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Diffstat (limited to 'Master/tlpkg/tlperl/lib/Math/BigInt.pm')
| -rwxr-xr-x | Master/tlpkg/tlperl/lib/Math/BigInt.pm | 5115 |
1 files changed, 5115 insertions, 0 deletions
diff --git a/Master/tlpkg/tlperl/lib/Math/BigInt.pm b/Master/tlpkg/tlperl/lib/Math/BigInt.pm new file mode 100755 index 00000000000..b19b3dc92d7 --- /dev/null +++ b/Master/tlpkg/tlperl/lib/Math/BigInt.pm @@ -0,0 +1,5115 @@ +package Math::BigInt; + +# +# "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 $class = "Math::BigInt"; +use 5.006; + +$VERSION = '1.89'; + +@ISA = qw(Exporter); +@EXPORT_OK = qw(objectify bgcd blcm); + +# _trap_inf and _trap_nan are internal and should never be accessed from the +# outside +use vars qw/$round_mode $accuracy $precision $div_scale $rnd_mode + $upgrade $downgrade $_trap_nan $_trap_inf/; +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 is true. +# In some cases (like add, $x = $x + 2 is the same as $x = 2 + $x) this makes +# no difference, but in some cases it does. + +# 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 { $_[0]->copy(); }, + +# 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]); }, +'<<=' => sub { $_[0]->blsft($_[1]); }, +'>>=' => sub { $_[0]->brsft($_[1]); }, + +# not supported by Perl yet +'..' => \&_pointpoint, + +'<=>' => sub { my $rc = $_[2] ? + ref($_[0])->bcmp($_[1],$_[0]) : + $_[0]->bcmp($_[1]); + $rc = 1 unless defined $rc; + $rc <=> 0; + }, +# we need '>=' to get things like "1 >= NaN" right: +'>=' => sub { my $rc = $_[2] ? + ref($_[0])->bcmp($_[1],$_[0]) : + $_[0]->bcmp($_[1]); + # if there was a NaN involved, return false + return '' unless defined $rc; + $rc >= 0; + }, +'cmp' => sub { + $_[2] ? + "$_[1]" cmp $_[0]->bstr() : + $_[0]->bstr() cmp "$_[1]" }, + +'cos' => sub { $_[0]->copy->bcos(); }, +'sin' => sub { $_[0]->copy->bsin(); }, +'atan2' => sub { $_[2] ? + ref($_[0])->new($_[1])->batan2($_[0]) : + $_[0]->copy()->batan2($_[1]) }, + +# are not yet overloadable +#'hex' => sub { print "hex"; $_[0]; }, +#'oct' => sub { print "oct"; $_[0]; }, + +# log(N) is log(N, e), where e is Euler's number +'log' => sub { $_[0]->copy()->blog($_[1], undef); }, +'exp' => sub { $_[0]->copy()->bexp($_[1]); }, +'int' => sub { $_[0]->copy(); }, +'neg' => sub { $_[0]->copy()->bneg(); }, +'abs' => sub { $_[0]->copy()->babs(); }, +'sqrt' => sub { $_[0]->copy()->bsqrt(); }, +'~' => sub { $_[0]->copy()->bnot(); }, + +# for subtract it's a bit tricky to not modify b: b-a => -a+b +'-' => sub { my $c = $_[0]->copy; $_[2] ? + $c->bneg()->badd( $_[1]) : + $c->bsub( $_[1]) }, +'+' => sub { $_[0]->copy()->badd($_[1]); }, +'*' => sub { $_[0]->copy()->bmul($_[1]); }, + +'/' => sub { + $_[2] ? ref($_[0])->new($_[1])->bdiv($_[0]) : $_[0]->copy->bdiv($_[1]); + }, +'%' => sub { + $_[2] ? ref($_[0])->new($_[1])->bmod($_[0]) : $_[0]->copy->bmod($_[1]); + }, +'**' => sub { + $_[2] ? ref($_[0])->new($_[1])->bpow($_[0]) : $_[0]->copy->bpow($_[1]); + }, +'<<' => sub { + $_[2] ? ref($_[0])->new($_[1])->blsft($_[0]) : $_[0]->copy->blsft($_[1]); + }, +'>>' => sub { + $_[2] ? ref($_[0])->new($_[1])->brsft($_[0]) : $_[0]->copy->brsft($_[1]); + }, +'&' => sub { + $_[2] ? ref($_[0])->new($_[1])->band($_[0]) : $_[0]->copy->band($_[1]); + }, +'|' => sub { + $_[2] ? ref($_[0])->new($_[1])->bior($_[0]) : $_[0]->copy->bior($_[1]); + }, +'^' => sub { + $_[2] ? ref($_[0])->new($_[1])->bxor($_[0]) : $_[0]->copy->bxor($_[1]); + }, + +# can modify arg of ++ and --, so avoid a copy() for speed, but don't +# use $_[0]->bone(), it would modify $_[0] to be 1! +'++' => sub { $_[0]->binc() }, +'--' => sub { $_[0]->bdec() }, + +# if overloaded, O(1) instead of O(N) and twice as fast for small numbers +'bool' => sub { + # this kludge is needed for perl prior 5.6.0 since returning 0 here fails :-/ + # v5.6.1 dumps on this: return !$_[0]->is_zero() || undef; :-( + my $t = undef; + $t = 1 if !$_[0]->is_zero(); + $t; + }, + +# the original qw() does not work with the TIESCALAR below, why? +# Order of arguments unsignificant +'""' => sub { $_[0]->bstr(); }, +'0+' => sub { $_[0]->numify(); } +; + +############################################################################## +# global constants, flags and accessory + +# These vars are public, but their direct usage is not recommended, use the +# accessor methods instead + +$round_mode = 'even'; # one of 'even', 'odd', '+inf', '-inf', 'zero', 'trunc' or 'common' +$accuracy = undef; +$precision = undef; +$div_scale = 40; + +$upgrade = undef; # default is no upgrade +$downgrade = undef; # default is no downgrade + +# These are internally, and not to be used from the outside at all + +$_trap_nan = 0; # are NaNs ok? set w/ config() +$_trap_inf = 0; # are infs ok? set w/ config() +my $nan = 'NaN'; # constants for easier life + +my $CALC = 'Math::BigInt::FastCalc'; # module to do the low level math + # default is FastCalc.pm +my $IMPORT = 0; # was import() called yet? + # used to make require work +my %WARN; # warn only once for low-level libs +my %CAN; # cache for $CALC->can(...) +my %CALLBACKS; # callbacks to notify on lib loads +my $EMU_LIB = 'Math/BigInt/CalcEmu.pm'; # emulate low-level math + +############################################################################## +# 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 to enable $rnd_mode to work transparently + tie $rnd_mode, 'Math::BigInt'; + + # set up some handy alias names + *as_int = \&as_number; + *is_pos = \&is_positive; + *is_neg = \&is_negative; + } + +############################################################################## + +sub round_mode + { + no strict 'refs'; + # make Class->round_mode() work + my $self = shift; + my $class = ref($self) || $self || __PACKAGE__; + if (defined $_[0]) + { + my $m = shift; + if ($m !~ /^(even|odd|\+inf|\-inf|zero|trunc|common)$/) + { + require Carp; Carp::croak ("Unknown round mode '$m'"); + } + return ${"${class}::round_mode"} = $m; + } + ${"${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) + { + return ${"${class}::upgrade"} = $_[0]; + } + ${"${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) + { + return ${"${class}::downgrade"} = $_[0]; + } + ${"${class}::downgrade"}; + } + +sub div_scale + { + no strict 'refs'; + # make Class->div_scale() work + my $self = shift; + my $class = ref($self) || $self || __PACKAGE__; + if (defined $_[0]) + { + if ($_[0] < 0) + { + require Carp; Carp::croak ('div_scale must be greater than zero'); + } + ${"${class}::div_scale"} = $_[0]; + } + ${"${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; + # convert objects to scalars to avoid deep recursion. If object doesn't + # have numify(), then hopefully it will have overloading for int() and + # boolean test without wandering into a deep recursion path... + $a = $a->numify() if ref($a) && $a->can('numify'); + + if (defined $a) + { + # also croak on non-numerical + if (!$a || $a <= 0) + { + require Carp; + Carp::croak ('Argument to accuracy must be greater than zero'); + } + if (int($a) != $a) + { + require Carp; + Carp::croak ('Argument to accuracy must be an integer'); + } + } + if (ref($x)) + { + # $object->accuracy() or fallback to global + $x->bround($a) if $a; # not for undef, 0 + $x->{_a} = $a; # set/overwrite, even if not rounded + delete $x->{_p}; # clear P + $a = ${"${class}::accuracy"} unless defined $a; # proper return value + } + else + { + ${"${class}::accuracy"} = $a; # set global A + ${"${class}::precision"} = undef; # clear global P + } + return $a; # shortcut + } + + my $a; + # $object->accuracy() or fallback to global + $a = $x->{_a} if ref($x); + # but don't return global undef, when $x's accuracy is 0! + $a = ${"${class}::accuracy"} if !defined $a; + $a; + } + +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'; + if (@_ > 0) + { + my $p = shift; + # convert objects to scalars to avoid deep recursion. If object doesn't + # have numify(), then hopefully it will have overloading for int() and + # boolean test without wandering into a deep recursion path... + $p = $p->numify() if ref($p) && $p->can('numify'); + if ((defined $p) && (int($p) != $p)) + { + require Carp; Carp::croak ('Argument to precision must be an integer'); + } + if (ref($x)) + { + # $object->precision() or fallback to global + $x->bfround($p) if $p; # not for undef, 0 + $x->{_p} = $p; # set/overwrite, even if not rounded + delete $x->{_a}; # clear A + $p = ${"${class}::precision"} unless defined $p; # proper return value + } + else + { + ${"${class}::precision"} = $p; # set global P + ${"${class}::accuracy"} = undef; # clear global A + } + return $p; # shortcut + } + + my $p; + # $object->precision() or fallback to global + $p = $x->{_p} if ref($x); + # but don't return global undef, when $x's precision is 0! + $p = ${"${class}::precision"} if !defined $p; + $p; + } + +sub config + { + # return (or set) configuration data as hash ref + my $class = shift || 'Math::BigInt'; + + no strict 'refs'; + if (@_ > 1 || (@_ == 1 && (ref($_[0]) eq 'HASH'))) + { + # try to set given options as arguments from hash + + my $args = $_[0]; + if (ref($args) ne 'HASH') + { + $args = { @_ }; + } + # these values can be "set" + my $set_args = {}; + foreach my $key ( + qw/trap_inf trap_nan + upgrade downgrade precision accuracy round_mode div_scale/ + ) + { + $set_args->{$key} = $args->{$key} if exists $args->{$key}; + delete $args->{$key}; + } + if (keys %$args > 0) + { + require Carp; + Carp::croak ("Illegal key(s) '", + join("','",keys %$args),"' passed to $class\->config()"); + } + foreach my $key (keys %$set_args) + { + if ($key =~ /^trap_(inf|nan)\z/) + { + ${"${class}::_trap_$1"} = ($set_args->{"trap_$1"} ? 1 : 0); + next; + } + # use a call instead of just setting the $variable to check argument + $class->$key($set_args->{$key}); + } + } + + # now return actual configuration + + my $cfg = { + lib => $CALC, + lib_version => ${"${CALC}::VERSION"}, + class => $class, + trap_nan => ${"${class}::_trap_nan"}, + trap_inf => ${"${class}::_trap_inf"}, + version => ${"${class}::VERSION"}, + }; + foreach my $key (qw/ + upgrade downgrade precision accuracy round_mode div_scale + /) + { + $cfg->{$key} = ${"${class}::$key"}; + }; + if (@_ == 1 && (ref($_[0]) ne 'HASH')) + { + # calls of the style config('lib') return just this value + return $cfg->{$_[0]}; + } + $cfg; + } + +sub _scale_a + { + # select accuracy parameter based on precedence, + # used by bround() and bfround(), may return undef for scale (means no op) + my ($x,$scale,$mode) = @_; + + $scale = $x->{_a} unless defined $scale; + + no strict 'refs'; + my $class = ref($x); + + $scale = ${ $class . '::accuracy' } unless defined $scale; + $mode = ${ $class . '::round_mode' } unless defined $mode; + + if (defined $scale) + { + $scale = $scale->can('numify') ? $scale->numify() : "$scale" if ref($scale); + $scale = int($scale); + } + + ($scale,$mode); + } + +sub _scale_p + { + # select precision parameter based on precedence, + # used by bround() and bfround(), may return undef for scale (means no op) + my ($x,$scale,$mode) = @_; + + $scale = $x->{_p} unless defined $scale; + + no strict 'refs'; + my $class = ref($x); + + $scale = ${ $class . '::precision' } unless defined $scale; + $mode = ${ $class . '::round_mode' } unless defined $mode; + + if (defined $scale) + { + $scale = $scale->can('numify') ? $scale->numify() : "$scale" if ref($scale); + $scale = int($scale); + } + + ($scale,$mode); + } + +############################################################################## +# constructors + +sub copy + { + # if two arguments, the first one is the class to "swallow" subclasses + if (@_ > 1) + { + my $self = bless { + sign => $_[1]->{sign}, + value => $CALC->_copy($_[1]->{value}), + }, $_[0] if @_ > 1; + + $self->{_a} = $_[1]->{_a} if defined $_[1]->{_a}; + $self->{_p} = $_[1]->{_p} if defined $_[1]->{_p}; + return $self; + } + + my $self = bless { + sign => $_[0]->{sign}, + value => $CALC->_copy($_[0]->{value}), + }, ref($_[0]); + + $self->{_a} = $_[0]->{_a} if defined $_[0]->{_a}; + $self->{_p} = $_[0]->{_p} if defined $_[0]->{_p}; + $self; + } + +sub new + { + # create a new BigInt object from a string or another BigInt object. + # see hash keys documented at top + + # the argument could be an object, so avoid ||, && etc on it, this would + # cause costly overloaded code to be called. The only allowed ops are + # ref() and defined. + + my ($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} = $1 || '+'; + + if ($wanted =~ /^[+-]/) + { + # remove sign without touching wanted to make it work with constants + my $t = $wanted; $t =~ s/^[+-]//; + $self->{value} = $CALC->_new($t); + } + else + { + $self->{value} = $CALC->_new($wanted); + } + 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\z/) + { + $self->{sign} = $wanted; # set a default sign for bstr() + return $self->binf($wanted); + } + # 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 $mis) + { + if ($_trap_nan) + { + require Carp; Carp::croak("$wanted is not a number in $class"); + } + $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 ($diff < 0) # Not integer + { + if ($_trap_nan) + { + require Carp; Carp::croak("$wanted not an integer in $class"); + } + #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 ($$mfv ne '') # e <= 0 + { + # fraction and negative/zero E => NOI + if ($_trap_nan) + { + require Carp; Carp::croak("$wanted not an integer in $class"); + } + #print "NOI 2 \$\$mfv '$$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 ($$miv !~ s/0{$e}$//) # can strip so many zero's? + { + if ($_trap_nan) + { + require Carp; Carp::croak("$wanted not an integer in $class"); + } + #print "NOI 3\n"; + return $upgrade->new($wanted,$a,$p,$r) if defined $upgrade; + $self->{sign} = $nan; + } + } + } + $self->{sign} = '+' if $$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; + } + no strict 'refs'; + if (${"${class}::_trap_nan"}) + { + require Carp; + Carp::croak ("Tried to set $self to NaN in $class\::bnan()"); + } + $self->import() if $IMPORT == 0; # make require work + return if $self->modify('bnan'); + 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 + $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; + } + no strict 'refs'; + if (${"${class}::_trap_inf"}) + { + require Carp; + Carp::croak ("Tried to set $self to +-inf in $class\::binf()"); + } + $self->import() if $IMPORT == 0; # make require work + return if $self->modify('binf'); + 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 + $self; + } + +sub bzero + { + # create a bigint '+0', if given a BigInt, set it to 0 + my $self = shift; + $self = __PACKAGE__ 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, respectively + 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 + { + # call like: $x->bone($sign,$a,$p,$r); + $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 ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_); + + if ($x->{sign} !~ /^[+-]$/) + { + return $x->{sign} unless $x->{sign} eq '+inf'; # -inf, NaN + return 'inf'; # +inf + } + my ($m,$e) = $x->parts(); + #$m->bstr() . 'e+' . $e->bstr(); # e can only be positive in BigInt + # 'e+' because E can only be positive in BigInt + $m->bstr() . 'e+' . $CALC->_str($e->{value}); + } + +sub bstr + { + # make a string from bigint object + my ($self,$x) = ref($_[0]) ? (undef,$_[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 '-'; + $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->bstr() 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]) ? (undef,$_[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. + + # !!!!!!! If you change this, remember to change round(), too! !!!!!!!!!! + + # 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(). + + # returns ($self) or ($self,$a,$p,$r) - sets $self to NaN of both A and P + # were requested/defined (locally or globally or both) + + 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) + + my $c = ref($self); # find out class of argument(s) + no strict 'refs'; + + # convert to normal scalar for speed and correctness in inner parts + $a = $a->can('numify') ? $a->numify() : "$a" if defined $a && ref($a); + $p = $p->can('numify') ? $p->numify() : "$p" if defined $p && ref($p); + + # 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; + + # A == 0 is useless, so undef it to signal no rounding + $a = undef if defined $a && $a == 0; + + # 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; # error + + $r = ${"$c\::round_mode"} unless defined $r; + if ($r !~ /^(even|odd|\+inf|\-inf|zero|trunc|common)$/) + { + require Carp; Carp::croak ("Unknown round mode '$r'"); + } + + $a = int($a) if defined $a; + $p = int($p) if defined $p; + + ($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) + + 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; + + # A == 0 is useless, so undef it to signal no rounding + $a = undef if defined $a && $a == 0; + + # 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; + if ($r !~ /^(even|odd|\+inf|\-inf|zero|trunc|common)$/) + { + require Carp; Carp::croak ("Unknown round mode '$r'"); + } + + # now round, by calling either fround or ffround: + if (defined $a) + { + $self->bround(int($a),$r) if !defined $self->{_a} || $self->{_a} >= $a; + } + else # both can't be undefined due to early out + { + $self->bfround(int($p),$r) if !defined $self->{_p} || $self->{_p} <= $p; + } + # bround() or bfround() already callled bnorm() if nec. + $self; + } + +sub bnorm + { + # (numstr or BINT) return BINT + # Normalize number -- no-op here + my ($self,$x) = ref($_[0]) ? (undef,$_[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]) ? (undef,$_[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]) ? (undef,$_[0]) : objectify(1,@_); + + return $x if $x->modify('bneg'); + + # for +0 dont negate (to have always normalized +0). Does nothing for 'NaN' + $x->{sign} =~ tr/+-/-+/ unless ($x->{sign} eq '+' && $CALC->_is_zero($x->{value})); + $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,@_); + } + + return $upgrade->bcmp($x,$y) if defined $upgrade && + ((!$x->isa($self)) || (!$y->isa($self))); + + 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 acmp (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,@_); + } + + return $upgrade->bacmp($x,$y) if defined $upgrade && + ((!$x->isa($self)) || (!$y->isa($self))); + + 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 if $x->{sign} =~ /^[+-]inf$/ && $y->{sign} !~ /^[+-]inf$/; + return -1; + } + $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($upgrade->new($x),$upgrade->new($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 + } + else + { + my $a = $CALC->_acmp ($y->{value},$x->{value}); # absolute compare + if ($a > 0) + { + $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 + $x->{value} = $CALC->_zero(); + $x->{sign} = '+'; + } + else # a < 0 + { + $x->{value} = $CALC->_sub($x->{value}, $y->{value}); # abs sub + } + } + $x->round(@r); + } + +sub bsub + { + # (BINT or num_str, BINT or num_str) return BINT + # 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'); + + return $upgrade->new($x)->bsub($upgrade->new($y),@r) if defined $upgrade && + ((!$x->isa($self)) || (!$y->isa($self))); + + return $x->round(@r) if $y->is_zero(); + + # To correctly handle the lone special case $x->bsub($x), we note the sign + # of $x, then flip the sign from $y, and if the sign of $x did change, too, + # then we caught the special case: + my $xsign = $x->{sign}; + $y->{sign} =~ tr/+\-/-+/; # does nothing for NaN + if ($xsign ne $x->{sign}) + { + # special case of $x->bsub($x) results in 0 + return $x->bzero(@r) if $xsign =~ /^[+-]$/; + return $x->bnan(); # NaN, -inf, +inf + } + $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 nec. + } + +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}); + return $x->round($a,$p,$r); + } + elsif ($x->{sign} eq '-') + { + $x->{value} = $CALC->_dec($x->{value}); + $x->{sign} = '+' if $CALC->_is_zero($x->{value}); # -1 +1 => -0 => +0 + return $x->round($a,$p,$r); + } + # inf, nan handling etc + $x->badd($self->bone(),$a,$p,$r); # badd does round + } + +sub bdec + { + # decrement arg by one + my ($self,$x,@r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_); + return $x if $x->modify('bdec'); + + if ($x->{sign} eq '-') + { + # x already < 0 + $x->{value} = $CALC->_inc($x->{value}); + } + else + { + return $x->badd($self->bone('-'),@r) unless $x->{sign} eq '+'; # inf or NaN + # >= 0 + if ($CALC->_is_zero($x->{value})) + { + # == 0 + $x->{value} = $CALC->_one(); $x->{sign} = '-'; # 0 => -1 + } + else + { + # > 0 + $x->{value} = $CALC->_dec($x->{value}); + } + } + $x->round(@r); + } + +sub blog + { + # calculate $x = $a ** $base + $b and return $a (e.g. the log() to base + # $base of $x) + + # set up parameters + my ($self,$x,$base,@r) = (undef,@_); + # objectify is costly, so avoid it + if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1]))) + { + ($self,$x,$base,@r) = objectify(1,ref($x),@_); + } + + return $x if $x->modify('blog'); + + $base = $self->new($base) if defined $base && !ref $base; + + # inf, -inf, NaN, <0 => NaN + return $x->bnan() + if $x->{sign} ne '+' || (defined $base && $base->{sign} ne '+'); + + return $upgrade->blog($upgrade->new($x),$base,@r) if + defined $upgrade; + + # fix for bug #24969: + # the default base is e (Euler's number) which is not an integer + if (!defined $base) + { + require Math::BigFloat; + my $u = Math::BigFloat->blog(Math::BigFloat->new($x))->as_int(); + # modify $x in place + $x->{value} = $u->{value}; + $x->{sign} = $u->{sign}; + return $x; + } + + my ($rc,$exact) = $CALC->_log_int($x->{value},$base->{value}); + return $x->bnan() unless defined $rc; # not possible to take log? + $x->{value} = $rc; + $x->round(@r); + } + +sub bnok + { + # Calculate n over k (binomial coefficient or "choose" function) as integer. + # 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('bnok'); + return $x->bnan() if $x->{sign} eq 'NaN' || $y->{sign} eq 'NaN'; + return $x->binf() if $x->{sign} eq '+inf'; + + # k > n or k < 0 => 0 + my $cmp = $x->bacmp($y); + return $x->bzero() if $cmp < 0 || $y->{sign} =~ /^-/; + # k == n => 1 + return $x->bone(@r) if $cmp == 0; + + if ($CALC->can('_nok')) + { + $x->{value} = $CALC->_nok($x->{value},$y->{value}); + } + else + { + # ( 7 ) 7! 7*6*5 * 4*3*2*1 7 * 6 * 5 + # ( - ) = --------- = --------------- = --------- + # ( 3 ) 3! (7-3)! 3*2*1 * 4*3*2*1 3 * 2 * 1 + + # compute n - k + 2 (so we start with 5 in the example above) + my $z = $x - $y; + if (!$z->is_one()) + { + $z->binc(); + my $r = $z->copy(); $z->binc(); + my $d = $self->new(2); + while ($z->bacmp($x) <= 0) # f < x ? + { + $r->bmul($z); $r->bdiv($d); + $z->binc(); $d->binc(); + } + $x->{value} = $r->{value}; $x->{sign} = '+'; + } + else { $x->bone(); } + } + $x->round(@r); + } + +sub bexp + { + # Calculate e ** $x (Euler's number to the power of X), truncated to + # an integer value. + my ($self,$x,@r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_); + return $x if $x->modify('bexp'); + + # inf, -inf, NaN, <0 => NaN + return $x->bnan() if $x->{sign} eq 'NaN'; + return $x->bone() if $x->is_zero(); + return $x if $x->{sign} eq '+inf'; + return $x->bzero() if $x->{sign} eq '-inf'; + + my $u; + { + # run through Math::BigFloat unless told otherwise + require Math::BigFloat unless defined $upgrade; + local $upgrade = 'Math::BigFloat' unless defined $upgrade; + # calculate result, truncate it to integer + $u = $upgrade->bexp($upgrade->new($x),@r); + } + + if (!defined $upgrade) + { + $u = $u->as_int(); + # modify $x in place + $x->{value} = $u->{value}; + $x->round(@r); + } + else { $x = $u; } + } + +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); + } + my $self = ref($x); + while (@_) + { + my $y = shift; $y = $self->new($y) if !ref ($y); + $x = __lcm($x,$y); + } + $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 = $class->new($y) if !ref($y); + my $self = ref($y); + my $x = $y->copy()->babs(); # keep arguments + return $x->bnan() if $x->{sign} !~ /^[+-]$/; # x NaN? + + while (@_) + { + $y = shift; $y = $self->new($y) if !ref($y); + return $x->bnan() if $y->{sign} !~ /^[+-]$/; # y NaN? + $x->{value} = $CALC->_gcd($x->{value},$y->{value}); + last if $CALC->_is_one($x->{value}); + } + $x; + } + +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->binc()->bneg(); # binc already does round + } + +############################################################################## +# is_foo test routines +# we don't need $self, so undef instead of ref($_[0]) make it slightly faster + +sub is_zero + { + # return true if arg (BINT or num_str) is zero (array '+', '0') + 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]) ? (undef,$_[0]) : objectify(1,@_); + + $x->{sign} eq $nan ? 1 : 0; + } + +sub is_inf + { + # return true if arg (BINT or num_str) is +-inf + my ($self,$x,$sign) = ref($_[0]) ? (undef,@_) : objectify(1,@_); + + if (defined $sign) + { + $sign = '[+-]inf' if $sign eq ''; # +- doesn't matter, only that's inf + $sign = "[$1]inf" if $sign =~ /^([+-])(inf)?$/; # extract '+' or '-' + return $x->{sign} =~ /^$sign$/ ? 1 : 0; + } + $x->{sign} =~ /^[+-]inf$/ ? 1 : 0; # only +-inf is infinity + } + +sub is_one + { + # return true if arg (BINT or num_str) is +1, or -1 if sign is given + my ($self,$x,$sign) = ref($_[0]) ? (undef,@_) : objectify(1,@_); + + $sign = '+' if !defined $sign || $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 + 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 + 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) + my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_); + + return 1 if $x->{sign} eq '+inf'; # +inf is positive + + # 0+ is neither positive nor negative + ($x->{sign} eq '+' && !$x->is_zero()) ? 1 : 0; + } + +sub is_negative + { + # return true when arg (BINT or num_str) is negative (< 0) + my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_); + + $x->{sign} =~ /^-/ ? 1 : 0; # -inf is negative, but NaN is not + } + +sub is_int + { + # return true when arg (BINT or num_str) is an integer + # always true for BigInt, but different for BigFloats + my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_); + + $x->{sign} =~ /^[+-]$/ ? 1 : 0; # inf/-inf/NaN aren't + } + +############################################################################### + +sub bmul + { + # multiply the first number by the second number + # (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,$upgrade->new($y),@r) + if defined $upgrade && !$y->isa($self); + + $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); + } + +sub bmuladd + { + # multiply two numbers and then add the third to the result + # (BINT or num_str, BINT or num_str, BINT or num_str) return BINT + + # set up parameters + my ($self,$x,$y,$z,@r) = (ref($_[0]),@_); + # objectify is costly, so avoid it + if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1]))) + { + ($self,$x,$y,$z,@r) = objectify(3,@_); + } + + return $x if $x->modify('bmuladd'); + + return $x->bnan() if ($x->{sign} eq $nan) || + ($y->{sign} eq $nan) || + ($z->{sign} eq $nan); + + # inf handling of x and y + 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('-'); + } + # inf handling x*y and z + if (($z->{sign} =~ /^[+-]inf$/)) + { + # something +-inf => +-inf + $x->{sign} = $z->{sign}, return $x if $z->{sign} =~ /^[+-]inf$/; + } + + return $upgrade->bmuladd($x,$upgrade->new($y),$upgrade->new($z),@r) + if defined $upgrade && (!$y->isa($self) || !$z->isa($self) || !$x->isa($self)); + + # TODO: what if $y and $z have A or P set? + $r[3] = $z; # 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 + + my ($sx, $sz) = ( $x->{sign}, $z->{sign} ); # get signs + + if ($sx eq $sz) + { + $x->{value} = $CALC->_add($x->{value},$z->{value}); # same sign, abs add + } + else + { + my $a = $CALC->_acmp ($z->{value},$x->{value}); # absolute compare + if ($a > 0) + { + $x->{value} = $CALC->_sub($z->{value},$x->{value},1); # abs sub w/ swap + $x->{sign} = $sz; + } + elsif ($a == 0) + { + # speedup, if equal, set result to 0 + $x->{value} = $CALC->_zero(); + $x->{sign} = '+'; + } + else # a < 0 + { + $x->{value} = $CALC->_sub($x->{value}, $z->{value}); # abs sub + } + } + $x->round(@r); + } + +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),$upgrade->new($y),@r) + if defined $upgrade; + + $r[3] = $y; # no push! + + # calc new sign and in case $y == +/- 1, return $x + my $xsign = $x->{sign}; # keep + $x->{sign} = ($x->{sign} ne $y->{sign} ? '-' : '+'); + + 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->copy()->bsub($rem) if $xsign ne $y->{sign}; # one of them '-' + } + else + { + $rem->{sign} = '+'; # dont leave -0 + } + $rem->round(@r); + return ($x,$rem); + } + + $x->{value} = $CALC->_div($x->{value},$y->{value}); + $x->{sign} = '+' if $CALC->_is_zero($x->{value}); + + $x->round(@r); + } + +############################################################################### +# 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); + } + + # calc new sign and in case $y == +/- 1, return $x + $x->{value} = $CALC->_mod($x->{value},$y->{value}); + if (!$CALC->_is_zero($x->{value})) + { + $x->{value} = $CALC->_sub($y->{value},$x->{value},1) # $y-$x + if ($x->{sign} ne $y->{sign}); + $x->{sign} = $y->{sign}; + } + else + { + $x->{sign} = '+'; # dont leave -0 + } + $x->round(@r); + } + +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) = (undef,@_); + # 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 '-'; + + my $sign; + ($x->{value},$sign) = $CALC->_modinv($x->{value},$y->{value}); + return $x->bnan() if !defined $x->{value}; # in case no GCD found + return $x if !defined $sign; # already real result + $x->{sign} = $sign; # flip/flop see below + $x->bmod($y); # calc real result + $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} !~ /^[+-]$/; + + # $mod is positive, sign on $exp is ignored, result also positive + $num->{value} = $CALC->_modpow($num->{value},$exp->{value},$mod->{value}); + $num; + } + +############################################################################### + +sub bfac + { + # (BINT or num_str, BINT or num_str) return BINT + # compute factorial number from $x, modify $x in place + my ($self,$x,@r) = ref($_[0]) ? (undef,@_) : objectify(1,@_); + + return $x if $x->modify('bfac') || $x->{sign} eq '+inf'; # inf => inf + return $x->bnan() if $x->{sign} ne '+'; # NaN, <0 etc => NaN + + $x->{value} = $CALC->_fac($x->{value}); + $x->round(@r); + } + +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 $x->bnan() if $x->{sign} eq $nan || $y->{sign} eq $nan; + + # inf handling + if (($x->{sign} =~ /^[+-]inf$/) || ($y->{sign} =~ /^[+-]inf$/)) + { + if (($x->{sign} =~ /^[+-]inf$/) && ($y->{sign} =~ /^[+-]inf$/)) + { + # +-inf ** +-inf + return $x->bnan(); + } + # +-inf ** Y + if ($x->{sign} =~ /^[+-]inf/) + { + # +inf ** 0 => NaN + return $x->bnan() if $y->is_zero(); + # -inf ** -1 => 1/inf => 0 + return $x->bzero() if $y->is_one('-') && $x->is_negative(); + + # +inf ** Y => inf + return $x if $x->{sign} eq '+inf'; + + # -inf ** Y => -inf if Y is odd + return $x if $y->is_odd(); + return $x->babs(); + } + # X ** +-inf + + # 1 ** +inf => 1 + return $x if $x->is_one(); + + # 0 ** inf => 0 + return $x if $x->is_zero() && $y->{sign} =~ /^[+]/; + + # 0 ** -inf => inf + return $x->binf() if $x->is_zero(); + + # -1 ** -inf => NaN + return $x->bnan() if $x->is_one('-') && $y->{sign} =~ /^[-]/; + + # -X ** -inf => 0 + return $x->bzero() if $x->{sign} eq '-' && $y->{sign} =~ /^[-]/; + + # -1 ** inf => NaN + return $x->bnan() if $x->{sign} eq '-'; + + # X ** inf => inf + return $x->binf() if $y->{sign} =~ /^[+]/; + # X ** -inf => 0 + return $x->bzero(); + } + + return $upgrade->bpow($upgrade->new($x),$y,@r) + if defined $upgrade && (!$y->isa($self) || $y->{sign} eq '-'); + + $r[3] = $y; # no push! + + # cases 0 ** Y, X ** 0, X ** 1, 1 ** Y are handled by Calc or Emu + + my $new_sign = '+'; + $new_sign = $y->is_odd() ? '-' : '+' if ($x->{sign} ne '+'); + + # 0 ** -7 => ( 1 / (0 ** 7)) => 1 / 0 => +inf + return $x->binf() + if $y->{sign} eq '-' && $x->{sign} eq '+' && $CALC->_is_zero($x->{value}); + # 1 ** -y => 1 / (1 ** |y|) + # so do test for negative $y after above's clause + return $x->bnan() if $y->{sign} eq '-' && !$CALC->_is_one($x->{value}); + + $x->{value} = $CALC->_pow($x->{value},$y->{value}); + $x->{sign} = $new_sign; + $x->{sign} = '+' if $CALC->_is_zero($y->{value}); + $x->round(@r); + } + +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 '-'; + + $x->{value} = $CALC->_lsft($x->{value},$y->{value},$n); + $x->round(@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 ($y >= CORE::length($bin)) + { + $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 + } + 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 < 0, n == 2, y == 1 + $x->bdec(); # n == 2, but $y == 1: this fixes it + } + + $x->{value} = $CALC->_rsft($x->{value},$y->{value},$n); + $x->round(@r); + } + +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! + + return $x->bnan() if ($x->{sign} !~ /^[+-]$/ || $y->{sign} !~ /^[+-]$/); + + my $sx = $x->{sign} eq '+' ? 1 : -1; + my $sy = $y->{sign} eq '+' ? 1 : -1; + + if ($sx == 1 && $sy == 1) + { + $x->{value} = $CALC->_and($x->{value},$y->{value}); + return $x->round(@r); + } + + if ($CAN{signed_and}) + { + $x->{value} = $CALC->_signed_and($x->{value},$y->{value},$sx,$sy); + return $x->round(@r); + } + + require $EMU_LIB; + __emu_band($self,$x,$y,$sx,$sy,@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! + + return $x->bnan() if ($x->{sign} !~ /^[+-]$/ || $y->{sign} !~ /^[+-]$/); + + my $sx = $x->{sign} eq '+' ? 1 : -1; + my $sy = $y->{sign} eq '+' ? 1 : -1; + + # the sign of X follows the sign of X, e.g. sign of Y irrelevant for bior() + + # don't use lib for negative values + if ($sx == 1 && $sy == 1) + { + $x->{value} = $CALC->_or($x->{value},$y->{value}); + return $x->round(@r); + } + + # if lib can do negative values, let it handle this + if ($CAN{signed_or}) + { + $x->{value} = $CALC->_signed_or($x->{value},$y->{value},$sx,$sy); + return $x->round(@r); + } + + require $EMU_LIB; + __emu_bior($self,$x,$y,$sx,$sy,@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! + + return $x->bnan() if ($x->{sign} !~ /^[+-]$/ || $y->{sign} !~ /^[+-]$/); + + my $sx = $x->{sign} eq '+' ? 1 : -1; + my $sy = $y->{sign} eq '+' ? 1 : -1; + + # don't use lib for negative values + if ($sx == 1 && $sy == 1) + { + $x->{value} = $CALC->_xor($x->{value},$y->{value}); + return $x->round(@r); + } + + # if lib can do negative values, let it handle this + if ($CAN{signed_xor}) + { + $x->{value} = $CALC->_signed_xor($x->{value},$y->{value},$sx,$sy); + return $x->round(@r); + } + + require $EMU_LIB; + __emu_bxor($self,$x,$y,$sx,$sy,@r); + } + +sub length + { + my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_); + + my $e = $CALC->_len($x->{value}); + wantarray ? ($e,0) : $e; + } + +sub digit + { + # return the nth decimal digit, negative values count backward, 0 is right + my ($self,$x,$n) = ref($_[0]) ? (undef,@_) : objectify(1,@_); + + $n = $n->numify() if ref($n); + $CALC->_digit($x->{value},$n||0); + } + +sub _trailing_zeros + { + # return the amount of trailing zeros in $x (as scalar) + my $x = shift; + $x = $class->new($x) unless ref $x; + + return 0 if $x->{sign} !~ /^[+-]$/; # NaN, inf, -inf etc + + $CALC->_zeros($x->{value}); # must handle odd values, 0 etc + } + +sub bsqrt + { + # calculate square root of $x + my ($self,$x,@r) = ref($_[0]) ? (undef,@_) : objectify(1,@_); + + return $x if $x->modify('bsqrt'); + + return $x->bnan() if $x->{sign} !~ /^\+/; # -x or -inf or NaN => NaN + return $x if $x->{sign} eq '+inf'; # sqrt(+inf) == inf + + return $upgrade->bsqrt($x,@r) if defined $upgrade; + + $x->{value} = $CALC->_sqrt($x->{value}); + $x->round(@r); + } + +sub broot + { + # calculate $y'th root of $x + + # set up parameters + my ($self,$x,$y,@r) = (ref($_[0]),@_); + + $y = $self->new(2) unless defined $y; + + # objectify is costly, so avoid it + if ((!ref($x)) || (ref($x) ne ref($y))) + { + ($self,$x,$y,@r) = objectify(2,$self || $class,@_); + } + + return $x if $x->modify('broot'); + + # NaN handling: $x ** 1/0, x or y NaN, or y inf/-inf or y == 0 + return $x->bnan() if $x->{sign} !~ /^\+/ || $y->is_zero() || + $y->{sign} !~ /^\+$/; + + return $x->round(@r) + if $x->is_zero() || $x->is_one() || $x->is_inf() || $y->is_one(); + + return $upgrade->new($x)->broot($upgrade->new($y),@r) if defined $upgrade; + + $x->{value} = $CALC->_root($x->{value},$y->{value}); + $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/^[+-]//; # NaN, -inf,+inf => NaN or inf + return $self->new($s); + } + return $self->bone() if $x->is_zero(); + + # 12300 => 2 trailing zeros => exponent is 2 + $self->new( $CALC->_zeros($x->{value}) ); + } + +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} !~ /^[+-]$/) + { + # for NaN, +inf, -inf: keep the sign + return $self->new($x->{sign}); + } + my $m = $x->copy(); delete $m->{_p}; delete $m->{_a}; + + # that's a bit inefficient: + my $zeros = $CALC->_zeros($m->{value}); + $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]) ? (undef,$_[0]) : objectify(1,@_); + + ($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; my $self = ref($x) || $x; $x = $self->new($x) unless ref $x; + + my ($scale,$mode) = $x->_scale_p(@_); + + return $x if !defined $scale || $x->modify('bfround'); # no-op + + # no-op for BigInts if $n <= 0 + $x->bround( $x->length()-$scale, $mode) if $scale > 0; + + delete $x->{_a}; # delete to save memory + $x->{_p} = $scale; # store new _p + $x; + } + +sub _scan_for_nonzero + { + # internal, used by bround() to scan for non-zeros after a '5' + my ($x,$pad,$xs,$len) = @_; + + return 0 if $len == 1; # "5" is trailed by invisible zeros + my $follow = $pad - 1; + return 0 if $follow > $len || $follow < 1; + + # use the string form to check whether only '0's follow or not + substr ($xs,-$follow) =~ /[^0]/ ? 1 : 0; + } + +sub fround + { + # Exists to make life easier for switch between MBF and MBI (should we + # autoload fxxx() like MBF does for bxxx()?) + my $x = shift; $x = $class->new($x) unless ref $x; + $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(@_); + return $x if !defined $scale || $x->modify('bround'); # no-op + + 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(); + # convert $scale to a scalar in case it is an object (put's a limit on the + # number length, but this would already limited by memory constraints), makes + # it faster + $scale = $scale->numify() if ref ($scale); + + # 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 very costly for binary => decimal + # getting the entire string is also costly, but we need to do it only once + 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,$len) == 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; # replace with '00...' + $put_back = 1; # need to put back + } + elsif ($pad > $len) + { + $x->bzero(); # round to '0' + } + + if ($round_up) # what gave test above? + { + $put_back = 1; # need to put back + $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, 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; no-op since it's already integer + my ($self,$x,@r) = ref($_[0]) ? (undef,@_) : objectify(1,@_); + + $x->round(@r); + } + +sub bceil + { + # return integer greater or equal then number; no-op since it's already int + my ($self,$x,@r) = ref($_[0]) ? (undef,@_) : objectify(1,@_); + + $x->round(@r); + } + +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. + $_[0]->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 + + my $s = ''; + $s = $x->{sign} if $x->{sign} eq '-'; + $s . $CALC->_as_hex($x->{value}); + } + +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 + + my $s = ''; $s = $x->{sign} if $x->{sign} eq '-'; + return $s . $CALC->_as_bin($x->{value}); + } + +sub as_oct + { + # return as octal string, with prefixed 0 + my $x = shift; $x = $class->new($x) if !ref($x); + + return $x->bstr() if $x->{sign} !~ /^[+-]$/; # inf, nan etc + + my $s = ''; $s = $x->{sign} if $x->{sign} eq '-'; + return $s . $CALC->_as_oct($x->{value}); + } + +############################################################################## +# private stuff (internal use only) + +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 being 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 STDERR "# Now in objectify, my class is today $a[0], count = $count\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); + } + 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); + } + push @a,$k; + } + push @a,@_; # return other params, too + } + if (! wantarray) + { + require Carp; Carp::croak ("$class objectify needs list context"); + } + ${"$a[0]::downgrade"} = $d; + @a; + } + +sub _register_callback + { + my ($class,$callback) = @_; + + if (ref($callback) ne 'CODE') + { + require Carp; + Carp::croak ("$callback is not a coderef"); + } + $CALLBACKS{$class} = $callback; + } + +sub import + { + my $self = shift; + + $IMPORT++; # remember we did import() + my @a; my $l = scalar @_; + my $warn_or_die = 0; # 0 - no warn, 1 - warn, 2 - die + 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) }, + binary => sub { $self->new(shift) }; + } + elsif ($_[$i] eq 'upgrade') + { + # this causes upgrading + $upgrade = $_[$i+1]; # or undef to disable + $i++; + } + elsif ($_[$i] =~ /^(lib|try|only)\z/) + { + # this causes a different low lib to take care... + $CALC = $_[$i+1] || ''; + # lib => 1 (warn on fallback), try => 0 (no warn), only => 2 (die on fallback) + $warn_or_die = 1 if $_[$i] eq 'lib'; + $warn_or_die = 2 if $_[$i] eq 'only'; + $i++; + } + else + { + push @a, $_[$i]; + } + } + # any non :constant stuff is handled by our parent, Exporter + if (@a > 0) + { + require Exporter; + + $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; + foreach (@c) + { + $_ =~ tr/a-zA-Z0-9://cd; # limit to sane characters + } + push @c, \'FastCalc', \'Calc' # if all fail, try these + if $warn_or_die < 2; # but not for "only" + $CALC = ''; # signal error + foreach my $l (@c) + { + # fallback libraries are "marked" as \'string', extract string if nec. + my $lib = $l; $lib = $$l if ref($l); + + next if ($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 $lib qw/@c/;"; + } + if ($@ eq '') + { + my $ok = 1; + # loaded it ok, see if the api_version() is high enough + if ($lib->can('api_version') && $lib->api_version() >= 1.0) + { + $ok = 0; + # api_version matches, check if it really provides anything we need + for my $method (qw/ + one two ten + str num + add mul div sub dec inc + acmp len digit is_one is_zero is_even is_odd + is_two is_ten + zeros new copy check + from_hex from_oct from_bin as_hex as_bin as_oct + rsft lsft xor and or + mod sqrt root fac pow modinv modpow log_int gcd + /) + { + if (!$lib->can("_$method")) + { + if (($WARN{$lib}||0) < 2) + { + require Carp; + Carp::carp ("$lib is missing method '_$method'"); + $WARN{$lib} = 1; # still warn about the lib + } + $ok++; last; + } + } + } + if ($ok == 0) + { + $CALC = $lib; + if ($warn_or_die > 0 && ref($l)) + { + require Carp; + my $msg = "Math::BigInt: couldn't load specified math lib(s), fallback to $lib"; + Carp::carp ($msg) if $warn_or_die == 1; + Carp::croak ($msg) if $warn_or_die == 2; + } + last; # found a usable one, break + } + else + { + if (($WARN{$lib}||0) < 2) + { + my $ver = eval "\$$lib\::VERSION" || 'unknown'; + require Carp; + Carp::carp ("Cannot load outdated $lib v$ver, please upgrade"); + $WARN{$lib} = 2; # never warn again + } + } + } + } + if ($CALC eq '') + { + require Carp; + if ($warn_or_die == 2) + { + Carp::croak ("Couldn't load specified math lib(s) and fallback disallowed"); + } + else + { + Carp::croak ("Couldn't load any math lib(s), not even fallback to Calc.pm"); + } + } + + # notify callbacks + foreach my $class (keys %CALLBACKS) + { + &{$CALLBACKS{$class}}($CALC); + } + + # Fill $CAN with the results of $CALC->can(...) for emulating lower math lib + # functions + + %CAN = (); + for my $method (qw/ signed_and signed_or signed_xor /) + { + $CAN{$method} = $CALC->can("_$method") ? 1 : 0; + } + + # import done + } + +sub from_hex + { + # create a bigint from a hexadecimal string + my ($self, $hs) = @_; + + my $rc = __from_hex($hs); + + return $self->bnan() unless defined $rc; + + $rc; + } + +sub from_bin + { + # create a bigint from a hexadecimal string + my ($self, $bs) = @_; + + my $rc = __from_bin($bs); + + return $self->bnan() unless defined $rc; + + $rc; + } + +sub from_oct + { + # create a bigint from a hexadecimal string + my ($self, $os) = @_; + + my $x = $self->bzero(); + + # strip underscores + $os =~ s/([0-7])_([0-7])/$1$2/g; + $os =~ s/([0-7])_([0-7])/$1$2/g; + + return $x->bnan() if $os !~ /^[\-\+]?0[0-7]+\z/; + + my $sign = '+'; $sign = '-' if $os =~ /^-/; + + $os =~ s/^[+-]//; # strip sign + $x->{value} = $CALC->_from_oct($os); + $x->{sign} = $sign unless $CALC->_is_zero($x->{value}); # no '-0' + $x; + } + +sub __from_hex + { + # internal + # 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 + $x->{value} = $CALC->_from_hex($hs); + $x->{sign} = $sign unless $CALC->_is_zero($x->{value}); # no '-0' + $x; + } + +sub __from_bin + { + # internal + # 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 + + $x->{value} = $CALC->_from_bin($bs); + $x->{sign} = $sign unless $CALC->_is_zero($x->{value}); # no '-0' + $x; + } + +sub _split + { + # input: num_str; output: undef for invalid or + # (\$mantissa_sign,\$mantissa_value,\$mantissa_fraction,\$exp_sign,\$exp_value) + # 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 =~ /^[+-]?[0-9]+\z/) + { + $x =~ s/^([+-])0*([0-9])/$2/; my $sign = $1 || '+'; + return (\$sign, \$x, \'', \'', \0); + } + + # invalid starting char? + return if $x !~ /^[+-]?(\.?[0-9]|0b[0-1]|0x[0-9a-fA-F])/; + + return __from_hex($x) if $x =~ /^[\-\+]?0x/; # hex string + return __from_bin($x) if $x =~ /^[\-\+]?0b/; # binary string + + # strip underscores between digits + $x =~ s/([0-9])_([0-9])/$1$2/g; + $x =~ s/([0-9])_([0-9])/$1$2/g; # do twice for 1_2_3 + + # some possible inputs: + # 2.1234 # 0.12 # 1 # 1E1 # 2.134E1 # 434E-10 # 1.02009E-2 + # .2 # 1_2_3.4_5_6 # 1.4E1_2_3 # 1e3 # +.2 # 0e999 + + my ($m,$e,$last) = split /[Ee]/,$x; + return if defined $last; # last defined => 1e2E3 or others + $e = '0' if !defined $e || $e eq ""; + + # sign,value for exponent,mantint,mantfrac + my ($es,$ev,$mis,$miv,$mfv); + # valid exponent? + if ($e =~ /^([+-]?)0*([0-9]+)$/) # strip leading zeros + { + $es = $1; $ev = $2; + # valid mantissa? + return if $m eq '.' || $m eq ''; + my ($mi,$mf,$lastf) = split /\./,$m; + return if defined $lastf; # lastf 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*([0-9]+)$/) # strip leading zeros + { + $mis = $1||'+'; $miv = $2; + return unless ($mf =~ /^([0-9]*?)0*$/); # strip trailing zeros + $mfv = $1; + # handle the 0e999 case here + $ev = 0 if $miv eq '0' && $mfv eq ''; + return (\$mis,\$miv,\$mfv,\$es,\$ev); + } + } + return; # NaN, not a number + } + +############################################################################## +# internal calculation routines (others are in Math::BigInt::Calc etc) + +sub __lcm + { + # (BINT or num_str, BINT or num_str) return BINT + # does modify first argument + # LCM + + my ($x,$ty) = @_; + return $x->bnan() if ($x->{sign} eq $nan) || ($ty->{sign} eq $nan); + my $method = ref($x) . '::bgcd'; + no strict 'refs'; + $x * $ty / &$method($x,$ty); + } + +############################################################################### +# trigonometric functions + +sub bpi + { + # Calculate PI to N digits. Unless upgrading is in effect, returns the + # result truncated to an integer, that is, always returns '3'. + my ($self,$n) = @_; + if (@_ == 1) + { + # called like Math::BigInt::bpi(10); + $n = $self; $self = $class; + } + $self = ref($self) if ref($self); + + return $upgrade->new($n) if defined $upgrade; + + # hard-wired to "3" + $self->new(3); + } + +sub bcos + { + # Calculate cosinus(x) to N digits. Unless upgrading is in effect, returns the + # result truncated to an integer. + my ($self,$x,@r) = ref($_[0]) ? (undef,@_) : objectify(1,@_); + + return $x if $x->modify('bcos'); + + return $x->bnan() if $x->{sign} !~ /^[+-]\z/; # -inf +inf or NaN => NaN + + return $upgrade->new($x)->bcos(@r) if defined $upgrade; + + require Math::BigFloat; + # calculate the result and truncate it to integer + my $t = Math::BigFloat->new($x)->bcos(@r)->as_int(); + + $x->bone() if $t->is_one(); + $x->bzero() if $t->is_zero(); + $x->round(@r); + } + +sub bsin + { + # Calculate sinus(x) to N digits. Unless upgrading is in effect, returns the + # result truncated to an integer. + my ($self,$x,@r) = ref($_[0]) ? (undef,@_) : objectify(1,@_); + + return $x if $x->modify('bsin'); + + return $x->bnan() if $x->{sign} !~ /^[+-]\z/; # -inf +inf or NaN => NaN + + return $upgrade->new($x)->bsin(@r) if defined $upgrade; + + require Math::BigFloat; + # calculate the result and truncate it to integer + my $t = Math::BigFloat->new($x)->bsin(@r)->as_int(); + + $x->bone() if $t->is_one(); + $x->bzero() if $t->is_zero(); + $x->round(@r); + } + +sub batan2 + { + # calculate arcus tangens of ($y/$x) + + # set up parameters + my ($self,$y,$x,@r) = (ref($_[0]),@_); + # objectify is costly, so avoid it + if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1]))) + { + ($self,$y,$x,@r) = objectify(2,@_); + } + + return $y if $y->modify('batan2'); + + return $y->bnan() if ($y->{sign} eq $nan) || ($x->{sign} eq $nan); + + # Y X + # != 0 -inf result is +- pi + if ($x->is_inf() || $y->is_inf()) + { + # upgrade to BigFloat etc. + return $upgrade->new($y)->batan2($upgrade->new($x),@r) if defined $upgrade; + if ($y->is_inf()) + { + if ($x->{sign} eq '-inf') + { + # calculate 3 pi/4 => 2.3.. => 2 + $y->bone( substr($y->{sign},0,1) ); + $y->bmul($self->new(2)); + } + elsif ($x->{sign} eq '+inf') + { + # calculate pi/4 => 0.7 => 0 + $y->bzero(); + } + else + { + # calculate pi/2 => 1.5 => 1 + $y->bone( substr($y->{sign},0,1) ); + } + } + else + { + if ($x->{sign} eq '+inf') + { + # calculate pi/4 => 0.7 => 0 + $y->bzero(); + } + else + { + # PI => 3.1415.. => 3 + $y->bone( substr($y->{sign},0,1) ); + $y->bmul($self->new(3)); + } + } + return $y; + } + + return $upgrade->new($y)->batan2($upgrade->new($x),@r) if defined $upgrade; + + require Math::BigFloat; + my $r = Math::BigFloat->new($y)->batan2(Math::BigFloat->new($x),@r)->as_int(); + + $x->{value} = $r->{value}; + $x->{sign} = $r->{sign}; + + $x; + } + +sub batan + { + # Calculate arcus tangens of x to N digits. Unless upgrading is in effect, returns the + # result truncated to an integer. + my ($self,$x,@r) = ref($_[0]) ? (undef,@_) : objectify(1,@_); + + return $x if $x->modify('batan'); + + return $x->bnan() if $x->{sign} !~ /^[+-]\z/; # -inf +inf or NaN => NaN + + return $upgrade->new($x)->batan(@r) if defined $upgrade; + + # calculate the result and truncate it to integer + my $t = Math::BigFloat->new($x)->batan(@r); + + $x->{value} = $CALC->_new( $x->as_int()->bstr() ); + $x->round(@r); + } + +############################################################################### +# this method returns 0 if the object can be modified, or 1 if not. +# We use a fast constant sub() here, to avoid costly calls. Subclasses +# may override it with special code (f.i. Math::BigInt::Constant does so) + +sub modify () { 0; } + +1; +__END__ + +=pod + +=head1 NAME + +Math::BigInt - Arbitrary size integer/float math package + +=head1 SYNOPSIS + + use Math::BigInt; + + # or make it faster with huge numbers: install (optional) + # Math::BigInt::GMP and always use (it will fall back to + # pure Perl if the GMP library is not installed): + # (See also the L<MATH LIBRARY> section!) + + # will warn if Math::BigInt::GMP cannot be found + use Math::BigInt lib => 'GMP'; + + # to supress the warning use this: + # use Math::BigInt try => 'GMP'; + + # dies if GMP cannot be loaded: + # use Math::BigInt only => 'GMP'; + + my $str = '1234567890'; + my @values = (64,74,18); + my $n = 1; my $sign = '-'; + + # Number creation + my $x = Math::BigInt->new($str); # defaults to 0 + my $y = $x->copy(); # make a true copy + my $nan = Math::BigInt->bnan(); # create a NotANumber + my $zero = Math::BigInt->bzero(); # create a +0 + my $inf = Math::BigInt->binf(); # create a +inf + my $inf = Math::BigInt->binf('-'); # create a -inf + my $one = Math::BigInt->bone(); # create a +1 + my $mone = Math::BigInt->bone('-'); # create a -1 + + my $pi = Math::BigInt->bpi(); # returns '3' + # see Math::BigFloat::bpi() + + $h = Math::BigInt->new('0x123'); # from hexadecimal + $b = Math::BigInt->new('0b101'); # from binary + $o = Math::BigInt->from_oct('0101'); # from octal + + # Testing (don't modify their arguments) + # (return true if the condition is met, otherwise false) + + $x->is_zero(); # if $x is +0 + $x->is_nan(); # if $x is NaN + $x->is_one(); # if $x is +1 + $x->is_one('-'); # if $x is -1 + $x->is_odd(); # if $x is odd + $x->is_even(); # if $x is even + $x->is_pos(); # if $x >= 0 + $x->is_neg(); # if $x < 0 + $x->is_inf($sign); # if $x is +inf, or -inf (sign is default '+') + $x->is_int(); # if $x is an integer (not a float) + + # comparing and digit/sign extraction + $x->bcmp($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. If you want to preserve + # $x, use $z = $x->copy()->bXXX($y); See under L<CAVEATS> for why this is + # necessary when mixing $a = $b assignments with non-overloaded math. + + $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 in BigInt) + $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 + + $x->bmuladd($y,$z); # $x = $x * $y + $z + + $x->bmod($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 in base 2 + $x->brsft($y); # right shift in base 2 + # returns (quo,rem) or quo if in scalar context + $x->blsft($y,$n); # left shift by $y places in base $n + $x->brsft($y,$n); # right shift by $y places in base $n + # returns (quo,rem) or quo if in scalar context + + $x->band($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 + $x->broot($y); # $y'th root of $x (e.g. $y == 3 => cubic root) + $x->bfac(); # factorial of $x (1*2*3*4*..$x) + + $x->bnok($y); # x over y (binomial coefficient n over k) + + $x->blog(); # logarithm of $x to base e (Euler's number) + $x->blog($base); # logarithm of $x to base $base (f.i. 2) + $x->bexp(); # calculate e ** $x where e is Euler's number + + $x->round($A,$P,$mode); # round to accuracy or precision using mode $mode + $x->bround($n); # accuracy: preserve $n digits + $x->bfround($n); # $n > 0: round $nth digits, + # $n < 0: round to the $nth digit after the + # dot, no-op for BigInts + + # The following do not modify their arguments in BigInt (are no-ops), + # but do so 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: + + # greatest common divisor (no OO style) + my $gcd = Math::BigInt::bgcd(@values); + # lowest common multiplicator (no OO style) + my $lcm = Math::BigInt::blcm(@values); + + $x->length(); # return number of digits in number + ($xl,$f) = $x->length(); # length of number and length of fraction part, + # latter is always 0 digits long for BigInts + + $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_int(); # return as BigInt (in BigInt: same as copy()) + $x->numify(); # return as scalar (might overflow!) + + # conversation to string (do not modify their argument) + $x->bstr(); # normalized string (e.g. '3') + $x->bsstr(); # norm. string in scientific notation (e.g. '3E0') + $x->as_hex(); # as signed hexadecimal string with prefixed 0x + $x->as_bin(); # as signed binary string with prefixed 0b + $x->as_oct(); # as signed octal string with prefixed 0 + + + # 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 + + # Global methods + Math::BigInt->precision(); # get/set global P for all BigInt objects + Math::BigInt->accuracy(); # get/set global A for all BigInt objects + Math::BigInt->round_mode(); # get/set global round mode, one of + # 'even', 'odd', '+inf', '-inf', 'zero', 'trunc' or 'common' + Math::BigInt->config(); # return hash containing configuration + +=head1 DESCRIPTION + +All operators (including basic math operations) are overloaded if you +declare your big integers as + + $i = new Math::BigInt '123_456_789_123_456_789'; + +Operations with overloaded operators preserve the arguments which is +exactly what you expect. + +=over 2 + +=item Input + +Input values to these routines may be any string, that looks like a number +and results in an integer, including hexadecimal and binary numbers. + +Scalars holding numbers may also be passed, but note that non-integer numbers +may already have lost precision due to the conversation to float. Quote +your input if you want BigInt to see all the digits: + + $x = Math::BigInt->new(12345678890123456789); # bad + $x = Math::BigInt->new('12345678901234567890'); # good + +You can include one underscore between any two digits. + +This means integer values like 1.01E2 or even 1000E-2 are also accepted. +Non-integer values result in NaN. + +Hexadecimal (prefixed with "0x") and binary numbers (prefixed with "0b") +are accepted, too. Please note that octal numbers are not recognized +by new(), so the following will print "123": + + perl -MMath::BigInt -le 'print Math::BigInt->new("0123")' + +To convert an octal number, use from_oct(); + + perl -MMath::BigInt -le 'print Math::BigInt->from_oct("0123")' + +Currently, Math::BigInt::new() defaults to 0, while Math::BigInt::new('') +results in 'NaN'. This might change in the future, so use always the following +explicit forms to get a zero or NaN: + + $zero = Math::BigInt->bzero(); + $nan = Math::BigInt->bnan(); + +C<bnorm()> on a BigInt object is now effectively a no-op, since the numbers +are always stored in normalized form. If passed a string, creates a BigInt +object from the input. + +=item Output + +Output values are BigInt objects (normalized), except for the methods which +return a string (see L<SYNOPSIS>). + +Some routines (C<is_odd()>, C<is_even()>, C<is_zero()>, C<is_one()>, +C<is_nan()>, etc.) return true or false, while others (C<bcmp()>, C<bacmp()>) +return either undef (if NaN is involved), <0, 0 or >0 and are suited for sort. + +=back + +=head1 METHODS + +Each of the methods below (except config(), accuracy() and precision()) +accepts three additional parameters. These arguments C<$A>, C<$P> and C<$R> +are C<accuracy>, C<precision> and C<round_mode>. Please see the section about +L<ACCURACY and PRECISION> for more information. + +=head2 config() + + use Data::Dumper; + + print Dumper ( Math::BigInt->config() ); + print Math::BigInt->config()->{lib},"\n"; + +Returns a hash containing the configuration, e.g. the version number, lib +loaded etc. The following hash keys are currently filled in with the +appropriate information. + + key Description + Example + ============================================================ + lib Name of the low-level math library + Math::BigInt::Calc + lib_version Version of low-level math library (see 'lib') + 0.30 + class The class name of config() you just called + Math::BigInt + upgrade To which class math operations might be upgraded + Math::BigFloat + downgrade To which class math operations might be downgraded + undef + precision Global precision + undef + accuracy Global accuracy + undef + round_mode Global round mode + even + version version number of the class you used + 1.61 + div_scale Fallback accuracy for div + 40 + trap_nan If true, traps creation of NaN via croak() + 1 + trap_inf If true, traps creation of +inf/-inf via croak() + 1 + +The following values can be set by passing C<config()> a reference to a hash: + + trap_inf trap_nan + upgrade downgrade precision accuracy round_mode div_scale + +Example: + + $new_cfg = Math::BigInt->config( { trap_inf => 1, precision => 5 } ); + +=head2 accuracy() + + $x->accuracy(5); # local for $x + CLASS->accuracy(5); # global for all members of CLASS + # Note: This also applies to new()! + + $A = $x->accuracy(); # read out accuracy that affects $x + $A = CLASS->accuracy(); # read out global accuracy + +Set or get the global or local accuracy, aka how many significant digits the +results have. If you set a global accuracy, then this also applies to new()! + +Warning! The accuracy I<sticks>, e.g. once you created a number under the +influence of C<< CLASS->accuracy($A) >>, all results from math operations with +that number will also be rounded. + +In most cases, you should probably round the results explicitly using one of +L<round()>, L<bround()> or L<bfround()> or by passing the desired accuracy +to the math operation as additional parameter: + + my $x = Math::BigInt->new(30000); + my $y = Math::BigInt->new(7); + print scalar $x->copy()->bdiv($y, 2); # print 4300 + print scalar $x->copy()->bdiv($y)->bround(2); # print 4300 + +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); # $x will be automatically rounded! + print "$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 + +Note: Works also for subclasses like Math::BigFloat. Each class has it's own +globals separated from Math::BigInt, but it is possible to subclass +Math::BigInt and make the globals of the subclass aliases to the ones from +Math::BigInt. + +=head2 precision() + + $x->precision(-2); # local for $x, round at the second digit right of the dot + $x->precision(2); # ditto, round at the second digit left of the dot + + CLASS->precision(5); # Global for all members of CLASS + # This also applies to new()! + CLASS->precision(-5); # ditto + + $P = CLASS->precision(); # read out global precision + $P = $x->precision(); # read out precision that affects $x + +Note: You probably want to use L<accuracy()> instead. With L<accuracy> you +set the number of digits each result should have, with L<precision> you +set the place where to round! + +C<precision()> sets or gets the global or local precision, aka at which digit +before or after the dot to round all results. A set global precision also +applies to all newly created numbers! + +In Math::BigInt, passing a negative number precision has no effect since no +numbers have digits after the dot. In L<Math::BigFloat>, it will round all +results to P digits after the dot. + +Please see the section about L<ACCURACY AND PRECISION> for further details. + +Pass an undef value to disable it: + + $x->precision(undef); + Math::BigInt->precision(undef); + +Returns the current precision. For C<$x->precision()> it will return either the +local precision of $x, or if not defined, the global. This means the return +value represents the prevision that will be in effect for $x: + + $y = Math::BigInt->new(1234567); # unrounded + print Math::BigInt->precision(4),"\n"; # set 4, print 4 + $x = Math::BigInt->new(123456); # will be automatically rounded + print $x; # print "120000"! + +Note: Works also for subclasses like L<Math::BigFloat>. Each class has its +own globals separated from Math::BigInt, but it is possible to subclass +Math::BigInt and make the globals of the subclass aliases to the ones from +Math::BigInt. + +=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 scalar or another BigInt object. The +input is accepted as decimal, hex (with leading '0x') or binary (with leading +'0b'). + +See L<Input> for more info on accepted input formats. + +=head2 from_oct() + + $x = Math::BigInt->from_oct("0775"); # input is octal + +=head2 from_hex() + + $x = Math::BigInt->from_hex("0xcafe"); # input is hexadecimal + +=head2 from_bin() + + $x = Math::BigInt->from_oct("0x10011"); # input is binary + +=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 + $x->is_inf('-'); # true if -inf (sign is default '+') + +These methods all test the BigInt for being one specific value and return +true or false depending on the input. These are faster than doing something +like: + + if ($x == 0) + +=head2 is_pos()/is_neg()/is_positive()/is_negative() + + $x->is_pos(); # true if > 0 + $x->is_neg(); # 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 neither positive nor negative. + +These methods are only testing the sign, and not the value. + +C<is_positive()> and C<is_negative()> are aliases to C<is_pos()> and +C<is_neg()>, respectively. C<is_positive()> and C<is_negative()> were +introduced in v1.36, while C<is_pos()> and C<is_neg()> were only introduced +in v1.68. + +=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. + +In BigInt, all numbers except C<NaN>, C<+inf> and C<-inf> are integers. + +=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. + +If you want $x to have a certain sign, use one of the following methods: + + $x->babs(); # '+' + $x->babs()->bneg(); # '-' + $x->bnan(); # 'NaN' + $x->binf(); # '+inf' + $x->binf('-'); # '-inf' + +=head2 digit() + + $x->digit($n); # return the nth digit, counting from right + +If C<$n> is negative, returns the digit counting from left. + +=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 its 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 (bitwise not). This is equivalent to + + $x->binc()->bneg(); + +but faster. + +=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 bmuladd() + + $x->bmuladd($y,$z); + +Multiply $x by $y, and then add $z to the result, + +This method was added in v1.87 of Math::BigInt (June 2007). + +=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 % $mod + +because it is much faster - it reduces internal variables into +the modulus whenever possible, so it operates on smaller numbers. + +C<bmodpow> also supports negative exponents. + + bmodpow($num, -1, $mod) + +is exactly equivalent to + + bmodinv($num, $mod) + +=head2 bpow() + + $x->bpow($y); # power of arguments (x ** y) + +=head2 blog() + + $x->blog($base, $accuracy); # logarithm of x to the base $base + +If C<$base> is not defined, Euler's number (e) is used: + + print $x->blog(undef, 100); # log(x) to 100 digits + +=head2 bexp() + + $x->bexp($accuracy); # calculate e ** X + +Calculates the expression C<e ** $x> where C<e> is Euler's number. + +This method was added in v1.82 of Math::BigInt (April 2007). + +See also L<blog()>. + +=head2 bnok() + + $x->bnok($y); # x over y (binomial coefficient n over k) + +Calculates the binomial coefficient n over k, also called the "choose" +function. The result is equivalent to: + + ( n ) n! + | - | = ------- + ( k ) k!(n-k)! + +This method was added in v1.84 of Math::BigInt (April 2007). + +=head2 bpi() + + print Math::BigInt->bpi(100), "\n"; # 3 + +Returns PI truncated to an integer, with the argument being ignored. This means +under BigInt this always returns C<3>. + +If upgrading is in effect, returns PI, rounded to N digits with the +current rounding mode: + + use Math::BigFloat; + use Math::BigInt upgrade => Math::BigFloat; + print Math::BigInt->bpi(3), "\n"; # 3.14 + print Math::BigInt->bpi(100), "\n"; # 3.1415.... + +This method was added in v1.87 of Math::BigInt (June 2007). + +=head2 bcos() + + my $x = Math::BigInt->new(1); + print $x->bcos(100), "\n"; + +Calculate the cosinus of $x, modifying $x in place. + +In BigInt, unless upgrading is in effect, the result is truncated to an +integer. + +This method was added in v1.87 of Math::BigInt (June 2007). + +=head2 bsin() + + my $x = Math::BigInt->new(1); + print $x->bsin(100), "\n"; + +Calculate the sinus of $x, modifying $x in place. + +In BigInt, unless upgrading is in effect, the result is truncated to an +integer. + +This method was added in v1.87 of Math::BigInt (June 2007). + +=head2 batan2() + + my $x = Math::BigInt->new(1); + my $y = Math::BigInt->new(1); + print $y->batan2($x), "\n"; + +Calculate the arcus tangens of C<$y> divided by C<$x>, modifying $y in place. + +In BigInt, unless upgrading is in effect, the result is truncated to an +integer. + +This method was added in v1.87 of Math::BigInt (June 2007). + +=head2 batan() + + my $x = Math::BigFloat->new(0.5); + print $x->batan(100), "\n"; + +Calculate the arcus tangens of $x, modifying $x in place. + +In BigInt, unless upgrading is in effect, the result is truncated to an +integer. + +This method was added in v1.87 of Math::BigInt (June 2007). + +=head2 blsft() + + $x->blsft($y); # left shift in base 2 + $x->blsft($y,$n); # left shift, in base $n (like 10) + +=head2 brsft() + + $x->brsft($y); # right shift in base 2 + $x->brsft($y,$n); # right shift, in 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 broot() + + $x->broot($N); + +Calculates the N'th root of C<$x>. + +=head2 bfac() + + $x->bfac(); # factorial of $x (1*2*3*4*..$x) + +=head2 round() + + $x->round($A,$P,$round_mode); + +Round $x to accuracy C<$A> or precision C<$P> using the round mode +C<$round_mode>. + +=head2 bround() + + $x->bround($N); # accuracy: preserve $N digits + +=head2 bfround() + + $x->bfround($N); + +If N is > 0, rounds to the Nth digit from the left. If N < 0, rounds to +the Nth digit after the dot. Since BigInts are integers, the case N < 0 +is a no-op for them. + +Examples: + + Input N Result + =================================================== + 123456.123456 3 123500 + 123456.123456 2 123450 + 123456.123456 -2 123456.12 + 123456.123456 -3 123456.123 + +=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_int()/as_number() + + $x->as_int(); + +Returns $x as a BigInt (truncated towards zero). In BigInt this is the same as +C<copy()>. + +C<as_number()> is an alias to this method. C<as_number> was introduced in +v1.22, while C<as_int()> was only introduced in v1.68. + +=head2 bstr() + + $x->bstr(); + +Returns a normalized string representation of C<$x>. + +=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 + +=head2 as_oct() + + $x->as_oct(); # as signed octal string with prefixed 0 + +=head2 numify() + + print $x->numify(); + +This returns a normal Perl scalar from $x. It is used automatically +whenever a scalar is needed, for instance in array index operations. + +This loses precision, to avoid this use L<as_int()> instead. + +=head2 modify() + + $x->modify('bpowd'); + +This method returns 0 if the object can be modified with the given +peration, or 1 if not. + +This is used for instance by L<Math::BigInt::Constant>. + +=head2 upgrade()/downgrade() + +Set/get the class for downgrade/upgrade operations. Thuis is used +for instance by L<bignum>. The defaults are '', thus the following +operation will create a BigInt, not a BigFloat: + + my $i = Math::BigInt->new(123); + my $f = Math::BigFloat->new('123.1'); + + print $i + $f,"\n"; # print 246 + +=head2 div_scale() + +Set/get the number of digits for the default precision in divide +operations. + +=head2 round_mode() + +Set/get the current round mode. + +=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. + +=item 'common' + +round up if the digit immediately to the right of the rounding place +is 5 or greater, otherwise round down. E.g., 0.15 becomes 0.2 and +0.149 becomes 0.1. + +=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(divisor) - 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 (oops) and actually gets 2 for '+100' and 4 for '+101'. Oops + 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 C<< Math::BigInt->accuracy() >> or + C<< Math::BigFloat->accuracy() >> or whatever class you are using. + * You can also set P globally by using C<< Math::SomeClass->precision() >> + likewise. + * Globals are classwide, and not inherited by subclasses. + * to undefine A, use C<< Math::SomeCLass->accuracy(undef); >> + * to undefine P, use C<< Math::SomeClass->precision(undef); >> + * Setting C<< Math::SomeClass->accuracy() >> clears automatically + C<< 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, use C<< Math::SomeClass->accuracy() >> + * to find out the current global P, use C<< Math::SomeClass->precision() >> + * use C<< $x->accuracy() >> respective C<< $x->precision() >> for the local + setting of C<< $x >>. + * Please note that C<< $x->accuracy() >> respective C<< $x->precision() >> + return eventually defined global A or P, when C<< $x >>'s A or P is not + set. + +=item Creating numbers + + * When you create a number, you can give the 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 its own + globals enforced upon creation of a number by using + C<< $x = Math::BigInt->new($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::BigInt internally, but setting A or P inside + Math::BigInt as globals does not tamper with the parts of a BigFloat. + 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. + If you set either A or P on one object, or globally, the other one will + be automatically cleared. + * 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 precedence over P (Hint: A comes before P). + 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 two modi: + + never round (this is the default): + This is done by setting A and P to undef. No math operation + will round the result, with fdiv() and fsqrt() as exceptions to guard + against overflows. You must explicitly 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 or P locally by using C<< $x->accuracy() >> or + C<< $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. + * C<< $x->accuracy() >> clears C<< $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', 'common' + * you can set/get the global R by using C<< Math::SomeClass->round_mode() >> + or by setting C<< $Math::SomeClass::round_mode >> + * after each operation, C<< $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 C<< $x->round($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 + +=head1 Infinity and Not a Number + +While BigInt has extensive handling of inf and NaN, certain quirks remain. + +=over 2 + +=item oct()/hex() + +These perl routines currently (as of Perl v.5.8.6) cannot handle passed +inf. + + te@linux:~> perl -wle 'print 2 ** 3333' + inf + te@linux:~> perl -wle 'print 2 ** 3333 == 2 ** 3333' + 1 + te@linux:~> perl -wle 'print oct(2 ** 3333)' + 0 + te@linux:~> perl -wle 'print hex(2 ** 3333)' + Illegal hexadecimal digit 'i' ignored at -e line 1. + 0 + +The same problems occur if you pass them Math::BigInt->binf() objects. Since +overloading these routines is not possible, this cannot be fixed from BigInt. + +=item ==, !=, <, >, <=, >= with NaNs + +BigInt's bcmp() routine currently returns undef to signal that a NaN was +involved in a comparison. However, the overload code turns that into +either 1 or '' and thus operations like C<< NaN != NaN >> might return +wrong values. + +=item log(-inf) + +C<< log(-inf) >> is highly weird. Since log(-x)=pi*i+log(x), then +log(-inf)=pi*i+inf. However, since the imaginary part is finite, the real +infinity "overshadows" it, so the number might as well just be infinity. +However, the result is a complex number, and since BigInt/BigFloat can only +have real numbers as results, the result is NaN. + +=item exp(), cos(), sin(), atan2() + +These all might have problems handling infinity right. + +=back + +=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 B<ONLY> method calls like C<< $x->sign(); >> +instead relying on the internal representation. + +=head2 MATH LIBRARY + +Math with the numbers is done (by default) by a module called +C<Math::BigInt::Calc>. This is equivalent to saying: + + use Math::BigInt try => 'Calc'; + +You can change this backend library by using: + + use Math::BigInt try => 'GMP'; + +B<Note>: General purpose packages should not be explicit about the library +to use; let the script author decide which is best. + +If your script works with huge numbers and Calc is too slow for them, +you can also for the loading of one of these libraries and if none +of them can be used, the code will die: + + use Math::BigInt only => 'GMP,Pari'; + +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 try => 'Foo,Math::BigInt::Bar'; + +The library that is loaded last will be used. Note that this can be +overwritten at any time by loading a different library, and numbers +constructed with different libraries cannot be used in math operations +together. + +=head3 What library to use? + +B<Note>: General purpose packages should not be explicit about the library +to use; let the script author decide which is best. + +L<Math::BigInt::GMP> and L<Math::BigInt::Pari> are in cases involving big +numbers much faster than Calc, however it is slower when dealing with very +small numbers (less than about 20 digits) and when converting very large +numbers to decimal (for instance for printing, rounding, calculating their +length in decimal etc). + +So please select carefully what libary you want to use. + +Different low-level libraries use different formats to store the numbers. +However, you should B<NOT> depend on the number having a specific format +internally. + +See the respective math library module documentation for further details. + +=head2 SIGN + +The sign is either '+', '-', 'NaN', '+inf' or '-inf'. + +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> is always 0, except +inf and -inf, where it is +C<+inf>; and for NaN, where it is C<NaN>; and for C<$x == 0>, where it is C<1> +(to be compatible with Math::BigFloat's internal representation of a zero as +C<0E1>). + +C<$m> is currently just a copy of the original number. The relation between +C<$e> and C<$m> will stay always the same, though their real values might +change. + +=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: + + 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 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 $x takes +more time then the actual addition. + +With a technique called copy-on-write, the cost of copying with overload could +be minimized or even completely avoided. A test implementation of COW did show +performance gains for overloaded math, but introduced a performance loss due +to a constant overhead for all other operations. So Math::BigInt does currently +not COW. + +The rewritten version of this module (vs. v0.01) is slower on certain +operations, like C<new()>, C<bstr()> and C<numify()>. The reason are that it +does now more work and handles much more cases. The time spent in these +operations is usually gained in the other math operations so that code on +the average should get (much) faster. If they don't, please contact the author. + +Some operations may be slower for small numbers, but are significantly faster +for big numbers. Other operations are now constant (O(1), like C<bneg()>, +C<babs()> etc), instead of O(N) and thus nearly always take much less time. +These optimizations were done on purpose. + +If you find the Calc module to slow, try to install any of the replacement +modules and see if they help you. + +=head2 Alternative math libraries + +You can use an alternative library to drive Math::BigInt. See the section +L<MATH LIBRARY> for more information. + +For more benchmark results see L<http://bloodgate.com/perl/benchmarks.html>. + +=head2 SUBCLASSING + +=head1 Subclassing Math::BigInt + +The basic design of Math::BigInt allows simple subclasses with very little +work, as long as a few simple rules are followed: + +=over 2 + +=item * + +The public API must remain consistent, i.e. if a sub-class is overloading +addition, the sub-class must use the same name, in this case badd(). The +reason for this is that Math::BigInt is optimized to call the object methods +directly. + +=item * + +The private object hash keys like C<$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 its 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>: + + use bignum; + +Also good for oneliners: + + perl -Mbignum -le 'print 2 ** 255' + +This makes it possible to mix arguments of different classes (as in 2.5 + 2) +as well es preserve accuracy (as in sqrt(3)). + +Beware: This feature is not fully implemented yet. + +=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() + +=item bexp() + +=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 EXPORTS + +C<Math::BigInt> exports nothing by default, but can export the following methods: + + bgcd + blcm + +=head1 CAVEATS + +Some things might not work as you expect them. Below is documented what is +known to be troublesome: + +=over 1 + +=item bstr(), bsstr() and 'cmp' + +Both C<bstr()> and C<bsstr()> as well as automated stringify via overload now +drop the leading '+'. The old code would return '+3', the new returns '3'. +This is to be consistent with Perl and to make C<cmp> (especially with +overloading) to work as you expect. It also solves problems with C<Test.pm>, +because its C<ok()> uses 'eq' internally. + +Mark Biggar said, when asked about to drop the '+' altogether, or make only +C<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 comparison, but Perl will represent some numbers as 100 and others +as 1e+308. If in doubt, convert both arguments to Math::BigInt before +comparing them as strings: + + 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 C<< <=> >> for comparisons, this 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. + +See also the section about L<Infinity and Not a Number> for problems in +comparing NaNs. + +=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()> or C<as_int> for the same +effect: + + $x = Math::BigFloat->new(123.45); + $y = $x->as_number(); # BigInt 123 + $y = $x->as_int(); # ditto + +This also works for other subclasses, like Math::String. + +If you want a real Perl scalar, use C<numify()>: + + $y = $x->numify(); # 123 as scalar + +This is seldom necessary, though, because this is done automatically, like +when you access an array: + + $z = $array[$x]; # does work automatically + +=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 careful. 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 behind 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). + +=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 autoupgrading|upgrading. See the +pragmas L<bignum>, L<bigint> and L<bigrat> for an easy way to do this. + +=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. The reason is that the result is always truncated to an integer. + +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>, L<Math::BigRat> and L<Math::Big> as well as +L<Math::BigInt::BitVect>, L<Math::BigInt::Pari> and L<Math::BigInt::GMP>. + +The pragmas L<bignum>, L<bigint> and L<bigrat> also might be of interest +because they solve the autoupgrading/downgrading issue, at least partly. + +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 AUTHORS + +Original code by Mark Biggar, overloaded interface by Ilya Zakharevich. +Completely rewritten by Tels http://bloodgate.com in late 2000, 2001 - 2006 +and still at it in 2007. + +Many people contributed in one or more ways to the final beast, see the file +CREDITS for an (incomplete) list. If you miss your name, please drop me a +mail. Thank you! + +=cut |