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 do not 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
1506 $self->_log($x,$scale); # need to do the "normal" way
1507 $x->badd($l_10) if defined $l_10; # correct it by ln(10)
1508 $x->badd($l_2) if defined $l_2; # and maybe by ln(2)
1510 # all done, $x contains now the result
1516 # (BFLOAT or num_str, BFLOAT or num_str) return BFLOAT
1517 # does not modify arguments, but returns new object
1518 # Lowest Common Multiplicator
1520 my ($self,@arg) = objectify(0,@_);
1521 my $x = $self->new(shift @arg);
1522 while (@arg) { $x = Math::BigInt::__lcm($x,shift @arg); }
1528 # (BINT or num_str, BINT or num_str) return BINT
1529 # does not modify arguments, but returns new object
1532 $y = __PACKAGE__->new($y) if !ref($y);
1534 my $x = $y->copy()->babs(); # keep arguments
1536 return $x->bnan() if $x->{sign} !~ /^[+-]$/ # x NaN?
1537 || !$x->is_int(); # only for integers now
1541 my $t = shift; $t = $self->new($t) if !ref($t);
1542 $y = $t->copy()->babs();
1544 return $x->bnan() if $y->{sign} !~ /^[+-]$/ # y NaN?
1545 || !$y->is_int(); # only for integers now
1547 # greatest common divisor
1548 while (! $y->is_zero())
1550 ($x,$y) = ($y->copy(), $x->copy()->bmod($y));
1553 last if $x->is_one();
1558 ##############################################################################
1562 # Internal helper sub to take two positive integers and their signs and
1563 # then add them. Input ($CALC,$CALC,('+'|'-'),('+'|'-')),
1564 # output ($CALC,('+'|'-'))
1565 my ($x,$y,$xs,$ys) = @_;
1567 # if the signs are equal we can add them (-5 + -3 => -(5 + 3) => -8)
1570 $x = $MBI->_add ($x, $y ); # a+b
1571 # the sign follows $xs
1575 my $a = $MBI->_acmp($x,$y);
1578 $x = $MBI->_sub ($x , $y); # abs sub
1582 $x = $MBI->_zero(); # result is 0
1587 $x = $MBI->_sub ( $y, $x, 1 ); # abs sub
1595 # Internal helper sub to take two positive integers and their signs and
1596 # then subtract them. Input ($CALC,$CALC,('+'|'-'),('+'|'-')),
1597 # output ($CALC,('+'|'-'))
1598 my ($x,$y,$xs,$ys) = @_;
1602 _e_add($x,$y,$xs,$ys); # call add (does subtract now)
1605 ###############################################################################
1606 # is_foo methods (is_negative, is_positive are inherited from BigInt)
1610 # return true if arg (BFLOAT or num_str) is an integer
1611 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1613 (($x->{sign} =~ /^[+-]$/) && # NaN and +-inf aren't
1614 ($x->{_es} eq '+')) ? 1 : 0; # 1e-1 => no integer
1619 # return true if arg (BFLOAT or num_str) is zero
1620 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1622 ($x->{sign} eq '+' && $MBI->_is_zero($x->{_m})) ? 1 : 0;
1627 # return true if arg (BFLOAT or num_str) is +1 or -1 if signis given
1628 my ($self,$x,$sign) = ref($_[0]) ? (undef,@_) : objectify(1,@_);
1630 $sign = '+' if !defined $sign || $sign ne '-';
1632 ($x->{sign} eq $sign &&
1633 $MBI->_is_zero($x->{_e}) &&
1634 $MBI->_is_one($x->{_m}) ) ? 1 : 0;
1639 # return true if arg (BFLOAT or num_str) is odd or false if even
1640 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1642 (($x->{sign} =~ /^[+-]$/) && # NaN & +-inf aren't
1643 ($MBI->_is_zero($x->{_e})) &&
1644 ($MBI->_is_odd($x->{_m}))) ? 1 : 0;
1649 # return true if arg (BINT or num_str) is even or false if odd
1650 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1652 (($x->{sign} =~ /^[+-]$/) && # NaN & +-inf aren't
1653 ($x->{_es} eq '+') && # 123.45 isn't
1654 ($MBI->_is_even($x->{_m}))) ? 1 : 0; # but 1200 is
1659 # multiply two numbers
1662 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1663 # objectify is costly, so avoid it
1664 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1666 ($self,$x,$y,@r) = objectify(2,@_);
1669 return $x if $x->modify('bmul');
1671 return $x->bnan() if (($x->{sign} eq $nan) || ($y->{sign} eq $nan));
1674 if (($x->{sign} =~ /^[+-]inf$/) || ($y->{sign} =~ /^[+-]inf$/))
1676 return $x->bnan() if $x->is_zero() || $y->is_zero();
1677 # result will always be +-inf:
1678 # +inf * +/+inf => +inf, -inf * -/-inf => +inf
1679 # +inf * -/-inf => -inf, -inf * +/+inf => -inf
1680 return $x->binf() if ($x->{sign} =~ /^\+/ && $y->{sign} =~ /^\+/);
1681 return $x->binf() if ($x->{sign} =~ /^-/ && $y->{sign} =~ /^-/);
1682 return $x->binf('-');
1685 return $upgrade->bmul($x,$y,@r) if defined $upgrade &&
1686 ((!$x->isa($self)) || (!$y->isa($self)));
1688 # aEb * cEd = (a*c)E(b+d)
1689 $MBI->_mul($x->{_m},$y->{_m});
1690 ($x->{_e}, $x->{_es}) = _e_add($x->{_e}, $y->{_e}, $x->{_es}, $y->{_es});
1692 $r[3] = $y; # no push!
1695 $x->{sign} = $x->{sign} ne $y->{sign} ? '-' : '+';
1696 $x->bnorm->round(@r);
1701 # multiply two numbers and add the third to the result
1704 my ($self,$x,$y,$z,@r) = objectify(3,@_);
1706 return $x if $x->modify('bmuladd');
1708 return $x->bnan() if (($x->{sign} eq $nan) ||
1709 ($y->{sign} eq $nan) ||
1710 ($z->{sign} eq $nan));
1713 if (($x->{sign} =~ /^[+-]inf$/) || ($y->{sign} =~ /^[+-]inf$/))
1715 return $x->bnan() if $x->is_zero() || $y->is_zero();
1716 # result will always be +-inf:
1717 # +inf * +/+inf => +inf, -inf * -/-inf => +inf
1718 # +inf * -/-inf => -inf, -inf * +/+inf => -inf
1719 return $x->binf() if ($x->{sign} =~ /^\+/ && $y->{sign} =~ /^\+/);
1720 return $x->binf() if ($x->{sign} =~ /^-/ && $y->{sign} =~ /^-/);
1721 return $x->binf('-');
1724 return $upgrade->bmul($x,$y,@r) if defined $upgrade &&
1725 ((!$x->isa($self)) || (!$y->isa($self)));
1727 # aEb * cEd = (a*c)E(b+d)
1728 $MBI->_mul($x->{_m},$y->{_m});
1729 ($x->{_e}, $x->{_es}) = _e_add($x->{_e}, $y->{_e}, $x->{_es}, $y->{_es});
1731 $r[3] = $y; # no push!
1734 $x->{sign} = $x->{sign} ne $y->{sign} ? '-' : '+';
1736 # z=inf handling (z=NaN handled above)
1737 $x->{sign} = $z->{sign}, return $x if $z->{sign} =~ /^[+-]inf$/;
1739 # take lower of the two e's and adapt m1 to it to match m2
1741 $e = $MBI->_zero() if !defined $e; # if no BFLOAT?
1742 $e = $MBI->_copy($e); # make copy (didn't do it yet)
1746 ($e,$es) = _e_sub($e, $x->{_e}, $z->{_es} || '+', $x->{_es});
1748 my $add = $MBI->_copy($z->{_m});
1750 if ($es eq '-') # < 0
1752 $MBI->_lsft( $x->{_m}, $e, 10);
1753 ($x->{_e},$x->{_es}) = _e_add($x->{_e}, $e, $x->{_es}, $es);
1755 elsif (!$MBI->_is_zero($e)) # > 0
1757 $MBI->_lsft($add, $e, 10);
1759 # else: both e are the same, so just leave them
1761 if ($x->{sign} eq $z->{sign})
1764 $x->{_m} = $MBI->_add($x->{_m}, $add);
1768 ($x->{_m}, $x->{sign}) =
1769 _e_add($x->{_m}, $add, $x->{sign}, $z->{sign});
1772 # delete trailing zeros, then round
1773 $x->bnorm()->round(@r);
1778 # (dividend: BFLOAT or num_str, divisor: BFLOAT or num_str) return
1779 # (BFLOAT,BFLOAT) (quo,rem) or BFLOAT (only rem)
1782 my ($self,$x,$y,$a,$p,$r) = (ref($_[0]),@_);
1783 # objectify is costly, so avoid it
1784 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1786 ($self,$x,$y,$a,$p,$r) = objectify(2,@_);
1789 return $x if $x->modify('bdiv');
1791 return $self->_div_inf($x,$y)
1792 if (($x->{sign} !~ /^[+-]$/) || ($y->{sign} !~ /^[+-]$/) || $y->is_zero());
1794 # x== 0 # also: or y == 1 or y == -1
1795 return wantarray ? ($x,$self->bzero()) : $x if $x->is_zero();
1798 return $upgrade->bdiv($upgrade->new($x),$y,$a,$p,$r) if defined $upgrade;
1800 # we need to limit the accuracy to protect against overflow
1802 my (@params,$scale);
1803 ($x,@params) = $x->_find_round_parameters($a,$p,$r,$y);
1805 return $x if $x->is_nan(); # error in _find_round_parameters?
1807 # no rounding at all, so must use fallback
1808 if (scalar @params == 0)
1810 # simulate old behaviour
1811 $params[0] = $self->div_scale(); # and round to it as accuracy
1812 $scale = $params[0]+4; # at least four more for proper round
1813 $params[2] = $r; # round mode by caller or undef
1814 $fallback = 1; # to clear a/p afterwards
1818 # the 4 below is empirical, and there might be cases where it is not
1820 $scale = abs($params[0] || $params[1]) + 4; # take whatever is defined
1823 my $rem; $rem = $self->bzero() if wantarray;
1825 $y = $self->new($y) unless $y->isa('Math::BigFloat');
1827 my $lx = $MBI->_len($x->{_m}); my $ly = $MBI->_len($y->{_m});
1828 $scale = $lx if $lx > $scale;
1829 $scale = $ly if $ly > $scale;
1830 my $diff = $ly - $lx;
1831 $scale += $diff if $diff > 0; # if lx << ly, but not if ly << lx!
1833 # already handled inf/NaN/-inf above:
1835 # check that $y is not 1 nor -1 and cache the result:
1836 my $y_not_one = !($MBI->_is_zero($y->{_e}) && $MBI->_is_one($y->{_m}));
1838 # flipping the sign of $y will also flip the sign of $x for the special
1839 # case of $x->bsub($x); so we can catch it below:
1840 my $xsign = $x->{sign};
1841 $y->{sign} =~ tr/+-/-+/;
1843 if ($xsign ne $x->{sign})
1845 # special case of $x /= $x results in 1
1846 $x->bone(); # "fixes" also sign of $y, since $x is $y
1850 # correct $y's sign again
1851 $y->{sign} =~ tr/+-/-+/;
1852 # continue with normal div code:
1854 # make copy of $x in case of list context for later remainder calculation
1855 if (wantarray && $y_not_one)
1860 $x->{sign} = $x->{sign} ne $y->sign() ? '-' : '+';
1862 # check for / +-1 ( +/- 1E0)
1865 # promote BigInts and it's subclasses (except when already a BigFloat)
1866 $y = $self->new($y) unless $y->isa('Math::BigFloat');
1868 # calculate the result to $scale digits and then round it
1869 # a * 10 ** b / c * 10 ** d => a/c * 10 ** (b-d)
1870 $MBI->_lsft($x->{_m},$MBI->_new($scale),10);
1871 $MBI->_div ($x->{_m},$y->{_m}); # a/c
1873 # correct exponent of $x
1874 ($x->{_e},$x->{_es}) = _e_sub($x->{_e}, $y->{_e}, $x->{_es}, $y->{_es});
1875 # correct for 10**scale
1876 ($x->{_e},$x->{_es}) = _e_sub($x->{_e}, $MBI->_new($scale), $x->{_es}, '+');
1877 $x->bnorm(); # remove trailing 0's
1879 } # end else $x != $y
1881 # shortcut to not run through _find_round_parameters again
1882 if (defined $params[0])
1884 delete $x->{_a}; # clear before round
1885 $x->bround($params[0],$params[2]); # then round accordingly
1889 delete $x->{_p}; # clear before round
1890 $x->bfround($params[1],$params[2]); # then round accordingly
1894 # clear a/p after round, since user did not request it
1895 delete $x->{_a}; delete $x->{_p};
1902 $rem->bmod($y,@params); # copy already done
1906 # clear a/p after round, since user did not request it
1907 delete $rem->{_a}; delete $rem->{_p};
1916 # (dividend: BFLOAT or num_str, divisor: BFLOAT or num_str) return remainder
1919 my ($self,$x,$y,$a,$p,$r) = (ref($_[0]),@_);
1920 # objectify is costly, so avoid it
1921 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1923 ($self,$x,$y,$a,$p,$r) = objectify(2,@_);
1926 return $x if $x->modify('bmod');
1928 # handle NaN, inf, -inf
1929 if (($x->{sign} !~ /^[+-]$/) || ($y->{sign} !~ /^[+-]$/))
1931 my ($d,$re) = $self->SUPER::_div_inf($x,$y);
1932 $x->{sign} = $re->{sign};
1933 $x->{_e} = $re->{_e};
1934 $x->{_m} = $re->{_m};
1935 return $x->round($a,$p,$r,$y);
1939 return $x->bnan() if $x->is_zero();
1943 return $x->bzero() if $x->is_zero()
1945 # check that $y == +1 or $y == -1:
1946 ($MBI->_is_zero($y->{_e}) && $MBI->_is_one($y->{_m})));
1948 my $cmp = $x->bacmp($y); # equal or $x < $y?
1949 return $x->bzero($a,$p) if $cmp == 0; # $x == $y => result 0
1951 # only $y of the operands negative?
1952 my $neg = 0; $neg = 1 if $x->{sign} ne $y->{sign};
1954 $x->{sign} = $y->{sign}; # calc sign first
1955 return $x->round($a,$p,$r) if $cmp < 0 && $neg == 0; # $x < $y => result $x
1957 my $ym = $MBI->_copy($y->{_m});
1960 $MBI->_lsft( $ym, $y->{_e}, 10)
1961 if $y->{_es} eq '+' && !$MBI->_is_zero($y->{_e});
1963 # if $y has digits after dot
1964 my $shifty = 0; # correct _e of $x by this
1965 if ($y->{_es} eq '-') # has digits after dot
1967 # 123 % 2.5 => 1230 % 25 => 5 => 0.5
1968 $shifty = $MBI->_num($y->{_e}); # no more digits after dot
1969 $MBI->_lsft($x->{_m}, $y->{_e}, 10);# 123 => 1230, $y->{_m} is already 25
1971 # $ym is now mantissa of $y based on exponent 0
1973 my $shiftx = 0; # correct _e of $x by this
1974 if ($x->{_es} eq '-') # has digits after dot
1976 # 123.4 % 20 => 1234 % 200
1977 $shiftx = $MBI->_num($x->{_e}); # no more digits after dot
1978 $MBI->_lsft($ym, $x->{_e}, 10); # 123 => 1230
1980 # 123e1 % 20 => 1230 % 20
1981 if ($x->{_es} eq '+' && !$MBI->_is_zero($x->{_e}))
1983 $MBI->_lsft( $x->{_m}, $x->{_e},10); # es => '+' here
1986 $x->{_e} = $MBI->_new($shiftx);
1988 $x->{_es} = '-' if $shiftx != 0 || $shifty != 0;
1989 $MBI->_add( $x->{_e}, $MBI->_new($shifty)) if $shifty != 0;
1991 # now mantissas are equalized, exponent of $x is adjusted, so calc result
1993 $x->{_m} = $MBI->_mod( $x->{_m}, $ym);
1995 $x->{sign} = '+' if $MBI->_is_zero($x->{_m}); # fix sign for -0
1998 if ($neg != 0) # one of them negative => correct in place
2001 $x->{_m} = $r->{_m};
2002 $x->{_e} = $r->{_e};
2003 $x->{_es} = $r->{_es};
2004 $x->{sign} = '+' if $MBI->_is_zero($x->{_m}); # fix sign for -0
2008 $x->round($a,$p,$r,$y); # round and return
2013 # calculate $y'th root of $x
2016 my ($self,$x,$y,$a,$p,$r) = (ref($_[0]),@_);
2017 # objectify is costly, so avoid it
2018 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
2020 ($self,$x,$y,$a,$p,$r) = objectify(2,@_);
2023 return $x if $x->modify('broot');
2025 # NaN handling: $x ** 1/0, x or y NaN, or y inf/-inf or y == 0
2026 return $x->bnan() if $x->{sign} !~ /^\+/ || $y->is_zero() ||
2027 $y->{sign} !~ /^\+$/;
2029 return $x if $x->is_zero() || $x->is_one() || $x->is_inf() || $y->is_one();
2031 # we need to limit the accuracy to protect against overflow
2033 my (@params,$scale);
2034 ($x,@params) = $x->_find_round_parameters($a,$p,$r);
2036 return $x if $x->is_nan(); # error in _find_round_parameters?
2038 # no rounding at all, so must use fallback
2039 if (scalar @params == 0)
2041 # simulate old behaviour
2042 $params[0] = $self->div_scale(); # and round to it as accuracy
2043 $scale = $params[0]+4; # at least four more for proper round
2044 $params[2] = $r; # round mode by caller or undef
2045 $fallback = 1; # to clear a/p afterwards
2049 # the 4 below is empirical, and there might be cases where it is not
2051 $scale = abs($params[0] || $params[1]) + 4; # take whatever is defined
2054 # when user set globals, they would interfere with our calculation, so
2055 # disable them and later re-enable them
2057 my $abr = "$self\::accuracy"; my $ab = $$abr; $$abr = undef;
2058 my $pbr = "$self\::precision"; my $pb = $$pbr; $$pbr = undef;
2059 # we also need to disable any set A or P on $x (_find_round_parameters took
2060 # them already into account), since these would interfere, too
2061 delete $x->{_a}; delete $x->{_p};
2062 # need to disable $upgrade in BigInt, to avoid deep recursion
2063 local $Math::BigInt::upgrade = undef; # should be really parent class vs MBI
2065 # remember sign and make $x positive, since -4 ** (1/2) => -2
2066 my $sign = 0; $sign = 1 if $x->{sign} eq '-'; $x->{sign} = '+';
2069 if ($y->isa('Math::BigFloat'))
2071 $is_two = ($y->{sign} eq '+' && $MBI->_is_two($y->{_m}) && $MBI->_is_zero($y->{_e}));
2075 $is_two = ($y == 2);
2078 # normal square root if $y == 2:
2081 $x->bsqrt($scale+4);
2083 elsif ($y->is_one('-'))
2086 my $u = $self->bone()->bdiv($x,$scale);
2087 # copy private parts over
2088 $x->{_m} = $u->{_m};
2089 $x->{_e} = $u->{_e};
2090 $x->{_es} = $u->{_es};
2094 # calculate the broot() as integer result first, and if it fits, return
2095 # it rightaway (but only if $x and $y are integer):
2097 my $done = 0; # not yet
2098 if ($y->is_int() && $x->is_int())
2100 my $i = $MBI->_copy( $x->{_m} );
2101 $MBI->_lsft( $i, $x->{_e}, 10 ) unless $MBI->_is_zero($x->{_e});
2102 my $int = Math::BigInt->bzero();
2104 $int->broot($y->as_number());
2106 if ($int->copy()->bpow($y) == $x)
2108 # found result, return it
2109 $x->{_m} = $int->{value};
2110 $x->{_e} = $MBI->_zero();
2118 my $u = $self->bone()->bdiv($y,$scale+4);
2119 delete $u->{_a}; delete $u->{_p}; # otherwise it conflicts
2120 $x->bpow($u,$scale+4); # el cheapo
2123 $x->bneg() if $sign == 1;
2125 # shortcut to not run through _find_round_parameters again
2126 if (defined $params[0])
2128 $x->bround($params[0],$params[2]); # then round accordingly
2132 $x->bfround($params[1],$params[2]); # then round accordingly
2136 # clear a/p after round, since user did not request it
2137 delete $x->{_a}; delete $x->{_p};
2140 $$abr = $ab; $$pbr = $pb;
2146 # calculate square root
2147 my ($self,$x,$a,$p,$r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_);
2149 return $x if $x->modify('bsqrt');
2151 return $x->bnan() if $x->{sign} !~ /^[+]/; # NaN, -inf or < 0
2152 return $x if $x->{sign} eq '+inf'; # sqrt(inf) == inf
2153 return $x->round($a,$p,$r) if $x->is_zero() || $x->is_one();
2155 # we need to limit the accuracy to protect against overflow
2157 my (@params,$scale);
2158 ($x,@params) = $x->_find_round_parameters($a,$p,$r);
2160 return $x if $x->is_nan(); # error in _find_round_parameters?
2162 # no rounding at all, so must use fallback
2163 if (scalar @params == 0)
2165 # simulate old behaviour
2166 $params[0] = $self->div_scale(); # and round to it as accuracy
2167 $scale = $params[0]+4; # at least four more for proper round
2168 $params[2] = $r; # round mode by caller or undef
2169 $fallback = 1; # to clear a/p afterwards
2173 # the 4 below is empirical, and there might be cases where it is not
2175 $scale = abs($params[0] || $params[1]) + 4; # take whatever is defined
2178 # when user set globals, they would interfere with our calculation, so
2179 # disable them and later re-enable them
2181 my $abr = "$self\::accuracy"; my $ab = $$abr; $$abr = undef;
2182 my $pbr = "$self\::precision"; my $pb = $$pbr; $$pbr = undef;
2183 # we also need to disable any set A or P on $x (_find_round_parameters took
2184 # them already into account), since these would interfere, too
2185 delete $x->{_a}; delete $x->{_p};
2186 # need to disable $upgrade in BigInt, to avoid deep recursion
2187 local $Math::BigInt::upgrade = undef; # should be really parent class vs MBI
2189 my $i = $MBI->_copy( $x->{_m} );
2190 $MBI->_lsft( $i, $x->{_e}, 10 ) unless $MBI->_is_zero($x->{_e});
2191 my $xas = Math::BigInt->bzero();
2194 my $gs = $xas->copy()->bsqrt(); # some guess
2196 if (($x->{_es} ne '-') # guess can't be accurate if there are
2197 # digits after the dot
2198 && ($xas->bacmp($gs * $gs) == 0)) # guess hit the nail on the head?
2200 # exact result, copy result over to keep $x
2201 $x->{_m} = $gs->{value}; $x->{_e} = $MBI->_zero(); $x->{_es} = '+';
2203 # shortcut to not run through _find_round_parameters again
2204 if (defined $params[0])
2206 $x->bround($params[0],$params[2]); # then round accordingly
2210 $x->bfround($params[1],$params[2]); # then round accordingly
2214 # clear a/p after round, since user did not request it
2215 delete $x->{_a}; delete $x->{_p};
2217 # re-enable A and P, upgrade is taken care of by "local"
2218 ${"$self\::accuracy"} = $ab; ${"$self\::precision"} = $pb;
2222 # sqrt(2) = 1.4 because sqrt(2*100) = 1.4*10; so we can increase the accuracy
2223 # of the result by multiplying the input by 100 and then divide the integer
2224 # result of sqrt(input) by 10. Rounding afterwards returns the real result.
2226 # The following steps will transform 123.456 (in $x) into 123456 (in $y1)
2227 my $y1 = $MBI->_copy($x->{_m});
2229 my $length = $MBI->_len($y1);
2231 # Now calculate how many digits the result of sqrt(y1) would have
2232 my $digits = int($length / 2);
2234 # But we need at least $scale digits, so calculate how many are missing
2235 my $shift = $scale - $digits;
2237 # This happens if the input had enough digits
2238 # (we take care of integer guesses above)
2239 $shift = 0 if $shift < 0;
2241 # Multiply in steps of 100, by shifting left two times the "missing" digits
2242 my $s2 = $shift * 2;
2244 # We now make sure that $y1 has the same odd or even number of digits than
2245 # $x had. So when _e of $x is odd, we must shift $y1 by one digit left,
2246 # because we always must multiply by steps of 100 (sqrt(100) is 10) and not
2247 # steps of 10. The length of $x does not count, since an even or odd number
2248 # of digits before the dot is not changed by adding an even number of digits
2249 # after the dot (the result is still odd or even digits long).
2250 $s2++ if $MBI->_is_odd($x->{_e});
2252 $MBI->_lsft( $y1, $MBI->_new($s2), 10);
2254 # now take the square root and truncate to integer
2255 $y1 = $MBI->_sqrt($y1);
2257 # By "shifting" $y1 right (by creating a negative _e) we calculate the final
2258 # result, which is than later rounded to the desired scale.
2260 # calculate how many zeros $x had after the '.' (or before it, depending
2261 # on sign of $dat, the result should have half as many:
2262 my $dat = $MBI->_num($x->{_e});
2263 $dat = -$dat if $x->{_es} eq '-';
2268 # no zeros after the dot (e.g. 1.23, 0.49 etc)
2269 # preserve half as many digits before the dot than the input had
2270 # (but round this "up")
2271 $dat = int(($dat+1)/2);
2275 $dat = int(($dat)/2);
2277 $dat -= $MBI->_len($y1);
2281 $x->{_e} = $MBI->_new( $dat );
2286 $x->{_e} = $MBI->_new( $dat );
2292 # shortcut to not run through _find_round_parameters again
2293 if (defined $params[0])
2295 $x->bround($params[0],$params[2]); # then round accordingly
2299 $x->bfround($params[1],$params[2]); # then round accordingly
2303 # clear a/p after round, since user did not request it
2304 delete $x->{_a}; delete $x->{_p};
2307 $$abr = $ab; $$pbr = $pb;
2313 # (BFLOAT or num_str, BFLOAT or num_str) return BFLOAT
2314 # compute factorial number, modifies first argument
2317 my ($self,$x,@r) = (ref($_[0]),@_);
2318 # objectify is costly, so avoid it
2319 ($self,$x,@r) = objectify(1,@_) if !ref($x);
2322 return $x if $x->modify('bfac') || $x->{sign} eq '+inf';
2325 if (($x->{sign} ne '+') || # inf, NaN, <0 etc => NaN
2326 ($x->{_es} ne '+')); # digits after dot?
2328 # use BigInt's bfac() for faster calc
2329 if (! $MBI->_is_zero($x->{_e}))
2331 $MBI->_lsft($x->{_m}, $x->{_e},10); # change 12e1 to 120e0
2332 $x->{_e} = $MBI->_zero(); # normalize
2335 $MBI->_fac($x->{_m}); # calculate factorial
2336 $x->bnorm()->round(@r); # norm again and round result
2341 # Calculate a power where $y is a non-integer, like 2 ** 0.3
2345 # if $y == 0.5, it is sqrt($x)
2346 $HALF = $self->new($HALF) unless ref($HALF);
2347 return $x->bsqrt(@r,$y) if $y->bcmp($HALF) == 0;
2350 # a ** x == e ** (x * ln a)
2354 # Taylor: | u u^2 u^3 |
2355 # x ** y = 1 + | --- + --- + ----- + ... |
2358 # we need to limit the accuracy to protect against overflow
2360 my ($scale,@params);
2361 ($x,@params) = $x->_find_round_parameters(@r);
2363 return $x if $x->is_nan(); # error in _find_round_parameters?
2365 # no rounding at all, so must use fallback
2366 if (scalar @params == 0)
2368 # simulate old behaviour
2369 $params[0] = $self->div_scale(); # and round to it as accuracy
2370 $params[1] = undef; # disable P
2371 $scale = $params[0]+4; # at least four more for proper round
2372 $params[2] = $r[2]; # round mode by caller or undef
2373 $fallback = 1; # to clear a/p afterwards
2377 # the 4 below is empirical, and there might be cases where it is not
2379 $scale = abs($params[0] || $params[1]) + 4; # take whatever is defined
2382 # when user set globals, they would interfere with our calculation, so
2383 # disable them and later re-enable them
2385 my $abr = "$self\::accuracy"; my $ab = $$abr; $$abr = undef;
2386 my $pbr = "$self\::precision"; my $pb = $$pbr; $$pbr = undef;
2387 # we also need to disable any set A or P on $x (_find_round_parameters took
2388 # them already into account), since these would interfere, too
2389 delete $x->{_a}; delete $x->{_p};
2390 # need to disable $upgrade in BigInt, to avoid deep recursion
2391 local $Math::BigInt::upgrade = undef;
2393 my ($limit,$v,$u,$below,$factor,$next,$over);
2395 $u = $x->copy()->blog(undef,$scale)->bmul($y);
2396 $v = $self->bone(); # 1
2397 $factor = $self->new(2); # 2
2398 $x->bone(); # first term: 1
2400 $below = $v->copy();
2403 $limit = $self->new("1E-". ($scale-1));
2407 # we calculate the next term, and add it to the last
2408 # when the next term is below our limit, it won't affect the outcome
2409 # anymore, so we stop:
2410 $next = $over->copy()->bdiv($below,$scale);
2411 last if $next->bacmp($limit) <= 0;
2413 # calculate things for the next term
2414 $over *= $u; $below *= $factor; $factor->binc();
2416 last if $x->{sign} !~ /^[-+]$/;
2421 # shortcut to not run through _find_round_parameters again
2422 if (defined $params[0])
2424 $x->bround($params[0],$params[2]); # then round accordingly
2428 $x->bfround($params[1],$params[2]); # then round accordingly
2432 # clear a/p after round, since user did not request it
2433 delete $x->{_a}; delete $x->{_p};
2436 $$abr = $ab; $$pbr = $pb;
2442 # (BFLOAT or num_str, BFLOAT or num_str) return BFLOAT
2443 # compute power of two numbers, second arg is used as integer
2444 # modifies first argument
2447 my ($self,$x,$y,$a,$p,$r) = (ref($_[0]),@_);
2448 # objectify is costly, so avoid it
2449 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
2451 ($self,$x,$y,$a,$p,$r) = objectify(2,@_);
2454 return $x if $x->modify('bpow');
2456 return $x->bnan() if $x->{sign} eq $nan || $y->{sign} eq $nan;
2457 return $x if $x->{sign} =~ /^[+-]inf$/;
2459 # cache the result of is_zero
2460 my $y_is_zero = $y->is_zero();
2461 return $x->bone() if $y_is_zero;
2462 return $x if $x->is_one() || $y->is_one();
2464 my $x_is_zero = $x->is_zero();
2465 return $x->_pow($y,$a,$p,$r) if !$x_is_zero && !$y->is_int(); # non-integer power
2467 my $y1 = $y->as_number()->{value}; # make MBI part
2470 if ($x->{sign} eq '-' && $MBI->_is_one($x->{_m}) && $MBI->_is_zero($x->{_e}))
2472 # if $x == -1 and odd/even y => +1/-1 because +-1 ^ (+-1) => +-1
2473 return $MBI->_is_odd($y1) ? $x : $x->babs(1);
2477 return $x if $y->{sign} eq '+'; # 0**y => 0 (if not y <= 0)
2478 # 0 ** -y => 1 / (0 ** y) => 1 / 0! (1 / 0 => +inf)
2483 $new_sign = $MBI->_is_odd($y1) ? '-' : '+' if $x->{sign} ne '+';
2485 # calculate $x->{_m} ** $y and $x->{_e} * $y separately (faster)
2486 $x->{_m} = $MBI->_pow( $x->{_m}, $y1);
2487 $x->{_e} = $MBI->_mul ($x->{_e}, $y1);
2489 $x->{sign} = $new_sign;
2491 if ($y->{sign} eq '-')
2493 # modify $x in place!
2494 my $z = $x->copy(); $x->bone();
2495 return scalar $x->bdiv($z,$a,$p,$r); # round in one go (might ignore y's A!)
2497 $x->round($a,$p,$r,$y);
2502 # takes a very large number to a very large exponent in a given very
2503 # large modulus, quickly, thanks to binary exponentiation. Supports
2504 # negative exponents.
2505 my ($self,$num,$exp,$mod,@r) = objectify(3,@_);
2507 return $num if $num->modify('bmodpow');
2509 # check modulus for valid values
2510 return $num->bnan() if ($mod->{sign} ne '+' # NaN, - , -inf, +inf
2511 || $mod->is_zero());
2513 # check exponent for valid values
2514 if ($exp->{sign} =~ /\w/)
2516 # i.e., if it's NaN, +inf, or -inf...
2517 return $num->bnan();
2520 $num->bmodinv ($mod) if ($exp->{sign} eq '-');
2522 # check num for valid values (also NaN if there was no inverse but $exp < 0)
2523 return $num->bnan() if $num->{sign} !~ /^[+-]$/;
2525 # $mod is positive, sign on $exp is ignored, result also positive
2527 # XXX TODO: speed it up when all three numbers are integers
2528 $num->bpow($exp)->bmod($mod);
2531 ###############################################################################
2532 # trigonometric functions
2534 # helper function for bpi() and batan2(), calculates arcus tanges (1/x)
2538 # return a/b so that a/b approximates atan(1/x) to at least limit digits
2539 my ($self, $x, $limit) = @_;
2541 # Taylor: x^3 x^5 x^7 x^9
2542 # atan = x - --- + --- - --- + --- - ...
2546 # atan 1/x = - - ------- + ------- - ------- + ...
2547 # x x^3 * 3 x^5 * 5 x^7 * 7
2550 # atan 1/x = - - --------- + ---------- - ----------- + ...
2551 # 5 3 * 125 5 * 3125 7 * 78125
2553 # Subtraction/addition of a rational:
2556 # - +- - = ----------
2561 # a 1 a * d * c +- b
2562 # ----- +- ------------------ = ----------------
2565 # since b1 = b0 * (d-2) * c
2567 # a 1 a * d +- b / c
2568 # ----- +- ------------------ = ----------------
2575 # stop if length($u) > limit
2582 my $a = $MBI->_one();
2583 my $b = $MBI->_copy($x);
2585 my $x2 = $MBI->_mul( $MBI->_copy($x), $b); # x2 = x * x
2586 my $d = $MBI->_new( 3 ); # d = 3
2587 my $c = $MBI->_mul( $MBI->_copy($x), $x2); # c = x ^ 3
2588 my $two = $MBI->_new( 2 );
2590 # run the first step unconditionally
2591 my $u = $MBI->_mul( $MBI->_copy($d), $c);
2592 $a = $MBI->_mul($a, $u);
2593 $a = $MBI->_sub($a, $b);
2594 $b = $MBI->_mul($b, $u);
2595 $d = $MBI->_add($d, $two);
2596 $c = $MBI->_mul($c, $x2);
2598 # a is now a * (d-3) * c
2599 # b is now b * (d-2) * c
2601 # run the second step unconditionally
2602 $u = $MBI->_mul( $MBI->_copy($d), $c);
2603 $a = $MBI->_mul($a, $u);
2604 $a = $MBI->_add($a, $b);
2605 $b = $MBI->_mul($b, $u);
2606 $d = $MBI->_add($d, $two);
2607 $c = $MBI->_mul($c, $x2);
2609 # a is now a * (d-3) * (d-5) * c * c
2610 # b is now b * (d-2) * (d-4) * c * c
2612 # so we can remove c * c from both a and b to shorten the numbers involved:
2613 $a = $MBI->_div($a, $x2);
2614 $b = $MBI->_div($b, $x2);
2615 $a = $MBI->_div($a, $x2);
2616 $b = $MBI->_div($b, $x2);
2619 my $sign = 0; # 0 => -, 1 => +
2623 # if (($i++ % 100) == 0)
2625 # print "a=",$MBI->_str($a),"\n";
2626 # print "b=",$MBI->_str($b),"\n";
2628 # print "d=",$MBI->_str($d),"\n";
2629 # print "x2=",$MBI->_str($x2),"\n";
2630 # print "c=",$MBI->_str($c),"\n";
2632 my $u = $MBI->_mul( $MBI->_copy($d), $c);
2633 # use _alen() for libs like GMP where _len() would be O(N^2)
2634 last if $MBI->_alen($u) > $limit;
2635 my ($bc,$r) = $MBI->_div( $MBI->_copy($b), $c);
2636 if ($MBI->_is_zero($r))
2638 # b / c is an integer, so we can remove c from all terms
2639 # this happens almost every time:
2640 $a = $MBI->_mul($a, $d);
2641 $a = $MBI->_sub($a, $bc) if $sign == 0;
2642 $a = $MBI->_add($a, $bc) if $sign == 1;
2643 $b = $MBI->_mul($b, $d);
2647 # b / c is not an integer, so we keep c in the terms
2648 # this happens very rarely, for instance for x = 5, this happens only
2649 # at the following steps:
2650 # 1, 5, 14, 32, 72, 157, 340, ...
2651 $a = $MBI->_mul($a, $u);
2652 $a = $MBI->_sub($a, $b) if $sign == 0;
2653 $a = $MBI->_add($a, $b) if $sign == 1;
2654 $b = $MBI->_mul($b, $u);
2656 $d = $MBI->_add($d, $two);
2657 $c = $MBI->_mul($c, $x2);
2662 # print "Took $step steps for ", $MBI->_str($x),"\n";
2663 # print "a=",$MBI->_str($a),"\n"; print "b=",$MBI->_str($b),"\n";
2664 # return a/b so that a/b approximates atan(1/x)
2677 # called like Math::BigFloat::bpi(10);
2678 $n = $self; $self = $class;
2679 # called like Math::BigFloat->bpi();
2680 $n = undef if $n eq 'Math::BigFloat';
2682 $self = ref($self) if ref($self);
2683 my $fallback = defined $n ? 0 : 1;
2684 $n = 40 if !defined $n || $n < 1;
2686 # after 黃見利 (Hwang Chien-Lih) (1997)
2687 # pi/4 = 183 * atan(1/239) + 32 * atan(1/1023) – 68 * atan(1/5832)
2688 # + 12 * atan(1/110443) - 12 * atan(1/4841182) - 100 * atan(1/6826318)
2690 # a few more to prevent rounding errors
2693 my ($a,$b) = $self->_atan_inv( $MBI->_new(239),$n);
2694 my ($c,$d) = $self->_atan_inv( $MBI->_new(1023),$n);
2695 my ($e,$f) = $self->_atan_inv( $MBI->_new(5832),$n);
2696 my ($g,$h) = $self->_atan_inv( $MBI->_new(110443),$n);
2697 my ($i,$j) = $self->_atan_inv( $MBI->_new(4841182),$n);
2698 my ($k,$l) = $self->_atan_inv( $MBI->_new(6826318),$n);
2700 $MBI->_mul($a, $MBI->_new(732));
2701 $MBI->_mul($c, $MBI->_new(128));
2702 $MBI->_mul($e, $MBI->_new(272));
2703 $MBI->_mul($g, $MBI->_new(48));
2704 $MBI->_mul($i, $MBI->_new(48));
2705 $MBI->_mul($k, $MBI->_new(400));
2707 my $x = $self->bone(); $x->{_m} = $a; my $x_d = $self->bone(); $x_d->{_m} = $b;
2708 my $y = $self->bone(); $y->{_m} = $c; my $y_d = $self->bone(); $y_d->{_m} = $d;
2709 my $z = $self->bone(); $z->{_m} = $e; my $z_d = $self->bone(); $z_d->{_m} = $f;
2710 my $u = $self->bone(); $u->{_m} = $g; my $u_d = $self->bone(); $u_d->{_m} = $h;
2711 my $v = $self->bone(); $v->{_m} = $i; my $v_d = $self->bone(); $v_d->{_m} = $j;
2712 my $w = $self->bone(); $w->{_m} = $k; my $w_d = $self->bone(); $w_d->{_m} = $l;
2720 delete $x->{_a}; delete $y->{_a}; delete $z->{_a};
2721 delete $u->{_a}; delete $v->{_a}; delete $w->{_a};
2722 $x->badd($y)->bsub($z)->badd($u)->bsub($v)->bsub($w);
2725 delete $x->{_a} if $fallback == 1;
2731 # Calculate a cosinus of x.
2732 my ($self,$x,@r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_);
2734 # Taylor: x^2 x^4 x^6 x^8
2735 # cos = 1 - --- + --- - --- + --- ...
2738 # we need to limit the accuracy to protect against overflow
2740 my ($scale,@params);
2741 ($x,@params) = $x->_find_round_parameters(@r);
2743 # constant object or error in _find_round_parameters?
2744 return $x if $x->modify('bcos') || $x->is_nan();
2746 return $x->bone(@r) if $x->is_zero();
2748 # no rounding at all, so must use fallback
2749 if (scalar @params == 0)
2751 # simulate old behaviour
2752 $params[0] = $self->div_scale(); # and round to it as accuracy
2753 $params[1] = undef; # disable P
2754 $scale = $params[0]+4; # at least four more for proper round
2755 $params[2] = $r[2]; # round mode by caller or undef
2756 $fallback = 1; # to clear a/p afterwards
2760 # the 4 below is empirical, and there might be cases where it is not
2762 $scale = abs($params[0] || $params[1]) + 4; # take whatever is defined
2765 # when user set globals, they would interfere with our calculation, so
2766 # disable them and later re-enable them
2768 my $abr = "$self\::accuracy"; my $ab = $$abr; $$abr = undef;
2769 my $pbr = "$self\::precision"; my $pb = $$pbr; $$pbr = undef;
2770 # we also need to disable any set A or P on $x (_find_round_parameters took
2771 # them already into account), since these would interfere, too
2772 delete $x->{_a}; delete $x->{_p};
2773 # need to disable $upgrade in BigInt, to avoid deep recursion
2774 local $Math::BigInt::upgrade = undef;
2777 my $over = $x * $x; # X ^ 2
2778 my $x2 = $over->copy(); # X ^ 2; difference between terms
2779 my $sign = 1; # start with -=
2780 my $below = $self->new(2); my $factorial = $self->new(3);
2781 $x->bone(); delete $x->{_a}; delete $x->{_p};
2783 my $limit = $self->new("1E-". ($scale-1));
2787 # we calculate the next term, and add it to the last
2788 # when the next term is below our limit, it won't affect the outcome
2789 # anymore, so we stop:
2790 my $next = $over->copy()->bdiv($below,$scale);
2791 last if $next->bacmp($limit) <= 0;
2801 $sign = 1-$sign; # alternate
2802 # calculate things for the next term
2803 $over->bmul($x2); # $x*$x
2804 $below->bmul($factorial); $factorial->binc(); # n*(n+1)
2805 $below->bmul($factorial); $factorial->binc(); # n*(n+1)
2808 # shortcut to not run through _find_round_parameters again
2809 if (defined $params[0])
2811 $x->bround($params[0],$params[2]); # then round accordingly
2815 $x->bfround($params[1],$params[2]); # then round accordingly
2819 # clear a/p after round, since user did not request it
2820 delete $x->{_a}; delete $x->{_p};
2823 $$abr = $ab; $$pbr = $pb;
2829 # Calculate a sinus of x.
2830 my ($self,$x,@r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_);
2832 # taylor: x^3 x^5 x^7 x^9
2833 # sin = x - --- + --- - --- + --- ...
2836 # we need to limit the accuracy to protect against overflow
2838 my ($scale,@params);
2839 ($x,@params) = $x->_find_round_parameters(@r);
2841 # constant object or error in _find_round_parameters?
2842 return $x if $x->modify('bsin') || $x->is_nan();
2844 return $x->bzero(@r) if $x->is_zero();
2846 # no rounding at all, so must use fallback
2847 if (scalar @params == 0)
2849 # simulate old behaviour
2850 $params[0] = $self->div_scale(); # and round to it as accuracy
2851 $params[1] = undef; # disable P
2852 $scale = $params[0]+4; # at least four more for proper round
2853 $params[2] = $r[2]; # round mode by caller or undef
2854 $fallback = 1; # to clear a/p afterwards
2858 # the 4 below is empirical, and there might be cases where it is not
2860 $scale = abs($params[0] || $params[1]) + 4; # take whatever is defined
2863 # when user set globals, they would interfere with our calculation, so
2864 # disable them and later re-enable them
2866 my $abr = "$self\::accuracy"; my $ab = $$abr; $$abr = undef;
2867 my $pbr = "$self\::precision"; my $pb = $$pbr; $$pbr = undef;
2868 # we also need to disable any set A or P on $x (_find_round_parameters took
2869 # them already into account), since these would interfere, too
2870 delete $x->{_a}; delete $x->{_p};
2871 # need to disable $upgrade in BigInt, to avoid deep recursion
2872 local $Math::BigInt::upgrade = undef;
2875 my $over = $x * $x; # X ^ 2
2876 my $x2 = $over->copy(); # X ^ 2; difference between terms
2877 $over->bmul($x); # X ^ 3 as starting value
2878 my $sign = 1; # start with -=
2879 my $below = $self->new(6); my $factorial = $self->new(4);
2880 delete $x->{_a}; delete $x->{_p};
2882 my $limit = $self->new("1E-". ($scale-1));
2886 # we calculate the next term, and add it to the last
2887 # when the next term is below our limit, it won't affect the outcome
2888 # anymore, so we stop:
2889 my $next = $over->copy()->bdiv($below,$scale);
2890 last if $next->bacmp($limit) <= 0;
2900 $sign = 1-$sign; # alternate
2901 # calculate things for the next term
2902 $over->bmul($x2); # $x*$x
2903 $below->bmul($factorial); $factorial->binc(); # n*(n+1)
2904 $below->bmul($factorial); $factorial->binc(); # n*(n+1)
2907 # shortcut to not run through _find_round_parameters again
2908 if (defined $params[0])
2910 $x->bround($params[0],$params[2]); # then round accordingly
2914 $x->bfround($params[1],$params[2]); # then round accordingly
2918 # clear a/p after round, since user did not request it
2919 delete $x->{_a}; delete $x->{_p};
2922 $$abr = $ab; $$pbr = $pb;
2928 # calculate arcus tangens of ($y/$x)
2931 my ($self,$y,$x,@r) = (ref($_[0]),@_);
2932 # objectify is costly, so avoid it
2933 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
2935 ($self,$y,$x,@r) = objectify(2,@_);
2938 return $y if $y->modify('batan2');
2940 return $y->bnan() if ($y->{sign} eq $nan) || ($x->{sign} eq $nan);
2946 return $y->bzero(@r) if ($x->is_inf('+') && !$y->is_inf()) || ($y->is_zero() && $x->{sign} eq '+');
2949 # != 0 -inf result is +- pi
2950 if ($x->is_inf() || $y->is_inf())
2953 my $pi = $self->bpi(@r);
2956 # upgrade to BigRat etc.
2957 return $upgrade->new($y)->batan2($upgrade->new($x),@r) if defined $upgrade;
2958 if ($x->{sign} eq '-inf')
2961 $MBI->_mul($pi->{_m}, $MBI->_new(3));
2962 $MBI->_div($pi->{_m}, $MBI->_new(4));
2964 elsif ($x->{sign} eq '+inf')
2967 $MBI->_div($pi->{_m}, $MBI->_new(4));
2972 $MBI->_div($pi->{_m}, $MBI->_new(2));
2974 $y->{sign} = substr($y->{sign},0,1); # keep +/-
2976 # modify $y in place
2977 $y->{_m} = $pi->{_m};
2978 $y->{_e} = $pi->{_e};
2979 $y->{_es} = $pi->{_es};
2980 # keep the sign of $y
2984 return $upgrade->new($y)->batan2($upgrade->new($x),@r) if defined $upgrade;
2991 my $pi = $self->bpi(@r);
2992 # modify $y in place
2993 $y->{_m} = $pi->{_m};
2994 $y->{_e} = $pi->{_e};
2995 $y->{_es} = $pi->{_es};
3001 # +y 0 result is PI/2
3002 # -y 0 result is -PI/2
3006 my $pi = $self->bpi(@r);
3007 # modify $y in place
3008 $y->{_m} = $pi->{_m};
3009 $y->{_e} = $pi->{_e};
3010 $y->{_es} = $pi->{_es};
3011 # -y => -PI/2, +y => PI/2
3012 $MBI->_div($y->{_m}, $MBI->_new(2));
3016 # we need to limit the accuracy to protect against overflow
3018 my ($scale,@params);
3019 ($y,@params) = $y->_find_round_parameters(@r);
3021 # error in _find_round_parameters?
3022 return $y if $y->is_nan();
3024 # no rounding at all, so must use fallback
3025 if (scalar @params == 0)
3027 # simulate old behaviour
3028 $params[0] = $self->div_scale(); # and round to it as accuracy
3029 $params[1] = undef; # disable P
3030 $scale = $params[0]+4; # at least four more for proper round
3031 $params[2] = $r[2]; # round mode by caller or undef
3032 $fallback = 1; # to clear a/p afterwards
3036 # the 4 below is empirical, and there might be cases where it is not
3038 $scale = abs($params[0] || $params[1]) + 4; # take whatever is defined
3041 # inlined is_one() && is_one('-')
3042 if ($MBI->_is_one($y->{_m}) && $MBI->_is_zero($y->{_e}))
3044 # shortcut: 1 1 result is PI/4
3045 # inlined is_one() && is_one('-')
3046 if ($MBI->_is_one($x->{_m}) && $MBI->_is_zero($x->{_e}))
3049 my $pi_4 = $self->bpi( $scale - 3);
3050 # modify $y in place
3051 $y->{_m} = $pi_4->{_m};
3052 $y->{_e} = $pi_4->{_e};
3053 $y->{_es} = $pi_4->{_es};
3058 $y->{sign} = $x->{sign} eq $y->{sign} ? '+' : '-';
3059 $MBI->_div($y->{_m}, $MBI->_new(4));
3062 # shortcut: 1 int(X) result is _atan_inv(X)
3065 if ($x->{_es} eq '+')
3067 my $x1 = $MBI->_copy($x->{_m});
3068 $MBI->_lsft($x1, $x->{_e},10) unless $MBI->_is_zero($x->{_e});
3070 my ($a,$b) = $self->_atan_inv($x1, $scale);
3071 my $y_sign = $y->{sign};
3073 $y->bone(); $y->{_m} = $a; my $y_d = $self->bone(); $y_d->{_m} = $b;
3075 $y->{sign} = $y_sign;
3080 # handle all other cases
3085 # -x -y -PI/2 to -PI
3087 my $y_sign = $y->{sign};
3090 $y->bdiv($x, $scale) unless $x->is_one();
3094 $y->{sign} = $y_sign;
3101 # Calculate a arcus tangens of x.
3105 # taylor: x^3 x^5 x^7 x^9
3106 # atan = x - --- + --- - --- + --- ...
3109 # we need to limit the accuracy to protect against overflow
3111 my ($scale,@params);
3112 ($x,@params) = $x->_find_round_parameters(@r);
3114 # constant object or error in _find_round_parameters?
3115 return $x if $x->modify('batan') || $x->is_nan();
3117 if ($x->{sign} =~ /^[+-]inf\z/)
3119 # +inf result is PI/2
3120 # -inf result is -PI/2
3122 my $pi = $self->bpi(@r);
3123 # modify $x in place
3124 $x->{_m} = $pi->{_m};
3125 $x->{_e} = $pi->{_e};
3126 $x->{_es} = $pi->{_es};
3127 # -y => -PI/2, +y => PI/2
3128 $x->{sign} = substr($x->{sign},0,1); # +inf => +
3129 $MBI->_div($x->{_m}, $MBI->_new(2));
3133 return $x->bzero(@r) if $x->is_zero();
3135 # no rounding at all, so must use fallback
3136 if (scalar @params == 0)
3138 # simulate old behaviour
3139 $params[0] = $self->div_scale(); # and round to it as accuracy
3140 $params[1] = undef; # disable P
3141 $scale = $params[0]+4; # at least four more for proper round
3142 $params[2] = $r[2]; # round mode by caller or undef
3143 $fallback = 1; # to clear a/p afterwards
3147 # the 4 below is empirical, and there might be cases where it is not
3149 $scale = abs($params[0] || $params[1]) + 4; # take whatever is defined
3153 # inlined is_one() && is_one('-')
3154 if ($MBI->_is_one($x->{_m}) && $MBI->_is_zero($x->{_e}))
3156 my $pi = $self->bpi($scale - 3);
3157 # modify $x in place
3158 $x->{_m} = $pi->{_m};
3159 $x->{_e} = $pi->{_e};
3160 $x->{_es} = $pi->{_es};
3161 # leave the sign of $x alone (+1 => +PI/4, -1 => -PI/4)
3162 $MBI->_div($x->{_m}, $MBI->_new(4));
3166 # This series is only valid if -1 < x < 1, so for other x we need to
3167 # to calculate PI/2 - atan(1/x):
3168 my $one = $MBI->_new(1);
3170 if ($x->{_es} eq '+' && ($MBI->_acmp($x->{_m},$one) >= 0))
3173 $pi = $self->bpi($scale - 3);
3174 $MBI->_div($pi->{_m}, $MBI->_new(2));
3176 my $x_copy = $x->copy();
3177 # modify $x in place
3178 $x->bone(); $x->bdiv($x_copy,$scale);
3181 # when user set globals, they would interfere with our calculation, so
3182 # disable them and later re-enable them
3184 my $abr = "$self\::accuracy"; my $ab = $$abr; $$abr = undef;
3185 my $pbr = "$self\::precision"; my $pb = $$pbr; $$pbr = undef;
3186 # we also need to disable any set A or P on $x (_find_round_parameters took
3187 # them already into account), since these would interfere, too
3188 delete $x->{_a}; delete $x->{_p};
3189 # need to disable $upgrade in BigInt, to avoid deep recursion
3190 local $Math::BigInt::upgrade = undef;
3193 my $over = $x * $x; # X ^ 2
3194 my $x2 = $over->copy(); # X ^ 2; difference between terms
3195 $over->bmul($x); # X ^ 3 as starting value
3196 my $sign = 1; # start with -=
3197 my $below = $self->new(3);
3198 my $two = $self->new(2);
3199 delete $x->{_a}; delete $x->{_p};
3201 my $limit = $self->new("1E-". ($scale-1));
3205 # we calculate the next term, and add it to the last
3206 # when the next term is below our limit, it won't affect the outcome
3207 # anymore, so we stop:
3208 my $next = $over->copy()->bdiv($below,$scale);
3209 last if $next->bacmp($limit) <= 0;
3219 $sign = 1-$sign; # alternate
3220 # calculate things for the next term
3221 $over->bmul($x2); # $x*$x
3222 $below->badd($two); # n += 2
3227 my $x_copy = $x->copy();
3228 # modify $x in place
3229 $x->{_m} = $pi->{_m};
3230 $x->{_e} = $pi->{_e};
3231 $x->{_es} = $pi->{_es};
3236 # shortcut to not run through _find_round_parameters again
3237 if (defined $params[0])
3239 $x->bround($params[0],$params[2]); # then round accordingly
3243 $x->bfround($params[1],$params[2]); # then round accordingly
3247 # clear a/p after round, since user did not request it
3248 delete $x->{_a}; delete $x->{_p};
3251 $$abr = $ab; $$pbr = $pb;
3255 ###############################################################################
3256 # rounding functions
3260 # precision: round to the $Nth digit left (+$n) or right (-$n) from the '.'
3261 # $n == 0 means round to integer
3262 # expects and returns normalized numbers!
3263 my $x = shift; my $self = ref($x) || $x; $x = $self->new(shift) if !ref($x);
3265 my ($scale,$mode) = $x->_scale_p(@_);
3266 return $x if !defined $scale || $x->modify('bfround'); # no-op
3268 # never round a 0, +-inf, NaN
3271 $x->{_p} = $scale if !defined $x->{_p} || $x->{_p} < $scale; # -3 < -2
3274 return $x if $x->{sign} !~ /^[+-]$/;
3276 # don't round if x already has lower precision
3277 return $x if (defined $x->{_p} && $x->{_p} < 0 && $scale < $x->{_p});
3279 $x->{_p} = $scale; # remember round in any case
3280 delete $x->{_a}; # and clear A
3283 # round right from the '.'
3285 return $x if $x->{_es} eq '+'; # e >= 0 => nothing to round
3287 $scale = -$scale; # positive for simplicity
3288 my $len = $MBI->_len($x->{_m}); # length of mantissa
3290 # the following poses a restriction on _e, but if _e is bigger than a
3291 # scalar, you got other problems (memory etc) anyway
3292 my $dad = -(0+ ($x->{_es}.$MBI->_num($x->{_e}))); # digits after dot
3293 my $zad = 0; # zeros after dot
3294 $zad = $dad - $len if (-$dad < -$len); # for 0.00..00xxx style
3296 # print "scale $scale dad $dad zad $zad len $len\n";
3297 # number bsstr len zad dad
3298 # 0.123 123e-3 3 0 3
3299 # 0.0123 123e-4 3 1 4
3302 # 1.2345 12345e-4 5 0 4
3304 # do not round after/right of the $dad
3305 return $x if $scale > $dad; # 0.123, scale >= 3 => exit
3307 # round to zero if rounding inside the $zad, but not for last zero like:
3308 # 0.0065, scale -2, round last '0' with following '65' (scale == zad case)
3309 return $x->bzero() if $scale < $zad;
3310 if ($scale == $zad) # for 0.006, scale -3 and trunc
3316 # adjust round-point to be inside mantissa
3319 $scale = $scale-$zad;
3323 my $dbd = $len - $dad; $dbd = 0 if $dbd < 0; # digits before dot
3324 $scale = $dbd+$scale;
3330 # round left from the '.'
3332 # 123 => 100 means length(123) = 3 - $scale (2) => 1
3334 my $dbt = $MBI->_len($x->{_m});
3336 my $dbd = $dbt + ($x->{_es} . $MBI->_num($x->{_e}));
3337 # should be the same, so treat it as this
3338 $scale = 1 if $scale == 0;
3339 # shortcut if already integer
3340 return $x if $scale == 1 && $dbt <= $dbd;
3341 # maximum digits before dot
3346 # not enough digits before dot, so round to zero
3349 elsif ( $scale == $dbd )
3356 $scale = $dbd - $scale;
3359 # pass sign to bround for rounding modes '+inf' and '-inf'
3360 my $m = bless { sign => $x->{sign}, value => $x->{_m} }, 'Math::BigInt';
3361 $m->bround($scale,$mode);
3362 $x->{_m} = $m->{value}; # get our mantissa back
3368 # accuracy: preserve $N digits, and overwrite the rest with 0's
3369 my $x = shift; my $self = ref($x) || $x; $x = $self->new(shift) if !ref($x);
3371 if (($_[0] || 0) < 0)
3373 require Carp; Carp::croak ('bround() needs positive accuracy');
3376 my ($scale,$mode) = $x->_scale_a(@_);
3377 return $x if !defined $scale || $x->modify('bround'); # no-op
3379 # scale is now either $x->{_a}, $accuracy, or the user parameter
3380 # test whether $x already has lower accuracy, do nothing in this case
3381 # but do round if the accuracy is the same, since a math operation might
3382 # want to round a number with A=5 to 5 digits afterwards again
3383 return $x if defined $x->{_a} && $x->{_a} < $scale;
3385 # scale < 0 makes no sense
3386 # scale == 0 => keep all digits
3387 # never round a +-inf, NaN
3388 return $x if ($scale <= 0) || $x->{sign} !~ /^[+-]$/;
3390 # 1: never round a 0
3391 # 2: if we should keep more digits than the mantissa has, do nothing
3392 if ($x->is_zero() || $MBI->_len($x->{_m}) <= $scale)
3394 $x->{_a} = $scale if !defined $x->{_a} || $x->{_a} > $scale;
3398 # pass sign to bround for '+inf' and '-inf' rounding modes
3399 my $m = bless { sign => $x->{sign}, value => $x->{_m} }, 'Math::BigInt';
3401 $m->bround($scale,$mode); # round mantissa
3402 $x->{_m} = $m->{value}; # get our mantissa back
3403 $x->{_a} = $scale; # remember rounding
3404 delete $x->{_p}; # and clear P
3405 $x->bnorm(); # del trailing zeros gen. by bround()
3410 # round towards minus infinity
3411 my ($self,$x,$a,$p,$r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_);
3413 return $x if $x->modify('bfloor');
3415 return $x if $x->{sign} !~ /^[+-]$/; # nan, +inf, -inf
3417 # if $x has digits after dot
3418 if ($x->{_es} eq '-')
3420 $x->{_m} = $MBI->_rsft($x->{_m},$x->{_e},10); # cut off digits after dot
3421 $x->{_e} = $MBI->_zero(); # trunc/norm
3422 $x->{_es} = '+'; # abs e
3423 $MBI->_inc($x->{_m}) if $x->{sign} eq '-'; # increment if negative
3425 $x->round($a,$p,$r);
3430 # round towards plus infinity
3431 my ($self,$x,$a,$p,$r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_);
3433 return $x if $x->modify('bceil');
3434 return $x if $x->{sign} !~ /^[+-]$/; # nan, +inf, -inf
3436 # if $x has digits after dot
3437 if ($x->{_es} eq '-')
3439 $x->{_m} = $MBI->_rsft($x->{_m},$x->{_e},10); # cut off digits after dot
3440 $x->{_e} = $MBI->_zero(); # trunc/norm
3441 $x->{_es} = '+'; # abs e
3442 $MBI->_inc($x->{_m}) if $x->{sign} eq '+'; # increment if positive
3444 $x->round($a,$p,$r);
3449 # round towards zero
3450 my ($self,$x,$a,$p,$r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_);
3452 return $x if $x->modify('bint');
3453 return $x if $x->{sign} !~ /^[+-]$/; # nan, +inf, -inf
3455 # if $x has digits after the decimal point
3456 if ($x->{_es} eq '-')
3458 $x->{_m} = $MBI->_rsft($x->{_m},$x->{_e},10); # cut off digits after dot
3459 $x->{_e} = $MBI->_zero(); # truncate/normalize
3460 $x->{_es} = '+'; # abs e
3462 $x->round($a,$p,$r);
3467 # shift right by $y (divide by power of $n)
3470 my ($self,$x,$y,$n,$a,$p,$r) = (ref($_[0]),@_);
3471 # objectify is costly, so avoid it
3472 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
3474 ($self,$x,$y,$n,$a,$p,$r) = objectify(2,@_);
3477 return $x if $x->modify('brsft');
3478 return $x if $x->{sign} !~ /^[+-]$/; # nan, +inf, -inf
3480 $n = 2 if !defined $n; $n = $self->new($n);
3483 return $x->blsft($y->copy()->babs(),$n) if $y->{sign} =~ /^-/;
3485 # the following call to bdiv() will return either quo or (quo,remainder):
3486 $x->bdiv($n->bpow($y),$a,$p,$r,$y);
3491 # shift left by $y (multiply by power of $n)
3494 my ($self,$x,$y,$n,$a,$p,$r) = (ref($_[0]),@_);
3495 # objectify is costly, so avoid it
3496 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
3498 ($self,$x,$y,$n,$a,$p,$r) = objectify(2,@_);
3501 return $x if $x->modify('blsft');
3502 return $x if $x->{sign} !~ /^[+-]$/; # nan, +inf, -inf
3504 $n = 2 if !defined $n; $n = $self->new($n);
3507 return $x->brsft($y->copy()->babs(),$n) if $y->{sign} =~ /^-/;
3509 $x->bmul($n->bpow($y),$a,$p,$r,$y);
3512 ###############################################################################
3516 # going through AUTOLOAD for every DESTROY is costly, avoid it by empty sub
3521 # make fxxx and bxxx both work by selectively mapping fxxx() to MBF::bxxx()
3522 # or falling back to MBI::bxxx()
3523 my $name = $AUTOLOAD;
3525 $name =~ s/(.*):://; # split package
3526 my $c = $1 || $class;
3528 $c->import() if $IMPORT == 0;
3529 if (!_method_alias($name))
3533 # delayed load of Carp and avoid recursion
3535 Carp::croak ("$c: Can't call a method without name");
3537 if (!_method_hand_up($name))
3539 # delayed load of Carp and avoid recursion
3541 Carp::croak ("Can't call $c\-\>$name, not a valid method");
3543 # try one level up, but subst. bxxx() for fxxx() since MBI only got bxxx()
3545 return &{"Math::BigInt"."::$name"}(@_);
3547 my $bname = $name; $bname =~ s/^f/b/;
3555 # return a copy of the exponent
3556 my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
3558 if ($x->{sign} !~ /^[+-]$/)
3560 my $s = $x->{sign}; $s =~ s/^[+-]//;
3561 return Math::BigInt->new($s); # -inf, +inf => +inf
3563 Math::BigInt->new( $x->{_es} . $MBI->_str($x->{_e}));
3568 # return a copy of the mantissa
3569 my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
3571 if ($x->{sign} !~ /^[+-]$/)
3573 my $s = $x->{sign}; $s =~ s/^[+]//;
3574 return Math::BigInt->new($s); # -inf, +inf => +inf
3576 my $m = Math::BigInt->new( $MBI->_str($x->{_m}));
3577 $m->bneg() if $x->{sign} eq '-';
3584 # return a copy of both the exponent and the mantissa
3585 my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
3587 if ($x->{sign} !~ /^[+-]$/)
3589 my $s = $x->{sign}; $s =~ s/^[+]//; my $se = $s; $se =~ s/^[-]//;
3590 return ($self->new($s),$self->new($se)); # +inf => inf and -inf,+inf => inf
3592 my $m = Math::BigInt->bzero();
3593 $m->{value} = $MBI->_copy($x->{_m});
3594 $m->bneg() if $x->{sign} eq '-';
3595 ($m, Math::BigInt->new( $x->{_es} . $MBI->_num($x->{_e}) ));
3598 ##############################################################################
3599 # private stuff (internal use only)
3605 my $lib = ''; my @a;
3606 my $lib_kind = 'try';
3608 for ( my $i = 0; $i < $l ; $i++)
3610 if ( $_[$i] eq ':constant' )
3612 # This causes overlord er load to step in. 'binary' and 'integer'
3613 # are handled by BigInt.
3614 overload::constant float => sub { $self->new(shift); };
3616 elsif ($_[$i] eq 'upgrade')
3618 # this causes upgrading
3619 $upgrade = $_[$i+1]; # or undef to disable
3622 elsif ($_[$i] eq 'downgrade')
3624 # this causes downgrading
3625 $downgrade = $_[$i+1]; # or undef to disable
3628 elsif ($_[$i] =~ /^(lib|try|only)\z/)
3630 # alternative library
3631 $lib = $_[$i+1] || ''; # default Calc
3632 $lib_kind = $1; # lib, try or only
3635 elsif ($_[$i] eq 'with')
3637 # alternative class for our private parts()
3638 # XXX: no longer supported
3639 # $MBI = $_[$i+1] || 'Math::BigInt';
3648 $lib =~ tr/a-zA-Z0-9,://cd; # restrict to sane characters
3649 # let use Math::BigInt lib => 'GMP'; use Math::BigFloat; still work
3650 my $mbilib = eval { Math::BigInt->config()->{lib} };
3651 if ((defined $mbilib) && ($MBI eq 'Math::BigInt::Calc'))
3653 # MBI already loaded
3654 Math::BigInt->import( $lib_kind, "$lib,$mbilib", 'objectify');
3658 # MBI not loaded, or with ne "Math::BigInt::Calc"
3659 $lib .= ",$mbilib" if defined $mbilib;
3660 $lib =~ s/^,//; # don't leave empty
3662 # replacement library can handle lib statement, but also could ignore it
3664 # Perl < 5.6.0 dies with "out of memory!" when eval() and ':constant' is
3665 # used in the same script, or eval inside import(). So we require MBI:
3666 require Math::BigInt;
3667 Math::BigInt->import( $lib_kind => $lib, 'objectify' );
3671 require Carp; Carp::croak ("Couldn't load $lib: $! $@");
3673 # find out which one was actually loaded
3674 $MBI = Math::BigInt->config()->{lib};
3676 # register us with MBI to get notified of future lib changes
3677 Math::BigInt::_register_callback( $self, sub { $MBI = $_[0]; } );
3679 $self->export_to_level(1,$self,@a); # export wanted functions
3684 # adjust m and e so that m is smallest possible
3685 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
3687 return $x if $x->{sign} !~ /^[+-]$/; # inf, nan etc
3689 my $zeros = $MBI->_zeros($x->{_m}); # correct for trailing zeros
3692 my $z = $MBI->_new($zeros);
3693 $x->{_m} = $MBI->_rsft ($x->{_m}, $z, 10);
3694 if ($x->{_es} eq '-')
3696 if ($MBI->_acmp($x->{_e},$z) >= 0)
3698 $x->{_e} = $MBI->_sub ($x->{_e}, $z);
3699 $x->{_es} = '+' if $MBI->_is_zero($x->{_e});
3703 $x->{_e} = $MBI->_sub ( $MBI->_copy($z), $x->{_e});
3709 $x->{_e} = $MBI->_add ($x->{_e}, $z);
3714 # $x can only be 0Ey if there are no trailing zeros ('0' has 0 trailing
3715 # zeros). So, for something like 0Ey, set y to 1, and -0 => +0
3716 $x->{sign} = '+', $x->{_es} = '+', $x->{_e} = $MBI->_one()
3717 if $MBI->_is_zero($x->{_m});
3720 $x; # MBI bnorm is no-op, so do not call it
3723 ##############################################################################
3727 # return number as hexadecimal string (only for integers defined)
3728 my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
3730 return $x->bstr() if $x->{sign} !~ /^[+-]$/; # inf, nan etc
3731 return '0x0' if $x->is_zero();
3733 return $nan if $x->{_es} ne '+'; # how to do 1e-1 in hex!?
3735 my $z = $MBI->_copy($x->{_m});
3736 if (! $MBI->_is_zero($x->{_e})) # > 0
3738 $MBI->_lsft( $z, $x->{_e},10);
3740 $z = Math::BigInt->new( $x->{sign} . $MBI->_num($z));
3746 # return number as binary digit string (only for integers defined)
3747 my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
3749 return $x->bstr() if $x->{sign} !~ /^[+-]$/; # inf, nan etc
3750 return '0b0' if $x->is_zero();
3752 return $nan if $x->{_es} ne '+'; # how to do 1e-1 in hex!?
3754 my $z = $MBI->_copy($x->{_m});
3755 if (! $MBI->_is_zero($x->{_e})) # > 0
3757 $MBI->_lsft( $z, $x->{_e},10);
3759 $z = Math::BigInt->new( $x->{sign} . $MBI->_num($z));
3765 # return number as octal digit string (only for integers defined)
3766 my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
3768 return $x->bstr() if $x->{sign} !~ /^[+-]$/; # inf, nan etc
3769 return '0' if $x->is_zero();
3771 return $nan if $x->{_es} ne '+'; # how to do 1e-1 in hex!?
3773 my $z = $MBI->_copy($x->{_m});
3774 if (! $MBI->_is_zero($x->{_e})) # > 0
3776 $MBI->_lsft( $z, $x->{_e},10);
3778 $z = Math::BigInt->new( $x->{sign} . $MBI->_num($z));
3784 # return copy as a bigint representation of this BigFloat number
3785 my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
3787 return $x if $x->modify('as_number');
3789 if (!$x->isa('Math::BigFloat'))
3791 # if the object can as_number(), use it
3792 return $x->as_number() if $x->can('as_number');
3793 # otherwise, get us a float and then a number
3794 $x = $x->can('as_float') ? $x->as_float() : $self->new(0+"$x");
3797 return Math::BigInt->binf($x->sign()) if $x->is_inf();
3798 return Math::BigInt->bnan() if $x->is_nan();
3800 my $z = $MBI->_copy($x->{_m});
3801 if ($x->{_es} eq '-') # < 0
3803 $MBI->_rsft( $z, $x->{_e},10);
3805 elsif (! $MBI->_is_zero($x->{_e})) # > 0
3807 $MBI->_lsft( $z, $x->{_e},10);
3809 $z = Math::BigInt->new( $x->{sign} . $MBI->_str($z));
3816 my $class = ref($x) || $x;
3817 $x = $class->new(shift) unless ref($x);
3819 return 1 if $MBI->_is_zero($x->{_m});
3821 my $len = $MBI->_len($x->{_m});
3822 $len += $MBI->_num($x->{_e}) if $x->{_es} eq '+';
3826 $t = $MBI->_num($x->{_e}) if $x->{_es} eq '-';
3840 Math::BigFloat - Arbitrary size floating point math package
3847 my $x = Math::BigFloat->new($str); # defaults to 0
3848 my $y = $x->copy(); # make a true copy
3849 my $nan = Math::BigFloat->bnan(); # create a NotANumber
3850 my $zero = Math::BigFloat->bzero(); # create a +0
3851 my $inf = Math::BigFloat->binf(); # create a +inf
3852 my $inf = Math::BigFloat->binf('-'); # create a -inf
3853 my $one = Math::BigFloat->bone(); # create a +1
3854 my $mone = Math::BigFloat->bone('-'); # create a -1
3856 my $pi = Math::BigFloat->bpi(100); # PI to 100 digits
3858 # the following examples compute their result to 100 digits accuracy:
3859 my $cos = Math::BigFloat->new(1)->bcos(100); # cosinus(1)
3860 my $sin = Math::BigFloat->new(1)->bsin(100); # sinus(1)
3861 my $atan = Math::BigFloat->new(1)->batan(100); # arcus tangens(1)
3863 my $atan2 = Math::BigFloat->new( 1 )->batan2( 1 ,100); # batan(1)
3864 my $atan2 = Math::BigFloat->new( 1 )->batan2( 8 ,100); # batan(1/8)
3865 my $atan2 = Math::BigFloat->new( -2 )->batan2( 1 ,100); # batan(-2)
3868 $x->is_zero(); # true if arg is +0
3869 $x->is_nan(); # true if arg is NaN
3870 $x->is_one(); # true if arg is +1
3871 $x->is_one('-'); # true if arg is -1
3872 $x->is_odd(); # true if odd, false for even
3873 $x->is_even(); # true if even, false for odd
3874 $x->is_pos(); # true if >= 0
3875 $x->is_neg(); # true if < 0
3876 $x->is_inf(sign); # true if +inf, or -inf (default is '+')
3878 $x->bcmp($y); # compare numbers (undef,<0,=0,>0)
3879 $x->bacmp($y); # compare absolutely (undef,<0,=0,>0)
3880 $x->sign(); # return the sign, either +,- or NaN
3881 $x->digit($n); # return the nth digit, counting from right
3882 $x->digit(-$n); # return the nth digit, counting from left
3884 # The following all modify their first argument. If you want to pre-
3885 # serve $x, use $z = $x->copy()->bXXX($y); See under L</CAVEATS> for
3886 # necessary when mixing $a = $b assignments with non-overloaded math.
3889 $x->bzero(); # set $i to 0
3890 $x->bnan(); # set $i to NaN
3891 $x->bone(); # set $x to +1
3892 $x->bone('-'); # set $x to -1
3893 $x->binf(); # set $x to inf
3894 $x->binf('-'); # set $x to -inf
3896 $x->bneg(); # negation
3897 $x->babs(); # absolute value
3898 $x->bnorm(); # normalize (no-op)
3899 $x->bnot(); # two's complement (bit wise not)
3900 $x->binc(); # increment x by 1
3901 $x->bdec(); # decrement x by 1
3903 $x->badd($y); # addition (add $y to $x)
3904 $x->bsub($y); # subtraction (subtract $y from $x)
3905 $x->bmul($y); # multiplication (multiply $x by $y)
3906 $x->bdiv($y); # divide, set $x to quotient
3907 # return (quo,rem) or quo if scalar
3909 $x->bmod($y); # modulus ($x % $y)
3910 $x->bpow($y); # power of arguments ($x ** $y)
3911 $x->bmodpow($exp,$mod); # modular exponentiation (($num**$exp) % $mod))
3912 $x->blsft($y, $n); # left shift by $y places in base $n
3913 $x->brsft($y, $n); # right shift by $y places in base $n
3914 # returns (quo,rem) or quo if in scalar context
3916 $x->blog(); # logarithm of $x to base e (Euler's number)
3917 $x->blog($base); # logarithm of $x to base $base (f.i. 2)
3918 $x->bexp(); # calculate e ** $x where e is Euler's number
3920 $x->band($y); # bit-wise and
3921 $x->bior($y); # bit-wise inclusive or
3922 $x->bxor($y); # bit-wise exclusive or
3923 $x->bnot(); # bit-wise not (two's complement)
3925 $x->bsqrt(); # calculate square-root
3926 $x->broot($y); # $y'th root of $x (e.g. $y == 3 => cubic root)
3927 $x->bfac(); # factorial of $x (1*2*3*4*..$x)
3929 $x->bround($N); # accuracy: preserve $N digits
3930 $x->bfround($N); # precision: round to the $Nth digit
3932 $x->bfloor(); # return integer less or equal than $x
3933 $x->bceil(); # return integer greater or equal than $x
3934 $x->bint(); # round towards zero
3936 # The following do not modify their arguments:
3938 bgcd(@values); # greatest common divisor
3939 blcm(@values); # lowest common multiplicator
3941 $x->bstr(); # return string
3942 $x->bsstr(); # return string in scientific notation
3944 $x->as_int(); # return $x as BigInt
3945 $x->exponent(); # return exponent as BigInt
3946 $x->mantissa(); # return mantissa as BigInt
3947 $x->parts(); # return (mantissa,exponent) as BigInt
3949 $x->length(); # number of digits (w/o sign and '.')
3950 ($l,$f) = $x->length(); # number of digits, and length of fraction
3952 $x->precision(); # return P of $x (or global, if P of $x undef)
3953 $x->precision($n); # set P of $x to $n
3954 $x->accuracy(); # return A of $x (or global, if A of $x undef)
3955 $x->accuracy($n); # set A $x to $n
3957 # these get/set the appropriate global value for all BigFloat objects
3958 Math::BigFloat->precision(); # Precision
3959 Math::BigFloat->accuracy(); # Accuracy
3960 Math::BigFloat->round_mode(); # rounding mode
3964 All operators (including basic math operations) are overloaded if you
3965 declare your big floating point numbers as
3967 $i = new Math::BigFloat '12_3.456_789_123_456_789E-2';
3969 Operations with overloaded operators preserve the arguments, which is
3970 exactly what you expect.
3974 Input to these routines are either BigFloat objects, or strings of the
3975 following four forms:
3989 C</^[+-]\d+E[+-]?\d+$/>
3993 C</^[+-]\d*\.\d+E[+-]?\d+$/>
3997 all with optional leading and trailing zeros and/or spaces. Additionally,
3998 numbers are allowed to have an underscore between any two digits.
4000 Empty strings as well as other illegal numbers results in 'NaN'.
4002 bnorm() on a BigFloat object is now effectively a no-op, since the numbers
4003 are always stored in normalized form. On a string, it creates a BigFloat
4008 Output values are BigFloat objects (normalized), except for bstr() and bsstr().
4010 The string output will always have leading and trailing zeros stripped and drop
4011 a plus sign. C<bstr()> will give you always the form with a decimal point,
4012 while C<bsstr()> (s for scientific) gives you the scientific notation.
4014 Input bstr() bsstr()
4016 ' -123 123 123' '-123123123' '-123123123E0'
4017 '00.0123' '0.0123' '123E-4'
4018 '123.45E-2' '1.2345' '12345E-4'
4019 '10E+3' '10000' '1E4'
4021 Some routines (C<is_odd()>, C<is_even()>, C<is_zero()>, C<is_one()>,
4022 C<is_nan()>) return true or false, while others (C<bcmp()>, C<bacmp()>)
4023 return either undef, <0, 0 or >0 and are suited for sort.
4025 Actual math is done by using the class defined with C<< with => Class; >>
4026 (which defaults to BigInts) to represent the mantissa and exponent.
4028 The sign C</^[+-]$/> is stored separately. The string 'NaN' is used to
4029 represent the result when input arguments are not numbers, as well as
4030 the result of dividing by zero.
4032 =head2 mantissa(), exponent() and parts()
4034 mantissa() and exponent() return the said parts of the BigFloat
4035 as BigInts such that:
4037 $m = $x->mantissa();
4038 $e = $x->exponent();
4039 $y = $m * ( 10 ** $e );
4040 print "ok\n" if $x == $y;
4042 C<< ($m,$e) = $x->parts(); >> is just a shortcut giving you both of them.
4044 A zero is represented and returned as C<0E1>, B<not> C<0E0> (after Knuth).
4046 Currently the mantissa is reduced as much as possible, favouring higher
4047 exponents over lower ones (e.g. returning 1e7 instead of 10e6 or 10000000e0).
4048 This might change in the future, so do not depend on it.
4050 =head2 Accuracy vs. Precision
4052 See also: L<Rounding|/Rounding>.
4054 Math::BigFloat supports both precision (rounding to a certain place before or
4055 after the dot) and accuracy (rounding to a certain number of digits). For a
4056 full documentation, examples and tips on these topics please see the large
4057 section about rounding in L<Math::BigInt>.
4059 Since things like C<sqrt(2)> or C<1 / 3> must presented with a limited
4060 accuracy lest a operation consumes all resources, each operation produces
4061 no more than the requested number of digits.
4063 If there is no global precision or accuracy set, B<and> the operation in
4064 question was not called with a requested precision or accuracy, B<and> the
4065 input $x has no accuracy or precision set, then a fallback parameter will
4066 be used. For historical reasons, it is called C<div_scale> and can be accessed
4069 $d = Math::BigFloat->div_scale(); # query
4070 Math::BigFloat->div_scale($n); # set to $n digits
4072 The default value for C<div_scale> is 40.
4074 In case the result of one operation has more digits than specified,
4075 it is rounded. The rounding mode taken is either the default mode, or the one
4076 supplied to the operation after the I<scale>:
4078 $x = Math::BigFloat->new(2);
4079 Math::BigFloat->accuracy(5); # 5 digits max
4080 $y = $x->copy()->bdiv(3); # will give 0.66667
4081 $y = $x->copy()->bdiv(3,6); # will give 0.666667
4082 $y = $x->copy()->bdiv(3,6,undef,'odd'); # will give 0.666667
4083 Math::BigFloat->round_mode('zero');
4084 $y = $x->copy()->bdiv(3,6); # will also give 0.666667
4086 Note that C<< Math::BigFloat->accuracy() >> and C<< Math::BigFloat->precision() >>
4087 set the global variables, and thus B<any> newly created number will be subject
4088 to the global rounding B<immediately>. This means that in the examples above, the
4089 C<3> as argument to C<bdiv()> will also get an accuracy of B<5>.
4091 It is less confusing to either calculate the result fully, and afterwards
4092 round it explicitly, or use the additional parameters to the math
4096 $x = Math::BigFloat->new(2);
4097 $y = $x->copy()->bdiv(3);
4098 print $y->bround(5),"\n"; # will give 0.66667
4103 $x = Math::BigFloat->new(2);
4104 $y = $x->copy()->bdiv(3,5); # will give 0.66667
4111 =item ffround ( +$scale )
4113 Rounds to the $scale'th place left from the '.', counting from the dot.
4114 The first digit is numbered 1.
4116 =item ffround ( -$scale )
4118 Rounds to the $scale'th place right from the '.', counting from the dot.
4122 Rounds to an integer.
4124 =item fround ( +$scale )
4126 Preserves accuracy to $scale digits from the left (aka significant digits)
4127 and pads the rest with zeros. If the number is between 1 and -1, the
4128 significant digits count from the first non-zero after the '.'
4130 =item fround ( -$scale ) and fround ( 0 )
4132 These are effectively no-ops.
4136 All rounding functions take as a second parameter a rounding mode from one of
4137 the following: 'even', 'odd', '+inf', '-inf', 'zero', 'trunc' or 'common'.
4139 The default rounding mode is 'even'. By using
4140 C<< Math::BigFloat->round_mode($round_mode); >> you can get and set the default
4141 mode for subsequent rounding. The usage of C<$Math::BigFloat::$round_mode> is
4142 no longer supported.
4143 The second parameter to the round functions then overrides the default
4146 The C<as_number()> function returns a BigInt from a Math::BigFloat. It uses
4147 'trunc' as rounding mode to make it equivalent to:
4152 You can override this by passing the desired rounding mode as parameter to
4155 $x = Math::BigFloat->new(2.5);
4156 $y = $x->as_number('odd'); # $y = 3
4160 Math::BigFloat supports all methods that Math::BigInt supports, except it
4161 calculates non-integer results when possible. Please see L<Math::BigInt>
4162 for a full description of each method. Below are just the most important
4169 $x->accuracy(5); # local for $x
4170 CLASS->accuracy(5); # global for all members of CLASS
4171 # Note: This also applies to new()!
4173 $A = $x->accuracy(); # read out accuracy that affects $x
4174 $A = CLASS->accuracy(); # read out global accuracy
4176 Set or get the global or local accuracy, aka how many significant digits the
4177 results have. If you set a global accuracy, then this also applies to new()!
4179 Warning! The accuracy I<sticks>, e.g. once you created a number under the
4180 influence of C<< CLASS->accuracy($A) >>, all results from math operations with
4181 that number will also be rounded.
4183 In most cases, you should probably round the results explicitly using one of
4184 L<Math::BigInt/round()>, L<Math::BigInt/bround()> or L<Math::BigInt/bfround()> or by passing the desired accuracy
4185 to the math operation as additional parameter:
4187 my $x = Math::BigInt->new(30000);
4188 my $y = Math::BigInt->new(7);
4189 print scalar $x->copy()->bdiv($y, 2); # print 4300
4190 print scalar $x->copy()->bdiv($y)->bround(2); # print 4300
4194 $x->precision(-2); # local for $x, round at the second
4195 # digit right of the dot
4196 $x->precision(2); # ditto, round at the second digit
4199 CLASS->precision(5); # Global for all members of CLASS
4200 # This also applies to new()!
4201 CLASS->precision(-5); # ditto
4203 $P = CLASS->precision(); # read out global precision
4204 $P = $x->precision(); # read out precision that affects $x
4206 Note: You probably want to use L</accuracy()> instead. With L</accuracy()> you
4207 set the number of digits each result should have, with L</precision()> you
4208 set the place where to round!
4212 $x->bexp($accuracy); # calculate e ** X
4214 Calculates the expression C<e ** $x> where C<e> is Euler's number.
4216 This method was added in v1.82 of Math::BigInt (April 2007).
4220 $x->bnok($y); # x over y (binomial coefficient n over k)
4222 Calculates the binomial coefficient n over k, also called the "choose"
4223 function. The result is equivalent to:
4229 This method was added in v1.84 of Math::BigInt (April 2007).
4233 print Math::BigFloat->bpi(100), "\n";
4235 Calculate PI to N digits (including the 3 before the dot). The result is
4236 rounded according to the current rounding mode, which defaults to "even".
4238 This method was added in v1.87 of Math::BigInt (June 2007).
4242 my $x = Math::BigFloat->new(1);
4243 print $x->bcos(100), "\n";
4245 Calculate the cosinus of $x, modifying $x in place.
4247 This method was added in v1.87 of Math::BigInt (June 2007).
4251 my $x = Math::BigFloat->new(1);
4252 print $x->bsin(100), "\n";
4254 Calculate the sinus of $x, modifying $x in place.
4256 This method was added in v1.87 of Math::BigInt (June 2007).
4260 my $y = Math::BigFloat->new(2);
4261 my $x = Math::BigFloat->new(3);
4262 print $y->batan2($x), "\n";
4264 Calculate the arcus tanges of C<$y> divided by C<$x>, modifying $y in place.
4265 See also L</batan()>.
4267 This method was added in v1.87 of Math::BigInt (June 2007).
4271 my $x = Math::BigFloat->new(1);
4272 print $x->batan(100), "\n";
4274 Calculate the arcus tanges of $x, modifying $x in place. See also L</batan2()>.
4276 This method was added in v1.87 of Math::BigInt (June 2007).
4282 Multiply $x by $y, and then add $z to the result.
4284 This method was added in v1.87 of Math::BigInt (June 2007).
4288 =head1 Autocreating constants
4290 After C<use Math::BigFloat ':constant'> all the floating point constants
4291 in the given scope are converted to C<Math::BigFloat>. This conversion
4292 happens at compile time.