diff options
| author |   millert <millert@openbsd.org> | 2002-10-27 22:25:13 +0000 |
|---|---|---|
| committer | millert <millert@openbsd.org> | 2002-10-27 22:25:13 +0000 |
| commit | 79cd0b9ae197e67390710f96587afb9169e5346d (patch) | |
| tree | 8952f7a8f773436ffd1169eb9ac0d56c7ce1118f /gnu/usr.bin/perl/lib/Math/BigInt.pm | |
| parent | stock perl 5.8.0 from CPAN (diff) | |
| download | wireguard-openbsd-79cd0b9ae197e67390710f96587afb9169e5346d.tar.xz wireguard-openbsd-79cd0b9ae197e67390710f96587afb9169e5346d.zip | |
Resolve conflicts, remove old files, merge local changes
Diffstat (limited to 'gnu/usr.bin/perl/lib/Math/BigInt.pm')
| -rw-r--r-- | gnu/usr.bin/perl/lib/Math/BigInt.pm | 4643 |
1 files changed, 4188 insertions, 455 deletions
diff --git a/gnu/usr.bin/perl/lib/Math/BigInt.pm b/gnu/usr.bin/perl/lib/Math/BigInt.pm index 066577d4cc1..5a1385d5193 100644 --- a/gnu/usr.bin/perl/lib/Math/BigInt.pm +++ b/gnu/usr.bin/perl/lib/Math/BigInt.pm @@ -1,430 +1,2697 @@ package Math::BigInt; -$VERSION='0.01'; + +# +# "Mike had an infinite amount to do and a negative amount of time in which +# to do it." - Before and After +# + +# The following hash values are used: +# 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 + +# Remember not to take shortcuts ala $xs = $x->{value}; $CALC->foo($xs); since +# underlying lib might change the reference! + +my $class = "Math::BigInt"; +require 5.005; + +# This is a patched v1.60, containing a fix for the "1234567890\n" bug +$VERSION = '1.60'; +use Exporter; +@ISA = qw( Exporter ); +@EXPORT_OK = qw( objectify _swap bgcd blcm); +use vars qw/$round_mode $accuracy $precision $div_scale $rnd_mode/; +use vars qw/$upgrade $downgrade/; +use strict; + +# Inside overload, the first arg is always an object. If the original code had +# it reversed (like $x = 2 * $y), then the third paramater indicates this +# swapping. To make it work, we use a helper routine which not only reswaps the +# params, but also makes a new object in this case. See _swap() for details, +# especially the cases of operators with different classes. + +# For overloaded ops with only one argument we simple use $_[0]->copy() to +# preserve the argument. + +# Thus inheritance of overload operators becomes possible and transparent for +# our subclasses without the need to repeat the entire overload section there. use overload -'+' => sub {new Math::BigInt &badd}, -'-' => sub {new Math::BigInt - $_[2]? bsub($_[1],${$_[0]}) : bsub(${$_[0]},$_[1])}, -'<=>' => sub {$_[2]? bcmp($_[1],${$_[0]}) : bcmp(${$_[0]},$_[1])}, -'cmp' => sub {$_[2]? ($_[1] cmp ${$_[0]}) : (${$_[0]} cmp $_[1])}, -'*' => sub {new Math::BigInt &bmul}, -'/' => sub {new Math::BigInt - $_[2]? scalar bdiv($_[1],${$_[0]}) : - scalar bdiv(${$_[0]},$_[1])}, -'%' => sub {new Math::BigInt - $_[2]? bmod($_[1],${$_[0]}) : bmod(${$_[0]},$_[1])}, -'**' => sub {new Math::BigInt - $_[2]? bpow($_[1],${$_[0]}) : bpow(${$_[0]},$_[1])}, -'neg' => sub {new Math::BigInt &bneg}, -'abs' => sub {new Math::BigInt &babs}, -'<<' => sub {new Math::BigInt - $_[2]? blsft($_[1],${$_[0]}) : blsft(${$_[0]},$_[1])}, -'>>' => sub {new Math::BigInt - $_[2]? brsft($_[1],${$_[0]}) : brsft(${$_[0]},$_[1])}, -'&' => sub {new Math::BigInt &band}, -'|' => sub {new Math::BigInt &bior}, -'^' => sub {new Math::BigInt &bxor}, -'~' => sub {new Math::BigInt &bnot}, - -qw( -"" stringify -0+ numify) # Order of arguments unsignificant +'=' => sub { $_[0]->copy(); }, + +# '+' and '-' do not use _swap, since it is a triffle slower. If you want to +# override _swap (if ever), then override overload of '+' and '-', too! +# for sub it is a bit tricky to keep b: b-a => -a+b +'-' => sub { my $c = $_[0]->copy; $_[2] ? + $c->bneg()->badd($_[1]) : + $c->bsub( $_[1]) }, +'+' => sub { $_[0]->copy()->badd($_[1]); }, + +# some shortcuts for speed (assumes that reversed order of arguments is routed +# to normal '+' and we thus can always modify first arg. If this is changed, +# this breaks and must be adjusted.) +'+=' => sub { $_[0]->badd($_[1]); }, +'-=' => sub { $_[0]->bsub($_[1]); }, +'*=' => sub { $_[0]->bmul($_[1]); }, +'/=' => sub { scalar $_[0]->bdiv($_[1]); }, +'%=' => sub { $_[0]->bmod($_[1]); }, +'^=' => sub { $_[0]->bxor($_[1]); }, +'&=' => sub { $_[0]->band($_[1]); }, +'|=' => sub { $_[0]->bior($_[1]); }, +'**=' => sub { $_[0]->bpow($_[1]); }, + +# not supported by Perl yet +'..' => \&_pointpoint, + +'<=>' => sub { $_[2] ? + ref($_[0])->bcmp($_[1],$_[0]) : + $_[0]->bcmp($_[1])}, +'cmp' => sub { + $_[2] ? + "$_[1]" cmp $_[0]->bstr() : + $_[0]->bstr() cmp "$_[1]" }, + +'log' => sub { $_[0]->copy()->blog(); }, +'int' => sub { $_[0]->copy(); }, +'neg' => sub { $_[0]->copy()->bneg(); }, +'abs' => sub { $_[0]->copy()->babs(); }, +'sqrt' => sub { $_[0]->copy()->bsqrt(); }, +'~' => sub { $_[0]->copy()->bnot(); }, + +'*' => sub { my @a = ref($_[0])->_swap(@_); $a[0]->bmul($a[1]); }, +'/' => sub { my @a = ref($_[0])->_swap(@_);scalar $a[0]->bdiv($a[1]);}, +'%' => sub { my @a = ref($_[0])->_swap(@_); $a[0]->bmod($a[1]); }, +'**' => sub { my @a = ref($_[0])->_swap(@_); $a[0]->bpow($a[1]); }, +'<<' => sub { my @a = ref($_[0])->_swap(@_); $a[0]->blsft($a[1]); }, +'>>' => sub { my @a = ref($_[0])->_swap(@_); $a[0]->brsft($a[1]); }, + +'&' => sub { my @a = ref($_[0])->_swap(@_); $a[0]->band($a[1]); }, +'|' => sub { my @a = ref($_[0])->_swap(@_); $a[0]->bior($a[1]); }, +'^' => sub { my @a = ref($_[0])->_swap(@_); $a[0]->bxor($a[1]); }, + +# can modify arg of ++ and --, so avoid a new-copy for speed, but don't +# use $_[0]->__one(), it modifies $_[0] to be 1! +'++' => sub { $_[0]->binc() }, +'--' => sub { $_[0]->bdec() }, + +# if overloaded, O(1) instead of O(N) and twice as fast for small numbers +'bool' => sub { + # this kludge is needed for perl prior 5.6.0 since returning 0 here fails :-/ + # v5.6.1 dumps on that: return !$_[0]->is_zero() || undef; :-( + my $t = !$_[0]->is_zero(); + undef $t if $t == 0; + $t; + }, + +# the original qw() does not work with the TIESCALAR below, why? +# Order of arguments unsignificant +'""' => sub { $_[0]->bstr(); }, +'0+' => sub { $_[0]->numify(); } ; -$NaNOK=1; - -sub new { - my($class) = shift; - my($foo) = bnorm(shift); - die "Not a number initialized to Math::BigInt" if !$NaNOK && $foo eq "NaN"; - bless \$foo, $class; -} -sub stringify { "${$_[0]}" } -sub numify { 0 + "${$_[0]}" } # Not needed, additional overhead - # comparing to direct compilation based on - # stringify -sub import { - shift; - return unless @_; - die "unknown import: @_" unless @_ == 1 and $_[0] eq ':constant'; - overload::constant integer => sub {Math::BigInt->new(shift)}; -} - -$zero = 0; - -# overcome a floating point problem on certain osnames (posix-bc, os390) -BEGIN { - my $x = 100000.0; - my $use_mult = int($x*1e-5)*1e5 == $x ? 1 : 0; -} - -# normalize string form of number. Strip leading zeros. Strip any -# white space and add a sign, if missing. -# Strings that are not numbers result the value 'NaN'. - -sub bnorm { #(num_str) return num_str - local($_) = @_; - s/\s+//g; # strip white space - if (s/^([+-]?)0*(\d+)$/$1$2/) { # test if number - substr($_,$[,0) = '+' unless $1; # Add missing sign - s/^-0/+0/; - $_; - } else { - 'NaN'; - } -} - -# Convert a number from string format to internal base 100000 format. -# Assumes normalized value as input. -sub internal { #(num_str) return int_num_array - local($d) = @_; - ($is,$il) = (substr($d,$[,1),length($d)-2); - substr($d,$[,1) = ''; - ($is, reverse(unpack("a" . ($il%5+1) . ("a5" x ($il/5)), $d))); -} - -# Convert a number from internal base 100000 format to string format. -# This routine scribbles all over input array. -sub external { #(int_num_array) return num_str - $es = shift; - grep($_ > 9999 || ($_ = substr('0000'.$_,-5)), @_); # zero pad - &bnorm(join('', $es, reverse(@_))); # reverse concat and normalize -} - -# Negate input value. -sub bneg { #(num_str) return num_str - local($_) = &bnorm(@_); - return $_ if $_ eq '+0' or $_ eq 'NaN'; - vec($_,0,8) ^= ord('+') ^ ord('-'); - $_; -} - -# Returns the absolute value of the input. -sub babs { #(num_str) return num_str - &abs(&bnorm(@_)); -} - -sub abs { # post-normalized abs for internal use - local($_) = @_; - s/^-/+/; - $_; -} - -# Compares 2 values. Returns one of undef, <0, =0, >0. (suitable for sort) -sub bcmp { #(num_str, num_str) return cond_code - local($x,$y) = (&bnorm($_[$[]),&bnorm($_[$[+1])); - if ($x eq 'NaN') { - undef; - } elsif ($y eq 'NaN') { - undef; - } else { - &cmp($x,$y) <=> 0; - } -} - -sub cmp { # post-normalized compare for internal use - local($cx, $cy) = @_; - - return 0 if ($cx eq $cy); - - local($sx, $sy) = (substr($cx, 0, 1), substr($cy, 0, 1)); - local($ld); - - if ($sx eq '+') { - return 1 if ($sy eq '-' || $cy eq '+0'); - $ld = length($cx) - length($cy); - return $ld if ($ld); - return $cx cmp $cy; - } else { # $sx eq '-' - return -1 if ($sy eq '+'); - $ld = length($cy) - length($cx); - return $ld if ($ld); - return $cy cmp $cx; - } -} - -sub badd { #(num_str, num_str) return num_str - local(*x, *y); ($x, $y) = (&bnorm($_[$[]),&bnorm($_[$[+1])); - if ($x eq 'NaN') { - 'NaN'; - } elsif ($y eq 'NaN') { - 'NaN'; - } else { - @x = &internal($x); # convert to internal form - @y = &internal($y); - local($sx, $sy) = (shift @x, shift @y); # get signs - if ($sx eq $sy) { - &external($sx, &add(*x, *y)); # if same sign add - } else { - ($x, $y) = (&abs($x),&abs($y)); # make abs - if (&cmp($y,$x) > 0) { - &external($sy, &sub(*y, *x)); - } else { - &external($sx, &sub(*x, *y)); - } +############################################################################## +# global constants, flags and accessory + +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 +my $IMPORT = 0; # did import() yet? + +$round_mode = 'even'; # one of 'even', 'odd', '+inf', '-inf', 'zero' or 'trunc' +$accuracy = undef; +$precision = undef; +$div_scale = 40; + +$upgrade = undef; # default is no upgrade +$downgrade = undef; # default is no downgrade + +############################################################################## +# the old code had $rnd_mode, so we need to support it, too + +$rnd_mode = 'even'; +sub TIESCALAR { my ($class) = @_; bless \$round_mode, $class; } +sub FETCH { return $round_mode; } +sub STORE { $rnd_mode = $_[0]->round_mode($_[1]); } + +BEGIN { tie $rnd_mode, 'Math::BigInt'; } + +############################################################################## + +sub round_mode + { + no strict 'refs'; + # make Class->round_mode() work + my $self = shift; + my $class = ref($self) || $self || __PACKAGE__; + if (defined $_[0]) + { + my $m = shift; + die "Unknown round mode $m" + if $m !~ /^(even|odd|\+inf|\-inf|zero|trunc)$/; + return ${"${class}::round_mode"} = $m; + } + return ${"${class}::round_mode"}; + } + +sub upgrade + { + no strict 'refs'; + # make Class->upgrade() work + my $self = shift; + my $class = ref($self) || $self || __PACKAGE__; + # need to set new value? + if (@_ > 0) + { + my $u = shift; + return ${"${class}::upgrade"} = $u; + } + return ${"${class}::upgrade"}; + } + +sub downgrade + { + no strict 'refs'; + # make Class->downgrade() work + my $self = shift; + my $class = ref($self) || $self || __PACKAGE__; + # need to set new value? + if (@_ > 0) + { + my $u = shift; + return ${"${class}::downgrade"} = $u; + } + return ${"${class}::downgrade"}; + } + +sub div_scale + { + no strict 'refs'; + # make Class->round_mode() work + my $self = shift; + my $class = ref($self) || $self || __PACKAGE__; + if (defined $_[0]) + { + die ('div_scale must be greater than zero') if $_[0] < 0; + ${"${class}::div_scale"} = shift; + } + return ${"${class}::div_scale"}; + } + +sub accuracy + { + # $x->accuracy($a); ref($x) $a + # $x->accuracy(); ref($x) + # Class->accuracy(); class + # Class->accuracy($a); class $a + + my $x = shift; + my $class = ref($x) || $x || __PACKAGE__; + + no strict 'refs'; + # need to set new value? + if (@_ > 0) + { + my $a = shift; + die ('accuracy must not be zero') if defined $a && $a == 0; + if (ref($x)) + { + # $object->accuracy() or fallback to global + $x->bround($a) if defined $a; + $x->{_a} = $a; # set/overwrite, even if not rounded + $x->{_p} = undef; # clear P + } + else + { + # set global + ${"${class}::accuracy"} = $a; + ${"${class}::precision"} = undef; # clear P + } + return $a; # shortcut + } + + my $r; + # $object->accuracy() or fallback to global + $r = $x->{_a} if ref($x); + # but don't return global undef, when $x's accuracy is 0! + $r = ${"${class}::accuracy"} if !defined $r; + $r; + } + +sub precision + { + # $x->precision($p); ref($x) $p + # $x->precision(); ref($x) + # Class->precision(); class + # Class->precision($p); class $p + + my $x = shift; + my $class = ref($x) || $x || __PACKAGE__; + + no strict 'refs'; + # need to set new value? + if (@_ > 0) + { + my $p = shift; + if (ref($x)) + { + # $object->precision() or fallback to global + $x->bfround($p) if defined $p; + $x->{_p} = $p; # set/overwrite, even if not rounded + $x->{_a} = undef; # clear A + } + else + { + # set global + ${"${class}::precision"} = $p; + ${"${class}::accuracy"} = undef; # clear A + } + return $p; # shortcut + } + + my $r; + # $object->precision() or fallback to global + $r = $x->{_p} if ref($x); + # but don't return global undef, when $x's precision is 0! + $r = ${"${class}::precision"} if !defined $r; + $r; + } + +sub config + { + # return (later set?) configuration data as hash ref + my $class = shift || 'Math::BigInt'; + + no strict 'refs'; + my $lib = $CALC; + my $cfg = { + lib => $lib, + lib_version => ${"${lib}::VERSION"}, + class => $class, + }; + foreach ( + qw/upgrade downgrade precision accuracy round_mode VERSION div_scale/) + { + $cfg->{lc($_)} = ${"${class}::$_"}; + }; + $cfg; + } + +sub _scale_a + { + # select accuracy parameter based on precedence, + # used by bround() and bfround(), may return undef for scale (means no op) + my ($x,$s,$m,$scale,$mode) = @_; + $scale = $x->{_a} if !defined $scale; + $scale = $s if (!defined $scale); + $mode = $m if !defined $mode; + return ($scale,$mode); + } + +sub _scale_p + { + # select precision parameter based on precedence, + # used by bround() and bfround(), may return undef for scale (means no op) + my ($x,$s,$m,$scale,$mode) = @_; + $scale = $x->{_p} if !defined $scale; + $scale = $s if (!defined $scale); + $mode = $m if !defined $mode; + return ($scale,$mode); + } + +############################################################################## +# constructors + +sub copy + { + my ($c,$x); + if (@_ > 1) + { + # if two arguments, the first one is the class to "swallow" subclasses + ($c,$x) = @_; + } + else + { + $x = shift; + $c = ref($x); + } + return unless ref($x); # only for objects + + my $self = {}; bless $self,$c; + my $r; + foreach my $k (keys %$x) + { + if ($k eq 'value') + { + $self->{value} = $CALC->_copy($x->{value}); next; + } + if (!($r = ref($x->{$k}))) + { + $self->{$k} = $x->{$k}; next; + } + if ($r eq 'SCALAR') + { + $self->{$k} = \${$x->{$k}}; + } + elsif ($r eq 'ARRAY') + { + $self->{$k} = [ @{$x->{$k}} ]; + } + elsif ($r eq 'HASH') + { + # only one level deep! + foreach my $h (keys %{$x->{$k}}) + { + $self->{$k}->{$h} = $x->{$k}->{$h}; + } + } + else # normal ref + { + my $xk = $x->{$k}; + if ($xk->can('copy')) + { + $self->{$k} = $xk->copy(); + } + else + { + $self->{$k} = $xk->new($xk); } + } } -} - -sub bsub { #(num_str, num_str) return num_str - &badd($_[$[],&bneg($_[$[+1])); -} - -# GCD -- Euclids algorithm Knuth Vol 2 pg 296 -sub bgcd { #(num_str, num_str) return num_str - local($x,$y) = (&bnorm($_[$[]),&bnorm($_[$[+1])); - if ($x eq 'NaN' || $y eq 'NaN') { - 'NaN'; - } else { - ($x, $y) = ($y,&bmod($x,$y)) while $y ne '+0'; - $x; - } -} - -# 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 -sub add { #(int_num_array, int_num_array) return int_num_array - local(*x, *y) = @_; - $car = 0; - for $x (@x) { - last unless @y || $car; - $x -= 1e5 if $car = (($x += (@y ? shift(@y) : 0) + $car) >= 1e5) ? 1 : 0; - } - for $y (@y) { - last unless $car; - $y -= 1e5 if $car = (($y += $car) >= 1e5) ? 1 : 0; - } - (@x, @y, $car); -} - -# subtract base 1e5 numbers -- stolen from Knuth Vol 2 pg 232, $x > $y -sub sub { #(int_num_array, int_num_array) return int_num_array - local(*sx, *sy) = @_; - $bar = 0; - for $sx (@sx) { - last unless @sy || $bar; - $sx += 1e5 if $bar = (($sx -= (@sy ? shift(@sy) : 0) + $bar) < 0); - } - @sx; -} - -# multiply two numbers -- stolen from Knuth Vol 2 pg 233 -sub bmul { #(num_str, num_str) return num_str - local(*x, *y); ($x, $y) = (&bnorm($_[$[]), &bnorm($_[$[+1])); - if ($x eq 'NaN') { - 'NaN'; - } elsif ($y eq 'NaN') { - 'NaN'; - } else { - @x = &internal($x); - @y = &internal($y); - &external(&mul(*x,*y)); - } -} - -# multiply two numbers in internal representation -# destroys the arguments, supposes that two arguments are different -sub mul { #(*int_num_array, *int_num_array) return int_num_array - local(*x, *y) = (shift, shift); - local($signr) = (shift @x ne shift @y) ? '-' : '+'; - @prod = (); - for $x (@x) { - ($car, $cty) = (0, $[); - for $y (@y) { - $prod = $x * $y + ($prod[$cty] || 0) + $car; - if ($use_mult) { - $prod[$cty++] = - $prod - ($car = int($prod * 1e-5)) * 1e5; + $self; + } + +sub new + { + # create a new BigInt object from a string or another BigInt object. + # see hash keys documented at top + + # the argument could be an object, so avoid ||, && etc on it, this would + # cause costly overloaded code to be called. The only allowed ops are + # ref() and defined. + + my ($class,$wanted,$a,$p,$r) = @_; + + # avoid numify-calls by not using || on $wanted! + return $class->bzero($a,$p) if !defined $wanted; # default to 0 + return $class->copy($wanted,$a,$p,$r) + if ref($wanted) && $wanted->isa($class); # MBI or subclass + + $class->import() if $IMPORT == 0; # make require work + + my $self = bless {}, $class; + + # shortcut for "normal" numbers + if ((!ref $wanted) && ($wanted =~ /^([+-]?)[1-9][0-9]*\z/)) + { + $self->{sign} = $1 || '+'; + my $ref = \$wanted; + if ($wanted =~ /^[+-]/) + { + # remove sign without touching wanted + my $t = $wanted; $t =~ s/^[+-]//; $ref = \$t; + } + $self->{value} = $CALC->_new($ref); + no strict 'refs'; + if ( (defined $a) || (defined $p) + || (defined ${"${class}::precision"}) + || (defined ${"${class}::accuracy"}) + ) + { + $self->round($a,$p,$r) unless (@_ == 4 && !defined $a && !defined $p); + } + return $self; + } + + # handle '+inf', '-inf' first + if ($wanted =~ /^[+-]?inf$/) + { + $self->{value} = $CALC->_zero(); + $self->{sign} = $wanted; $self->{sign} = '+inf' if $self->{sign} eq 'inf'; + return $self; + } + # split str in m mantissa, e exponent, i integer, f fraction, v value, s sign + my ($mis,$miv,$mfv,$es,$ev) = _split(\$wanted); + if (!ref $mis) + { + die "$wanted is not a number initialized to $class" if !$NaNOK; + #print "NaN 1\n"; + $self->{value} = $CALC->_zero(); + $self->{sign} = $nan; + return $self; + } + if (!ref $miv) + { + # _from_hex or _from_bin + $self->{value} = $mis->{value}; + $self->{sign} = $mis->{sign}; + return $self; # throw away $mis + } + # make integer from mantissa by adjusting exp, then convert to bigint + $self->{sign} = $$mis; # store sign + $self->{value} = $CALC->_zero(); # for all the NaN cases + my $e = int("$$es$$ev"); # exponent (avoid recursion) + if ($e > 0) + { + my $diff = $e - CORE::length($$mfv); + if ($diff < 0) # Not integer + { + #print "NOI 1\n"; + return $upgrade->new($wanted,$a,$p,$r) if defined $upgrade; + $self->{sign} = $nan; + } + else # diff >= 0 + { + # adjust fraction and add it to value + # print "diff > 0 $$miv\n"; + $$miv = $$miv . ($$mfv . '0' x $diff); + } + } + else + { + if ($$mfv ne '') # e <= 0 + { + # fraction and negative/zero E => NOI + #print "NOI 2 \$\$mfv '$$mfv'\n"; + return $upgrade->new($wanted,$a,$p,$r) if defined $upgrade; + $self->{sign} = $nan; + } + elsif ($e < 0) + { + # xE-y, and empty mfv + #print "xE-y\n"; + $e = abs($e); + if ($$miv !~ s/0{$e}$//) # can strip so many zero's? + { + #print "NOI 3\n"; + return $upgrade->new($wanted,$a,$p,$r) if defined $upgrade; + $self->{sign} = $nan; } - else { - $prod[$cty++] = - $prod - ($car = int($prod / 1e5)) * 1e5; + } + } + $self->{sign} = '+' if $$miv eq '0'; # normalize -0 => +0 + $self->{value} = $CALC->_new($miv) if $self->{sign} =~ /^[+-]$/; + # if any of the globals is set, use them to round and store them inside $self + # do not round for new($x,undef,undef) since that is used by MBF to signal + # no rounding + $self->round($a,$p,$r) unless @_ == 4 && !defined $a && !defined $p; + $self; + } + +sub bnan + { + # create a bigint 'NaN', if given a BigInt, set it to 'NaN' + my $self = shift; + $self = $class if !defined $self; + if (!ref($self)) + { + my $c = $self; $self = {}; bless $self, $c; + } + $self->import() if $IMPORT == 0; # make require work + return if $self->modify('bnan'); + my $c = ref($self); + if ($self->can('_bnan')) + { + # use subclass to initialize + $self->_bnan(); + } + else + { + # otherwise do our own thing + $self->{value} = $CALC->_zero(); + } + $self->{sign} = $nan; + delete $self->{_a}; delete $self->{_p}; # rounding NaN is silly + return $self; + } + +sub binf + { + # create a bigint '+-inf', if given a BigInt, set it to '+-inf' + # the sign is either '+', or if given, used from there + my $self = shift; + my $sign = shift; $sign = '+' if !defined $sign || $sign !~ /^-(inf)?$/; + $self = $class if !defined $self; + if (!ref($self)) + { + my $c = $self; $self = {}; bless $self, $c; + } + $self->import() if $IMPORT == 0; # make require work + return if $self->modify('binf'); + my $c = ref($self); + if ($self->can('_binf')) + { + # use subclass to initialize + $self->_binf(); + } + else + { + # otherwise do our own thing + $self->{value} = $CALC->_zero(); + } + $sign = $sign . 'inf' if $sign !~ /inf$/; # - => -inf + $self->{sign} = $sign; + ($self->{_a},$self->{_p}) = @_; # take over requested rounding + return $self; + } + +sub bzero + { + # create a bigint '+0', if given a BigInt, set it to 0 + my $self = shift; + $self = $class if !defined $self; + + if (!ref($self)) + { + my $c = $self; $self = {}; bless $self, $c; + } + $self->import() if $IMPORT == 0; # make require work + return if $self->modify('bzero'); + + if ($self->can('_bzero')) + { + # use subclass to initialize + $self->_bzero(); + } + else + { + # otherwise do our own thing + $self->{value} = $CALC->_zero(); + } + $self->{sign} = '+'; + if (@_ > 0) + { + if (@_ > 3) + { + # call like: $x->bzero($a,$p,$r,$y); + ($self,$self->{_a},$self->{_p}) = $self->_find_round_parameters(@_); + } + else + { + $self->{_a} = $_[0] + if ( (!defined $self->{_a}) || (defined $_[0] && $_[0] > $self->{_a})); + $self->{_p} = $_[1] + if ( (!defined $self->{_p}) || (defined $_[1] && $_[1] > $self->{_p})); + } + } + $self; + } + +sub bone + { + # create a bigint '+1' (or -1 if given sign '-'), + # if given a BigInt, set it to +1 or -1, respecively + my $self = shift; + my $sign = shift; $sign = '+' if !defined $sign || $sign ne '-'; + $self = $class if !defined $self; + + if (!ref($self)) + { + my $c = $self; $self = {}; bless $self, $c; + } + $self->import() if $IMPORT == 0; # make require work + return if $self->modify('bone'); + + if ($self->can('_bone')) + { + # use subclass to initialize + $self->_bone(); + } + else + { + # otherwise do our own thing + $self->{value} = $CALC->_one(); + } + $self->{sign} = $sign; + if (@_ > 0) + { + if (@_ > 3) + { + # call like: $x->bone($sign,$a,$p,$r,$y); + ($self,$self->{_a},$self->{_p}) = $self->_find_round_parameters(@_); + } + else + { + $self->{_a} = $_[0] + if ( (!defined $self->{_a}) || (defined $_[0] && $_[0] > $self->{_a})); + $self->{_p} = $_[1] + if ( (!defined $self->{_p}) || (defined $_[1] && $_[1] > $self->{_p})); + } + } + $self; + } + +############################################################################## +# string conversation + +sub bsstr + { + # (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") + my $x = shift; $class = ref($x) || $x; $x = $class->new(shift) if !ref($x); + # my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_); + + if ($x->{sign} !~ /^[+-]$/) + { + return $x->{sign} unless $x->{sign} eq '+inf'; # -inf, NaN + return 'inf'; # +inf + } + my ($m,$e) = $x->parts(); + # e can only be positive + my $sign = 'e+'; + # MBF: my $s = $e->{sign}; $s = '' if $s eq '-'; my $sep = 'e'.$s; + return $m->bstr().$sign.$e->bstr(); + } + +sub bstr + { + # make a string from bigint object + my $x = shift; $class = ref($x) || $x; $x = $class->new(shift) if !ref($x); + # my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_); + + if ($x->{sign} !~ /^[+-]$/) + { + return $x->{sign} unless $x->{sign} eq '+inf'; # -inf, NaN + return 'inf'; # +inf + } + my $es = ''; $es = $x->{sign} if $x->{sign} eq '-'; + return $es.${$CALC->_str($x->{value})}; + } + +sub numify + { + # Make a "normal" scalar from a BigInt object + my $x = shift; $x = $class->new($x) unless ref $x; + return $x->{sign} if $x->{sign} !~ /^[+-]$/; + my $num = $CALC->_num($x->{value}); + return -$num if $x->{sign} eq '-'; + $num; + } + +############################################################################## +# public stuff (usually prefixed with "b") + +sub sign + { + # return the sign of the number: +/-/-inf/+inf/NaN + my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_); + + $x->{sign}; + } + +sub _find_round_parameters + { + # After any operation or when calling round(), the result is rounded by + # regarding the A & P from arguments, local parameters, or globals. + + # This procedure finds the round parameters, but it is for speed reasons + # duplicated in round. Otherwise, it is tested by the testsuite and used + # by fdiv(). + + my ($self,$a,$p,$r,@args) = @_; + # $a accuracy, if given by caller + # $p precision, if given by caller + # $r round_mode, if given by caller + # @args all 'other' arguments (0 for unary, 1 for binary ops) + + # leave bigfloat parts alone + return ($self) if exists $self->{_f} && $self->{_f} & MB_NEVER_ROUND != 0; + + my $c = ref($self); # find out class of argument(s) + no strict 'refs'; + + # now pick $a or $p, but only if we have got "arguments" + if (!defined $a) + { + foreach ($self,@args) + { + # take the defined one, or if both defined, the one that is smaller + $a = $_->{_a} if (defined $_->{_a}) && (!defined $a || $_->{_a} < $a); + } + } + if (!defined $p) + { + # even if $a is defined, take $p, to signal error for both defined + foreach ($self,@args) + { + # take the defined one, or if both defined, the one that is bigger + # -2 > -3, and 3 > 2 + $p = $_->{_p} if (defined $_->{_p}) && (!defined $p || $_->{_p} > $p); + } + } + # if still none defined, use globals (#2) + $a = ${"$c\::accuracy"} unless defined $a; + $p = ${"$c\::precision"} unless defined $p; + + # no rounding today? + return ($self) unless defined $a || defined $p; # early out + + # set A and set P is an fatal error + return ($self->bnan()) if defined $a && defined $p; + + $r = ${"$c\::round_mode"} unless defined $r; + die "Unknown round mode '$r'" if $r !~ /^(even|odd|\+inf|\-inf|zero|trunc)$/; + + return ($self,$a,$p,$r); + } + +sub round + { + # Round $self according to given parameters, or given second argument's + # parameters or global defaults + + # for speed reasons, _find_round_parameters is embeded here: + + my ($self,$a,$p,$r,@args) = @_; + # $a accuracy, if given by caller + # $p precision, if given by caller + # $r round_mode, if given by caller + # @args all 'other' arguments (0 for unary, 1 for binary ops) + + # leave bigfloat parts alone + return ($self) if exists $self->{_f} && $self->{_f} & MB_NEVER_ROUND != 0; + + my $c = ref($self); # find out class of argument(s) + no strict 'refs'; + + # now pick $a or $p, but only if we have got "arguments" + if (!defined $a) + { + foreach ($self,@args) + { + # take the defined one, or if both defined, the one that is smaller + $a = $_->{_a} if (defined $_->{_a}) && (!defined $a || $_->{_a} < $a); + } + } + if (!defined $p) + { + # even if $a is defined, take $p, to signal error for both defined + foreach ($self,@args) + { + # take the defined one, or if both defined, the one that is bigger + # -2 > -3, and 3 > 2 + $p = $_->{_p} if (defined $_->{_p}) && (!defined $p || $_->{_p} > $p); + } + } + # if still none defined, use globals (#2) + $a = ${"$c\::accuracy"} unless defined $a; + $p = ${"$c\::precision"} unless defined $p; + + # no rounding today? + return $self unless defined $a || defined $p; # early out + + # set A and set P is an fatal error + return $self->bnan() if defined $a && defined $p; + + $r = ${"$c\::round_mode"} unless defined $r; + die "Unknown round mode '$r'" if $r !~ /^(even|odd|\+inf|\-inf|zero|trunc)$/; + + # now round, by calling either fround or ffround: + if (defined $a) + { + $self->bround($a,$r) if !defined $self->{_a} || $self->{_a} >= $a; + } + else # both can't be undefined due to early out + { + $self->bfround($p,$r) if !defined $self->{_p} || $self->{_p} <= $p; + } + $self->bnorm(); # after round, normalize + } + +sub bnorm + { + # (numstr or BINT) return BINT + # Normalize number -- no-op here + my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_); + $x; + } + +sub babs + { + # (BINT or num_str) return BINT + # make number absolute, or return absolute BINT from string + my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_); + + return $x if $x->modify('babs'); + # post-normalized abs for internal use (does nothing for NaN) + $x->{sign} =~ s/^-/+/; + $x; + } + +sub bneg + { + # (BINT or num_str) return BINT + # negate number or make a negated number from string + my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_); + + return $x if $x->modify('bneg'); + + # for +0 dont negate (to have always normalized) + $x->{sign} =~ tr/+-/-+/ if !$x->is_zero(); # does nothing for NaN + $x; + } + +sub bcmp + { + # Compares 2 values. Returns one of undef, <0, =0, >0. (suitable for sort) + # (BINT or num_str, BINT or num_str) return cond_code + + # set up parameters + my ($self,$x,$y) = (ref($_[0]),@_); + + # objectify is costly, so avoid it + if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1]))) + { + ($self,$x,$y) = objectify(2,@_); + } + + if (($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; + } + # check sign for speed first + return 1 if $x->{sign} eq '+' && $y->{sign} eq '-'; # does also 0 <=> -y + return -1 if $x->{sign} eq '-' && $y->{sign} eq '+'; # does also -x <=> 0 + + # have same sign, so compare absolute values. Don't make tests for zero here + # because it's actually slower than testin in Calc (especially w/ Pari et al) + + # post-normalized compare for internal use (honors signs) + if ($x->{sign} eq '+') + { + # $x and $y both > 0 + return $CALC->_acmp($x->{value},$y->{value}); + } + + # $x && $y both < 0 + $CALC->_acmp($y->{value},$x->{value}); # swaped (lib returns 0,1,-1) + } + +sub bacmp + { + # Compares 2 values, ignoring their signs. + # Returns one of undef, <0, =0, >0. (suitable for sort) + # (BINT, BINT) return cond_code + + # set up parameters + my ($self,$x,$y) = (ref($_[0]),@_); + # objectify is costly, so avoid it + if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1]))) + { + ($self,$x,$y) = objectify(2,@_); + } + + if (($x->{sign} !~ /^[+-]$/) || ($y->{sign} !~ /^[+-]$/)) + { + # handle +-inf and NaN + return undef if (($x->{sign} eq $nan) || ($y->{sign} eq $nan)); + return 0 if $x->{sign} =~ /^[+-]inf$/ && $y->{sign} =~ /^[+-]inf$/; + return +1; # inf is always bigger + } + $CALC->_acmp($x->{value},$y->{value}); # lib does only 0,1,-1 + } + +sub badd + { + # add second arg (BINT or string) to first (BINT) (modifies first) + # return result as BINT + + # set up parameters + my ($self,$x,$y,@r) = (ref($_[0]),@_); + # objectify is costly, so avoid it + if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1]))) + { + ($self,$x,$y,@r) = objectify(2,@_); + } + + return $x if $x->modify('badd'); + return $upgrade->badd($x,$y,@r) if defined $upgrade && + ((!$x->isa($self)) || (!$y->isa($self))); + + $r[3] = $y; # no push! + # inf and NaN handling + if (($x->{sign} !~ /^[+-]$/) || ($y->{sign} !~ /^[+-]$/)) + { + # NaN first + return $x->bnan() if (($x->{sign} eq $nan) || ($y->{sign} eq $nan)); + # inf handling + if (($x->{sign} =~ /^[+-]inf$/) && ($y->{sign} =~ /^[+-]inf$/)) + { + # +inf++inf or -inf+-inf => same, rest is NaN + return $x if $x->{sign} eq $y->{sign}; + return $x->bnan(); + } + # +-inf + something => +inf + # something +-inf => +-inf + $x->{sign} = $y->{sign}, return $x if $y->{sign} =~ /^[+-]inf$/; + return $x; + } + + my ($sx, $sy) = ( $x->{sign}, $y->{sign} ); # get signs + + if ($sx eq $sy) + { + $x->{value} = $CALC->_add($x->{value},$y->{value}); # same sign, abs add + $x->{sign} = $sx; + } + else + { + my $a = $CALC->_acmp ($y->{value},$x->{value}); # absolute compare + if ($a > 0) + { + #print "swapped sub (a=$a)\n"; + $x->{value} = $CALC->_sub($y->{value},$x->{value},1); # abs sub w/ swap + $x->{sign} = $sy; + } + elsif ($a == 0) + { + # speedup, if equal, set result to 0 + #print "equal sub, result = 0\n"; + $x->{value} = $CALC->_zero(); + $x->{sign} = '+'; + } + else # a < 0 + { + #print "unswapped sub (a=$a)\n"; + $x->{value} = $CALC->_sub($x->{value}, $y->{value}); # abs sub + $x->{sign} = $sx; + } + } + $x->round(@r) if !exists $x->{_f} || $x->{_f} & MB_NEVER_ROUND == 0; + $x; + } + +sub bsub + { + # (BINT or num_str, BINT or num_str) return num_str + # subtract second arg from first, modify first + + # set up parameters + my ($self,$x,$y,@r) = (ref($_[0]),@_); + # objectify is costly, so avoid it + if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1]))) + { + ($self,$x,$y,@r) = objectify(2,@_); + } + + return $x if $x->modify('bsub'); + +# upgrade done by badd(): +# return $upgrade->badd($x,$y,@r) if defined $upgrade && +# ((!$x->isa($self)) || (!$y->isa($self))); + + if ($y->is_zero()) + { + $x->round(@r) if !exists $x->{_f} || $x->{_f} & MB_NEVER_ROUND == 0; + return $x; + } + + $y->{sign} =~ tr/+\-/-+/; # does nothing for NaN + $x->badd($y,@r); # badd does not leave internal zeros + $y->{sign} =~ tr/+\-/-+/; # refix $y (does nothing for NaN) + $x; # already rounded by badd() or no round necc. + } + +sub binc + { + # increment arg by one + my ($self,$x,$a,$p,$r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_); + return $x if $x->modify('binc'); + + if ($x->{sign} eq '+') + { + $x->{value} = $CALC->_inc($x->{value}); + $x->round($a,$p,$r) if !exists $x->{_f} || $x->{_f} & MB_NEVER_ROUND == 0; + return $x; + } + elsif ($x->{sign} eq '-') + { + $x->{value} = $CALC->_dec($x->{value}); + $x->{sign} = '+' if $CALC->_is_zero($x->{value}); # -1 +1 => -0 => +0 + $x->round($a,$p,$r) if !exists $x->{_f} || $x->{_f} & MB_NEVER_ROUND == 0; + return $x; + } + # inf, nan handling etc + $x->badd($self->__one(),$a,$p,$r); # badd does round + } + +sub bdec + { + # decrement arg by one + my ($self,$x,$a,$p,$r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_); + return $x if $x->modify('bdec'); + + my $zero = $CALC->_is_zero($x->{value}) && $x->{sign} eq '+'; + # <= 0 + if (($x->{sign} eq '-') || $zero) + { + $x->{value} = $CALC->_inc($x->{value}); + $x->{sign} = '-' if $zero; # 0 => 1 => -1 + $x->{sign} = '+' if $CALC->_is_zero($x->{value}); # -1 +1 => -0 => +0 + $x->round($a,$p,$r) if !exists $x->{_f} || $x->{_f} & MB_NEVER_ROUND == 0; + return $x; + } + # > 0 + elsif ($x->{sign} eq '+') + { + $x->{value} = $CALC->_dec($x->{value}); + $x->round($a,$p,$r) if !exists $x->{_f} || $x->{_f} & MB_NEVER_ROUND == 0; + return $x; + } + # inf, nan handling etc + $x->badd($self->__one('-'),$a,$p,$r); # badd does round + } + +sub blog + { + # not implemented yet + my ($self,$x,$base,$a,$p,$r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_); + + return $upgrade->blog($x,$base,$a,$p,$r) if defined $upgrade; + + return $x->bnan(); + } + +sub blcm + { + # (BINT or num_str, BINT or num_str) return BINT + # does not modify arguments, but returns new object + # Lowest Common Multiplicator + + my $y = shift; my ($x); + if (ref($y)) + { + $x = $y->copy(); + } + else + { + $x = $class->new($y); + } + while (@_) { $x = __lcm($x,shift); } + $x; + } + +sub bgcd + { + # (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) + + my $y = shift; + $y = __PACKAGE__->new($y) if !ref($y); + my $self = ref($y); + my $x = $y->copy(); # keep arguments + if ($CALC->can('_gcd')) + { + while (@_) + { + $y = shift; $y = $self->new($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 (@_) + { + $y = shift; $y = $self->new($y) if !ref($y); + $x = __gcd($x,$y->copy()); last if $x->is_one(); # _gcd handles NaN + } + } + $x->babs(); + } + +sub bnot + { + # (num_str or BINT) return BINT + # represent ~x as twos-complement number + # we don't need $self, so undef instead of ref($_[0]) make it slightly faster + my ($self,$x,$a,$p,$r) = ref($_[0]) ? (undef,@_) : objectify(1,@_); + + return $x if $x->modify('bnot'); + $x->bneg()->bdec(); # bdec already does round + } + +# is_foo test routines + +sub is_zero + { + # return true if arg (BINT or num_str) is zero (array '+', '0') + # we don't need $self, so undef instead of ref($_[0]) make it slightly faster + my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_); + + return 0 if $x->{sign} !~ /^\+$/; # -, NaN & +-inf aren't + $CALC->_is_zero($x->{value}); + } + +sub is_nan + { + # return true if arg (BINT or num_str) is NaN + my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_); + + return 1 if $x->{sign} eq $nan; + 0; + } + +sub is_inf + { + # return true if arg (BINT or num_str) is +-inf + my ($self,$x,$sign) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_); + + $sign = '' if !defined $sign; + return 1 if $sign eq $x->{sign}; # match ("+inf" eq "+inf") + return 0 if $sign !~ /^([+-]|)$/; + + if ($sign eq '') + { + return 1 if ($x->{sign} =~ /^[+-]inf$/); + return 0; + } + $sign = quotemeta($sign.'inf'); + return 1 if ($x->{sign} =~ /^$sign$/); + 0; + } + +sub is_one + { + # return true if arg (BINT or num_str) is +1 + # or -1 if sign is given + # we don't need $self, so undef instead of ref($_[0]) make it slightly faster + my ($self,$x,$sign) = ref($_[0]) ? (undef,@_) : objectify(1,@_); + + $sign = '' if !defined $sign; $sign = '+' if $sign ne '-'; + + return 0 if $x->{sign} ne $sign; # -1 != +1, NaN, +-inf aren't either + $CALC->_is_one($x->{value}); + } + +sub is_odd + { + # return true when arg (BINT or num_str) is odd, false for even + # we don't need $self, so undef instead of ref($_[0]) make it slightly faster + my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_); + + return 0 if $x->{sign} !~ /^[+-]$/; # NaN & +-inf aren't + $CALC->_is_odd($x->{value}); + } + +sub is_even + { + # return true when arg (BINT or num_str) is even, false for odd + # we don't need $self, so undef instead of ref($_[0]) make it slightly faster + my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_); + + return 0 if $x->{sign} !~ /^[+-]$/; # NaN & +-inf aren't + $CALC->_is_even($x->{value}); + } + +sub is_positive + { + # return true when arg (BINT or num_str) is positive (>= 0) + # we don't need $self, so undef instead of ref($_[0]) make it slightly faster + my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_); + + return 1 if $x->{sign} =~ /^\+/; + 0; + } + +sub is_negative + { + # return true when arg (BINT or num_str) is negative (< 0) + # we don't need $self, so undef instead of ref($_[0]) make it slightly faster + my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_); + + return 1 if ($x->{sign} =~ /^-/); + 0; + } + +sub is_int + { + # return true when arg (BINT or num_str) is an integer + # always true for BigInt, but different for Floats + # we don't need $self, so undef instead of ref($_[0]) make it slightly faster + my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_); + + $x->{sign} =~ /^[+-]$/ ? 1 : 0; # inf/-inf/NaN aren't + } + +############################################################################### + +sub bmul + { + # multiply two numbers -- stolen from Knuth Vol 2 pg 233 + # (BINT or num_str, BINT or num_str) return BINT + + # set up parameters + my ($self,$x,$y,@r) = (ref($_[0]),@_); + # objectify is costly, so avoid it + if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1]))) + { + ($self,$x,$y,@r) = objectify(2,@_); + } + + return $x if $x->modify('bmul'); + + return $x->bnan() if (($x->{sign} eq $nan) || ($y->{sign} eq $nan)); + + # inf handling + if (($x->{sign} =~ /^[+-]inf$/) || ($y->{sign} =~ /^[+-]inf$/)) + { + return $x->bnan() if $x->is_zero() || $y->is_zero(); + # result will always be +-inf: + # +inf * +/+inf => +inf, -inf * -/-inf => +inf + # +inf * -/-inf => -inf, -inf * +/+inf => -inf + return $x->binf() if ($x->{sign} =~ /^\+/ && $y->{sign} =~ /^\+/); + return $x->binf() if ($x->{sign} =~ /^-/ && $y->{sign} =~ /^-/); + return $x->binf('-'); + } + + return $upgrade->bmul($x,$y,@r) + if defined $upgrade && $y->isa($upgrade); + + $r[3] = $y; # no push here + + $x->{sign} = $x->{sign} eq $y->{sign} ? '+' : '-'; # +1 * +1 or -1 * -1 => + + + $x->{value} = $CALC->_mul($x->{value},$y->{value}); # do actual math + $x->{sign} = '+' if $CALC->_is_zero($x->{value}); # no -0 + + $x->round(@r) if !exists $x->{_f} || $x->{_f} & MB_NEVER_ROUND == 0; + $x; + } + +sub _div_inf + { + # helper function that handles +-inf cases for bdiv()/bmod() to reuse code + my ($self,$x,$y) = @_; + + # NaN if x == NaN or y == NaN or x==y==0 + return wantarray ? ($x->bnan(),$self->bnan()) : $x->bnan() + if (($x->is_nan() || $y->is_nan()) || + ($x->is_zero() && $y->is_zero())); + + # +-inf / +-inf == NaN, reminder also NaN + if (($x->{sign} =~ /^[+-]inf$/) && ($y->{sign} =~ /^[+-]inf$/)) + { + return wantarray ? ($x->bnan(),$self->bnan()) : $x->bnan(); + } + # x / +-inf => 0, remainder x (works even if x == 0) + if ($y->{sign} =~ /^[+-]inf$/) + { + my $t = $x->copy(); # bzero clobbers up $x + return wantarray ? ($x->bzero(),$t) : $x->bzero() + } + + # 5 / 0 => +inf, -6 / 0 => -inf + # +inf / 0 = inf, inf, and -inf / 0 => -inf, -inf + # exception: -8 / 0 has remainder -8, not 8 + # exception: -inf / 0 has remainder -inf, not inf + if ($y->is_zero()) + { + # +-inf / 0 => special case for -inf + return wantarray ? ($x,$x->copy()) : $x if $x->is_inf(); + if (!$x->is_zero() && !$x->is_inf()) + { + my $t = $x->copy(); # binf clobbers up $x + return wantarray ? + ($x->binf($x->{sign}),$t) : $x->binf($x->{sign}) + } + } + + # last case: +-inf / ordinary number + my $sign = '+inf'; + $sign = '-inf' if substr($x->{sign},0,1) ne $y->{sign}; + $x->{sign} = $sign; + return wantarray ? ($x,$self->bzero()) : $x; + } + +sub bdiv + { + # (dividend: BINT or num_str, divisor: BINT or num_str) return + # (BINT,BINT) (quo,rem) or BINT (only rem) + + # set up parameters + my ($self,$x,$y,@r) = (ref($_[0]),@_); + # objectify is costly, so avoid it + if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1]))) + { + ($self,$x,$y,@r) = objectify(2,@_); + } + + return $x if $x->modify('bdiv'); + + return $self->_div_inf($x,$y) + if (($x->{sign} !~ /^[+-]$/) || ($y->{sign} !~ /^[+-]$/) || $y->is_zero()); + + return $upgrade->bdiv($upgrade->new($x),$y,@r) + if defined $upgrade && !$y->isa($self); + + $r[3] = $y; # no push! + + # 0 / something + return + wantarray ? ($x->round(@r),$self->bzero(@r)):$x->round(@r) if $x->is_zero(); + + # Is $x in the interval [0, $y) (aka $x <= $y) ? + my $cmp = $CALC->_acmp($x->{value},$y->{value}); + if (($cmp < 0) and (($x->{sign} eq $y->{sign}) or !wantarray)) + { + return $upgrade->bdiv($upgrade->new($x),$upgrade->new($y),@r) + if defined $upgrade; + + return $x->bzero()->round(@r) unless wantarray; + my $t = $x->copy(); # make copy first, because $x->bzero() clobbers $x + return ($x->bzero()->round(@r),$t); + } + elsif ($cmp == 0) + { + # shortcut, both are the same, so set to +/- 1 + $x->__one( ($x->{sign} ne $y->{sign} ? '-' : '+') ); + return $x unless wantarray; + return ($x->round(@r),$self->bzero(@r)); + } + return $upgrade->bdiv($upgrade->new($x),$upgrade->new($y),@r) + if defined $upgrade; + + # calc new sign and in case $y == +/- 1, return $x + my $xsign = $x->{sign}; # keep + $x->{sign} = ($x->{sign} ne $y->{sign} ? '-' : '+'); + # check for / +-1 (cant use $y->is_one due to '-' + if ($CALC->_is_one($y->{value})) + { + return wantarray ? ($x->round(@r),$self->bzero(@r)) : $x->round(@r); + } + + if (wantarray) + { + my $rem = $self->bzero(); + ($x->{value},$rem->{value}) = $CALC->_div($x->{value},$y->{value}); + $x->{sign} = '+' if $CALC->_is_zero($x->{value}); + $rem->{_a} = $x->{_a}; + $rem->{_p} = $x->{_p}; + $x->round(@r); + if (! $CALC->_is_zero($rem->{value})) + { + $rem->{sign} = $y->{sign}; + $rem = $y-$rem if $xsign ne $y->{sign}; # one of them '-' + } + else + { + $rem->{sign} = '+'; # dont leave -0 + } + return ($x,$rem->round(@r)); + } + + $x->{value} = $CALC->_div($x->{value},$y->{value}); + $x->{sign} = '+' if $CALC->_is_zero($x->{value}); + + $x->round(@r) if !exists $x->{_f} || $x->{_f} & MB_NEVER_ROUND == 0; + $x; + } + +############################################################################### +# modulus functions + +sub bmod + { + # modulus (or remainder) + # (BINT or num_str, BINT or num_str) return BINT + + # set up parameters + my ($self,$x,$y,@r) = (ref($_[0]),@_); + # objectify is costly, so avoid it + if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1]))) + { + ($self,$x,$y,@r) = objectify(2,@_); + } + + return $x if $x->modify('bmod'); + $r[3] = $y; # no push! + if (($x->{sign} !~ /^[+-]$/) || ($y->{sign} !~ /^[+-]$/) || $y->is_zero()) + { + my ($d,$r) = $self->_div_inf($x,$y); + $x->{sign} = $r->{sign}; + $x->{value} = $r->{value}; + return $x->round(@r); + } + + if ($CALC->can('_mod')) + { + # calc new sign and in case $y == +/- 1, return $x + $x->{value} = $CALC->_mod($x->{value},$y->{value}); + if (!$CALC->_is_zero($x->{value})) + { + my $xsign = $x->{sign}; + $x->{sign} = $y->{sign}; + if ($xsign ne $y->{sign}) + { + my $t = $CALC->_copy($x->{value}); # copy $x + $x->{value} = $CALC->_copy($y->{value}); # copy $y to $x + $x->{value} = $CALC->_sub($y->{value},$t,1); # $y-$x } } - $prod[$cty] += $car if $car; - $x = shift @prod; - } - ($signr, @x, @prod); -} - -# modulus -sub bmod { #(num_str, num_str) return num_str - (&bdiv(@_))[$[+1]; -} - -sub bdiv { #(dividend: num_str, divisor: num_str) return num_str - local (*x, *y); ($x, $y) = (&bnorm($_[$[]), &bnorm($_[$[+1])); - return wantarray ? ('NaN','NaN') : 'NaN' - if ($x eq 'NaN' || $y eq 'NaN' || $y eq '+0'); - return wantarray ? ('+0',$x) : '+0' if (&cmp(&abs($x),&abs($y)) < 0); - @x = &internal($x); @y = &internal($y); - $srem = $y[$[]; - $sr = (shift @x ne shift @y) ? '-' : '+'; - $car = $bar = $prd = 0; - if (($dd = int(1e5/($y[$#y]+1))) != 1) { - for $x (@x) { - $x = $x * $dd + $car; - if ($use_mult) { - $x -= ($car = int($x * 1e-5)) * 1e5; - } - else { - $x -= ($car = int($x / 1e5)) * 1e5; - } - } - push(@x, $car); $car = 0; - for $y (@y) { - $y = $y * $dd + $car; - if ($use_mult) { - $y -= ($car = int($y * 1e-5)) * 1e5; - } - else { - $y -= ($car = int($y / 1e5)) * 1e5; - } - } + else + { + $x->{sign} = '+'; # dont leave -0 + } + $x->round(@r) if !exists $x->{_f} || $x->{_f} & MB_NEVER_ROUND == 0; + return $x; } - else { - push(@x, 0); - } - @q = (); ($v2,$v1) = @y[-2,-1]; - $v2 = 0 unless $v2; - while ($#x > $#y) { - ($u2,$u1,$u0) = @x[-3..-1]; - $u2 = 0 unless $u2; - $q = (($u0 == $v1) ? 99999 : int(($u0*1e5+$u1)/$v1)); - --$q while ($v2*$q > ($u0*1e5+$u1-$q*$v1)*1e5+$u2); - if ($q) { - ($car, $bar) = (0,0); - for ($y = $[, $x = $#x-$#y+$[-1; $y <= $#y; ++$y,++$x) { - $prd = $q * $y[$y] + $car; - if ($use_mult) { - $prd -= ($car = int($prd * 1e-5)) * 1e5; - } - else { - $prd -= ($car = int($prd / 1e5)) * 1e5; - } - $x[$x] += 1e5 if ($bar = (($x[$x] -= $prd + $bar) < 0)); - } - if ($x[$#x] < $car + $bar) { - $car = 0; --$q; - for ($y = $[, $x = $#x-$#y+$[-1; $y <= $#y; ++$y,++$x) { - $x[$x] -= 1e5 - if ($car = (($x[$x] += $y[$y] + $car) > 1e5)); - } - } - } - pop(@x); unshift(@q, $q); - } - if (wantarray) { - @d = (); - if ($dd != 1) { - $car = 0; - for $x (reverse @x) { - $prd = $car * 1e5 + $x; - $car = $prd - ($tmp = int($prd / $dd)) * $dd; - unshift(@d, $tmp); - } - } - else { - @d = @x; - } - (&external($sr, @q), &external($srem, @d, $zero)); - } else { - &external($sr, @q); - } -} - -# compute power of two numbers -- stolen from Knuth Vol 2 pg 233 -sub bpow { #(num_str, num_str) return num_str - local(*x, *y); ($x, $y) = (&bnorm($_[$[]), &bnorm($_[$[+1])); - if ($x eq 'NaN') { - 'NaN'; - } elsif ($y eq 'NaN') { - 'NaN'; - } elsif ($x eq '+1') { - '+1'; - } elsif ($x eq '-1') { - &bmod($x,2) ? '-1': '+1'; - } elsif ($y =~ /^-/) { - 'NaN'; - } elsif ($x eq '+0' && $y eq '+0') { - 'NaN'; - } else { - @x = &internal($x); - local(@pow2)=@x; - local(@pow)=&internal("+1"); - local($y1,$res,@tmp1,@tmp2)=(1); # need tmp to send to mul - while ($y ne '+0') { - ($y,$res)=&bdiv($y,2); - if ($res ne '+0') {@tmp=@pow2; @pow=&mul(*pow,*tmp);} - if ($y ne '+0') {@tmp=@pow2;@pow2=&mul(*pow2,*tmp);} - } - &external(@pow); - } -} - -# compute x << y, y >= 0 -sub blsft { #(num_str, num_str) return num_str - &bmul($_[$[], &bpow(2, $_[$[+1])); -} - -# compute x >> y, y >= 0 -sub brsft { #(num_str, num_str) return num_str - &bdiv($_[$[], &bpow(2, $_[$[+1])); -} - -# compute x & y -sub band { #(num_str, num_str) return num_str - local($x,$y,$r,$m,$xr,$yr) = (&bnorm($_[$[]),&bnorm($_[$[+1]),0,1); - if ($x eq 'NaN' || $y eq 'NaN') { - 'NaN'; - } else { - while ($x ne '+0' && $y ne '+0') { - ($x, $xr) = &bdiv($x, 0x10000); - ($y, $yr) = &bdiv($y, 0x10000); - $r = &badd(&bmul(int $xr & $yr, $m), $r); - $m = &bmul($m, 0x10000); + my ($t,$rem) = $self->bdiv($x->copy(),$y,@r); # slow way (also rounds) + # modify in place + foreach (qw/value sign _a _p/) + { + $x->{$_} = $rem->{$_}; + } + $x; + } + +sub bmodinv + { + # modular inverse. given a number which is (hopefully) relatively + # prime to the modulus, calculate its inverse using Euclid's + # alogrithm. if the number is not relatively prime to the modulus + # (i.e. their gcd is not one) then NaN is returned. + + # set up parameters + my ($self,$x,$y,@r) = (ref($_[0]),@_); + # objectify is costly, so avoid it + if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1]))) + { + ($self,$x,$y,@r) = objectify(2,@_); + } + + return $x if $x->modify('bmodinv'); + + return $x->bnan() + if ($y->{sign} ne '+' # -, NaN, +inf, -inf + || $x->is_zero() # or num == 0 + || $x->{sign} !~ /^[+-]$/ # or num NaN, inf, -inf + ); + + # put least residue into $x if $x was negative, and thus make it positive + $x->bmod($y) if $x->{sign} eq '-'; + + if ($CALC->can('_modinv')) + { + $x->{value} = $CALC->_modinv($x->{value},$y->{value}); + $x->bnan() if !defined $x->{value} ; # in case there was none + return $x; + } + + my ($u, $u1) = ($self->bzero(), $self->bone()); + my ($a, $b) = ($y->copy(), $x->copy()); + + # first step need always be done since $num (and thus $b) is never 0 + # Note that the loop is aligned so that the check occurs between #2 and #1 + # thus saving us one step #2 at the loop end. Typical loop count is 1. Even + # a case with 28 loops still gains about 3% with this layout. + my $q; + ($a, $q, $b) = ($b, $a->bdiv($b)); # step #1 + # Euclid's Algorithm + while (!$b->is_zero()) + { + ($u, $u1) = ($u1, $u->bsub($u1->copy()->bmul($q))); # step #2 + ($a, $q, $b) = ($b, $a->bdiv($b)); # step #1 again + } + + # if the gcd is not 1, then return NaN! It would be pointless to + # have called bgcd to check this first, because we would then be performing + # the same Euclidean Algorithm *twice* + return $x->bnan() unless $a->is_one(); + + $u1->bmod($y); + $x->{value} = $u1->{value}; + $x->{sign} = $u1->{sign}; + $x; + } + +sub bmodpow + { + # takes a very large number to a very large exponent in a given very + # large modulus, quickly, thanks to binary exponentation. supports + # negative exponents. + my ($self,$num,$exp,$mod,@r) = objectify(3,@_); + + return $num if $num->modify('bmodpow'); + + # check modulus for valid values + return $num->bnan() if ($mod->{sign} ne '+' # NaN, - , -inf, +inf + || $mod->is_zero()); + + # check exponent for valid values + if ($exp->{sign} =~ /\w/) + { + # i.e., if it's NaN, +inf, or -inf... + return $num->bnan(); + } + + $num->bmodinv ($mod) if ($exp->{sign} eq '-'); + + # check num for valid values (also NaN if there was no inverse but $exp < 0) + return $num->bnan() if $num->{sign} !~ /^[+-]$/; + + if ($CALC->can('_modpow')) + { + # $mod is positive, sign on $exp is ignored, result also positive + $num->{value} = $CALC->_modpow($num->{value},$exp->{value},$mod->{value}); + return $num; + } + + # in the trivial case, + return $num->bzero(@r) if $mod->is_one(); + return $num->bone('+',@r) if $num->is_zero() or $num->is_one(); + + # $num->bmod($mod); # if $x is large, make it smaller first + my $acc = $num->copy(); # but this is not really faster... + + $num->bone(); # keep ref to $num + + my $expbin = $exp->as_bin(); $expbin =~ s/^[-]?0b//; # ignore sign and prefix + my $len = length($expbin); + while (--$len >= 0) + { + if( substr($expbin,$len,1) eq '1') + { + $num->bmul($acc)->bmod($mod); + } + $acc->bmul($acc)->bmod($mod); + } + + $num; + } + +############################################################################### + +sub bfac + { + # (BINT or num_str, BINT or num_str) return BINT + # compute factorial numbers + # modifies first argument + my ($self,$x,@r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_); + + return $x if $x->modify('bfac'); + + return $x->bnan() if $x->{sign} ne '+'; # inf, NnN, <0 etc => NaN + return $x->bone('+',@r) if $x->is_zero() || $x->is_one(); # 0 or 1 => 1 + + if ($CALC->can('_fac')) + { + $x->{value} = $CALC->_fac($x->{value}); + return $x->round(@r); + } + + my $n = $x->copy(); + $x->bone(); + # seems we need not to temp. clear A/P of $x since the result is the same + my $f = $self->new(2); + while ($f->bacmp($n) < 0) + { + $x->bmul($f); $f->binc(); + } + $x->bmul($f,@r); # last step and also round + } + +sub bpow + { + # (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 + + # set up parameters + my ($self,$x,$y,@r) = (ref($_[0]),@_); + # objectify is costly, so avoid it + if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1]))) + { + ($self,$x,$y,@r) = objectify(2,@_); + } + + return $x if $x->modify('bpow'); + + return $upgrade->bpow($upgrade->new($x),$y,@r) + if defined $upgrade && !$y->isa($self); + + $r[3] = $y; # no push! + return $x if $x->{sign} =~ /^[+-]inf$/; # -inf/+inf ** x + return $x->bnan() if $x->{sign} eq $nan || $y->{sign} eq $nan; + return $x->bone('+',@r) if $y->is_zero(); + return $x->round(@r) if $x->is_one() || $y->is_one(); + if ($x->{sign} eq '-' && $CALC->_is_one($x->{value})) + { + # if $x == -1 and odd/even y => +1/-1 + return $y->is_odd() ? $x->round(@r) : $x->babs()->round(@r); + # my Casio FX-5500L has a bug here: -1 ** 2 is -1, but -1 * -1 is 1; + } + # 1 ** -y => 1 / (1 ** |y|) + # so do test for negative $y after above's clause + return $x->bnan() if $y->{sign} eq '-'; + return $x->round(@r) if $x->is_zero(); # 0**y => 0 (if not y <= 0) + + if ($CALC->can('_pow')) + { + $x->{value} = $CALC->_pow($x->{value},$y->{value}); + $x->round(@r) if !exists $x->{_f} || $x->{_f} & MB_NEVER_ROUND == 0; + return $x; + } + +# 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 since brsft/blsft use bpow sometimes. +# 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(@r); +# } + + my $pow2 = $self->__one(); + my $y_bin = $y->as_bin(); $y_bin =~ s/^0b//; + my $len = length($y_bin); + while (--$len > 0) + { + $pow2->bmul($x) if substr($y_bin,$len,1) eq '1'; # is odd? + $x->bmul($x); + } + $x->bmul($pow2); + $x->round(@r) if !exists $x->{_f} || $x->{_f} & MB_NEVER_ROUND == 0; + $x; + } + +sub blsft + { + # (BINT or num_str, BINT or num_str) return BINT + # compute x << y, base n, y >= 0 + + # set up parameters + my ($self,$x,$y,$n,@r) = (ref($_[0]),@_); + # objectify is costly, so avoid it + if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1]))) + { + ($self,$x,$y,$n,@r) = objectify(2,@_); + } + + return $x if $x->modify('blsft'); + return $x->bnan() if ($x->{sign} !~ /^[+-]$/ || $y->{sign} !~ /^[+-]$/); + return $x->round(@r) if $y->is_zero(); + + $n = 2 if !defined $n; return $x->bnan() if $n <= 0 || $y->{sign} eq '-'; + + my $t; $t = $CALC->_lsft($x->{value},$y->{value},$n) if $CALC->can('_lsft'); + if (defined $t) + { + $x->{value} = $t; return $x->round(@r); + } + # fallback + return $x->bmul( $self->bpow($n, $y, @r), @r ); + } + +sub brsft + { + # (BINT or num_str, BINT or num_str) return BINT + # compute x >> y, base n, y >= 0 + + # set up parameters + my ($self,$x,$y,$n,@r) = (ref($_[0]),@_); + # objectify is costly, so avoid it + if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1]))) + { + ($self,$x,$y,$n,@r) = objectify(2,@_); + } + + return $x if $x->modify('brsft'); + return $x->bnan() if ($x->{sign} !~ /^[+-]$/ || $y->{sign} !~ /^[+-]$/); + return $x->round(@r) if $y->is_zero(); + return $x->bzero(@r) if $x->is_zero(); # 0 => 0 + + $n = 2 if !defined $n; return $x->bnan() if $n <= 0 || $y->{sign} eq '-'; + + # this only works for negative numbers when shifting in base 2 + if (($x->{sign} eq '-') && ($n == 2)) + { + return $x->round(@r) if $x->is_one('-'); # -1 => -1 + if (!$y->is_one()) + { + # although this is O(N*N) in calc (as_bin!) it is O(N) in Pari et al + # but perhaps there is a better emulation for two's complement shift... + # if $y != 1, we must simulate it by doing: + # convert to bin, flip all bits, shift, and be done + $x->binc(); # -3 => -2 + my $bin = $x->as_bin(); + $bin =~ s/^-0b//; # strip '-0b' prefix + $bin =~ tr/10/01/; # flip bits + # now shift + if (CORE::length($bin) <= $y) + { + $bin = '0'; # shifting to far right creates -1 + # 0, because later increment makes + # that 1, attached '-' makes it '-1' + # because -1 >> x == -1 ! + } + else + { + $bin =~ s/.{$y}$//; # cut off at the right side + $bin = '1' . $bin; # extend left side by one dummy '1' + $bin =~ tr/10/01/; # flip bits back } - $r; - } -} - -# compute x | y -sub bior { #(num_str, num_str) return num_str - local($x,$y,$r,$m,$xr,$yr) = (&bnorm($_[$[]),&bnorm($_[$[+1]),0,1); - if ($x eq 'NaN' || $y eq 'NaN') { - 'NaN'; - } else { - while ($x ne '+0' || $y ne '+0') { - ($x, $xr) = &bdiv($x, 0x10000); - ($y, $yr) = &bdiv($y, 0x10000); - $r = &badd(&bmul(int $xr | $yr, $m), $r); - $m = &bmul($m, 0x10000); + my $res = $self->new('0b'.$bin); # add prefix and convert back + $res->binc(); # remember to increment + $x->{value} = $res->{value}; # take over value + return $x->round(@r); # we are done now, magic, isn't? + } + $x->bdec(); # n == 2, but $y == 1: this fixes it + } + + my $t; $t = $CALC->_rsft($x->{value},$y->{value},$n) if $CALC->can('_rsft'); + if (defined $t) + { + $x->{value} = $t; + return $x->round(@r); + } + # fallback + $x->bdiv($self->bpow($n,$y, @r), @r); + $x; + } + +sub band + { + #(BINT or num_str, BINT or num_str) return BINT + # compute x & y + + # set up parameters + my ($self,$x,$y,@r) = (ref($_[0]),@_); + # objectify is costly, so avoid it + if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1]))) + { + ($self,$x,$y,@r) = objectify(2,@_); + } + + return $x if $x->modify('band'); + + $r[3] = $y; # no push! + local $Math::BigInt::upgrade = undef; + + return $x->bnan() if ($x->{sign} !~ /^[+-]$/ || $y->{sign} !~ /^[+-]$/); + return $x->bzero(@r) if $y->is_zero() || $x->is_zero(); + + my $sign = 0; # sign of result + $sign = 1 if ($x->{sign} eq '-') && ($y->{sign} eq '-'); + my $sx = 1; $sx = -1 if $x->{sign} eq '-'; + my $sy = 1; $sy = -1 if $y->{sign} eq '-'; + + if ($CALC->can('_and') && $sx == 1 && $sy == 1) + { + $x->{value} = $CALC->_and($x->{value},$y->{value}); + return $x->round(@r); + } + + my $m = $self->bone(); my ($xr,$yr); + my $x10000 = $self->new (0x1000); + my $y1 = copy(ref($x),$y); # make copy + $y1->babs(); # and positive + my $x1 = $x->copy()->babs(); $x->bzero(); # modify x in place! + use integer; # need this for negative bools + while (!$x1->is_zero() && !$y1->is_zero()) + { + ($x1, $xr) = bdiv($x1, $x10000); + ($y1, $yr) = bdiv($y1, $x10000); + # make both op's numbers! + $x->badd( bmul( $class->new( + abs($sx*int($xr->numify()) & $sy*int($yr->numify()))), + $m)); + $m->bmul($x10000); + } + $x->bneg() if $sign; + $x->round(@r); + } + +sub bior + { + #(BINT or num_str, BINT or num_str) return BINT + # compute x | y + + # set up parameters + my ($self,$x,$y,@r) = (ref($_[0]),@_); + # objectify is costly, so avoid it + if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1]))) + { + ($self,$x,$y,@r) = objectify(2,@_); + } + + return $x if $x->modify('bior'); + $r[3] = $y; # no push! + + local $Math::BigInt::upgrade = undef; + + return $x->bnan() if ($x->{sign} !~ /^[+-]$/ || $y->{sign} !~ /^[+-]$/); + return $x->round(@r) if $y->is_zero(); + + my $sign = 0; # sign of result + $sign = 1 if ($x->{sign} eq '-') || ($y->{sign} eq '-'); + my $sx = 1; $sx = -1 if $x->{sign} eq '-'; + my $sy = 1; $sy = -1 if $y->{sign} eq '-'; + + # don't use lib for negative values + if ($CALC->can('_or') && $sx == 1 && $sy == 1) + { + $x->{value} = $CALC->_or($x->{value},$y->{value}); + return $x->round(@r); + } + + my $m = $self->bone(); my ($xr,$yr); + my $x10000 = $self->new(0x10000); + my $y1 = copy(ref($x),$y); # make copy + $y1->babs(); # and positive + my $x1 = $x->copy()->babs(); $x->bzero(); # modify x in place! + use integer; # need this for negative bools + while (!$x1->is_zero() || !$y1->is_zero()) + { + ($x1, $xr) = bdiv($x1,$x10000); + ($y1, $yr) = bdiv($y1,$x10000); + # make both op's numbers! + $x->badd( bmul( $class->new( + abs($sx*int($xr->numify()) | $sy*int($yr->numify()))), + $m)); + $m->bmul($x10000); + } + $x->bneg() if $sign; + $x->round(@r); + } + +sub bxor + { + #(BINT or num_str, BINT or num_str) return BINT + # compute x ^ y + + # set up parameters + my ($self,$x,$y,@r) = (ref($_[0]),@_); + # objectify is costly, so avoid it + if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1]))) + { + ($self,$x,$y,@r) = objectify(2,@_); + } + + return $x if $x->modify('bxor'); + $r[3] = $y; # no push! + + local $Math::BigInt::upgrade = undef; + + return $x->bnan() if ($x->{sign} !~ /^[+-]$/ || $y->{sign} !~ /^[+-]$/); + return $x->round(@r) if $y->is_zero(); + + my $sign = 0; # sign of result + $sign = 1 if $x->{sign} ne $y->{sign}; + my $sx = 1; $sx = -1 if $x->{sign} eq '-'; + my $sy = 1; $sy = -1 if $y->{sign} eq '-'; + + # don't use lib for negative values + if ($CALC->can('_xor') && $sx == 1 && $sy == 1) + { + $x->{value} = $CALC->_xor($x->{value},$y->{value}); + return $x->round(@r); + } + + my $m = $self->bone(); my ($xr,$yr); + my $x10000 = $self->new(0x10000); + my $y1 = copy(ref($x),$y); # make copy + $y1->babs(); # and positive + my $x1 = $x->copy()->babs(); $x->bzero(); # modify x in place! + use integer; # need this for negative bools + while (!$x1->is_zero() || !$y1->is_zero()) + { + ($x1, $xr) = bdiv($x1, $x10000); + ($y1, $yr) = bdiv($y1, $x10000); + # make both op's numbers! + $x->badd( bmul( $class->new( + abs($sx*int($xr->numify()) ^ $sy*int($yr->numify()))), + $m)); + $m->bmul($x10000); + } + $x->bneg() if $sign; + $x->round(@r); + } + +sub length + { + my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_); + + my $e = $CALC->_len($x->{value}); + return wantarray ? ($e,0) : $e; + } + +sub digit + { + # return the nth decimal digit, negative values count backward, 0 is right + my ($self,$x,$n) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_); + + $CALC->_digit($x->{value},$n||0); + } + +sub _trailing_zeros + { + # return the amount of trailing zeros in $x + my $x = shift; + $x = $class->new($x) unless ref $x; + + return 0 if $x->is_zero() || $x->is_odd() || $x->{sign} !~ /^[+-]$/; + + return $CALC->_zeros($x->{value}) if $CALC->can('_zeros'); + + # if not: since we do not know underlying internal representation: + my $es = "$x"; $es =~ /([0]*)$/; + return 0 if !defined $1; # no zeros + CORE::length("$1"); # as string, not as +0! + } + +sub bsqrt + { + my ($self,$x,@r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_); + + return $x if $x->modify('bsqrt'); + + return $x->bnan() if $x->{sign} ne '+'; # -x or inf or NaN => NaN + return $x->bzero(@r) if $x->is_zero(); # 0 => 0 + return $x->round(@r) if $x->is_one(); # 1 => 1 + + return $upgrade->bsqrt($x,@r) if defined $upgrade; + + if ($CALC->can('_sqrt')) + { + $x->{value} = $CALC->_sqrt($x->{value}); + return $x->round(@r); + } + + return $x->bone('+',@r) if $x < 4; # 2,3 => 1 + my $y = $x->copy(); + my $l = int($x->length()/2); + + $x->bone(); # keep ref($x), but modify it + $x->blsft($l,10); + + my $last = $self->bzero(); + my $two = $self->new(2); + my $lastlast = $x+$two; + while ($last != $x && $lastlast != $x) + { + $lastlast = $last; $last = $x; + $x += $y / $x; + $x /= $two; + } + $x-- if $x * $x > $y; # overshot? + $x->round(@r); + } + +sub exponent + { + # return a copy of the exponent (here always 0, NaN or 1 for $m == 0) + my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_); + + if ($x->{sign} !~ /^[+-]$/) + { + my $s = $x->{sign}; $s =~ s/^[+-]//; + return $self->new($s); # -inf,+inf => inf + } + my $e = $class->bzero(); + return $e->binc() if $x->is_zero(); + $e += $x->_trailing_zeros(); + $e; + } + +sub mantissa + { + # return the mantissa (compatible to Math::BigFloat, e.g. reduced) + my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_); + + if ($x->{sign} !~ /^[+-]$/) + { + return $self->new($x->{sign}); # keep + or - sign + } + my $m = $x->copy(); + # that's inefficient + my $zeros = $m->_trailing_zeros(); + $m->brsft($zeros,10) if $zeros != 0; + $m; + } + +sub parts + { + # return a copy of both the exponent and the mantissa + my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_); + + return ($x->mantissa(),$x->exponent()); + } + +############################################################################## +# rounding functions + +sub bfround + { + # precision: round to the $Nth digit left (+$n) or right (-$n) from the '.' + # $n == 0 || $n == 1 => round to integer + my $x = shift; $x = $class->new($x) unless ref $x; + my ($scale,$mode) = $x->_scale_p($x->precision(),$x->round_mode(),@_); + return $x if !defined $scale; # no-op + return $x if $x->modify('bfround'); + + # no-op for BigInts if $n <= 0 + if ($scale <= 0) + { + $x->{_a} = undef; # clear an eventual set A + $x->{_p} = $scale; return $x; + } + + $x->bround( $x->length()-$scale, $mode); + $x->{_a} = undef; # bround sets {_a} + $x->{_p} = $scale; # so correct it + $x; + } + +sub _scan_for_nonzero + { + 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; + + # since we do not know underlying represention of $x, use decimal string + #my $r = substr ($$xs,-$follow); + my $r = substr ("$x",-$follow); + return 1 if $r =~ /[^0]/; + 0; + } + +sub fround + { + # to make life easier for switch between MBF and MBI (autoload fxxx() + # like MBF does for bxxx()?) + my $x = shift; + return $x->bround(@_); + } + +sub bround + { + # accuracy: +$n preserve $n digits from left, + # -$n preserve $n digits from right (f.i. for 0.1234 style in MBF) + # no-op for $n == 0 + # and overwrite the rest with 0's, return normalized number + # do not return $x->bnorm(), but $x + + my $x = shift; $x = $class->new($x) unless ref $x; + my ($scale,$mode) = $x->_scale_a($x->accuracy(),$x->round_mode(),@_); + return $x if !defined $scale; # no-op + return $x if $x->modify('bround'); + + if ($x->is_zero() || $scale == 0) + { + $x->{_a} = $scale if !defined $x->{_a} || $x->{_a} > $scale; # 3 > 2 + return $x; + } + return $x if $x->{sign} !~ /^[+-]$/; # inf, NaN + + # we have fewer digits than we want to scale to + my $len = $x->length(); + # scale < 0, but > -len (not >=!) + if (($scale < 0 && $scale < -$len-1) || ($scale >= $len)) + { + $x->{_a} = $scale if !defined $x->{_a} || $x->{_a} > $scale; # 3 > 2 + return $x; + } + + # count of 0's to pad, from left (+) or right (-): 9 - +6 => 3, or |-6| => 6 + my ($pad,$digit_round,$digit_after); + $pad = $len - $scale; + $pad = abs($scale-1) if $scale < 0; + + # do not use digit(), it is costly for binary => decimal + + my $xs = $CALC->_str($x->{value}); + my $pl = -$pad-1; + + # pad: 123: 0 => -1, at 1 => -2, at 2 => -3, at 3 => -4 + # pad+1: 123: 0 => 0, at 1 => -1, at 2 => -2, at 3 => -3 + $digit_round = '0'; $digit_round = substr($$xs,$pl,1) if $pad <= $len; + $pl++; $pl ++ if $pad >= $len; + $digit_after = '0'; $digit_after = substr($$xs,$pl,1) if $pad > 0; + + # in case of 01234 we round down, for 6789 up, and only in case 5 we look + # closer at the remaining digits of the original $x, remember decision + my $round_up = 1; # default round up + $round_up -- if + ($mode eq 'trunc') || # trunc by round down + ($digit_after =~ /[01234]/) || # round down anyway, + # 6789 => round up + ($digit_after eq '5') && # not 5000...0000 + ($x->_scan_for_nonzero($pad,$xs) == 0) && + ( + ($mode eq 'even') && ($digit_round =~ /[24680]/) || + ($mode eq 'odd') && ($digit_round =~ /[13579]/) || + ($mode eq '+inf') && ($x->{sign} eq '-') || + ($mode eq '-inf') && ($x->{sign} eq '+') || + ($mode eq 'zero') # round down if zero, sign adjusted below + ); + my $put_back = 0; # not yet modified + + if (($pad > 0) && ($pad <= $len)) + { + substr($$xs,-$pad,$pad) = '0' x $pad; + $put_back = 1; + } + elsif ($pad > $len) + { + $x->bzero(); # round to '0' + } + + if ($round_up) # what gave test above? + { + $put_back = 1; + $pad = $len, $$xs = '0'x$pad if $scale < 0; # tlr: whack 0.51=>1.0 + + # we modify directly the string variant instead of creating a number and + # adding it, since that is faster (we already have the string) + my $c = 0; $pad ++; # for $pad == $len case + while ($pad <= $len) + { + $c = substr($$xs,-$pad,1) + 1; $c = '0' if $c eq '10'; + substr($$xs,-$pad,1) = $c; $pad++; + last if $c != 0; # no overflow => early out + } + $$xs = '1'.$$xs if $c == 0; + + } + $x->{value} = $CALC->_new($xs) if $put_back == 1; # put back in if needed + + $x->{_a} = $scale if $scale >= 0; + if ($scale < 0) + { + $x->{_a} = $len+$scale; + $x->{_a} = 0 if $scale < -$len; + } + $x; + } + +sub bfloor + { + # return integer less or equal then number, since it is already integer, + # always returns $self + my ($self,$x,@r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_); + + $x->round(@r); + } + +sub bceil + { + # return integer greater or equal then number, since it is already integer, + # always returns $self + my ($self,$x,@r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_); + + $x->round(@r); + } + +############################################################################## +# private stuff (internal use only) + +sub __one + { + # internal speedup, set argument to 1, or create a +/- 1 + my $self = shift; + my $x = $self->bone(); # $x->{value} = $CALC->_one(); + $x->{sign} = shift || '+'; + $x; + } + +sub _swap + { + # Overload will swap params if first one is no object ref so that the first + # one is always an object ref. In this case, third param is true. + # This routine is to overcome the effect of scalar,$object creating an object + # of the class of this package, instead of the second param $object. This + # happens inside overload, when the overload section of this package is + # inherited by sub classes. + # For overload cases (and this is used only there), we need to preserve the + # args, hence the copy(). + # You can override this method in a subclass, the overload section will call + # $object->_swap() to make sure it arrives at the proper subclass, with some + # exceptions like '+' and '-'. To make '+' and '-' work, you also need to + # specify your own overload for them. + + # object, (object|scalar) => preserve first and make copy + # scalar, object => swapped, re-swap and create new from first + # (using class of second object, not $class!!) + my $self = shift; # for override in subclass + if ($_[2]) + { + my $c = ref ($_[0]) || $class; # fallback $class should not happen + return ( $c->new($_[1]), $_[0] ); + } + return ( $_[0]->copy(), $_[1] ); + } + +sub objectify + { + # check for strings, if yes, return objects instead + + # the first argument is number of args objectify() should look at it will + # return $count+1 elements, the first will be a classname. This is because + # overloaded '""' calls bstr($object,undef,undef) and this would result in + # useless objects beeing created and thrown away. So we cannot simple loop + # over @_. If the given count is 0, all arguments will be used. + + # If the second arg is a ref, use it as class. + # If not, try to use it as classname, unless undef, then use $class + # (aka Math::BigInt). The latter shouldn't happen,though. + + # caller: gives us: + # $x->badd(1); => ref x, scalar y + # Class->badd(1,2); => classname x (scalar), scalar x, scalar y + # Class->badd( Class->(1),2); => classname x (scalar), ref x, scalar y + # Math::BigInt::badd(1,2); => scalar x, scalar y + # In the last case we check number of arguments to turn it silently into + # $class,1,2. (We can not take '1' as class ;o) + # badd($class,1) is not supported (it should, eventually, try to add undef) + # currently it tries 'Math::BigInt' + 1, which will not work. + + # some shortcut for the common cases + # $x->unary_op(); + return (ref($_[1]),$_[1]) if (@_ == 2) && ($_[0]||0 == 1) && ref($_[1]); + + my $count = abs(shift || 0); + + my (@a,$k,$d); # resulting array, temp, and downgrade + if (ref $_[0]) + { + # okay, got object as first + $a[0] = ref $_[0]; + } + else + { + # nope, got 1,2 (Class->xxx(1) => Class,1 and not supported) + $a[0] = $class; + $a[0] = shift if $_[0] =~ /^[A-Z].*::/; # classname as first? + } + + no strict 'refs'; + # disable downgrading, because Math::BigFLoat->foo('1.0','2.0') needs floats + if (defined ${"$a[0]::downgrade"}) + { + $d = ${"$a[0]::downgrade"}; + ${"$a[0]::downgrade"} = undef; + } + + my $up = ${"$a[0]::upgrade"}; + # print "Now in objectify, my class is today $a[0]\n"; + if ($count == 0) + { + while (@_) + { + $k = shift; + if (!ref($k)) + { + $k = $a[0]->new($k); + } + elsif (!defined $up && ref($k) ne $a[0]) + { + # foreign object, try to convert to integer + $k->can('as_number') ? $k = $k->as_number() : $k = $a[0]->new($k); } - $r; - } -} - -# compute x ^ y -sub bxor { #(num_str, num_str) return num_str - local($x,$y,$r,$m,$xr,$yr) = (&bnorm($_[$[]),&bnorm($_[$[+1]),0,1); - if ($x eq 'NaN' || $y eq 'NaN') { - 'NaN'; - } else { - while ($x ne '+0' || $y ne '+0') { - ($x, $xr) = &bdiv($x, 0x10000); - ($y, $yr) = &bdiv($y, 0x10000); - $r = &badd(&bmul(int $xr ^ $yr, $m), $r); - $m = &bmul($m, 0x10000); + push @a,$k; + } + } + else + { + while ($count > 0) + { + $count--; + $k = shift; + if (!ref($k)) + { + $k = $a[0]->new($k); + } + elsif (!defined $up && ref($k) ne $a[0]) + { + # foreign object, try to convert to integer + $k->can('as_number') ? $k = $k->as_number() : $k = $a[0]->new($k); } - $r; + push @a,$k; + } + push @a,@_; # return other params, too + } + die "$class objectify needs list context" unless wantarray; + ${"$a[0]::downgrade"} = $d; + @a; + } + +sub import + { + my $self = shift; + + $IMPORT++; + my @a; my $l = scalar @_; + for ( my $i = 0; $i < $l ; $i++ ) + { + if ($_[$i] eq ':constant') + { + # this causes overlord er load to step in + overload::constant integer => sub { $self->new(shift) }; + overload::constant binary => sub { $self->new(shift) }; + } + elsif ($_[$i] eq 'upgrade') + { + # this causes upgrading + $upgrade = $_[$i+1]; # or undef to disable + $i++; + } + elsif ($_[$i] =~ /^lib$/i) + { + # this causes a different low lib to take care... + $CALC = $_[$i+1] || ''; + $i++; + } + else + { + push @a, $_[$i]; + } + } + # any non :constant stuff is handled by our parent, Exporter + # even if @_ is empty, to give it a chance + $self->SUPER::import(@a); # need it for subclasses + $self->export_to_level(1,$self,@a); # need it for MBF + + # try to load core math lib + my @c = split /\s*,\s*/,$CALC; + push @c,'Calc'; # if all fail, try this + $CALC = ''; # signal error + foreach my $lib (@c) + { + next if ($lib || '') eq ''; + $lib = 'Math::BigInt::'.$lib if $lib !~ /^Math::BigInt/i; + $lib =~ s/\.pm$//; + if ($] < 5.006) + { + # Perl < 5.6.0 dies with "out of memory!" when eval() and ':constant' is + # used in the same script, or eval inside import(). + my @parts = split /::/, $lib; # Math::BigInt => Math BigInt + my $file = pop @parts; $file .= '.pm'; # BigInt => BigInt.pm + require File::Spec; + $file = File::Spec->catfile (@parts, $file); + eval { require "$file"; $lib->import( @c ); } + } + else + { + eval "use $lib qw/@c/;"; + } + $CALC = $lib, last if $@ eq ''; # no error in loading lib? + } + die "Couldn't load any math lib, not even the default" if $CALC eq ''; + } + +sub __from_hex + { + # convert a (ref to) big hex string to BigInt, return undef for error + my $hs = shift; + + my $x = Math::BigInt->bzero(); + + # strip underscores + $$hs =~ s/([0-9a-fA-F])_([0-9a-fA-F])/$1$2/g; + $$hs =~ s/([0-9a-fA-F])_([0-9a-fA-F])/$1$2/g; + + return $x->bnan() if $$hs !~ /^[\-\+]?0x[0-9A-Fa-f]+$/; + + my $sign = '+'; $sign = '-' if ($$hs =~ /^-/); + + $$hs =~ s/^[+-]//; # strip sign + if ($CALC->can('_from_hex')) + { + $x->{value} = $CALC->_from_hex($hs); + } + else + { + # 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 + $i -= 4; $len --; + $x += $mul * $val if $val != 0; + $mul *= $x65536 if $len >= 0; # skip last mul + } + } + $x->{sign} = $sign unless $CALC->_is_zero($x->{value}); # no '-0' + $x; + } + +sub __from_bin + { + # convert a (ref to) big binary string to BigInt, return undef for error + my $bs = shift; + + my $x = Math::BigInt->bzero(); + # strip underscores + $$bs =~ s/([01])_([01])/$1$2/g; + $$bs =~ s/([01])_([01])/$1$2/g; + return $x->bnan() if $$bs !~ /^[+-]?0b[01]+$/; + + my $sign = '+'; $sign = '-' if ($$bs =~ /^\-/); + $$bs =~ s/^[+-]//; # strip sign + if ($CALC->can('_from_bin')) + { + $x->{value} = $CALC->_from_bin($bs); } -} + else + { + my $mul = Math::BigInt->bzero(); $mul++; + my $x256 = Math::BigInt->new(256); + my $len = CORE::length($$bs)-2; + $len = int($len/8); # 8-digit parts, w/o '0b' + my $val; my $i = -8; + while ($len >= 0) + { + $val = substr($$bs,$i,8); + $val =~ s/^[+-]?0b// if $len == 0; # for last part only + #$val = oct('0b'.$val); # does not work on Perl prior to 5.6.0 + # slower: + # $val = ('0' x (8-CORE::length($val))).$val if CORE::length($val) < 8; + $val = ord(pack('B8',substr('00000000'.$val,-8,8))); + $i -= 8; $len --; + $x += $mul * $val if $val != 0; + $mul *= $x256 if $len >= 0; # skip last mul + } + } + $x->{sign} = $sign unless $CALC->_is_zero($x->{value}); # no '-0' + $x; + } + +sub _split + { + # (ref to num_str) return num_str + # internal, take apart a string and return the pieces + # strip leading/trailing whitespace, leading zeros, underscore and reject + # invalid input + my $x = shift; + + # strip white space at front, also extranous leading zeros + $$x =~ s/^\s*([-]?)0*([0-9])/$1$2/g; # will not strip ' .2' + $$x =~ s/^\s+//; # but this will + $$x =~ s/\s+$//g; # strip white space at end + + # shortcut, if nothing to split, return early + if ($$x =~ /^[+-]?\d+\z/) + { + $$x =~ s/^([+-])0*([0-9])/$2/; my $sign = $1 || '+'; + return (\$sign, $x, \'', \'', \0); + } + + # invalid starting char? + return if $$x !~ /^[+-]?(\.?[0-9]|0b[0-1]|0x[0-9a-fA-F])/; -# represent ~x as twos-complement number -sub bnot { #(num_str) return num_str - &bsub(-1,$_[$[]); -} + return __from_hex($x) if $$x =~ /^[\-\+]?0x/; # hex string + return __from_bin($x) if $$x =~ /^[\-\+]?0b/; # binary string + + # strip underscores between digits + $$x =~ s/(\d)_(\d)/$1$2/g; + $$x =~ s/(\d)_(\d)/$1$2/g; # do twice for 1_2_3 + + # some possible inputs: + # 2.1234 # 0.12 # 1 # 1E1 # 2.134E1 # 434E-10 # 1.02009E-2 + # .2 # 1_2_3.4_5_6 # 1.4E1_2_3 # 1e3 # +.2 + + return if $$x =~ /[Ee].*[Ee]/; # more than one E => error + + my ($m,$e) = split /[Ee]/,$$x; + $e = '0' if !defined $e || $e eq ""; + # sign,value for exponent,mantint,mantfrac + my ($es,$ev,$mis,$miv,$mfv); + # valid exponent? + if ($e =~ /^([+-]?)0*(\d+)$/) # strip leading zeros + { + $es = $1; $ev = $2; + # valid mantissa? + return if $m eq '.' || $m eq ''; + my ($mi,$mf,$last) = split /\./,$m; + return if defined $last; # last defined => 1.2.3 or others + $mi = '0' if !defined $mi; + $mi .= '0' if $mi =~ /^[\-\+]?$/; + $mf = '0' if !defined $mf || $mf eq ''; + if ($mi =~ /^([+-]?)0*(\d+)$/) # strip leading zeros + { + $mis = $1||'+'; $miv = $2; + return unless ($mf =~ /^(\d*?)0*$/); # strip trailing zeros + $mfv = $1; + return (\$mis,\$miv,\$mfv,\$es,\$ev); + } + } + return; # NaN, not a number + } + +sub as_number + { + # an object might be asked to return itself as bigint on certain overloaded + # operations, this does exactly this, so that sub classes can simple inherit + # it or override with their own integer conversion routine + my $self = shift; + + $self->copy(); + } + +sub as_hex + { + # return as hex string, with prefixed 0x + my $x = shift; $x = $class->new($x) if !ref($x); + + return $x->bstr() if $x->{sign} !~ /^[+-]$/; # inf, nan etc + return '0x0' if $x->is_zero(); + + my $es = ''; my $s = ''; + $s = $x->{sign} if $x->{sign} eq '-'; + if ($CALC->can('_as_hex')) + { + $es = ${$CALC->_as_hex($x->{value})}; + } + else + { + my $x1 = $x->copy()->babs(); my ($xr,$x10000,$h); + if ($] >= 5.006) + { + $x10000 = Math::BigInt->new (0x10000); $h = 'h4'; + } + else + { + $x10000 = Math::BigInt->new (0x1000); $h = 'h3'; + } + while (!$x1->is_zero()) + { + ($x1, $xr) = bdiv($x1,$x10000); + $es .= unpack($h,pack('v',$xr->numify())); + } + $es = reverse $es; + $es =~ s/^[0]+//; # strip leading zeros + $s .= '0x'; + } + $s . $es; + } + +sub as_bin + { + # return as binary string, with prefixed 0b + my $x = shift; $x = $class->new($x) if !ref($x); + + return $x->bstr() if $x->{sign} !~ /^[+-]$/; # inf, nan etc + return '0b0' if $x->is_zero(); + + my $es = ''; my $s = ''; + $s = $x->{sign} if $x->{sign} eq '-'; + if ($CALC->can('_as_bin')) + { + $es = ${$CALC->_as_bin($x->{value})}; + } + else + { + my $x1 = $x->copy()->babs(); my ($xr,$x10000,$b); + if ($] >= 5.006) + { + $x10000 = Math::BigInt->new (0x10000); $b = 'b16'; + } + else + { + $x10000 = Math::BigInt->new (0x1000); $b = 'b12'; + } + while (!$x1->is_zero()) + { + ($x1, $xr) = bdiv($x1,$x10000); + $es .= unpack($b,pack('v',$xr->numify())); + } + $es = reverse $es; + $es =~ s/^[0]+//; # strip leading zeros + $s .= '0b'; + } + $s . $es; + } + +############################################################################## +# internal calculation routines (others are in Math::BigInt::Calc etc) + +sub __lcm + { + # (BINT or num_str, BINT or num_str) return BINT + # does modify first argument + # LCM + + my $x = shift; my $ty = shift; + return $x->bnan() if ($x->{sign} eq $nan) || ($ty->{sign} eq $nan); + return $x * $ty / bgcd($x,$ty); + } + +sub __gcd + { + # (BINT or num_str, BINT or num_str) return BINT + # does modify both arguments + # GCD -- Euclids algorithm E, Knuth Vol 2 pg 296 + my ($x,$ty) = @_; + + return $x->bnan() if $x->{sign} !~ /^[+-]$/ || $ty->{sign} !~ /^[+-]$/; + + while (!$ty->is_zero()) + { + ($x, $ty) = ($ty,bmod($x,$ty)); + } + $x; + } + +############################################################################### +# 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) + +sub modify () { 0; } 1; __END__ @@ -436,84 +2703,1550 @@ Math::BigInt - Arbitrary size integer math package =head1 SYNOPSIS use Math::BigInt; - $i = Math::BigInt->new($string); - - $i->bneg return BINT negation - $i->babs return BINT absolute value - $i->bcmp(BINT) return CODE compare numbers (undef,<0,=0,>0) - $i->badd(BINT) return BINT addition - $i->bsub(BINT) return BINT subtraction - $i->bmul(BINT) return BINT multiplication - $i->bdiv(BINT) return (BINT,BINT) division (quo,rem) just quo if scalar - $i->bmod(BINT) return BINT modulus - $i->bgcd(BINT) return BINT greatest common divisor - $i->bnorm return BINT normalization - $i->blsft(BINT) return BINT left shift - $i->brsft(BINT) return (BINT,BINT) right shift (quo,rem) just quo if scalar - $i->band(BINT) return BINT bit-wise and - $i->bior(BINT) return BINT bit-wise inclusive or - $i->bxor(BINT) return BINT bit-wise exclusive or - $i->bnot return BINT bit-wise not + + # Number creation + $x = Math::BigInt->new($str); # defaults to 0 + $nan = Math::BigInt->bnan(); # create a NotANumber + $zero = Math::BigInt->bzero(); # create a +0 + $inf = Math::BigInt->binf(); # create a +inf + $inf = Math::BigInt->binf('-'); # create a -inf + $one = Math::BigInt->bone(); # create a +1 + $one = Math::BigInt->bone('-'); # create a -1 + + # Testing + $x->is_zero(); # true if arg is +0 + $x->is_nan(); # true if arg is NaN + $x->is_one(); # true if arg is +1 + $x->is_one('-'); # true if arg is -1 + $x->is_odd(); # true if odd, false for even + $x->is_even(); # true if even, false for odd + $x->is_positive(); # true if >= 0 + $x->is_negative(); # true if < 0 + $x->is_inf(sign); # true if +inf, or -inf (sign is default '+') + $x->is_int(); # true if $x is an integer (not a float) + + $x->bcmp($y); # compare numbers (undef,<0,=0,>0) + $x->bacmp($y); # compare absolutely (undef,<0,=0,>0) + $x->sign(); # return the sign, either +,- or NaN + $x->digit($n); # return the nth digit, counting from right + $x->digit(-$n); # return the nth digit, counting from left + + # The following all modify their first argument: + + # set + $x->bzero(); # set $x to 0 + $x->bnan(); # set $x to NaN + $x->bone(); # set $x to +1 + $x->bone('-'); # set $x to -1 + $x->binf(); # set $x to inf + $x->binf('-'); # set $x to -inf + + $x->bneg(); # negation + $x->babs(); # absolute value + $x->bnorm(); # normalize (no-op) + $x->bnot(); # two's complement (bit wise not) + $x->binc(); # increment x by 1 + $x->bdec(); # decrement x by 1 + + $x->badd($y); # addition (add $y to $x) + $x->bsub($y); # subtraction (subtract $y from $x) + $x->bmul($y); # multiplication (multiply $x by $y) + $x->bdiv($y); # divide, set $x to quotient + # return (quo,rem) or quo if scalar + + $x->bmod($y); # modulus (x % y) + $x->bmodpow($exp,$mod); # modular exponentation (($num**$exp) % $mod)) + $x->bmodinv($mod); # the inverse of $x in the given modulus $mod + + $x->bpow($y); # power of arguments (x ** y) + $x->blsft($y); # left shift + $x->brsft($y); # right shift + $x->blsft($y,$n); # left shift, by base $n (like 10) + $x->brsft($y,$n); # right shift, by base $n (like 10) + + $x->band($y); # bitwise and + $x->bior($y); # bitwise inclusive or + $x->bxor($y); # bitwise exclusive or + $x->bnot(); # bitwise not (two's complement) + + $x->bsqrt(); # calculate square-root + $x->bfac(); # factorial of $x (1*2*3*4*..$x) + + $x->round($A,$P,$round_mode); # round to accuracy or precision using mode $r + $x->bround($N); # accuracy: preserve $N digits + $x->bfround($N); # round to $Nth digit, no-op for BigInts + + # The following do not modify their arguments in BigInt, but do in BigFloat: + $x->bfloor(); # return integer less or equal than $x + $x->bceil(); # return integer greater or equal than $x + + # The following do not modify their arguments: + + bgcd(@values); # greatest common divisor (no OO style) + blcm(@values); # lowest common multiplicator (no OO style) + + $x->length(); # return number of digits in number + ($x,$f) = $x->length(); # length of number and length of fraction part, + # latter is always 0 digits long for BigInt's + + $x->exponent(); # return exponent as BigInt + $x->mantissa(); # return (signed) mantissa as BigInt + $x->parts(); # return (mantissa,exponent) as BigInt + $x->copy(); # make a true copy of $x (unlike $y = $x;) + $x->as_number(); # return as BigInt (in BigInt: same as copy()) + + # conversation to string + $x->bstr(); # normalized string + $x->bsstr(); # normalized string in scientific notation + $x->as_hex(); # as signed hexadecimal string with prefixed 0x + $x->as_bin(); # as signed binary string with prefixed 0b + + Math::BigInt->config(); # return hash containing configuration/version + + # precision and accuracy (see section about rounding for more) + $x->precision(); # return P of $x (or global, if P of $x undef) + $x->precision($n); # set P of $x to $n + $x->accuracy(); # return A of $x (or global, if A of $x undef) + $x->accuracy($n); # set A $x to $n + + Math::BigInt->precision(); # get/set global P for all BigInt objects + Math::BigInt->accuracy(); # get/set global A for all BigInt objects =head1 DESCRIPTION -All basic math operations are overloaded if you declare your big -integers as +All operators (inlcuding basic math operations) are overloaded if you +declare your big integers as - $i = new Math::BigInt '123 456 789 123 456 789'; + $i = new Math::BigInt '123_456_789_123_456_789'; +Operations with overloaded operators preserve the arguments which is +exactly what you expect. =over 2 =item Canonical notation -Big integer value are strings of the form C</^[+-]\d+$/> with leading +Big integer values are strings of the form C</^[+-]\d+$/> with leading zeros suppressed. + '-0' canonical value '-0', normalized '0' + ' -123_123_123' canonical value '-123123123' + '1_23_456_7890' canonical value '1234567890' + =item Input -Input values to these routines may be strings of the form -C</^\s*[+-]?[\d\s]+$/>. +Input values to these routines may be either Math::BigInt objects or +strings of the form C</^[+-]?[\d]+\.?[\d]*E?[+-]?[\d]*$/>. + +You can include one underscore between any two digits. The input string may +have leading and trailing whitespace, which will be ignored. In later +versions, a more strict (no whitespace at all) or more lax (whitespace +allowed everywhere) input checking will also be possible. + +This means integer values like 1.01E2 or even 1000E-2 are also accepted. +Non integer values result in NaN. + +Math::BigInt::new() defaults to 0, while Math::BigInt::new('') results +in 'NaN'. + +bnorm() on a BigInt object is now effectively a no-op, since the numbers +are always stored in normalized form. On a string, it creates a BigInt +object. =item Output -Output values always always in canonical form +Output values are BigInt objects (normalized), except for bstr(), which +returns a string in normalized form. +Some routines (C<is_odd()>, C<is_even()>, C<is_zero()>, C<is_one()>, +C<is_nan()>) return true or false, while others (C<bcmp()>, C<bacmp()>) +return either undef, <0, 0 or >0 and are suited for sort. + +=back + +=head1 METHODS + +Each of the methods below accepts three additional parameters. These arguments +$A, $P and $R are accuracy, precision and round_mode. Please see more in the +section about ACCURACY and ROUNDIND. + +=head2 config + + use Data::Dumper; + + print Dumper ( Math::BigInt->config() ); + +Returns a hash containing the configuration, e.g. the version number, lib +loaded etc. + +=head2 accuracy + + $x->accuracy(5); # local for $x + $class->accuracy(5); # global for all members of $class + +Set or get the global or local accuracy, aka how many significant digits the +results have. Please see the section about L<ACCURACY AND PRECISION> for +further details. + +Value must be greater than zero. Pass an undef value to disable it: + + $x->accuracy(undef); + Math::BigInt->accuracy(undef); + +Returns the current accuracy. For C<$x->accuracy()> it will return either the +local accuracy, or if not defined, the global. This means the return value +represents the accuracy that will be in effect for $x: + + $y = Math::BigInt->new(1234567); # unrounded + print Math::BigInt->accuracy(4),"\n"; # set 4, print 4 + $x = Math::BigInt->new(123456); # will be automatically rounded + print "$x $y\n"; # '123500 1234567' + print $x->accuracy(),"\n"; # will be 4 + print $y->accuracy(),"\n"; # also 4, since global is 4 + print Math::BigInt->accuracy(5),"\n"; # set to 5, print 5 + print $x->accuracy(),"\n"; # still 4 + print $y->accuracy(),"\n"; # 5, since global is 5 + +=head2 brsft + + $x->brsft($y,$n); + +Shifts $x right by $y in base $n. Default is base 2, used are usually 10 and +2, but others work, too. + +Right shifting usually amounts to dividing $x by $n ** $y and truncating the +result: + + + $x = Math::BigInt->new(10); + $x->brsft(1); # same as $x >> 1: 5 + $x = Math::BigInt->new(1234); + $x->brsft(2,10); # result 12 + +There is one exception, and that is base 2 with negative $x: + + + $x = Math::BigInt->new(-5); + print $x->brsft(1); + +This will print -3, not -2 (as it would if you divide -5 by 2 and truncate the +result). + +=head2 new + + $x = Math::BigInt->new($str,$A,$P,$R); + +Creates a new BigInt object from a string or another BigInt object. The +input is accepted as decimal, hex (with leading '0x') or binary (with leading +'0b'). + +=head2 bnan + + $x = Math::BigInt->bnan(); + +Creates a new BigInt object representing NaN (Not A Number). +If used on an object, it will set it to NaN: + + $x->bnan(); + +=head2 bzero + + $x = Math::BigInt->bzero(); + +Creates a new BigInt object representing zero. +If used on an object, it will set it to zero: + + $x->bzero(); + +=head2 binf + + $x = Math::BigInt->binf($sign); + +Creates a new BigInt object representing infinity. The optional argument is +either '-' or '+', indicating whether you want infinity or minus infinity. +If used on an object, it will set it to infinity: + + $x->binf(); + $x->binf('-'); + +=head2 bone + + $x = Math::BigInt->binf($sign); + +Creates a new BigInt object representing one. The optional argument is +either '-' or '+', indicating whether you want one or minus one. +If used on an object, it will set it to one: + + $x->bone(); # +1 + $x->bone('-'); # -1 + +=head2 is_one()/is_zero()/is_nan()/is_inf() + + + $x->is_zero(); # true if arg is +0 + $x->is_nan(); # true if arg is NaN + $x->is_one(); # true if arg is +1 + $x->is_one('-'); # true if arg is -1 + $x->is_inf(); # true if +inf + $x->is_inf('-'); # true if -inf (sign is default '+') + +These methods all test the BigInt for beeing one specific value and return +true or false depending on the input. These are faster than doing something +like: + + if ($x == 0) + +=head2 is_positive()/is_negative() + + $x->is_positive(); # true if >= 0 + $x->is_negative(); # true if < 0 + +The methods return true if the argument is positive or negative, respectively. +C<NaN> is neither positive nor negative, while C<+inf> counts as positive, and +C<-inf> is negative. A C<zero> is positive. + +These methods are only testing the sign, and not the value. + +=head2 is_odd()/is_even()/is_int() + + $x->is_odd(); # true if odd, false for even + $x->is_even(); # true if even, false for odd + $x->is_int(); # true if $x is an integer + +The return true when the argument satisfies the condition. C<NaN>, C<+inf>, +C<-inf> are not integers and are neither odd nor even. + +=head2 bcmp + + $x->bcmp($y); + +Compares $x with $y and takes the sign into account. +Returns -1, 0, 1 or undef. + +=head2 bacmp + + $x->bacmp($y); + +Compares $x with $y while ignoring their. Returns -1, 0, 1 or undef. + +=head2 sign + + $x->sign(); + +Return the sign, of $x, meaning either C<+>, C<->, C<-inf>, C<+inf> or NaN. + +=head2 bcmp + + $x->digit($n); # return the nth digit, counting from right + +=head2 bneg + + $x->bneg(); + +Negate the number, e.g. change the sign between '+' and '-', or between '+inf' +and '-inf', respectively. Does nothing for NaN or zero. + +=head2 babs + + $x->babs(); + +Set the number to it's absolute value, e.g. change the sign from '-' to '+' +and from '-inf' to '+inf', respectively. Does nothing for NaN or positive +numbers. + +=head2 bnorm + + $x->bnorm(); # normalize (no-op) + +=head2 bnot + + $x->bnot(); # two's complement (bit wise not) + +=head2 binc + + $x->binc(); # increment x by 1 + +=head2 bdec + + $x->bdec(); # decrement x by 1 + +=head2 badd + + $x->badd($y); # addition (add $y to $x) + +=head2 bsub + + $x->bsub($y); # subtraction (subtract $y from $x) + +=head2 bmul + + $x->bmul($y); # multiplication (multiply $x by $y) + +=head2 bdiv + + $x->bdiv($y); # divide, set $x to quotient + # return (quo,rem) or quo if scalar + +=head2 bmod + + $x->bmod($y); # modulus (x % y) + +=head2 bmodinv + + $num->bmodinv($mod); # modular inverse + +Returns the inverse of C<$num> in the given modulus C<$mod>. 'C<NaN>' is +returned unless C<$num> is relatively prime to C<$mod>, i.e. unless +C<bgcd($num, $mod)==1>. + +=head2 bmodpow + + $num->bmodpow($exp,$mod); # modular exponentation ($num**$exp % $mod) + +Returns the value of C<$num> taken to the power C<$exp> in the modulus +C<$mod> using binary exponentation. C<bmodpow> is far superior to +writing + + $num ** $exp % $mod + +because C<bmodpow> is much faster--it reduces internal variables into +the modulus whenever possible, so it operates on smaller numbers. + +C<bmodpow> also supports negative exponents. + + bmodpow($num, -1, $mod) + +is exactly equivalent to + + bmodinv($num, $mod) + +=head2 bpow + + $x->bpow($y); # power of arguments (x ** y) + +=head2 blsft + + $x->blsft($y); # left shift + $x->blsft($y,$n); # left shift, by base $n (like 10) + +=head2 brsft + + $x->brsft($y); # right shift + $x->brsft($y,$n); # right shift, by base $n (like 10) + +=head2 band + + $x->band($y); # bitwise and + +=head2 bior + + $x->bior($y); # bitwise inclusive or + +=head2 bxor + + $x->bxor($y); # bitwise exclusive or + +=head2 bnot + + $x->bnot(); # bitwise not (two's complement) + +=head2 bsqrt + + $x->bsqrt(); # calculate square-root + +=head2 bfac + + $x->bfac(); # factorial of $x (1*2*3*4*..$x) + +=head2 round + + $x->round($A,$P,$round_mode); # round to accuracy or precision using mode $r + +=head2 bround + + $x->bround($N); # accuracy: preserve $N digits + +=head2 bfround + + $x->bfround($N); # round to $Nth digit, no-op for BigInts + +=head2 bfloor + + $x->bfloor(); + +Set $x to the integer less or equal than $x. This is a no-op in BigInt, but +does change $x in BigFloat. + +=head2 bceil + + $x->bceil(); + +Set $x to the integer greater or equal than $x. This is a no-op in BigInt, but +does change $x in BigFloat. + +=head2 bgcd + + bgcd(@values); # greatest common divisor (no OO style) + +=head2 blcm + + blcm(@values); # lowest common multiplicator (no OO style) + +head2 length + + $x->length(); + ($xl,$fl) = $x->length(); + +Returns the number of digits in the decimal representation of the number. +In list context, returns the length of the integer and fraction part. For +BigInt's, the length of the fraction part will always be 0. + +=head2 exponent + + $x->exponent(); + +Return the exponent of $x as BigInt. + +=head2 mantissa + + $x->mantissa(); + +Return the signed mantissa of $x as BigInt. + +=head2 parts + + $x->parts(); # return (mantissa,exponent) as BigInt + +=head2 copy + + $x->copy(); # make a true copy of $x (unlike $y = $x;) + +=head2 as_number + + $x->as_number(); # return as BigInt (in BigInt: same as copy()) + +=head2 bsrt + + $x->bstr(); # normalized string + +=head2 bsstr + + $x->bsstr(); # normalized string in scientific notation + +=head2 as_hex + + $x->as_hex(); # as signed hexadecimal string with prefixed 0x + +=head2 as_bin + + $x->as_bin(); # as signed binary string with prefixed 0b + +=head1 ACCURACY and PRECISION + +Since version v1.33, Math::BigInt and Math::BigFloat have full support for +accuracy and precision based rounding, both automatically after every +operation as well as manually. + +This section describes the accuracy/precision handling in Math::Big* as it +used to be and as it is now, complete with an explanation of all terms and +abbreviations. + +Not yet implemented things (but with correct description) are marked with '!', +things that need to be answered are marked with '?'. + +In the next paragraph follows a short description of terms used here (because +these may differ from terms used by others people or documentation). + +During the rest of this document, the shortcuts A (for accuracy), P (for +precision), F (fallback) and R (rounding mode) will be used. + +=head2 Precision P + +A fixed number of digits before (positive) or after (negative) +the decimal point. For example, 123.45 has a precision of -2. 0 means an +integer like 123 (or 120). A precision of 2 means two digits to the left +of the decimal point are zero, so 123 with P = 1 becomes 120. Note that +numbers with zeros before the decimal point may have different precisions, +because 1200 can have p = 0, 1 or 2 (depending on what the inital value +was). It could also have p < 0, when the digits after the decimal point +are zero. + +The string output (of floating point numbers) will be padded with zeros: + + Initial value P A Result String + ------------------------------------------------------------ + 1234.01 -3 1000 1000 + 1234 -2 1200 1200 + 1234.5 -1 1230 1230 + 1234.001 1 1234 1234.0 + 1234.01 0 1234 1234 + 1234.01 2 1234.01 1234.01 + 1234.01 5 1234.01 1234.01000 + +For BigInts, no padding occurs. + +=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. For example, 123.456 has +A of 6, 10203 has 5, 123.0506 has 7, 123.450000 has 8 and 0.000123 has 3. + +The string output (of floating point numbers) will be padded with zeros: + + Initial value P A Result String + ------------------------------------------------------------ + 1234.01 3 1230 1230 + 1234.01 6 1234.01 1234.01 + 1234.1 8 1234.1 1234.1000 + +For BigInts, no padding occurs. + +=head2 Fallback F + +When both A and P are undefined, this is used as a fallback accuracy when +dividing numbers. + +=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.) + +=over 2 + +=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. + +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, +and 0.4501 also 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 Precision + + * ffround($p) is able to round to $p number of digits after the decimal + point + * otherwise P is unused + +=item Accuracy (significant digits) + + * fround($a) rounds to $a significant digits + * only fdiv() and fsqrt() take A as (optional) paramater + + other operations simply create the same number (fneg etc), or more (fmul) + of digits + + rounding/truncating is only done when explicitly calling one of fround + or ffround, and never for BigInt (not implemented) + * fsqrt() simply hands its accuracy argument over to fdiv. + * the documentation and the comment in the code indicate two different ways + on how fdiv() determines the maximum number of digits it should calculate, + and the actual code does yet another thing + POD: + max($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. + +=back + +This is how it works now: + +=over 2 + +=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); + * 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. + +=item Creating numbers + + * When you create a number, you can give it's desired A or P via: + $x = Math::BigInt->new($number,$A,$P); + * Only one of A or P can be defined, otherwise the result is NaN + * If no A or P is give ($x = Math::BigInt->new($number) form), then the + globals (if set) will be used. Thus changing the global defaults later on + will not change the A or P of previously created numbers (i.e., A and P of + $x will be what was in effect when $x was created) + * If given undef for A and P, B<no> rounding will occur, and the globals will + B<not> be used. This is used by subclasses to create numbers without + suffering rounding in the parent. Thus a subclass is able to have it's own + globals enforced upon creation of a number by using + $x = Math::BigInt->new($number,undef,undef): + + use Math::Bigint::SomeSubclass; + use Math::BigInt; + + Math::BigInt->accuracy(2); + Math::BigInt::SomeSubClass->accuracy(3); + $x = Math::BigInt::SomeSubClass->new(1234); + + $x is now 1230, and not 1200. A subclass might choose to implement + this otherwise, e.g. falling back to the parent's A and P. + +=item Usage + + * If A or P are enabled/defined, they are used to round the result of each + operation according to the rules below + * Negative P is ignored in Math::BigInt, since BigInts never have digits + after the decimal point + * Math::BigFloat uses Math::BigInts internally, but setting A or P inside + Math::BigInt as globals should not tamper with the parts of a BigFloat. + Thus a flag is used to mark all Math::BigFloat numbers as 'never round' + +=item Precedence + + * It only makes sense that a number has only one of A or P at a time. + Since you can set/get both A and P, there is a rule that will practically + enforce only A or P to be in effect at a time, even if both are set. + This is called precedence. + * If two objects are involved in an operation, and one of them has A in + effect, and the other P, this results in an error (NaN). + * A takes precendence over P (Hint: A comes before P). If A is defined, it + is used, otherwise P is used. If neither of them is defined, nothing is + used, i.e. the result will have as many digits as it can (with an + exception for fdiv/fsqrt) and will not be rounded. + * There is another setting for fdiv() (and thus for fsqrt()). If neither of + A or P is defined, fdiv() will use a fallback (F) of $div_scale digits. + If either the dividend's or the divisor's mantissa has more digits than + the value of F, the higher value will be used instead of F. + This is to limit the digits (A) of the result (just consider what would + happen with unlimited A and P in the case of 1/3 :-) + * fdiv will calculate (at least) 4 more digits than required (determined by + A, P or F), and, if F is not used, round the result + (this will still fail in the case of a result like 0.12345000000001 with A + or P of 5, but this can not be helped - or can it?) + * Thus you can have the math done by on Math::Big* class in three modes: + + never round (this is the default): + This is done by setting A and P to undef. No math operation + will round the result, with fdiv() and fsqrt() as exceptions to guard + against overflows. You must explicitely call bround(), bfround() or + round() (the latter with parameters). + Note: Once you have rounded a number, the settings will 'stick' on it + and 'infect' all other numbers engaged in math operations with it, since + local settings have the highest precedence. So, to get SaferRound[tm], + use a copy() before rounding like this: + + $x = Math::BigFloat->new(12.34); + $y = Math::BigFloat->new(98.76); + $z = $x * $y; # 1218.6984 + print $x->copy()->fround(3); # 12.3 (but A is now 3!) + $z = $x * $y; # still 1218.6984, without + # copy would have been 1210! + + + round after each op: + After each single operation (except for testing like is_zero()), the + method round() is called and the result is rounded appropriately. By + setting proper values for A and P, you can have all-the-same-A or + all-the-same-P modes. For example, Math::Currency might set A to undef, + and P to -2, globally. + + ?Maybe an extra option that forbids local A & P settings would be in order, + ?so that intermediate rounding does not 'poison' further math? + +=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 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 + +=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 immediately rounds $x to the new value. + * $x->accuracy() clears $x->precision(), and vice versa. + +=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::round_mode + * after each operation, $result->round() is called, and the result may + eventually be rounded (that is, if A or P were set either locally, + globally or as parameter to the operation) + * to manually round a number, call $x->round($A,$P,$round_mode); + this will round the number by using the appropriate rounding function + and then normalize it. + * rounding modifies the local settings of the number: + + $x = Math::BigFloat->new(123.456); + $x->accuracy(5); + $x->bround(4); + + Here 4 takes precedence over 5, so 123.5 is the result and $x->accuracy() + will be 4 from now on. + +=item Default values + + * R: 'even' + * F: 40 + * A: undef + * P: undef + +=item Remarks + + * The defaults are set up so that the new code gives the same results as + the old code (except in a few cases on fdiv): + + Both A and P are undefined and thus will not be used for rounding + after each operation. + + round() is thus a no-op, unless given extra parameters A and P =back -Actual math is done in an internal format consisting of an array -whose first element is the sign (/^[+-]$/) and whose remaining -elements are base 100000 digits with the least significant digit first. -The string 'NaN' is used to represent the result when input arguments -are not numbers, as well as the result of dividing by zero. +=head1 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 C<< $x->sign(); >> +instead relying on the internal hash keys like in C<< $x->{sign}; >>. + +=head2 MATH LIBRARY + +Math with the numbers is done (by default) by a module called +Math::BigInt::Calc. This is equivalent to saying: + + use Math::BigInt lib => 'Calc'; + +You can change this by using: + + use Math::BigInt lib => 'BitVect'; + +The following would first try to find Math::BigInt::Foo, then +Math::BigInt::Bar, and when this also fails, revert to Math::BigInt::Calc: + + use Math::BigInt lib => 'Foo,Math::BigInt::Bar'; + +Calc.pm uses as internal format an array of elements of some decimal base +(usually 1e5 or 1e7) with the least significant digit first, while BitVect.pm +uses a bit vector of base 2, most significant bit first. Other modules might +use even different means of representing the numbers. See the respective +module documentation for further details. + +=head2 SIGN + +The sign is either '+', '-', 'NaN', '+inf' or '-inf' and stored seperately. + +A sign of 'NaN' is used to represent the result when input arguments are not +numbers or as a result of 0/0. '+inf' and '-inf' represent plus respectively +minus infinity. You will get '+inf' when dividing a positive number by 0, and +'-inf' when dividing any negative number by 0. + +=head2 mantissa(), exponent() and parts() + +C<mantissa()> and C<exponent()> return the said parts of the BigInt such +that: + + $m = $x->mantissa(); + $e = $x->exponent(); + $y = $m * ( 10 ** $e ); + print "ok\n" if $x == $y; + +C<< ($m,$e) = $x->parts() >> is just a shortcut that gives you both of them +in one go. Both the returned mantissa and exponent have a sign. + +Currently, for BigInts C<$e> will be always 0, except for NaN, +inf and -inf, +where it will be NaN; and for $x == 0, where it will be 1 +(to be compatible with Math::BigFloat's internal representation of a zero as +C<0E1>). + +C<$m> will always be a copy of the original number. The relation between $e +and $m might change in the future, but will always be equivalent in a +numerical sense, e.g. $m might get minimized. =head1 EXAMPLES + + use Math::BigInt; + + sub bint { Math::BigInt->new(shift); } + + $x = Math::BigInt->bstr("1234") # string "1234" + $x = "$x"; # same as bstr() + $x = Math::BigInt->bneg("1234"); # Bigint "-1234" + $x = Math::BigInt->babs("-12345"); # Bigint "12345" + $x = Math::BigInt->bnorm("-0 00"); # BigInt "0" + $x = bint(1) + bint(2); # BigInt "3" + $x = bint(1) + "2"; # ditto (auto-BigIntify of "2") + $x = bint(1); # BigInt "1" + $x = $x + 5 / 2; # BigInt "3" + $x = $x ** 3; # BigInt "27" + $x *= 2; # BigInt "54" + $x = Math::BigInt->new(0); # BigInt "0" + $x--; # BigInt "-1" + $x = Math::BigInt->badd(4,5) # BigInt "9" + print $x->bsstr(); # 9e+0 + +Examples for rounding: - '+0' canonical zero value - ' -123 123 123' canonical value '-123123123' - '1 23 456 7890' canonical value '+1234567890' + use Math::BigFloat; + use Test; + $x = Math::BigFloat->new(123.4567); + $y = Math::BigFloat->new(123.456789); + Math::BigFloat->accuracy(4); # no more A than 4 + + ok ($x->copy()->fround(),123.4); # even rounding + print $x->copy()->fround(),"\n"; # 123.4 + Math::BigFloat->round_mode('odd'); # round to odd + print $x->copy()->fround(),"\n"; # 123.5 + Math::BigFloat->accuracy(5); # no more A than 5 + Math::BigFloat->round_mode('odd'); # round to odd + print $x->copy()->fround(),"\n"; # 123.46 + $y = $x->copy()->fround(4),"\n"; # A = 4: 123.4 + print "$y, ",$y->accuracy(),"\n"; # 123.4, 4 + + Math::BigFloat->accuracy(undef); # A not important now + Math::BigFloat->precision(2); # P important + print $x->copy()->bnorm(),"\n"; # 123.46 + print $x->copy()->fround(),"\n"; # 123.46 + +Examples for converting: + + my $x = Math::BigInt->new('0b1'.'01' x 123); + print "bin: ",$x->as_bin()," hex:",$x->as_hex()," dec: ",$x,"\n"; =head1 Autocreating constants -After C<use Math::BigInt ':constant'> all the integer decimal constants -in the given scope are converted to C<Math::BigInt>. This conversion -happens at compile time. +After C<use Math::BigInt ':constant'> all the B<integer> decimal, hexadecimal +and binary constants in the given scope are converted to C<Math::BigInt>. +This conversion happens at compile time. + +In particular, + + 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 perl scalar. + +Please note that strings and floating point constants are not affected, +so that + + use Math::BigInt qw/:constant/; + + $x = 1234567890123456789012345678901234567890 + + 123456789123456789; + $y = '1234567890123456789012345678901234567890' + + '123456789123456789'; + +do not work. You need an explicit Math::BigInt->new() around one of the +operands. You should also quote large constants to protect loss of precision: + + use Math::Bigint; + + $x = Math::BigInt->new('1234567889123456789123456789123456789'); + +Without the quotes Perl would convert the large number to a floating point +constant at compile time and then hand the result to BigInt, which results in +an truncated result or a NaN. + +This also applies to integers that look like floating point constants: + + use Math::BigInt ':constant'; + + print ref(123e2),"\n"; + print ref(123.2e2),"\n"; + +will print nothing but newlines. Use either L<bignum> or L<Math::BigFloat> +to get this to work. + +=head1 PERFORMANCE + +Using the form $x += $y; etc over $x = $x + $y is faster, since a copy of $x +must be made in the second case. For long numbers, the copy can eat up to 20% +of the work (in the case of addition/subtraction, less for +multiplication/division). If $y is very small compared to $x, the form +$x += $y is MUCH faster than $x = $x + $y since making the copy of $x takes +more time then the actual addition. + +With a technique called copy-on-write, the cost of copying with overload could +be minimized or even completely avoided. A test implementation of COW did show +performance gains for overloaded math, but introduced a performance loss due +to a constant overhead for all other operatons. + +The rewritten version of this module is slower on certain operations, like +new(), bstr() and numify(). The reason are that it does now more work and +handles more cases. The time spent in these operations is usually gained in +the other operations so that programs on the average should get faster. If +they don't, please contect the author. + +Some operations may be slower for small numbers, but are significantly faster +for big numbers. Other operations are now constant (O(1), like bneg(), babs() +etc), instead of O(N) and thus nearly always take much less time. These +optimizations were done on purpose. + +If you find the Calc module to slow, try to install any of the replacement +modules and see if they help you. + +=head2 Alternative math libraries + +You can use an alternative library to drive Math::BigInt via: + + use Math::BigInt lib => 'Module'; + +See L<MATH LIBRARY> for more information. + +For more benchmark results see L<http://bloodgate.com/perl/benchmarks.html>. + +=head2 SUBCLASSING + +=head1 Subclassing Math::BigInt + +The basic design of Math::BigInt allows simple subclasses with very little +work, as long as a few simple rules are followed: + +=over 2 + +=item * + +The public API must remain consistent, i.e. if a sub-class is overloading +addition, the sub-class must use the same name, in this case badd(). The +reason for this is that Math::BigInt is optimized to call the object methods +directly. + +=item * + +The private object hash keys like C<$x->{sign}> may not be changed, but +additional keys can be added, like C<$x->{_custom}>. + +=item * + +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. + +=back + +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 C<bstr()>. + +All other object methods and overloaded functions can be directly inherited +from the parent class. + +At the very minimum, any subclass will need to provide it's own C<new()> and can +store additional hash keys in the object. There are also some package globals +that must be defined, e.g.: + + # Globals + $accuracy = undef; + $precision = -2; # round to 2 decimal places + $round_mode = 'even'; + $div_scale = 40; + +Additionally, you might want to provide the following two globals to allow +auto-upgrading and auto-downgrading to work correctly: + + $upgrade = undef; + $downgrade = undef; + +This allows Math::BigInt to correctly retrieve package globals from the +subclass, like C<$SubClass::precision>. See t/Math/BigInt/Subclass.pm or +t/Math/BigFloat/SubClass.pm completely functional subclass examples. + +Don't forget to + + use overload; + +in your subclass to automatically inherit the overloading from the parent. If +you like, you can change part of the overloading, look at Math::String for an +example. + +=head1 UPGRADING + +When used like this: + + use Math::BigInt upgrade => 'Foo::Bar'; + +certain operations will 'upgrade' their calculation and thus the result to +the class Foo::Bar. Usually this is used in conjunction with Math::BigFloat: + + use Math::BigInt upgrade => 'Math::BigFloat'; + +As a shortcut, you can use the module C<bignum>: -In particular + use bignum; - perl -MMath::BigInt=:constant -e 'print 2**100' +Also good for oneliners: -print the integer value of C<2**100>. Note that without conversion of -constants the expression 2**100 will be calculated as floating point number. + perl -Mbignum -le 'print 2 ** 255' + +This makes it possible to mix arguments of different classes (as in 2.5 + 2) +as well es preserve accuracy (as in sqrt(3)). + +Beware: This feature is not fully implemented yet. + +=head2 Auto-upgrade + +The following methods upgrade themselves unconditionally; that is if upgrade +is in effect, they will always hand up their work: + +=over 2 + +=item bsqrt() + +=item div() + +=item blog() + +=back + +Beware: This list is not complete. + +All other methods upgrade themselves only when one (or all) of their +arguments are of the class mentioned in $upgrade (This might change in later +versions to a more sophisticated scheme): =head1 BUGS -The current version of this module is a preliminary version of the -real thing that is currently (as of perl5.002) under development. +=over 2 + +=item Out of Memory! + +Under Perl prior to 5.6.0 having an C<use Math::BigInt ':constant';> and +C<eval()> in your code will crash with "Out of memory". This is probably an +overload/exporter bug. You can workaround by not having C<eval()> +and ':constant' at the same time or upgrade your Perl to a newer version. + +=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. + +=back + +=head1 CAVEATS + +Some things might not work as you expect them. Below is documented what is +known to be troublesome: + +=over 1 + +=item stringify, bstr(), bsstr() and 'cmp' + +Both stringify and bstr() now drop the leading '+'. The old code would return +'+3', the new returns '3'. This is to be consistent with Perl and to make +cmp (especially with overloading) to work as you expect. It also solves +problems with Test.pm, it's ok() uses 'eq' internally. + +Mark said, when asked about to drop the '+' altogether, or make only cmp work: + + I agree (with the first alternative), don't add the '+' on positive + numbers. It's not as important anymore with the new internal + form for numbers. It made doing things like abs and neg easier, + but those have to be done differently now anyway. + +So, the following examples will now work all as expected: + + use Test; + BEGIN { plan tests => 1 } + use Math::BigInt; + + my $x = new Math::BigInt 3*3; + my $y = new Math::BigInt 3*3; + + ok ($x,3*3); + print "$x eq 9" if $x eq $y; + print "$x eq 9" if $x eq '9'; + print "$x eq 9" if $x eq 3*3; + +Additionally, the following still works: + + print "$x == 9" if $x == $y; + print "$x == 9" if $x == 9; + print "$x == 9" if $x == 3*3; + +There is now a C<bsstr()> method to get the string in scientific notation aka +C<1e+2> instead of C<100>. Be advised that overloaded 'eq' always uses bstr() +for comparisation, but Perl will represent some numbers as 100 and others +as 1e+308. If in doubt, convert both arguments to Math::BigInt before doing eq: + + use Test; + BEGIN { plan tests => 3 } + use Math::BigInt; + + $x = Math::BigInt->new('1e56'); $y = 1e56; + ok ($x,$y); # will fail + ok ($x->bsstr(),$y); # okay + $y = Math::BigInt->new($y); + ok ($x,$y); # okay + +Alternatively, simple use <=> for comparisations, that will get it always +right. There is not yet a way to get a number automatically represented as +a string that matches exactly the way Perl represents it. + +=item int() + +C<int()> will return (at least for Perl v5.7.1 and up) another BigInt, not a +Perl scalar: + + $x = Math::BigInt->new(123); + $y = int($x); # BigInt 123 + $x = Math::BigFloat->new(123.45); + $y = int($x); # BigInt 123 + +In all Perl versions you can use C<as_number()> for the same effect: + + $x = Math::BigFloat->new(123.45); + $y = $x->as_number(); # BigInt 123 + +This also works for other subclasses, like Math::String. + +It is yet unlcear whether overloaded int() should return a scalar or a BigInt. + +=item length + +The following will probably not do what you expect: + + $c = Math::BigInt->new(123); + print $c->length(),"\n"; # prints 30 + +It prints both the number of digits in the number and in the fraction part +since print calls C<length()> in list context. Use something like: + + print scalar $c->length(),"\n"; # prints 3 + +=item bdiv + +The following will probably not do what you expect: + + print $c->bdiv(10000),"\n"; + +It prints both quotient and remainder since print calls C<bdiv()> in list +context. Also, C<bdiv()> will modify $c, so be carefull. You probably want +to use + + print $c / 10000,"\n"; + print scalar $c->bdiv(10000),"\n"; # or if you want to modify $c + +instead. + +The quotient is always the greatest integer less than or equal to the +real-valued quotient of the two operands, and the remainder (when it is +nonzero) always has the same sign as the second operand; so, for +example, + + 1 / 4 => ( 0, 1) + 1 / -4 => (-1,-3) + -3 / 4 => (-1, 1) + -3 / -4 => ( 0,-3) + -11 / 2 => (-5,1) + 11 /-2 => (-5,-1) + +As a consequence, the behavior of the operator % agrees with the +behavior of Perl's built-in % operator (as documented in the perlop +manpage), and the equation + + $x == ($x / $y) * $y + ($x % $y) + +holds true for any $x and $y, which justifies calling the two return +values of bdiv() the quotient and remainder. The only exception to this rule +are when $y == 0 and $x is negative, then the remainder will also be +negative. See below under "infinity handling" for the reasoning behing this. + +Perl's 'use integer;' changes the behaviour of % and / for scalars, but will +not change BigInt's way to do things. This is because under 'use integer' Perl +will do what the underlying C thinks is right and this is different for each +system. If you need BigInt's behaving exactly like Perl's 'use integer', bug +the author to implement it ;) + +=item infinity handling + +Here are some examples that explain the reasons why certain results occur while +handling infinity: + +The following table shows the result of the division and the remainder, so that +the equation above holds true. Some "ordinary" cases are strewn in to show more +clearly the reasoning: + + A / B = C, R so that C * B + R = A + ========================================================= + 5 / 8 = 0, 5 0 * 8 + 5 = 5 + 0 / 8 = 0, 0 0 * 8 + 0 = 0 + 0 / inf = 0, 0 0 * inf + 0 = 0 + 0 /-inf = 0, 0 0 * -inf + 0 = 0 + 5 / inf = 0, 5 0 * inf + 5 = 5 + 5 /-inf = 0, 5 0 * -inf + 5 = 5 + -5/ inf = 0, -5 0 * inf + -5 = -5 + -5/-inf = 0, -5 0 * -inf + -5 = -5 + inf/ 5 = inf, 0 inf * 5 + 0 = inf + -inf/ 5 = -inf, 0 -inf * 5 + 0 = -inf + inf/ -5 = -inf, 0 -inf * -5 + 0 = inf + -inf/ -5 = inf, 0 inf * -5 + 0 = -inf + 5/ 5 = 1, 0 1 * 5 + 0 = 5 + -5/ -5 = 1, 0 1 * -5 + 0 = -5 + inf/ inf = 1, 0 1 * inf + 0 = inf + -inf/-inf = 1, 0 1 * -inf + 0 = -inf + inf/-inf = -1, 0 -1 * -inf + 0 = inf + -inf/ inf = -1, 0 1 * -inf + 0 = -inf + 8/ 0 = inf, 8 inf * 0 + 8 = 8 + inf/ 0 = inf, inf inf * 0 + inf = inf + 0/ 0 = NaN + +These cases below violate the "remainder has the sign of the second of the two +arguments", since they wouldn't match up otherwise. + + A / B = C, R so that C * B + R = A + ======================================================== + -inf/ 0 = -inf, -inf -inf * 0 + inf = -inf + -8/ 0 = -inf, -8 -inf * 0 + 8 = -8 + +=item Modifying and = + +Beware of: + + $x = Math::BigFloat->new(5); + $y = $x; + +It will not do what you think, e.g. making a copy of $x. Instead it just makes +a second reference to the B<same> object and stores it in $y. Thus anything +that modifies $x (except overloaded operators) will modify $y, and vice versa. +Or in other words, C<=> is only safe if you modify your BigInts only via +overloaded math. As soon as you use a method call it breaks: + + $x->bmul(2); + print "$x, $y\n"; # prints '10, 10' + +If you want a true copy of $x, use: + + $y = $x->copy(); + +You can also chain the calls like this, this will make first a copy and then +multiply it by 2: + + $y = $x->copy()->bmul(2); + +See also the documentation for overload.pm regarding C<=>. + +=item bpow + +C<bpow()> (and the rounding functions) now modifies the first argument and +returns it, unlike the old code which left it alone and only returned the +result. This is to be consistent with C<badd()> etc. The first three will +modify $x, the last one won't: + + print bpow($x,$i),"\n"; # modify $x + print $x->bpow($i),"\n"; # ditto + print $x **= $i,"\n"; # the same + print $x ** $i,"\n"; # leave $x alone + +The form C<$x **= $y> is faster than C<$x = $x ** $y;>, though. + +=item Overloading -$x + +The following: + + $x = -$x; + +is slower than + + $x->bneg(); + +since overload calls C<sub($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). + +With Copy-On-Write, this issue would be gone, but C-o-W is not implemented +since it is slower for all other things. + +=item Mixing different object types + +In Perl you will get a floating point value if you do one of the following: + + $float = 5.0 + 2; + $float = 2 + 5.0; + $float = 5 / 2; + +With overloaded math, only the first two variants will result in a BigFloat: + + use Math::BigInt; + use Math::BigFloat; + + $mbf = Math::BigFloat->new(5); + $mbi2 = Math::BigInteger->new(5); + $mbi = Math::BigInteger->new(2); + + # what actually gets called: + $float = $mbf + $mbi; # $mbf->badd() + $float = $mbf / $mbi; # $mbf->bdiv() + $integer = $mbi + $mbf; # $mbi->badd() + $integer = $mbi2 / $mbi; # $mbi2->bdiv() + $integer = $mbi2 / $mbf; # $mbi2->bdiv() + +This is because math with overloaded operators follows the first (dominating) +operand, and the operation of that is called and returns thus the result. So, +Math::BigInt::bdiv() will always return a Math::BigInt, regardless whether +the result should be a Math::BigFloat or the second operant is one. + +To get a Math::BigFloat you either need to call the operation manually, +make sure the operands are already of the proper type or casted to that type +via Math::BigFloat->new(): + + $float = Math::BigFloat->new($mbi2) / $mbi; # = 2.5 + +Beware of simple "casting" the entire expression, this would only convert +the already computed result: + + $float = Math::BigFloat->new($mbi2 / $mbi); # = 2.0 thus wrong! + +Beware also of the order of more complicated expressions like: + + $integer = ($mbi2 + $mbi) / $mbf; # int / float => int + $integer = $mbi2 / Math::BigFloat->new($mbi); # ditto + +If in doubt, break the expression into simpler terms, or cast all operands +to the desired resulting type. + +Scalar values are a bit different, since: + + $float = 2 + $mbf; + $float = $mbf + 2; + +will both result in the proper type due to the way the overloaded math works. + +This section also applies to other overloaded math packages, like Math::String. + +One solution to you problem might be L<autoupgrading|upgrading>. + +=item bsqrt() + +C<bsqrt()> works only good if the result is a big integer, e.g. the square +root of 144 is 12, but from 12 the square root is 3, regardless of rounding +mode. + +If you want a better approximation of the square root, then use: + + $x = Math::BigFloat->new(12); + Math::BigFloat->precision(0); + Math::BigFloat->round_mode('even'); + print $x->copy->bsqrt(),"\n"; # 4 + + Math::BigFloat->precision(2); + print $x->bsqrt(),"\n"; # 3.46 + print $x->bsqrt(3),"\n"; # 3.464 + +=item brsft() + +For negative numbers in base see also L<brsft|brsft>. + +=back + +=head1 LICENSE + +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> as well as L<Math::BigInt::BitVect>, +L<Math::BigInt::Pari> and L<Math::BigInt::GMP>. + +The package at +L<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. -=head1 AUTHOR +=head1 AUTHORS -Mark Biggar, overloaded interface by Ilya Zakharevich. +Original code by Mark Biggar, overloaded interface by Ilya Zakharevich. +Completely rewritten by Tels http://bloodgate.com in late 2000, 2001. =cut |