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920dae64 AT |
1 | package Math::BigFloat; |
2 | ||
3 | # | |
4 | # Mike grinned. 'Two down, infinity to go' - Mike Nostrus in 'Before and After' | |
5 | # | |
6 | ||
7 | # The following hash values are internally used: | |
8 | # _e : exponent (ref to $CALC object) | |
9 | # _m : mantissa (ref to $CALC object) | |
10 | # _es : sign of _e | |
11 | # sign : +,-,+inf,-inf, or "NaN" if not a number | |
12 | # _a : accuracy | |
13 | # _p : precision | |
14 | ||
15 | $VERSION = '1.51'; | |
16 | require 5.005; | |
17 | ||
18 | require Exporter; | |
19 | @ISA = qw(Exporter Math::BigInt); | |
20 | ||
21 | use strict; | |
22 | # $_trap_inf/$_trap_nan are internal and should never be accessed from outside | |
23 | use vars qw/$AUTOLOAD $accuracy $precision $div_scale $round_mode $rnd_mode | |
24 | $upgrade $downgrade $_trap_nan $_trap_inf/; | |
25 | my $class = "Math::BigFloat"; | |
26 | ||
27 | use overload | |
28 | '<=>' => sub { $_[2] ? | |
29 | ref($_[0])->bcmp($_[1],$_[0]) : | |
30 | ref($_[0])->bcmp($_[0],$_[1])}, | |
31 | 'int' => sub { $_[0]->as_number() }, # 'trunc' to bigint | |
32 | ; | |
33 | ||
34 | ############################################################################## | |
35 | # global constants, flags and assorted stuff | |
36 | ||
37 | # the following are public, but their usage is not recommended. Use the | |
38 | # accessor methods instead. | |
39 | ||
40 | # class constants, use Class->constant_name() to access | |
41 | $round_mode = 'even'; # one of 'even', 'odd', '+inf', '-inf', 'zero' or 'trunc' | |
42 | $accuracy = undef; | |
43 | $precision = undef; | |
44 | $div_scale = 40; | |
45 | ||
46 | $upgrade = undef; | |
47 | $downgrade = undef; | |
48 | # the package we are using for our private parts, defaults to: | |
49 | # Math::BigInt->config()->{lib} | |
50 | my $MBI = 'Math::BigInt::FastCalc'; | |
51 | ||
52 | # are NaNs ok? (otherwise it dies when encountering an NaN) set w/ config() | |
53 | $_trap_nan = 0; | |
54 | # the same for infinity | |
55 | $_trap_inf = 0; | |
56 | ||
57 | # constant for easier life | |
58 | my $nan = 'NaN'; | |
59 | ||
60 | my $IMPORT = 0; # was import() called yet? used to make require work | |
61 | ||
62 | # some digits of accuracy for blog(undef,10); which we use in blog() for speed | |
63 | my $LOG_10 = | |
64 | '2.3025850929940456840179914546843642076011014886287729760333279009675726097'; | |
65 | my $LOG_10_A = length($LOG_10)-1; | |
66 | # ditto for log(2) | |
67 | my $LOG_2 = | |
68 | '0.6931471805599453094172321214581765680755001343602552541206800094933936220'; | |
69 | my $LOG_2_A = length($LOG_2)-1; | |
70 | my $HALF = '0.5'; # made into an object if necc. | |
71 | ||
72 | ############################################################################## | |
73 | # the old code had $rnd_mode, so we need to support it, too | |
74 | ||
75 | sub TIESCALAR { my ($class) = @_; bless \$round_mode, $class; } | |
76 | sub FETCH { return $round_mode; } | |
77 | sub STORE { $rnd_mode = $_[0]->round_mode($_[1]); } | |
78 | ||
79 | BEGIN | |
80 | { | |
81 | # when someone set's $rnd_mode, we catch this and check the value to see | |
82 | # whether it is valid or not. | |
83 | $rnd_mode = 'even'; tie $rnd_mode, 'Math::BigFloat'; | |
84 | } | |
85 | ||
86 | ############################################################################## | |
87 | ||
88 | { | |
89 | # valid method aliases for AUTOLOAD | |
90 | my %methods = map { $_ => 1 } | |
91 | qw / fadd fsub fmul fdiv fround ffround fsqrt fmod fstr fsstr fpow fnorm | |
92 | fint facmp fcmp fzero fnan finf finc fdec flog ffac fneg | |
93 | fceil ffloor frsft flsft fone flog froot | |
94 | /; | |
95 | # valid method's that can be hand-ed up (for AUTOLOAD) | |
96 | my %hand_ups = map { $_ => 1 } | |
97 | qw / is_nan is_inf is_negative is_positive is_pos is_neg | |
98 | accuracy precision div_scale round_mode fabs fnot | |
99 | objectify upgrade downgrade | |
100 | bone binf bnan bzero | |
101 | /; | |
102 | ||
103 | sub method_alias { exists $methods{$_[0]||''}; } | |
104 | sub method_hand_up { exists $hand_ups{$_[0]||''}; } | |
105 | } | |
106 | ||
107 | ############################################################################## | |
108 | # constructors | |
109 | ||
110 | sub new | |
111 | { | |
112 | # create a new BigFloat object from a string or another bigfloat object. | |
113 | # _e: exponent | |
114 | # _m: mantissa | |
115 | # sign => sign (+/-), or "NaN" | |
116 | ||
117 | my ($class,$wanted,@r) = @_; | |
118 | ||
119 | # avoid numify-calls by not using || on $wanted! | |
120 | return $class->bzero() if !defined $wanted; # default to 0 | |
121 | return $wanted->copy() if UNIVERSAL::isa($wanted,'Math::BigFloat'); | |
122 | ||
123 | $class->import() if $IMPORT == 0; # make require work | |
124 | ||
125 | my $self = {}; bless $self, $class; | |
126 | # shortcut for bigints and its subclasses | |
127 | if ((ref($wanted)) && (ref($wanted) ne $class)) | |
128 | { | |
129 | $self->{_m} = $wanted->as_number()->{value}; # get us a bigint copy | |
130 | $self->{_e} = $MBI->_zero(); | |
131 | $self->{_es} = '+'; | |
132 | $self->{sign} = $wanted->sign(); | |
133 | return $self->bnorm(); | |
134 | } | |
135 | # else: got a string | |
136 | ||
137 | # handle '+inf', '-inf' first | |
138 | if ($wanted =~ /^[+-]?inf\z/) | |
139 | { | |
140 | return $downgrade->new($wanted) if $downgrade; | |
141 | ||
142 | $self->{sign} = $wanted; # set a default sign for bstr() | |
143 | return $self->binf($wanted); | |
144 | } | |
145 | ||
146 | # shortcut for simple forms like '12' that neither have trailing nor leading | |
147 | # zeros | |
148 | if ($wanted =~ /^([+-]?)([1-9][0-9]*[1-9])$/) | |
149 | { | |
150 | $self->{_e} = $MBI->_zero(); | |
151 | $self->{_es} = '+'; | |
152 | $self->{sign} = $1 || '+'; | |
153 | $self->{_m} = $MBI->_new($2); | |
154 | return $self->round(@r) if !$downgrade; | |
155 | } | |
156 | ||
157 | my ($mis,$miv,$mfv,$es,$ev) = Math::BigInt::_split($wanted); | |
158 | if (!ref $mis) | |
159 | { | |
160 | if ($_trap_nan) | |
161 | { | |
162 | require Carp; | |
163 | Carp::croak ("$wanted is not a number initialized to $class"); | |
164 | } | |
165 | ||
166 | return $downgrade->bnan() if $downgrade; | |
167 | ||
168 | $self->{_e} = $MBI->_zero(); | |
169 | $self->{_es} = '+'; | |
170 | $self->{_m} = $MBI->_zero(); | |
171 | $self->{sign} = $nan; | |
172 | } | |
173 | else | |
174 | { | |
175 | # make integer from mantissa by adjusting exp, then convert to int | |
176 | $self->{_e} = $MBI->_new($$ev); # exponent | |
177 | $self->{_es} = $$es || '+'; | |
178 | my $mantissa = "$$miv$$mfv"; # create mant. | |
179 | $mantissa =~ s/^0+(\d)/$1/; # strip leading zeros | |
180 | $self->{_m} = $MBI->_new($mantissa); # create mant. | |
181 | ||
182 | # 3.123E0 = 3123E-3, and 3.123E-2 => 3123E-5 | |
183 | if (CORE::length($$mfv) != 0) | |
184 | { | |
185 | my $len = $MBI->_new( CORE::length($$mfv)); | |
186 | ($self->{_e}, $self->{_es}) = | |
187 | _e_sub ($self->{_e}, $len, $self->{_es}, '+'); | |
188 | } | |
189 | # we can only have trailing zeros on the mantissa if $$mfv eq '' | |
190 | else | |
191 | { | |
192 | # Use a regexp to count the trailing zeros in $$miv instead of _zeros() | |
193 | # because that is faster, especially when _m is not stored in base 10. | |
194 | my $zeros = 0; $zeros = CORE::length($1) if $$miv =~ /[1-9](0*)$/; | |
195 | if ($zeros != 0) | |
196 | { | |
197 | my $z = $MBI->_new($zeros); | |
198 | # turn '120e2' into '12e3' | |
199 | $MBI->_rsft ( $self->{_m}, $z, 10); | |
200 | ($self->{_e}, $self->{_es}) = | |
201 | _e_add ( $self->{_e}, $z, $self->{_es}, '+'); | |
202 | } | |
203 | } | |
204 | $self->{sign} = $$mis; | |
205 | ||
206 | # for something like 0Ey, set y to 1, and -0 => +0 | |
207 | # Check $$miv for beeing '0' and $$mfv eq '', because otherwise _m could not | |
208 | # have become 0. That's faster than to call $MBI->_is_zero(). | |
209 | $self->{sign} = '+', $self->{_e} = $MBI->_one() | |
210 | if $$miv eq '0' and $$mfv eq ''; | |
211 | ||
212 | return $self->round(@r) if !$downgrade; | |
213 | } | |
214 | # if downgrade, inf, NaN or integers go down | |
215 | ||
216 | if ($downgrade && $self->{_es} eq '+') | |
217 | { | |
218 | if ($MBI->_is_zero( $self->{_e} )) | |
219 | { | |
220 | return $downgrade->new($$mis . $MBI->_str( $self->{_m} )); | |
221 | } | |
222 | return $downgrade->new($self->bsstr()); | |
223 | } | |
224 | $self->bnorm()->round(@r); # first normalize, then round | |
225 | } | |
226 | ||
227 | sub copy | |
228 | { | |
229 | my ($c,$x); | |
230 | if (@_ > 1) | |
231 | { | |
232 | # if two arguments, the first one is the class to "swallow" subclasses | |
233 | ($c,$x) = @_; | |
234 | } | |
235 | else | |
236 | { | |
237 | $x = shift; | |
238 | $c = ref($x); | |
239 | } | |
240 | return unless ref($x); # only for objects | |
241 | ||
242 | my $self = {}; bless $self,$c; | |
243 | ||
244 | $self->{sign} = $x->{sign}; | |
245 | $self->{_es} = $x->{_es}; | |
246 | $self->{_m} = $MBI->_copy($x->{_m}); | |
247 | $self->{_e} = $MBI->_copy($x->{_e}); | |
248 | $self->{_a} = $x->{_a} if defined $x->{_a}; | |
249 | $self->{_p} = $x->{_p} if defined $x->{_p}; | |
250 | $self; | |
251 | } | |
252 | ||
253 | sub _bnan | |
254 | { | |
255 | # used by parent class bone() to initialize number to NaN | |
256 | my $self = shift; | |
257 | ||
258 | if ($_trap_nan) | |
259 | { | |
260 | require Carp; | |
261 | my $class = ref($self); | |
262 | Carp::croak ("Tried to set $self to NaN in $class\::_bnan()"); | |
263 | } | |
264 | ||
265 | $IMPORT=1; # call our import only once | |
266 | $self->{_m} = $MBI->_zero(); | |
267 | $self->{_e} = $MBI->_zero(); | |
268 | $self->{_es} = '+'; | |
269 | } | |
270 | ||
271 | sub _binf | |
272 | { | |
273 | # used by parent class bone() to initialize number to +-inf | |
274 | my $self = shift; | |
275 | ||
276 | if ($_trap_inf) | |
277 | { | |
278 | require Carp; | |
279 | my $class = ref($self); | |
280 | Carp::croak ("Tried to set $self to +-inf in $class\::_binf()"); | |
281 | } | |
282 | ||
283 | $IMPORT=1; # call our import only once | |
284 | $self->{_m} = $MBI->_zero(); | |
285 | $self->{_e} = $MBI->_zero(); | |
286 | $self->{_es} = '+'; | |
287 | } | |
288 | ||
289 | sub _bone | |
290 | { | |
291 | # used by parent class bone() to initialize number to 1 | |
292 | my $self = shift; | |
293 | $IMPORT=1; # call our import only once | |
294 | $self->{_m} = $MBI->_one(); | |
295 | $self->{_e} = $MBI->_zero(); | |
296 | $self->{_es} = '+'; | |
297 | } | |
298 | ||
299 | sub _bzero | |
300 | { | |
301 | # used by parent class bone() to initialize number to 0 | |
302 | my $self = shift; | |
303 | $IMPORT=1; # call our import only once | |
304 | $self->{_m} = $MBI->_zero(); | |
305 | $self->{_e} = $MBI->_one(); | |
306 | $self->{_es} = '+'; | |
307 | } | |
308 | ||
309 | sub isa | |
310 | { | |
311 | my ($self,$class) = @_; | |
312 | return if $class =~ /^Math::BigInt/; # we aren't one of these | |
313 | UNIVERSAL::isa($self,$class); | |
314 | } | |
315 | ||
316 | sub config | |
317 | { | |
318 | # return (later set?) configuration data as hash ref | |
319 | my $class = shift || 'Math::BigFloat'; | |
320 | ||
321 | my $cfg = $class->SUPER::config(@_); | |
322 | ||
323 | # now we need only to override the ones that are different from our parent | |
324 | $cfg->{class} = $class; | |
325 | $cfg->{with} = $MBI; | |
326 | $cfg; | |
327 | } | |
328 | ||
329 | ############################################################################## | |
330 | # string conversation | |
331 | ||
332 | sub bstr | |
333 | { | |
334 | # (ref to BFLOAT or num_str ) return num_str | |
335 | # Convert number from internal format to (non-scientific) string format. | |
336 | # internal format is always normalized (no leading zeros, "-0" => "+0") | |
337 | my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_); | |
338 | ||
339 | if ($x->{sign} !~ /^[+-]$/) | |
340 | { | |
341 | return $x->{sign} unless $x->{sign} eq '+inf'; # -inf, NaN | |
342 | return 'inf'; # +inf | |
343 | } | |
344 | ||
345 | my $es = '0'; my $len = 1; my $cad = 0; my $dot = '.'; | |
346 | ||
347 | # $x is zero? | |
348 | my $not_zero = !($x->{sign} eq '+' && $MBI->_is_zero($x->{_m})); | |
349 | if ($not_zero) | |
350 | { | |
351 | $es = $MBI->_str($x->{_m}); | |
352 | $len = CORE::length($es); | |
353 | my $e = $MBI->_num($x->{_e}); | |
354 | $e = -$e if $x->{_es} eq '-'; | |
355 | if ($e < 0) | |
356 | { | |
357 | $dot = ''; | |
358 | # if _e is bigger than a scalar, the following will blow your memory | |
359 | if ($e <= -$len) | |
360 | { | |
361 | my $r = abs($e) - $len; | |
362 | $es = '0.'. ('0' x $r) . $es; $cad = -($len+$r); | |
363 | } | |
364 | else | |
365 | { | |
366 | substr($es,$e,0) = '.'; $cad = $MBI->_num($x->{_e}); | |
367 | $cad = -$cad if $x->{_es} eq '-'; | |
368 | } | |
369 | } | |
370 | elsif ($e > 0) | |
371 | { | |
372 | # expand with zeros | |
373 | $es .= '0' x $e; $len += $e; $cad = 0; | |
374 | } | |
375 | } # if not zero | |
376 | ||
377 | $es = '-'.$es if $x->{sign} eq '-'; | |
378 | # if set accuracy or precision, pad with zeros on the right side | |
379 | if ((defined $x->{_a}) && ($not_zero)) | |
380 | { | |
381 | # 123400 => 6, 0.1234 => 4, 0.001234 => 4 | |
382 | my $zeros = $x->{_a} - $cad; # cad == 0 => 12340 | |
383 | $zeros = $x->{_a} - $len if $cad != $len; | |
384 | $es .= $dot.'0' x $zeros if $zeros > 0; | |
385 | } | |
386 | elsif ((($x->{_p} || 0) < 0)) | |
387 | { | |
388 | # 123400 => 6, 0.1234 => 4, 0.001234 => 6 | |
389 | my $zeros = -$x->{_p} + $cad; | |
390 | $es .= $dot.'0' x $zeros if $zeros > 0; | |
391 | } | |
392 | $es; | |
393 | } | |
394 | ||
395 | sub bsstr | |
396 | { | |
397 | # (ref to BFLOAT or num_str ) return num_str | |
398 | # Convert number from internal format to scientific string format. | |
399 | # internal format is always normalized (no leading zeros, "-0E0" => "+0E0") | |
400 | my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_); | |
401 | ||
402 | if ($x->{sign} !~ /^[+-]$/) | |
403 | { | |
404 | return $x->{sign} unless $x->{sign} eq '+inf'; # -inf, NaN | |
405 | return 'inf'; # +inf | |
406 | } | |
407 | my $sep = 'e'.$x->{_es}; | |
408 | my $sign = $x->{sign}; $sign = '' if $sign eq '+'; | |
409 | $sign . $MBI->_str($x->{_m}) . $sep . $MBI->_str($x->{_e}); | |
410 | } | |
411 | ||
412 | sub numify | |
413 | { | |
414 | # Make a number from a BigFloat object | |
415 | # simple return a string and let Perl's atoi()/atof() handle the rest | |
416 | my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_); | |
417 | $x->bsstr(); | |
418 | } | |
419 | ||
420 | ############################################################################## | |
421 | # public stuff (usually prefixed with "b") | |
422 | ||
423 | sub bneg | |
424 | { | |
425 | # (BINT or num_str) return BINT | |
426 | # negate number or make a negated number from string | |
427 | my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_); | |
428 | ||
429 | return $x if $x->modify('bneg'); | |
430 | ||
431 | # for +0 dont negate (to have always normalized +0). Does nothing for 'NaN' | |
432 | $x->{sign} =~ tr/+-/-+/ unless ($x->{sign} eq '+' && $MBI->_is_zero($x->{_m})); | |
433 | $x; | |
434 | } | |
435 | ||
436 | # tels 2001-08-04 | |
437 | # XXX TODO this must be overwritten and return NaN for non-integer values | |
438 | # band(), bior(), bxor(), too | |
439 | #sub bnot | |
440 | # { | |
441 | # $class->SUPER::bnot($class,@_); | |
442 | # } | |
443 | ||
444 | sub bcmp | |
445 | { | |
446 | # Compares 2 values. Returns one of undef, <0, =0, >0. (suitable for sort) | |
447 | ||
448 | # set up parameters | |
449 | my ($self,$x,$y) = (ref($_[0]),@_); | |
450 | # objectify is costly, so avoid it | |
451 | if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1]))) | |
452 | { | |
453 | ($self,$x,$y) = objectify(2,@_); | |
454 | } | |
455 | ||
456 | return $upgrade->bcmp($x,$y) if defined $upgrade && | |
457 | ((!$x->isa($self)) || (!$y->isa($self))); | |
458 | ||
459 | if (($x->{sign} !~ /^[+-]$/) || ($y->{sign} !~ /^[+-]$/)) | |
460 | { | |
461 | # handle +-inf and NaN | |
462 | return undef if (($x->{sign} eq $nan) || ($y->{sign} eq $nan)); | |
463 | return 0 if ($x->{sign} eq $y->{sign}) && ($x->{sign} =~ /^[+-]inf$/); | |
464 | return +1 if $x->{sign} eq '+inf'; | |
465 | return -1 if $x->{sign} eq '-inf'; | |
466 | return -1 if $y->{sign} eq '+inf'; | |
467 | return +1; | |
468 | } | |
469 | ||
470 | # check sign for speed first | |
471 | return 1 if $x->{sign} eq '+' && $y->{sign} eq '-'; # does also 0 <=> -y | |
472 | return -1 if $x->{sign} eq '-' && $y->{sign} eq '+'; # does also -x <=> 0 | |
473 | ||
474 | # shortcut | |
475 | my $xz = $x->is_zero(); | |
476 | my $yz = $y->is_zero(); | |
477 | return 0 if $xz && $yz; # 0 <=> 0 | |
478 | return -1 if $xz && $y->{sign} eq '+'; # 0 <=> +y | |
479 | return 1 if $yz && $x->{sign} eq '+'; # +x <=> 0 | |
480 | ||
481 | # adjust so that exponents are equal | |
482 | my $lxm = $MBI->_len($x->{_m}); | |
483 | my $lym = $MBI->_len($y->{_m}); | |
484 | # the numify somewhat limits our length, but makes it much faster | |
485 | my ($xes,$yes) = (1,1); | |
486 | $xes = -1 if $x->{_es} ne '+'; | |
487 | $yes = -1 if $y->{_es} ne '+'; | |
488 | my $lx = $lxm + $xes * $MBI->_num($x->{_e}); | |
489 | my $ly = $lym + $yes * $MBI->_num($y->{_e}); | |
490 | my $l = $lx - $ly; $l = -$l if $x->{sign} eq '-'; | |
491 | return $l <=> 0 if $l != 0; | |
492 | ||
493 | # lengths (corrected by exponent) are equal | |
494 | # so make mantissa equal length by padding with zero (shift left) | |
495 | my $diff = $lxm - $lym; | |
496 | my $xm = $x->{_m}; # not yet copy it | |
497 | my $ym = $y->{_m}; | |
498 | if ($diff > 0) | |
499 | { | |
500 | $ym = $MBI->_copy($y->{_m}); | |
501 | $ym = $MBI->_lsft($ym, $MBI->_new($diff), 10); | |
502 | } | |
503 | elsif ($diff < 0) | |
504 | { | |
505 | $xm = $MBI->_copy($x->{_m}); | |
506 | $xm = $MBI->_lsft($xm, $MBI->_new(-$diff), 10); | |
507 | } | |
508 | my $rc = $MBI->_acmp($xm,$ym); | |
509 | $rc = -$rc if $x->{sign} eq '-'; # -124 < -123 | |
510 | $rc <=> 0; | |
511 | } | |
512 | ||
513 | sub bacmp | |
514 | { | |
515 | # Compares 2 values, ignoring their signs. | |
516 | # Returns one of undef, <0, =0, >0. (suitable for sort) | |
517 | ||
518 | # set up parameters | |
519 | my ($self,$x,$y) = (ref($_[0]),@_); | |
520 | # objectify is costly, so avoid it | |
521 | if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1]))) | |
522 | { | |
523 | ($self,$x,$y) = objectify(2,@_); | |
524 | } | |
525 | ||
526 | return $upgrade->bacmp($x,$y) if defined $upgrade && | |
527 | ((!$x->isa($self)) || (!$y->isa($self))); | |
528 | ||
529 | # handle +-inf and NaN's | |
530 | if ($x->{sign} !~ /^[+-]$/ || $y->{sign} !~ /^[+-]$/) | |
531 | { | |
532 | return undef if (($x->{sign} eq $nan) || ($y->{sign} eq $nan)); | |
533 | return 0 if ($x->is_inf() && $y->is_inf()); | |
534 | return 1 if ($x->is_inf() && !$y->is_inf()); | |
535 | return -1; | |
536 | } | |
537 | ||
538 | # shortcut | |
539 | my $xz = $x->is_zero(); | |
540 | my $yz = $y->is_zero(); | |
541 | return 0 if $xz && $yz; # 0 <=> 0 | |
542 | return -1 if $xz && !$yz; # 0 <=> +y | |
543 | return 1 if $yz && !$xz; # +x <=> 0 | |
544 | ||
545 | # adjust so that exponents are equal | |
546 | my $lxm = $MBI->_len($x->{_m}); | |
547 | my $lym = $MBI->_len($y->{_m}); | |
548 | my ($xes,$yes) = (1,1); | |
549 | $xes = -1 if $x->{_es} ne '+'; | |
550 | $yes = -1 if $y->{_es} ne '+'; | |
551 | # the numify somewhat limits our length, but makes it much faster | |
552 | my $lx = $lxm + $xes * $MBI->_num($x->{_e}); | |
553 | my $ly = $lym + $yes * $MBI->_num($y->{_e}); | |
554 | my $l = $lx - $ly; | |
555 | return $l <=> 0 if $l != 0; | |
556 | ||
557 | # lengths (corrected by exponent) are equal | |
558 | # so make mantissa equal-length by padding with zero (shift left) | |
559 | my $diff = $lxm - $lym; | |
560 | my $xm = $x->{_m}; # not yet copy it | |
561 | my $ym = $y->{_m}; | |
562 | if ($diff > 0) | |
563 | { | |
564 | $ym = $MBI->_copy($y->{_m}); | |
565 | $ym = $MBI->_lsft($ym, $MBI->_new($diff), 10); | |
566 | } | |
567 | elsif ($diff < 0) | |
568 | { | |
569 | $xm = $MBI->_copy($x->{_m}); | |
570 | $xm = $MBI->_lsft($xm, $MBI->_new(-$diff), 10); | |
571 | } | |
572 | $MBI->_acmp($xm,$ym); | |
573 | } | |
574 | ||
575 | sub badd | |
576 | { | |
577 | # add second arg (BFLOAT or string) to first (BFLOAT) (modifies first) | |
578 | # return result as BFLOAT | |
579 | ||
580 | # set up parameters | |
581 | my ($self,$x,$y,$a,$p,$r) = (ref($_[0]),@_); | |
582 | # objectify is costly, so avoid it | |
583 | if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1]))) | |
584 | { | |
585 | ($self,$x,$y,$a,$p,$r) = objectify(2,@_); | |
586 | } | |
587 | ||
588 | # inf and NaN handling | |
589 | if (($x->{sign} !~ /^[+-]$/) || ($y->{sign} !~ /^[+-]$/)) | |
590 | { | |
591 | # NaN first | |
592 | return $x->bnan() if (($x->{sign} eq $nan) || ($y->{sign} eq $nan)); | |
593 | # inf handling | |
594 | if (($x->{sign} =~ /^[+-]inf$/) && ($y->{sign} =~ /^[+-]inf$/)) | |
595 | { | |
596 | # +inf++inf or -inf+-inf => same, rest is NaN | |
597 | return $x if $x->{sign} eq $y->{sign}; | |
598 | return $x->bnan(); | |
599 | } | |
600 | # +-inf + something => +inf; something +-inf => +-inf | |
601 | $x->{sign} = $y->{sign}, return $x if $y->{sign} =~ /^[+-]inf$/; | |
602 | return $x; | |
603 | } | |
604 | ||
605 | return $upgrade->badd($x,$y,$a,$p,$r) if defined $upgrade && | |
606 | ((!$x->isa($self)) || (!$y->isa($self))); | |
607 | ||
608 | # speed: no add for 0+y or x+0 | |
609 | return $x->bround($a,$p,$r) if $y->is_zero(); # x+0 | |
610 | if ($x->is_zero()) # 0+y | |
611 | { | |
612 | # make copy, clobbering up x (modify in place!) | |
613 | $x->{_e} = $MBI->_copy($y->{_e}); | |
614 | $x->{_es} = $y->{_es}; | |
615 | $x->{_m} = $MBI->_copy($y->{_m}); | |
616 | $x->{sign} = $y->{sign} || $nan; | |
617 | return $x->round($a,$p,$r,$y); | |
618 | } | |
619 | ||
620 | # take lower of the two e's and adapt m1 to it to match m2 | |
621 | my $e = $y->{_e}; | |
622 | $e = $MBI->_zero() if !defined $e; # if no BFLOAT? | |
623 | $e = $MBI->_copy($e); # make copy (didn't do it yet) | |
624 | ||
625 | my $es; | |
626 | ||
627 | ($e,$es) = _e_sub($e, $x->{_e}, $y->{_es} || '+', $x->{_es}); | |
628 | ||
629 | my $add = $MBI->_copy($y->{_m}); | |
630 | ||
631 | if ($es eq '-') # < 0 | |
632 | { | |
633 | $MBI->_lsft( $x->{_m}, $e, 10); | |
634 | ($x->{_e},$x->{_es}) = _e_add($x->{_e}, $e, $x->{_es}, $es); | |
635 | } | |
636 | elsif (!$MBI->_is_zero($e)) # > 0 | |
637 | { | |
638 | $MBI->_lsft($add, $e, 10); | |
639 | } | |
640 | # else: both e are the same, so just leave them | |
641 | ||
642 | if ($x->{sign} eq $y->{sign}) | |
643 | { | |
644 | # add | |
645 | $x->{_m} = $MBI->_add($x->{_m}, $add); | |
646 | } | |
647 | else | |
648 | { | |
649 | ($x->{_m}, $x->{sign}) = | |
650 | _e_add($x->{_m}, $add, $x->{sign}, $y->{sign}); | |
651 | } | |
652 | ||
653 | # delete trailing zeros, then round | |
654 | $x->bnorm()->round($a,$p,$r,$y); | |
655 | } | |
656 | ||
657 | # sub bsub is inherited from Math::BigInt! | |
658 | ||
659 | sub binc | |
660 | { | |
661 | # increment arg by one | |
662 | my ($self,$x,@r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_); | |
663 | ||
664 | if ($x->{_es} eq '-') | |
665 | { | |
666 | return $x->badd($self->bone(),@r); # digits after dot | |
667 | } | |
668 | ||
669 | if (!$MBI->_is_zero($x->{_e})) # _e == 0 for NaN, inf, -inf | |
670 | { | |
671 | # 1e2 => 100, so after the shift below _m has a '0' as last digit | |
672 | $x->{_m} = $MBI->_lsft($x->{_m}, $x->{_e},10); # 1e2 => 100 | |
673 | $x->{_e} = $MBI->_zero(); # normalize | |
674 | $x->{_es} = '+'; | |
675 | # we know that the last digit of $x will be '1' or '9', depending on the | |
676 | # sign | |
677 | } | |
678 | # now $x->{_e} == 0 | |
679 | if ($x->{sign} eq '+') | |
680 | { | |
681 | $MBI->_inc($x->{_m}); | |
682 | return $x->bnorm()->bround(@r); | |
683 | } | |
684 | elsif ($x->{sign} eq '-') | |
685 | { | |
686 | $MBI->_dec($x->{_m}); | |
687 | $x->{sign} = '+' if $MBI->_is_zero($x->{_m}); # -1 +1 => -0 => +0 | |
688 | return $x->bnorm()->bround(@r); | |
689 | } | |
690 | # inf, nan handling etc | |
691 | $x->badd($self->bone(),@r); # badd() does round | |
692 | } | |
693 | ||
694 | sub bdec | |
695 | { | |
696 | # decrement arg by one | |
697 | my ($self,$x,@r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_); | |
698 | ||
699 | if ($x->{_es} eq '-') | |
700 | { | |
701 | return $x->badd($self->bone('-'),@r); # digits after dot | |
702 | } | |
703 | ||
704 | if (!$MBI->_is_zero($x->{_e})) | |
705 | { | |
706 | $x->{_m} = $MBI->_lsft($x->{_m}, $x->{_e},10); # 1e2 => 100 | |
707 | $x->{_e} = $MBI->_zero(); # normalize | |
708 | $x->{_es} = '+'; | |
709 | } | |
710 | # now $x->{_e} == 0 | |
711 | my $zero = $x->is_zero(); | |
712 | # <= 0 | |
713 | if (($x->{sign} eq '-') || $zero) | |
714 | { | |
715 | $MBI->_inc($x->{_m}); | |
716 | $x->{sign} = '-' if $zero; # 0 => 1 => -1 | |
717 | $x->{sign} = '+' if $MBI->_is_zero($x->{_m}); # -1 +1 => -0 => +0 | |
718 | return $x->bnorm()->round(@r); | |
719 | } | |
720 | # > 0 | |
721 | elsif ($x->{sign} eq '+') | |
722 | { | |
723 | $MBI->_dec($x->{_m}); | |
724 | return $x->bnorm()->round(@r); | |
725 | } | |
726 | # inf, nan handling etc | |
727 | $x->badd($self->bone('-'),@r); # does round | |
728 | } | |
729 | ||
730 | sub DEBUG () { 0; } | |
731 | ||
732 | sub blog | |
733 | { | |
734 | my ($self,$x,$base,$a,$p,$r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_); | |
735 | ||
736 | # $base > 0, $base != 1; if $base == undef default to $base == e | |
737 | # $x >= 0 | |
738 | ||
739 | # we need to limit the accuracy to protect against overflow | |
740 | my $fallback = 0; | |
741 | my ($scale,@params); | |
742 | ($x,@params) = $x->_find_round_parameters($a,$p,$r); | |
743 | ||
744 | # also takes care of the "error in _find_round_parameters?" case | |
745 | return $x->bnan() if $x->{sign} ne '+' || $x->is_zero(); | |
746 | ||
747 | ||
748 | # no rounding at all, so must use fallback | |
749 | if (scalar @params == 0) | |
750 | { | |
751 | # simulate old behaviour | |
752 | $params[0] = $self->div_scale(); # and round to it as accuracy | |
753 | $params[1] = undef; # P = undef | |
754 | $scale = $params[0]+4; # at least four more for proper round | |
755 | $params[2] = $r; # round mode by caller or undef | |
756 | $fallback = 1; # to clear a/p afterwards | |
757 | } | |
758 | else | |
759 | { | |
760 | # the 4 below is empirical, and there might be cases where it is not | |
761 | # enough... | |
762 | $scale = abs($params[0] || $params[1]) + 4; # take whatever is defined | |
763 | } | |
764 | ||
765 | return $x->bzero(@params) if $x->is_one(); | |
766 | # base not defined => base == Euler's constant e | |
767 | if (defined $base) | |
768 | { | |
769 | # make object, since we don't feed it through objectify() to still get the | |
770 | # case of $base == undef | |
771 | $base = $self->new($base) unless ref($base); | |
772 | # $base > 0; $base != 1 | |
773 | return $x->bnan() if $base->is_zero() || $base->is_one() || | |
774 | $base->{sign} ne '+'; | |
775 | # if $x == $base, we know the result must be 1.0 | |
776 | if ($x->bcmp($base) == 0) | |
777 | { | |
778 | $x->bone('+',@params); | |
779 | if ($fallback) | |
780 | { | |
781 | # clear a/p after round, since user did not request it | |
782 | delete $x->{_a}; delete $x->{_p}; | |
783 | } | |
784 | return $x; | |
785 | } | |
786 | } | |
787 | ||
788 | # when user set globals, they would interfere with our calculation, so | |
789 | # disable them and later re-enable them | |
790 | no strict 'refs'; | |
791 | my $abr = "$self\::accuracy"; my $ab = $$abr; $$abr = undef; | |
792 | my $pbr = "$self\::precision"; my $pb = $$pbr; $$pbr = undef; | |
793 | # we also need to disable any set A or P on $x (_find_round_parameters took | |
794 | # them already into account), since these would interfere, too | |
795 | delete $x->{_a}; delete $x->{_p}; | |
796 | # need to disable $upgrade in BigInt, to avoid deep recursion | |
797 | local $Math::BigInt::upgrade = undef; | |
798 | local $Math::BigFloat::downgrade = undef; | |
799 | ||
800 | # upgrade $x if $x is not a BigFloat (handle BigInt input) | |
801 | if (!$x->isa('Math::BigFloat')) | |
802 | { | |
803 | $x = Math::BigFloat->new($x); | |
804 | $self = ref($x); | |
805 | } | |
806 | ||
807 | my $done = 0; | |
808 | ||
809 | # If the base is defined and an integer, try to calculate integer result | |
810 | # first. This is very fast, and in case the real result was found, we can | |
811 | # stop right here. | |
812 | if (defined $base && $base->is_int() && $x->is_int()) | |
813 | { | |
814 | my $i = $MBI->_copy( $x->{_m} ); | |
815 | $MBI->_lsft( $i, $x->{_e}, 10 ) unless $MBI->_is_zero($x->{_e}); | |
816 | my $int = Math::BigInt->bzero(); | |
817 | $int->{value} = $i; | |
818 | $int->blog($base->as_number()); | |
819 | # if ($exact) | |
820 | if ($base->as_number()->bpow($int) == $x) | |
821 | { | |
822 | # found result, return it | |
823 | $x->{_m} = $int->{value}; | |
824 | $x->{_e} = $MBI->_zero(); | |
825 | $x->{_es} = '+'; | |
826 | $x->bnorm(); | |
827 | $done = 1; | |
828 | } | |
829 | } | |
830 | ||
831 | if ($done == 0) | |
832 | { | |
833 | # first calculate the log to base e (using reduction by 10 (and probably 2)) | |
834 | $self->_log_10($x,$scale); | |
835 | ||
836 | # and if a different base was requested, convert it | |
837 | if (defined $base) | |
838 | { | |
839 | $base = Math::BigFloat->new($base) unless $base->isa('Math::BigFloat'); | |
840 | # not ln, but some other base (don't modify $base) | |
841 | $x->bdiv( $base->copy()->blog(undef,$scale), $scale ); | |
842 | } | |
843 | } | |
844 | ||
845 | # shortcut to not run through _find_round_parameters again | |
846 | if (defined $params[0]) | |
847 | { | |
848 | $x->bround($params[0],$params[2]); # then round accordingly | |
849 | } | |
850 | else | |
851 | { | |
852 | $x->bfround($params[1],$params[2]); # then round accordingly | |
853 | } | |
854 | if ($fallback) | |
855 | { | |
856 | # clear a/p after round, since user did not request it | |
857 | delete $x->{_a}; delete $x->{_p}; | |
858 | } | |
859 | # restore globals | |
860 | $$abr = $ab; $$pbr = $pb; | |
861 | ||
862 | $x; | |
863 | } | |
864 | ||
865 | sub _log | |
866 | { | |
867 | # internal log function to calculate ln() based on Taylor series. | |
868 | # Modifies $x in place. | |
869 | my ($self,$x,$scale) = @_; | |
870 | ||
871 | # in case of $x == 1, result is 0 | |
872 | return $x->bzero() if $x->is_one(); | |
873 | ||
874 | # http://www.efunda.com/math/taylor_series/logarithmic.cfm?search_string=log | |
875 | ||
876 | # u = x-1, v = x+1 | |
877 | # _ _ | |
878 | # Taylor: | u 1 u^3 1 u^5 | | |
879 | # ln (x) = 2 | --- + - * --- + - * --- + ... | x > 0 | |
880 | # |_ v 3 v^3 5 v^5 _| | |
881 | ||
882 | # This takes much more steps to calculate the result and is thus not used | |
883 | # u = x-1 | |
884 | # _ _ | |
885 | # Taylor: | u 1 u^2 1 u^3 | | |
886 | # ln (x) = 2 | --- + - * --- + - * --- + ... | x > 1/2 | |
887 | # |_ x 2 x^2 3 x^3 _| | |
888 | ||
889 | my ($limit,$v,$u,$below,$factor,$two,$next,$over,$f); | |
890 | ||
891 | $v = $x->copy(); $v->binc(); # v = x+1 | |
892 | $x->bdec(); $u = $x->copy(); # u = x-1; x = x-1 | |
893 | $x->bdiv($v,$scale); # first term: u/v | |
894 | $below = $v->copy(); | |
895 | $over = $u->copy(); | |
896 | $u *= $u; $v *= $v; # u^2, v^2 | |
897 | $below->bmul($v); # u^3, v^3 | |
898 | $over->bmul($u); | |
899 | $factor = $self->new(3); $f = $self->new(2); | |
900 | ||
901 | my $steps = 0 if DEBUG; | |
902 | $limit = $self->new("1E-". ($scale-1)); | |
903 | while (3 < 5) | |
904 | { | |
905 | # we calculate the next term, and add it to the last | |
906 | # when the next term is below our limit, it won't affect the outcome | |
907 | # anymore, so we stop | |
908 | ||
909 | # calculating the next term simple from over/below will result in quite | |
910 | # a time hog if the input has many digits, since over and below will | |
911 | # accumulate more and more digits, and the result will also have many | |
912 | # digits, but in the end it is rounded to $scale digits anyway. So if we | |
913 | # round $over and $below first, we save a lot of time for the division | |
914 | # (not with log(1.2345), but try log (123**123) to see what I mean. This | |
915 | # can introduce a rounding error if the division result would be f.i. | |
916 | # 0.1234500000001 and we round it to 5 digits it would become 0.12346, but | |
917 | # if we truncated $over and $below we might get 0.12345. Does this matter | |
918 | # for the end result? So we give $over and $below 4 more digits to be | |
919 | # on the safe side (unscientific error handling as usual... :+D | |
920 | ||
921 | $next = $over->copy->bround($scale+4)->bdiv( | |
922 | $below->copy->bmul($factor)->bround($scale+4), | |
923 | $scale); | |
924 | ||
925 | ## old version: | |
926 | ## $next = $over->copy()->bdiv($below->copy()->bmul($factor),$scale); | |
927 | ||
928 | last if $next->bacmp($limit) <= 0; | |
929 | ||
930 | delete $next->{_a}; delete $next->{_p}; | |
931 | $x->badd($next); | |
932 | # calculate things for the next term | |
933 | $over *= $u; $below *= $v; $factor->badd($f); | |
934 | if (DEBUG) | |
935 | { | |
936 | $steps++; print "step $steps = $x\n" if $steps % 10 == 0; | |
937 | } | |
938 | } | |
939 | $x->bmul($f); # $x *= 2 | |
940 | print "took $steps steps\n" if DEBUG; | |
941 | } | |
942 | ||
943 | sub _log_10 | |
944 | { | |
945 | # Internal log function based on reducing input to the range of 0.1 .. 9.99 | |
946 | # and then "correcting" the result to the proper one. Modifies $x in place. | |
947 | my ($self,$x,$scale) = @_; | |
948 | ||
949 | # taking blog() from numbers greater than 10 takes a *very long* time, so we | |
950 | # break the computation down into parts based on the observation that: | |
951 | # blog(x*y) = blog(x) + blog(y) | |
952 | # We set $y here to multiples of 10 so that $x is below 1 (the smaller $x is | |
953 | # the faster it get's, especially because 2*$x takes about 10 times as long, | |
954 | # so by dividing $x by 10 we make it at least factor 100 faster...) | |
955 | ||
956 | # The same observation is valid for numbers smaller than 0.1 (e.g. computing | |
957 | # log(1) is fastest, and the farther away we get from 1, the longer it takes) | |
958 | # so we also 'break' this down by multiplying $x with 10 and subtract the | |
959 | # log(10) afterwards to get the correct result. | |
960 | ||
961 | # calculate nr of digits before dot | |
962 | my $dbd = $MBI->_num($x->{_e}); | |
963 | $dbd = -$dbd if $x->{_es} eq '-'; | |
964 | $dbd += $MBI->_len($x->{_m}); | |
965 | ||
966 | # more than one digit (e.g. at least 10), but *not* exactly 10 to avoid | |
967 | # infinite recursion | |
968 | ||
969 | my $calc = 1; # do some calculation? | |
970 | ||
971 | # disable the shortcut for 10, since we need log(10) and this would recurse | |
972 | # infinitely deep | |
973 | if ($x->{_es} eq '+' && $MBI->_is_one($x->{_e}) && $MBI->_is_one($x->{_m})) | |
974 | { | |
975 | $dbd = 0; # disable shortcut | |
976 | # we can use the cached value in these cases | |
977 | if ($scale <= $LOG_10_A) | |
978 | { | |
979 | $x->bzero(); $x->badd($LOG_10); | |
980 | $calc = 0; # no need to calc, but round | |
981 | } | |
982 | } | |
983 | else | |
984 | { | |
985 | # disable the shortcut for 2, since we maybe have it cached | |
986 | if (($MBI->_is_zero($x->{_e}) && $MBI->_is_two($x->{_m}))) | |
987 | { | |
988 | $dbd = 0; # disable shortcut | |
989 | # we can use the cached value in these cases | |
990 | if ($scale <= $LOG_2_A) | |
991 | { | |
992 | $x->bzero(); $x->badd($LOG_2); | |
993 | $calc = 0; # no need to calc, but round | |
994 | } | |
995 | } | |
996 | } | |
997 | ||
998 | # if $x = 0.1, we know the result must be 0-log(10) | |
999 | if ($calc != 0 && $x->{_es} eq '-' && $MBI->_is_one($x->{_e}) && | |
1000 | $MBI->_is_one($x->{_m})) | |
1001 | { | |
1002 | $dbd = 0; # disable shortcut | |
1003 | # we can use the cached value in these cases | |
1004 | if ($scale <= $LOG_10_A) | |
1005 | { | |
1006 | $x->bzero(); $x->bsub($LOG_10); | |
1007 | $calc = 0; # no need to calc, but round | |
1008 | } | |
1009 | } | |
1010 | ||
1011 | return if $calc == 0; # already have the result | |
1012 | ||
1013 | # default: these correction factors are undef and thus not used | |
1014 | my $l_10; # value of ln(10) to A of $scale | |
1015 | my $l_2; # value of ln(2) to A of $scale | |
1016 | ||
1017 | # $x == 2 => 1, $x == 13 => 2, $x == 0.1 => 0, $x == 0.01 => -1 | |
1018 | # so don't do this shortcut for 1 or 0 | |
1019 | if (($dbd > 1) || ($dbd < 0)) | |
1020 | { | |
1021 | # convert our cached value to an object if not already (avoid doing this | |
1022 | # at import() time, since not everybody needs this) | |
1023 | $LOG_10 = $self->new($LOG_10,undef,undef) unless ref $LOG_10; | |
1024 | ||
1025 | #print "x = $x, dbd = $dbd, calc = $calc\n"; | |
1026 | # got more than one digit before the dot, or more than one zero after the | |
1027 | # dot, so do: | |
1028 | # log(123) == log(1.23) + log(10) * 2 | |
1029 | # log(0.0123) == log(1.23) - log(10) * 2 | |
1030 | ||
1031 | if ($scale <= $LOG_10_A) | |
1032 | { | |
1033 | # use cached value | |
1034 | $l_10 = $LOG_10->copy(); # copy for mul | |
1035 | } | |
1036 | else | |
1037 | { | |
1038 | # else: slower, compute it (but don't cache it, because it could be big) | |
1039 | # also disable downgrade for this code path | |
1040 | local $Math::BigFloat::downgrade = undef; | |
1041 | $l_10 = $self->new(10)->blog(undef,$scale); # scale+4, actually | |
1042 | } | |
1043 | $dbd-- if ($dbd > 1); # 20 => dbd=2, so make it dbd=1 | |
1044 | $l_10->bmul( $self->new($dbd)); # log(10) * (digits_before_dot-1) | |
1045 | my $dbd_sign = '+'; | |
1046 | if ($dbd < 0) | |
1047 | { | |
1048 | $dbd = -$dbd; | |
1049 | $dbd_sign = '-'; | |
1050 | } | |
1051 | ($x->{_e}, $x->{_es}) = | |
1052 | _e_sub( $x->{_e}, $MBI->_new($dbd), $x->{_es}, $dbd_sign); # 123 => 1.23 | |
1053 | ||
1054 | } | |
1055 | ||
1056 | # Now: 0.1 <= $x < 10 (and possible correction in l_10) | |
1057 | ||
1058 | ### Since $x in the range 0.5 .. 1.5 is MUCH faster, we do a repeated div | |
1059 | ### or mul by 2 (maximum times 3, since x < 10 and x > 0.1) | |
1060 | ||
1061 | $HALF = $self->new($HALF) unless ref($HALF); | |
1062 | ||
1063 | my $twos = 0; # default: none (0 times) | |
1064 | my $two = $self->new(2); | |
1065 | while ($x->bacmp($HALF) <= 0) | |
1066 | { | |
1067 | $twos--; $x->bmul($two); | |
1068 | } | |
1069 | while ($x->bacmp($two) >= 0) | |
1070 | { | |
1071 | $twos++; $x->bdiv($two,$scale+4); # keep all digits | |
1072 | } | |
1073 | # $twos > 0 => did mul 2, < 0 => did div 2 (never both) | |
1074 | # calculate correction factor based on ln(2) | |
1075 | if ($twos != 0) | |
1076 | { | |
1077 | $LOG_2 = $self->new($LOG_2,undef,undef) unless ref $LOG_2; | |
1078 | if ($scale <= $LOG_2_A) | |
1079 | { | |
1080 | # use cached value | |
1081 | $l_2 = $LOG_2->copy(); # copy for mul | |
1082 | } | |
1083 | else | |
1084 | { | |
1085 | # else: slower, compute it (but don't cache it, because it could be big) | |
1086 | # also disable downgrade for this code path | |
1087 | local $Math::BigFloat::downgrade = undef; | |
1088 | $l_2 = $two->blog(undef,$scale); # scale+4, actually | |
1089 | } | |
1090 | $l_2->bmul($twos); # * -2 => subtract, * 2 => add | |
1091 | } | |
1092 | ||
1093 | $self->_log($x,$scale); # need to do the "normal" way | |
1094 | $x->badd($l_10) if defined $l_10; # correct it by ln(10) | |
1095 | $x->badd($l_2) if defined $l_2; # and maybe by ln(2) | |
1096 | # all done, $x contains now the result | |
1097 | } | |
1098 | ||
1099 | sub blcm | |
1100 | { | |
1101 | # (BFLOAT or num_str, BFLOAT or num_str) return BFLOAT | |
1102 | # does not modify arguments, but returns new object | |
1103 | # Lowest Common Multiplicator | |
1104 | ||
1105 | my ($self,@arg) = objectify(0,@_); | |
1106 | my $x = $self->new(shift @arg); | |
1107 | while (@arg) { $x = Math::BigInt::__lcm($x,shift @arg); } | |
1108 | $x; | |
1109 | } | |
1110 | ||
1111 | sub bgcd | |
1112 | { | |
1113 | # (BINT or num_str, BINT or num_str) return BINT | |
1114 | # does not modify arguments, but returns new object | |
1115 | ||
1116 | my $y = shift; | |
1117 | $y = __PACKAGE__->new($y) if !ref($y); | |
1118 | my $self = ref($y); | |
1119 | my $x = $y->copy()->babs(); # keep arguments | |
1120 | ||
1121 | return $x->bnan() if $x->{sign} !~ /^[+-]$/ # x NaN? | |
1122 | || !$x->is_int(); # only for integers now | |
1123 | ||
1124 | while (@_) | |
1125 | { | |
1126 | my $t = shift; $t = $self->new($t) if !ref($t); | |
1127 | $y = $t->copy()->babs(); | |
1128 | ||
1129 | return $x->bnan() if $y->{sign} !~ /^[+-]$/ # y NaN? | |
1130 | || !$y->is_int(); # only for integers now | |
1131 | ||
1132 | # greatest common divisor | |
1133 | while (! $y->is_zero()) | |
1134 | { | |
1135 | ($x,$y) = ($y->copy(), $x->copy()->bmod($y)); | |
1136 | } | |
1137 | ||
1138 | last if $x->is_one(); | |
1139 | } | |
1140 | $x; | |
1141 | } | |
1142 | ||
1143 | ############################################################################## | |
1144 | ||
1145 | sub _e_add | |
1146 | { | |
1147 | # Internal helper sub to take two positive integers and their signs and | |
1148 | # then add them. Input ($CALC,$CALC,('+'|'-'),('+'|'-')), | |
1149 | # output ($CALC,('+'|'-')) | |
1150 | my ($x,$y,$xs,$ys) = @_; | |
1151 | ||
1152 | # if the signs are equal we can add them (-5 + -3 => -(5 + 3) => -8) | |
1153 | if ($xs eq $ys) | |
1154 | { | |
1155 | $x = $MBI->_add ($x, $y ); # a+b | |
1156 | # the sign follows $xs | |
1157 | return ($x, $xs); | |
1158 | } | |
1159 | ||
1160 | my $a = $MBI->_acmp($x,$y); | |
1161 | if ($a > 0) | |
1162 | { | |
1163 | $x = $MBI->_sub ($x , $y); # abs sub | |
1164 | } | |
1165 | elsif ($a == 0) | |
1166 | { | |
1167 | $x = $MBI->_zero(); # result is 0 | |
1168 | $xs = '+'; | |
1169 | } | |
1170 | else # a < 0 | |
1171 | { | |
1172 | $x = $MBI->_sub ( $y, $x, 1 ); # abs sub | |
1173 | $xs = $ys; | |
1174 | } | |
1175 | ($x,$xs); | |
1176 | } | |
1177 | ||
1178 | sub _e_sub | |
1179 | { | |
1180 | # Internal helper sub to take two positive integers and their signs and | |
1181 | # then subtract them. Input ($CALC,$CALC,('+'|'-'),('+'|'-')), | |
1182 | # output ($CALC,('+'|'-')) | |
1183 | my ($x,$y,$xs,$ys) = @_; | |
1184 | ||
1185 | # flip sign | |
1186 | $ys =~ tr/+-/-+/; | |
1187 | _e_add($x,$y,$xs,$ys); # call add (does subtract now) | |
1188 | } | |
1189 | ||
1190 | ############################################################################### | |
1191 | # is_foo methods (is_negative, is_positive are inherited from BigInt) | |
1192 | ||
1193 | sub is_int | |
1194 | { | |
1195 | # return true if arg (BFLOAT or num_str) is an integer | |
1196 | my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_); | |
1197 | ||
1198 | return 1 if ($x->{sign} =~ /^[+-]$/) && # NaN and +-inf aren't | |
1199 | $x->{_es} eq '+'; # 1e-1 => no integer | |
1200 | 0; | |
1201 | } | |
1202 | ||
1203 | sub is_zero | |
1204 | { | |
1205 | # return true if arg (BFLOAT or num_str) is zero | |
1206 | my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_); | |
1207 | ||
1208 | return 1 if $x->{sign} eq '+' && $MBI->_is_zero($x->{_m}); | |
1209 | 0; | |
1210 | } | |
1211 | ||
1212 | sub is_one | |
1213 | { | |
1214 | # return true if arg (BFLOAT or num_str) is +1 or -1 if signis given | |
1215 | my ($self,$x,$sign) = ref($_[0]) ? (undef,@_) : objectify(1,@_); | |
1216 | ||
1217 | $sign = '+' if !defined $sign || $sign ne '-'; | |
1218 | return 1 | |
1219 | if ($x->{sign} eq $sign && | |
1220 | $MBI->_is_zero($x->{_e}) && $MBI->_is_one($x->{_m})); | |
1221 | 0; | |
1222 | } | |
1223 | ||
1224 | sub is_odd | |
1225 | { | |
1226 | # return true if arg (BFLOAT or num_str) is odd or false if even | |
1227 | my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_); | |
1228 | ||
1229 | return 1 if ($x->{sign} =~ /^[+-]$/) && # NaN & +-inf aren't | |
1230 | ($MBI->_is_zero($x->{_e}) && $MBI->_is_odd($x->{_m})); | |
1231 | 0; | |
1232 | } | |
1233 | ||
1234 | sub is_even | |
1235 | { | |
1236 | # return true if arg (BINT or num_str) is even or false if odd | |
1237 | my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_); | |
1238 | ||
1239 | return 0 if $x->{sign} !~ /^[+-]$/; # NaN & +-inf aren't | |
1240 | return 1 if ($x->{_es} eq '+' # 123.45 is never | |
1241 | && $MBI->_is_even($x->{_m})); # but 1200 is | |
1242 | 0; | |
1243 | } | |
1244 | ||
1245 | sub bmul | |
1246 | { | |
1247 | # multiply two numbers -- stolen from Knuth Vol 2 pg 233 | |
1248 | # (BINT or num_str, BINT or num_str) return BINT | |
1249 | ||
1250 | # set up parameters | |
1251 | my ($self,$x,$y,$a,$p,$r) = (ref($_[0]),@_); | |
1252 | # objectify is costly, so avoid it | |
1253 | if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1]))) | |
1254 | { | |
1255 | ($self,$x,$y,$a,$p,$r) = objectify(2,@_); | |
1256 | } | |
1257 | ||
1258 | return $x->bnan() if (($x->{sign} eq $nan) || ($y->{sign} eq $nan)); | |
1259 | ||
1260 | # inf handling | |
1261 | if (($x->{sign} =~ /^[+-]inf$/) || ($y->{sign} =~ /^[+-]inf$/)) | |
1262 | { | |
1263 | return $x->bnan() if $x->is_zero() || $y->is_zero(); | |
1264 | # result will always be +-inf: | |
1265 | # +inf * +/+inf => +inf, -inf * -/-inf => +inf | |
1266 | # +inf * -/-inf => -inf, -inf * +/+inf => -inf | |
1267 | return $x->binf() if ($x->{sign} =~ /^\+/ && $y->{sign} =~ /^\+/); | |
1268 | return $x->binf() if ($x->{sign} =~ /^-/ && $y->{sign} =~ /^-/); | |
1269 | return $x->binf('-'); | |
1270 | } | |
1271 | # handle result = 0 | |
1272 | return $x->bzero() if $x->is_zero() || $y->is_zero(); | |
1273 | ||
1274 | return $upgrade->bmul($x,$y,$a,$p,$r) if defined $upgrade && | |
1275 | ((!$x->isa($self)) || (!$y->isa($self))); | |
1276 | ||
1277 | # aEb * cEd = (a*c)E(b+d) | |
1278 | $MBI->_mul($x->{_m},$y->{_m}); | |
1279 | ($x->{_e}, $x->{_es}) = _e_add($x->{_e}, $y->{_e}, $x->{_es}, $y->{_es}); | |
1280 | ||
1281 | # adjust sign: | |
1282 | $x->{sign} = $x->{sign} ne $y->{sign} ? '-' : '+'; | |
1283 | return $x->bnorm()->round($a,$p,$r,$y); | |
1284 | } | |
1285 | ||
1286 | sub bdiv | |
1287 | { | |
1288 | # (dividend: BFLOAT or num_str, divisor: BFLOAT or num_str) return | |
1289 | # (BFLOAT,BFLOAT) (quo,rem) or BFLOAT (only rem) | |
1290 | ||
1291 | # set up parameters | |
1292 | my ($self,$x,$y,$a,$p,$r) = (ref($_[0]),@_); | |
1293 | # objectify is costly, so avoid it | |
1294 | if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1]))) | |
1295 | { | |
1296 | ($self,$x,$y,$a,$p,$r) = objectify(2,@_); | |
1297 | } | |
1298 | ||
1299 | return $self->_div_inf($x,$y) | |
1300 | if (($x->{sign} !~ /^[+-]$/) || ($y->{sign} !~ /^[+-]$/) || $y->is_zero()); | |
1301 | ||
1302 | # x== 0 # also: or y == 1 or y == -1 | |
1303 | return wantarray ? ($x,$self->bzero()) : $x if $x->is_zero(); | |
1304 | ||
1305 | # upgrade ? | |
1306 | return $upgrade->bdiv($upgrade->new($x),$y,$a,$p,$r) if defined $upgrade; | |
1307 | ||
1308 | # we need to limit the accuracy to protect against overflow | |
1309 | my $fallback = 0; | |
1310 | my (@params,$scale); | |
1311 | ($x,@params) = $x->_find_round_parameters($a,$p,$r,$y); | |
1312 | ||
1313 | return $x if $x->is_nan(); # error in _find_round_parameters? | |
1314 | ||
1315 | # no rounding at all, so must use fallback | |
1316 | if (scalar @params == 0) | |
1317 | { | |
1318 | # simulate old behaviour | |
1319 | $params[0] = $self->div_scale(); # and round to it as accuracy | |
1320 | $scale = $params[0]+4; # at least four more for proper round | |
1321 | $params[2] = $r; # round mode by caller or undef | |
1322 | $fallback = 1; # to clear a/p afterwards | |
1323 | } | |
1324 | else | |
1325 | { | |
1326 | # the 4 below is empirical, and there might be cases where it is not | |
1327 | # enough... | |
1328 | $scale = abs($params[0] || $params[1]) + 4; # take whatever is defined | |
1329 | } | |
1330 | ||
1331 | my $rem; $rem = $self->bzero() if wantarray; | |
1332 | ||
1333 | $y = $self->new($y) unless $y->isa('Math::BigFloat'); | |
1334 | ||
1335 | my $lx = $MBI->_len($x->{_m}); my $ly = $MBI->_len($y->{_m}); | |
1336 | $scale = $lx if $lx > $scale; | |
1337 | $scale = $ly if $ly > $scale; | |
1338 | my $diff = $ly - $lx; | |
1339 | $scale += $diff if $diff > 0; # if lx << ly, but not if ly << lx! | |
1340 | ||
1341 | # already handled inf/NaN/-inf above: | |
1342 | ||
1343 | # check that $y is not 1 nor -1 and cache the result: | |
1344 | my $y_not_one = !($MBI->_is_zero($y->{_e}) && $MBI->_is_one($y->{_m})); | |
1345 | ||
1346 | # flipping the sign of $y will also flip the sign of $x for the special | |
1347 | # case of $x->bsub($x); so we can catch it below: | |
1348 | my $xsign = $x->{sign}; | |
1349 | $y->{sign} =~ tr/+-/-+/; | |
1350 | ||
1351 | if ($xsign ne $x->{sign}) | |
1352 | { | |
1353 | # special case of $x /= $x results in 1 | |
1354 | $x->bone(); # "fixes" also sign of $y, since $x is $y | |
1355 | } | |
1356 | else | |
1357 | { | |
1358 | # correct $y's sign again | |
1359 | $y->{sign} =~ tr/+-/-+/; | |
1360 | # continue with normal div code: | |
1361 | ||
1362 | # make copy of $x in case of list context for later reminder calculation | |
1363 | if (wantarray && $y_not_one) | |
1364 | { | |
1365 | $rem = $x->copy(); | |
1366 | } | |
1367 | ||
1368 | $x->{sign} = $x->{sign} ne $y->sign() ? '-' : '+'; | |
1369 | ||
1370 | # check for / +-1 ( +/- 1E0) | |
1371 | if ($y_not_one) | |
1372 | { | |
1373 | # promote BigInts and it's subclasses (except when already a BigFloat) | |
1374 | $y = $self->new($y) unless $y->isa('Math::BigFloat'); | |
1375 | ||
1376 | # calculate the result to $scale digits and then round it | |
1377 | # a * 10 ** b / c * 10 ** d => a/c * 10 ** (b-d) | |
1378 | $MBI->_lsft($x->{_m},$MBI->_new($scale),10); | |
1379 | $MBI->_div ($x->{_m},$y->{_m}); # a/c | |
1380 | ||
1381 | # correct exponent of $x | |
1382 | ($x->{_e},$x->{_es}) = _e_sub($x->{_e}, $y->{_e}, $x->{_es}, $y->{_es}); | |
1383 | # correct for 10**scale | |
1384 | ($x->{_e},$x->{_es}) = _e_sub($x->{_e}, $MBI->_new($scale), $x->{_es}, '+'); | |
1385 | $x->bnorm(); # remove trailing 0's | |
1386 | } | |
1387 | } # ende else $x != $y | |
1388 | ||
1389 | # shortcut to not run through _find_round_parameters again | |
1390 | if (defined $params[0]) | |
1391 | { | |
1392 | delete $x->{_a}; # clear before round | |
1393 | $x->bround($params[0],$params[2]); # then round accordingly | |
1394 | } | |
1395 | else | |
1396 | { | |
1397 | delete $x->{_p}; # clear before round | |
1398 | $x->bfround($params[1],$params[2]); # then round accordingly | |
1399 | } | |
1400 | if ($fallback) | |
1401 | { | |
1402 | # clear a/p after round, since user did not request it | |
1403 | delete $x->{_a}; delete $x->{_p}; | |
1404 | } | |
1405 | ||
1406 | if (wantarray) | |
1407 | { | |
1408 | if ($y_not_one) | |
1409 | { | |
1410 | $rem->bmod($y,@params); # copy already done | |
1411 | } | |
1412 | if ($fallback) | |
1413 | { | |
1414 | # clear a/p after round, since user did not request it | |
1415 | delete $rem->{_a}; delete $rem->{_p}; | |
1416 | } | |
1417 | return ($x,$rem); | |
1418 | } | |
1419 | $x; | |
1420 | } | |
1421 | ||
1422 | sub bmod | |
1423 | { | |
1424 | # (dividend: BFLOAT or num_str, divisor: BFLOAT or num_str) return reminder | |
1425 | ||
1426 | # set up parameters | |
1427 | my ($self,$x,$y,$a,$p,$r) = (ref($_[0]),@_); | |
1428 | # objectify is costly, so avoid it | |
1429 | if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1]))) | |
1430 | { | |
1431 | ($self,$x,$y,$a,$p,$r) = objectify(2,@_); | |
1432 | } | |
1433 | ||
1434 | # handle NaN, inf, -inf | |
1435 | if (($x->{sign} !~ /^[+-]$/) || ($y->{sign} !~ /^[+-]$/)) | |
1436 | { | |
1437 | my ($d,$re) = $self->SUPER::_div_inf($x,$y); | |
1438 | $x->{sign} = $re->{sign}; | |
1439 | $x->{_e} = $re->{_e}; | |
1440 | $x->{_m} = $re->{_m}; | |
1441 | return $x->round($a,$p,$r,$y); | |
1442 | } | |
1443 | if ($y->is_zero()) | |
1444 | { | |
1445 | return $x->bnan() if $x->is_zero(); | |
1446 | return $x; | |
1447 | } | |
1448 | ||
1449 | return $x->bzero() if $x->is_zero() | |
1450 | || ($x->is_int() && | |
1451 | # check that $y == +1 or $y == -1: | |
1452 | ($MBI->_is_zero($y->{_e}) && $MBI->_is_one($y->{_m}))); | |
1453 | ||
1454 | my $cmp = $x->bacmp($y); # equal or $x < $y? | |
1455 | return $x->bzero($a,$p) if $cmp == 0; # $x == $y => result 0 | |
1456 | ||
1457 | # only $y of the operands negative? | |
1458 | my $neg = 0; $neg = 1 if $x->{sign} ne $y->{sign}; | |
1459 | ||
1460 | $x->{sign} = $y->{sign}; # calc sign first | |
1461 | return $x->round($a,$p,$r) if $cmp < 0 && $neg == 0; # $x < $y => result $x | |
1462 | ||
1463 | my $ym = $MBI->_copy($y->{_m}); | |
1464 | ||
1465 | # 2e1 => 20 | |
1466 | $MBI->_lsft( $ym, $y->{_e}, 10) | |
1467 | if $y->{_es} eq '+' && !$MBI->_is_zero($y->{_e}); | |
1468 | ||
1469 | # if $y has digits after dot | |
1470 | my $shifty = 0; # correct _e of $x by this | |
1471 | if ($y->{_es} eq '-') # has digits after dot | |
1472 | { | |
1473 | # 123 % 2.5 => 1230 % 25 => 5 => 0.5 | |
1474 | $shifty = $MBI->_num($y->{_e}); # no more digits after dot | |
1475 | $MBI->_lsft($x->{_m}, $y->{_e}, 10);# 123 => 1230, $y->{_m} is already 25 | |
1476 | } | |
1477 | # $ym is now mantissa of $y based on exponent 0 | |
1478 | ||
1479 | my $shiftx = 0; # correct _e of $x by this | |
1480 | if ($x->{_es} eq '-') # has digits after dot | |
1481 | { | |
1482 | # 123.4 % 20 => 1234 % 200 | |
1483 | $shiftx = $MBI->_num($x->{_e}); # no more digits after dot | |
1484 | $MBI->_lsft($ym, $x->{_e}, 10); # 123 => 1230 | |
1485 | } | |
1486 | # 123e1 % 20 => 1230 % 20 | |
1487 | if ($x->{_es} eq '+' && !$MBI->_is_zero($x->{_e})) | |
1488 | { | |
1489 | $MBI->_lsft( $x->{_m}, $x->{_e},10); # es => '+' here | |
1490 | } | |
1491 | ||
1492 | $x->{_e} = $MBI->_new($shiftx); | |
1493 | $x->{_es} = '+'; | |
1494 | $x->{_es} = '-' if $shiftx != 0 || $shifty != 0; | |
1495 | $MBI->_add( $x->{_e}, $MBI->_new($shifty)) if $shifty != 0; | |
1496 | ||
1497 | # now mantissas are equalized, exponent of $x is adjusted, so calc result | |
1498 | ||
1499 | $x->{_m} = $MBI->_mod( $x->{_m}, $ym); | |
1500 | ||
1501 | $x->{sign} = '+' if $MBI->_is_zero($x->{_m}); # fix sign for -0 | |
1502 | $x->bnorm(); | |
1503 | ||
1504 | if ($neg != 0) # one of them negative => correct in place | |
1505 | { | |
1506 | my $r = $y - $x; | |
1507 | $x->{_m} = $r->{_m}; | |
1508 | $x->{_e} = $r->{_e}; | |
1509 | $x->{_es} = $r->{_es}; | |
1510 | $x->{sign} = '+' if $MBI->_is_zero($x->{_m}); # fix sign for -0 | |
1511 | $x->bnorm(); | |
1512 | } | |
1513 | ||
1514 | $x->round($a,$p,$r,$y); # round and return | |
1515 | } | |
1516 | ||
1517 | sub broot | |
1518 | { | |
1519 | # calculate $y'th root of $x | |
1520 | ||
1521 | # set up parameters | |
1522 | my ($self,$x,$y,$a,$p,$r) = (ref($_[0]),@_); | |
1523 | # objectify is costly, so avoid it | |
1524 | if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1]))) | |
1525 | { | |
1526 | ($self,$x,$y,$a,$p,$r) = objectify(2,@_); | |
1527 | } | |
1528 | ||
1529 | # NaN handling: $x ** 1/0, x or y NaN, or y inf/-inf or y == 0 | |
1530 | return $x->bnan() if $x->{sign} !~ /^\+/ || $y->is_zero() || | |
1531 | $y->{sign} !~ /^\+$/; | |
1532 | ||
1533 | return $x if $x->is_zero() || $x->is_one() || $x->is_inf() || $y->is_one(); | |
1534 | ||
1535 | # we need to limit the accuracy to protect against overflow | |
1536 | my $fallback = 0; | |
1537 | my (@params,$scale); | |
1538 | ($x,@params) = $x->_find_round_parameters($a,$p,$r); | |
1539 | ||
1540 | return $x if $x->is_nan(); # error in _find_round_parameters? | |
1541 | ||
1542 | # no rounding at all, so must use fallback | |
1543 | if (scalar @params == 0) | |
1544 | { | |
1545 | # simulate old behaviour | |
1546 | $params[0] = $self->div_scale(); # and round to it as accuracy | |
1547 | $scale = $params[0]+4; # at least four more for proper round | |
1548 | $params[2] = $r; # iound mode by caller or undef | |
1549 | $fallback = 1; # to clear a/p afterwards | |
1550 | } | |
1551 | else | |
1552 | { | |
1553 | # the 4 below is empirical, and there might be cases where it is not | |
1554 | # enough... | |
1555 | $scale = abs($params[0] || $params[1]) + 4; # take whatever is defined | |
1556 | } | |
1557 | ||
1558 | # when user set globals, they would interfere with our calculation, so | |
1559 | # disable them and later re-enable them | |
1560 | no strict 'refs'; | |
1561 | my $abr = "$self\::accuracy"; my $ab = $$abr; $$abr = undef; | |
1562 | my $pbr = "$self\::precision"; my $pb = $$pbr; $$pbr = undef; | |
1563 | # we also need to disable any set A or P on $x (_find_round_parameters took | |
1564 | # them already into account), since these would interfere, too | |
1565 | delete $x->{_a}; delete $x->{_p}; | |
1566 | # need to disable $upgrade in BigInt, to avoid deep recursion | |
1567 | local $Math::BigInt::upgrade = undef; # should be really parent class vs MBI | |
1568 | ||
1569 | # remember sign and make $x positive, since -4 ** (1/2) => -2 | |
1570 | my $sign = 0; $sign = 1 if $x->{sign} eq '-'; $x->{sign} = '+'; | |
1571 | ||
1572 | my $is_two = 0; | |
1573 | if ($y->isa('Math::BigFloat')) | |
1574 | { | |
1575 | $is_two = ($y->{sign} eq '+' && $MBI->_is_two($y->{_m}) && $MBI->_is_zero($y->{_e})); | |
1576 | } | |
1577 | else | |
1578 | { | |
1579 | $is_two = ($y == 2); | |
1580 | } | |
1581 | ||
1582 | # normal square root if $y == 2: | |
1583 | if ($is_two) | |
1584 | { | |
1585 | $x->bsqrt($scale+4); | |
1586 | } | |
1587 | elsif ($y->is_one('-')) | |
1588 | { | |
1589 | # $x ** -1 => 1/$x | |
1590 | my $u = $self->bone()->bdiv($x,$scale); | |
1591 | # copy private parts over | |
1592 | $x->{_m} = $u->{_m}; | |
1593 | $x->{_e} = $u->{_e}; | |
1594 | $x->{_es} = $u->{_es}; | |
1595 | } | |
1596 | else | |
1597 | { | |
1598 | # calculate the broot() as integer result first, and if it fits, return | |
1599 | # it rightaway (but only if $x and $y are integer): | |
1600 | ||
1601 | my $done = 0; # not yet | |
1602 | if ($y->is_int() && $x->is_int()) | |
1603 | { | |
1604 | my $i = $MBI->_copy( $x->{_m} ); | |
1605 | $MBI->_lsft( $i, $x->{_e}, 10 ) unless $MBI->_is_zero($x->{_e}); | |
1606 | my $int = Math::BigInt->bzero(); | |
1607 | $int->{value} = $i; | |
1608 | $int->broot($y->as_number()); | |
1609 | # if ($exact) | |
1610 | if ($int->copy()->bpow($y) == $x) | |
1611 | { | |
1612 | # found result, return it | |
1613 | $x->{_m} = $int->{value}; | |
1614 | $x->{_e} = $MBI->_zero(); | |
1615 | $x->{_es} = '+'; | |
1616 | $x->bnorm(); | |
1617 | $done = 1; | |
1618 | } | |
1619 | } | |
1620 | if ($done == 0) | |
1621 | { | |
1622 | my $u = $self->bone()->bdiv($y,$scale+4); | |
1623 | delete $u->{_a}; delete $u->{_p}; # otherwise it conflicts | |
1624 | $x->bpow($u,$scale+4); # el cheapo | |
1625 | } | |
1626 | } | |
1627 | $x->bneg() if $sign == 1; | |
1628 | ||
1629 | # shortcut to not run through _find_round_parameters again | |
1630 | if (defined $params[0]) | |
1631 | { | |
1632 | $x->bround($params[0],$params[2]); # then round accordingly | |
1633 | } | |
1634 | else | |
1635 | { | |
1636 | $x->bfround($params[1],$params[2]); # then round accordingly | |
1637 | } | |
1638 | if ($fallback) | |
1639 | { | |
1640 | # clear a/p after round, since user did not request it | |
1641 | delete $x->{_a}; delete $x->{_p}; | |
1642 | } | |
1643 | # restore globals | |
1644 | $$abr = $ab; $$pbr = $pb; | |
1645 | $x; | |
1646 | } | |
1647 | ||
1648 | sub bsqrt | |
1649 | { | |
1650 | # calculate square root | |
1651 | my ($self,$x,$a,$p,$r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_); | |
1652 | ||
1653 | return $x->bnan() if $x->{sign} !~ /^[+]/; # NaN, -inf or < 0 | |
1654 | return $x if $x->{sign} eq '+inf'; # sqrt(inf) == inf | |
1655 | return $x->round($a,$p,$r) if $x->is_zero() || $x->is_one(); | |
1656 | ||
1657 | # we need to limit the accuracy to protect against overflow | |
1658 | my $fallback = 0; | |
1659 | my (@params,$scale); | |
1660 | ($x,@params) = $x->_find_round_parameters($a,$p,$r); | |
1661 | ||
1662 | return $x if $x->is_nan(); # error in _find_round_parameters? | |
1663 | ||
1664 | # no rounding at all, so must use fallback | |
1665 | if (scalar @params == 0) | |
1666 | { | |
1667 | # simulate old behaviour | |
1668 | $params[0] = $self->div_scale(); # and round to it as accuracy | |
1669 | $scale = $params[0]+4; # at least four more for proper round | |
1670 | $params[2] = $r; # round mode by caller or undef | |
1671 | $fallback = 1; # to clear a/p afterwards | |
1672 | } | |
1673 | else | |
1674 | { | |
1675 | # the 4 below is empirical, and there might be cases where it is not | |
1676 | # enough... | |
1677 | $scale = abs($params[0] || $params[1]) + 4; # take whatever is defined | |
1678 | } | |
1679 | ||
1680 | # when user set globals, they would interfere with our calculation, so | |
1681 | # disable them and later re-enable them | |
1682 | no strict 'refs'; | |
1683 | my $abr = "$self\::accuracy"; my $ab = $$abr; $$abr = undef; | |
1684 | my $pbr = "$self\::precision"; my $pb = $$pbr; $$pbr = undef; | |
1685 | # we also need to disable any set A or P on $x (_find_round_parameters took | |
1686 | # them already into account), since these would interfere, too | |
1687 | delete $x->{_a}; delete $x->{_p}; | |
1688 | # need to disable $upgrade in BigInt, to avoid deep recursion | |
1689 | local $Math::BigInt::upgrade = undef; # should be really parent class vs MBI | |
1690 | ||
1691 | my $i = $MBI->_copy( $x->{_m} ); | |
1692 | $MBI->_lsft( $i, $x->{_e}, 10 ) unless $MBI->_is_zero($x->{_e}); | |
1693 | my $xas = Math::BigInt->bzero(); | |
1694 | $xas->{value} = $i; | |
1695 | ||
1696 | my $gs = $xas->copy()->bsqrt(); # some guess | |
1697 | ||
1698 | if (($x->{_es} ne '-') # guess can't be accurate if there are | |
1699 | # digits after the dot | |
1700 | && ($xas->bacmp($gs * $gs) == 0)) # guess hit the nail on the head? | |
1701 | { | |
1702 | # exact result, copy result over to keep $x | |
1703 | $x->{_m} = $gs->{value}; $x->{_e} = $MBI->_zero(); $x->{_es} = '+'; | |
1704 | $x->bnorm(); | |
1705 | # shortcut to not run through _find_round_parameters again | |
1706 | if (defined $params[0]) | |
1707 | { | |
1708 | $x->bround($params[0],$params[2]); # then round accordingly | |
1709 | } | |
1710 | else | |
1711 | { | |
1712 | $x->bfround($params[1],$params[2]); # then round accordingly | |
1713 | } | |
1714 | if ($fallback) | |
1715 | { | |
1716 | # clear a/p after round, since user did not request it | |
1717 | delete $x->{_a}; delete $x->{_p}; | |
1718 | } | |
1719 | # re-enable A and P, upgrade is taken care of by "local" | |
1720 | ${"$self\::accuracy"} = $ab; ${"$self\::precision"} = $pb; | |
1721 | return $x; | |
1722 | } | |
1723 | ||
1724 | # sqrt(2) = 1.4 because sqrt(2*100) = 1.4*10; so we can increase the accuracy | |
1725 | # of the result by multipyling the input by 100 and then divide the integer | |
1726 | # result of sqrt(input) by 10. Rounding afterwards returns the real result. | |
1727 | ||
1728 | # The following steps will transform 123.456 (in $x) into 123456 (in $y1) | |
1729 | my $y1 = $MBI->_copy($x->{_m}); | |
1730 | ||
1731 | my $length = $MBI->_len($y1); | |
1732 | ||
1733 | # Now calculate how many digits the result of sqrt(y1) would have | |
1734 | my $digits = int($length / 2); | |
1735 | ||
1736 | # But we need at least $scale digits, so calculate how many are missing | |
1737 | my $shift = $scale - $digits; | |
1738 | ||
1739 | # That should never happen (we take care of integer guesses above) | |
1740 | # $shift = 0 if $shift < 0; | |
1741 | ||
1742 | # Multiply in steps of 100, by shifting left two times the "missing" digits | |
1743 | my $s2 = $shift * 2; | |
1744 | ||
1745 | # We now make sure that $y1 has the same odd or even number of digits than | |
1746 | # $x had. So when _e of $x is odd, we must shift $y1 by one digit left, | |
1747 | # because we always must multiply by steps of 100 (sqrt(100) is 10) and not | |
1748 | # steps of 10. The length of $x does not count, since an even or odd number | |
1749 | # of digits before the dot is not changed by adding an even number of digits | |
1750 | # after the dot (the result is still odd or even digits long). | |
1751 | $s2++ if $MBI->_is_odd($x->{_e}); | |
1752 | ||
1753 | $MBI->_lsft( $y1, $MBI->_new($s2), 10); | |
1754 | ||
1755 | # now take the square root and truncate to integer | |
1756 | $y1 = $MBI->_sqrt($y1); | |
1757 | ||
1758 | # By "shifting" $y1 right (by creating a negative _e) we calculate the final | |
1759 | # result, which is than later rounded to the desired scale. | |
1760 | ||
1761 | # calculate how many zeros $x had after the '.' (or before it, depending | |
1762 | # on sign of $dat, the result should have half as many: | |
1763 | my $dat = $MBI->_num($x->{_e}); | |
1764 | $dat = -$dat if $x->{_es} eq '-'; | |
1765 | $dat += $length; | |
1766 | ||
1767 | if ($dat > 0) | |
1768 | { | |
1769 | # no zeros after the dot (e.g. 1.23, 0.49 etc) | |
1770 | # preserve half as many digits before the dot than the input had | |
1771 | # (but round this "up") | |
1772 | $dat = int(($dat+1)/2); | |
1773 | } | |
1774 | else | |
1775 | { | |
1776 | $dat = int(($dat)/2); | |
1777 | } | |
1778 | $dat -= $MBI->_len($y1); | |
1779 | if ($dat < 0) | |
1780 | { | |
1781 | $dat = abs($dat); | |
1782 | $x->{_e} = $MBI->_new( $dat ); | |
1783 | $x->{_es} = '-'; | |
1784 | } | |
1785 | else | |
1786 | { | |
1787 | $x->{_e} = $MBI->_new( $dat ); | |
1788 | $x->{_es} = '+'; | |
1789 | } | |
1790 | $x->{_m} = $y1; | |
1791 | $x->bnorm(); | |
1792 | ||
1793 | # shortcut to not run through _find_round_parameters again | |
1794 | if (defined $params[0]) | |
1795 | { | |
1796 | $x->bround($params[0],$params[2]); # then round accordingly | |
1797 | } | |
1798 | else | |
1799 | { | |
1800 | $x->bfround($params[1],$params[2]); # then round accordingly | |
1801 | } | |
1802 | if ($fallback) | |
1803 | { | |
1804 | # clear a/p after round, since user did not request it | |
1805 | delete $x->{_a}; delete $x->{_p}; | |
1806 | } | |
1807 | # restore globals | |
1808 | $$abr = $ab; $$pbr = $pb; | |
1809 | $x; | |
1810 | } | |
1811 | ||
1812 | sub bfac | |
1813 | { | |
1814 | # (BFLOAT or num_str, BFLOAT or num_str) return BFLOAT | |
1815 | # compute factorial number, modifies first argument | |
1816 | ||
1817 | # set up parameters | |
1818 | my ($self,$x,@r) = (ref($_[0]),@_); | |
1819 | # objectify is costly, so avoid it | |
1820 | ($self,$x,@r) = objectify(1,@_) if !ref($x); | |
1821 | ||
1822 | return $x if $x->{sign} eq '+inf'; # inf => inf | |
1823 | return $x->bnan() | |
1824 | if (($x->{sign} ne '+') || # inf, NaN, <0 etc => NaN | |
1825 | ($x->{_es} ne '+')); # digits after dot? | |
1826 | ||
1827 | # use BigInt's bfac() for faster calc | |
1828 | if (! $MBI->_is_zero($x->{_e})) | |
1829 | { | |
1830 | $MBI->_lsft($x->{_m}, $x->{_e},10); # change 12e1 to 120e0 | |
1831 | $x->{_e} = $MBI->_zero(); # normalize | |
1832 | $x->{_es} = '+'; | |
1833 | } | |
1834 | $MBI->_fac($x->{_m}); # calculate factorial | |
1835 | $x->bnorm()->round(@r); # norm again and round result | |
1836 | } | |
1837 | ||
1838 | sub _pow | |
1839 | { | |
1840 | # Calculate a power where $y is a non-integer, like 2 ** 0.5 | |
1841 | my ($x,$y,$a,$p,$r) = @_; | |
1842 | my $self = ref($x); | |
1843 | ||
1844 | # if $y == 0.5, it is sqrt($x) | |
1845 | $HALF = $self->new($HALF) unless ref($HALF); | |
1846 | return $x->bsqrt($a,$p,$r,$y) if $y->bcmp($HALF) == 0; | |
1847 | ||
1848 | # Using: | |
1849 | # a ** x == e ** (x * ln a) | |
1850 | ||
1851 | # u = y * ln x | |
1852 | # _ _ | |
1853 | # Taylor: | u u^2 u^3 | | |
1854 | # x ** y = 1 + | --- + --- + ----- + ... | | |
1855 | # |_ 1 1*2 1*2*3 _| | |
1856 | ||
1857 | # we need to limit the accuracy to protect against overflow | |
1858 | my $fallback = 0; | |
1859 | my ($scale,@params); | |
1860 | ($x,@params) = $x->_find_round_parameters($a,$p,$r); | |
1861 | ||
1862 | return $x if $x->is_nan(); # error in _find_round_parameters? | |
1863 | ||
1864 | # no rounding at all, so must use fallback | |
1865 | if (scalar @params == 0) | |
1866 | { | |
1867 | # simulate old behaviour | |
1868 | $params[0] = $self->div_scale(); # and round to it as accuracy | |
1869 | $params[1] = undef; # disable P | |
1870 | $scale = $params[0]+4; # at least four more for proper round | |
1871 | $params[2] = $r; # round mode by caller or undef | |
1872 | $fallback = 1; # to clear a/p afterwards | |
1873 | } | |
1874 | else | |
1875 | { | |
1876 | # the 4 below is empirical, and there might be cases where it is not | |
1877 | # enough... | |
1878 | $scale = abs($params[0] || $params[1]) + 4; # take whatever is defined | |
1879 | } | |
1880 | ||
1881 | # when user set globals, they would interfere with our calculation, so | |
1882 | # disable them and later re-enable them | |
1883 | no strict 'refs'; | |
1884 | my $abr = "$self\::accuracy"; my $ab = $$abr; $$abr = undef; | |
1885 | my $pbr = "$self\::precision"; my $pb = $$pbr; $$pbr = undef; | |
1886 | # we also need to disable any set A or P on $x (_find_round_parameters took | |
1887 | # them already into account), since these would interfere, too | |
1888 | delete $x->{_a}; delete $x->{_p}; | |
1889 | # need to disable $upgrade in BigInt, to avoid deep recursion | |
1890 | local $Math::BigInt::upgrade = undef; | |
1891 | ||
1892 | my ($limit,$v,$u,$below,$factor,$next,$over); | |
1893 | ||
1894 | $u = $x->copy()->blog(undef,$scale)->bmul($y); | |
1895 | $v = $self->bone(); # 1 | |
1896 | $factor = $self->new(2); # 2 | |
1897 | $x->bone(); # first term: 1 | |
1898 | ||
1899 | $below = $v->copy(); | |
1900 | $over = $u->copy(); | |
1901 | ||
1902 | $limit = $self->new("1E-". ($scale-1)); | |
1903 | #my $steps = 0; | |
1904 | while (3 < 5) | |
1905 | { | |
1906 | # we calculate the next term, and add it to the last | |
1907 | # when the next term is below our limit, it won't affect the outcome | |
1908 | # anymore, so we stop | |
1909 | $next = $over->copy()->bdiv($below,$scale); | |
1910 | last if $next->bacmp($limit) <= 0; | |
1911 | $x->badd($next); | |
1912 | # calculate things for the next term | |
1913 | $over *= $u; $below *= $factor; $factor->binc(); | |
1914 | ||
1915 | last if $x->{sign} !~ /^[-+]$/; | |
1916 | ||
1917 | #$steps++; | |
1918 | } | |
1919 | ||
1920 | # shortcut to not run through _find_round_parameters again | |
1921 | if (defined $params[0]) | |
1922 | { | |
1923 | $x->bround($params[0],$params[2]); # then round accordingly | |
1924 | } | |
1925 | else | |
1926 | { | |
1927 | $x->bfround($params[1],$params[2]); # then round accordingly | |
1928 | } | |
1929 | if ($fallback) | |
1930 | { | |
1931 | # clear a/p after round, since user did not request it | |
1932 | delete $x->{_a}; delete $x->{_p}; | |
1933 | } | |
1934 | # restore globals | |
1935 | $$abr = $ab; $$pbr = $pb; | |
1936 | $x; | |
1937 | } | |
1938 | ||
1939 | sub bpow | |
1940 | { | |
1941 | # (BFLOAT or num_str, BFLOAT or num_str) return BFLOAT | |
1942 | # compute power of two numbers, second arg is used as integer | |
1943 | # modifies first argument | |
1944 | ||
1945 | # set up parameters | |
1946 | my ($self,$x,$y,$a,$p,$r) = (ref($_[0]),@_); | |
1947 | # objectify is costly, so avoid it | |
1948 | if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1]))) | |
1949 | { | |
1950 | ($self,$x,$y,$a,$p,$r) = objectify(2,@_); | |
1951 | } | |
1952 | ||
1953 | return $x->bnan() if $x->{sign} eq $nan || $y->{sign} eq $nan; | |
1954 | return $x if $x->{sign} =~ /^[+-]inf$/; | |
1955 | ||
1956 | # -2 ** -2 => NaN | |
1957 | return $x->bnan() if $x->{sign} eq '-' && $y->{sign} eq '-'; | |
1958 | ||
1959 | # cache the result of is_zero | |
1960 | my $y_is_zero = $y->is_zero(); | |
1961 | return $x->bone() if $y_is_zero; | |
1962 | return $x if $x->is_one() || $y->is_one(); | |
1963 | ||
1964 | my $x_is_zero = $x->is_zero(); | |
1965 | return $x->_pow($y,$a,$p,$r) if !$x_is_zero && !$y->is_int(); # non-integer power | |
1966 | ||
1967 | my $y1 = $y->as_number()->{value}; # make MBI part | |
1968 | ||
1969 | # if ($x == -1) | |
1970 | if ($x->{sign} eq '-' && $MBI->_is_one($x->{_m}) && $MBI->_is_zero($x->{_e})) | |
1971 | { | |
1972 | # if $x == -1 and odd/even y => +1/-1 because +-1 ^ (+-1) => +-1 | |
1973 | return $MBI->_is_odd($y1) ? $x : $x->babs(1); | |
1974 | } | |
1975 | if ($x_is_zero) | |
1976 | { | |
1977 | return $x->bone() if $y_is_zero; | |
1978 | return $x if $y->{sign} eq '+'; # 0**y => 0 (if not y <= 0) | |
1979 | # 0 ** -y => 1 / (0 ** y) => 1 / 0! (1 / 0 => +inf) | |
1980 | return $x->binf(); | |
1981 | } | |
1982 | ||
1983 | my $new_sign = '+'; | |
1984 | $new_sign = $MBI->_is_odd($y1) ? '-' : '+' if $x->{sign} ne '+'; | |
1985 | ||
1986 | # calculate $x->{_m} ** $y and $x->{_e} * $y separately (faster) | |
1987 | $x->{_m} = $MBI->_pow( $x->{_m}, $y1); | |
1988 | $x->{_e} = $MBI->_mul ($x->{_e}, $y1); | |
1989 | ||
1990 | $x->{sign} = $new_sign; | |
1991 | $x->bnorm(); | |
1992 | if ($y->{sign} eq '-') | |
1993 | { | |
1994 | # modify $x in place! | |
1995 | my $z = $x->copy(); $x->bone(); | |
1996 | return $x->bdiv($z,$a,$p,$r); # round in one go (might ignore y's A!) | |
1997 | } | |
1998 | $x->round($a,$p,$r,$y); | |
1999 | } | |
2000 | ||
2001 | ############################################################################### | |
2002 | # rounding functions | |
2003 | ||
2004 | sub bfround | |
2005 | { | |
2006 | # precision: round to the $Nth digit left (+$n) or right (-$n) from the '.' | |
2007 | # $n == 0 means round to integer | |
2008 | # expects and returns normalized numbers! | |
2009 | my $x = shift; my $self = ref($x) || $x; $x = $self->new(shift) if !ref($x); | |
2010 | ||
2011 | my ($scale,$mode) = $x->_scale_p(@_); | |
2012 | return $x if !defined $scale || $x->modify('bfround'); # no-op | |
2013 | ||
2014 | # never round a 0, +-inf, NaN | |
2015 | if ($x->is_zero()) | |
2016 | { | |
2017 | $x->{_p} = $scale if !defined $x->{_p} || $x->{_p} < $scale; # -3 < -2 | |
2018 | return $x; | |
2019 | } | |
2020 | return $x if $x->{sign} !~ /^[+-]$/; | |
2021 | ||
2022 | # don't round if x already has lower precision | |
2023 | return $x if (defined $x->{_p} && $x->{_p} < 0 && $scale < $x->{_p}); | |
2024 | ||
2025 | $x->{_p} = $scale; # remember round in any case | |
2026 | delete $x->{_a}; # and clear A | |
2027 | if ($scale < 0) | |
2028 | { | |
2029 | # round right from the '.' | |
2030 | ||
2031 | return $x if $x->{_es} eq '+'; # e >= 0 => nothing to round | |
2032 | ||
2033 | $scale = -$scale; # positive for simplicity | |
2034 | my $len = $MBI->_len($x->{_m}); # length of mantissa | |
2035 | ||
2036 | # the following poses a restriction on _e, but if _e is bigger than a | |
2037 | # scalar, you got other problems (memory etc) anyway | |
2038 | my $dad = -(0+ ($x->{_es}.$MBI->_num($x->{_e}))); # digits after dot | |
2039 | my $zad = 0; # zeros after dot | |
2040 | $zad = $dad - $len if (-$dad < -$len); # for 0.00..00xxx style | |
2041 | ||
2042 | # p rint "scale $scale dad $dad zad $zad len $len\n"; | |
2043 | # number bsstr len zad dad | |
2044 | # 0.123 123e-3 3 0 3 | |
2045 | # 0.0123 123e-4 3 1 4 | |
2046 | # 0.001 1e-3 1 2 3 | |
2047 | # 1.23 123e-2 3 0 2 | |
2048 | # 1.2345 12345e-4 5 0 4 | |
2049 | ||
2050 | # do not round after/right of the $dad | |
2051 | return $x if $scale > $dad; # 0.123, scale >= 3 => exit | |
2052 | ||
2053 | # round to zero if rounding inside the $zad, but not for last zero like: | |
2054 | # 0.0065, scale -2, round last '0' with following '65' (scale == zad case) | |
2055 | return $x->bzero() if $scale < $zad; | |
2056 | if ($scale == $zad) # for 0.006, scale -3 and trunc | |
2057 | { | |
2058 | $scale = -$len; | |
2059 | } | |
2060 | else | |
2061 | { | |
2062 | # adjust round-point to be inside mantissa | |
2063 | if ($zad != 0) | |
2064 | { | |
2065 | $scale = $scale-$zad; | |
2066 | } | |
2067 | else | |
2068 | { | |
2069 | my $dbd = $len - $dad; $dbd = 0 if $dbd < 0; # digits before dot | |
2070 | $scale = $dbd+$scale; | |
2071 | } | |
2072 | } | |
2073 | } | |
2074 | else | |
2075 | { | |
2076 | # round left from the '.' | |
2077 | ||
2078 | # 123 => 100 means length(123) = 3 - $scale (2) => 1 | |
2079 | ||
2080 | my $dbt = $MBI->_len($x->{_m}); | |
2081 | # digits before dot | |
2082 | my $dbd = $dbt + ($x->{_es} . $MBI->_num($x->{_e})); | |
2083 | # should be the same, so treat it as this | |
2084 | $scale = 1 if $scale == 0; | |
2085 | # shortcut if already integer | |
2086 | return $x if $scale == 1 && $dbt <= $dbd; | |
2087 | # maximum digits before dot | |
2088 | ++$dbd; | |
2089 | ||
2090 | if ($scale > $dbd) | |
2091 | { | |
2092 | # not enough digits before dot, so round to zero | |
2093 | return $x->bzero; | |
2094 | } | |
2095 | elsif ( $scale == $dbd ) | |
2096 | { | |
2097 | # maximum | |
2098 | $scale = -$dbt; | |
2099 | } | |
2100 | else | |
2101 | { | |
2102 | $scale = $dbd - $scale; | |
2103 | } | |
2104 | } | |
2105 | # pass sign to bround for rounding modes '+inf' and '-inf' | |
2106 | my $m = bless { sign => $x->{sign}, value => $x->{_m} }, 'Math::BigInt'; | |
2107 | $m->bround($scale,$mode); | |
2108 | $x->{_m} = $m->{value}; # get our mantissa back | |
2109 | $x->bnorm(); | |
2110 | } | |
2111 | ||
2112 | sub bround | |
2113 | { | |
2114 | # accuracy: preserve $N digits, and overwrite the rest with 0's | |
2115 | my $x = shift; my $self = ref($x) || $x; $x = $self->new(shift) if !ref($x); | |
2116 | ||
2117 | if (($_[0] || 0) < 0) | |
2118 | { | |
2119 | require Carp; Carp::croak ('bround() needs positive accuracy'); | |
2120 | } | |
2121 | ||
2122 | my ($scale,$mode) = $x->_scale_a(@_); | |
2123 | return $x if !defined $scale || $x->modify('bround'); # no-op | |
2124 | ||
2125 | # scale is now either $x->{_a}, $accuracy, or the user parameter | |
2126 | # test whether $x already has lower accuracy, do nothing in this case | |
2127 | # but do round if the accuracy is the same, since a math operation might | |
2128 | # want to round a number with A=5 to 5 digits afterwards again | |
2129 | return $x if defined $x->{_a} && $x->{_a} < $scale; | |
2130 | ||
2131 | # scale < 0 makes no sense | |
2132 | # scale == 0 => keep all digits | |
2133 | # never round a +-inf, NaN | |
2134 | return $x if ($scale <= 0) || $x->{sign} !~ /^[+-]$/; | |
2135 | ||
2136 | # 1: never round a 0 | |
2137 | # 2: if we should keep more digits than the mantissa has, do nothing | |
2138 | if ($x->is_zero() || $MBI->_len($x->{_m}) <= $scale) | |
2139 | { | |
2140 | $x->{_a} = $scale if !defined $x->{_a} || $x->{_a} > $scale; | |
2141 | return $x; | |
2142 | } | |
2143 | ||
2144 | # pass sign to bround for '+inf' and '-inf' rounding modes | |
2145 | my $m = bless { sign => $x->{sign}, value => $x->{_m} }, 'Math::BigInt'; | |
2146 | ||
2147 | $m->bround($scale,$mode); # round mantissa | |
2148 | $x->{_m} = $m->{value}; # get our mantissa back | |
2149 | $x->{_a} = $scale; # remember rounding | |
2150 | delete $x->{_p}; # and clear P | |
2151 | $x->bnorm(); # del trailing zeros gen. by bround() | |
2152 | } | |
2153 | ||
2154 | sub bfloor | |
2155 | { | |
2156 | # return integer less or equal then $x | |
2157 | my ($self,$x,$a,$p,$r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_); | |
2158 | ||
2159 | return $x if $x->modify('bfloor'); | |
2160 | ||
2161 | return $x if $x->{sign} !~ /^[+-]$/; # nan, +inf, -inf | |
2162 | ||
2163 | # if $x has digits after dot | |
2164 | if ($x->{_es} eq '-') | |
2165 | { | |
2166 | $x->{_m} = $MBI->_rsft($x->{_m},$x->{_e},10); # cut off digits after dot | |
2167 | $x->{_e} = $MBI->_zero(); # trunc/norm | |
2168 | $x->{_es} = '+'; # abs e | |
2169 | $MBI->_inc($x->{_m}) if $x->{sign} eq '-'; # increment if negative | |
2170 | } | |
2171 | $x->round($a,$p,$r); | |
2172 | } | |
2173 | ||
2174 | sub bceil | |
2175 | { | |
2176 | # return integer greater or equal then $x | |
2177 | my ($self,$x,$a,$p,$r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_); | |
2178 | ||
2179 | return $x if $x->modify('bceil'); | |
2180 | return $x if $x->{sign} !~ /^[+-]$/; # nan, +inf, -inf | |
2181 | ||
2182 | # if $x has digits after dot | |
2183 | if ($x->{_es} eq '-') | |
2184 | { | |
2185 | $x->{_m} = $MBI->_rsft($x->{_m},$x->{_e},10); # cut off digits after dot | |
2186 | $x->{_e} = $MBI->_zero(); # trunc/norm | |
2187 | $x->{_es} = '+'; # abs e | |
2188 | $MBI->_inc($x->{_m}) if $x->{sign} eq '+'; # increment if positive | |
2189 | } | |
2190 | $x->round($a,$p,$r); | |
2191 | } | |
2192 | ||
2193 | sub brsft | |
2194 | { | |
2195 | # shift right by $y (divide by power of $n) | |
2196 | ||
2197 | # set up parameters | |
2198 | my ($self,$x,$y,$n,$a,$p,$r) = (ref($_[0]),@_); | |
2199 | # objectify is costly, so avoid it | |
2200 | if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1]))) | |
2201 | { | |
2202 | ($self,$x,$y,$n,$a,$p,$r) = objectify(2,@_); | |
2203 | } | |
2204 | ||
2205 | return $x if $x->modify('brsft'); | |
2206 | return $x if $x->{sign} !~ /^[+-]$/; # nan, +inf, -inf | |
2207 | ||
2208 | $n = 2 if !defined $n; $n = $self->new($n); | |
2209 | $x->bdiv($n->bpow($y),$a,$p,$r,$y); | |
2210 | } | |
2211 | ||
2212 | sub blsft | |
2213 | { | |
2214 | # shift left by $y (multiply by power of $n) | |
2215 | ||
2216 | # set up parameters | |
2217 | my ($self,$x,$y,$n,$a,$p,$r) = (ref($_[0]),@_); | |
2218 | # objectify is costly, so avoid it | |
2219 | if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1]))) | |
2220 | { | |
2221 | ($self,$x,$y,$n,$a,$p,$r) = objectify(2,@_); | |
2222 | } | |
2223 | ||
2224 | return $x if $x->modify('blsft'); | |
2225 | return $x if $x->{sign} !~ /^[+-]$/; # nan, +inf, -inf | |
2226 | ||
2227 | $n = 2 if !defined $n; $n = $self->new($n); | |
2228 | $x->bmul($n->bpow($y),$a,$p,$r,$y); | |
2229 | } | |
2230 | ||
2231 | ############################################################################### | |
2232 | ||
2233 | sub DESTROY | |
2234 | { | |
2235 | # going through AUTOLOAD for every DESTROY is costly, avoid it by empty sub | |
2236 | } | |
2237 | ||
2238 | sub AUTOLOAD | |
2239 | { | |
2240 | # make fxxx and bxxx both work by selectively mapping fxxx() to MBF::bxxx() | |
2241 | # or falling back to MBI::bxxx() | |
2242 | my $name = $AUTOLOAD; | |
2243 | ||
2244 | $name =~ s/(.*):://; # split package | |
2245 | my $c = $1 || $class; | |
2246 | no strict 'refs'; | |
2247 | $c->import() if $IMPORT == 0; | |
2248 | if (!method_alias($name)) | |
2249 | { | |
2250 | if (!defined $name) | |
2251 | { | |
2252 | # delayed load of Carp and avoid recursion | |
2253 | require Carp; | |
2254 | Carp::croak ("$c: Can't call a method without name"); | |
2255 | } | |
2256 | if (!method_hand_up($name)) | |
2257 | { | |
2258 | # delayed load of Carp and avoid recursion | |
2259 | require Carp; | |
2260 | Carp::croak ("Can't call $c\-\>$name, not a valid method"); | |
2261 | } | |
2262 | # try one level up, but subst. bxxx() for fxxx() since MBI only got bxxx() | |
2263 | $name =~ s/^f/b/; | |
2264 | return &{"Math::BigInt"."::$name"}(@_); | |
2265 | } | |
2266 | my $bname = $name; $bname =~ s/^f/b/; | |
2267 | $c .= "::$name"; | |
2268 | *{$c} = \&{$bname}; | |
2269 | &{$c}; # uses @_ | |
2270 | } | |
2271 | ||
2272 | sub exponent | |
2273 | { | |
2274 | # return a copy of the exponent | |
2275 | my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_); | |
2276 | ||
2277 | if ($x->{sign} !~ /^[+-]$/) | |
2278 | { | |
2279 | my $s = $x->{sign}; $s =~ s/^[+-]//; | |
2280 | return Math::BigInt->new($s); # -inf, +inf => +inf | |
2281 | } | |
2282 | Math::BigInt->new( $x->{_es} . $MBI->_str($x->{_e})); | |
2283 | } | |
2284 | ||
2285 | sub mantissa | |
2286 | { | |
2287 | # return a copy of the mantissa | |
2288 | my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_); | |
2289 | ||
2290 | if ($x->{sign} !~ /^[+-]$/) | |
2291 | { | |
2292 | my $s = $x->{sign}; $s =~ s/^[+]//; | |
2293 | return Math::BigInt->new($s); # -inf, +inf => +inf | |
2294 | } | |
2295 | my $m = Math::BigInt->new( $MBI->_str($x->{_m})); | |
2296 | $m->bneg() if $x->{sign} eq '-'; | |
2297 | ||
2298 | $m; | |
2299 | } | |
2300 | ||
2301 | sub parts | |
2302 | { | |
2303 | # return a copy of both the exponent and the mantissa | |
2304 | my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_); | |
2305 | ||
2306 | if ($x->{sign} !~ /^[+-]$/) | |
2307 | { | |
2308 | my $s = $x->{sign}; $s =~ s/^[+]//; my $se = $s; $se =~ s/^[-]//; | |
2309 | return ($self->new($s),$self->new($se)); # +inf => inf and -inf,+inf => inf | |
2310 | } | |
2311 | my $m = Math::BigInt->bzero(); | |
2312 | $m->{value} = $MBI->_copy($x->{_m}); | |
2313 | $m->bneg() if $x->{sign} eq '-'; | |
2314 | ($m, Math::BigInt->new( $x->{_es} . $MBI->_num($x->{_e}) )); | |
2315 | } | |
2316 | ||
2317 | ############################################################################## | |
2318 | # private stuff (internal use only) | |
2319 | ||
2320 | sub import | |
2321 | { | |
2322 | my $self = shift; | |
2323 | my $l = scalar @_; | |
2324 | my $lib = ''; my @a; | |
2325 | $IMPORT=1; | |
2326 | for ( my $i = 0; $i < $l ; $i++) | |
2327 | { | |
2328 | if ( $_[$i] eq ':constant' ) | |
2329 | { | |
2330 | # This causes overlord er load to step in. 'binary' and 'integer' | |
2331 | # are handled by BigInt. | |
2332 | overload::constant float => sub { $self->new(shift); }; | |
2333 | } | |
2334 | elsif ($_[$i] eq 'upgrade') | |
2335 | { | |
2336 | # this causes upgrading | |
2337 | $upgrade = $_[$i+1]; # or undef to disable | |
2338 | $i++; | |
2339 | } | |
2340 | elsif ($_[$i] eq 'downgrade') | |
2341 | { | |
2342 | # this causes downgrading | |
2343 | $downgrade = $_[$i+1]; # or undef to disable | |
2344 | $i++; | |
2345 | } | |
2346 | elsif ($_[$i] eq 'lib') | |
2347 | { | |
2348 | # alternative library | |
2349 | $lib = $_[$i+1] || ''; # default Calc | |
2350 | $i++; | |
2351 | } | |
2352 | elsif ($_[$i] eq 'with') | |
2353 | { | |
2354 | # alternative class for our private parts() | |
2355 | # XXX: no longer supported | |
2356 | # $MBI = $_[$i+1] || 'Math::BigInt'; | |
2357 | $i++; | |
2358 | } | |
2359 | else | |
2360 | { | |
2361 | push @a, $_[$i]; | |
2362 | } | |
2363 | } | |
2364 | ||
2365 | $lib =~ tr/a-zA-Z0-9,://cd; # restrict to sane characters | |
2366 | # let use Math::BigInt lib => 'GMP'; use Math::BigFloat; still work | |
2367 | my $mbilib = eval { Math::BigInt->config()->{lib} }; | |
2368 | if ((defined $mbilib) && ($MBI eq 'Math::BigInt::Calc')) | |
2369 | { | |
2370 | # MBI already loaded | |
2371 | Math::BigInt->import('lib',"$lib,$mbilib", 'objectify'); | |
2372 | } | |
2373 | else | |
2374 | { | |
2375 | # MBI not loaded, or with ne "Math::BigInt::Calc" | |
2376 | $lib .= ",$mbilib" if defined $mbilib; | |
2377 | $lib =~ s/^,//; # don't leave empty | |
2378 | ||
2379 | # replacement library can handle lib statement, but also could ignore it | |
2380 | ||
2381 | # Perl < 5.6.0 dies with "out of memory!" when eval() and ':constant' is | |
2382 | # used in the same script, or eval inside import(). So we require MBI: | |
2383 | require Math::BigInt; | |
2384 | Math::BigInt->import( lib => $lib, 'objectify' ); | |
2385 | } | |
2386 | if ($@) | |
2387 | { | |
2388 | require Carp; Carp::croak ("Couldn't load $lib: $! $@"); | |
2389 | } | |
2390 | # find out which one was actually loaded | |
2391 | $MBI = Math::BigInt->config()->{lib}; | |
2392 | ||
2393 | # register us with MBI to get notified of future lib changes | |
2394 | Math::BigInt::_register_callback( $self, sub { $MBI = $_[0]; } ); | |
2395 | ||
2396 | # any non :constant stuff is handled by our parent, Exporter | |
2397 | # even if @_ is empty, to give it a chance | |
2398 | $self->SUPER::import(@a); # for subclasses | |
2399 | $self->export_to_level(1,$self,@a); # need this, too | |
2400 | } | |
2401 | ||
2402 | sub bnorm | |
2403 | { | |
2404 | # adjust m and e so that m is smallest possible | |
2405 | # round number according to accuracy and precision settings | |
2406 | my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_); | |
2407 | ||
2408 | return $x if $x->{sign} !~ /^[+-]$/; # inf, nan etc | |
2409 | ||
2410 | my $zeros = $MBI->_zeros($x->{_m}); # correct for trailing zeros | |
2411 | if ($zeros != 0) | |
2412 | { | |
2413 | my $z = $MBI->_new($zeros); | |
2414 | $x->{_m} = $MBI->_rsft ($x->{_m}, $z, 10); | |
2415 | if ($x->{_es} eq '-') | |
2416 | { | |
2417 | if ($MBI->_acmp($x->{_e},$z) >= 0) | |
2418 | { | |
2419 | $x->{_e} = $MBI->_sub ($x->{_e}, $z); | |
2420 | $x->{_es} = '+' if $MBI->_is_zero($x->{_e}); | |
2421 | } | |
2422 | else | |
2423 | { | |
2424 | $x->{_e} = $MBI->_sub ( $MBI->_copy($z), $x->{_e}); | |
2425 | $x->{_es} = '+'; | |
2426 | } | |
2427 | } | |
2428 | else | |
2429 | { | |
2430 | $x->{_e} = $MBI->_add ($x->{_e}, $z); | |
2431 | } | |
2432 | } | |
2433 | else | |
2434 | { | |
2435 | # $x can only be 0Ey if there are no trailing zeros ('0' has 0 trailing | |
2436 | # zeros). So, for something like 0Ey, set y to 1, and -0 => +0 | |
2437 | $x->{sign} = '+', $x->{_es} = '+', $x->{_e} = $MBI->_one() | |
2438 | if $MBI->_is_zero($x->{_m}); | |
2439 | } | |
2440 | ||
2441 | $x; # MBI bnorm is no-op, so dont call it | |
2442 | } | |
2443 | ||
2444 | ############################################################################## | |
2445 | ||
2446 | sub as_hex | |
2447 | { | |
2448 | # return number as hexadecimal string (only for integers defined) | |
2449 | my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_); | |
2450 | ||
2451 | return $x->bstr() if $x->{sign} !~ /^[+-]$/; # inf, nan etc | |
2452 | return '0x0' if $x->is_zero(); | |
2453 | ||
2454 | return $nan if $x->{_es} ne '+'; # how to do 1e-1 in hex!? | |
2455 | ||
2456 | my $z = $MBI->_copy($x->{_m}); | |
2457 | if (! $MBI->_is_zero($x->{_e})) # > 0 | |
2458 | { | |
2459 | $MBI->_lsft( $z, $x->{_e},10); | |
2460 | } | |
2461 | $z = Math::BigInt->new( $x->{sign} . $MBI->_num($z)); | |
2462 | $z->as_hex(); | |
2463 | } | |
2464 | ||
2465 | sub as_bin | |
2466 | { | |
2467 | # return number as binary digit string (only for integers defined) | |
2468 | my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_); | |
2469 | ||
2470 | return $x->bstr() if $x->{sign} !~ /^[+-]$/; # inf, nan etc | |
2471 | return '0b0' if $x->is_zero(); | |
2472 | ||
2473 | return $nan if $x->{_es} ne '+'; # how to do 1e-1 in hex!? | |
2474 | ||
2475 | my $z = $MBI->_copy($x->{_m}); | |
2476 | if (! $MBI->_is_zero($x->{_e})) # > 0 | |
2477 | { | |
2478 | $MBI->_lsft( $z, $x->{_e},10); | |
2479 | } | |
2480 | $z = Math::BigInt->new( $x->{sign} . $MBI->_num($z)); | |
2481 | $z->as_bin(); | |
2482 | } | |
2483 | ||
2484 | sub as_number | |
2485 | { | |
2486 | # return copy as a bigint representation of this BigFloat number | |
2487 | my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_); | |
2488 | ||
2489 | my $z = $MBI->_copy($x->{_m}); | |
2490 | if ($x->{_es} eq '-') # < 0 | |
2491 | { | |
2492 | $MBI->_rsft( $z, $x->{_e},10); | |
2493 | } | |
2494 | elsif (! $MBI->_is_zero($x->{_e})) # > 0 | |
2495 | { | |
2496 | $MBI->_lsft( $z, $x->{_e},10); | |
2497 | } | |
2498 | $z = Math::BigInt->new( $x->{sign} . $MBI->_num($z)); | |
2499 | $z; | |
2500 | } | |
2501 | ||
2502 | sub length | |
2503 | { | |
2504 | my $x = shift; | |
2505 | my $class = ref($x) || $x; | |
2506 | $x = $class->new(shift) unless ref($x); | |
2507 | ||
2508 | return 1 if $MBI->_is_zero($x->{_m}); | |
2509 | ||
2510 | my $len = $MBI->_len($x->{_m}); | |
2511 | $len += $MBI->_num($x->{_e}) if $x->{_es} eq '+'; | |
2512 | if (wantarray()) | |
2513 | { | |
2514 | my $t = 0; | |
2515 | $t = $MBI->_num($x->{_e}) if $x->{_es} eq '-'; | |
2516 | return ($len, $t); | |
2517 | } | |
2518 | $len; | |
2519 | } | |
2520 | ||
2521 | 1; | |
2522 | __END__ | |
2523 | ||
2524 | =head1 NAME | |
2525 | ||
2526 | Math::BigFloat - Arbitrary size floating point math package | |
2527 | ||
2528 | =head1 SYNOPSIS | |
2529 | ||
2530 | use Math::BigFloat; | |
2531 | ||
2532 | # Number creation | |
2533 | $x = Math::BigFloat->new($str); # defaults to 0 | |
2534 | $nan = Math::BigFloat->bnan(); # create a NotANumber | |
2535 | $zero = Math::BigFloat->bzero(); # create a +0 | |
2536 | $inf = Math::BigFloat->binf(); # create a +inf | |
2537 | $inf = Math::BigFloat->binf('-'); # create a -inf | |
2538 | $one = Math::BigFloat->bone(); # create a +1 | |
2539 | $one = Math::BigFloat->bone('-'); # create a -1 | |
2540 | ||
2541 | # Testing | |
2542 | $x->is_zero(); # true if arg is +0 | |
2543 | $x->is_nan(); # true if arg is NaN | |
2544 | $x->is_one(); # true if arg is +1 | |
2545 | $x->is_one('-'); # true if arg is -1 | |
2546 | $x->is_odd(); # true if odd, false for even | |
2547 | $x->is_even(); # true if even, false for odd | |
2548 | $x->is_pos(); # true if >= 0 | |
2549 | $x->is_neg(); # true if < 0 | |
2550 | $x->is_inf(sign); # true if +inf, or -inf (default is '+') | |
2551 | ||
2552 | $x->bcmp($y); # compare numbers (undef,<0,=0,>0) | |
2553 | $x->bacmp($y); # compare absolutely (undef,<0,=0,>0) | |
2554 | $x->sign(); # return the sign, either +,- or NaN | |
2555 | $x->digit($n); # return the nth digit, counting from right | |
2556 | $x->digit(-$n); # return the nth digit, counting from left | |
2557 | ||
2558 | # The following all modify their first argument. If you want to preserve | |
2559 | # $x, use $z = $x->copy()->bXXX($y); See under L<CAVEATS> for why this is | |
2560 | # neccessary when mixing $a = $b assigments with non-overloaded math. | |
2561 | ||
2562 | # set | |
2563 | $x->bzero(); # set $i to 0 | |
2564 | $x->bnan(); # set $i to NaN | |
2565 | $x->bone(); # set $x to +1 | |
2566 | $x->bone('-'); # set $x to -1 | |
2567 | $x->binf(); # set $x to inf | |
2568 | $x->binf('-'); # set $x to -inf | |
2569 | ||
2570 | $x->bneg(); # negation | |
2571 | $x->babs(); # absolute value | |
2572 | $x->bnorm(); # normalize (no-op) | |
2573 | $x->bnot(); # two's complement (bit wise not) | |
2574 | $x->binc(); # increment x by 1 | |
2575 | $x->bdec(); # decrement x by 1 | |
2576 | ||
2577 | $x->badd($y); # addition (add $y to $x) | |
2578 | $x->bsub($y); # subtraction (subtract $y from $x) | |
2579 | $x->bmul($y); # multiplication (multiply $x by $y) | |
2580 | $x->bdiv($y); # divide, set $x to quotient | |
2581 | # return (quo,rem) or quo if scalar | |
2582 | ||
2583 | $x->bmod($y); # modulus ($x % $y) | |
2584 | $x->bpow($y); # power of arguments ($x ** $y) | |
2585 | $x->blsft($y); # left shift | |
2586 | $x->brsft($y); # right shift | |
2587 | # return (quo,rem) or quo if scalar | |
2588 | ||
2589 | $x->blog(); # logarithm of $x to base e (Euler's number) | |
2590 | $x->blog($base); # logarithm of $x to base $base (f.i. 2) | |
2591 | ||
2592 | $x->band($y); # bit-wise and | |
2593 | $x->bior($y); # bit-wise inclusive or | |
2594 | $x->bxor($y); # bit-wise exclusive or | |
2595 | $x->bnot(); # bit-wise not (two's complement) | |
2596 | ||
2597 | $x->bsqrt(); # calculate square-root | |
2598 | $x->broot($y); # $y'th root of $x (e.g. $y == 3 => cubic root) | |
2599 | $x->bfac(); # factorial of $x (1*2*3*4*..$x) | |
2600 | ||
2601 | $x->bround($N); # accuracy: preserve $N digits | |
2602 | $x->bfround($N); # precision: round to the $Nth digit | |
2603 | ||
2604 | $x->bfloor(); # return integer less or equal than $x | |
2605 | $x->bceil(); # return integer greater or equal than $x | |
2606 | ||
2607 | # The following do not modify their arguments: | |
2608 | ||
2609 | bgcd(@values); # greatest common divisor | |
2610 | blcm(@values); # lowest common multiplicator | |
2611 | ||
2612 | $x->bstr(); # return string | |
2613 | $x->bsstr(); # return string in scientific notation | |
2614 | ||
2615 | $x->as_int(); # return $x as BigInt | |
2616 | $x->exponent(); # return exponent as BigInt | |
2617 | $x->mantissa(); # return mantissa as BigInt | |
2618 | $x->parts(); # return (mantissa,exponent) as BigInt | |
2619 | ||
2620 | $x->length(); # number of digits (w/o sign and '.') | |
2621 | ($l,$f) = $x->length(); # number of digits, and length of fraction | |
2622 | ||
2623 | $x->precision(); # return P of $x (or global, if P of $x undef) | |
2624 | $x->precision($n); # set P of $x to $n | |
2625 | $x->accuracy(); # return A of $x (or global, if A of $x undef) | |
2626 | $x->accuracy($n); # set A $x to $n | |
2627 | ||
2628 | # these get/set the appropriate global value for all BigFloat objects | |
2629 | Math::BigFloat->precision(); # Precision | |
2630 | Math::BigFloat->accuracy(); # Accuracy | |
2631 | Math::BigFloat->round_mode(); # rounding mode | |
2632 | ||
2633 | =head1 DESCRIPTION | |
2634 | ||
2635 | All operators (inlcuding basic math operations) are overloaded if you | |
2636 | declare your big floating point numbers as | |
2637 | ||
2638 | $i = new Math::BigFloat '12_3.456_789_123_456_789E-2'; | |
2639 | ||
2640 | Operations with overloaded operators preserve the arguments, which is | |
2641 | exactly what you expect. | |
2642 | ||
2643 | =head2 Canonical notation | |
2644 | ||
2645 | Input to these routines are either BigFloat objects, or strings of the | |
2646 | following four forms: | |
2647 | ||
2648 | =over 2 | |
2649 | ||
2650 | =item * | |
2651 | ||
2652 | C</^[+-]\d+$/> | |
2653 | ||
2654 | =item * | |
2655 | ||
2656 | C</^[+-]\d+\.\d*$/> | |
2657 | ||
2658 | =item * | |
2659 | ||
2660 | C</^[+-]\d+E[+-]?\d+$/> | |
2661 | ||
2662 | =item * | |
2663 | ||
2664 | C</^[+-]\d*\.\d+E[+-]?\d+$/> | |
2665 | ||
2666 | =back | |
2667 | ||
2668 | all with optional leading and trailing zeros and/or spaces. Additonally, | |
2669 | numbers are allowed to have an underscore between any two digits. | |
2670 | ||
2671 | Empty strings as well as other illegal numbers results in 'NaN'. | |
2672 | ||
2673 | bnorm() on a BigFloat object is now effectively a no-op, since the numbers | |
2674 | are always stored in normalized form. On a string, it creates a BigFloat | |
2675 | object. | |
2676 | ||
2677 | =head2 Output | |
2678 | ||
2679 | Output values are BigFloat objects (normalized), except for bstr() and bsstr(). | |
2680 | ||
2681 | The string output will always have leading and trailing zeros stripped and drop | |
2682 | a plus sign. C<bstr()> will give you always the form with a decimal point, | |
2683 | while C<bsstr()> (s for scientific) gives you the scientific notation. | |
2684 | ||
2685 | Input bstr() bsstr() | |
2686 | '-0' '0' '0E1' | |
2687 | ' -123 123 123' '-123123123' '-123123123E0' | |
2688 | '00.0123' '0.0123' '123E-4' | |
2689 | '123.45E-2' '1.2345' '12345E-4' | |
2690 | '10E+3' '10000' '1E4' | |
2691 | ||
2692 | Some routines (C<is_odd()>, C<is_even()>, C<is_zero()>, C<is_one()>, | |
2693 | C<is_nan()>) return true or false, while others (C<bcmp()>, C<bacmp()>) | |
2694 | return either undef, <0, 0 or >0 and are suited for sort. | |
2695 | ||
2696 | Actual math is done by using the class defined with C<with => Class;> (which | |
2697 | defaults to BigInts) to represent the mantissa and exponent. | |
2698 | ||
2699 | The sign C</^[+-]$/> is stored separately. The string 'NaN' is used to | |
2700 | represent the result when input arguments are not numbers, as well as | |
2701 | the result of dividing by zero. | |
2702 | ||
2703 | =head2 C<mantissa()>, C<exponent()> and C<parts()> | |
2704 | ||
2705 | C<mantissa()> and C<exponent()> return the said parts of the BigFloat | |
2706 | as BigInts such that: | |
2707 | ||
2708 | $m = $x->mantissa(); | |
2709 | $e = $x->exponent(); | |
2710 | $y = $m * ( 10 ** $e ); | |
2711 | print "ok\n" if $x == $y; | |
2712 | ||
2713 | C<< ($m,$e) = $x->parts(); >> is just a shortcut giving you both of them. | |
2714 | ||
2715 | A zero is represented and returned as C<0E1>, B<not> C<0E0> (after Knuth). | |
2716 | ||
2717 | Currently the mantissa is reduced as much as possible, favouring higher | |
2718 | exponents over lower ones (e.g. returning 1e7 instead of 10e6 or 10000000e0). | |
2719 | This might change in the future, so do not depend on it. | |
2720 | ||
2721 | =head2 Accuracy vs. Precision | |
2722 | ||
2723 | See also: L<Rounding|Rounding>. | |
2724 | ||
2725 | Math::BigFloat supports both precision (rounding to a certain place before or | |
2726 | after the dot) and accuracy (rounding to a certain number of digits). For a | |
2727 | full documentation, examples and tips on these topics please see the large | |
2728 | section about rounding in L<Math::BigInt>. | |
2729 | ||
2730 | Since things like C<sqrt(2)> or C<1 / 3> must presented with a limited | |
2731 | accuracy lest a operation consumes all resources, each operation produces | |
2732 | no more than the requested number of digits. | |
2733 | ||
2734 | If there is no gloabl precision or accuracy set, B<and> the operation in | |
2735 | question was not called with a requested precision or accuracy, B<and> the | |
2736 | input $x has no accuracy or precision set, then a fallback parameter will | |
2737 | be used. For historical reasons, it is called C<div_scale> and can be accessed | |
2738 | via: | |
2739 | ||
2740 | $d = Math::BigFloat->div_scale(); # query | |
2741 | Math::BigFloat->div_scale($n); # set to $n digits | |
2742 | ||
2743 | The default value for C<div_scale> is 40. | |
2744 | ||
2745 | In case the result of one operation has more digits than specified, | |
2746 | it is rounded. The rounding mode taken is either the default mode, or the one | |
2747 | supplied to the operation after the I<scale>: | |
2748 | ||
2749 | $x = Math::BigFloat->new(2); | |
2750 | Math::BigFloat->accuracy(5); # 5 digits max | |
2751 | $y = $x->copy()->bdiv(3); # will give 0.66667 | |
2752 | $y = $x->copy()->bdiv(3,6); # will give 0.666667 | |
2753 | $y = $x->copy()->bdiv(3,6,undef,'odd'); # will give 0.666667 | |
2754 | Math::BigFloat->round_mode('zero'); | |
2755 | $y = $x->copy()->bdiv(3,6); # will also give 0.666667 | |
2756 | ||
2757 | Note that C<< Math::BigFloat->accuracy() >> and C<< Math::BigFloat->precision() >> | |
2758 | set the global variables, and thus B<any> newly created number will be subject | |
2759 | to the global rounding B<immidiately>. This means that in the examples above, the | |
2760 | C<3> as argument to C<bdiv()> will also get an accuracy of B<5>. | |
2761 | ||
2762 | It is less confusing to either calculate the result fully, and afterwards | |
2763 | round it explicitely, or use the additional parameters to the math | |
2764 | functions like so: | |
2765 | ||
2766 | use Math::BigFloat; | |
2767 | $x = Math::BigFloat->new(2); | |
2768 | $y = $x->copy()->bdiv(3); | |
2769 | print $y->bround(5),"\n"; # will give 0.66667 | |
2770 | ||
2771 | or | |
2772 | ||
2773 | use Math::BigFloat; | |
2774 | $x = Math::BigFloat->new(2); | |
2775 | $y = $x->copy()->bdiv(3,5); # will give 0.66667 | |
2776 | print "$y\n"; | |
2777 | ||
2778 | =head2 Rounding | |
2779 | ||
2780 | =over 2 | |
2781 | ||
2782 | =item ffround ( +$scale ) | |
2783 | ||
2784 | Rounds to the $scale'th place left from the '.', counting from the dot. | |
2785 | The first digit is numbered 1. | |
2786 | ||
2787 | =item ffround ( -$scale ) | |
2788 | ||
2789 | Rounds to the $scale'th place right from the '.', counting from the dot. | |
2790 | ||
2791 | =item ffround ( 0 ) | |
2792 | ||
2793 | Rounds to an integer. | |
2794 | ||
2795 | =item fround ( +$scale ) | |
2796 | ||
2797 | Preserves accuracy to $scale digits from the left (aka significant digits) | |
2798 | and pads the rest with zeros. If the number is between 1 and -1, the | |
2799 | significant digits count from the first non-zero after the '.' | |
2800 | ||
2801 | =item fround ( -$scale ) and fround ( 0 ) | |
2802 | ||
2803 | These are effectively no-ops. | |
2804 | ||
2805 | =back | |
2806 | ||
2807 | All rounding functions take as a second parameter a rounding mode from one of | |
2808 | the following: 'even', 'odd', '+inf', '-inf', 'zero' or 'trunc'. | |
2809 | ||
2810 | The default rounding mode is 'even'. By using | |
2811 | C<< Math::BigFloat->round_mode($round_mode); >> you can get and set the default | |
2812 | mode for subsequent rounding. The usage of C<$Math::BigFloat::$round_mode> is | |
2813 | no longer supported. | |
2814 | The second parameter to the round functions then overrides the default | |
2815 | temporarily. | |
2816 | ||
2817 | The C<as_number()> function returns a BigInt from a Math::BigFloat. It uses | |
2818 | 'trunc' as rounding mode to make it equivalent to: | |
2819 | ||
2820 | $x = 2.5; | |
2821 | $y = int($x) + 2; | |
2822 | ||
2823 | You can override this by passing the desired rounding mode as parameter to | |
2824 | C<as_number()>: | |
2825 | ||
2826 | $x = Math::BigFloat->new(2.5); | |
2827 | $y = $x->as_number('odd'); # $y = 3 | |
2828 | ||
2829 | =head1 METHODS | |
2830 | ||
2831 | =head2 accuracy | |
2832 | ||
2833 | $x->accuracy(5); # local for $x | |
2834 | CLASS->accuracy(5); # global for all members of CLASS | |
2835 | # Note: This also applies to new()! | |
2836 | ||
2837 | $A = $x->accuracy(); # read out accuracy that affects $x | |
2838 | $A = CLASS->accuracy(); # read out global accuracy | |
2839 | ||
2840 | Set or get the global or local accuracy, aka how many significant digits the | |
2841 | results have. If you set a global accuracy, then this also applies to new()! | |
2842 | ||
2843 | Warning! The accuracy I<sticks>, e.g. once you created a number under the | |
2844 | influence of C<< CLASS->accuracy($A) >>, all results from math operations with | |
2845 | that number will also be rounded. | |
2846 | ||
2847 | In most cases, you should probably round the results explicitely using one of | |
2848 | L<round()>, L<bround()> or L<bfround()> or by passing the desired accuracy | |
2849 | to the math operation as additional parameter: | |
2850 | ||
2851 | my $x = Math::BigInt->new(30000); | |
2852 | my $y = Math::BigInt->new(7); | |
2853 | print scalar $x->copy()->bdiv($y, 2); # print 4300 | |
2854 | print scalar $x->copy()->bdiv($y)->bround(2); # print 4300 | |
2855 | ||
2856 | =head2 precision() | |
2857 | ||
2858 | $x->precision(-2); # local for $x, round at the second digit right of the dot | |
2859 | $x->precision(2); # ditto, round at the second digit left of the dot | |
2860 | ||
2861 | CLASS->precision(5); # Global for all members of CLASS | |
2862 | # This also applies to new()! | |
2863 | CLASS->precision(-5); # ditto | |
2864 | ||
2865 | $P = CLASS->precision(); # read out global precision | |
2866 | $P = $x->precision(); # read out precision that affects $x | |
2867 | ||
2868 | Note: You probably want to use L<accuracy()> instead. With L<accuracy> you | |
2869 | set the number of digits each result should have, with L<precision> you | |
2870 | set the place where to round! | |
2871 | ||
2872 | =head1 Autocreating constants | |
2873 | ||
2874 | After C<use Math::BigFloat ':constant'> all the floating point constants | |
2875 | in the given scope are converted to C<Math::BigFloat>. This conversion | |
2876 | happens at compile time. | |
2877 | ||
2878 | In particular | |
2879 | ||
2880 | perl -MMath::BigFloat=:constant -e 'print 2E-100,"\n"' | |
2881 | ||
2882 | prints the value of C<2E-100>. Note that without conversion of | |
2883 | constants the expression 2E-100 will be calculated as normal floating point | |
2884 | number. | |
2885 | ||
2886 | Please note that ':constant' does not affect integer constants, nor binary | |
2887 | nor hexadecimal constants. Use L<bignum> or L<Math::BigInt> to get this to | |
2888 | work. | |
2889 | ||
2890 | =head2 Math library | |
2891 | ||
2892 | Math with the numbers is done (by default) by a module called | |
2893 | Math::BigInt::Calc. This is equivalent to saying: | |
2894 | ||
2895 | use Math::BigFloat lib => 'Calc'; | |
2896 | ||
2897 | You can change this by using: | |
2898 | ||
2899 | use Math::BigFloat lib => 'BitVect'; | |
2900 | ||
2901 | The following would first try to find Math::BigInt::Foo, then | |
2902 | Math::BigInt::Bar, and when this also fails, revert to Math::BigInt::Calc: | |
2903 | ||
2904 | use Math::BigFloat lib => 'Foo,Math::BigInt::Bar'; | |
2905 | ||
2906 | Calc.pm uses as internal format an array of elements of some decimal base | |
2907 | (usually 1e7, but this might be differen for some systems) with the least | |
2908 | significant digit first, while BitVect.pm uses a bit vector of base 2, most | |
2909 | significant bit first. Other modules might use even different means of | |
2910 | representing the numbers. See the respective module documentation for further | |
2911 | details. | |
2912 | ||
2913 | Please note that Math::BigFloat does B<not> use the denoted library itself, | |
2914 | but it merely passes the lib argument to Math::BigInt. So, instead of the need | |
2915 | to do: | |
2916 | ||
2917 | use Math::BigInt lib => 'GMP'; | |
2918 | use Math::BigFloat; | |
2919 | ||
2920 | you can roll it all into one line: | |
2921 | ||
2922 | use Math::BigFloat lib => 'GMP'; | |
2923 | ||
2924 | It is also possible to just require Math::BigFloat: | |
2925 | ||
2926 | require Math::BigFloat; | |
2927 | ||
2928 | This will load the neccessary things (like BigInt) when they are needed, and | |
2929 | automatically. | |
2930 | ||
2931 | Use the lib, Luke! And see L<Using Math::BigInt::Lite> for more details than | |
2932 | you ever wanted to know about loading a different library. | |
2933 | ||
2934 | =head2 Using Math::BigInt::Lite | |
2935 | ||
2936 | It is possible to use L<Math::BigInt::Lite> with Math::BigFloat: | |
2937 | ||
2938 | # 1 | |
2939 | use Math::BigFloat with => 'Math::BigInt::Lite'; | |
2940 | ||
2941 | There is no need to "use Math::BigInt" or "use Math::BigInt::Lite", but you | |
2942 | can combine these if you want. For instance, you may want to use | |
2943 | Math::BigInt objects in your main script, too. | |
2944 | ||
2945 | # 2 | |
2946 | use Math::BigInt; | |
2947 | use Math::BigFloat with => 'Math::BigInt::Lite'; | |
2948 | ||
2949 | Of course, you can combine this with the C<lib> parameter. | |
2950 | ||
2951 | # 3 | |
2952 | use Math::BigFloat with => 'Math::BigInt::Lite', lib => 'GMP,Pari'; | |
2953 | ||
2954 | There is no need for a "use Math::BigInt;" statement, even if you want to | |
2955 | use Math::BigInt's, since Math::BigFloat will needs Math::BigInt and thus | |
2956 | always loads it. But if you add it, add it B<before>: | |
2957 | ||
2958 | # 4 | |
2959 | use Math::BigInt; | |
2960 | use Math::BigFloat with => 'Math::BigInt::Lite', lib => 'GMP,Pari'; | |
2961 | ||
2962 | Notice that the module with the last C<lib> will "win" and thus | |
2963 | it's lib will be used if the lib is available: | |
2964 | ||
2965 | # 5 | |
2966 | use Math::BigInt lib => 'Bar,Baz'; | |
2967 | use Math::BigFloat with => 'Math::BigInt::Lite', lib => 'Foo'; | |
2968 | ||
2969 | That would try to load Foo, Bar, Baz and Calc (in that order). Or in other | |
2970 | words, Math::BigFloat will try to retain previously loaded libs when you | |
2971 | don't specify it onem but if you specify one, it will try to load them. | |
2972 | ||
2973 | Actually, the lib loading order would be "Bar,Baz,Calc", and then | |
2974 | "Foo,Bar,Baz,Calc", but independend of which lib exists, the result is the | |
2975 | same as trying the latter load alone, except for the fact that one of Bar or | |
2976 | Baz might be loaded needlessly in an intermidiate step (and thus hang around | |
2977 | and waste memory). If neither Bar nor Baz exist (or don't work/compile), they | |
2978 | will still be tried to be loaded, but this is not as time/memory consuming as | |
2979 | actually loading one of them. Still, this type of usage is not recommended due | |
2980 | to these issues. | |
2981 | ||
2982 | The old way (loading the lib only in BigInt) still works though: | |
2983 | ||
2984 | # 6 | |
2985 | use Math::BigInt lib => 'Bar,Baz'; | |
2986 | use Math::BigFloat; | |
2987 | ||
2988 | You can even load Math::BigInt afterwards: | |
2989 | ||
2990 | # 7 | |
2991 | use Math::BigFloat; | |
2992 | use Math::BigInt lib => 'Bar,Baz'; | |
2993 | ||
2994 | But this has the same problems like #5, it will first load Calc | |
2995 | (Math::BigFloat needs Math::BigInt and thus loads it) and then later Bar or | |
2996 | Baz, depending on which of them works and is usable/loadable. Since this | |
2997 | loads Calc unnecc., it is not recommended. | |
2998 | ||
2999 | Since it also possible to just require Math::BigFloat, this poses the question | |
3000 | about what libary this will use: | |
3001 | ||
3002 | require Math::BigFloat; | |
3003 | my $x = Math::BigFloat->new(123); $x += 123; | |
3004 | ||
3005 | It will use Calc. Please note that the call to import() is still done, but | |
3006 | only when you use for the first time some Math::BigFloat math (it is triggered | |
3007 | via any constructor, so the first time you create a Math::BigFloat, the load | |
3008 | will happen in the background). This means: | |
3009 | ||
3010 | require Math::BigFloat; | |
3011 | Math::BigFloat->import ( lib => 'Foo,Bar' ); | |
3012 | ||
3013 | would be the same as: | |
3014 | ||
3015 | use Math::BigFloat lib => 'Foo, Bar'; | |
3016 | ||
3017 | But don't try to be clever to insert some operations in between: | |
3018 | ||
3019 | require Math::BigFloat; | |
3020 | my $x = Math::BigFloat->bone() + 4; # load BigInt and Calc | |
3021 | Math::BigFloat->import( lib => 'Pari' ); # load Pari, too | |
3022 | $x = Math::BigFloat->bone()+4; # now use Pari | |
3023 | ||
3024 | While this works, it loads Calc needlessly. But maybe you just wanted that? | |
3025 | ||
3026 | B<Examples #3 is highly recommended> for daily usage. | |
3027 | ||
3028 | =head1 BUGS | |
3029 | ||
3030 | Please see the file BUGS in the CPAN distribution Math::BigInt for known bugs. | |
3031 | ||
3032 | =head1 CAVEATS | |
3033 | ||
3034 | =over 1 | |
3035 | ||
3036 | =item stringify, bstr() | |
3037 | ||
3038 | Both stringify and bstr() now drop the leading '+'. The old code would return | |
3039 | '+1.23', the new returns '1.23'. See the documentation in L<Math::BigInt> for | |
3040 | reasoning and details. | |
3041 | ||
3042 | =item bdiv | |
3043 | ||
3044 | The following will probably not do what you expect: | |
3045 | ||
3046 | print $c->bdiv(123.456),"\n"; | |
3047 | ||
3048 | It prints both quotient and reminder since print works in list context. Also, | |
3049 | bdiv() will modify $c, so be carefull. You probably want to use | |
3050 | ||
3051 | print $c / 123.456,"\n"; | |
3052 | print scalar $c->bdiv(123.456),"\n"; # or if you want to modify $c | |
3053 | ||
3054 | instead. | |
3055 | ||
3056 | =item Modifying and = | |
3057 | ||
3058 | Beware of: | |
3059 | ||
3060 | $x = Math::BigFloat->new(5); | |
3061 | $y = $x; | |
3062 | ||
3063 | It will not do what you think, e.g. making a copy of $x. Instead it just makes | |
3064 | a second reference to the B<same> object and stores it in $y. Thus anything | |
3065 | that modifies $x will modify $y (except overloaded math operators), and vice | |
3066 | versa. See L<Math::BigInt> for details and how to avoid that. | |
3067 | ||
3068 | =item bpow | |
3069 | ||
3070 | C<bpow()> now modifies the first argument, unlike the old code which left | |
3071 | it alone and only returned the result. This is to be consistent with | |
3072 | C<badd()> etc. The first will modify $x, the second one won't: | |
3073 | ||
3074 | print bpow($x,$i),"\n"; # modify $x | |
3075 | print $x->bpow($i),"\n"; # ditto | |
3076 | print $x ** $i,"\n"; # leave $x alone | |
3077 | ||
3078 | =item precision() vs. accuracy() | |
3079 | ||
3080 | A common pitfall is to use L<precision()> when you want to round a result to | |
3081 | a certain number of digits: | |
3082 | ||
3083 | use Math::BigFloat; | |
3084 | ||
3085 | Math::BigFloat->precision(4); # does not do what you think it does | |
3086 | my $x = Math::BigFloat->new(12345); # rounds $x to "12000"! | |
3087 | print "$x\n"; # print "12000" | |
3088 | my $y = Math::BigFloat->new(3); # rounds $y to "0"! | |
3089 | print "$y\n"; # print "0" | |
3090 | $z = $x / $y; # 12000 / 0 => NaN! | |
3091 | print "$z\n"; | |
3092 | print $z->precision(),"\n"; # 4 | |
3093 | ||
3094 | Replacing L<precision> with L<accuracy> is probably not what you want, either: | |
3095 | ||
3096 | use Math::BigFloat; | |
3097 | ||
3098 | Math::BigFloat->accuracy(4); # enables global rounding: | |
3099 | my $x = Math::BigFloat->new(123456); # rounded immidiately to "12350" | |
3100 | print "$x\n"; # print "123500" | |
3101 | my $y = Math::BigFloat->new(3); # rounded to "3 | |
3102 | print "$y\n"; # print "3" | |
3103 | print $z = $x->copy()->bdiv($y),"\n"; # 41170 | |
3104 | print $z->accuracy(),"\n"; # 4 | |
3105 | ||
3106 | What you want to use instead is: | |
3107 | ||
3108 | use Math::BigFloat; | |
3109 | ||
3110 | my $x = Math::BigFloat->new(123456); # no rounding | |
3111 | print "$x\n"; # print "123456" | |
3112 | my $y = Math::BigFloat->new(3); # no rounding | |
3113 | print "$y\n"; # print "3" | |
3114 | print $z = $x->copy()->bdiv($y,4),"\n"; # 41150 | |
3115 | print $z->accuracy(),"\n"; # undef | |
3116 | ||
3117 | In addition to computing what you expected, the last example also does B<not> | |
3118 | "taint" the result with an accuracy or precision setting, which would | |
3119 | influence any further operation. | |
3120 | ||
3121 | =back | |
3122 | ||
3123 | =head1 SEE ALSO | |
3124 | ||
3125 | L<Math::BigInt>, L<Math::BigRat> and L<Math::Big> as well as | |
3126 | L<Math::BigInt::BitVect>, L<Math::BigInt::Pari> and L<Math::BigInt::GMP>. | |
3127 | ||
3128 | The pragmas L<bignum>, L<bigint> and L<bigrat> might also be of interest | |
3129 | because they solve the autoupgrading/downgrading issue, at least partly. | |
3130 | ||
3131 | The package at | |
3132 | L<http://search.cpan.org/search?mode=module&query=Math%3A%3ABigInt> contains | |
3133 | more documentation including a full version history, testcases, empty | |
3134 | subclass files and benchmarks. | |
3135 | ||
3136 | =head1 LICENSE | |
3137 | ||
3138 | This program is free software; you may redistribute it and/or modify it under | |
3139 | the same terms as Perl itself. | |
3140 | ||
3141 | =head1 AUTHORS | |
3142 | ||
3143 | Mark Biggar, overloaded interface by Ilya Zakharevich. | |
3144 | Completely rewritten by Tels L<http://bloodgate.com> in 2001 - 2004, and still | |
3145 | at it in 2005. | |
3146 | ||
3147 | =cut |