#!/usr/bin/perl -w
-# mark.biggar@TrustedSysLabs.com
-# eay@mincom.com is dead (math::BigInteger)
-# see: http://www.cypherspace.org/~adam/rsa/pureperl.html (contacted c. adam
-# on 2000/11/13 - but email is dead
-
-# todo:
-# - fully remove funky $# stuff (maybe)
-# - use integer; vs 1e7 as base
-# - speed issues (XS? Bit::Vector?)
-# - split out actual math code to Math::BigNumber
-
# Qs: what exactly happens on numify of HUGE numbers? overflow?
# $a = -$a is much slower (making copy of $a) than $a->bneg(), hm!?
# (copy_on_write will help there, but that is not yet implemented)
# The following hash values are used:
-# value: the internal array, base 100000
+# value: unsigned int with actual value (as a Math::BigInt::Calc or similiar)
# sign : +,-,NaN,+inf,-inf
# _a : accuracy
# _p : precision
+# _f : flags, used by MBF to flag parts of a float as untouchable
# _cow : copy on write: number of objects that share the data (NRY)
-# Internally the numbers are stored in an array with at least 1 element, no
-# leading zero parts (except the first) and in base 100000
-
-# USE_MUL: due to problems on certain os (os390, posix-bc) "* 1e-5" is used
-# instead of "/ 1e5" at some places, (marked with USE_MUL). But instead of
-# using the reverse only on problematic machines, I used it everytime to avoid
-# the costly comparisations. This _should_ work everywhere. Thanx Peter Prymmer
package Math::BigInt;
my $class = "Math::BigInt";
+require 5.005;
-$VERSION = 1.35;
+$VERSION = 1.36;
use Exporter;
@ISA = qw( Exporter );
@EXPORT_OK = qw( bneg babs bcmp badd bmul bdiv bmod bnorm bsub
blsft brsft band bior bxor bnot bpow bnan bzero
bacmp bstr bsstr binc bdec bint binf bfloor bceil
is_odd is_even is_zero is_one is_nan is_inf sign
+ is_positive is_negative
length as_number
- trace objectify _swap
+ objectify _swap
);
#@EXPORT = qw( );
##############################################################################
# global constants, flags and accessory
-# are NaNs ok?
-my $NaNOK=1;
-# set to 1 for tracing
-my $trace = 0;
-# constants for easier life
-my $nan = 'NaN';
-my $BASE_LEN = 5;
-my $BASE = int("1e".$BASE_LEN); # var for trying to change it to 1e7
-my $RBASE = 1e-5; # see USE_MUL
-
-# Rounding modes one of 'even', 'odd', '+inf', '-inf', 'zero' or 'trunc'
+use constant MB_NEVER_ROUND => 0x0001;
+
+my $NaNOK=1; # are NaNs ok?
+my $nan = 'NaN'; # constants for easier life
+
+my $CALC = 'Math::BigInt::Calc'; # module to do low level math
+sub _core_lib () { return $CALC; } # for test suite
+
+# Rounding modes, one of 'even', 'odd', '+inf', '-inf', 'zero' or 'trunc'
$rnd_mode = 'even';
$accuracy = undef;
$precision = undef;
my $self = {}; bless $self,$c;
foreach my $k (keys %$x)
{
- if (ref($x->{$k}) eq 'ARRAY')
+ if ($k eq 'value')
+ {
+ $self->{$k} = $CALC->_copy($x->{$k});
+ }
+ elsif (ref($x->{$k}) eq 'SCALAR')
+ {
+ $self->{$k} = \${$x->{$k}};
+ }
+ elsif (ref($x->{$k}) eq 'ARRAY')
{
$self->{$k} = [ @{$x->{$k}} ];
}
sub new
{
- # create a new BigInts object from a string or another bigint object.
- # value => internal array representation
- # sign => sign (+/-), or "NaN"
+ # create a new BigInt object from a string or another BigIint object.
+ # see hash keys documented at top
# the argument could be an object, so avoid ||, && etc on it, this would
# cause costly overloaded code to be called. The only allowed op are ref()
# and definend.
- trace (@_);
my $class = shift;
my $wanted = shift; # avoid numify call by not using || here
# handle '+inf', '-inf' first
if ($wanted =~ /^[+-]inf$/)
{
- $self->{value} = [ 0 ];
+ $self->{value} = $CALC->_zero();
$self->{sign} = $wanted;
return $self;
}
my ($mis,$miv,$mfv,$es,$ev) = _split(\$wanted);
if (ref $mis && !ref $miv)
{
- # _from_hex
+ # _from_hex or _from_bin
$self->{value} = $mis->{value};
$self->{sign} = $mis->{sign};
- return $self;
+ return $self; # throw away $mis
}
if (!ref $mis)
{
die "$wanted is not a number initialized to $class" if !$NaNOK;
#print "NaN 1\n";
- $self->{value} = [ 0 ];
+ $self->{value} = $CALC->_zero();
$self->{sign} = $nan;
return $self;
}
# make integer from mantissa by adjusting exp, then convert to bigint
$self->{sign} = $$mis; # store sign
- $self->{value} = [ 0 ]; # for all the NaN cases
+ $self->{value} = $CALC->_zero(); # for all the NaN cases
my $e = int("$$es$$ev"); # exponent (avoid recursion)
if ($e > 0)
{
}
}
$self->{sign} = '+' if $$miv eq '0'; # normalize -0 => +0
- $self->_internal($miv) if $self->{sign} ne $nan; # as internal array
- #print "$wanted => $self->{sign} $self->{value}->[0]\n";
- # if any of the globals is set, round to them and thus store them insid $self
+ $self->{value} = $CALC->_new($miv) if $self->{sign} =~ /^[+-]$/;
+ #print "$wanted => $self->{sign}\n";
+ # if any of the globals is set, use them to round and store them inside $self
$self->round($accuracy,$precision,$rnd_mode)
if defined $accuracy || defined $precision;
return $self;
sub bint
{
# exportable version of new
- trace(@_);
return $class->new(@_);
}
my $c = $self; $self = {}; bless $self, $c;
}
return if $self->modify('bnan');
- $self->{value} = [ 0 ];
+ $self->{value} = $CALC->_zero();
$self->{sign} = $nan;
- trace('NaN');
return $self;
}
my $c = $self; $self = {}; bless $self, $c;
}
return if $self->modify('binf');
- $self->{value} = [ 0 ];
+ $self->{value} = $CALC->_zero();
$self->{sign} = $sign.'inf';
- trace('inf');
return $self;
}
# create a bigint '+0', if given a BigInt, set it to 0
my $self = shift;
$self = $class if !defined $self;
+ #print "bzero $self\n";
+
if (!ref($self))
{
my $c = $self; $self = {}; bless $self, $c;
}
return if $self->modify('bzero');
- $self->{value} = [ 0 ];
+ $self->{value} = $CALC->_zero();
$self->{sign} = '+';
- trace('0');
+ #print "result: $self\n";
return $self;
}
# (ref to BFLOAT or num_str ) return num_str
# Convert number from internal format to scientific string format.
# internal format is always normalized (no leading zeros, "-0E0" => "+0E0")
- trace(@_);
my ($self,$x) = objectify(1,@_);
return $x->{sign} if $x->{sign} !~ /^[+-]$/;
sub bstr
{
- # (ref to BINT or num_str ) return num_str
- # Convert number from internal base 100000 format to string format.
- # internal format is always normalized (no leading zeros, "-0" => "+0")
- trace(@_);
+ # make a string from bigint object
my $x = shift; $x = $class->new($x) unless ref $x;
- # my ($self,$x) = objectify(1,@_);
-
return $x->{sign} if $x->{sign} !~ /^[+-]$/;
- my $ar = $x->{value} || return $nan; # should not happen
- my $es = "";
- $es = $x->{sign} if $x->{sign} eq '-'; # get sign, but not '+'
- my $l = scalar @$ar; # number of parts
- return $nan if $l < 1; # should not happen
- # handle first one different to strip leading zeros from it (there are no
- # leading zero parts in internal representation)
- $l --; $es .= $ar->[$l]; $l--;
- # Interestingly, the pre-padd method uses more time
- # the old grep variant takes longer (14 to 10 sec)
- while ($l >= 0)
- {
- $es .= substr('0000'.$ar->[$l],-5); # fastest way I could think of
- $l--;
- }
- return $es;
+ my $es = ''; $es = $x->{sign} if $x->{sign} eq '-';
+ return $es.${$CALC->_str($x->{value})};
}
sub numify
{
# Make a number from a BigInt object
- # old: simple return string and let Perl's atoi() handle the rest
- # new: calc because it is faster than bstr()+atoi()
- #trace (@_);
- #my ($self,$x) = objectify(1,@_);
- #return $x->bstr(); # ref($x);
my $x = shift; $x = $class->new($x) unless ref $x;
-
- return $nan if $x->{sign} eq $nan;
- my $fac = 1; $fac = -1 if $x->{sign} eq '-';
- return $fac*$x->{value}->[0] if @{$x->{value}} == 1; # below $BASE
- my $num = 0;
- foreach (@{$x->{value}})
- {
- $num += $fac*$_; $fac *= $BASE;
- }
+ return $x->{sign} if $x->{sign} !~ /^[+-]$/;
+ my $num = $CALC->_num($x->{value});
+ return -$num if $x->{sign} eq '-';
return $num;
}
my $r = shift; # round_mode, if given by caller
my @args = @_; # all 'other' arguments (0 for unary, 1 for binary ops)
+ # leave bigfloat parts alone
+ return $self if exists $self->{_f} && $self->{_f} & MB_NEVER_ROUND != 0;
+
unshift @args,$self; # add 'first' argument
$self = new($self) unless ref($self); # if not object, make one
# Compares 2 values. Returns one of undef, <0, =0, >0. (suitable for sort)
# (BINT or num_str, BINT or num_str) return cond_code
my ($self,$x,$y) = objectify(2,@_);
- return undef if (($x->{sign} eq $nan) || ($y->{sign} eq $nan));
+
+ if (($x->{sign} !~ /^[+-]$/) || ($y->{sign} !~ /^[+-]$/))
+ {
+ # handle +-inf and NaN
+ return undef if (($x->{sign} eq $nan) || ($y->{sign} eq $nan));
+ return 0 if ($x->{sign} eq $y->{sign}) && ($x->{sign} =~ /^[+-]inf$/);
+ return +1 if $x->{sign} eq '+inf';
+ return -1 if $x->{sign} eq '-inf';
+ return -1 if $y->{sign} eq '+inf';
+ return +1 if $y->{sign} eq '-inf';
+ }
+ # normal compare now
&cmp($x->{value},$y->{value},$x->{sign},$y->{sign}) <=> 0;
}
# (BINT, BINT) return cond_code
my ($self,$x,$y) = objectify(2,@_);
return undef if (($x->{sign} eq $nan) || ($y->{sign} eq $nan));
- acmp($x->{value},$y->{value}) <=> 0;
+ #acmp($x->{value},$y->{value}) <=> 0;
+ $CALC->_acmp($x->{value},$y->{value}) <=> 0;
}
sub badd
{
# add second arg (BINT or string) to first (BINT) (modifies first)
# return result as BINT
- trace(@_);
my ($self,$x,$y,$a,$p,$r) = objectify(2,@_);
return $x if $x->modify('badd');
- return $x->bnan() if (($x->{sign} eq $nan) || ($y->{sign} eq $nan));
+ return $x->bnan() if (($x->{sign} !~ /^[+-]$/) || ($y->{sign} !~ /^[+-]$/));
- # for round calls, make array
- my @bn = ($a,$p,$r,$y);
+ my @bn = ($a,$p,$r,$y); # make array for round calls
# speed: no add for 0+y or x+0
- return $x->bnorm(@bn) if $y->is_zero(); # x+0
+ return $x->round(@bn) if $y->is_zero(); # x+0
if ($x->is_zero()) # 0+y
{
# make copy, clobbering up x
- $x->{value} = [ @{$y->{value}} ];
+ $x->{value} = $CALC->_copy($y->{value});
+ #$x->{value} = [ @{$y->{value}} ];
$x->{sign} = $y->{sign} || $nan;
return $x->round(@bn);
}
if ($sx eq $sy)
{
- add($xv,$yv); # if same sign, absolute add
+ $CALC->_add($xv,$yv); # if same sign, absolute add
$x->{sign} = $sx;
}
else
{
- my $a = acmp ($yv,$xv); # absolute compare
+ my $a = $CALC->_acmp ($yv,$xv); # absolute compare
if ($a > 0)
{
#print "swapped sub (a=$a)\n";
- &sub($yv,$xv,1); # absolute sub w/ swapped params
+ $CALC->_sub($yv,$xv,1); # absolute sub w/ swapped params
$x->{sign} = $sy;
}
elsif ($a == 0)
{
# speedup, if equal, set result to 0
- $x->{value} = [ 0 ];
+ #print "equal sub, result = 0\n";
+ $x->{value} = $CALC->_zero();
$x->{sign} = '+';
}
else # a < 0
{
#print "unswapped sub (a=$a)\n";
- &sub($xv, $yv); # absolute sub
+ $CALC->_sub($xv, $yv); # absolute sub
$x->{sign} = $sx;
}
}
# subtract second arg from first, modify first
my ($self,$x,$y,$a,$p,$r) = objectify(2,@_);
- trace(@_);
return $x if $x->modify('bsub');
$x->badd($y->bneg()); # badd does not leave internal zeros
$y->bneg(); # refix y, assumes no one reads $y in between
# increment arg by one
my ($self,$x,$a,$p,$r) = objectify(1,@_);
# my $x = shift; $x = $class->new($x) unless ref $x; my $self = ref($x);
- trace(@_);
return $x if $x->modify('binc');
$x->badd($self->_one())->round($a,$p,$r);
}
{
# decrement arg by one
my ($self,$x,$a,$p,$r) = objectify(1,@_);
- trace(@_);
return $x if $x->modify('bdec');
$x->badd($self->_one('-'))->round($a,$p,$r);
}
# (BINT or num_str, BINT or num_str) return BINT
# does not modify arguments, but returns new object
# Lowest Common Multiplicator
- trace(@_);
- my ($self,@arg) = objectify(0,@_);
- my $x = $self->new(shift @arg);
- while (@arg) { $x = _lcm($x,shift @arg); }
+ my $y = shift; my ($x);
+ if (ref($y))
+ {
+ $x = $y->copy();
+ }
+ else
+ {
+ $x = $class->new($y);
+ }
+ while (@_) { $x = _lcm($x,shift); }
$x;
}
# (BINT or num_str, BINT or num_str) return BINT
# does not modify arguments, but returns new object
# GCD -- Euclids algorithm, variant C (Knuth Vol 3, pg 341 ff)
- trace(@_);
-
- my ($self,@arg) = objectify(0,@_);
- my $x = $self->new(shift @arg);
- while (@arg)
+
+ my $y = shift; my ($x);
+ if (ref($y))
{
- #$x = _gcd($x,shift @arg); last if $x->is_one(); # new fast, but is slower
- $x = _gcd_old($x,shift @arg); last if $x->is_one(); # old, slow, but faster
- }
- $x;
+ $x = $y->copy();
+ }
+ else
+ {
+ $x = $class->new($y);
+ }
+
+ if ($CALC->can('_gcd'))
+ {
+ while (@_)
+ {
+ $y = shift; $y = $class->new($y) if !ref($y);
+ next if $y->is_zero();
+ return $x->bnan() if $y->{sign} !~ /^[+-]$/; # y NaN?
+ $x->{value} = $CALC->_gcd($x->{value},$y->{value}); last if $x->is_one();
+ }
+ }
+ else
+ {
+ while (@_)
+ {
+ $x = _gcd($x,shift); last if $x->is_one(); # _gcd handles NaN
+ }
+ }
+ $x->babs();
}
sub bmod
{
# return true if arg (BINT or num_str) is zero (array '+', '0')
#my ($self,$x) = objectify(1,@_);
- #trace(@_);
my $x = shift; $x = $class->new($x) unless ref $x;
- return (@{$x->{value}} == 1) && ($x->{sign} eq '+')
- && ($x->{value}->[0] == 0);
+
+ return 0 if $x->{sign} !~ /^[+-]$/;
+ return $CALC->_is_zero($x->{value});
+ #return (@{$x->{value}} == 1) && ($x->{sign} eq '+')
+ # && ($x->{value}->[0] == 0);
}
sub is_nan
{
# return true if arg (BINT or num_str) is NaN
#my ($self,$x) = objectify(1,@_);
- #trace(@_);
my $x = shift; $x = $class->new($x) unless ref $x;
return ($x->{sign} eq $nan);
}
{
# return true if arg (BINT or num_str) is +-inf
#my ($self,$x) = objectify(1,@_);
- #trace(@_);
my $x = shift; $x = $class->new($x) unless ref $x;
my $sign = shift || '';
- return $x->{sign} =~ /^[+-]inf/ if $sign eq '';
- return $x->{sign} =~ /^[$sign]inf/;
+ return $x->{sign} =~ /^[+-]inf$/ if $sign eq '';
+ return $x->{sign} =~ /^[$sign]inf$/;
}
sub is_one
# or -1 if signis given
#my ($self,$x) = objectify(1,@_);
my $x = shift; $x = $class->new($x) unless ref $x;
- my $sign = shift || '+'; #$_[2] || '+';
- return (@{$x->{value}} == 1) && ($x->{sign} eq $sign)
- && ($x->{value}->[0] == 1);
+ my $sign = shift || '+';
+
+ # catch also NaN, +inf, -inf
+ return 0 if $x->{sign} ne $sign || $x->{sign} !~ /^[+-]$/;
+ return $CALC->_is_one($x->{value});
+ #return (@{$x->{value}} == 1) && ($x->{sign} eq $sign)
+ # && ($x->{value}->[0] == 1);
}
sub is_odd
# return true when arg (BINT or num_str) is odd, false for even
my $x = shift; $x = $class->new($x) unless ref $x;
#my ($self,$x) = objectify(1,@_);
- return (($x->{sign} ne $nan) && ($x->{value}->[0] & 1));
+
+ return 0 if ($x->{sign} !~ /^[+-]$/);
+ return $CALC->_is_odd($x->{value});
+ #return (($x->{sign} ne $nan) && ($x->{value}->[0] & 1));
}
sub is_even
# return true when arg (BINT or num_str) is even, false for odd
my $x = shift; $x = $class->new($x) unless ref $x;
#my ($self,$x) = objectify(1,@_);
- return (($x->{sign} ne $nan) && (!($x->{value}->[0] & 1)));
+
+ return 0 if ($x->{sign} !~ /^[+-]$/);
+ return $CALC->_is_even($x->{value});
+ #return (($x->{sign} ne $nan) && (!($x->{value}->[0] & 1)));
+ #return (($x->{sign} !~ /^[+-]$/) && ($CALC->_is_even($x->{value})));
+ }
+
+sub is_positive
+ {
+ # return true when arg (BINT or num_str) is positive (>= 0)
+ my $x = shift; $x = $class->new($x) unless ref $x;
+ return ($x->{sign} =~ /^[\+]/);
+ }
+
+sub is_negative
+ {
+ # return true when arg (BINT or num_str) is negative (< 0)
+ my $x = shift; $x = $class->new($x) unless ref $x;
+ return ($x->{sign} =~ /^[\-]/);
}
+###############################################################################
+
sub bmul
{
# multiply two numbers -- stolen from Knuth Vol 2 pg 233
# (BINT or num_str, BINT or num_str) return BINT
my ($self,$x,$y,$a,$p,$r) = objectify(2,@_);
- #print "$self bmul $x ",ref($x)," $y ",ref($y),"\n";
- trace(@_);
+
return $x if $x->modify('bmul');
- return $x->bnan() if (($x->{sign} eq $nan) || ($y->{sign} eq $nan));
+ return $x->bnan() if (($x->{sign} !~ /^[+-]$/) || ($y->{sign} !~ /^[+-]$/));
- mul($x,$y); # do actual math
+ return $x->bzero() if $x->is_zero() || $y->is_zero(); # handle result = 0
+ $x->{sign} = $x->{sign} eq $y->{sign} ? '+' : '-'; # +1 * +1 or -1 * -1 => +
+ $CALC->_mul($x->{value},$y->{value}); # do actual math
return $x->round($a,$p,$r,$y);
}
{
# (dividend: BINT or num_str, divisor: BINT or num_str) return
# (BINT,BINT) (quo,rem) or BINT (only rem)
- trace(@_);
my ($self,$x,$y,$a,$p,$r) = objectify(2,@_);
return $x if $x->modify('bdiv');
+ # 5 / 0 => +inf, -6 / 0 => -inf (0 /0 => 1 or +inf?)
+ #return wantarray
+ # ? ($x->binf($x->{sign}),binf($x->{sign})) : $x->binf($x->{sign})
+ # if ($x->{sign} =~ /^[+-]$/ && $y->is_zero());
+
# NaN?
return wantarray ? ($x->bnan(),bnan()) : $x->bnan()
- if ($x->{sign} eq $nan || $y->{sign} eq $nan || $y->is_zero());
+ if ($x->{sign} !~ /^[+-]$/ || $y->{sign} !~ /^[+-]$/ || $y->is_zero());
# 0 / something
return wantarray ? ($x,$self->bzero()) : $x if $x->is_zero();
# Is $x in the interval [0, $y) ?
- my $cmp = acmp($x->{value},$y->{value});
+ my $cmp = $CALC->_acmp($x->{value},$y->{value});
if (($cmp < 0) and ($x->{sign} eq $y->{sign}))
{
return $x->bzero() unless wantarray;
# calc new sign and in case $y == +/- 1, return $x
$x->{sign} = ($x->{sign} ne $y->{sign} ? '-' : '+');
# check for / +-1 (cant use $y->is_one due to '-'
- if ((@{$y->{value}} == 1) && ($y->{value}->[0] == 1))
+ if (($y == 1) || ($y == -1)) # slow!
+ #if ((@{$y->{value}} == 1) && ($y->{value}->[0] == 1))
{
return wantarray ? ($x,$self->bzero()) : $x;
}
# call div here
my $rem = $self->bzero();
$rem->{sign} = $y->{sign};
- ($x->{value},$rem->{value}) = div($x->{value},$y->{value});
+ #($x->{value},$rem->{value}) = div($x->{value},$y->{value});
+ ($x->{value},$rem->{value}) = $CALC->_div($x->{value},$y->{value});
# do not leave rest "-0";
- $rem->{sign} = '+' if (@{$rem->{value}} == 1) && ($rem->{value}->[0] == 0);
+ # $rem->{sign} = '+' if (@{$rem->{value}} == 1) && ($rem->{value}->[0] == 0);
+ $rem->{sign} = '+' if $CALC->_is_zero($rem->{value});
if (($x->{sign} eq '-') and (!$rem->is_zero()))
{
$x->bdec();
# (BINT or num_str, BINT or num_str) return BINT
# compute power of two numbers -- stolen from Knuth Vol 2 pg 233
# modifies first argument
- #trace(@_);
my ($self,$x,$y,$a,$p,$r) = objectify(2,@_);
return $x if $x->modify('bpow');
+ return $x if $x->{sign} =~ /^[+-]inf$/; # -inf/+inf ** x
return $x->bnan() if $x->{sign} eq $nan || $y->{sign} eq $nan;
return $x->_one() if $y->is_zero();
return $x if $x->is_one() || $y->is_one();
- if ($x->{sign} eq '-' && @{$x->{value}} == 1 && $x->{value}->[0] == 1)
+ #if ($x->{sign} eq '-' && @{$x->{value}} == 1 && $x->{value}->[0] == 1)
+ if ($x->{sign} eq '-' && $CALC->_is_one($x->{value}))
{
# if $x == -1 and odd/even y => +1/-1
- return $y->is_odd() ? $x : $x->_set(1); # $x->babs() would work to
- # my Casio FX-5500L has here a bug, -1 ** 2 is -1, but -1 * -1 is 1 LOL
- }
- # shortcut for $x ** 2
- if ($y->{sign} eq '+' && @{$y->{value}} == 1 && $y->{value}->[0] == 2)
- {
- return $x->bmul($x)->bround($a,$p,$r);
+ return $y->is_odd() ? $x : $x->babs();
+ # my Casio FX-5500L has here a bug, -1 ** 2 is -1, but -1 * -1 is 1; LOL
}
# 1 ** -y => 1 / (1**y), so do test for negative $y after above's clause
return $x->bnan() if $y->{sign} eq '-';
return $x if $x->is_zero(); # 0**y => 0 (if not y <= 0)
- # tels: 10**x is special (actually 100**x etc is special, too) but not here
- #if ((@{$x->{value}} == 1) && ($x->{value}->[0] == 10))
- # {
- # # 10**2
- # my $yi = int($y); my $yi5 = int($yi/5);
- # $x->{value} = [];
- # my $v = $x->{value};
- # if ($yi5 > 0)
- # {
- # # $x->{value}->[$yi5-1] = 0; # pre-padd array (no use)
- # for (my $i = 0; $i < $yi5; $i++)
- # {
- # $v->[$i] = 0;
- # }
- # }
- # push @{$v}, int( '1'.'0' x ($yi % 5));
- # if ($x->{sign} eq '-')
- # {
- # $x->{sign} = $y->is_odd() ? '-' : '+'; # -10**2 = 100, -10**3 = -1000
- # }
- # return $x;
- # }
-
- # based on the assumption that shifting in base 10 is fast, and that bpow()
- # works faster if numbers are small: we count trailing zeros (this step is
- # O(1)..O(N), but in case of O(N) we save much more time), stripping them
- # out of the multiplication, and add $count * $y zeros afterwards:
- # 300 ** 3 == 300*300*300 == 3*3*3 . '0' x 2 * 3 == 27 . '0' x 6
- my $zeros = $x->_trailing_zeros();
- if ($zeros > 0)
+ if ($CALC->can('_pow'))
{
- $x->brsft($zeros,10); # remove zeros
- $x->bpow($y); # recursion (will not branch into here again)
- $zeros = $y * $zeros; # real number of zeros to add
- $x->blsft($zeros,10);
- return $x;
+ $CALC->_pow($x->{value},$y->{value});
+ return $x->round($a,$p,$r);
}
+ # based on the assumption that shifting in base 10 is fast, and that mul
+ # works faster if numbers are small: we count trailing zeros (this step is
+ # O(1)..O(N), but in case of O(N) we save much more time due to this),
+ # stripping them out of the multiplication, and add $count * $y zeros
+ # afterwards like this:
+ # 300 ** 3 == 300*300*300 == 3*3*3 . '0' x 2 * 3 == 27 . '0' x 6
+ # creates deep recursion?
+ #my $zeros = $x->_trailing_zeros();
+ #if ($zeros > 0)
+ # {
+ # $x->brsft($zeros,10); # remove zeros
+ # $x->bpow($y); # recursion (will not branch into here again)
+ # $zeros = $y * $zeros; # real number of zeros to add
+ # $x->blsft($zeros,10);
+ # return $x->round($a,$p,$r);
+ # }
my $pow2 = $self->_one();
my $y1 = $class->new($y);
($y1,$res)=&bdiv($y1,2);
if (!$res->is_zero()) { &bmul($pow2,$x); }
if (!$y1->is_zero()) { &bmul($x,$x); }
+ #print "$x $y\n";
}
#print "bpow: e p2: $pow2 x: $x y: $y1 r: $res\n";
&bmul($x,$pow2) if (!$pow2->is_one());
$n = 2 if !defined $n; return $x if $n == 0;
return $x->bnan() if $n < 0 || $y->{sign} eq '-';
- if ($n != 10)
- {
+ #if ($n != 10)
+ # {
$x->bmul( $self->bpow($n, $y) );
- }
- else
- {
- # shortcut (faster) for shifting by 10) since we are in base 10eX
- # multiples of 5:
- my $src = scalar @{$x->{value}}; # source
- my $len = $y->numify(); # shift-len as normal int
- my $rem = $len % 5; # reminder to shift
- my $dst = $src + int($len/5); # destination
-
- my $v = $x->{value}; # speed-up
- my $vd; # further speedup
- #print "src $src:",$v->[$src]||0," dst $dst:",$v->[$dst]||0," rem $rem\n";
- $v->[$src] = 0; # avoid first ||0 for speed
- while ($src >= 0)
- {
- $vd = $v->[$src]; $vd = '00000'.$vd;
- #print "s $src d $dst '$vd' ";
- $vd = substr($vd,-5+$rem,5-$rem);
- #print "'$vd' ";
- $vd .= $src > 0 ? substr('00000'.$v->[$src-1],-5,$rem) : '0' x $rem;
- #print "'$vd' ";
- $vd = substr($vd,-5,5) if length($vd) > 5;
- #print "'$vd'\n";
- $v->[$dst] = int($vd);
- $dst--; $src--;
- }
- # set lowest parts to 0
- while ($dst >= 0) { $v->[$dst--] = 0; }
- # fix spurios last zero element
- splice @$v,-1 if $v->[-1] == 0;
- #print "elems: "; my $i = 0;
- #foreach (reverse @$v) { print "$i $_ "; $i++; } print "\n";
- # old way: $x->bmul( $self->bpow($n, $y) );
- }
+ # }
+ #else
+ # {
+ # # shortcut (faster) for shifting by 10) since we are in base 10eX
+ # # multiples of 5:
+ # my $src = scalar @{$x->{value}}; # source
+ # my $len = $y->numify(); # shift-len as normal int
+ # my $rem = $len % 5; # reminder to shift
+ # my $dst = $src + int($len/5); # destination
+ #
+ # my $v = $x->{value}; # speed-up
+ # my $vd; # further speedup
+ # #print "src $src:",$v->[$src]||0," dst $dst:",$v->[$dst]||0," rem $rem\n";
+ # $v->[$src] = 0; # avoid first ||0 for speed
+ # while ($src >= 0)
+ # {
+ # $vd = $v->[$src]; $vd = '00000'.$vd;
+ # #print "s $src d $dst '$vd' ";
+ # $vd = substr($vd,-5+$rem,5-$rem);
+ # #print "'$vd' ";
+ # $vd .= $src > 0 ? substr('00000'.$v->[$src-1],-5,$rem) : '0' x $rem;
+ # #print "'$vd' ";
+ # $vd = substr($vd,-5,5) if length($vd) > 5;
+ # #print "'$vd'\n";
+ # $v->[$dst] = int($vd);
+ # $dst--; $src--;
+ # }
+ # # set lowest parts to 0
+ # while ($dst >= 0) { $v->[$dst--] = 0; }
+ # # fix spurios last zero element
+ # splice @$v,-1 if $v->[-1] == 0;
+ # #print "elems: "; my $i = 0;
+ # #foreach (reverse @$v) { print "$i $_ "; $i++; } print "\n";
+ # # old way: $x->bmul( $self->bpow($n, $y) );
+ # }
return $x;
}
return $x->bnan() if ($x->{sign} !~ /^[+-]$/ || $y->{sign} !~ /^[+-]$/);
$n = 2 if !defined $n; return $x->bnan() if $n <= 0 || $y->{sign} eq '-';
- if ($n != 10)
- {
+ #if ($n != 10)
+ # {
scalar bdiv($x, $self->bpow($n, $y));
- }
- else
- {
- # shortcut (faster) for shifting by 10)
- # multiples of 5:
- my $dst = 0; # destination
- my $src = $y->numify(); # as normal int
- my $rem = $src % 5; # reminder to shift
- $src = int($src / 5); # source
- my $len = scalar @{$x->{value}} - $src; # elems to go
- my $v = $x->{value}; # speed-up
- if ($rem == 0)
- {
- splice (@$v,0,$src); # even faster, 38.4 => 39.3
- }
- else
- {
- my $vd;
- $v->[scalar @$v] = 0; # avoid || 0 test inside loop
- while ($dst < $len)
- {
- $vd = '00000'.$v->[$src];
- #print "$dst $src '$vd' ";
- $vd = substr($vd,-5,5-$rem);
- #print "'$vd' ";
- $src++;
- $vd = substr('00000'.$v->[$src],-$rem,$rem) . $vd;
- #print "'$vd1' ";
- #print "'$vd'\n";
- $vd = substr($vd,-5,5) if length($vd) > 5;
- $v->[$dst] = int($vd);
- $dst++;
- }
- splice (@$v,$dst) if $dst > 0; # kill left-over array elems
- pop @$v if $v->[-1] == 0; # kill last element
- } # else rem == 0
- # old way: scalar bdiv($x, $self->bpow($n, $y));
- }
+ # }
+ #else
+ # {
+ # # shortcut (faster) for shifting by 10)
+ # # multiples of 5:
+ # my $dst = 0; # destination
+ # my $src = $y->numify(); # as normal int
+ # my $rem = $src % 5; # reminder to shift
+ # $src = int($src / 5); # source
+ # my $len = scalar @{$x->{value}} - $src; # elems to go
+ # my $v = $x->{value}; # speed-up
+ # if ($rem == 0)
+ # {
+ # splice (@$v,0,$src); # even faster, 38.4 => 39.3
+ # }
+ # else
+ # {
+ # my $vd;
+ # $v->[scalar @$v] = 0; # avoid || 0 test inside loop
+ # while ($dst < $len)
+ # {
+ # $vd = '00000'.$v->[$src];
+ # #print "$dst $src '$vd' ";
+ # $vd = substr($vd,-5,5-$rem);
+ # #print "'$vd' ";
+ # $src++;
+ # $vd = substr('00000'.$v->[$src],-$rem,$rem) . $vd;
+ # #print "'$vd1' ";
+ # #print "'$vd'\n";
+ # $vd = substr($vd,-5,5) if length($vd) > 5;
+ # $v->[$dst] = int($vd);
+ # $dst++;
+ # }
+ # splice (@$v,$dst) if $dst > 0; # kill left-over array elems
+ # pop @$v if $v->[-1] == 0; # kill last element
+ # } # else rem == 0
+ # # old way: scalar bdiv($x, $self->bpow($n, $y));
+ # }
return $x;
}
{
#(BINT or num_str, BINT or num_str) return BINT
# compute x & y
- trace(@_);
- my ($self,$x,$y) = objectify(2,@_);
+ my ($self,$x,$y,$a,$p,$r) = objectify(2,@_);
return $x if $x->modify('band');
return $x->bnan() if ($x->{sign} !~ /^[+-]$/ || $y->{sign} !~ /^[+-]$/);
return $x->bzero() if $y->is_zero();
- my $r = $self->bzero(); my $m = new Math::BigInt 1; my ($xr,$yr);
+
+ if ($CALC->can('_and'))
+ {
+ $CALC->_and($x->{value},$y->{value});
+ return $x->round($a,$p,$r);
+ }
+
+ my $m = new Math::BigInt 1; my ($xr,$yr);
my $x10000 = new Math::BigInt (0x10000);
my $y1 = copy(ref($x),$y); # make copy
- while (!$x->is_zero() && !$y1->is_zero())
+ my $x1 = $x->copy(); $x->bzero(); # modify x in place!
+ while (!$x1->is_zero() && !$y1->is_zero())
{
- ($x, $xr) = bdiv($x, $x10000);
+ ($x1, $xr) = bdiv($x1, $x10000);
($y1, $yr) = bdiv($y1, $x10000);
- $r->badd( bmul( new Math::BigInt ( $xr->numify() & $yr->numify()), $m ));
+ #print ref($xr), " $xr ", $xr->numify(),"\n";
+ #print ref($yr), " $yr ", $yr->numify(),"\n";
+ #print "res: ",$yr->numify() & $xr->numify(),"\n";
+ my $u = bmul( $class->new( $xr->numify() & $yr->numify() ), $m);
+ #print "res: $u\n";
+ $x->badd( bmul( $class->new( $xr->numify() & $yr->numify() ), $m));
$m->bmul($x10000);
}
- $x = $r;
+ return $x->round($a,$p,$r);
}
sub bior
{
#(BINT or num_str, BINT or num_str) return BINT
# compute x | y
- trace(@_);
- my ($self,$x,$y) = objectify(2,@_);
+ my ($self,$x,$y,$a,$p,$r) = objectify(2,@_);
return $x if $x->modify('bior');
return $x->bnan() if ($x->{sign} !~ /^[+-]$/ || $y->{sign} !~ /^[+-]$/);
return $x if $y->is_zero();
- my $r = $self->bzero(); my $m = new Math::BigInt 1; my ($xr,$yr);
+ if ($CALC->can('_or'))
+ {
+ $CALC->_or($x->{value},$y->{value});
+ return $x->round($a,$p,$r);
+ }
+
+ my $m = new Math::BigInt 1; my ($xr,$yr);
my $x10000 = new Math::BigInt (0x10000);
my $y1 = copy(ref($x),$y); # make copy
- while (!$x->is_zero() || !$y1->is_zero())
+ my $x1 = $x->copy(); $x->bzero(); # modify x in place!
+ while (!$x1->is_zero() || !$y1->is_zero())
{
- ($x, $xr) = bdiv($x,$x10000);
+ ($x1, $xr) = bdiv($x1,$x10000);
($y1, $yr) = bdiv($y1,$x10000);
- $r->badd( bmul( new Math::BigInt ( $xr->numify() | $yr->numify()), $m ));
+ $x->badd( bmul( $class->new( $xr->numify() | $yr->numify() ), $m));
$m->bmul($x10000);
}
- $x = $r;
+ return $x->round($a,$p,$r);
}
sub bxor
{
#(BINT or num_str, BINT or num_str) return BINT
# compute x ^ y
- my ($self,$x,$y) = objectify(2,@_);
+ my ($self,$x,$y,$a,$p,$r) = objectify(2,@_);
return $x if $x->modify('bxor');
- return $x->bnan() if ($x->{sign} eq $nan || $y->{sign} eq $nan);
+ return $x->bnan() if ($x->{sign} !~ /^[+-]$/ || $y->{sign} !~ /^[+-]$/);
return $x if $y->is_zero();
return $x->bzero() if $x == $y; # shortcut
- my $r = $self->bzero(); my $m = new Math::BigInt 1; my ($xr,$yr);
+
+ if ($CALC->can('_xor'))
+ {
+ $CALC->_xor($x->{value},$y->{value});
+ return $x->round($a,$p,$r);
+ }
+
+ my $m = new Math::BigInt 1; my ($xr,$yr);
my $x10000 = new Math::BigInt (0x10000);
my $y1 = copy(ref($x),$y); # make copy
- while (!$x->is_zero() || !$y1->is_zero())
+ my $x1 = $x->copy(); $x->bzero(); # modify x in place!
+ while (!$x1->is_zero() || !$y1->is_zero())
{
- ($x, $xr) = bdiv($x, $x10000);
+ ($x1, $xr) = bdiv($x1, $x10000);
($y1, $yr) = bdiv($y1, $x10000);
- $r->badd( bmul( new Math::BigInt ( $xr->numify() ^ $yr->numify()), $m ));
+ $x->badd( bmul( $class->new( $xr->numify() ^ $yr->numify() ), $m));
$m->bmul($x10000);
}
- $x = $r;
+ return $x->round($a,$p,$r);
}
sub length
{
my ($self,$x) = objectify(1,@_);
- return (_digits($x->{value}), 0) if wantarray;
- _digits($x->{value});
+ my $e = $CALC->_len($x->{value});
+ # # fallback, since we do not know the underlying representation
+ #my $es = "$x"; my $c = 0; $c = 1 if $es =~ /^[+-]/; # if lib returns '+123'
+ #my $e = CORE::length($es)-$c;
+ return wantarray ? ($e,0) : $e;
}
sub digit
{
- # return the nth digit, negative values count backward
+ # return the nth decimal digit, negative values count backward, 0 is right
my $x = shift;
my $n = shift || 0;
- my $len = $x->length();
-
- $n = $len+$n if $n < 0; # -1 last, -2 second-to-last
- $n = abs($n); # if negatives are to big
- $len--; $n = $len if $n > $len; # n to big?
-
- my $elem = int($n / 5); # which array element
- my $digit = $n % 5; # which digit in this element
- $elem = '0000'.$x->{value}->[$elem]; # get element padded with 0's
- return substr($elem,-$digit-1,1);
+ return $CALC->_digit($x->{value},$n);
}
sub _trailing_zeros
my $x = shift;
$x = $class->new($x) unless ref $x;
- return 0 if $x->is_zero() || $x->is_nan();
- # check each array elem in _m for having 0 at end as long as elem == 0
- # Upon finding a elem != 0, stop
- my $zeros = 0; my $elem;
- foreach my $e (@{$x->{value}})
- {
- if ($e != 0)
- {
- $elem = "$e"; # preserve x
- $elem =~ s/.*?(0*$)/$1/; # strip anything not zero
- $zeros *= 5; # elems * 5
- $zeros += CORE::length($elem); # count trailing zeros
- last; # early out
- }
- $zeros ++; # real else branch: 50% slower!
- }
- return $zeros;
+ return 0 if $x->is_zero() || $x->is_nan() || $x->is_inf();
+
+ return $CALC->_zeros($x->{value}) if $CALC->can('_zeros');
+
+ # if not: since we do not know underlying internal represantation:
+ my $es = "$x"; $es =~ /([0]*)$/;
+
+ return 0 if !defined $1; # no zeros
+ return CORE::length("$1"); # as string, not as +0!
}
sub bsqrt
{
my $x = shift;
my $pad = shift;
+ my $xs = shift;
my $len = $x->length();
return 0 if $len == 1; # '5' is trailed by invisible zeros
my $follow = $pad - 1;
return 0 if $follow > $len || $follow < 1;
#print "checking $x $r\n";
- # old, slow way checking string for non-zero characters
+
+ # since we do not know underlying represantion of $x, use decimal string
+ #my $r = substr ($$xs,-$follow);
my $r = substr ("$x",-$follow);
return 1 if $r =~ /[^0]/; return 0;
-
- # faster way checking array contents; it is actually not faster (even in a
- # rounding-only-shoutout, so I leave the simpler code in)
- #my $rem = $follow % 5; my $div = $follow / 5; my $v = $x->{value};
- # pad with zeros and extract
- #print "last part : ",'00000'.$v->[$div]," $rem = '";
- #print substr('00000'.$v->[$div],-$rem,5),"'\n";
- #my $r1 = substr ('00000'.$v->[$div],-$rem,5);
- #print "$r1\n";
- #return 1 if $r1 =~ /[^0]/;
- #
- #for (my $j = $div-1; $j >= 0; $j --)
- # {
- # #print "part $v->[$j]\n";
- # return 1 if $v->[$j] != 0;
- # }
- #return 0;
}
sub fround
my ($pad,$digit_round,$digit_after);
$pad = $len - $scale;
$pad = abs($scale)+1 if $scale < 0;
- $digit_round = '0'; $digit_round = $x->digit($pad) if $pad < $len;
- $digit_after = '0'; $digit_after = $x->digit($pad-1) if $pad > 0;
- # print "r $x: pos:$pad l:$len s:$scale r:$digit_round a:$digit_after m: $mode\n";
+ # do not use digit(), it is costly for binary => decimal
+ #$digit_round = '0'; $digit_round = $x->digit($pad) if $pad < $len;
+ #$digit_after = '0'; $digit_after = $x->digit($pad-1) if $pad > 0;
+ my $xs = $CALC->_str($x->{value});
+ my $pl = -$pad-1;
+ # pad: 123: 0 => -1, at 1 => -2, at 2 => -3, at 3 => -4
+ # pad+1: 123: 0 => 0, at 1 => -1, at 2 => -2, at 3 => -3
+ $digit_round = '0'; $digit_round = substr($$xs,$pl,1) if $pad <= $len;
+ $pl++; $pl ++ if $pad >= $len;
+ $digit_after = '0'; $digit_after = substr($$xs,$pl,1)
+ if $pad > 0;
+
+ #my $d_round = '0'; $d_round = $x->digit($pad) if $pad < $len;
+ #my $d_after = '0'; $d_after = $x->digit($pad-1) if $pad > 0;
+ # print "$pad $pl $$xs $digit_round:$d_round $digit_after:$d_after\n";
# in case of 01234 we round down, for 6789 up, and only in case 5 we look
# closer at the remaining digits of the original $x, remember decision
($digit_after =~ /[01234]/) || # round down anyway,
# 6789 => round up
($digit_after eq '5') && # not 5000...0000
- ($x->_scan_for_nonzero($pad) == 0) &&
+ ($x->_scan_for_nonzero($pad,$xs) == 0) &&
(
($mode eq 'even') && ($digit_round =~ /[24680]/) ||
($mode eq 'odd') && ($digit_round =~ /[13579]/) ||
# this is triggering warnings, and buggy for $scale < 0
#if (-$scale != $len)
{
- # split mantissa at $scale and then pad with zeros
- my $s5 = int($pad / 5);
- my $i = 0;
- while ($i < $s5)
+ # old code, depend on internal represantation
+ # split mantissa at $pad and then pad with zeros
+ #my $s5 = int($pad / 5);
+ #my $i = 0;
+ #while ($i < $s5)
+ # {
+ # $x->{value}->[$i++] = 0; # replace with 5 x 0
+ # }
+ #$x->{value}->[$s5] = '00000'.$x->{value}->[$s5]; # pad with 0
+ #my $rem = $pad % 5; # so much left over
+ #if ($rem > 0)
+ # {
+ # #print "remainder $rem\n";
+ ## #print "elem $x->{value}->[$s5]\n";
+ # substr($x->{value}->[$s5],-$rem,$rem) = '0' x $rem; # stamp w/ '0'
+ # }
+ #$x->{value}->[$s5] = int ($x->{value}->[$s5]); # str '05' => int '5'
+ #print ${$CALC->_str($pad->{value})}," $len\n";
+ if (($pad > 0) && ($pad <= $len))
{
- $x->{value}->[$i++] = 0; # replace with 5 x 0
+ substr($$xs,-$pad,$pad) = '0' x $pad;
+ $x->{value} = $CALC->_new($xs); # put back in
}
- $x->{value}->[$s5] = '00000'.$x->{value}->[$s5]; # pad with 0
- my $rem = $pad % 5; # so much left over
- if ($rem > 0)
+ elsif ($pad > $len)
{
- #print "remainder $rem\n";
- #print "elem $x->{value}->[$s5]\n";
- substr($x->{value}->[$s5],-$rem,$rem) = '0' x $rem; # stamp w/ '0'
+ $x->{value} = $CALC->_zero(); # round to '0'
}
- $x->{value}->[$s5] = int ($x->{value}->[$s5]); # str '05' => int '5'
+ #print "res $$xs\n";
}
+ # move this later on after the inc of the string
+ #$x->{value} = $CALC->_new($xs); # put back in
if ($round_up) # what gave test above?
{
$pad = $len if $scale < 0; # tlr: whack 0.51=>1.0
# modify $x in place, undef, undef to avoid rounding
- $x->badd( Math::BigInt->new($x->{sign}.'1'.'0'x$pad),
- undef,undef );
# str creation much faster than 10 ** something
+ $x->badd( Math::BigInt->new($x->{sign}.'1'.'0'x$pad) );
+ # increment string in place, to avoid dec=>hex for the '1000...000'
+ # $xs ...blah foo
}
+ # to here:
+ #$x->{value} = $CALC->_new($xs); # put back in
$x;
}
##############################################################################
# private stuff (internal use only)
-sub trace
- {
- # print out a number without using bstr (avoid deep recurse) for trace/debug
- return unless $trace;
-
- my ($package,$file,$line,$sub) = caller(1);
- print "'$sub' called from '$package' line $line:\n ";
-
- foreach my $x (@_)
- {
- if (!defined $x)
- {
- print "undef, "; next;
- }
- if (!ref($x))
- {
- print "'$x' "; next;
- }
- next if (ref($x) ne "HASH");
- print "$x->{sign} ";
- foreach (@{$x->{value}})
- {
- print "$_ ";
- }
- print ", ";
- }
- print "\n";
- }
-
-sub _set
- {
- # internal set routine to set X fast to an integer value < [+-]100000
- my $self = shift;
- my $wanted = shift || 0;
-
- $self->{sign} = $nan, return if $wanted !~ /^[+-]?[0-9]+$/;
- $self->{sign} = '-'; $self->{sign} = '+' if $wanted >= 0;
- $self->{value} = [ abs($wanted) ];
- return $self;
- }
-
sub _one
{
# internal speedup, set argument to 1, or create a +/- 1
my $self = shift;
- my $x = $self->bzero(); $x->{value} = [ 1 ]; $x->{sign} = shift || '+'; $x;
+ #my $x = $self->bzero(); $x->{value} = [ 1 ]; $x->{sign} = shift || '+'; $x;
+ my $x = $self->bzero(); $x->{value} = $CALC->_one();
+ $x->{sign} = shift || '+';
+ return $x;
}
sub _swap
# badd($class,1) is not supported (it should, eventually, try to add undef)
# currently it tries 'Math::BigInt' + 1, which will not work.
- trace(@_);
my $count = abs(shift || 0);
#print caller(),"\n";
{
my $self = shift;
#print "import $self @_\n";
- for ( my $i = 0; $i < @_ ; $i++ )
+ my @a = @_; my $l = scalar @_; my $j = 0;
+ for ( my $i = 0; $i < $l ; $i++,$j++ )
{
- if ( $_[$i] eq ':constant' )
+ if ($_[$i] eq ':constant')
{
- # this rest causes overlord er load to step in
+ # this causes overlord er load to step in
overload::constant integer => sub { $self->new(shift) };
- splice @_, $i, 1; last;
+ splice @a, $j, 1; $j --;
+ }
+ elsif ($_[$i] =~ /^lib$/i)
+ {
+ # this causes a different low lib to take care...
+ $CALC = $_[$i+1] || $CALC;
+ my $s = 2; $s = 1 if @a-$j < 2; # avoid "can not modify non-existant..."
+ splice @a, $j, $s; $j -= $s;
}
}
# any non :constant stuff is handled by our parent, Exporter
# even if @_ is empty, to give it a chance
- #$self->SUPER::import(@_); # does not work
- $self->export_to_level(1,$self,@_); # need this instead
- }
+ #$self->SUPER::import(@a); # does not work
+ $self->export_to_level(1,$self,@a); # need this instead
-sub _internal
- {
- # (ref to self, ref to string) return ref to num_array
- # Convert a number from string format to internal base 100000 format.
- # Assumes normalized value as input.
- my ($s,$d) = @_;
- my $il = CORE::length($$d)-1;
- # these leaves '00000' instead of int 0 and will be corrected after any op
- $s->{value} = [ reverse(unpack("a" . ($il%5+1) . ("a5" x ($il/5)), $$d)) ];
- $s;
+ # load core math lib
+ $CALC = 'Math::BigInt::'.$CALC if $CALC !~ /^Math::BigInt/i;
+ my $c = $CALC; $c =~ s/::/\//g; $c .= '.pm' if $c !~ /\.pm$/;
+ require $c;
}
sub _strip_zeros
{
# internal normalization function that strips leading zeros from the array
# args: ref to array
- #trace(@_);
my $s = shift;
my $cnt = scalar @$s; # get count of parts
my $x = Math::BigInt->bzero();
return $x->bnan() if $$hs !~ /^[\-\+]?0x[0-9A-Fa-f]+$/;
- my $mul = Math::BigInt->bzero(); $mul++;
- my $x65536 = Math::BigInt->new(65536);
+ my $sign = '+'; $sign = '-' if ($$hs =~ /^\-/);
- my $len = CORE::length($$hs)-2; my $sign = '+';
- if ($$hs =~ /^\-/)
+ $$hs =~ s/^[+-]?//; # strip sign
+ if ($CALC->can('_from_hex'))
{
- $sign = '-'; $len--;
+ $x->{value} = $CALC->_from_hex($hs);
}
- $len = int($len/4); # 4-digit parts, w/o '0x'
- my $val; my $i = -4;
- while ($len >= 0)
+ else
{
- $val = substr($$hs,$i,4);
- $val =~ s/^[\-\+]?0x// if $len == 0; # for last part only because
- $val = hex($val); # hex does not like wrong chars
- # print "$val ",substr($$hs,$i,4),"\n";
- $i -= 4; $len --;
- $x += $mul * $val if $val != 0;
- $mul *= $x65536 if $len >= 0; # skip last mul
+ # fallback to pure perl
+ my $mul = Math::BigInt->bzero(); $mul++;
+ my $x65536 = Math::BigInt->new(65536);
+ my $len = CORE::length($$hs)-2;
+ $len = int($len/4); # 4-digit parts, w/o '0x'
+ my $val; my $i = -4;
+ while ($len >= 0)
+ {
+ $val = substr($$hs,$i,4);
+ $val =~ s/^[\-\+]?0x// if $len == 0; # for last part only because
+ $val = hex($val); # hex does not like wrong chars
+ # print "$val ",substr($$hs,$i,4),"\n";
+ $i -= 4; $len --;
+ $x += $mul * $val if $val != 0;
+ $mul *= $x65536 if $len >= 0; # skip last mul
+ }
}
- $x->{sign} = $sign if !$x->is_zero();
+ $x->{sign} = $sign if !$x->is_zero(); # no '-0'
return $x;
}
my $mul = Math::BigInt->bzero(); $mul++;
my $x256 = Math::BigInt->new(256);
- my $len = CORE::length($$bs)-2; my $sign = '+';
- if ($$bs =~ /^\-/)
+ my $sign = '+'; $sign = '-' if ($$bs =~ /^\-/);
+ $$bs =~ s/^[+-]?//; # strip sign
+ if ($CALC->can('_from_bin'))
{
- $sign = '-'; $len--;
+ $x->{value} = $CALC->_from_bin($bs);
}
- $len = int($len/8); # 8-digit parts, w/o '0b'
- my $val; my $i = -8;
- while ($len >= 0)
+ else
{
- $val = substr($$bs,$i,8);
- $val =~ s/^[\-\+]?0b// if $len == 0; # for last part only
- #$val = oct('0b'.$val); # does not work on Perl prior 5.6.0
- $val = ('0' x (8-CORE::length($val))).$val if CORE::length($val) < 8;
- $val = ord(pack('B8',$val));
- # print "$val ",substr($$bs,$i,16),"\n";
- $i -= 8; $len --;
- $x += $mul * $val if $val != 0;
- $mul *= $x256 if $len >= 0; # skip last mul
+ my $len = CORE::length($$bs)-2;
+ $len = int($len/8); # 8-digit parts, w/o '0b'
+ my $val; my $i = -8;
+ while ($len >= 0)
+ {
+ $val = substr($$bs,$i,8);
+ $val =~ s/^[\-\+]?0b// if $len == 0; # for last part only
+ #$val = oct('0b'.$val); # does not work on Perl prior 5.6.0
+ $val = ('0' x (8-CORE::length($val))).$val if CORE::length($val) < 8;
+ $val = ord(pack('B8',$val));
+ # print "$val ",substr($$bs,$i,16),"\n";
+ $i -= 8; $len --;
+ $x += $mul * $val if $val != 0;
+ $mul *= $x256 if $len >= 0; # skip last mul
+ }
}
$x->{sign} = $sign if !$x->is_zero();
return $x;
if ($mi =~ /^([+-]?)0*(\d+)$/) # strip leading zeros
{
$mis = $1||'+'; $miv = $2;
- #print "$mis $miv";
+ # print "$mis $miv";
# valid, existing fraction part of mantissa?
return unless ($mf =~ /^(\d*?)0*$/); # strip trailing zeros
$mfv = $1;
return; # NaN, not a number
}
-sub _digits
- {
- # computer number of digits in bigint, minus the sign
- # int() because add/sub leaves sometimes strings (like '00005') instead of
- # int ('5') in this place, causing length to fail
- my $cx = shift;
-
- #print "len: ",(@$cx-1)*5+CORE::length(int($cx->[-1])),"\n";
- return (@$cx-1)*5+CORE::length(int($cx->[-1]));
- }
-
sub as_number
{
# an object might be asked to return itself as bigint on certain overloaded
}
##############################################################################
-# internal calculation routines
-
-sub acmp
- {
- # internal absolute post-normalized compare (ignore signs)
- # ref to array, ref to array, return <0, 0, >0
- # arrays must have at least on entry, this is not checked for
-
- my ($cx, $cy) = @_;
-
- #print "$cx $cy\n";
- my ($i,$a,$x,$y,$k);
- # calculate length based on digits, not parts
- $x = _digits($cx); $y = _digits($cy);
- # print "length: ",($x-$y),"\n";
- return $x-$y if ($x - $y); # if different in length
- #print "full compare\n";
- $i = 0; $a = 0;
- # first way takes 5.49 sec instead of 4.87, but has the early out advantage
- # so grep is slightly faster, but more unflexible. hm. $_ instead if $k
- # yields 5.6 instead of 5.5 sec huh?
- # manual way (abort if unequal, good for early ne)
- my $j = scalar @$cx - 1;
- while ($j >= 0)
- {
- # print "$cx->[$j] $cy->[$j] $a",$cx->[$j]-$cy->[$j],"\n";
- last if ($a = $cx->[$j] - $cy->[$j]); $j--;
- }
- return $a;
- # while it early aborts, it is even slower than the manual variant
- #grep { return $a if ($a = $_ - $cy->[$i++]); } @$cx;
- # grep way, go trough all (bad for early ne)
- #grep { $a = $_ - $cy->[$i++]; } @$cx;
- #return $a;
- }
+# internal calculation routines (others are in Math::BigInt::Calc etc)
sub cmp
{
# post-normalized compare for internal use (honors signs)
- # ref to array, ref to array, return < 0, 0, >0
+ # input: ref to value, ref to value, sign, sign
+ # output: <0, 0, >0
my ($cx,$cy,$sx,$sy) = @_;
- #return 0 if (is0($cx,$sx) && is0($cy,$sy));
-
if ($sx eq '+')
{
return 1 if $sy eq '-'; # 0 check handled above
- return acmp($cx,$cy);
+ #return acmp($cx,$cy);
+ return $CALC->_acmp($cx,$cy);
}
else
{
# $sx eq '-'
- return -1 if ($sy eq '+');
- return acmp($cy,$cx);
+ return -1 if $sy eq '+';
+ #return acmp($cy,$cx);
+ return $CALC->_acmp($cy,$cx);
}
return 0; # equal
}
-sub add
- {
- # (ref to int_num_array, ref to int_num_array)
- # routine to add two base 1e5 numbers
- # stolen from Knuth Vol 2 Algorithm A pg 231
- # there are separate routines to add and sub as per Kunth pg 233
- # This routine clobbers up array x, but not y.
-
- my ($x,$y) = @_;
-
- # for each in Y, add Y to X and carry. If after that, something is left in
- # X, foreach in X add carry to X and then return X, carry
- # Trades one "$j++" for having to shift arrays, $j could be made integer
- # but this would impose a limit to number-length to 2**32.
- my $i; my $car = 0; my $j = 0;
- for $i (@$y)
- {
- $x->[$j] -= $BASE
- if $car = (($x->[$j] += $i + $car) >= $BASE) ? 1 : 0;
- $j++;
- }
- while ($car != 0)
- {
- $x->[$j] -= $BASE if $car = (($x->[$j] += $car) >= $BASE) ? 1 : 0; $j++;
- }
- }
-
-sub sub
- {
- # (ref to int_num_array, ref to int_num_array)
- # subtract base 1e5 numbers -- stolen from Knuth Vol 2 pg 232, $x > $y
- # subtract Y from X (X is always greater/equal!) by modifiyng x in place
- my ($sx,$sy,$s) = @_;
-
- my $car = 0; my $i; my $j = 0;
- if (!$s)
- {
- #print "case 2\n";
- for $i (@$sx)
- {
- last unless defined $sy->[$j] || $car;
- #print "x: $i y: $sy->[$j] c: $car\n";
- $i += $BASE if $car = (($i -= ($sy->[$j] || 0) + $car) < 0); $j++;
- #print "x: $i y: $sy->[$j-1] c: $car\n";
- }
- # might leave leading zeros, so fix that
- _strip_zeros($sx);
- return $sx;
- }
- else
- {
- #print "case 1 (swap)\n";
- for $i (@$sx)
- {
- last unless defined $sy->[$j] || $car;
- #print "$sy->[$j] $i $car => $sx->[$j]\n";
- $sy->[$j] += $BASE
- if $car = (($sy->[$j] = $i-($sy->[$j]||0) - $car) < 0);
- #print "$sy->[$j] $i $car => $sy->[$j]\n";
- $j++;
- }
- # might leave leading zeros, so fix that
- _strip_zeros($sy);
- return $sy;
- }
- }
-
-sub mul
- {
- # (BINT, BINT) return nothing
- # multiply two numbers in internal representation
- # modifies first arg, second needs not be different from first
- my ($x,$y) = @_;
-
- $x->{sign} = $x->{sign} ne $y->{sign} ? '-' : '+';
- my @prod = (); my ($prod,$car,$cty,$xi,$yi);
- my $xv = $x->{value};
- my $yv = $y->{value};
- # since multiplying $x with $x fails, make copy in this case
- $yv = [@$xv] if "$xv" eq "$yv";
- for $xi (@$xv)
- {
- $car = 0; $cty = 0;
- for $yi (@$yv)
- {
- $prod = $xi * $yi + ($prod[$cty] || 0) + $car;
- $prod[$cty++] =
- $prod - ($car = int($prod * 1e-5)) * $BASE; # see USE_MUL
- }
- $prod[$cty] += $car if $car; # need really to check for 0?
- $xi = shift @prod;
- }
- push @$xv, @prod;
- _strip_zeros($x->{value});
- # normalize (handled last to save check for $y->is_zero()
- $x->{sign} = '+' if @$xv == 1 && $xv->[0] == 0; # not is_zero due to '-'
- }
-
-sub div
- {
- # ref to array, ref to array, modify first array and return reminder if
- # in list context
- # does no longer handle sign
- my ($x,$yorg) = @_;
- my ($car,$bar,$prd,$dd,$xi,$yi,@q,$v2,$v1);
-
- my (@d,$tmp,$q,$u2,$u1,$u0);
-
- $car = $bar = $prd = 0;
-
- my $y = [ @$yorg ];
- if (($dd = int($BASE/($y->[-1]+1))) != 1)
- {
- for $xi (@$x)
- {
- $xi = $xi * $dd + $car;
- $xi -= ($car = int($xi * $RBASE)) * $BASE; # see USE_MUL
- }
- push(@$x, $car); $car = 0;
- for $yi (@$y)
- {
- $yi = $yi * $dd + $car;
- $yi -= ($car = int($yi * $RBASE)) * $BASE; # see USE_MUL
- }
- }
- else
- {
- push(@$x, 0);
- }
- @q = (); ($v2,$v1) = @$y[-2,-1];
- $v2 = 0 unless $v2;
- while ($#$x > $#$y)
- {
- ($u2,$u1,$u0) = @$x[-3..-1];
- $u2 = 0 unless $u2;
- print "oups v1 is 0, u0: $u0 $y->[-2] $y->[-1] l ",scalar @$y,"\n"
- if $v1 == 0;
- $q = (($u0 == $v1) ? 99999 : int(($u0*$BASE+$u1)/$v1));
- --$q while ($v2*$q > ($u0*1e5+$u1-$q*$v1)*$BASE+$u2);
- if ($q)
- {
- ($car, $bar) = (0,0);
- for ($yi = 0, $xi = $#$x-$#$y-1; $yi <= $#$y; ++$yi,++$xi)
- {
- $prd = $q * $y->[$yi] + $car;
- $prd -= ($car = int($prd * $RBASE)) * $BASE; # see USE_MUL
- $x->[$xi] += 1e5 if ($bar = (($x->[$xi] -= $prd + $bar) < 0));
- }
- if ($x->[-1] < $car + $bar)
- {
- $car = 0; --$q;
- for ($yi = 0, $xi = $#$x-$#$y-1; $yi <= $#$y; ++$yi,++$xi)
- {
- $x->[$xi] -= 1e5
- if ($car = (($x->[$xi] += $y->[$yi] + $car) > $BASE));
- }
- }
- }
- pop(@$x); unshift(@q, $q);
- }
- if (wantarray)
- {
- @d = ();
- if ($dd != 1)
- {
- $car = 0;
- for $xi (reverse @$x)
- {
- $prd = $car * $BASE + $xi;
- $car = $prd - ($tmp = int($prd / $dd)) * $dd; # see USE_MUL
- unshift(@d, $tmp);
- }
- }
- else
- {
- @d = @$x;
- }
- @$x = @q;
- _strip_zeros($x);
- _strip_zeros(\@d);
- return ($x,\@d);
- }
- @$x = @q;
- _strip_zeros($x);
- return $x;
- }
-
sub _lcm
{
# (BINT or num_str, BINT or num_str) return BINT
return $x * $ty / bgcd($x,$ty);
}
-sub _gcd_old
+sub _gcd
{
# (BINT or num_str, BINT or num_str) return BINT
# does modify first arg
# GCD -- Euclids algorithm E, Knuth Vol 2 pg 296
- trace(@_);
- my $x = shift; my $ty = $class->new(shift); # preserve y
- return $x->bnan() if ($x->{sign} eq $nan) || ($ty->{sign} eq $nan);
+ my $x = shift; my $ty = $class->new(shift); # preserve y, but make class
+ return $x->bnan() if $x->{sign} !~ /^[+-]$/ || $ty->{sign} !~ /^[+-]$/;
while (!$ty->is_zero())
{
$x;
}
-sub _gcd
- {
- # (BINT or num_str, BINT or num_str) return BINT
- # does not modify arguments
- # GCD -- Euclids algorithm, variant L (Lehmer), Knuth Vol 3 pg 347 ff
- # unfortunately, it is slower and also seems buggy (the A=0, B=1, C=1, D=0
- # case..)
- trace(@_);
-
- my $u = $class->new(shift); my $v = $class->new(shift); # preserve u,v
- return $u->bnan() if ($u->{sign} eq $nan) || ($v->{sign} eq $nan);
-
- $u->babs(); $v->babs(); # Euclid is valid for |u| and |v|
-
- my ($U,$V,$A,$B,$C,$D,$T,$Q); # single precision variables
- my ($t); # multiprecision variables
-
- while ($v > $BASE)
- {
- #sleep 1;
- ($u,$v) = ($v,$u) if ($u < $v); # make sure that u >= v
- #print "gcd: $u $v\n";
- # step L1, initialize
- $A = 1; $B = 0; $C = 0; $D = 1;
- $U = $u->{value}->[-1]; # leading digits of u
- $V = $v->{value}->[-1]; # leading digits of v
-
- # step L2, test quotient
- if (($V + $C != 0) && ($V + $D != 0)) # div by zero => go to L4
- {
- $Q = int(($U + $A)/($V + $C)); # quotient
- #print "L1 A=$A B=$B C=$C D=$D U=$U V=$V Q=$Q\n";
- # do L3? (div by zero => go to L4)
- while ($Q == int(($U + $B)/($V + $D)))
- {
- # step L3, emulate Euclid
- #print "L3a A=$A B=$B C=$C D=$D U=$U V=$V Q=$Q\n";
- $T = $A - $Q*$C; $A = $C; $C = $T;
- $T = $B - $Q*$D; $B = $D; $D = $T;
- $T = $U - $Q*$V; $U = $V; $V = $T;
- last if ($V + $D == 0) || ($V + $C == 0); # div by zero => L4
- $Q = int(($U + $A)/($V + $C)); # quotient for next test
- #print "L3b A=$A B=$B C=$C D=$D U=$U V=$V Q=$Q\n";
- }
- }
- # step L4, multiprecision step
- # was if ($B == 0)
- # in case A = 0, B = 1, C = 0 and D = 1, this case would simple swap u & v
- # and loop endless. Not sure why this happens, Knuth does not make a
- # remark about this special case. bug?
- if (($B == 0) || (($A == 0) && ($C == 1) && ($D == 0)))
- {
- #print "L4b1: u=$u v=$v\n";
- ($u,$v) = ($v,bmod($u,$v));
- #$t = $u % $v; $u = $v->copy(); $v = $t;
- #print "L4b12: u=$u v=$v\n";
- }
- else
- {
- #print "L4b: $u $v $A $B $C $D\n";
- $t = $A*$u + $B*$v; $v *= $D; $v += $C*$u; $u = $t;
- #print "L4b2: $u $v\n";
- }
- } # back to L1
-
- return _gcd_old($u,$v) if $v != 1; # v too small
- return $v; # 1
- }
-
###############################################################################
# this method return 0 if the object can be modified, or 1 for not
# We use a fast use constant statement here, to avoid costly calls. Subclasses
# may override it with special code (f.i. Math::BigInt::Constant does so)
-use constant modify => 0;
-
-#sub modify
-# {
-# my $self = shift;
-# my $method = shift;
-# print "original $self modify by $method\n";
-# return 0; # $self;
-# }
+sub modify () { 0; }
1;
__END__
# Testing
$x->is_zero(); # return whether arg is zero or not
$x->is_nan(); # return whether arg is NaN or not
- $x->is_one(); # return true if arg is +1
- $x->is_one('-'); # return true if arg is -1
- $x->is_odd(); # return true if odd, false for even
- $x->is_even(); # return true if even, false for odd
+ $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->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())
=head1 DESCRIPTION
=back
-=head2 Rounding
+=head1 ACCURACY and PRECISION
+
+Since version v1.33 Math::BigInt and Math::BigFloat do have full support for
+accuracy and precision based rounding, both automatically after every
+operation as manual.
+
+This section describes the accuracy/precision handling in Math::Big* as it
+used to be and is now, completed with an explanation of all terms and
+abbreviations.
+
+Not yet implemented things (but with correct description) are marked with '!',
+things that need to be answered are marked with '?'.
+
+In the next paragraph follows a short description of terms used here (because
+these may differ from terms used by others people or documentations).
+
+During the rest of this document the shortcuts A (for accuracy), P (for
+precision), F (fallback) and R (rounding mode) will be used.
+
+=head2 Precision P
+
+A fixed number of digits before (positive) or after (negative)
+the dot. F.i. 123.45 has a precision of -2. 0 means an integer like 123
+(or 120). A precision of 2 means two digits left of the dot are zero, so
+123 with P = 1 becomes 120. Note that numbers with zeros before the dot 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 dot 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
+
+=head2 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. F.i. 123.456 has A of 6,
+10203 has 5, 123.0506 has 7, 123.450000 has 8, and 0.000123 has 3.
+
+=head2 Fallback F
+
+When both A and P are undefined, this is used as a fallback accuracy.
+
+=head2 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.)
-Only C<bround()> is defined for BigInts, for further details of rounding see
-L<Math::BigFloat>.
+=over 2
+
+=item 'trunc'
+
+truncation invariably removes all digits following the
+rounding place, replacing them with zeros. Thus, 987.65 rounded
+to tenths (P=1) becomes 980, and rounded to the fourth sigdig
+becomes 987.6 (A=4). 123.456 rounded to the second place after the
+dot (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:
+
+=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.
+
+=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.
+
+=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,
+but 0.4501 becomes 0.5.
+
+=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.
+
+=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.
+
+=back
+
+The handling of A & P in MBI/MBF (the old core code shipped with Perl
+versions <= 5.7.2) is like this:
=over 2
-=item bfround ( +$scale ) rounds to the $scale'th place left from the '.'
+=item Precision
+
+ * ffround($p) is able to round to $p number of digits after the dot
+ * otherwise P is unused
+
+=item Accuracy (significant digits)
+
+ * fround($a) rounds to $a significant digits
+ * only fdiv() and fsqrt() take A as (optional) paramater
+ + other operations simple create the same amount (fneg etc), or more (fmul)
+ of digits
+ + rounding/truncating is only done when explicitly calling one of fround
+ or ffround, and never for BigInt (not implemented)
+ * fsqrt() simple hands it's accuracy argument over to fdiv.
+ * the documentation and the comment in the code indicate two different ways
+ on how fdiv() determines the maximum number of digits it should calculate,
+ and the actual code does yet another thing
+ POD:
+ max($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 be actually 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 $y to determine
+ wether it must round up or down.
+ ? I have no idea which is the right way. Thats why I used scheme a bit more
+ ? simple and tweaked the few failing the testcases to match it.
+
+=back
+
+This is how it works now:
-=item bround ( +$scale ) preserves accuracy to $scale sighnificant digits counted from the left and paddes the number with zeros
+=over 2
-=item bround ( -$scale ) preserves accuracy to $scale significant digits counted from the right and paddes the number with zeros.
+=item 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's place right of the dot,
+ positive values mean left from the dot. 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
+
+=item Creating numbers
+
+ !* When you create a number, there should be a way to define it's 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 immidiately and also
+ stored locally at the number. Thus changing the global defaults later on
+ will not change the A or P of previously created numbers (aka A and P of
+ $x will be what was in effect when $x was created)
+
+=item Usage
+
+ * If A or P are enabled/defined, the are used to round the result of each
+ operation according to the rules below
+ * Negative P are ignored in Math::BigInt, since it never has digits after
+ the dot
+ !* Since Math::BigFloat uses Math::BigInts internally, setting A or P inside
+ ! Math::BigInt as globals should not hamper with the parts of a BigFloat.
+ ! Thus a flag is used to mark all Math::BigFloat numbers as 'do never round'
+
+=item Precedence
+
+ * It makes only sense that a number has only 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 engaged 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 none of them is defined, nothing is used,
+ e.g. 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 none of A
+ or P are defined, fdiv() will use a fallback (F) of $div_scale digits.
+ If either the dividend or the divisors mantissa have more digits than the
+ F, the higher value will be used instead as F.
+ This is to limit the digits (A) of the result (just think if what happens
+ with unlimited A and P in case of 1/3 :-)
+ * fdiv will calculate 1 more digits than required (determined by
+ A, P or F), and, if F is not used, round the result
+ (this will still fail in 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 modi:
+ + never round (this is the default):
+ This is done by setting A and P to undef. No math operation
+ will round the result, with fdiv() and fsqrt() as exception to guard
+ against overflows. You must explicitely call bround(), bfround() or
+ round() (the latter with with parameters).
+ Note: Once you 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 appropriately rounded. By
+ setting proper values for A and P, you can have all-the-same-A or
+ all-the-same-P modi. F.i. Math::Current 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 intermidiate rounding does not 'poison' further math?
+
+=item 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 one (A) is undefined. These three parameters override the
+ globals in the order detailed as follows, aka the first defined value
+ wins:
+ (local: per object, global: globally 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 it's arguments to fdiv(), as it used to, only now for two
+ arguments (A and P) instead of one
+
+=item 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 immidiately rounds $x to the new value.
+
+=item 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 local, global
+ 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 does modify the local settings of the number, so that
+
+ $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 $x->accuracy()
+ will be 4 from now on.
+
+=item Default values
+
+ * R: 'even'
+ * F: 40
+ * A: undef
+ * P: undef
+
+=item Remarks
+
+ * The defaults are set up so that the new code gives the same results as
+ the old code (except in a few cases on fdiv):
+ + Both A and P are undefined and thus will not be used for rounding
+ after each operation.
+ + round() is thus a no-op, unless given extra parameters A and P
=back
-C<bfround()> does nothing in case of negative C<$scale>. Both C<bround()> and
-C<bfround()> are a no-ops for a scale of 0.
+=head1 INTERNALS
+
+The actual numbers are stored as unsigned big integers, and math with them
+done (by default) by a module called Math::BigInt::Calc. This is equivalent to:
-All rounding functions take as a second parameter a rounding mode from one of
-the following: 'even', 'odd', '+inf', '-inf', 'zero' or 'trunc'.
+ use Math::BigInt lib => 'calc';
-The default is 'even'. By using C<< Math::BigInt->round_mode($rnd_mode); >>
-you can get and set the default round mode for subsequent rounding.
+You can change this by using:
-The second parameter to the round functions than overrides the default
-temporarily.
+ use Math::BigInt lib => 'BitVect';
-=head2 Internals
+('Math::BitInt::BitVect' works, too.)
+
+Calc.pm uses as internal format an array of elements of base 100000 digits
+with the least significant digit first, BitVect.pm uses a bit vector of base 2,
+most significant bit first.
-Actual math is done in an internal format consisting of an array of
-elements of base 100000 digits with the least significant digit first.
The sign C</^[+-]$/> is stored separately. The string 'NaN' is used to
represent the result when input arguments are not numbers, as well as
-the result of dividing by zero.
+the result of dividing by zero. '+inf' or '-inf' represent infinity.
-You sould neither care nor depend on the internal represantation, it might
+You sould neither care nor depend on the internal representation, it might
change without notice. Use only method calls like C<< $x->sign(); >> instead
relying on the internal hash keys like in C<< $x->{sign}; >>.
C<$m> will always be a copy of the original number. The relation between $e
and $m might change in the future, but will be always equivalent in a
-numerical sense, e.g. $m might get minized.
-
+numerical sense, e.g. $m might get minimized.
+
=head1 EXAMPLES
use Math::BigInt qw(bstr bint);
$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
+
=head1 Autocreating constants
After C<use Math::BigInt ':constant'> all the B<integer> decimal constants
perl -MMath::BigInt=:constant -e 'print 2**100,"\n"'
prints the integer value of C<2**100>. Note that without conversion of
-constants the expression 2**100 will be calculated as floating point
-number.
+constants the expression 2**100 will be calculated as perl scalar.
Please note that strings and floating point constants are not affected,
so that
For more benchmark results see http://bloodgate.com/perl/benchmarks.html
+=head2 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 consist 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
+L<http://search.cpan.org/>:
+
+ use Math::BigInt lib => 'BitVect';
+
+ my $x = Math::BigInt->new(2);
+ print $x ** (1024*1024);
+
=head1 BUGS
=over 2
$y = $x->copy();
-See also the documentation in L<overload> regarding C<=>.
+See also the documentation in for overload.pm regarding C<=>.
=item bpow
since overload calls C<sub($x,0,1);> instead of C<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).
+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...
$float = Math::BigFloat->new($mbi2 / $mbi); # = 2.0 thus wrong!
-Beware of the order of more complicated expressions like:
+Beware also of the order of more complicated expressions like:
$integer = ($mbi2 + $mbi) / $mbf; # int / float => int
$integer = $mbi2 / Math::BigFloat->new($mbi); # ditto
This program is free software; you may redistribute it and/or modify it under
the same terms as Perl itself.
+=head1 SEE ALSO
+
+L<Math::BigFloat> and L<Math::Big>.
+
=head1 AUTHORS
Original code by Mark Biggar, overloaded interface by Ilya Zakharevich.
https://perl5.git.perl.org / perl5.git / blobdiff