BigFloat->accuracy(5);          # no more A than 5
\&  Math::BigFloat->round_mode('odd');    # round to odd
\&  print $x->copy()->fround(),"\en";      # 123.46
\&  $y = $x->copy()->fround(4),"\en";      # A = 4: 123.4
\&  print "$y, ",$y->accuracy(),"\en";     # 123.4, 4
.Ve
.PP
.Vb 4
\&  Math::BigFloat->accuracy(undef);      # A not important now
\&  Math::BigFloat->precision(2);         # P important
\&  print $x->copy()->bnorm(),"\en";       # 123.46
\&  print $x->copy()->fround(),"\en";      # 123.46
.Ve
.PP
Examples for converting:
.PP
.Vb 2
\&  my $x = Math::BigInt->new('0b1'.'01' x 123);
\&  print "bin: ",$x->as_bin()," hex:",$x->as_hex()," dec: ",$x,"\en";
.Ve
.SH "Autocreating constants"
.IX Header "Autocreating constants"
After \f(CW\*(C`use Math::BigInt ':constant'\*(C'\fR all the \fBinteger\fR decimal, hexadecimal
and binary constants in the given scope are converted to \f(CW\*(C`Math::BigInt\*(C'\fR.
This conversion happens at compile time. 
.PP
In particular,
.PP
.Vb 1
\&  perl -MMath::BigInt=:constant -e 'print 2**100,"\en"'
.Ve
.PP
prints the integer value of \f(CW\*(C`2**100\*(C'\fR. Note that without conversion of 
constants the expression 2**100 will be calculated as perl scalar.
.PP
Please note that strings and floating point constants are not affected,
so that
.PP
.Vb 1
\&        use Math::BigInt qw/:constant/;
.Ve
.PP
.Vb 4
\&        $x = 1234567890123456789012345678901234567890
\&                + 123456789123456789;
\&        $y = '1234567890123456789012345678901234567890'
\&                + '123456789123456789';
.Ve
.PP
do not work. You need an explicit Math::BigInt\->\fInew()\fR around one of the
operands. You should also quote large constants to protect loss of precision:
.PP
.Vb 1
\&        use Math::Bigint;
.Ve
.PP
.Vb 1
\&        $x = Math::BigInt->new('1234567889123456789123456789123456789');
.Ve
.PP
Without the quotes Perl would convert the large number to a floating point
constant at compile time and then hand the result to BigInt, which results in
an truncated result or a NaN.
.PP
This also applies to integers that look like floating point constants:
.PP
.Vb 1
\&        use Math::BigInt ':constant';
.Ve
.PP
.Vb 2
\&        print ref(123e2),"\en";
\&        print ref(123.2e2),"\en";
.Ve
.PP
will print nothing but newlines. Use either bignum or Math::BigFloat
to get this to work.
.SH "PERFORMANCE"
.IX Header "PERFORMANCE"
Using the form \f(CW$x\fR += \f(CW$y\fR; etc over \f(CW$x\fR = \f(CW$x\fR + \f(CW$y\fR is faster, since a copy of \f(CW$x\fR
must be made in the second case. For long numbers, the copy can eat up to 20%
of the work (in the case of addition/subtraction, less for
multiplication/division). If \f(CW$y\fR is very small compared to \f(CW$x\fR, the form
\&\f(CW$x\fR += \f(CW$y\fR is \s-1MUCH\s0 faster than \f(CW$x\fR = \f(CW$x\fR + \f(CW$y\fR since making the copy of \f(CW$x\fR takes
more time then the actual addition.
.PP
With a technique called copy\-on\-write, the cost of copying with overload could
be minimized or even completely avoided. A test implementation of \s-1COW\s0 did show
performance gains for overloaded math, but introduced a performance loss due
to a constant overhead for all other operatons.
.PP
The rewritten version of this module is slower on certain operations, like
\&\fInew()\fR, \fIbstr()\fR and \fInumify()\fR. 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.
.PP
Some operations may be slower for small numbers, but are significantly faster
for big numbers. Other operations are now constant (O(1), like \fIbneg()\fR, \fIbabs()\fR
etc), instead of O(N) and thus nearly always take much less time. These
optimizations were done on purpose.
.PP
If you find the Calc module to slow, try to install any of the replacement
modules and see if they help you. 
.Sh "Alternative math libraries"
.IX Subsection "Alternative math libraries"
You can use an alternative library to drive Math::BigInt via:
.PP
.Vb 1
\&        use Math::BigInt lib => 'Module';
.Ve
.PP
See \*(L"\s-1MATH\s0 \s-1LIBRARY\s0\*(R" for more information.
.PP
For more benchmark results see <http://bloodgate.com/perl/benchmarks.html>.
.Sh "\s-1SUBCLASSING\s0"
.IX Subsection "SUBCLASSING"
.SH "Subclassing Math::BigInt"
.IX Header "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:
.IP "\(bu" 2
The public \s-1API\s0 must remain consistent, i.e. if a sub-class is overloading
addition, the sub-class must use the same name, in this case \fIbadd()\fR. The
reason for this is that Math::BigInt is optimized to call the object methods
directly.
.IP "\(bu" 2
The private object hash keys like \f(CW\*(C`$x\-\*(C'\fR{sign}> may not be changed, but
additional keys can be added, like \f(CW\*(C`$x\-\*(C'\fR{_custom}>.
.IP "\(bu" 2
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.
.PP
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 \f(CW\*(C`bstr()\*(C'\fR. 
.PP
All other object methods and overloaded functions can be directly inherited
from the parent class.
.PP
At the very minimum, any subclass will need to provide it's own \f(CW\*(C`new()\*(C'\fR and can
store additional hash keys in the object. There are also some package globals
that must be defined, e.g.:
.PP
.Vb 5
\&  # Globals
\&  $accuracy = undef;
\&  $precision = -2;       # round to 2 decimal places
\&  $round_mode = 'even';
\&  $div_scale = 40;
.Ve
.PP
Additionally, you might want to provide the following two globals to allow
auto-upgrading and auto-downgrading to work correctly:
.PP
.Vb 2
\&  $upgrade = undef;
\&  $downgrade = undef;
.Ve
.PP
This allows Math::BigInt to correctly retrieve package globals from the 
subclass, like \f(CW$SubClass::precision\fR.  See t/Math/BigInt/Subclass.pm or
t/Math/BigFloat/SubClass.pm completely functional subclass examples.
.PP
Don't forget to 
.PP
.Vb 1
\&        use overload;
.Ve
.PP
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.
.SH "UPGRADING"
.IX Header "UPGRADING"
When used like this:
.PP
.Vb 1
\&        use Math::BigInt upgrade => 'Foo::Bar';
.Ve
.PP
certain operations will 'upgrade' their calculation and thus the result to
the class Foo::Bar. Usually this is used in conjunction with Math::BigFloat:
.PP
.Vb 1
\&        use Math::BigInt upgrade => 'Math::BigFloat';
.Ve
.PP
As a shortcut, you can use the module \f(CW\*(C`bignum\*(C'\fR:
.PP
.Vb 1
\&        use bignum;
.Ve
.PP
Also good for oneliners:
.PP
.Vb 1
\&        perl -Mbignum -le 'print 2 ** 255'
.Ve
.PP
This makes it possible to mix arguments of different classes (as in 2.5 + 2)
as well es preserve accuracy (as in \fIsqrt\fR\|(3)).
.PP
Beware: This feature is not fully implemented yet.
.Sh "Auto-upgrade"
.IX Subsection "Auto-upgrade"
The following methods upgrade themselves unconditionally; that is if upgrade
is in effect, they will always hand up their work:
.IP "\fIbsqrt()\fR" 2
.IX Item "bsqrt()"
.PD 0
.IP "\fIdiv()\fR" 2
.IX Item "div()"
.IP "\fIblog()\fR" 2
.IX Item "blog()"
.PD
.PP
Beware: This list is not complete.
.PP
All other methods upgrade themselves only when one (or all) of their
arguments are of the class mentioned in \f(CW$upgrade\fR (This might change in later
versions to a more sophisticated scheme):
.SH "BUGS"
.IX Header "BUGS"
.IP "Out of Memory!" 2
.IX Item "Out of Memory!"
Under Perl prior to 5.6.0 having an \f(CW\*(C`use Math::BigInt ':constant';\*(C'\fR and 
\&\f(CW\*(C`eval()\*(C'\fR in your code will crash with \*(L"Out of memory\*(R". This is probably an
overload/exporter bug. You can workaround by not having \f(CW\*(C`eval()\*(C'\fR 
and ':constant' at the same time or upgrade your Perl to a newer version.
.IP "Fails to load Calc on Perl prior 5.6.0" 2
.IX Item "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.  
.SH "CAVEATS"
.IX Header "CAVEATS"
Some things might not work as you expect them. Below is documented what is
known to be troublesome:
.IP "stringify, \fIbstr()\fR, \fIbsstr()\fR and 'cmp'" 1
.IX Item "stringify, bstr(), bsstr() and 'cmp'"
Both stringify and \fIbstr()\fR 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 \fIok()\fR uses 'eq' internally. 
.Sp
Mark said, when asked about to drop the '+' altogether, or make only cmp work:
.Sp
.Vb 4
\&        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.
.Ve
.Sp
So, the following examples will now work all as expected:
.Sp
.Vb 3
\&        use Test;
\&        BEGIN { plan tests => 1 }
\&        use Math::BigInt;
.Ve
.Sp
.Vb 2
\&        my $x = new Math::BigInt 3*3;
\&        my $y = new Math::BigInt 3*3;
.Ve
.Sp
.Vb 4
\&        ok ($x,3*3);
\&        print "$x eq 9" if $x eq $y;
\&        print "$x eq 9" if $x eq '9';
\&        print "$x eq 9" if $x eq 3*3;
.Ve
.Sp
Additionally, the following still works:
.Sp
.Vb 3
\&        print "$x == 9" if $x == $y;
\&        print "$x == 9" if $x == 9;
\&        print "$x == 9" if $x == 3*3;
.Ve
.Sp
There is now a \f(CW\*(C`bsstr()\*(C'\fR method to get the string in scientific notation aka
\&\f(CW1e+2\fR instead of \f(CW100\fR. Be advised that overloaded 'eq' always uses \fIbstr()\fR
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:
.Sp
.Vb 3
\&        use Test;
\&        BEGIN { plan tests => 3 }
\&        use Math::BigInt;
.Ve
.Sp
.Vb 5
\&        $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
.Ve
.Sp
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.
.IP "\fIint()\fR" 1
.IX Item "int()"
\&\f(CW\*(C`int()\*(C'\fR will return (at least for Perl v5.7.1 and up) another BigInt, not a 
Perl scalar:
.Sp
.Vb 4
\&        $x = Math::BigInt->new(123);
\&        $y = int($x);                           # BigInt 123
\&        $x = Math::BigFloat->new(123.45);
\&        $y = int($x);                           # BigInt 123
.Ve
.Sp
In all Perl versions you can use \f(CW\*(C`as_number()\*(C'\fR for the same effect:
.Sp
.Vb 2
\&        $x = Math::BigFloat->new(123.45);
\&        $y = $x->as_number();                   # BigInt 123
.Ve
.Sp
This also works for other subclasses, like Math::String.
.Sp
It is yet unlcear whether overloaded \fIint()\fR should return a scalar or a BigInt.
.IP "length" 1
.IX Item "length"
The following will probably not do what you expect:
.Sp
.Vb 2
\&        $c = Math::BigInt->new(123);
\&        print $c->length(),"\en";                # prints 30
.Ve
.Sp
It prints both the number of digits in the number and in the fraction part
since print calls \f(CW\*(C`length()\*(C'\fR in list context. Use something like: 
.Sp
.Vb 1
\&        print scalar $c->length(),"\en";         # prints 3
.Ve
.IP "bdiv" 1
.IX Item "bdiv"
The following will probably not do what you expect:
.Sp
.Vb 1
\&        print $c->bdiv(10000),"\en";
.Ve
.Sp
It prints both quotient and remainder since print calls \f(CW\*(C`bdiv()\*(C'\fR in list
context. Also, \f(CW\*(C`bdiv()\*(C'\fR will modify \f(CW$c\fR, so be carefull. You probably want
to use
.Sp
.Vb 2
\&        print $c / 10000,"\en";
\&        print scalar $c->bdiv(10000),"\en";  # or if you want to modify $c
.Ve
.Sp
instead.
.Sp
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,
.Sp
.Vb 6
\&          1 / 4  => ( 0, 1)
\&          1 / -4 => (-1,-3)
\&         -3 / 4  => (-1, 1)
\&         -3 / -4 => ( 0,-3)
\&        -11 / 2  => (-5,1)
\&         11 /-2  => (-5,-1)
.Ve
.Sp
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
.Sp
.Vb 1
\&        $x == ($x / $y) * $y + ($x % $y)
.Ve
.Sp
holds true for any \f(CW$x\fR and \f(CW$y\fR, which justifies calling the two return
values of \fIbdiv()\fR the quotient and remainder. The only exception to this rule
are when \f(CW$y\fR == 0 and \f(CW$x\fR is negative, then the remainder will also be
negative. See below under \*(L"infinity handling\*(R" for the reasoning behing this.
.Sp
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 ;)
.IP "infinity handling" 1
.IX Item "infinity handling"
Here are some examples that explain the reasons why certain results occur while
handling infinity:
.Sp
The following table shows the result of the division and the remainder, so that
the equation above holds true. Some \*(L"ordinary\*(R" cases are strewn in to show more
clearly the reasoning:
.Sp
.Vb 23
\&        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
.Ve
.Sp
These cases below violate the \*(L"remainder has the sign of the second of the two
arguments\*(R", since they wouldn't match up otherwise.
.Sp
.Vb 4
\&        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
.Ve
.IP "Modifying and =" 1
.IX Item "Modifying and ="
Beware of:
.Sp
.Vb 2
\&        $x = Math::BigFloat->new(5);
\&        $y = $x;
.Ve
.Sp
It will not do what you think, e.g. making a copy of \f(CW$x\fR. Instead it just makes
a second reference to the \fBsame\fR object and stores it in \f(CW$y\fR. Thus anything
that modifies \f(CW$x\fR (except overloaded operators) will modify \f(CW$y\fR, and vice versa.
Or in other words, \f(CW\*(C`=\*(C'\fR is only safe if you modify your BigInts only via
overloaded math. As soon as you use a method call it breaks:
.Sp
.Vb 2
\&        $x->bmul(2);
\&        print "$x, $y\en";       # prints '10, 10'
.Ve
.Sp
If you want a true copy of \f(CW$x\fR, use:
.Sp
.Vb 1
\&        $y = $x->copy();
.Ve
.Sp
You can also chain the calls like this, this will make first a copy and then
multiply it by 2:
.Sp
.Vb 1
\&        $y = $x->copy()->bmul(2);
.Ve
.Sp
See also the documentation for overload.pm regarding \f(CW\*(C`=\*(C'\fR.
.IP "bpow" 1
.IX Item "bpow"
\&\f(CW\*(C`bpow()\*(C'\fR (and the rounding functions) now modifies the first argument and
returns it, unlike the old code which left it alone and only returned the
result. This is to be consistent with \f(CW\*(C`badd()\*(C'\fR etc. The first three will
modify \f(CW$x\fR, the last one won't:
.Sp
.Vb 4
\&        print bpow($x,$i),"\en";         # modify $x
\&        print $x->bpow($i),"\en";        # ditto
\&        print $x **= $i,"\en";           # the same
\&        print $x ** $i,"\en";            # leave $x alone
.Ve
.Sp
The form \f(CW\*(C`$x **= $y\*(C'\fR is faster than \f(CW\*(C`$x = $x ** $y;\*(C'\fR, though.
.IP "Overloading \-$x" 1
.IX Item "Overloading -$x"
The following:
.Sp
.Vb 1
\&        $x = -$x;
.Ve
.Sp
is slower than
.Sp
.Vb 1
\&        $x->bneg();
.Ve
.Sp
since overload calls \f(CW\*(C`sub($x,0,1);\*(C'\fR instead of \f(CW\*(C`neg($x)\*(C'\fR. The first variant
needs to preserve \f(CW$x\fR since it does not know that it later will get overwritten.
This makes a copy of \f(CW$x\fR and takes O(N), but \f(CW$x\fR\->\fIbneg()\fR is O(1).
.Sp
With Copy\-On\-Write, this issue would be gone, but C\-o-W is not implemented
since it is slower for all other things.
.IP "Mixing different object types" 1
.IX Item "Mixing different object types"
In Perl you will get a floating point value if you do one of the following:
.Sp
.Vb 3
\&        $float = 5.0 + 2;
\&        $float = 2 + 5.0;
\&        $float = 5 / 2;
.Ve
.Sp
With overloaded math, only the first two variants will result in a BigFloat:
.Sp
.Vb 2
\&        use Math::BigInt;
\&        use Math::BigFloat;
.Ve
.Sp
.Vb 3
\&        $mbf = Math::BigFloat->new(5);
\&        $mbi2 = Math::BigInteger->new(5);
\&        $mbi = Math::BigInteger->new(2);
.Ve
.Sp
.Vb 6
\&                                        # 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()
.Ve
.Sp
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,
\&\fIMath::BigInt::bdiv()\fR will always return a Math::BigInt, regardless whether
the result should be a Math::BigFloat or the second operant is one.
.Sp
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\->\fInew()\fR:
.Sp
.Vb 1
\&        $float = Math::BigFloat->new($mbi2) / $mbi;     # = 2.5
.Ve
.Sp
Beware of simple \*(L"casting\*(R" the entire expression, this would only convert
the already computed result:
.Sp
.Vb 1
\&        $float = Math::BigFloat->new($mbi2 / $mbi);     # = 2.0 thus wrong!
.Ve
.Sp
Beware also of the order of more complicated expressions like:
.Sp
.Vb 2
\&        $integer = ($mbi2 + $mbi) / $mbf;               # int / float => int
\&        $integer = $mbi2 / Math::BigFloat->new($mbi);   # ditto
.Ve
.Sp
If in doubt, break the expression into simpler terms, or cast all operands
to the desired resulting type.
.Sp
Scalar values are a bit different, since:
.Sp
.Vb 2
\&        $float = 2 + $mbf;
\&        $float = $mbf + 2;
.Ve
.Sp
will both result in the proper type due to the way the overloaded math works.
.Sp
This section also applies to other overloaded math packages, like Math::String.
.Sp
One solution to you problem might be autoupgrading.
.IP "\fIbsqrt()\fR" 1
.IX Item "bsqrt()"
\&\f(CW\*(C`bsqrt()\*(C'\fR 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.
.Sp
If you want a better approximation of the square root, then use:
.Sp
.Vb 4
\&        $x = Math::BigFloat->new(12);
\&        Math::BigFloat->precision(0);
\&        Math::BigFloat->round_mode('even');
\&        print $x->copy->bsqrt(),"\en";           # 4
.Ve
.Sp
.Vb 3
\&        Math::BigFloat->precision(2);
\&        print $x->bsqrt(),"\en";                 # 3.46
\&        print $x->bsqrt(3),"\en";                # 3.464
.Ve
.IP "\fIbrsft()\fR" 1
.IX Item "brsft()"
For negative numbers in base see also brsft.
.SH "LICENSE"
.IX Header "LICENSE"
This program is free software; you may redistribute it and/or modify it under
the same terms as Perl itself.
.SH "SEE ALSO"
.IX Header "SEE ALSO"
Math::BigFloat and Math::Big as well as Math::BigInt::BitVect,
Math::BigInt::Pari and  Math::BigInt::GMP.
.PP
The package at
<http://search.cpan.org/search?mode=module&query=Math%3A%3ABigInt> contains
more documentation including a full version history, testcases, empty
subclass files and benchmarks.
.SH "AUTHORS"
.IX Header "AUTHORS"
Original code by Mark Biggar, overloaded interface by Ilya Zakharevich.
Completely rewritten by Tels http://bloodgate.com in late 2000, 2001.