1 package Math::BigFloat;
4 # Mike grinned. 'Two down, infinity to go' - Mike Nostrus in 'Before and After'
7 # The following hash values are internally used:
8 # _e : exponent (ref to $CALC object)
9 # _m : mantissa (ref to $CALC object)
11 # sign : +,-,+inf,-inf, or "NaN" if not a number
19 @ISA = qw/Math::BigInt/;
23 # $_trap_inf/$_trap_nan are internal and should never be accessed from outside
24 use vars qw/$AUTOLOAD $accuracy $precision $div_scale $round_mode $rnd_mode
25 $upgrade $downgrade $_trap_nan $_trap_inf/;
26 my $class = "Math::BigFloat";
29 '<=>' => sub { my $rc = $_[2] ?
30 ref($_[0])->bcmp($_[1],$_[0]) :
31 ref($_[0])->bcmp($_[0],$_[1]);
32 $rc = 1 unless defined $rc;
35 # we need '>=' to get things like "1 >= NaN" right:
36 '>=' => sub { my $rc = $_[2] ?
37 ref($_[0])->bcmp($_[1],$_[0]) :
38 ref($_[0])->bcmp($_[0],$_[1]);
39 # if there was a NaN involved, return false
40 return '' unless defined $rc;
43 'int' => sub { $_[0]->as_number() }, # 'trunc' to bigint
46 ##############################################################################
47 # global constants, flags and assorted stuff
49 # the following are public, but their usage is not recommended. Use the
50 # accessor methods instead.
52 # class constants, use Class->constant_name() to access
53 # one of 'even', 'odd', '+inf', '-inf', 'zero', 'trunc' or 'common'
61 # the package we are using for our private parts, defaults to:
62 # Math::BigInt->config()->{lib}
63 my $MBI = 'Math::BigInt::Calc';
65 # are NaNs ok? (otherwise it dies when encountering an NaN) set w/ config()
67 # the same for infinity
70 # constant for easier life
73 my $IMPORT = 0; # was import() called yet? used to make require work
75 # some digits of accuracy for blog(undef,10); which we use in blog() for speed
77 '2.3025850929940456840179914546843642076011014886287729760333279009675726097';
78 my $LOG_10_A = length($LOG_10)-1;
81 '0.6931471805599453094172321214581765680755001343602552541206800094933936220';
82 my $LOG_2_A = length($LOG_2)-1;
83 my $HALF = '0.5'; # made into an object if nec.
85 ##############################################################################
86 # the old code had $rnd_mode, so we need to support it, too
88 sub TIESCALAR { my ($class) = @_; bless \$round_mode, $class; }
89 sub FETCH { return $round_mode; }
90 sub STORE { $rnd_mode = $_[0]->round_mode($_[1]); }
94 # when someone sets $rnd_mode, we catch this and check the value to see
95 # whether it is valid or not.
96 $rnd_mode = 'even'; tie $rnd_mode, 'Math::BigFloat';
98 # we need both of them in this package:
99 *as_int = \&as_number;
102 ##############################################################################
105 # valid method aliases for AUTOLOAD
106 my %methods = map { $_ => 1 }
107 qw / fadd fsub fmul fdiv fround ffround fsqrt fmod fstr fsstr fpow fnorm
108 fint facmp fcmp fzero fnan finf finc fdec ffac fneg
109 fceil ffloor frsft flsft fone flog froot fexp
111 # valid methods that can be handed up (for AUTOLOAD)
112 my %hand_ups = map { $_ => 1 }
113 qw / is_nan is_inf is_negative is_positive is_pos is_neg
114 accuracy precision div_scale round_mode fabs fnot
115 objectify upgrade downgrade
120 sub _method_alias { exists $methods{$_[0]||''}; }
121 sub _method_hand_up { exists $hand_ups{$_[0]||''}; }
124 ##############################################################################
129 # create a new BigFloat object from a string or another bigfloat object.
132 # sign => sign (+/-), or "NaN"
134 my ($class,$wanted,@r) = @_;
136 # avoid numify-calls by not using || on $wanted!
137 return $class->bzero() if !defined $wanted; # default to 0
138 return $wanted->copy() if UNIVERSAL::isa($wanted,'Math::BigFloat');
140 $class->import() if $IMPORT == 0; # make require work
142 my $self = {}; bless $self, $class;
143 # shortcut for bigints and its subclasses
144 if ((ref($wanted)) && UNIVERSAL::can( $wanted, "as_number"))
146 $self->{_m} = $wanted->as_number()->{value}; # get us a bigint copy
147 $self->{_e} = $MBI->_zero();
149 $self->{sign} = $wanted->sign();
150 return $self->bnorm();
152 # else: got a string or something masquerading as number (with overload)
154 # handle '+inf', '-inf' first
155 if ($wanted =~ /^[+-]?inf\z/)
157 return $downgrade->new($wanted) if $downgrade;
159 $self->{sign} = $wanted; # set a default sign for bstr()
160 return $self->binf($wanted);
163 # shortcut for simple forms like '12' that neither have trailing nor leading
165 if ($wanted =~ /^([+-]?)([1-9][0-9]*[1-9])$/)
167 $self->{_e} = $MBI->_zero();
169 $self->{sign} = $1 || '+';
170 $self->{_m} = $MBI->_new($2);
171 return $self->round(@r) if !$downgrade;
174 my ($mis,$miv,$mfv,$es,$ev) = Math::BigInt::_split($wanted);
180 Carp::croak ("$wanted is not a number initialized to $class");
183 return $downgrade->bnan() if $downgrade;
185 $self->{_e} = $MBI->_zero();
187 $self->{_m} = $MBI->_zero();
188 $self->{sign} = $nan;
192 # make integer from mantissa by adjusting exp, then convert to int
193 $self->{_e} = $MBI->_new($$ev); # exponent
194 $self->{_es} = $$es || '+';
195 my $mantissa = "$$miv$$mfv"; # create mant.
196 $mantissa =~ s/^0+(\d)/$1/; # strip leading zeros
197 $self->{_m} = $MBI->_new($mantissa); # create mant.
199 # 3.123E0 = 3123E-3, and 3.123E-2 => 3123E-5
200 if (CORE::length($$mfv) != 0)
202 my $len = $MBI->_new( CORE::length($$mfv));
203 ($self->{_e}, $self->{_es}) =
204 _e_sub ($self->{_e}, $len, $self->{_es}, '+');
206 # we can only have trailing zeros on the mantissa if $$mfv eq ''
209 # Use a regexp to count the trailing zeros in $$miv instead of _zeros()
210 # because that is faster, especially when _m is not stored in base 10.
211 my $zeros = 0; $zeros = CORE::length($1) if $$miv =~ /[1-9](0*)$/;
214 my $z = $MBI->_new($zeros);
215 # turn '120e2' into '12e3'
216 $MBI->_rsft ( $self->{_m}, $z, 10);
217 ($self->{_e}, $self->{_es}) =
218 _e_add ( $self->{_e}, $z, $self->{_es}, '+');
221 $self->{sign} = $$mis;
223 # for something like 0Ey, set y to 1, and -0 => +0
224 # Check $$miv for being '0' and $$mfv eq '', because otherwise _m could not
225 # have become 0. That's faster than to call $MBI->_is_zero().
226 $self->{sign} = '+', $self->{_e} = $MBI->_one()
227 if $$miv eq '0' and $$mfv eq '';
229 return $self->round(@r) if !$downgrade;
231 # if downgrade, inf, NaN or integers go down
233 if ($downgrade && $self->{_es} eq '+')
235 if ($MBI->_is_zero( $self->{_e} ))
237 return $downgrade->new($$mis . $MBI->_str( $self->{_m} ));
239 return $downgrade->new($self->bsstr());
241 $self->bnorm()->round(@r); # first normalize, then round
246 # if two arguments, the first one is the class to "swallow" subclasses
250 sign => $_[1]->{sign},
252 _m => $MBI->_copy($_[1]->{_m}),
253 _e => $MBI->_copy($_[1]->{_e}),
256 $self->{_a} = $_[1]->{_a} if defined $_[1]->{_a};
257 $self->{_p} = $_[1]->{_p} if defined $_[1]->{_p};
262 sign => $_[0]->{sign},
264 _m => $MBI->_copy($_[0]->{_m}),
265 _e => $MBI->_copy($_[0]->{_e}),
268 $self->{_a} = $_[0]->{_a} if defined $_[0]->{_a};
269 $self->{_p} = $_[0]->{_p} if defined $_[0]->{_p};
275 # used by parent class bone() to initialize number to NaN
281 my $class = ref($self);
282 Carp::croak ("Tried to set $self to NaN in $class\::_bnan()");
285 $IMPORT=1; # call our import only once
286 $self->{_m} = $MBI->_zero();
287 $self->{_e} = $MBI->_zero();
293 # used by parent class bone() to initialize number to +-inf
299 my $class = ref($self);
300 Carp::croak ("Tried to set $self to +-inf in $class\::_binf()");
303 $IMPORT=1; # call our import only once
304 $self->{_m} = $MBI->_zero();
305 $self->{_e} = $MBI->_zero();
311 # used by parent class bone() to initialize number to 1
313 $IMPORT=1; # call our import only once
314 $self->{_m} = $MBI->_one();
315 $self->{_e} = $MBI->_zero();
321 # used by parent class bone() to initialize number to 0
323 $IMPORT=1; # call our import only once
324 $self->{_m} = $MBI->_zero();
325 $self->{_e} = $MBI->_one();
331 my ($self,$class) = @_;
332 return if $class =~ /^Math::BigInt/; # we aren't one of these
333 UNIVERSAL::isa($self,$class);
338 # return (later set?) configuration data as hash ref
339 my $class = shift || 'Math::BigFloat';
341 if (@_ == 1 && ref($_[0]) ne 'HASH')
343 my $cfg = $class->SUPER::config();
344 return $cfg->{$_[0]};
347 my $cfg = $class->SUPER::config(@_);
349 # now we need only to override the ones that are different from our parent
350 $cfg->{class} = $class;
355 ##############################################################################
360 # (ref to BFLOAT or num_str ) return num_str
361 # Convert number from internal format to (non-scientific) string format.
362 # internal format is always normalized (no leading zeros, "-0" => "+0")
363 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
365 if ($x->{sign} !~ /^[+-]$/)
367 return $x->{sign} unless $x->{sign} eq '+inf'; # -inf, NaN
371 my $es = '0'; my $len = 1; my $cad = 0; my $dot = '.';
374 my $not_zero = !($x->{sign} eq '+' && $MBI->_is_zero($x->{_m}));
377 $es = $MBI->_str($x->{_m});
378 $len = CORE::length($es);
379 my $e = $MBI->_num($x->{_e});
380 $e = -$e if $x->{_es} eq '-';
384 # if _e is bigger than a scalar, the following will blow your memory
387 my $r = abs($e) - $len;
388 $es = '0.'. ('0' x $r) . $es; $cad = -($len+$r);
392 substr($es,$e,0) = '.'; $cad = $MBI->_num($x->{_e});
393 $cad = -$cad if $x->{_es} eq '-';
399 $es .= '0' x $e; $len += $e; $cad = 0;
403 $es = '-'.$es if $x->{sign} eq '-';
404 # if set accuracy or precision, pad with zeros on the right side
405 if ((defined $x->{_a}) && ($not_zero))
407 # 123400 => 6, 0.1234 => 4, 0.001234 => 4
408 my $zeros = $x->{_a} - $cad; # cad == 0 => 12340
409 $zeros = $x->{_a} - $len if $cad != $len;
410 $es .= $dot.'0' x $zeros if $zeros > 0;
412 elsif ((($x->{_p} || 0) < 0))
414 # 123400 => 6, 0.1234 => 4, 0.001234 => 6
415 my $zeros = -$x->{_p} + $cad;
416 $es .= $dot.'0' x $zeros if $zeros > 0;
423 # (ref to BFLOAT or num_str ) return num_str
424 # Convert number from internal format to scientific string format.
425 # internal format is always normalized (no leading zeros, "-0E0" => "+0E0")
426 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
428 if ($x->{sign} !~ /^[+-]$/)
430 return $x->{sign} unless $x->{sign} eq '+inf'; # -inf, NaN
433 my $sep = 'e'.$x->{_es};
434 my $sign = $x->{sign}; $sign = '' if $sign eq '+';
435 $sign . $MBI->_str($x->{_m}) . $sep . $MBI->_str($x->{_e});
440 # Convert a Perl scalar number from a BigFloat object.
441 # Create a string and let Perl's atoi()/atof() handle the rest.
442 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
443 return 0 + $x->bsstr();
446 ##############################################################################
447 # public stuff (usually prefixed with "b")
451 # (BINT or num_str) return BINT
452 # negate number or make a negated number from string
453 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
455 return $x if $x->modify('bneg');
457 # for +0 dont negate (to have always normalized +0). Does nothing for 'NaN'
458 $x->{sign} =~ tr/+-/-+/ unless ($x->{sign} eq '+' && $MBI->_is_zero($x->{_m}));
463 # XXX TODO this must be overwritten and return NaN for non-integer values
464 # band(), bior(), bxor(), too
467 # $class->SUPER::bnot($class,@_);
472 # Compares 2 values. Returns one of undef, <0, =0, >0. (suitable for sort)
475 my ($self,$x,$y) = (ref($_[0]),@_);
477 # objectify is costly, so avoid it
478 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
480 ($self,$x,$y) = objectify(2,@_);
483 return $upgrade->bcmp($x,$y) if defined $upgrade &&
484 ((!$x->isa($self)) || (!$y->isa($self)));
486 # Handle all 'nan' cases.
488 return undef if ($x->{sign} eq $nan) || ($y->{sign} eq $nan);
490 # Handle all '+inf' and '-inf' cases.
492 return 0 if ($x->{sign} eq '+inf' && $y->{sign} eq '+inf' ||
493 $x->{sign} eq '-inf' && $y->{sign} eq '-inf');
494 return +1 if $x->{sign} eq '+inf'; # x = +inf and y < +inf
495 return -1 if $x->{sign} eq '-inf'; # x = -inf and y > -inf
496 return -1 if $y->{sign} eq '+inf'; # x < +inf and y = +inf
497 return +1 if $y->{sign} eq '-inf'; # x > -inf and y = -inf
499 # Handle all cases with opposite signs.
501 return +1 if $x->{sign} eq '+' && $y->{sign} eq '-'; # also does 0 <=> -y
502 return -1 if $x->{sign} eq '-' && $y->{sign} eq '+'; # also does -x <=> 0
504 # Handle all remaining zero cases.
506 my $xz = $x->is_zero();
507 my $yz = $y->is_zero();
508 return 0 if $xz && $yz; # 0 <=> 0
509 return -1 if $xz && $y->{sign} eq '+'; # 0 <=> +y
510 return +1 if $yz && $x->{sign} eq '+'; # +x <=> 0
512 # Both arguments are now finite, non-zero numbers with the same sign.
516 # The next step is to compare the exponents, but since each mantissa is an
517 # integer of arbitrary value, the exponents must be normalized by the length
518 # of the mantissas before we can compare them.
520 my $mxl = $MBI->_len($x->{_m});
521 my $myl = $MBI->_len($y->{_m});
523 # If the mantissas have the same length, there is no point in normalizing the
524 # exponents by the length of the mantissas, so treat that as a special case.
528 # First handle the two cases where the exponents have different signs.
530 if ($x->{_es} eq '+' && $y->{_es} eq '-') {
534 elsif ($x->{_es} eq '-' && $y->{_es} eq '+') {
538 # Then handle the case where the exponents have the same sign.
541 $cmp = $MBI->_acmp($x->{_e}, $y->{_e});
542 $cmp = -$cmp if $x->{_es} eq '-';
545 # Adjust for the sign, which is the same for x and y, and bail out if
548 $cmp = -$cmp if $x->{sign} eq '-'; # 124 > 123, but -124 < -123
553 # We must normalize each exponent by the length of the corresponding
554 # mantissa. Life is a lot easier if we first make both exponents
555 # non-negative. We do this by adding the same positive value to both
556 # exponent. This is safe, because when comparing the exponents, only the
557 # relative difference is important.
562 if ($x->{_es} eq '+') {
564 # If the exponent of x is >= 0 and the exponent of y is >= 0, there is no
565 # need to do anything special.
567 if ($y->{_es} eq '+') {
568 $ex = $MBI->_copy($x->{_e});
569 $ey = $MBI->_copy($y->{_e});
572 # If the exponent of x is >= 0 and the exponent of y is < 0, add the
573 # absolute value of the exponent of y to both.
576 $ex = $MBI->_copy($x->{_e});
577 $ex = $MBI->_add($ex, $y->{_e}); # ex + |ey|
578 $ey = $MBI->_zero(); # -ex + |ey| = 0
583 # If the exponent of x is < 0 and the exponent of y is >= 0, add the
584 # absolute value of the exponent of x to both.
586 if ($y->{_es} eq '+') {
587 $ex = $MBI->_zero(); # -ex + |ex| = 0
588 $ey = $MBI->_copy($y->{_e});
589 $ey = $MBI->_add($ey, $x->{_e}); # ey + |ex|
592 # If the exponent of x is < 0 and the exponent of y is < 0, add the
593 # absolute values of both exponents to both exponents.
596 $ex = $MBI->_copy($y->{_e}); # -ex + |ey| + |ex| = |ey|
597 $ey = $MBI->_copy($x->{_e}); # -ey + |ex| + |ey| = |ex|
602 # Now we can normalize the exponents by adding lengths of the mantissas.
604 $MBI->_add($ex, $MBI->_new($mxl));
605 $MBI->_add($ey, $MBI->_new($myl));
607 # We're done if the exponents are different.
609 $cmp = $MBI->_acmp($ex, $ey);
610 $cmp = -$cmp if $x->{sign} eq '-'; # 124 > 123, but -124 < -123
613 # Compare the mantissas, but first normalize them by padding the shorter
614 # mantissa with zeros (shift left) until it has the same length as the longer
621 $my = $MBI->_lsft($MBI->_copy($my), $MBI->_new($mxl - $myl), 10);
622 } elsif ($mxl < $myl) {
623 $mx = $MBI->_lsft($MBI->_copy($mx), $MBI->_new($myl - $mxl), 10);
626 $cmp = $MBI->_acmp($mx, $my);
627 $cmp = -$cmp if $x->{sign} eq '-'; # 124 > 123, but -124 < -123
634 # Compares 2 values, ignoring their signs.
635 # Returns one of undef, <0, =0, >0. (suitable for sort)
638 my ($self,$x,$y) = (ref($_[0]),@_);
639 # objectify is costly, so avoid it
640 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
642 ($self,$x,$y) = objectify(2,@_);
645 return $upgrade->bacmp($x,$y) if defined $upgrade &&
646 ((!$x->isa($self)) || (!$y->isa($self)));
648 # handle +-inf and NaN's
649 if ($x->{sign} !~ /^[+-]$/ || $y->{sign} !~ /^[+-]$/)
651 return undef if (($x->{sign} eq $nan) || ($y->{sign} eq $nan));
652 return 0 if ($x->is_inf() && $y->is_inf());
653 return 1 if ($x->is_inf() && !$y->is_inf());
658 my $xz = $x->is_zero();
659 my $yz = $y->is_zero();
660 return 0 if $xz && $yz; # 0 <=> 0
661 return -1 if $xz && !$yz; # 0 <=> +y
662 return 1 if $yz && !$xz; # +x <=> 0
664 # adjust so that exponents are equal
665 my $lxm = $MBI->_len($x->{_m});
666 my $lym = $MBI->_len($y->{_m});
667 my ($xes,$yes) = (1,1);
668 $xes = -1 if $x->{_es} ne '+';
669 $yes = -1 if $y->{_es} ne '+';
670 # the numify somewhat limits our length, but makes it much faster
671 my $lx = $lxm + $xes * $MBI->_num($x->{_e});
672 my $ly = $lym + $yes * $MBI->_num($y->{_e});
674 return $l <=> 0 if $l != 0;
676 # lengths (corrected by exponent) are equal
677 # so make mantissa equal-length by padding with zero (shift left)
678 my $diff = $lxm - $lym;
679 my $xm = $x->{_m}; # not yet copy it
683 $ym = $MBI->_copy($y->{_m});
684 $ym = $MBI->_lsft($ym, $MBI->_new($diff), 10);
688 $xm = $MBI->_copy($x->{_m});
689 $xm = $MBI->_lsft($xm, $MBI->_new(-$diff), 10);
691 $MBI->_acmp($xm,$ym);
696 # add second arg (BFLOAT or string) to first (BFLOAT) (modifies first)
697 # return result as BFLOAT
700 my ($self,$x,$y,@r) = (ref($_[0]),@_);
701 # objectify is costly, so avoid it
702 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
704 ($self,$x,$y,@r) = objectify(2,@_);
707 return $x if $x->modify('badd');
709 # inf and NaN handling
710 if (($x->{sign} !~ /^[+-]$/) || ($y->{sign} !~ /^[+-]$/))
713 return $x->bnan() if (($x->{sign} eq $nan) || ($y->{sign} eq $nan));
715 if (($x->{sign} =~ /^[+-]inf$/) && ($y->{sign} =~ /^[+-]inf$/))
717 # +inf++inf or -inf+-inf => same, rest is NaN
718 return $x if $x->{sign} eq $y->{sign};
721 # +-inf + something => +inf; something +-inf => +-inf
722 $x->{sign} = $y->{sign}, return $x if $y->{sign} =~ /^[+-]inf$/;
726 return $upgrade->badd($x,$y,@r) if defined $upgrade &&
727 ((!$x->isa($self)) || (!$y->isa($self)));
729 $r[3] = $y; # no push!
731 # speed: no add for 0+y or x+0
732 return $x->bround(@r) if $y->is_zero(); # x+0
733 if ($x->is_zero()) # 0+y
735 # make copy, clobbering up x (modify in place!)
736 $x->{_e} = $MBI->_copy($y->{_e});
737 $x->{_es} = $y->{_es};
738 $x->{_m} = $MBI->_copy($y->{_m});
739 $x->{sign} = $y->{sign} || $nan;
740 return $x->round(@r);
743 # take lower of the two e's and adapt m1 to it to match m2
745 $e = $MBI->_zero() if !defined $e; # if no BFLOAT?
746 $e = $MBI->_copy($e); # make copy (didn't do it yet)
750 ($e,$es) = _e_sub($e, $x->{_e}, $y->{_es} || '+', $x->{_es});
752 my $add = $MBI->_copy($y->{_m});
754 if ($es eq '-') # < 0
756 $MBI->_lsft( $x->{_m}, $e, 10);
757 ($x->{_e},$x->{_es}) = _e_add($x->{_e}, $e, $x->{_es}, $es);
759 elsif (!$MBI->_is_zero($e)) # > 0
761 $MBI->_lsft($add, $e, 10);
763 # else: both e are the same, so just leave them
765 if ($x->{sign} eq $y->{sign})
768 $x->{_m} = $MBI->_add($x->{_m}, $add);
772 ($x->{_m}, $x->{sign}) =
773 _e_add($x->{_m}, $add, $x->{sign}, $y->{sign});
776 # delete trailing zeros, then round
777 $x->bnorm()->round(@r);
780 # sub bsub is inherited from Math::BigInt!
784 # increment arg by one
785 my ($self,$x,@r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_);
787 return $x if $x->modify('binc');
789 if ($x->{_es} eq '-')
791 return $x->badd($self->bone(),@r); # digits after dot
794 if (!$MBI->_is_zero($x->{_e})) # _e == 0 for NaN, inf, -inf
796 # 1e2 => 100, so after the shift below _m has a '0' as last digit
797 $x->{_m} = $MBI->_lsft($x->{_m}, $x->{_e},10); # 1e2 => 100
798 $x->{_e} = $MBI->_zero(); # normalize
800 # we know that the last digit of $x will be '1' or '9', depending on the
804 if ($x->{sign} eq '+')
806 $MBI->_inc($x->{_m});
807 return $x->bnorm()->bround(@r);
809 elsif ($x->{sign} eq '-')
811 $MBI->_dec($x->{_m});
812 $x->{sign} = '+' if $MBI->_is_zero($x->{_m}); # -1 +1 => -0 => +0
813 return $x->bnorm()->bround(@r);
815 # inf, nan handling etc
816 $x->badd($self->bone(),@r); # badd() does round
821 # decrement arg by one
822 my ($self,$x,@r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_);
824 return $x if $x->modify('bdec');
826 if ($x->{_es} eq '-')
828 return $x->badd($self->bone('-'),@r); # digits after dot
831 if (!$MBI->_is_zero($x->{_e}))
833 $x->{_m} = $MBI->_lsft($x->{_m}, $x->{_e},10); # 1e2 => 100
834 $x->{_e} = $MBI->_zero(); # normalize
838 my $zero = $x->is_zero();
840 if (($x->{sign} eq '-') || $zero)
842 $MBI->_inc($x->{_m});
843 $x->{sign} = '-' if $zero; # 0 => 1 => -1
844 $x->{sign} = '+' if $MBI->_is_zero($x->{_m}); # -1 +1 => -0 => +0
845 return $x->bnorm()->round(@r);
848 elsif ($x->{sign} eq '+')
850 $MBI->_dec($x->{_m});
851 return $x->bnorm()->round(@r);
853 # inf, nan handling etc
854 $x->badd($self->bone('-'),@r); # does round
861 my ($self,$x,$base,$a,$p,$r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_);
863 return $x if $x->modify('blog');
865 # $base > 0, $base != 1; if $base == undef default to $base == e
868 # we need to limit the accuracy to protect against overflow
871 ($x,@params) = $x->_find_round_parameters($a,$p,$r);
873 # also takes care of the "error in _find_round_parameters?" case
874 return $x->bnan() if $x->{sign} ne '+' || $x->is_zero();
876 # no rounding at all, so must use fallback
877 if (scalar @params == 0)
879 # simulate old behaviour
880 $params[0] = $self->div_scale(); # and round to it as accuracy
881 $params[1] = undef; # P = undef
882 $scale = $params[0]+4; # at least four more for proper round
883 $params[2] = $r; # round mode by caller or undef
884 $fallback = 1; # to clear a/p afterwards
888 # the 4 below is empirical, and there might be cases where it is not
890 $scale = abs($params[0] || $params[1]) + 4; # take whatever is defined
893 return $x->bzero(@params) if $x->is_one();
894 # base not defined => base == Euler's number e
897 # make object, since we don't feed it through objectify() to still get the
898 # case of $base == undef
899 $base = $self->new($base) unless ref($base);
900 # $base > 0; $base != 1
901 return $x->bnan() if $base->is_zero() || $base->is_one() ||
902 $base->{sign} ne '+';
903 # if $x == $base, we know the result must be 1.0
904 if ($x->bcmp($base) == 0)
906 $x->bone('+',@params);
909 # clear a/p after round, since user did not request it
910 delete $x->{_a}; delete $x->{_p};
916 # when user set globals, they would interfere with our calculation, so
917 # disable them and later re-enable them
919 my $abr = "$self\::accuracy"; my $ab = $$abr; $$abr = undef;
920 my $pbr = "$self\::precision"; my $pb = $$pbr; $$pbr = undef;
921 # we also need to disable any set A or P on $x (_find_round_parameters took
922 # them already into account), since these would interfere, too
923 delete $x->{_a}; delete $x->{_p};
924 # need to disable $upgrade in BigInt, to avoid deep recursion
925 local $Math::BigInt::upgrade = undef;
926 local $Math::BigFloat::downgrade = undef;
928 # upgrade $x if $x is not a BigFloat (handle BigInt input)
930 if (!$x->isa('Math::BigFloat'))
932 $x = Math::BigFloat->new($x);
938 # If the base is defined and an integer, try to calculate integer result
939 # first. This is very fast, and in case the real result was found, we can
941 if (defined $base && $base->is_int() && $x->is_int())
943 my $i = $MBI->_copy( $x->{_m} );
944 $MBI->_lsft( $i, $x->{_e}, 10 ) unless $MBI->_is_zero($x->{_e});
945 my $int = Math::BigInt->bzero();
947 $int->blog($base->as_number());
949 if ($base->as_number()->bpow($int) == $x)
951 # found result, return it
952 $x->{_m} = $int->{value};
953 $x->{_e} = $MBI->_zero();
962 # base is undef, so base should be e (Euler's number), so first calculate the
963 # log to base e (using reduction by 10 (and probably 2)):
964 $self->_log_10($x,$scale);
966 # and if a different base was requested, convert it
969 $base = Math::BigFloat->new($base) unless $base->isa('Math::BigFloat');
970 # not ln, but some other base (don't modify $base)
971 $x->bdiv( $base->copy()->blog(undef,$scale), $scale );
975 # shortcut to not run through _find_round_parameters again
976 if (defined $params[0])
978 $x->bround($params[0],$params[2]); # then round accordingly
982 $x->bfround($params[1],$params[2]); # then round accordingly
986 # clear a/p after round, since user did not request it
987 delete $x->{_a}; delete $x->{_p};
990 $$abr = $ab; $$pbr = $pb;
997 # Given D (digits in decimal), compute N so that N! (N factorial) is
998 # at least D digits long. D should be at least 50.
1001 # two constants for the Ramanujan estimate of ln(N!)
1002 my $lg2 = log(2 * 3.14159265) / 2;
1005 # D = 50 => N => 42, so L = 40 and R = 50
1006 my $l = 40; my $r = $d;
1008 # Otherwise this does not work under -Mbignum and we do not yet have "no bignum;" :(
1009 $l = $l->numify if ref($l);
1010 $r = $r->numify if ref($r);
1011 $lg2 = $lg2->numify if ref($lg2);
1012 $lg10 = $lg10->numify if ref($lg10);
1014 # binary search for the right value (could this be written as the reverse of lg(n!)?)
1017 my $n = int(($r - $l) / 2) + $l;
1019 int(($n * log($n) - $n + log( $n * (1 + 4*$n*(1+2*$n)) ) / 6 + $lg2) / $lg10);
1020 $ramanujan > $d ? $r = $n : $l = $n;
1027 # Calculate n over k (binomial coefficient or "choose" function) as integer.
1029 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1031 # objectify is costly, so avoid it
1032 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1034 ($self,$x,$y,@r) = objectify(2,@_);
1037 return $x if $x->modify('bnok');
1039 return $x->bnan() if $x->is_nan() || $y->is_nan();
1040 return $x->binf() if $x->is_inf();
1042 my $u = $x->as_int();
1043 $u->bnok($y->as_int());
1045 $x->{_m} = $u->{value};
1046 $x->{_e} = $MBI->_zero();
1054 # Calculate e ** X (Euler's number to the power of X)
1055 my ($self,$x,$a,$p,$r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_);
1057 return $x if $x->modify('bexp');
1059 return $x->binf() if $x->{sign} eq '+inf';
1060 return $x->bzero() if $x->{sign} eq '-inf';
1062 # we need to limit the accuracy to protect against overflow
1064 my ($scale,@params);
1065 ($x,@params) = $x->_find_round_parameters($a,$p,$r);
1067 # also takes care of the "error in _find_round_parameters?" case
1068 return $x if $x->{sign} eq 'NaN';
1070 # no rounding at all, so must use fallback
1071 if (scalar @params == 0)
1073 # simulate old behaviour
1074 $params[0] = $self->div_scale(); # and round to it as accuracy
1075 $params[1] = undef; # P = undef
1076 $scale = $params[0]+4; # at least four more for proper round
1077 $params[2] = $r; # round mode by caller or undef
1078 $fallback = 1; # to clear a/p afterwards
1082 # the 4 below is empirical, and there might be cases where it's not enough...
1083 $scale = abs($params[0] || $params[1]) + 4; # take whatever is defined
1086 return $x->bone(@params) if $x->is_zero();
1088 if (!$x->isa('Math::BigFloat'))
1090 $x = Math::BigFloat->new($x);
1094 # when user set globals, they would interfere with our calculation, so
1095 # disable them and later re-enable them
1097 my $abr = "$self\::accuracy"; my $ab = $$abr; $$abr = undef;
1098 my $pbr = "$self\::precision"; my $pb = $$pbr; $$pbr = undef;
1099 # we also need to disable any set A or P on $x (_find_round_parameters took
1100 # them already into account), since these would interfere, too
1101 delete $x->{_a}; delete $x->{_p};
1102 # need to disable $upgrade in BigInt, to avoid deep recursion
1103 local $Math::BigInt::upgrade = undef;
1104 local $Math::BigFloat::downgrade = undef;
1106 my $x_org = $x->copy();
1108 # We use the following Taylor series:
1111 # e = 1 + --- + --- + --- + --- ...
1114 # The difference for each term is X and N, which would result in:
1115 # 2 copy, 2 mul, 2 add, 1 inc, 1 div operations per term
1117 # But it is faster to compute exp(1) and then raising it to the
1118 # given power, esp. if $x is really big and an integer because:
1120 # * The numerator is always 1, making the computation faster
1121 # * the series converges faster in the case of x == 1
1122 # * We can also easily check when we have reached our limit: when the
1123 # term to be added is smaller than "1E$scale", we can stop - f.i.
1124 # scale == 5, and we have 1/40320, then we stop since 1/40320 < 1E-5.
1125 # * we can compute the *exact* result by simulating bigrat math:
1127 # 1 1 gcd(3,4) = 1 1*24 + 1*6 5
1128 # - + - = ---------- = --
1131 # We do not compute the gcd() here, but simple do:
1133 # - + - = --------- = --
1137 # a c a*d + c*b and note that c is always 1 and d = (b*f)
1141 # This leads to: which can be reduced by b to:
1142 # a 1 a*b*f + b a*f + 1
1143 # - + - = --------- = -------
1146 # The first terms in the series are:
1148 # 1 1 1 1 1 1 1 1 13700
1149 # -- + -- + -- + -- + -- + --- + --- + ---- = -----
1150 # 1 1 2 6 24 120 720 5040 5040
1152 # Note that we cannot simple reduce 13700/5040 to 685/252, but must keep A and B!
1156 # set $x directly from a cached string form
1157 $x->{_m} = $MBI->_new(
1158 "27182818284590452353602874713526624977572470936999595749669676277240766303535476");
1161 $x->{_e} = $MBI->_new(79);
1165 # compute A and B so that e = A / B.
1167 # After some terms we end up with this, so we use it as a starting point:
1168 my $A = $MBI->_new("90933395208605785401971970164779391644753259799242");
1169 my $F = $MBI->_new(42); my $step = 42;
1171 # Compute how many steps we need to take to get $A and $B sufficiently big
1172 my $steps = _len_to_steps($scale - 4);
1173 # print STDERR "# Doing $steps steps for ", $scale-4, " digits\n";
1174 while ($step++ <= $steps)
1176 # calculate $a * $f + 1
1177 $A = $MBI->_mul($A, $F);
1178 $A = $MBI->_inc($A);
1180 $F = $MBI->_inc($F);
1182 # compute $B as factorial of $steps (this is faster than doing it manually)
1183 my $B = $MBI->_fac($MBI->_new($steps));
1185 # print "A ", $MBI->_str($A), "\nB ", $MBI->_str($B), "\n";
1187 # compute A/B with $scale digits in the result (truncate, not round)
1188 $A = $MBI->_lsft( $A, $MBI->_new($scale), 10);
1189 $A = $MBI->_div( $A, $B );
1194 $x->{_e} = $MBI->_new($scale);
1197 # $x contains now an estimate of e, with some surplus digits, so we can round
1198 if (!$x_org->is_one())
1200 # raise $x to the wanted power and round it in one step:
1201 $x->bpow($x_org, @params);
1205 # else just round the already computed result
1206 delete $x->{_a}; delete $x->{_p};
1207 # shortcut to not run through _find_round_parameters again
1208 if (defined $params[0])
1210 $x->bround($params[0],$params[2]); # then round accordingly
1214 $x->bfround($params[1],$params[2]); # then round accordingly
1219 # clear a/p after round, since user did not request it
1220 delete $x->{_a}; delete $x->{_p};
1223 $$abr = $ab; $$pbr = $pb;
1225 $x; # return modified $x
1230 # internal log function to calculate ln() based on Taylor series.
1231 # Modifies $x in place.
1232 my ($self,$x,$scale) = @_;
1234 # in case of $x == 1, result is 0
1235 return $x->bzero() if $x->is_one();
1237 # XXX TODO: rewrite this in a similar manner to bexp()
1239 # http://www.efunda.com/math/taylor_series/logarithmic.cfm?search_string=log
1243 # Taylor: | u 1 u^3 1 u^5 |
1244 # ln (x) = 2 | --- + - * --- + - * --- + ... | x > 0
1245 # |_ v 3 v^3 5 v^5 _|
1247 # This takes much more steps to calculate the result and is thus not used
1250 # Taylor: | u 1 u^2 1 u^3 |
1251 # ln (x) = 2 | --- + - * --- + - * --- + ... | x > 1/2
1252 # |_ x 2 x^2 3 x^3 _|
1254 my ($limit,$v,$u,$below,$factor,$two,$next,$over,$f);
1256 $v = $x->copy(); $v->binc(); # v = x+1
1257 $x->bdec(); $u = $x->copy(); # u = x-1; x = x-1
1258 $x->bdiv($v,$scale); # first term: u/v
1259 $below = $v->copy();
1261 $u *= $u; $v *= $v; # u^2, v^2
1262 $below->bmul($v); # u^3, v^3
1264 $factor = $self->new(3); $f = $self->new(2);
1266 my $steps = 0 if DEBUG;
1267 $limit = $self->new("1E-". ($scale-1));
1270 # we calculate the next term, and add it to the last
1271 # when the next term is below our limit, it won't affect the outcome
1272 # anymore, so we stop
1274 # calculating the next term simple from over/below will result in quite
1275 # a time hog if the input has many digits, since over and below will
1276 # accumulate more and more digits, and the result will also have many
1277 # digits, but in the end it is rounded to $scale digits anyway. So if we
1278 # round $over and $below first, we save a lot of time for the division
1279 # (not with log(1.2345), but try log (123**123) to see what I mean. This
1280 # can introduce a rounding error if the division result would be f.i.
1281 # 0.1234500000001 and we round it to 5 digits it would become 0.12346, but
1282 # if we truncated $over and $below we might get 0.12345. Does this matter
1283 # for the end result? So we give $over and $below 4 more digits to be
1284 # on the safe side (unscientific error handling as usual... :+D
1286 $next = $over->copy->bround($scale+4)->bdiv(
1287 $below->copy->bmul($factor)->bround($scale+4),
1291 ## $next = $over->copy()->bdiv($below->copy()->bmul($factor),$scale);
1293 last if $next->bacmp($limit) <= 0;
1295 delete $next->{_a}; delete $next->{_p};
1297 # calculate things for the next term
1298 $over *= $u; $below *= $v; $factor->badd($f);
1301 $steps++; print "step $steps = $x\n" if $steps % 10 == 0;
1304 print "took $steps steps\n" if DEBUG;
1305 $x->bmul($f); # $x *= 2
1310 # Internal log function based on reducing input to the range of 0.1 .. 9.99
1311 # and then "correcting" the result to the proper one. Modifies $x in place.
1312 my ($self,$x,$scale) = @_;
1314 # Taking blog() from numbers greater than 10 takes a *very long* time, so we
1315 # break the computation down into parts based on the observation that:
1316 # blog(X*Y) = blog(X) + blog(Y)
1317 # We set Y here to multiples of 10 so that $x becomes below 1 - the smaller
1318 # $x is the faster it gets. Since 2*$x takes about 10 times as
1319 # long, we make it faster by about a factor of 100 by dividing $x by 10.
1321 # The same observation is valid for numbers smaller than 0.1, e.g. computing
1322 # log(1) is fastest, and the further away we get from 1, the longer it takes.
1323 # So we also 'break' this down by multiplying $x with 10 and subtract the
1324 # log(10) afterwards to get the correct result.
1326 # To get $x even closer to 1, we also divide by 2 and then use log(2) to
1327 # correct for this. For instance if $x is 2.4, we use the formula:
1328 # blog(2.4 * 2) == blog (1.2) + blog(2)
1329 # and thus calculate only blog(1.2) and blog(2), which is faster in total
1330 # than calculating blog(2.4).
1332 # In addition, the values for blog(2) and blog(10) are cached.
1334 # Calculate nr of digits before dot:
1335 my $dbd = $MBI->_num($x->{_e});
1336 $dbd = -$dbd if $x->{_es} eq '-';
1337 $dbd += $MBI->_len($x->{_m});
1339 # more than one digit (e.g. at least 10), but *not* exactly 10 to avoid
1340 # infinite recursion
1342 my $calc = 1; # do some calculation?
1344 # disable the shortcut for 10, since we need log(10) and this would recurse
1346 if ($x->{_es} eq '+' && $MBI->_is_one($x->{_e}) && $MBI->_is_one($x->{_m}))
1348 $dbd = 0; # disable shortcut
1349 # we can use the cached value in these cases
1350 if ($scale <= $LOG_10_A)
1352 $x->bzero(); $x->badd($LOG_10); # modify $x in place
1353 $calc = 0; # no need to calc, but round
1355 # if we can't use the shortcut, we continue normally
1359 # disable the shortcut for 2, since we maybe have it cached
1360 if (($MBI->_is_zero($x->{_e}) && $MBI->_is_two($x->{_m})))
1362 $dbd = 0; # disable shortcut
1363 # we can use the cached value in these cases
1364 if ($scale <= $LOG_2_A)
1366 $x->bzero(); $x->badd($LOG_2); # modify $x in place
1367 $calc = 0; # no need to calc, but round
1369 # if we can't use the shortcut, we continue normally
1373 # if $x = 0.1, we know the result must be 0-log(10)
1374 if ($calc != 0 && $x->{_es} eq '-' && $MBI->_is_one($x->{_e}) &&
1375 $MBI->_is_one($x->{_m}))
1377 $dbd = 0; # disable shortcut
1378 # we can use the cached value in these cases
1379 if ($scale <= $LOG_10_A)
1381 $x->bzero(); $x->bsub($LOG_10);
1382 $calc = 0; # no need to calc, but round
1386 return if $calc == 0; # already have the result
1388 # default: these correction factors are undef and thus not used
1389 my $l_10; # value of ln(10) to A of $scale
1390 my $l_2; # value of ln(2) to A of $scale
1392 my $two = $self->new(2);
1394 # $x == 2 => 1, $x == 13 => 2, $x == 0.1 => 0, $x == 0.01 => -1
1395 # so don't do this shortcut for 1 or 0
1396 if (($dbd > 1) || ($dbd < 0))
1398 # convert our cached value to an object if not already (avoid doing this
1399 # at import() time, since not everybody needs this)
1400 $LOG_10 = $self->new($LOG_10,undef,undef) unless ref $LOG_10;
1402 #print "x = $x, dbd = $dbd, calc = $calc\n";
1403 # got more than one digit before the dot, or more than one zero after the
1405 # log(123) == log(1.23) + log(10) * 2
1406 # log(0.0123) == log(1.23) - log(10) * 2
1408 if ($scale <= $LOG_10_A)
1411 $l_10 = $LOG_10->copy(); # copy for mul
1415 # else: slower, compute and cache result
1416 # also disable downgrade for this code path
1417 local $Math::BigFloat::downgrade = undef;
1419 # shorten the time to calculate log(10) based on the following:
1420 # log(1.25 * 8) = log(1.25) + log(8)
1421 # = log(1.25) + log(2) + log(2) + log(2)
1423 # first get $l_2 (and possible compute and cache log(2))
1424 $LOG_2 = $self->new($LOG_2,undef,undef) unless ref $LOG_2;
1425 if ($scale <= $LOG_2_A)
1428 $l_2 = $LOG_2->copy(); # copy() for the mul below
1432 # else: slower, compute and cache result
1433 $l_2 = $two->copy(); $self->_log($l_2, $scale); # scale+4, actually
1434 $LOG_2 = $l_2->copy(); # cache the result for later
1435 # the copy() is for mul below
1439 # now calculate log(1.25):
1440 $l_10 = $self->new('1.25'); $self->_log($l_10, $scale); # scale+4, actually
1442 # log(1.25) + log(2) + log(2) + log(2):
1446 $LOG_10 = $l_10->copy(); # cache the result for later
1447 # the copy() is for mul below
1450 $dbd-- if ($dbd > 1); # 20 => dbd=2, so make it dbd=1
1451 $l_10->bmul( $self->new($dbd)); # log(10) * (digits_before_dot-1)
1458 ($x->{_e}, $x->{_es}) =
1459 _e_sub( $x->{_e}, $MBI->_new($dbd), $x->{_es}, $dbd_sign); # 123 => 1.23
1463 # Now: 0.1 <= $x < 10 (and possible correction in l_10)
1465 ### Since $x in the range 0.5 .. 1.5 is MUCH faster, we do a repeated div
1466 ### or mul by 2 (maximum times 3, since x < 10 and x > 0.1)
1468 $HALF = $self->new($HALF) unless ref($HALF);
1470 my $twos = 0; # default: none (0 times)
1471 while ($x->bacmp($HALF) <= 0) # X <= 0.5
1473 $twos--; $x->bmul($two);
1475 while ($x->bacmp($two) >= 0) # X >= 2
1477 $twos++; $x->bdiv($two,$scale+4); # keep all digits
1479 # $twos > 0 => did mul 2, < 0 => did div 2 (but we never did both)
1480 # So calculate correction factor based on ln(2):
1483 $LOG_2 = $self->new($LOG_2,undef,undef) unless ref $LOG_2;
1484 if ($scale <= $LOG_2_A)
1487 $l_2 = $LOG_2->copy(); # copy() for the mul below
1491 # else: slower, compute and cache result
1492 # also disable downgrade for this code path
1493 local $Math::BigFloat::downgrade = undef;
1494 $l_2 = $two->copy(); $self->_log($l_2, $scale); # scale+4, actually
1495 $LOG_2 = $l_2->copy(); # cache the result for later
1496 # the copy() is for mul below
1499 $l_2->bmul($twos); # * -2 => subtract, * 2 => add
1502 $self->_log($x,$scale); # need to do the "normal" way
1503 $x->badd($l_10) if defined $l_10; # correct it by ln(10)
1504 $x->badd($l_2) if defined $l_2; # and maybe by ln(2)
1506 # all done, $x contains now the result
1512 # (BFLOAT or num_str, BFLOAT or num_str) return BFLOAT
1513 # does not modify arguments, but returns new object
1514 # Lowest Common Multiplicator
1516 my ($self,@arg) = objectify(0,@_);
1517 my $x = $self->new(shift @arg);
1518 while (@arg) { $x = Math::BigInt::__lcm($x,shift @arg); }
1524 # (BINT or num_str, BINT or num_str) return BINT
1525 # does not modify arguments, but returns new object
1528 $y = __PACKAGE__->new($y) if !ref($y);
1530 my $x = $y->copy()->babs(); # keep arguments
1532 return $x->bnan() if $x->{sign} !~ /^[+-]$/ # x NaN?
1533 || !$x->is_int(); # only for integers now
1537 my $t = shift; $t = $self->new($t) if !ref($t);
1538 $y = $t->copy()->babs();
1540 return $x->bnan() if $y->{sign} !~ /^[+-]$/ # y NaN?
1541 || !$y->is_int(); # only for integers now
1543 # greatest common divisor
1544 while (! $y->is_zero())
1546 ($x,$y) = ($y->copy(), $x->copy()->bmod($y));
1549 last if $x->is_one();
1554 ##############################################################################
1558 # Internal helper sub to take two positive integers and their signs and
1559 # then add them. Input ($CALC,$CALC,('+'|'-'),('+'|'-')),
1560 # output ($CALC,('+'|'-'))
1561 my ($x,$y,$xs,$ys) = @_;
1563 # if the signs are equal we can add them (-5 + -3 => -(5 + 3) => -8)
1566 $x = $MBI->_add ($x, $y ); # a+b
1567 # the sign follows $xs
1571 my $a = $MBI->_acmp($x,$y);
1574 $x = $MBI->_sub ($x , $y); # abs sub
1578 $x = $MBI->_zero(); # result is 0
1583 $x = $MBI->_sub ( $y, $x, 1 ); # abs sub
1591 # Internal helper sub to take two positive integers and their signs and
1592 # then subtract them. Input ($CALC,$CALC,('+'|'-'),('+'|'-')),
1593 # output ($CALC,('+'|'-'))
1594 my ($x,$y,$xs,$ys) = @_;
1598 _e_add($x,$y,$xs,$ys); # call add (does subtract now)
1601 ###############################################################################
1602 # is_foo methods (is_negative, is_positive are inherited from BigInt)
1606 # return true if arg (BFLOAT or num_str) is an integer
1607 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1609 (($x->{sign} =~ /^[+-]$/) && # NaN and +-inf aren't
1610 ($x->{_es} eq '+')) ? 1 : 0; # 1e-1 => no integer
1615 # return true if arg (BFLOAT or num_str) is zero
1616 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1618 ($x->{sign} eq '+' && $MBI->_is_zero($x->{_m})) ? 1 : 0;
1623 # return true if arg (BFLOAT or num_str) is +1 or -1 if signis given
1624 my ($self,$x,$sign) = ref($_[0]) ? (undef,@_) : objectify(1,@_);
1626 $sign = '+' if !defined $sign || $sign ne '-';
1628 ($x->{sign} eq $sign &&
1629 $MBI->_is_zero($x->{_e}) &&
1630 $MBI->_is_one($x->{_m}) ) ? 1 : 0;
1635 # return true if arg (BFLOAT or num_str) is odd or false if even
1636 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1638 (($x->{sign} =~ /^[+-]$/) && # NaN & +-inf aren't
1639 ($MBI->_is_zero($x->{_e})) &&
1640 ($MBI->_is_odd($x->{_m}))) ? 1 : 0;
1645 # return true if arg (BINT or num_str) is even or false if odd
1646 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1648 (($x->{sign} =~ /^[+-]$/) && # NaN & +-inf aren't
1649 ($x->{_es} eq '+') && # 123.45 isn't
1650 ($MBI->_is_even($x->{_m}))) ? 1 : 0; # but 1200 is
1655 # multiply two numbers
1658 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1659 # objectify is costly, so avoid it
1660 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1662 ($self,$x,$y,@r) = objectify(2,@_);
1665 return $x if $x->modify('bmul');
1667 return $x->bnan() if (($x->{sign} eq $nan) || ($y->{sign} eq $nan));
1670 if (($x->{sign} =~ /^[+-]inf$/) || ($y->{sign} =~ /^[+-]inf$/))
1672 return $x->bnan() if $x->is_zero() || $y->is_zero();
1673 # result will always be +-inf:
1674 # +inf * +/+inf => +inf, -inf * -/-inf => +inf
1675 # +inf * -/-inf => -inf, -inf * +/+inf => -inf
1676 return $x->binf() if ($x->{sign} =~ /^\+/ && $y->{sign} =~ /^\+/);
1677 return $x->binf() if ($x->{sign} =~ /^-/ && $y->{sign} =~ /^-/);
1678 return $x->binf('-');
1681 return $upgrade->bmul($x,$y,@r) if defined $upgrade &&
1682 ((!$x->isa($self)) || (!$y->isa($self)));
1684 # aEb * cEd = (a*c)E(b+d)
1685 $MBI->_mul($x->{_m},$y->{_m});
1686 ($x->{_e}, $x->{_es}) = _e_add($x->{_e}, $y->{_e}, $x->{_es}, $y->{_es});
1688 $r[3] = $y; # no push!
1691 $x->{sign} = $x->{sign} ne $y->{sign} ? '-' : '+';
1692 $x->bnorm->round(@r);
1697 # multiply two numbers and add the third to the result
1700 my ($self,$x,$y,$z,@r) = objectify(3,@_);
1702 return $x if $x->modify('bmuladd');
1704 return $x->bnan() if (($x->{sign} eq $nan) ||
1705 ($y->{sign} eq $nan) ||
1706 ($z->{sign} eq $nan));
1709 if (($x->{sign} =~ /^[+-]inf$/) || ($y->{sign} =~ /^[+-]inf$/))
1711 return $x->bnan() if $x->is_zero() || $y->is_zero();
1712 # result will always be +-inf:
1713 # +inf * +/+inf => +inf, -inf * -/-inf => +inf
1714 # +inf * -/-inf => -inf, -inf * +/+inf => -inf
1715 return $x->binf() if ($x->{sign} =~ /^\+/ && $y->{sign} =~ /^\+/);
1716 return $x->binf() if ($x->{sign} =~ /^-/ && $y->{sign} =~ /^-/);
1717 return $x->binf('-');
1720 return $upgrade->bmul($x,$y,@r) if defined $upgrade &&
1721 ((!$x->isa($self)) || (!$y->isa($self)));
1723 # aEb * cEd = (a*c)E(b+d)
1724 $MBI->_mul($x->{_m},$y->{_m});
1725 ($x->{_e}, $x->{_es}) = _e_add($x->{_e}, $y->{_e}, $x->{_es}, $y->{_es});
1727 $r[3] = $y; # no push!
1730 $x->{sign} = $x->{sign} ne $y->{sign} ? '-' : '+';
1732 # z=inf handling (z=NaN handled above)
1733 $x->{sign} = $z->{sign}, return $x if $z->{sign} =~ /^[+-]inf$/;
1735 # take lower of the two e's and adapt m1 to it to match m2
1737 $e = $MBI->_zero() if !defined $e; # if no BFLOAT?
1738 $e = $MBI->_copy($e); # make copy (didn't do it yet)
1742 ($e,$es) = _e_sub($e, $x->{_e}, $z->{_es} || '+', $x->{_es});
1744 my $add = $MBI->_copy($z->{_m});
1746 if ($es eq '-') # < 0
1748 $MBI->_lsft( $x->{_m}, $e, 10);
1749 ($x->{_e},$x->{_es}) = _e_add($x->{_e}, $e, $x->{_es}, $es);
1751 elsif (!$MBI->_is_zero($e)) # > 0
1753 $MBI->_lsft($add, $e, 10);
1755 # else: both e are the same, so just leave them
1757 if ($x->{sign} eq $z->{sign})
1760 $x->{_m} = $MBI->_add($x->{_m}, $add);
1764 ($x->{_m}, $x->{sign}) =
1765 _e_add($x->{_m}, $add, $x->{sign}, $z->{sign});
1768 # delete trailing zeros, then round
1769 $x->bnorm()->round(@r);
1774 # (dividend: BFLOAT or num_str, divisor: BFLOAT or num_str) return
1775 # (BFLOAT,BFLOAT) (quo,rem) or BFLOAT (only rem)
1778 my ($self,$x,$y,$a,$p,$r) = (ref($_[0]),@_);
1779 # objectify is costly, so avoid it
1780 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1782 ($self,$x,$y,$a,$p,$r) = objectify(2,@_);
1785 return $x if $x->modify('bdiv');
1787 return $self->_div_inf($x,$y)
1788 if (($x->{sign} !~ /^[+-]$/) || ($y->{sign} !~ /^[+-]$/) || $y->is_zero());
1790 # x== 0 # also: or y == 1 or y == -1
1791 return wantarray ? ($x,$self->bzero()) : $x if $x->is_zero();
1794 return $upgrade->bdiv($upgrade->new($x),$y,$a,$p,$r) if defined $upgrade;
1796 # we need to limit the accuracy to protect against overflow
1798 my (@params,$scale);
1799 ($x,@params) = $x->_find_round_parameters($a,$p,$r,$y);
1801 return $x if $x->is_nan(); # error in _find_round_parameters?
1803 # no rounding at all, so must use fallback
1804 if (scalar @params == 0)
1806 # simulate old behaviour
1807 $params[0] = $self->div_scale(); # and round to it as accuracy
1808 $scale = $params[0]+4; # at least four more for proper round
1809 $params[2] = $r; # round mode by caller or undef
1810 $fallback = 1; # to clear a/p afterwards
1814 # the 4 below is empirical, and there might be cases where it is not
1816 $scale = abs($params[0] || $params[1]) + 4; # take whatever is defined
1819 my $rem; $rem = $self->bzero() if wantarray;
1821 $y = $self->new($y) unless $y->isa('Math::BigFloat');
1823 my $lx = $MBI->_len($x->{_m}); my $ly = $MBI->_len($y->{_m});
1824 $scale = $lx if $lx > $scale;
1825 $scale = $ly if $ly > $scale;
1826 my $diff = $ly - $lx;
1827 $scale += $diff if $diff > 0; # if lx << ly, but not if ly << lx!
1829 # already handled inf/NaN/-inf above:
1831 # check that $y is not 1 nor -1 and cache the result:
1832 my $y_not_one = !($MBI->_is_zero($y->{_e}) && $MBI->_is_one($y->{_m}));
1834 # flipping the sign of $y will also flip the sign of $x for the special
1835 # case of $x->bsub($x); so we can catch it below:
1836 my $xsign = $x->{sign};
1837 $y->{sign} =~ tr/+-/-+/;
1839 if ($xsign ne $x->{sign})
1841 # special case of $x /= $x results in 1
1842 $x->bone(); # "fixes" also sign of $y, since $x is $y
1846 # correct $y's sign again
1847 $y->{sign} =~ tr/+-/-+/;
1848 # continue with normal div code:
1850 # make copy of $x in case of list context for later remainder calculation
1851 if (wantarray && $y_not_one)
1856 $x->{sign} = $x->{sign} ne $y->sign() ? '-' : '+';
1858 # check for / +-1 ( +/- 1E0)
1861 # promote BigInts and it's subclasses (except when already a BigFloat)
1862 $y = $self->new($y) unless $y->isa('Math::BigFloat');
1864 # calculate the result to $scale digits and then round it
1865 # a * 10 ** b / c * 10 ** d => a/c * 10 ** (b-d)
1866 $MBI->_lsft($x->{_m},$MBI->_new($scale),10);
1867 $MBI->_div ($x->{_m},$y->{_m}); # a/c
1869 # correct exponent of $x
1870 ($x->{_e},$x->{_es}) = _e_sub($x->{_e}, $y->{_e}, $x->{_es}, $y->{_es});
1871 # correct for 10**scale
1872 ($x->{_e},$x->{_es}) = _e_sub($x->{_e}, $MBI->_new($scale), $x->{_es}, '+');
1873 $x->bnorm(); # remove trailing 0's
1875 } # ende else $x != $y
1877 # shortcut to not run through _find_round_parameters again
1878 if (defined $params[0])
1880 delete $x->{_a}; # clear before round
1881 $x->bround($params[0],$params[2]); # then round accordingly
1885 delete $x->{_p}; # clear before round
1886 $x->bfround($params[1],$params[2]); # then round accordingly
1890 # clear a/p after round, since user did not request it
1891 delete $x->{_a}; delete $x->{_p};
1898 $rem->bmod($y,@params); # copy already done
1902 # clear a/p after round, since user did not request it
1903 delete $rem->{_a}; delete $rem->{_p};
1912 # (dividend: BFLOAT or num_str, divisor: BFLOAT or num_str) return remainder
1915 my ($self,$x,$y,$a,$p,$r) = (ref($_[0]),@_);
1916 # objectify is costly, so avoid it
1917 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1919 ($self,$x,$y,$a,$p,$r) = objectify(2,@_);
1922 return $x if $x->modify('bmod');
1924 # handle NaN, inf, -inf
1925 if (($x->{sign} !~ /^[+-]$/) || ($y->{sign} !~ /^[+-]$/))
1927 my ($d,$re) = $self->SUPER::_div_inf($x,$y);
1928 $x->{sign} = $re->{sign};
1929 $x->{_e} = $re->{_e};
1930 $x->{_m} = $re->{_m};
1931 return $x->round($a,$p,$r,$y);
1935 return $x->bnan() if $x->is_zero();
1939 return $x->bzero() if $x->is_zero()
1941 # check that $y == +1 or $y == -1:
1942 ($MBI->_is_zero($y->{_e}) && $MBI->_is_one($y->{_m})));
1944 my $cmp = $x->bacmp($y); # equal or $x < $y?
1945 return $x->bzero($a,$p) if $cmp == 0; # $x == $y => result 0
1947 # only $y of the operands negative?
1948 my $neg = 0; $neg = 1 if $x->{sign} ne $y->{sign};
1950 $x->{sign} = $y->{sign}; # calc sign first
1951 return $x->round($a,$p,$r) if $cmp < 0 && $neg == 0; # $x < $y => result $x
1953 my $ym = $MBI->_copy($y->{_m});
1956 $MBI->_lsft( $ym, $y->{_e}, 10)
1957 if $y->{_es} eq '+' && !$MBI->_is_zero($y->{_e});
1959 # if $y has digits after dot
1960 my $shifty = 0; # correct _e of $x by this
1961 if ($y->{_es} eq '-') # has digits after dot
1963 # 123 % 2.5 => 1230 % 25 => 5 => 0.5
1964 $shifty = $MBI->_num($y->{_e}); # no more digits after dot
1965 $MBI->_lsft($x->{_m}, $y->{_e}, 10);# 123 => 1230, $y->{_m} is already 25
1967 # $ym is now mantissa of $y based on exponent 0
1969 my $shiftx = 0; # correct _e of $x by this
1970 if ($x->{_es} eq '-') # has digits after dot
1972 # 123.4 % 20 => 1234 % 200
1973 $shiftx = $MBI->_num($x->{_e}); # no more digits after dot
1974 $MBI->_lsft($ym, $x->{_e}, 10); # 123 => 1230
1976 # 123e1 % 20 => 1230 % 20
1977 if ($x->{_es} eq '+' && !$MBI->_is_zero($x->{_e}))
1979 $MBI->_lsft( $x->{_m}, $x->{_e},10); # es => '+' here
1982 $x->{_e} = $MBI->_new($shiftx);
1984 $x->{_es} = '-' if $shiftx != 0 || $shifty != 0;
1985 $MBI->_add( $x->{_e}, $MBI->_new($shifty)) if $shifty != 0;
1987 # now mantissas are equalized, exponent of $x is adjusted, so calc result
1989 $x->{_m} = $MBI->_mod( $x->{_m}, $ym);
1991 $x->{sign} = '+' if $MBI->_is_zero($x->{_m}); # fix sign for -0
1994 if ($neg != 0) # one of them negative => correct in place
1997 $x->{_m} = $r->{_m};
1998 $x->{_e} = $r->{_e};
1999 $x->{_es} = $r->{_es};
2000 $x->{sign} = '+' if $MBI->_is_zero($x->{_m}); # fix sign for -0
2004 $x->round($a,$p,$r,$y); # round and return
2009 # calculate $y'th root of $x
2012 my ($self,$x,$y,$a,$p,$r) = (ref($_[0]),@_);
2013 # objectify is costly, so avoid it
2014 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
2016 ($self,$x,$y,$a,$p,$r) = objectify(2,@_);
2019 return $x if $x->modify('broot');
2021 # NaN handling: $x ** 1/0, x or y NaN, or y inf/-inf or y == 0
2022 return $x->bnan() if $x->{sign} !~ /^\+/ || $y->is_zero() ||
2023 $y->{sign} !~ /^\+$/;
2025 return $x if $x->is_zero() || $x->is_one() || $x->is_inf() || $y->is_one();
2027 # we need to limit the accuracy to protect against overflow
2029 my (@params,$scale);
2030 ($x,@params) = $x->_find_round_parameters($a,$p,$r);
2032 return $x if $x->is_nan(); # error in _find_round_parameters?
2034 # no rounding at all, so must use fallback
2035 if (scalar @params == 0)
2037 # simulate old behaviour
2038 $params[0] = $self->div_scale(); # and round to it as accuracy
2039 $scale = $params[0]+4; # at least four more for proper round
2040 $params[2] = $r; # iound mode by caller or undef
2041 $fallback = 1; # to clear a/p afterwards
2045 # the 4 below is empirical, and there might be cases where it is not
2047 $scale = abs($params[0] || $params[1]) + 4; # take whatever is defined
2050 # when user set globals, they would interfere with our calculation, so
2051 # disable them and later re-enable them
2053 my $abr = "$self\::accuracy"; my $ab = $$abr; $$abr = undef;
2054 my $pbr = "$self\::precision"; my $pb = $$pbr; $$pbr = undef;
2055 # we also need to disable any set A or P on $x (_find_round_parameters took
2056 # them already into account), since these would interfere, too
2057 delete $x->{_a}; delete $x->{_p};
2058 # need to disable $upgrade in BigInt, to avoid deep recursion
2059 local $Math::BigInt::upgrade = undef; # should be really parent class vs MBI
2061 # remember sign and make $x positive, since -4 ** (1/2) => -2
2062 my $sign = 0; $sign = 1 if $x->{sign} eq '-'; $x->{sign} = '+';
2065 if ($y->isa('Math::BigFloat'))
2067 $is_two = ($y->{sign} eq '+' && $MBI->_is_two($y->{_m}) && $MBI->_is_zero($y->{_e}));
2071 $is_two = ($y == 2);
2074 # normal square root if $y == 2:
2077 $x->bsqrt($scale+4);
2079 elsif ($y->is_one('-'))
2082 my $u = $self->bone()->bdiv($x,$scale);
2083 # copy private parts over
2084 $x->{_m} = $u->{_m};
2085 $x->{_e} = $u->{_e};
2086 $x->{_es} = $u->{_es};
2090 # calculate the broot() as integer result first, and if it fits, return
2091 # it rightaway (but only if $x and $y are integer):
2093 my $done = 0; # not yet
2094 if ($y->is_int() && $x->is_int())
2096 my $i = $MBI->_copy( $x->{_m} );
2097 $MBI->_lsft( $i, $x->{_e}, 10 ) unless $MBI->_is_zero($x->{_e});
2098 my $int = Math::BigInt->bzero();
2100 $int->broot($y->as_number());
2102 if ($int->copy()->bpow($y) == $x)
2104 # found result, return it
2105 $x->{_m} = $int->{value};
2106 $x->{_e} = $MBI->_zero();
2114 my $u = $self->bone()->bdiv($y,$scale+4);
2115 delete $u->{_a}; delete $u->{_p}; # otherwise it conflicts
2116 $x->bpow($u,$scale+4); # el cheapo
2119 $x->bneg() if $sign == 1;
2121 # shortcut to not run through _find_round_parameters again
2122 if (defined $params[0])
2124 $x->bround($params[0],$params[2]); # then round accordingly
2128 $x->bfround($params[1],$params[2]); # then round accordingly
2132 # clear a/p after round, since user did not request it
2133 delete $x->{_a}; delete $x->{_p};
2136 $$abr = $ab; $$pbr = $pb;
2142 # calculate square root
2143 my ($self,$x,$a,$p,$r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_);
2145 return $x if $x->modify('bsqrt');
2147 return $x->bnan() if $x->{sign} !~ /^[+]/; # NaN, -inf or < 0
2148 return $x if $x->{sign} eq '+inf'; # sqrt(inf) == inf
2149 return $x->round($a,$p,$r) if $x->is_zero() || $x->is_one();
2151 # we need to limit the accuracy to protect against overflow
2153 my (@params,$scale);
2154 ($x,@params) = $x->_find_round_parameters($a,$p,$r);
2156 return $x if $x->is_nan(); # error in _find_round_parameters?
2158 # no rounding at all, so must use fallback
2159 if (scalar @params == 0)
2161 # simulate old behaviour
2162 $params[0] = $self->div_scale(); # and round to it as accuracy
2163 $scale = $params[0]+4; # at least four more for proper round
2164 $params[2] = $r; # round mode by caller or undef
2165 $fallback = 1; # to clear a/p afterwards
2169 # the 4 below is empirical, and there might be cases where it is not
2171 $scale = abs($params[0] || $params[1]) + 4; # take whatever is defined
2174 # when user set globals, they would interfere with our calculation, so
2175 # disable them and later re-enable them
2177 my $abr = "$self\::accuracy"; my $ab = $$abr; $$abr = undef;
2178 my $pbr = "$self\::precision"; my $pb = $$pbr; $$pbr = undef;
2179 # we also need to disable any set A or P on $x (_find_round_parameters took
2180 # them already into account), since these would interfere, too
2181 delete $x->{_a}; delete $x->{_p};
2182 # need to disable $upgrade in BigInt, to avoid deep recursion
2183 local $Math::BigInt::upgrade = undef; # should be really parent class vs MBI
2185 my $i = $MBI->_copy( $x->{_m} );
2186 $MBI->_lsft( $i, $x->{_e}, 10 ) unless $MBI->_is_zero($x->{_e});
2187 my $xas = Math::BigInt->bzero();
2190 my $gs = $xas->copy()->bsqrt(); # some guess
2192 if (($x->{_es} ne '-') # guess can't be accurate if there are
2193 # digits after the dot
2194 && ($xas->bacmp($gs * $gs) == 0)) # guess hit the nail on the head?
2196 # exact result, copy result over to keep $x
2197 $x->{_m} = $gs->{value}; $x->{_e} = $MBI->_zero(); $x->{_es} = '+';
2199 # shortcut to not run through _find_round_parameters again
2200 if (defined $params[0])
2202 $x->bround($params[0],$params[2]); # then round accordingly
2206 $x->bfround($params[1],$params[2]); # then round accordingly
2210 # clear a/p after round, since user did not request it
2211 delete $x->{_a}; delete $x->{_p};
2213 # re-enable A and P, upgrade is taken care of by "local"
2214 ${"$self\::accuracy"} = $ab; ${"$self\::precision"} = $pb;
2218 # sqrt(2) = 1.4 because sqrt(2*100) = 1.4*10; so we can increase the accuracy
2219 # of the result by multiplying the input by 100 and then divide the integer
2220 # result of sqrt(input) by 10. Rounding afterwards returns the real result.
2222 # The following steps will transform 123.456 (in $x) into 123456 (in $y1)
2223 my $y1 = $MBI->_copy($x->{_m});
2225 my $length = $MBI->_len($y1);
2227 # Now calculate how many digits the result of sqrt(y1) would have
2228 my $digits = int($length / 2);
2230 # But we need at least $scale digits, so calculate how many are missing
2231 my $shift = $scale - $digits;
2233 # This happens if the input had enough digits
2234 # (we take care of integer guesses above)
2235 $shift = 0 if $shift < 0;
2237 # Multiply in steps of 100, by shifting left two times the "missing" digits
2238 my $s2 = $shift * 2;
2240 # We now make sure that $y1 has the same odd or even number of digits than
2241 # $x had. So when _e of $x is odd, we must shift $y1 by one digit left,
2242 # because we always must multiply by steps of 100 (sqrt(100) is 10) and not
2243 # steps of 10. The length of $x does not count, since an even or odd number
2244 # of digits before the dot is not changed by adding an even number of digits
2245 # after the dot (the result is still odd or even digits long).
2246 $s2++ if $MBI->_is_odd($x->{_e});
2248 $MBI->_lsft( $y1, $MBI->_new($s2), 10);
2250 # now take the square root and truncate to integer
2251 $y1 = $MBI->_sqrt($y1);
2253 # By "shifting" $y1 right (by creating a negative _e) we calculate the final
2254 # result, which is than later rounded to the desired scale.
2256 # calculate how many zeros $x had after the '.' (or before it, depending
2257 # on sign of $dat, the result should have half as many:
2258 my $dat = $MBI->_num($x->{_e});
2259 $dat = -$dat if $x->{_es} eq '-';
2264 # no zeros after the dot (e.g. 1.23, 0.49 etc)
2265 # preserve half as many digits before the dot than the input had
2266 # (but round this "up")
2267 $dat = int(($dat+1)/2);
2271 $dat = int(($dat)/2);
2273 $dat -= $MBI->_len($y1);
2277 $x->{_e} = $MBI->_new( $dat );
2282 $x->{_e} = $MBI->_new( $dat );
2288 # shortcut to not run through _find_round_parameters again
2289 if (defined $params[0])
2291 $x->bround($params[0],$params[2]); # then round accordingly
2295 $x->bfround($params[1],$params[2]); # then round accordingly
2299 # clear a/p after round, since user did not request it
2300 delete $x->{_a}; delete $x->{_p};
2303 $$abr = $ab; $$pbr = $pb;
2309 # (BFLOAT or num_str, BFLOAT or num_str) return BFLOAT
2310 # compute factorial number, modifies first argument
2313 my ($self,$x,@r) = (ref($_[0]),@_);
2314 # objectify is costly, so avoid it
2315 ($self,$x,@r) = objectify(1,@_) if !ref($x);
2318 return $x if $x->modify('bfac') || $x->{sign} eq '+inf';
2321 if (($x->{sign} ne '+') || # inf, NaN, <0 etc => NaN
2322 ($x->{_es} ne '+')); # digits after dot?
2324 # use BigInt's bfac() for faster calc
2325 if (! $MBI->_is_zero($x->{_e}))
2327 $MBI->_lsft($x->{_m}, $x->{_e},10); # change 12e1 to 120e0
2328 $x->{_e} = $MBI->_zero(); # normalize
2331 $MBI->_fac($x->{_m}); # calculate factorial
2332 $x->bnorm()->round(@r); # norm again and round result
2337 # Calculate a power where $y is a non-integer, like 2 ** 0.3
2341 # if $y == 0.5, it is sqrt($x)
2342 $HALF = $self->new($HALF) unless ref($HALF);
2343 return $x->bsqrt(@r,$y) if $y->bcmp($HALF) == 0;
2346 # a ** x == e ** (x * ln a)
2350 # Taylor: | u u^2 u^3 |
2351 # x ** y = 1 + | --- + --- + ----- + ... |
2354 # we need to limit the accuracy to protect against overflow
2356 my ($scale,@params);
2357 ($x,@params) = $x->_find_round_parameters(@r);
2359 return $x if $x->is_nan(); # error in _find_round_parameters?
2361 # no rounding at all, so must use fallback
2362 if (scalar @params == 0)
2364 # simulate old behaviour
2365 $params[0] = $self->div_scale(); # and round to it as accuracy
2366 $params[1] = undef; # disable P
2367 $scale = $params[0]+4; # at least four more for proper round
2368 $params[2] = $r[2]; # round mode by caller or undef
2369 $fallback = 1; # to clear a/p afterwards
2373 # the 4 below is empirical, and there might be cases where it is not
2375 $scale = abs($params[0] || $params[1]) + 4; # take whatever is defined
2378 # when user set globals, they would interfere with our calculation, so
2379 # disable them and later re-enable them
2381 my $abr = "$self\::accuracy"; my $ab = $$abr; $$abr = undef;
2382 my $pbr = "$self\::precision"; my $pb = $$pbr; $$pbr = undef;
2383 # we also need to disable any set A or P on $x (_find_round_parameters took
2384 # them already into account), since these would interfere, too
2385 delete $x->{_a}; delete $x->{_p};
2386 # need to disable $upgrade in BigInt, to avoid deep recursion
2387 local $Math::BigInt::upgrade = undef;
2389 my ($limit,$v,$u,$below,$factor,$next,$over);
2391 $u = $x->copy()->blog(undef,$scale)->bmul($y);
2392 $v = $self->bone(); # 1
2393 $factor = $self->new(2); # 2
2394 $x->bone(); # first term: 1
2396 $below = $v->copy();
2399 $limit = $self->new("1E-". ($scale-1));
2403 # we calculate the next term, and add it to the last
2404 # when the next term is below our limit, it won't affect the outcome
2405 # anymore, so we stop:
2406 $next = $over->copy()->bdiv($below,$scale);
2407 last if $next->bacmp($limit) <= 0;
2409 # calculate things for the next term
2410 $over *= $u; $below *= $factor; $factor->binc();
2412 last if $x->{sign} !~ /^[-+]$/;
2417 # shortcut to not run through _find_round_parameters again
2418 if (defined $params[0])
2420 $x->bround($params[0],$params[2]); # then round accordingly
2424 $x->bfround($params[1],$params[2]); # then round accordingly
2428 # clear a/p after round, since user did not request it
2429 delete $x->{_a}; delete $x->{_p};
2432 $$abr = $ab; $$pbr = $pb;
2438 # (BFLOAT or num_str, BFLOAT or num_str) return BFLOAT
2439 # compute power of two numbers, second arg is used as integer
2440 # modifies first argument
2443 my ($self,$x,$y,$a,$p,$r) = (ref($_[0]),@_);
2444 # objectify is costly, so avoid it
2445 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
2447 ($self,$x,$y,$a,$p,$r) = objectify(2,@_);
2450 return $x if $x->modify('bpow');
2452 return $x->bnan() if $x->{sign} eq $nan || $y->{sign} eq $nan;
2453 return $x if $x->{sign} =~ /^[+-]inf$/;
2455 # cache the result of is_zero
2456 my $y_is_zero = $y->is_zero();
2457 return $x->bone() if $y_is_zero;
2458 return $x if $x->is_one() || $y->is_one();
2460 my $x_is_zero = $x->is_zero();
2461 return $x->_pow($y,$a,$p,$r) if !$x_is_zero && !$y->is_int(); # non-integer power
2463 my $y1 = $y->as_number()->{value}; # make MBI part
2466 if ($x->{sign} eq '-' && $MBI->_is_one($x->{_m}) && $MBI->_is_zero($x->{_e}))
2468 # if $x == -1 and odd/even y => +1/-1 because +-1 ^ (+-1) => +-1
2469 return $MBI->_is_odd($y1) ? $x : $x->babs(1);
2473 return $x if $y->{sign} eq '+'; # 0**y => 0 (if not y <= 0)
2474 # 0 ** -y => 1 / (0 ** y) => 1 / 0! (1 / 0 => +inf)
2479 $new_sign = $MBI->_is_odd($y1) ? '-' : '+' if $x->{sign} ne '+';
2481 # calculate $x->{_m} ** $y and $x->{_e} * $y separately (faster)
2482 $x->{_m} = $MBI->_pow( $x->{_m}, $y1);
2483 $x->{_e} = $MBI->_mul ($x->{_e}, $y1);
2485 $x->{sign} = $new_sign;
2487 if ($y->{sign} eq '-')
2489 # modify $x in place!
2490 my $z = $x->copy(); $x->bone();
2491 return scalar $x->bdiv($z,$a,$p,$r); # round in one go (might ignore y's A!)
2493 $x->round($a,$p,$r,$y);
2498 # takes a very large number to a very large exponent in a given very
2499 # large modulus, quickly, thanks to binary exponentiation. Supports
2500 # negative exponents.
2501 my ($self,$num,$exp,$mod,@r) = objectify(3,@_);
2503 return $num if $num->modify('bmodpow');
2505 # check modulus for valid values
2506 return $num->bnan() if ($mod->{sign} ne '+' # NaN, - , -inf, +inf
2507 || $mod->is_zero());
2509 # check exponent for valid values
2510 if ($exp->{sign} =~ /\w/)
2512 # i.e., if it's NaN, +inf, or -inf...
2513 return $num->bnan();
2516 $num->bmodinv ($mod) if ($exp->{sign} eq '-');
2518 # check num for valid values (also NaN if there was no inverse but $exp < 0)
2519 return $num->bnan() if $num->{sign} !~ /^[+-]$/;
2521 # $mod is positive, sign on $exp is ignored, result also positive
2523 # XXX TODO: speed it up when all three numbers are integers
2524 $num->bpow($exp)->bmod($mod);
2527 ###############################################################################
2528 # trigonometric functions
2530 # helper function for bpi() and batan2(), calculates arcus tanges (1/x)
2534 # return a/b so that a/b approximates atan(1/x) to at least limit digits
2535 my ($self, $x, $limit) = @_;
2537 # Taylor: x^3 x^5 x^7 x^9
2538 # atan = x - --- + --- - --- + --- - ...
2542 # atan 1/x = - - ------- + ------- - ------- + ...
2543 # x x^3 * 3 x^5 * 5 x^7 * 7
2546 # atan 1/x = - - --------- + ---------- - ----------- + ...
2547 # 5 3 * 125 5 * 3125 7 * 78125
2549 # Subtraction/addition of a rational:
2552 # - +- - = ----------
2557 # a 1 a * d * c +- b
2558 # ----- +- ------------------ = ----------------
2561 # since b1 = b0 * (d-2) * c
2563 # a 1 a * d +- b / c
2564 # ----- +- ------------------ = ----------------
2571 # stop if length($u) > limit
2578 my $a = $MBI->_one();
2579 my $b = $MBI->_copy($x);
2581 my $x2 = $MBI->_mul( $MBI->_copy($x), $b); # x2 = x * x
2582 my $d = $MBI->_new( 3 ); # d = 3
2583 my $c = $MBI->_mul( $MBI->_copy($x), $x2); # c = x ^ 3
2584 my $two = $MBI->_new( 2 );
2586 # run the first step unconditionally
2587 my $u = $MBI->_mul( $MBI->_copy($d), $c);
2588 $a = $MBI->_mul($a, $u);
2589 $a = $MBI->_sub($a, $b);
2590 $b = $MBI->_mul($b, $u);
2591 $d = $MBI->_add($d, $two);
2592 $c = $MBI->_mul($c, $x2);
2594 # a is now a * (d-3) * c
2595 # b is now b * (d-2) * c
2597 # run the second step unconditionally
2598 $u = $MBI->_mul( $MBI->_copy($d), $c);
2599 $a = $MBI->_mul($a, $u);
2600 $a = $MBI->_add($a, $b);
2601 $b = $MBI->_mul($b, $u);
2602 $d = $MBI->_add($d, $two);
2603 $c = $MBI->_mul($c, $x2);
2605 # a is now a * (d-3) * (d-5) * c * c
2606 # b is now b * (d-2) * (d-4) * c * c
2608 # so we can remove c * c from both a and b to shorten the numbers involved:
2609 $a = $MBI->_div($a, $x2);
2610 $b = $MBI->_div($b, $x2);
2611 $a = $MBI->_div($a, $x2);
2612 $b = $MBI->_div($b, $x2);
2615 my $sign = 0; # 0 => -, 1 => +
2619 # if (($i++ % 100) == 0)
2621 # print "a=",$MBI->_str($a),"\n";
2622 # print "b=",$MBI->_str($b),"\n";
2624 # print "d=",$MBI->_str($d),"\n";
2625 # print "x2=",$MBI->_str($x2),"\n";
2626 # print "c=",$MBI->_str($c),"\n";
2628 my $u = $MBI->_mul( $MBI->_copy($d), $c);
2629 # use _alen() for libs like GMP where _len() would be O(N^2)
2630 last if $MBI->_alen($u) > $limit;
2631 my ($bc,$r) = $MBI->_div( $MBI->_copy($b), $c);
2632 if ($MBI->_is_zero($r))
2634 # b / c is an integer, so we can remove c from all terms
2635 # this happens almost every time:
2636 $a = $MBI->_mul($a, $d);
2637 $a = $MBI->_sub($a, $bc) if $sign == 0;
2638 $a = $MBI->_add($a, $bc) if $sign == 1;
2639 $b = $MBI->_mul($b, $d);
2643 # b / c is not an integer, so we keep c in the terms
2644 # this happens very rarely, for instance for x = 5, this happens only
2645 # at the following steps:
2646 # 1, 5, 14, 32, 72, 157, 340, ...
2647 $a = $MBI->_mul($a, $u);
2648 $a = $MBI->_sub($a, $b) if $sign == 0;
2649 $a = $MBI->_add($a, $b) if $sign == 1;
2650 $b = $MBI->_mul($b, $u);
2652 $d = $MBI->_add($d, $two);
2653 $c = $MBI->_mul($c, $x2);
2658 # print "Took $step steps for ", $MBI->_str($x),"\n";
2659 # print "a=",$MBI->_str($a),"\n"; print "b=",$MBI->_str($b),"\n";
2660 # return a/b so that a/b approximates atan(1/x)
2673 # called like Math::BigFloat::bpi(10);
2674 $n = $self; $self = $class;
2675 # called like Math::BigFloat->bpi();
2676 $n = undef if $n eq 'Math::BigFloat';
2678 $self = ref($self) if ref($self);
2679 my $fallback = defined $n ? 0 : 1;
2680 $n = 40 if !defined $n || $n < 1;
2682 # after 黃見利 (Hwang Chien-Lih) (1997)
2683 # pi/4 = 183 * atan(1/239) + 32 * atan(1/1023) – 68 * atan(1/5832)
2684 # + 12 * atan(1/110443) - 12 * atan(1/4841182) - 100 * atan(1/6826318)
2686 # a few more to prevent rounding errors
2689 my ($a,$b) = $self->_atan_inv( $MBI->_new(239),$n);
2690 my ($c,$d) = $self->_atan_inv( $MBI->_new(1023),$n);
2691 my ($e,$f) = $self->_atan_inv( $MBI->_new(5832),$n);
2692 my ($g,$h) = $self->_atan_inv( $MBI->_new(110443),$n);
2693 my ($i,$j) = $self->_atan_inv( $MBI->_new(4841182),$n);
2694 my ($k,$l) = $self->_atan_inv( $MBI->_new(6826318),$n);
2696 $MBI->_mul($a, $MBI->_new(732));
2697 $MBI->_mul($c, $MBI->_new(128));
2698 $MBI->_mul($e, $MBI->_new(272));
2699 $MBI->_mul($g, $MBI->_new(48));
2700 $MBI->_mul($i, $MBI->_new(48));
2701 $MBI->_mul($k, $MBI->_new(400));
2703 my $x = $self->bone(); $x->{_m} = $a; my $x_d = $self->bone(); $x_d->{_m} = $b;
2704 my $y = $self->bone(); $y->{_m} = $c; my $y_d = $self->bone(); $y_d->{_m} = $d;
2705 my $z = $self->bone(); $z->{_m} = $e; my $z_d = $self->bone(); $z_d->{_m} = $f;
2706 my $u = $self->bone(); $u->{_m} = $g; my $u_d = $self->bone(); $u_d->{_m} = $h;
2707 my $v = $self->bone(); $v->{_m} = $i; my $v_d = $self->bone(); $v_d->{_m} = $j;
2708 my $w = $self->bone(); $w->{_m} = $k; my $w_d = $self->bone(); $w_d->{_m} = $l;
2716 delete $x->{_a}; delete $y->{_a}; delete $z->{_a};
2717 delete $u->{_a}; delete $v->{_a}; delete $w->{_a};
2718 $x->badd($y)->bsub($z)->badd($u)->bsub($v)->bsub($w);
2721 delete $x->{_a} if $fallback == 1;
2727 # Calculate a cosinus of x.
2728 my ($self,$x,@r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_);
2730 # Taylor: x^2 x^4 x^6 x^8
2731 # cos = 1 - --- + --- - --- + --- ...
2734 # we need to limit the accuracy to protect against overflow
2736 my ($scale,@params);
2737 ($x,@params) = $x->_find_round_parameters(@r);
2739 # constant object or error in _find_round_parameters?
2740 return $x if $x->modify('bcos') || $x->is_nan();
2742 return $x->bone(@r) if $x->is_zero();
2744 # no rounding at all, so must use fallback
2745 if (scalar @params == 0)
2747 # simulate old behaviour
2748 $params[0] = $self->div_scale(); # and round to it as accuracy
2749 $params[1] = undef; # disable P
2750 $scale = $params[0]+4; # at least four more for proper round
2751 $params[2] = $r[2]; # round mode by caller or undef
2752 $fallback = 1; # to clear a/p afterwards
2756 # the 4 below is empirical, and there might be cases where it is not
2758 $scale = abs($params[0] || $params[1]) + 4; # take whatever is defined
2761 # when user set globals, they would interfere with our calculation, so
2762 # disable them and later re-enable them
2764 my $abr = "$self\::accuracy"; my $ab = $$abr; $$abr = undef;
2765 my $pbr = "$self\::precision"; my $pb = $$pbr; $$pbr = undef;
2766 # we also need to disable any set A or P on $x (_find_round_parameters took
2767 # them already into account), since these would interfere, too
2768 delete $x->{_a}; delete $x->{_p};
2769 # need to disable $upgrade in BigInt, to avoid deep recursion
2770 local $Math::BigInt::upgrade = undef;
2773 my $over = $x * $x; # X ^ 2
2774 my $x2 = $over->copy(); # X ^ 2; difference between terms
2775 my $sign = 1; # start with -=
2776 my $below = $self->new(2); my $factorial = $self->new(3);
2777 $x->bone(); delete $x->{_a}; delete $x->{_p};
2779 my $limit = $self->new("1E-". ($scale-1));
2783 # we calculate the next term, and add it to the last
2784 # when the next term is below our limit, it won't affect the outcome
2785 # anymore, so we stop:
2786 my $next = $over->copy()->bdiv($below,$scale);
2787 last if $next->bacmp($limit) <= 0;
2797 $sign = 1-$sign; # alternate
2798 # calculate things for the next term
2799 $over->bmul($x2); # $x*$x
2800 $below->bmul($factorial); $factorial->binc(); # n*(n+1)
2801 $below->bmul($factorial); $factorial->binc(); # n*(n+1)
2804 # shortcut to not run through _find_round_parameters again
2805 if (defined $params[0])
2807 $x->bround($params[0],$params[2]); # then round accordingly
2811 $x->bfround($params[1],$params[2]); # then round accordingly
2815 # clear a/p after round, since user did not request it
2816 delete $x->{_a}; delete $x->{_p};
2819 $$abr = $ab; $$pbr = $pb;
2825 # Calculate a sinus of x.
2826 my ($self,$x,@r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_);
2828 # taylor: x^3 x^5 x^7 x^9
2829 # sin = x - --- + --- - --- + --- ...
2832 # we need to limit the accuracy to protect against overflow
2834 my ($scale,@params);
2835 ($x,@params) = $x->_find_round_parameters(@r);
2837 # constant object or error in _find_round_parameters?
2838 return $x if $x->modify('bsin') || $x->is_nan();
2840 return $x->bzero(@r) if $x->is_zero();
2842 # no rounding at all, so must use fallback
2843 if (scalar @params == 0)
2845 # simulate old behaviour
2846 $params[0] = $self->div_scale(); # and round to it as accuracy
2847 $params[1] = undef; # disable P
2848 $scale = $params[0]+4; # at least four more for proper round
2849 $params[2] = $r[2]; # round mode by caller or undef
2850 $fallback = 1; # to clear a/p afterwards
2854 # the 4 below is empirical, and there might be cases where it is not
2856 $scale = abs($params[0] || $params[1]) + 4; # take whatever is defined
2859 # when user set globals, they would interfere with our calculation, so
2860 # disable them and later re-enable them
2862 my $abr = "$self\::accuracy"; my $ab = $$abr; $$abr = undef;
2863 my $pbr = "$self\::precision"; my $pb = $$pbr; $$pbr = undef;
2864 # we also need to disable any set A or P on $x (_find_round_parameters took
2865 # them already into account), since these would interfere, too
2866 delete $x->{_a}; delete $x->{_p};
2867 # need to disable $upgrade in BigInt, to avoid deep recursion
2868 local $Math::BigInt::upgrade = undef;
2871 my $over = $x * $x; # X ^ 2
2872 my $x2 = $over->copy(); # X ^ 2; difference between terms
2873 $over->bmul($x); # X ^ 3 as starting value
2874 my $sign = 1; # start with -=
2875 my $below = $self->new(6); my $factorial = $self->new(4);
2876 delete $x->{_a}; delete $x->{_p};
2878 my $limit = $self->new("1E-". ($scale-1));
2882 # we calculate the next term, and add it to the last
2883 # when the next term is below our limit, it won't affect the outcome
2884 # anymore, so we stop:
2885 my $next = $over->copy()->bdiv($below,$scale);
2886 last if $next->bacmp($limit) <= 0;
2896 $sign = 1-$sign; # alternate
2897 # calculate things for the next term
2898 $over->bmul($x2); # $x*$x
2899 $below->bmul($factorial); $factorial->binc(); # n*(n+1)
2900 $below->bmul($factorial); $factorial->binc(); # n*(n+1)
2903 # shortcut to not run through _find_round_parameters again
2904 if (defined $params[0])
2906 $x->bround($params[0],$params[2]); # then round accordingly
2910 $x->bfround($params[1],$params[2]); # then round accordingly
2914 # clear a/p after round, since user did not request it
2915 delete $x->{_a}; delete $x->{_p};
2918 $$abr = $ab; $$pbr = $pb;
2924 # calculate arcus tangens of ($y/$x)
2927 my ($self,$y,$x,@r) = (ref($_[0]),@_);
2928 # objectify is costly, so avoid it
2929 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
2931 ($self,$y,$x,@r) = objectify(2,@_);
2934 return $y if $y->modify('batan2');
2936 return $y->bnan() if ($y->{sign} eq $nan) || ($x->{sign} eq $nan);
2942 return $y->bzero(@r) if ($x->is_inf('+') && !$y->is_inf()) || ($y->is_zero() && $x->{sign} eq '+');
2945 # != 0 -inf result is +- pi
2946 if ($x->is_inf() || $y->is_inf())
2949 my $pi = $self->bpi(@r);
2952 # upgrade to BigRat etc.
2953 return $upgrade->new($y)->batan2($upgrade->new($x),@r) if defined $upgrade;
2954 if ($x->{sign} eq '-inf')
2957 $MBI->_mul($pi->{_m}, $MBI->_new(3));
2958 $MBI->_div($pi->{_m}, $MBI->_new(4));
2960 elsif ($x->{sign} eq '+inf')
2963 $MBI->_div($pi->{_m}, $MBI->_new(4));
2968 $MBI->_div($pi->{_m}, $MBI->_new(2));
2970 $y->{sign} = substr($y->{sign},0,1); # keep +/-
2972 # modify $y in place
2973 $y->{_m} = $pi->{_m};
2974 $y->{_e} = $pi->{_e};
2975 $y->{_es} = $pi->{_es};
2976 # keep the sign of $y
2980 return $upgrade->new($y)->batan2($upgrade->new($x),@r) if defined $upgrade;
2987 my $pi = $self->bpi(@r);
2988 # modify $y in place
2989 $y->{_m} = $pi->{_m};
2990 $y->{_e} = $pi->{_e};
2991 $y->{_es} = $pi->{_es};
2997 # +y 0 result is PI/2
2998 # -y 0 result is -PI/2
3002 my $pi = $self->bpi(@r);
3003 # modify $y in place
3004 $y->{_m} = $pi->{_m};
3005 $y->{_e} = $pi->{_e};
3006 $y->{_es} = $pi->{_es};
3007 # -y => -PI/2, +y => PI/2
3008 $MBI->_div($y->{_m}, $MBI->_new(2));
3012 # we need to limit the accuracy to protect against overflow
3014 my ($scale,@params);
3015 ($y,@params) = $y->_find_round_parameters(@r);
3017 # error in _find_round_parameters?
3018 return $y if $y->is_nan();
3020 # no rounding at all, so must use fallback
3021 if (scalar @params == 0)
3023 # simulate old behaviour
3024 $params[0] = $self->div_scale(); # and round to it as accuracy
3025 $params[1] = undef; # disable P
3026 $scale = $params[0]+4; # at least four more for proper round
3027 $params[2] = $r[2]; # round mode by caller or undef
3028 $fallback = 1; # to clear a/p afterwards
3032 # the 4 below is empirical, and there might be cases where it is not
3034 $scale = abs($params[0] || $params[1]) + 4; # take whatever is defined
3037 # inlined is_one() && is_one('-')
3038 if ($MBI->_is_one($y->{_m}) && $MBI->_is_zero($y->{_e}))
3040 # shortcut: 1 1 result is PI/4
3041 # inlined is_one() && is_one('-')
3042 if ($MBI->_is_one($x->{_m}) && $MBI->_is_zero($x->{_e}))
3045 my $pi_4 = $self->bpi( $scale - 3);
3046 # modify $y in place
3047 $y->{_m} = $pi_4->{_m};
3048 $y->{_e} = $pi_4->{_e};
3049 $y->{_es} = $pi_4->{_es};
3054 $y->{sign} = $x->{sign} eq $y->{sign} ? '+' : '-';
3055 $MBI->_div($y->{_m}, $MBI->_new(4));
3058 # shortcut: 1 int(X) result is _atan_inv(X)
3061 if ($x->{_es} eq '+')
3063 my $x1 = $MBI->_copy($x->{_m});
3064 $MBI->_lsft($x1, $x->{_e},10) unless $MBI->_is_zero($x->{_e});
3066 my ($a,$b) = $self->_atan_inv($x1, $scale);
3067 my $y_sign = $y->{sign};
3069 $y->bone(); $y->{_m} = $a; my $y_d = $self->bone(); $y_d->{_m} = $b;
3071 $y->{sign} = $y_sign;
3076 # handle all other cases
3081 # -x -y -PI/2 to -PI
3083 my $y_sign = $y->{sign};
3086 $y->bdiv($x, $scale) unless $x->is_one();
3090 $y->{sign} = $y_sign;
3097 # Calculate a arcus tangens of x.
3101 # taylor: x^3 x^5 x^7 x^9
3102 # atan = x - --- + --- - --- + --- ...
3105 # we need to limit the accuracy to protect against overflow
3107 my ($scale,@params);
3108 ($x,@params) = $x->_find_round_parameters(@r);
3110 # constant object or error in _find_round_parameters?
3111 return $x if $x->modify('batan') || $x->is_nan();
3113 if ($x->{sign} =~ /^[+-]inf\z/)
3115 # +inf result is PI/2
3116 # -inf result is -PI/2
3118 my $pi = $self->bpi(@r);
3119 # modify $x in place
3120 $x->{_m} = $pi->{_m};
3121 $x->{_e} = $pi->{_e};
3122 $x->{_es} = $pi->{_es};
3123 # -y => -PI/2, +y => PI/2
3124 $x->{sign} = substr($x->{sign},0,1); # +inf => +
3125 $MBI->_div($x->{_m}, $MBI->_new(2));
3129 return $x->bzero(@r) if $x->is_zero();
3131 # no rounding at all, so must use fallback
3132 if (scalar @params == 0)
3134 # simulate old behaviour
3135 $params[0] = $self->div_scale(); # and round to it as accuracy
3136 $params[1] = undef; # disable P
3137 $scale = $params[0]+4; # at least four more for proper round
3138 $params[2] = $r[2]; # round mode by caller or undef
3139 $fallback = 1; # to clear a/p afterwards
3143 # the 4 below is empirical, and there might be cases where it is not
3145 $scale = abs($params[0] || $params[1]) + 4; # take whatever is defined
3149 # inlined is_one() && is_one('-')
3150 if ($MBI->_is_one($x->{_m}) && $MBI->_is_zero($x->{_e}))
3152 my $pi = $self->bpi($scale - 3);
3153 # modify $x in place
3154 $x->{_m} = $pi->{_m};
3155 $x->{_e} = $pi->{_e};
3156 $x->{_es} = $pi->{_es};
3157 # leave the sign of $x alone (+1 => +PI/4, -1 => -PI/4)
3158 $MBI->_div($x->{_m}, $MBI->_new(4));
3162 # This series is only valid if -1 < x < 1, so for other x we need to
3163 # to calculate PI/2 - atan(1/x):
3164 my $one = $MBI->_new(1);
3166 if ($x->{_es} eq '+' && ($MBI->_acmp($x->{_m},$one) >= 0))
3169 $pi = $self->bpi($scale - 3);
3170 $MBI->_div($pi->{_m}, $MBI->_new(2));
3172 my $x_copy = $x->copy();
3173 # modify $x in place
3174 $x->bone(); $x->bdiv($x_copy,$scale);
3177 # when user set globals, they would interfere with our calculation, so
3178 # disable them and later re-enable them
3180 my $abr = "$self\::accuracy"; my $ab = $$abr; $$abr = undef;
3181 my $pbr = "$self\::precision"; my $pb = $$pbr; $$pbr = undef;
3182 # we also need to disable any set A or P on $x (_find_round_parameters took
3183 # them already into account), since these would interfere, too
3184 delete $x->{_a}; delete $x->{_p};
3185 # need to disable $upgrade in BigInt, to avoid deep recursion
3186 local $Math::BigInt::upgrade = undef;
3189 my $over = $x * $x; # X ^ 2
3190 my $x2 = $over->copy(); # X ^ 2; difference between terms
3191 $over->bmul($x); # X ^ 3 as starting value
3192 my $sign = 1; # start with -=
3193 my $below = $self->new(3);
3194 my $two = $self->new(2);
3195 delete $x->{_a}; delete $x->{_p};
3197 my $limit = $self->new("1E-". ($scale-1));
3201 # we calculate the next term, and add it to the last
3202 # when the next term is below our limit, it won't affect the outcome
3203 # anymore, so we stop:
3204 my $next = $over->copy()->bdiv($below,$scale);
3205 last if $next->bacmp($limit) <= 0;
3215 $sign = 1-$sign; # alternate
3216 # calculate things for the next term
3217 $over->bmul($x2); # $x*$x
3218 $below->badd($two); # n += 2
3223 my $x_copy = $x->copy();
3224 # modify $x in place
3225 $x->{_m} = $pi->{_m};
3226 $x->{_e} = $pi->{_e};
3227 $x->{_es} = $pi->{_es};
3232 # shortcut to not run through _find_round_parameters again
3233 if (defined $params[0])
3235 $x->bround($params[0],$params[2]); # then round accordingly
3239 $x->bfround($params[1],$params[2]); # then round accordingly
3243 # clear a/p after round, since user did not request it
3244 delete $x->{_a}; delete $x->{_p};
3247 $$abr = $ab; $$pbr = $pb;
3251 ###############################################################################
3252 # rounding functions
3256 # precision: round to the $Nth digit left (+$n) or right (-$n) from the '.'
3257 # $n == 0 means round to integer
3258 # expects and returns normalized numbers!
3259 my $x = shift; my $self = ref($x) || $x; $x = $self->new(shift) if !ref($x);
3261 my ($scale,$mode) = $x->_scale_p(@_);
3262 return $x if !defined $scale || $x->modify('bfround'); # no-op
3264 # never round a 0, +-inf, NaN
3267 $x->{_p} = $scale if !defined $x->{_p} || $x->{_p} < $scale; # -3 < -2
3270 return $x if $x->{sign} !~ /^[+-]$/;
3272 # don't round if x already has lower precision
3273 return $x if (defined $x->{_p} && $x->{_p} < 0 && $scale < $x->{_p});
3275 $x->{_p} = $scale; # remember round in any case
3276 delete $x->{_a}; # and clear A
3279 # round right from the '.'
3281 return $x if $x->{_es} eq '+'; # e >= 0 => nothing to round
3283 $scale = -$scale; # positive for simplicity
3284 my $len = $MBI->_len($x->{_m}); # length of mantissa
3286 # the following poses a restriction on _e, but if _e is bigger than a
3287 # scalar, you got other problems (memory etc) anyway
3288 my $dad = -(0+ ($x->{_es}.$MBI->_num($x->{_e}))); # digits after dot
3289 my $zad = 0; # zeros after dot
3290 $zad = $dad - $len if (-$dad < -$len); # for 0.00..00xxx style
3292 # p rint "scale $scale dad $dad zad $zad len $len\n";
3293 # number bsstr len zad dad
3294 # 0.123 123e-3 3 0 3
3295 # 0.0123 123e-4 3 1 4
3298 # 1.2345 12345e-4 5 0 4
3300 # do not round after/right of the $dad
3301 return $x if $scale > $dad; # 0.123, scale >= 3 => exit
3303 # round to zero if rounding inside the $zad, but not for last zero like:
3304 # 0.0065, scale -2, round last '0' with following '65' (scale == zad case)
3305 return $x->bzero() if $scale < $zad;
3306 if ($scale == $zad) # for 0.006, scale -3 and trunc
3312 # adjust round-point to be inside mantissa
3315 $scale = $scale-$zad;
3319 my $dbd = $len - $dad; $dbd = 0 if $dbd < 0; # digits before dot
3320 $scale = $dbd+$scale;
3326 # round left from the '.'
3328 # 123 => 100 means length(123) = 3 - $scale (2) => 1
3330 my $dbt = $MBI->_len($x->{_m});
3332 my $dbd = $dbt + ($x->{_es} . $MBI->_num($x->{_e}));
3333 # should be the same, so treat it as this
3334 $scale = 1 if $scale == 0;
3335 # shortcut if already integer
3336 return $x if $scale == 1 && $dbt <= $dbd;
3337 # maximum digits before dot
3342 # not enough digits before dot, so round to zero
3345 elsif ( $scale == $dbd )
3352 $scale = $dbd - $scale;
3355 # pass sign to bround for rounding modes '+inf' and '-inf'
3356 my $m = bless { sign => $x->{sign}, value => $x->{_m} }, 'Math::BigInt';
3357 $m->bround($scale,$mode);
3358 $x->{_m} = $m->{value}; # get our mantissa back
3364 # accuracy: preserve $N digits, and overwrite the rest with 0's
3365 my $x = shift; my $self = ref($x) || $x; $x = $self->new(shift) if !ref($x);
3367 if (($_[0] || 0) < 0)
3369 require Carp; Carp::croak ('bround() needs positive accuracy');
3372 my ($scale,$mode) = $x->_scale_a(@_);
3373 return $x if !defined $scale || $x->modify('bround'); # no-op
3375 # scale is now either $x->{_a}, $accuracy, or the user parameter
3376 # test whether $x already has lower accuracy, do nothing in this case
3377 # but do round if the accuracy is the same, since a math operation might
3378 # want to round a number with A=5 to 5 digits afterwards again
3379 return $x if defined $x->{_a} && $x->{_a} < $scale;
3381 # scale < 0 makes no sense
3382 # scale == 0 => keep all digits
3383 # never round a +-inf, NaN
3384 return $x if ($scale <= 0) || $x->{sign} !~ /^[+-]$/;
3386 # 1: never round a 0
3387 # 2: if we should keep more digits than the mantissa has, do nothing
3388 if ($x->is_zero() || $MBI->_len($x->{_m}) <= $scale)
3390 $x->{_a} = $scale if !defined $x->{_a} || $x->{_a} > $scale;
3394 # pass sign to bround for '+inf' and '-inf' rounding modes
3395 my $m = bless { sign => $x->{sign}, value => $x->{_m} }, 'Math::BigInt';
3397 $m->bround($scale,$mode); # round mantissa
3398 $x->{_m} = $m->{value}; # get our mantissa back
3399 $x->{_a} = $scale; # remember rounding
3400 delete $x->{_p}; # and clear P
3401 $x->bnorm(); # del trailing zeros gen. by bround()
3406 # return integer less or equal then $x
3407 my ($self,$x,$a,$p,$r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_);
3409 return $x if $x->modify('bfloor');
3411 return $x if $x->{sign} !~ /^[+-]$/; # nan, +inf, -inf
3413 # if $x has digits after dot
3414 if ($x->{_es} eq '-')
3416 $x->{_m} = $MBI->_rsft($x->{_m},$x->{_e},10); # cut off digits after dot
3417 $x->{_e} = $MBI->_zero(); # trunc/norm
3418 $x->{_es} = '+'; # abs e
3419 $MBI->_inc($x->{_m}) if $x->{sign} eq '-'; # increment if negative
3421 $x->round($a,$p,$r);
3426 # return integer greater or equal then $x
3427 my ($self,$x,$a,$p,$r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_);
3429 return $x if $x->modify('bceil');
3430 return $x if $x->{sign} !~ /^[+-]$/; # nan, +inf, -inf
3432 # if $x has digits after dot
3433 if ($x->{_es} eq '-')
3435 $x->{_m} = $MBI->_rsft($x->{_m},$x->{_e},10); # cut off digits after dot
3436 $x->{_e} = $MBI->_zero(); # trunc/norm
3437 $x->{_es} = '+'; # abs e
3438 $MBI->_inc($x->{_m}) if $x->{sign} eq '+'; # increment if positive
3440 $x->round($a,$p,$r);
3445 # shift right by $y (divide by power of $n)
3448 my ($self,$x,$y,$n,$a,$p,$r) = (ref($_[0]),@_);
3449 # objectify is costly, so avoid it
3450 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
3452 ($self,$x,$y,$n,$a,$p,$r) = objectify(2,@_);
3455 return $x if $x->modify('brsft');
3456 return $x if $x->{sign} !~ /^[+-]$/; # nan, +inf, -inf
3458 $n = 2 if !defined $n; $n = $self->new($n);
3461 return $x->blsft($y->copy()->babs(),$n) if $y->{sign} =~ /^-/;
3463 # the following call to bdiv() will return either quo or (quo,remainder):
3464 $x->bdiv($n->bpow($y),$a,$p,$r,$y);
3469 # shift left by $y (multiply by power of $n)
3472 my ($self,$x,$y,$n,$a,$p,$r) = (ref($_[0]),@_);
3473 # objectify is costly, so avoid it
3474 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
3476 ($self,$x,$y,$n,$a,$p,$r) = objectify(2,@_);
3479 return $x if $x->modify('blsft');
3480 return $x if $x->{sign} !~ /^[+-]$/; # nan, +inf, -inf
3482 $n = 2 if !defined $n; $n = $self->new($n);
3485 return $x->brsft($y->copy()->babs(),$n) if $y->{sign} =~ /^-/;
3487 $x->bmul($n->bpow($y),$a,$p,$r,$y);
3490 ###############################################################################
3494 # going through AUTOLOAD for every DESTROY is costly, avoid it by empty sub
3499 # make fxxx and bxxx both work by selectively mapping fxxx() to MBF::bxxx()
3500 # or falling back to MBI::bxxx()
3501 my $name = $AUTOLOAD;
3503 $name =~ s/(.*):://; # split package
3504 my $c = $1 || $class;
3506 $c->import() if $IMPORT == 0;
3507 if (!_method_alias($name))
3511 # delayed load of Carp and avoid recursion
3513 Carp::croak ("$c: Can't call a method without name");
3515 if (!_method_hand_up($name))
3517 # delayed load of Carp and avoid recursion
3519 Carp::croak ("Can't call $c\-\>$name, not a valid method");
3521 # try one level up, but subst. bxxx() for fxxx() since MBI only got bxxx()
3523 return &{"Math::BigInt"."::$name"}(@_);
3525 my $bname = $name; $bname =~ s/^f/b/;
3533 # return a copy of the exponent
3534 my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
3536 if ($x->{sign} !~ /^[+-]$/)
3538 my $s = $x->{sign}; $s =~ s/^[+-]//;
3539 return Math::BigInt->new($s); # -inf, +inf => +inf
3541 Math::BigInt->new( $x->{_es} . $MBI->_str($x->{_e}));
3546 # return a copy of the mantissa
3547 my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
3549 if ($x->{sign} !~ /^[+-]$/)
3551 my $s = $x->{sign}; $s =~ s/^[+]//;
3552 return Math::BigInt->new($s); # -inf, +inf => +inf
3554 my $m = Math::BigInt->new( $MBI->_str($x->{_m}));
3555 $m->bneg() if $x->{sign} eq '-';
3562 # return a copy of both the exponent and the mantissa
3563 my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
3565 if ($x->{sign} !~ /^[+-]$/)
3567 my $s = $x->{sign}; $s =~ s/^[+]//; my $se = $s; $se =~ s/^[-]//;
3568 return ($self->new($s),$self->new($se)); # +inf => inf and -inf,+inf => inf
3570 my $m = Math::BigInt->bzero();
3571 $m->{value} = $MBI->_copy($x->{_m});
3572 $m->bneg() if $x->{sign} eq '-';
3573 ($m, Math::BigInt->new( $x->{_es} . $MBI->_num($x->{_e}) ));
3576 ##############################################################################
3577 # private stuff (internal use only)
3583 my $lib = ''; my @a;
3584 my $lib_kind = 'try';
3586 for ( my $i = 0; $i < $l ; $i++)
3588 if ( $_[$i] eq ':constant' )
3590 # This causes overlord er load to step in. 'binary' and 'integer'
3591 # are handled by BigInt.
3592 overload::constant float => sub { $self->new(shift); };
3594 elsif ($_[$i] eq 'upgrade')
3596 # this causes upgrading
3597 $upgrade = $_[$i+1]; # or undef to disable
3600 elsif ($_[$i] eq 'downgrade')
3602 # this causes downgrading
3603 $downgrade = $_[$i+1]; # or undef to disable
3606 elsif ($_[$i] =~ /^(lib|try|only)\z/)
3608 # alternative library
3609 $lib = $_[$i+1] || ''; # default Calc
3610 $lib_kind = $1; # lib, try or only
3613 elsif ($_[$i] eq 'with')
3615 # alternative class for our private parts()
3616 # XXX: no longer supported
3617 # $MBI = $_[$i+1] || 'Math::BigInt';
3626 $lib =~ tr/a-zA-Z0-9,://cd; # restrict to sane characters
3627 # let use Math::BigInt lib => 'GMP'; use Math::BigFloat; still work
3628 my $mbilib = eval { Math::BigInt->config()->{lib} };
3629 if ((defined $mbilib) && ($MBI eq 'Math::BigInt::Calc'))
3631 # MBI already loaded
3632 Math::BigInt->import( $lib_kind, "$lib,$mbilib", 'objectify');
3636 # MBI not loaded, or with ne "Math::BigInt::Calc"
3637 $lib .= ",$mbilib" if defined $mbilib;
3638 $lib =~ s/^,//; # don't leave empty
3640 # replacement library can handle lib statement, but also could ignore it
3642 # Perl < 5.6.0 dies with "out of memory!" when eval() and ':constant' is
3643 # used in the same script, or eval inside import(). So we require MBI:
3644 require Math::BigInt;
3645 Math::BigInt->import( $lib_kind => $lib, 'objectify' );
3649 require Carp; Carp::croak ("Couldn't load $lib: $! $@");
3651 # find out which one was actually loaded
3652 $MBI = Math::BigInt->config()->{lib};
3654 # register us with MBI to get notified of future lib changes
3655 Math::BigInt::_register_callback( $self, sub { $MBI = $_[0]; } );
3657 $self->export_to_level(1,$self,@a); # export wanted functions
3662 # adjust m and e so that m is smallest possible
3663 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
3665 return $x if $x->{sign} !~ /^[+-]$/; # inf, nan etc
3667 my $zeros = $MBI->_zeros($x->{_m}); # correct for trailing zeros
3670 my $z = $MBI->_new($zeros);
3671 $x->{_m} = $MBI->_rsft ($x->{_m}, $z, 10);
3672 if ($x->{_es} eq '-')
3674 if ($MBI->_acmp($x->{_e},$z) >= 0)
3676 $x->{_e} = $MBI->_sub ($x->{_e}, $z);
3677 $x->{_es} = '+' if $MBI->_is_zero($x->{_e});
3681 $x->{_e} = $MBI->_sub ( $MBI->_copy($z), $x->{_e});
3687 $x->{_e} = $MBI->_add ($x->{_e}, $z);
3692 # $x can only be 0Ey if there are no trailing zeros ('0' has 0 trailing
3693 # zeros). So, for something like 0Ey, set y to 1, and -0 => +0
3694 $x->{sign} = '+', $x->{_es} = '+', $x->{_e} = $MBI->_one()
3695 if $MBI->_is_zero($x->{_m});
3698 $x; # MBI bnorm is no-op, so dont call it
3701 ##############################################################################
3705 # return number as hexadecimal string (only for integers defined)
3706 my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
3708 return $x->bstr() if $x->{sign} !~ /^[+-]$/; # inf, nan etc
3709 return '0x0' if $x->is_zero();
3711 return $nan if $x->{_es} ne '+'; # how to do 1e-1 in hex!?
3713 my $z = $MBI->_copy($x->{_m});
3714 if (! $MBI->_is_zero($x->{_e})) # > 0
3716 $MBI->_lsft( $z, $x->{_e},10);
3718 $z = Math::BigInt->new( $x->{sign} . $MBI->_num($z));
3724 # return number as binary digit string (only for integers defined)
3725 my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
3727 return $x->bstr() if $x->{sign} !~ /^[+-]$/; # inf, nan etc
3728 return '0b0' if $x->is_zero();
3730 return $nan if $x->{_es} ne '+'; # how to do 1e-1 in hex!?
3732 my $z = $MBI->_copy($x->{_m});
3733 if (! $MBI->_is_zero($x->{_e})) # > 0
3735 $MBI->_lsft( $z, $x->{_e},10);
3737 $z = Math::BigInt->new( $x->{sign} . $MBI->_num($z));
3743 # return number as octal digit string (only for integers defined)
3744 my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
3746 return $x->bstr() if $x->{sign} !~ /^[+-]$/; # inf, nan etc
3747 return '0' if $x->is_zero();
3749 return $nan if $x->{_es} ne '+'; # how to do 1e-1 in hex!?
3751 my $z = $MBI->_copy($x->{_m});
3752 if (! $MBI->_is_zero($x->{_e})) # > 0
3754 $MBI->_lsft( $z, $x->{_e},10);
3756 $z = Math::BigInt->new( $x->{sign} . $MBI->_num($z));
3762 # return copy as a bigint representation of this BigFloat number
3763 my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
3765 return $x if $x->modify('as_number');
3767 if (!$x->isa('Math::BigFloat'))
3769 # if the object can as_number(), use it
3770 return $x->as_number() if $x->can('as_number');
3771 # otherwise, get us a float and then a number
3772 $x = $x->can('as_float') ? $x->as_float() : $self->new(0+"$x");
3775 return Math::BigInt->binf($x->sign()) if $x->is_inf();
3776 return Math::BigInt->bnan() if $x->is_nan();
3778 my $z = $MBI->_copy($x->{_m});
3779 if ($x->{_es} eq '-') # < 0
3781 $MBI->_rsft( $z, $x->{_e},10);
3783 elsif (! $MBI->_is_zero($x->{_e})) # > 0
3785 $MBI->_lsft( $z, $x->{_e},10);
3787 $z = Math::BigInt->new( $x->{sign} . $MBI->_str($z));
3794 my $class = ref($x) || $x;
3795 $x = $class->new(shift) unless ref($x);
3797 return 1 if $MBI->_is_zero($x->{_m});
3799 my $len = $MBI->_len($x->{_m});
3800 $len += $MBI->_num($x->{_e}) if $x->{_es} eq '+';
3804 $t = $MBI->_num($x->{_e}) if $x->{_es} eq '-';
3815 Math::BigFloat - Arbitrary size floating point math package
3822 my $x = Math::BigFloat->new($str); # defaults to 0
3823 my $y = $x->copy(); # make a true copy
3824 my $nan = Math::BigFloat->bnan(); # create a NotANumber
3825 my $zero = Math::BigFloat->bzero(); # create a +0
3826 my $inf = Math::BigFloat->binf(); # create a +inf
3827 my $inf = Math::BigFloat->binf('-'); # create a -inf
3828 my $one = Math::BigFloat->bone(); # create a +1
3829 my $mone = Math::BigFloat->bone('-'); # create a -1
3831 my $pi = Math::BigFloat->bpi(100); # PI to 100 digits
3833 # the following examples compute their result to 100 digits accuracy:
3834 my $cos = Math::BigFloat->new(1)->bcos(100); # cosinus(1)
3835 my $sin = Math::BigFloat->new(1)->bsin(100); # sinus(1)
3836 my $atan = Math::BigFloat->new(1)->batan(100); # arcus tangens(1)
3838 my $atan2 = Math::BigFloat->new( 1 )->batan2( 1 ,100); # batan(1)
3839 my $atan2 = Math::BigFloat->new( 1 )->batan2( 8 ,100); # batan(1/8)
3840 my $atan2 = Math::BigFloat->new( -2 )->batan2( 1 ,100); # batan(-2)
3843 $x->is_zero(); # true if arg is +0
3844 $x->is_nan(); # true if arg is NaN
3845 $x->is_one(); # true if arg is +1
3846 $x->is_one('-'); # true if arg is -1
3847 $x->is_odd(); # true if odd, false for even
3848 $x->is_even(); # true if even, false for odd
3849 $x->is_pos(); # true if >= 0
3850 $x->is_neg(); # true if < 0
3851 $x->is_inf(sign); # true if +inf, or -inf (default is '+')
3853 $x->bcmp($y); # compare numbers (undef,<0,=0,>0)
3854 $x->bacmp($y); # compare absolutely (undef,<0,=0,>0)
3855 $x->sign(); # return the sign, either +,- or NaN
3856 $x->digit($n); # return the nth digit, counting from right
3857 $x->digit(-$n); # return the nth digit, counting from left
3859 # The following all modify their first argument. If you want to pre-
3860 # serve $x, use $z = $x->copy()->bXXX($y); See under L<CAVEATS> for
3861 # necessary when mixing $a = $b assignments with non-overloaded math.
3864 $x->bzero(); # set $i to 0
3865 $x->bnan(); # set $i to NaN
3866 $x->bone(); # set $x to +1
3867 $x->bone('-'); # set $x to -1
3868 $x->binf(); # set $x to inf
3869 $x->binf('-'); # set $x to -inf
3871 $x->bneg(); # negation
3872 $x->babs(); # absolute value
3873 $x->bnorm(); # normalize (no-op)
3874 $x->bnot(); # two's complement (bit wise not)
3875 $x->binc(); # increment x by 1
3876 $x->bdec(); # decrement x by 1
3878 $x->badd($y); # addition (add $y to $x)
3879 $x->bsub($y); # subtraction (subtract $y from $x)
3880 $x->bmul($y); # multiplication (multiply $x by $y)
3881 $x->bdiv($y); # divide, set $x to quotient
3882 # return (quo,rem) or quo if scalar
3884 $x->bmod($y); # modulus ($x % $y)
3885 $x->bpow($y); # power of arguments ($x ** $y)
3886 $x->bmodpow($exp,$mod); # modular exponentiation (($num**$exp) % $mod))
3887 $x->blsft($y, $n); # left shift by $y places in base $n
3888 $x->brsft($y, $n); # right shift by $y places in base $n
3889 # returns (quo,rem) or quo if in scalar context
3891 $x->blog(); # logarithm of $x to base e (Euler's number)
3892 $x->blog($base); # logarithm of $x to base $base (f.i. 2)
3893 $x->bexp(); # calculate e ** $x where e is Euler's number
3895 $x->band($y); # bit-wise and
3896 $x->bior($y); # bit-wise inclusive or
3897 $x->bxor($y); # bit-wise exclusive or
3898 $x->bnot(); # bit-wise not (two's complement)
3900 $x->bsqrt(); # calculate square-root
3901 $x->broot($y); # $y'th root of $x (e.g. $y == 3 => cubic root)
3902 $x->bfac(); # factorial of $x (1*2*3*4*..$x)
3904 $x->bround($N); # accuracy: preserve $N digits
3905 $x->bfround($N); # precision: round to the $Nth digit
3907 $x->bfloor(); # return integer less or equal than $x
3908 $x->bceil(); # return integer greater or equal than $x
3910 # The following do not modify their arguments:
3912 bgcd(@values); # greatest common divisor
3913 blcm(@values); # lowest common multiplicator
3915 $x->bstr(); # return string
3916 $x->bsstr(); # return string in scientific notation
3918 $x->as_int(); # return $x as BigInt
3919 $x->exponent(); # return exponent as BigInt
3920 $x->mantissa(); # return mantissa as BigInt
3921 $x->parts(); # return (mantissa,exponent) as BigInt
3923 $x->length(); # number of digits (w/o sign and '.')
3924 ($l,$f) = $x->length(); # number of digits, and length of fraction
3926 $x->precision(); # return P of $x (or global, if P of $x undef)
3927 $x->precision($n); # set P of $x to $n
3928 $x->accuracy(); # return A of $x (or global, if A of $x undef)
3929 $x->accuracy($n); # set A $x to $n
3931 # these get/set the appropriate global value for all BigFloat objects
3932 Math::BigFloat->precision(); # Precision
3933 Math::BigFloat->accuracy(); # Accuracy
3934 Math::BigFloat->round_mode(); # rounding mode
3938 All operators (including basic math operations) are overloaded if you
3939 declare your big floating point numbers as
3941 $i = new Math::BigFloat '12_3.456_789_123_456_789E-2';
3943 Operations with overloaded operators preserve the arguments, which is
3944 exactly what you expect.
3946 =head2 Canonical notation
3948 Input to these routines are either BigFloat objects, or strings of the
3949 following four forms:
3963 C</^[+-]\d+E[+-]?\d+$/>
3967 C</^[+-]\d*\.\d+E[+-]?\d+$/>
3971 all with optional leading and trailing zeros and/or spaces. Additionally,
3972 numbers are allowed to have an underscore between any two digits.
3974 Empty strings as well as other illegal numbers results in 'NaN'.
3976 bnorm() on a BigFloat object is now effectively a no-op, since the numbers
3977 are always stored in normalized form. On a string, it creates a BigFloat
3982 Output values are BigFloat objects (normalized), except for bstr() and bsstr().
3984 The string output will always have leading and trailing zeros stripped and drop
3985 a plus sign. C<bstr()> will give you always the form with a decimal point,
3986 while C<bsstr()> (s for scientific) gives you the scientific notation.
3988 Input bstr() bsstr()
3990 ' -123 123 123' '-123123123' '-123123123E0'
3991 '00.0123' '0.0123' '123E-4'
3992 '123.45E-2' '1.2345' '12345E-4'
3993 '10E+3' '10000' '1E4'
3995 Some routines (C<is_odd()>, C<is_even()>, C<is_zero()>, C<is_one()>,
3996 C<is_nan()>) return true or false, while others (C<bcmp()>, C<bacmp()>)
3997 return either undef, <0, 0 or >0 and are suited for sort.
3999 Actual math is done by using the class defined with C<< with => Class; >> (which
4000 defaults to BigInts) to represent the mantissa and exponent.
4002 The sign C</^[+-]$/> is stored separately. The string 'NaN' is used to
4003 represent the result when input arguments are not numbers, as well as
4004 the result of dividing by zero.
4006 =head2 C<mantissa()>, C<exponent()> and C<parts()>
4008 C<mantissa()> and C<exponent()> return the said parts of the BigFloat
4009 as BigInts such that:
4011 $m = $x->mantissa();
4012 $e = $x->exponent();
4013 $y = $m * ( 10 ** $e );
4014 print "ok\n" if $x == $y;
4016 C<< ($m,$e) = $x->parts(); >> is just a shortcut giving you both of them.
4018 A zero is represented and returned as C<0E1>, B<not> C<0E0> (after Knuth).
4020 Currently the mantissa is reduced as much as possible, favouring higher
4021 exponents over lower ones (e.g. returning 1e7 instead of 10e6 or 10000000e0).
4022 This might change in the future, so do not depend on it.
4024 =head2 Accuracy vs. Precision
4026 See also: L<Rounding|Rounding>.
4028 Math::BigFloat supports both precision (rounding to a certain place before or
4029 after the dot) and accuracy (rounding to a certain number of digits). For a
4030 full documentation, examples and tips on these topics please see the large
4031 section about rounding in L<Math::BigInt>.
4033 Since things like C<sqrt(2)> or C<1 / 3> must presented with a limited
4034 accuracy lest a operation consumes all resources, each operation produces
4035 no more than the requested number of digits.
4037 If there is no global precision or accuracy set, B<and> the operation in
4038 question was not called with a requested precision or accuracy, B<and> the
4039 input $x has no accuracy or precision set, then a fallback parameter will
4040 be used. For historical reasons, it is called C<div_scale> and can be accessed
4043 $d = Math::BigFloat->div_scale(); # query
4044 Math::BigFloat->div_scale($n); # set to $n digits
4046 The default value for C<div_scale> is 40.
4048 In case the result of one operation has more digits than specified,
4049 it is rounded. The rounding mode taken is either the default mode, or the one
4050 supplied to the operation after the I<scale>:
4052 $x = Math::BigFloat->new(2);
4053 Math::BigFloat->accuracy(5); # 5 digits max
4054 $y = $x->copy()->bdiv(3); # will give 0.66667
4055 $y = $x->copy()->bdiv(3,6); # will give 0.666667
4056 $y = $x->copy()->bdiv(3,6,undef,'odd'); # will give 0.666667
4057 Math::BigFloat->round_mode('zero');
4058 $y = $x->copy()->bdiv(3,6); # will also give 0.666667
4060 Note that C<< Math::BigFloat->accuracy() >> and C<< Math::BigFloat->precision() >>
4061 set the global variables, and thus B<any> newly created number will be subject
4062 to the global rounding B<immediately>. This means that in the examples above, the
4063 C<3> as argument to C<bdiv()> will also get an accuracy of B<5>.
4065 It is less confusing to either calculate the result fully, and afterwards
4066 round it explicitly, or use the additional parameters to the math
4070 $x = Math::BigFloat->new(2);
4071 $y = $x->copy()->bdiv(3);
4072 print $y->bround(5),"\n"; # will give 0.66667
4077 $x = Math::BigFloat->new(2);
4078 $y = $x->copy()->bdiv(3,5); # will give 0.66667
4085 =item ffround ( +$scale )
4087 Rounds to the $scale'th place left from the '.', counting from the dot.
4088 The first digit is numbered 1.
4090 =item ffround ( -$scale )
4092 Rounds to the $scale'th place right from the '.', counting from the dot.
4096 Rounds to an integer.
4098 =item fround ( +$scale )
4100 Preserves accuracy to $scale digits from the left (aka significant digits)
4101 and pads the rest with zeros. If the number is between 1 and -1, the
4102 significant digits count from the first non-zero after the '.'
4104 =item fround ( -$scale ) and fround ( 0 )
4106 These are effectively no-ops.
4110 All rounding functions take as a second parameter a rounding mode from one of
4111 the following: 'even', 'odd', '+inf', '-inf', 'zero', 'trunc' or 'common'.
4113 The default rounding mode is 'even'. By using
4114 C<< Math::BigFloat->round_mode($round_mode); >> you can get and set the default
4115 mode for subsequent rounding. The usage of C<$Math::BigFloat::$round_mode> is
4116 no longer supported.
4117 The second parameter to the round functions then overrides the default
4120 The C<as_number()> function returns a BigInt from a Math::BigFloat. It uses
4121 'trunc' as rounding mode to make it equivalent to:
4126 You can override this by passing the desired rounding mode as parameter to
4129 $x = Math::BigFloat->new(2.5);
4130 $y = $x->as_number('odd'); # $y = 3
4134 Math::BigFloat supports all methods that Math::BigInt supports, except it
4135 calculates non-integer results when possible. Please see L<Math::BigInt>
4136 for a full description of each method. Below are just the most important
4141 $x->accuracy(5); # local for $x
4142 CLASS->accuracy(5); # global for all members of CLASS
4143 # Note: This also applies to new()!
4145 $A = $x->accuracy(); # read out accuracy that affects $x
4146 $A = CLASS->accuracy(); # read out global accuracy
4148 Set or get the global or local accuracy, aka how many significant digits the
4149 results have. If you set a global accuracy, then this also applies to new()!
4151 Warning! The accuracy I<sticks>, e.g. once you created a number under the
4152 influence of C<< CLASS->accuracy($A) >>, all results from math operations with
4153 that number will also be rounded.
4155 In most cases, you should probably round the results explicitly using one of
4156 L<round()>, L<bround()> or L<bfround()> or by passing the desired accuracy
4157 to the math operation as additional parameter:
4159 my $x = Math::BigInt->new(30000);
4160 my $y = Math::BigInt->new(7);
4161 print scalar $x->copy()->bdiv($y, 2); # print 4300
4162 print scalar $x->copy()->bdiv($y)->bround(2); # print 4300
4166 $x->precision(-2); # local for $x, round at the second
4167 # digit right of the dot
4168 $x->precision(2); # ditto, round at the second digit left
4171 CLASS->precision(5); # Global for all members of CLASS
4172 # This also applies to new()!
4173 CLASS->precision(-5); # ditto
4175 $P = CLASS->precision(); # read out global precision
4176 $P = $x->precision(); # read out precision that affects $x
4178 Note: You probably want to use L<accuracy()> instead. With L<accuracy> you
4179 set the number of digits each result should have, with L<precision> you
4180 set the place where to round!
4184 $x->bexp($accuracy); # calculate e ** X
4186 Calculates the expression C<e ** $x> where C<e> is Euler's number.
4188 This method was added in v1.82 of Math::BigInt (April 2007).
4192 $x->bnok($y); # x over y (binomial coefficient n over k)
4194 Calculates the binomial coefficient n over k, also called the "choose"
4195 function. The result is equivalent to:
4201 This method was added in v1.84 of Math::BigInt (April 2007).
4205 print Math::BigFloat->bpi(100), "\n";
4207 Calculate PI to N digits (including the 3 before the dot). The result is
4208 rounded according to the current rounding mode, which defaults to "even".
4210 This method was added in v1.87 of Math::BigInt (June 2007).
4214 my $x = Math::BigFloat->new(1);
4215 print $x->bcos(100), "\n";
4217 Calculate the cosinus of $x, modifying $x in place.
4219 This method was added in v1.87 of Math::BigInt (June 2007).
4223 my $x = Math::BigFloat->new(1);
4224 print $x->bsin(100), "\n";
4226 Calculate the sinus of $x, modifying $x in place.
4228 This method was added in v1.87 of Math::BigInt (June 2007).
4232 my $y = Math::BigFloat->new(2);
4233 my $x = Math::BigFloat->new(3);
4234 print $y->batan2($x), "\n";
4236 Calculate the arcus tanges of C<$y> divided by C<$x>, modifying $y in place.
4237 See also L<batan()>.
4239 This method was added in v1.87 of Math::BigInt (June 2007).
4243 my $x = Math::BigFloat->new(1);
4244 print $x->batan(100), "\n";
4246 Calculate the arcus tanges of $x, modifying $x in place. See also L<batan2()>.
4248 This method was added in v1.87 of Math::BigInt (June 2007).
4254 Multiply $x by $y, and then add $z to the result.
4256 This method was added in v1.87 of Math::BigInt (June 2007).
4258 =head1 Autocreating constants
4260 After C<use Math::BigFloat ':constant'> all the floating point constants
4261 in the given scope are converted to C<Math::BigFloat>. This conversion
4262 happens at compile time.
4266 perl -MMath::BigFloat=:constant -e 'print 2E-100,"\n"'
4268 prints the value of C<2E-100>. Note that without conversion of
4269 constants the expression 2E-100 will be calculated as normal floating point
4272 Please note that ':constant' does not affect integer constants, nor binary
4273 nor hexadecimal constants. Use L<bignum> or L<Math::BigInt> to get this to
4278 Math with the numbers is done (by default) by a module called
4279 Math::BigInt::Calc. This is equivalent to saying:
4281 use Math::BigFloat lib => 'Calc';
4283 You can change this by using:
4285 use Math::BigFloat lib => 'GMP';
4287 B<Note>: General purpose packages should not be explicit about the library
4288 to use; let the script author decide which is best.
4290 Note: The keyword 'lib' will warn when the requested library could not be
4291 loaded. To suppress the warning use 'try' instead:
4293 use Math::BigFloat try => 'GMP';
4295 If your script works with huge numbers and Calc is too slow for them,
4296 you can also for the loading of one of these libraries and if none
4297 of them can be used, the code will die:
4299 use Math::BigFloat only => 'GMP,Pari';
4301 The following would first try to find Math::BigInt::Foo, then
4302 Math::BigInt::Bar, and when this also fails, revert to Math::BigInt::Calc: