Commit | Line | Data |
---|---|---|
13a12e00 JH |
1 | package Math::BigFloat; |
2 | ||
3 | # | |
d614cd8b | 4 | # Mike grinned. 'Two down, infinity to go' - Mike Nostrus in 'Before and After' |
13a12e00 JH |
5 | # |
6 | ||
58cde26e | 7 | # The following hash values are internally used: |
9b924220 RGS |
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 | ||
79a0c134 | 15 | $VERSION = '1.99_01'; |
0d71d61a | 16 | require 5.006002; |
3a427a11 RGS |
17 | |
18 | require Exporter; | |
fdb4b05f T |
19 | @ISA = qw/Math::BigInt/; |
20 | @EXPORT_OK = qw/bpi/; | |
394e6ffb | 21 | |
58cde26e | 22 | use strict; |
03874afe | 23 | # $_trap_inf/$_trap_nan are internal and should never be accessed from outside |
b282a552 T |
24 | use vars qw/$AUTOLOAD $accuracy $precision $div_scale $round_mode $rnd_mode |
25 | $upgrade $downgrade $_trap_nan $_trap_inf/; | |
58cde26e | 26 | my $class = "Math::BigFloat"; |
a0d0e21e | 27 | |
a5f75d66 | 28 | use overload |
a0ac753d | 29 | '<=>' => sub { my $rc = $_[2] ? |
bd05a461 | 30 | ref($_[0])->bcmp($_[1],$_[0]) : |
a0ac753d T |
31 | ref($_[0])->bcmp($_[0],$_[1]); |
32 | $rc = 1 unless defined $rc; | |
33 | $rc <=> 0; | |
34 | }, | |
35 | # we need '>=' to get things like "1 >= NaN" right: | |
36 | '>=' => sub { my $rc = $_[2] ? | |
37 | ref($_[0])->bcmp($_[1],$_[0]) : | |
38 | ref($_[0])->bcmp($_[0],$_[1]); | |
39 | # if there was a NaN involved, return false | |
40 | return '' unless defined $rc; | |
41 | $rc >= 0; | |
42 | }, | |
0716bf9b | 43 | 'int' => sub { $_[0]->as_number() }, # 'trunc' to bigint |
a5f75d66 | 44 | ; |
a0d0e21e | 45 | |
0716bf9b | 46 | ############################################################################## |
990fb837 | 47 | # global constants, flags and assorted stuff |
0716bf9b | 48 | |
990fb837 RGS |
49 | # the following are public, but their usage is not recommended. Use the |
50 | # accessor methods instead. | |
58cde26e | 51 | |
ee15d750 | 52 | # class constants, use Class->constant_name() to access |
20e2035c T |
53 | # one of 'even', 'odd', '+inf', '-inf', 'zero', 'trunc' or 'common' |
54 | $round_mode = 'even'; | |
ee15d750 JH |
55 | $accuracy = undef; |
56 | $precision = undef; | |
57 | $div_scale = 40; | |
58cde26e | 58 | |
b3abae2a JH |
59 | $upgrade = undef; |
60 | $downgrade = undef; | |
9b924220 RGS |
61 | # the package we are using for our private parts, defaults to: |
62 | # Math::BigInt->config()->{lib} | |
233f7bc0 | 63 | my $MBI = 'Math::BigInt::FastCalc'; |
990fb837 RGS |
64 | |
65 | # are NaNs ok? (otherwise it dies when encountering an NaN) set w/ config() | |
66 | $_trap_nan = 0; | |
9b924220 | 67 | # the same for infinity |
990fb837 RGS |
68 | $_trap_inf = 0; |
69 | ||
70 | # constant for easier life | |
71 | my $nan = 'NaN'; | |
72 | ||
9b924220 | 73 | my $IMPORT = 0; # was import() called yet? used to make require work |
990fb837 RGS |
74 | |
75 | # some digits of accuracy for blog(undef,10); which we use in blog() for speed | |
76 | my $LOG_10 = | |
77 | '2.3025850929940456840179914546843642076011014886287729760333279009675726097'; | |
78 | my $LOG_10_A = length($LOG_10)-1; | |
79 | # ditto for log(2) | |
80 | my $LOG_2 = | |
81 | '0.6931471805599453094172321214581765680755001343602552541206800094933936220'; | |
82 | my $LOG_2_A = length($LOG_2)-1; | |
7b29e1e6 | 83 | my $HALF = '0.5'; # made into an object if nec. |
990fb837 | 84 | |
027dc388 JH |
85 | ############################################################################## |
86 | # the old code had $rnd_mode, so we need to support it, too | |
87 | ||
027dc388 JH |
88 | sub TIESCALAR { my ($class) = @_; bless \$round_mode, $class; } |
89 | sub FETCH { return $round_mode; } | |
90 | sub STORE { $rnd_mode = $_[0]->round_mode($_[1]); } | |
91 | ||
56b9c951 | 92 | BEGIN |
990fb837 | 93 | { |
7d193e39 | 94 | # when someone sets $rnd_mode, we catch this and check the value to see |
990fb837 | 95 | # whether it is valid or not. |
7b29e1e6 T |
96 | $rnd_mode = 'even'; tie $rnd_mode, 'Math::BigFloat'; |
97 | ||
98 | # we need both of them in this package: | |
99 | *as_int = \&as_number; | |
56b9c951 | 100 | } |
027dc388 JH |
101 | |
102 | ############################################################################## | |
103 | ||
58cde26e | 104 | { |
ee15d750 | 105 | # valid method aliases for AUTOLOAD |
58cde26e JH |
106 | my %methods = map { $_ => 1 } |
107 | qw / fadd fsub fmul fdiv fround ffround fsqrt fmod fstr fsstr fpow fnorm | |
7d193e39 T |
108 | fint facmp fcmp fzero fnan finf finc fdec ffac fneg |
109 | fceil ffloor frsft flsft fone flog froot fexp | |
ee15d750 | 110 | /; |
7b29e1e6 | 111 | # valid methods that can be handed up (for AUTOLOAD) |
ee15d750 | 112 | my %hand_ups = map { $_ => 1 } |
ef9466ea | 113 | qw / is_nan is_inf is_negative is_positive is_pos is_neg |
b68b7ab1 | 114 | accuracy precision div_scale round_mode fabs fnot |
28df3e88 | 115 | objectify upgrade downgrade |
13a12e00 | 116 | bone binf bnan bzero |
a0ac753d | 117 | bsub |
58cde26e JH |
118 | /; |
119 | ||
7b29e1e6 T |
120 | sub _method_alias { exists $methods{$_[0]||''}; } |
121 | sub _method_hand_up { exists $hand_ups{$_[0]||''}; } | |
a0d0e21e | 122 | } |
0e8b9368 | 123 | |
58cde26e JH |
124 | ############################################################################## |
125 | # constructors | |
a0d0e21e | 126 | |
58cde26e JH |
127 | sub new |
128 | { | |
129 | # create a new BigFloat object from a string or another bigfloat object. | |
130 | # _e: exponent | |
131 | # _m: mantissa | |
132 | # sign => sign (+/-), or "NaN" | |
a0d0e21e | 133 | |
61f5c3f5 | 134 | my ($class,$wanted,@r) = @_; |
b3abae2a | 135 | |
61f5c3f5 T |
136 | # avoid numify-calls by not using || on $wanted! |
137 | return $class->bzero() if !defined $wanted; # default to 0 | |
138 | return $wanted->copy() if UNIVERSAL::isa($wanted,'Math::BigFloat'); | |
a0d0e21e | 139 | |
990fb837 RGS |
140 | $class->import() if $IMPORT == 0; # make require work |
141 | ||
58cde26e | 142 | my $self = {}; bless $self, $class; |
b22b3e31 | 143 | # shortcut for bigints and its subclasses |
a0ac753d | 144 | if ((ref($wanted)) && UNIVERSAL::can( $wanted, "as_number")) |
58cde26e | 145 | { |
9b924220 RGS |
146 | $self->{_m} = $wanted->as_number()->{value}; # get us a bigint copy |
147 | $self->{_e} = $MBI->_zero(); | |
148 | $self->{_es} = '+'; | |
58cde26e | 149 | $self->{sign} = $wanted->sign(); |
0716bf9b | 150 | return $self->bnorm(); |
58cde26e | 151 | } |
9681bfa6 | 152 | # else: got a string or something masquerading as number (with overload) |
2d2b2744 | 153 | |
58cde26e | 154 | # handle '+inf', '-inf' first |
233f7bc0 | 155 | if ($wanted =~ /^[+-]?inf\z/) |
58cde26e | 156 | { |
28df3e88 JH |
157 | return $downgrade->new($wanted) if $downgrade; |
158 | ||
233f7bc0 T |
159 | $self->{sign} = $wanted; # set a default sign for bstr() |
160 | return $self->binf($wanted); | |
58cde26e | 161 | } |
b282a552 | 162 | |
2d2b2744 T |
163 | # shortcut for simple forms like '12' that neither have trailing nor leading |
164 | # zeros | |
165 | if ($wanted =~ /^([+-]?)([1-9][0-9]*[1-9])$/) | |
166 | { | |
167 | $self->{_e} = $MBI->_zero(); | |
168 | $self->{_es} = '+'; | |
169 | $self->{sign} = $1 || '+'; | |
170 | $self->{_m} = $MBI->_new($2); | |
171 | return $self->round(@r) if !$downgrade; | |
172 | } | |
173 | ||
9b924220 | 174 | my ($mis,$miv,$mfv,$es,$ev) = Math::BigInt::_split($wanted); |
58cde26e JH |
175 | if (!ref $mis) |
176 | { | |
990fb837 RGS |
177 | if ($_trap_nan) |
178 | { | |
179 | require Carp; | |
180 | Carp::croak ("$wanted is not a number initialized to $class"); | |
181 | } | |
28df3e88 JH |
182 | |
183 | return $downgrade->bnan() if $downgrade; | |
184 | ||
9b924220 RGS |
185 | $self->{_e} = $MBI->_zero(); |
186 | $self->{_es} = '+'; | |
187 | $self->{_m} = $MBI->_zero(); | |
58cde26e JH |
188 | $self->{sign} = $nan; |
189 | } | |
190 | else | |
191 | { | |
9b924220 RGS |
192 | # make integer from mantissa by adjusting exp, then convert to int |
193 | $self->{_e} = $MBI->_new($$ev); # exponent | |
194 | $self->{_es} = $$es || '+'; | |
195 | my $mantissa = "$$miv$$mfv"; # create mant. | |
196 | $mantissa =~ s/^0+(\d)/$1/; # strip leading zeros | |
197 | $self->{_m} = $MBI->_new($mantissa); # create mant. | |
b282a552 | 198 | |
58cde26e | 199 | # 3.123E0 = 3123E-3, and 3.123E-2 => 3123E-5 |
9b924220 RGS |
200 | if (CORE::length($$mfv) != 0) |
201 | { | |
202 | my $len = $MBI->_new( CORE::length($$mfv)); | |
203 | ($self->{_e}, $self->{_es}) = | |
204 | _e_sub ($self->{_e}, $len, $self->{_es}, '+'); | |
205 | } | |
2d2b2744 T |
206 | # we can only have trailing zeros on the mantissa if $$mfv eq '' |
207 | else | |
b282a552 | 208 | { |
2d2b2744 T |
209 | # Use a regexp to count the trailing zeros in $$miv instead of _zeros() |
210 | # because that is faster, especially when _m is not stored in base 10. | |
211 | my $zeros = 0; $zeros = CORE::length($1) if $$miv =~ /[1-9](0*)$/; | |
b282a552 T |
212 | if ($zeros != 0) |
213 | { | |
9b924220 | 214 | my $z = $MBI->_new($zeros); |
2d2b2744 | 215 | # turn '120e2' into '12e3' |
9b924220 | 216 | $MBI->_rsft ( $self->{_m}, $z, 10); |
7596a890 RGS |
217 | ($self->{_e}, $self->{_es}) = |
218 | _e_add ( $self->{_e}, $z, $self->{_es}, '+'); | |
b282a552 T |
219 | } |
220 | } | |
2d2b2744 T |
221 | $self->{sign} = $$mis; |
222 | ||
9b924220 | 223 | # for something like 0Ey, set y to 1, and -0 => +0 |
3c4b39be | 224 | # Check $$miv for being '0' and $$mfv eq '', because otherwise _m could not |
2d2b2744 | 225 | # have become 0. That's faster than to call $MBI->_is_zero(). |
9b924220 | 226 | $self->{sign} = '+', $self->{_e} = $MBI->_one() |
2d2b2744 T |
227 | if $$miv eq '0' and $$mfv eq ''; |
228 | ||
b282a552 | 229 | return $self->round(@r) if !$downgrade; |
58cde26e | 230 | } |
28df3e88 JH |
231 | # if downgrade, inf, NaN or integers go down |
232 | ||
9b924220 | 233 | if ($downgrade && $self->{_es} eq '+') |
28df3e88 | 234 | { |
9b924220 | 235 | if ($MBI->_is_zero( $self->{_e} )) |
28df3e88 | 236 | { |
9b924220 | 237 | return $downgrade->new($$mis . $MBI->_str( $self->{_m} )); |
28df3e88 | 238 | } |
8df1e0a2 | 239 | return $downgrade->new($self->bsstr()); |
28df3e88 | 240 | } |
990fb837 | 241 | $self->bnorm()->round(@r); # first normalize, then round |
58cde26e | 242 | } |
a0d0e21e | 243 | |
9b924220 RGS |
244 | sub copy |
245 | { | |
86f0d17a | 246 | # if two arguments, the first one is the class to "swallow" subclasses |
9b924220 RGS |
247 | if (@_ > 1) |
248 | { | |
86f0d17a T |
249 | my $self = bless { |
250 | sign => $_[1]->{sign}, | |
251 | _es => $_[1]->{_es}, | |
252 | _m => $MBI->_copy($_[1]->{_m}), | |
253 | _e => $MBI->_copy($_[1]->{_e}), | |
254 | }, $_[0] if @_ > 1; | |
255 | ||
256 | $self->{_a} = $_[1]->{_a} if defined $_[1]->{_a}; | |
257 | $self->{_p} = $_[1]->{_p} if defined $_[1]->{_p}; | |
258 | return $self; | |
9b924220 | 259 | } |
9b924220 | 260 | |
86f0d17a T |
261 | my $self = bless { |
262 | sign => $_[0]->{sign}, | |
263 | _es => $_[0]->{_es}, | |
264 | _m => $MBI->_copy($_[0]->{_m}), | |
265 | _e => $MBI->_copy($_[0]->{_e}), | |
266 | }, ref($_[0]); | |
9b924220 | 267 | |
86f0d17a T |
268 | $self->{_a} = $_[0]->{_a} if defined $_[0]->{_a}; |
269 | $self->{_p} = $_[0]->{_p} if defined $_[0]->{_p}; | |
9b924220 RGS |
270 | $self; |
271 | } | |
272 | ||
13a12e00 | 273 | sub _bnan |
58cde26e | 274 | { |
990fb837 | 275 | # used by parent class bone() to initialize number to NaN |
58cde26e | 276 | my $self = shift; |
990fb837 RGS |
277 | |
278 | if ($_trap_nan) | |
279 | { | |
280 | require Carp; | |
281 | my $class = ref($self); | |
282 | Carp::croak ("Tried to set $self to NaN in $class\::_bnan()"); | |
283 | } | |
284 | ||
285 | $IMPORT=1; # call our import only once | |
9b924220 RGS |
286 | $self->{_m} = $MBI->_zero(); |
287 | $self->{_e} = $MBI->_zero(); | |
288 | $self->{_es} = '+'; | |
58cde26e | 289 | } |
a0d0e21e | 290 | |
13a12e00 | 291 | sub _binf |
58cde26e | 292 | { |
990fb837 | 293 | # used by parent class bone() to initialize number to +-inf |
58cde26e | 294 | my $self = shift; |
990fb837 RGS |
295 | |
296 | if ($_trap_inf) | |
297 | { | |
298 | require Carp; | |
299 | my $class = ref($self); | |
300 | Carp::croak ("Tried to set $self to +-inf in $class\::_binf()"); | |
301 | } | |
302 | ||
303 | $IMPORT=1; # call our import only once | |
9b924220 RGS |
304 | $self->{_m} = $MBI->_zero(); |
305 | $self->{_e} = $MBI->_zero(); | |
306 | $self->{_es} = '+'; | |
58cde26e | 307 | } |
a0d0e21e | 308 | |
13a12e00 | 309 | sub _bone |
574bacfe | 310 | { |
13a12e00 | 311 | # used by parent class bone() to initialize number to 1 |
574bacfe | 312 | my $self = shift; |
990fb837 | 313 | $IMPORT=1; # call our import only once |
9b924220 RGS |
314 | $self->{_m} = $MBI->_one(); |
315 | $self->{_e} = $MBI->_zero(); | |
316 | $self->{_es} = '+'; | |
574bacfe JH |
317 | } |
318 | ||
13a12e00 | 319 | sub _bzero |
58cde26e | 320 | { |
990fb837 | 321 | # used by parent class bone() to initialize number to 0 |
58cde26e | 322 | my $self = shift; |
990fb837 | 323 | $IMPORT=1; # call our import only once |
9b924220 RGS |
324 | $self->{_m} = $MBI->_zero(); |
325 | $self->{_e} = $MBI->_one(); | |
326 | $self->{_es} = '+'; | |
58cde26e JH |
327 | } |
328 | ||
9393ace2 JH |
329 | sub isa |
330 | { | |
331 | my ($self,$class) = @_; | |
56b9c951 JH |
332 | return if $class =~ /^Math::BigInt/; # we aren't one of these |
333 | UNIVERSAL::isa($self,$class); | |
9393ace2 JH |
334 | } |
335 | ||
8f675a64 JH |
336 | sub config |
337 | { | |
338 | # return (later set?) configuration data as hash ref | |
339 | my $class = shift || 'Math::BigFloat'; | |
340 | ||
2ebb273f T |
341 | if (@_ == 1 && ref($_[0]) ne 'HASH') |
342 | { | |
343 | my $cfg = $class->SUPER::config(); | |
344 | return $cfg->{$_[0]}; | |
345 | } | |
346 | ||
990fb837 | 347 | my $cfg = $class->SUPER::config(@_); |
8f675a64 | 348 | |
990fb837 | 349 | # now we need only to override the ones that are different from our parent |
8f675a64 JH |
350 | $cfg->{class} = $class; |
351 | $cfg->{with} = $MBI; | |
8f675a64 JH |
352 | $cfg; |
353 | } | |
354 | ||
58cde26e | 355 | ############################################################################## |
9681bfa6 | 356 | # string conversion |
58cde26e JH |
357 | |
358 | sub bstr | |
359 | { | |
360 | # (ref to BFLOAT or num_str ) return num_str | |
361 | # Convert number from internal format to (non-scientific) string format. | |
362 | # internal format is always normalized (no leading zeros, "-0" => "+0") | |
b68b7ab1 | 363 | my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_); |
58cde26e | 364 | |
574bacfe | 365 | if ($x->{sign} !~ /^[+-]$/) |
58cde26e | 366 | { |
574bacfe JH |
367 | return $x->{sign} unless $x->{sign} eq '+inf'; # -inf, NaN |
368 | return 'inf'; # +inf | |
58cde26e | 369 | } |
c38b2de2 | 370 | |
574bacfe JH |
371 | my $es = '0'; my $len = 1; my $cad = 0; my $dot = '.'; |
372 | ||
c38b2de2 | 373 | # $x is zero? |
9b924220 | 374 | my $not_zero = !($x->{sign} eq '+' && $MBI->_is_zero($x->{_m})); |
574bacfe | 375 | if ($not_zero) |
58cde26e | 376 | { |
9b924220 | 377 | $es = $MBI->_str($x->{_m}); |
574bacfe | 378 | $len = CORE::length($es); |
9b924220 RGS |
379 | my $e = $MBI->_num($x->{_e}); |
380 | $e = -$e if $x->{_es} eq '-'; | |
c38b2de2 | 381 | if ($e < 0) |
58cde26e | 382 | { |
c38b2de2 JH |
383 | $dot = ''; |
384 | # if _e is bigger than a scalar, the following will blow your memory | |
385 | if ($e <= -$len) | |
574bacfe | 386 | { |
c38b2de2 JH |
387 | my $r = abs($e) - $len; |
388 | $es = '0.'. ('0' x $r) . $es; $cad = -($len+$r); | |
574bacfe JH |
389 | } |
390 | else | |
391 | { | |
9b924220 RGS |
392 | substr($es,$e,0) = '.'; $cad = $MBI->_num($x->{_e}); |
393 | $cad = -$cad if $x->{_es} eq '-'; | |
574bacfe | 394 | } |
82cf049f | 395 | } |
c38b2de2 JH |
396 | elsif ($e > 0) |
397 | { | |
398 | # expand with zeros | |
399 | $es .= '0' x $e; $len += $e; $cad = 0; | |
400 | } | |
574bacfe | 401 | } # if not zero |
9b924220 | 402 | |
c38b2de2 JH |
403 | $es = '-'.$es if $x->{sign} eq '-'; |
404 | # if set accuracy or precision, pad with zeros on the right side | |
574bacfe JH |
405 | if ((defined $x->{_a}) && ($not_zero)) |
406 | { | |
407 | # 123400 => 6, 0.1234 => 4, 0.001234 => 4 | |
408 | my $zeros = $x->{_a} - $cad; # cad == 0 => 12340 | |
409 | $zeros = $x->{_a} - $len if $cad != $len; | |
574bacfe | 410 | $es .= $dot.'0' x $zeros if $zeros > 0; |
82cf049f | 411 | } |
c38b2de2 | 412 | elsif ((($x->{_p} || 0) < 0)) |
58cde26e | 413 | { |
574bacfe JH |
414 | # 123400 => 6, 0.1234 => 4, 0.001234 => 6 |
415 | my $zeros = -$x->{_p} + $cad; | |
574bacfe | 416 | $es .= $dot.'0' x $zeros if $zeros > 0; |
58cde26e | 417 | } |
56b9c951 | 418 | $es; |
82cf049f | 419 | } |
f216259d | 420 | |
58cde26e JH |
421 | sub bsstr |
422 | { | |
423 | # (ref to BFLOAT or num_str ) return num_str | |
424 | # Convert number from internal format to scientific string format. | |
425 | # internal format is always normalized (no leading zeros, "-0E0" => "+0E0") | |
b68b7ab1 | 426 | my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_); |
a0d0e21e | 427 | |
574bacfe JH |
428 | if ($x->{sign} !~ /^[+-]$/) |
429 | { | |
430 | return $x->{sign} unless $x->{sign} eq '+inf'; # -inf, NaN | |
431 | return 'inf'; # +inf | |
432 | } | |
9b924220 | 433 | my $sep = 'e'.$x->{_es}; |
56d9de68 | 434 | my $sign = $x->{sign}; $sign = '' if $sign eq '+'; |
9b924220 | 435 | $sign . $MBI->_str($x->{_m}) . $sep . $MBI->_str($x->{_e}); |
58cde26e JH |
436 | } |
437 | ||
438 | sub numify | |
439 | { | |
440 | # Make a number from a BigFloat object | |
b282a552 T |
441 | # simple return a string and let Perl's atoi()/atof() handle the rest |
442 | my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_); | |
56b9c951 | 443 | $x->bsstr(); |
58cde26e | 444 | } |
a0d0e21e | 445 | |
58cde26e JH |
446 | ############################################################################## |
447 | # public stuff (usually prefixed with "b") | |
448 | ||
b68b7ab1 T |
449 | sub bneg |
450 | { | |
451 | # (BINT or num_str) return BINT | |
452 | # negate number or make a negated number from string | |
453 | my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_); | |
454 | ||
455 | return $x if $x->modify('bneg'); | |
456 | ||
457 | # for +0 dont negate (to have always normalized +0). Does nothing for 'NaN' | |
458 | $x->{sign} =~ tr/+-/-+/ unless ($x->{sign} eq '+' && $MBI->_is_zero($x->{_m})); | |
459 | $x; | |
460 | } | |
461 | ||
574bacfe | 462 | # tels 2001-08-04 |
b282a552 | 463 | # XXX TODO this must be overwritten and return NaN for non-integer values |
574bacfe | 464 | # band(), bior(), bxor(), too |
58cde26e JH |
465 | #sub bnot |
466 | # { | |
467 | # $class->SUPER::bnot($class,@_); | |
468 | # } | |
469 | ||
470 | sub bcmp | |
471 | { | |
472 | # Compares 2 values. Returns one of undef, <0, =0, >0. (suitable for sort) | |
f9a08e12 JH |
473 | |
474 | # set up parameters | |
475 | my ($self,$x,$y) = (ref($_[0]),@_); | |
476 | # objectify is costly, so avoid it | |
477 | if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1]))) | |
478 | { | |
479 | ($self,$x,$y) = objectify(2,@_); | |
480 | } | |
58cde26e | 481 | |
56d9de68 T |
482 | return $upgrade->bcmp($x,$y) if defined $upgrade && |
483 | ((!$x->isa($self)) || (!$y->isa($self))); | |
484 | ||
0716bf9b JH |
485 | if (($x->{sign} !~ /^[+-]$/) || ($y->{sign} !~ /^[+-]$/)) |
486 | { | |
487 | # handle +-inf and NaN | |
488 | return undef if (($x->{sign} eq $nan) || ($y->{sign} eq $nan)); | |
489 | return 0 if ($x->{sign} eq $y->{sign}) && ($x->{sign} =~ /^[+-]inf$/); | |
490 | return +1 if $x->{sign} eq '+inf'; | |
491 | return -1 if $x->{sign} eq '-inf'; | |
492 | return -1 if $y->{sign} eq '+inf'; | |
b3abae2a | 493 | return +1; |
0716bf9b JH |
494 | } |
495 | ||
496 | # check sign for speed first | |
574bacfe | 497 | return 1 if $x->{sign} eq '+' && $y->{sign} eq '-'; # does also 0 <=> -y |
58cde26e JH |
498 | return -1 if $x->{sign} eq '-' && $y->{sign} eq '+'; # does also -x <=> 0 |
499 | ||
574bacfe JH |
500 | # shortcut |
501 | my $xz = $x->is_zero(); | |
502 | my $yz = $y->is_zero(); | |
503 | return 0 if $xz && $yz; # 0 <=> 0 | |
504 | return -1 if $xz && $y->{sign} eq '+'; # 0 <=> +y | |
505 | return 1 if $yz && $x->{sign} eq '+'; # +x <=> 0 | |
58cde26e JH |
506 | |
507 | # adjust so that exponents are equal | |
9b924220 RGS |
508 | my $lxm = $MBI->_len($x->{_m}); |
509 | my $lym = $MBI->_len($y->{_m}); | |
28df3e88 | 510 | # the numify somewhat limits our length, but makes it much faster |
9b924220 RGS |
511 | my ($xes,$yes) = (1,1); |
512 | $xes = -1 if $x->{_es} ne '+'; | |
513 | $yes = -1 if $y->{_es} ne '+'; | |
514 | my $lx = $lxm + $xes * $MBI->_num($x->{_e}); | |
515 | my $ly = $lym + $yes * $MBI->_num($y->{_e}); | |
28df3e88 | 516 | my $l = $lx - $ly; $l = -$l if $x->{sign} eq '-'; |
bd05a461 | 517 | return $l <=> 0 if $l != 0; |
58cde26e | 518 | |
bd05a461 | 519 | # lengths (corrected by exponent) are equal |
28df3e88 | 520 | # so make mantissa equal length by padding with zero (shift left) |
bd05a461 JH |
521 | my $diff = $lxm - $lym; |
522 | my $xm = $x->{_m}; # not yet copy it | |
523 | my $ym = $y->{_m}; | |
524 | if ($diff > 0) | |
525 | { | |
9b924220 RGS |
526 | $ym = $MBI->_copy($y->{_m}); |
527 | $ym = $MBI->_lsft($ym, $MBI->_new($diff), 10); | |
bd05a461 JH |
528 | } |
529 | elsif ($diff < 0) | |
530 | { | |
9b924220 RGS |
531 | $xm = $MBI->_copy($x->{_m}); |
532 | $xm = $MBI->_lsft($xm, $MBI->_new(-$diff), 10); | |
bd05a461 | 533 | } |
9b924220 | 534 | my $rc = $MBI->_acmp($xm,$ym); |
58cde26e | 535 | $rc = -$rc if $x->{sign} eq '-'; # -124 < -123 |
b3abae2a | 536 | $rc <=> 0; |
58cde26e JH |
537 | } |
538 | ||
539 | sub bacmp | |
540 | { | |
541 | # Compares 2 values, ignoring their signs. | |
542 | # Returns one of undef, <0, =0, >0. (suitable for sort) | |
f9a08e12 JH |
543 | |
544 | # set up parameters | |
545 | my ($self,$x,$y) = (ref($_[0]),@_); | |
546 | # objectify is costly, so avoid it | |
547 | if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1]))) | |
548 | { | |
549 | ($self,$x,$y) = objectify(2,@_); | |
550 | } | |
ee15d750 | 551 | |
56d9de68 T |
552 | return $upgrade->bacmp($x,$y) if defined $upgrade && |
553 | ((!$x->isa($self)) || (!$y->isa($self))); | |
554 | ||
ee15d750 | 555 | # handle +-inf and NaN's |
abcfbf51 | 556 | if ($x->{sign} !~ /^[+-]$/ || $y->{sign} !~ /^[+-]$/) |
ee15d750 JH |
557 | { |
558 | return undef if (($x->{sign} eq $nan) || ($y->{sign} eq $nan)); | |
559 | return 0 if ($x->is_inf() && $y->is_inf()); | |
560 | return 1 if ($x->is_inf() && !$y->is_inf()); | |
b3abae2a | 561 | return -1; |
ee15d750 JH |
562 | } |
563 | ||
564 | # shortcut | |
565 | my $xz = $x->is_zero(); | |
566 | my $yz = $y->is_zero(); | |
567 | return 0 if $xz && $yz; # 0 <=> 0 | |
568 | return -1 if $xz && !$yz; # 0 <=> +y | |
569 | return 1 if $yz && !$xz; # +x <=> 0 | |
570 | ||
571 | # adjust so that exponents are equal | |
9b924220 RGS |
572 | my $lxm = $MBI->_len($x->{_m}); |
573 | my $lym = $MBI->_len($y->{_m}); | |
574 | my ($xes,$yes) = (1,1); | |
575 | $xes = -1 if $x->{_es} ne '+'; | |
576 | $yes = -1 if $y->{_es} ne '+'; | |
28df3e88 | 577 | # the numify somewhat limits our length, but makes it much faster |
9b924220 RGS |
578 | my $lx = $lxm + $xes * $MBI->_num($x->{_e}); |
579 | my $ly = $lym + $yes * $MBI->_num($y->{_e}); | |
394e6ffb | 580 | my $l = $lx - $ly; |
ee15d750 | 581 | return $l <=> 0 if $l != 0; |
58cde26e | 582 | |
ee15d750 | 583 | # lengths (corrected by exponent) are equal |
394e6ffb | 584 | # so make mantissa equal-length by padding with zero (shift left) |
ee15d750 JH |
585 | my $diff = $lxm - $lym; |
586 | my $xm = $x->{_m}; # not yet copy it | |
587 | my $ym = $y->{_m}; | |
588 | if ($diff > 0) | |
589 | { | |
9b924220 RGS |
590 | $ym = $MBI->_copy($y->{_m}); |
591 | $ym = $MBI->_lsft($ym, $MBI->_new($diff), 10); | |
ee15d750 JH |
592 | } |
593 | elsif ($diff < 0) | |
594 | { | |
9b924220 RGS |
595 | $xm = $MBI->_copy($x->{_m}); |
596 | $xm = $MBI->_lsft($xm, $MBI->_new(-$diff), 10); | |
ee15d750 | 597 | } |
9b924220 | 598 | $MBI->_acmp($xm,$ym); |
58cde26e | 599 | } |
a0d0e21e | 600 | |
58cde26e JH |
601 | sub badd |
602 | { | |
603 | # add second arg (BFLOAT or string) to first (BFLOAT) (modifies first) | |
604 | # return result as BFLOAT | |
f9a08e12 JH |
605 | |
606 | # set up parameters | |
80365507 | 607 | my ($self,$x,$y,@r) = (ref($_[0]),@_); |
f9a08e12 JH |
608 | # objectify is costly, so avoid it |
609 | if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1]))) | |
610 | { | |
80365507 | 611 | ($self,$x,$y,@r) = objectify(2,@_); |
f9a08e12 | 612 | } |
50109ad0 RGS |
613 | |
614 | return $x if $x->modify('badd'); | |
58cde26e | 615 | |
574bacfe JH |
616 | # inf and NaN handling |
617 | if (($x->{sign} !~ /^[+-]$/) || ($y->{sign} !~ /^[+-]$/)) | |
618 | { | |
619 | # NaN first | |
620 | return $x->bnan() if (($x->{sign} eq $nan) || ($y->{sign} eq $nan)); | |
13a12e00 | 621 | # inf handling |
574bacfe JH |
622 | if (($x->{sign} =~ /^[+-]inf$/) && ($y->{sign} =~ /^[+-]inf$/)) |
623 | { | |
13a12e00 JH |
624 | # +inf++inf or -inf+-inf => same, rest is NaN |
625 | return $x if $x->{sign} eq $y->{sign}; | |
626 | return $x->bnan(); | |
574bacfe | 627 | } |
56b9c951 | 628 | # +-inf + something => +inf; something +-inf => +-inf |
574bacfe JH |
629 | $x->{sign} = $y->{sign}, return $x if $y->{sign} =~ /^[+-]inf$/; |
630 | return $x; | |
631 | } | |
632 | ||
80365507 | 633 | return $upgrade->badd($x,$y,@r) if defined $upgrade && |
8f675a64 JH |
634 | ((!$x->isa($self)) || (!$y->isa($self))); |
635 | ||
80365507 T |
636 | $r[3] = $y; # no push! |
637 | ||
58cde26e | 638 | # speed: no add for 0+y or x+0 |
80365507 | 639 | return $x->bround(@r) if $y->is_zero(); # x+0 |
58cde26e JH |
640 | if ($x->is_zero()) # 0+y |
641 | { | |
642 | # make copy, clobbering up x (modify in place!) | |
9b924220 RGS |
643 | $x->{_e} = $MBI->_copy($y->{_e}); |
644 | $x->{_es} = $y->{_es}; | |
645 | $x->{_m} = $MBI->_copy($y->{_m}); | |
58cde26e | 646 | $x->{sign} = $y->{sign} || $nan; |
80365507 | 647 | return $x->round(@r); |
a0d0e21e | 648 | } |
58cde26e JH |
649 | |
650 | # take lower of the two e's and adapt m1 to it to match m2 | |
28df3e88 | 651 | my $e = $y->{_e}; |
9b924220 RGS |
652 | $e = $MBI->_zero() if !defined $e; # if no BFLOAT? |
653 | $e = $MBI->_copy($e); # make copy (didn't do it yet) | |
654 | ||
655 | my $es; | |
656 | ||
657 | ($e,$es) = _e_sub($e, $x->{_e}, $y->{_es} || '+', $x->{_es}); | |
658 | ||
659 | my $add = $MBI->_copy($y->{_m}); | |
660 | ||
661 | if ($es eq '-') # < 0 | |
58cde26e | 662 | { |
9b924220 RGS |
663 | $MBI->_lsft( $x->{_m}, $e, 10); |
664 | ($x->{_e},$x->{_es}) = _e_add($x->{_e}, $e, $x->{_es}, $es); | |
58cde26e | 665 | } |
9b924220 | 666 | elsif (!$MBI->_is_zero($e)) # > 0 |
58cde26e | 667 | { |
9b924220 | 668 | $MBI->_lsft($add, $e, 10); |
58cde26e | 669 | } |
61f5c3f5 | 670 | # else: both e are the same, so just leave them |
9b924220 RGS |
671 | |
672 | if ($x->{sign} eq $y->{sign}) | |
673 | { | |
674 | # add | |
675 | $x->{_m} = $MBI->_add($x->{_m}, $add); | |
676 | } | |
677 | else | |
678 | { | |
679 | ($x->{_m}, $x->{sign}) = | |
680 | _e_add($x->{_m}, $add, $x->{sign}, $y->{sign}); | |
681 | } | |
682 | ||
61f5c3f5 | 683 | # delete trailing zeros, then round |
80365507 | 684 | $x->bnorm()->round(@r); |
58cde26e JH |
685 | } |
686 | ||
03874afe | 687 | # sub bsub is inherited from Math::BigInt! |
58cde26e JH |
688 | |
689 | sub binc | |
690 | { | |
691 | # increment arg by one | |
b282a552 | 692 | my ($self,$x,@r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_); |
e745a66c | 693 | |
50109ad0 RGS |
694 | return $x if $x->modify('binc'); |
695 | ||
9b924220 | 696 | if ($x->{_es} eq '-') |
e745a66c | 697 | { |
b282a552 | 698 | return $x->badd($self->bone(),@r); # digits after dot |
e745a66c JH |
699 | } |
700 | ||
9b924220 | 701 | if (!$MBI->_is_zero($x->{_e})) # _e == 0 for NaN, inf, -inf |
e745a66c | 702 | { |
b282a552 | 703 | # 1e2 => 100, so after the shift below _m has a '0' as last digit |
9b924220 RGS |
704 | $x->{_m} = $MBI->_lsft($x->{_m}, $x->{_e},10); # 1e2 => 100 |
705 | $x->{_e} = $MBI->_zero(); # normalize | |
706 | $x->{_es} = '+'; | |
b282a552 T |
707 | # we know that the last digit of $x will be '1' or '9', depending on the |
708 | # sign | |
e745a66c JH |
709 | } |
710 | # now $x->{_e} == 0 | |
711 | if ($x->{sign} eq '+') | |
712 | { | |
9b924220 | 713 | $MBI->_inc($x->{_m}); |
b282a552 | 714 | return $x->bnorm()->bround(@r); |
e745a66c JH |
715 | } |
716 | elsif ($x->{sign} eq '-') | |
717 | { | |
9b924220 RGS |
718 | $MBI->_dec($x->{_m}); |
719 | $x->{sign} = '+' if $MBI->_is_zero($x->{_m}); # -1 +1 => -0 => +0 | |
b282a552 | 720 | return $x->bnorm()->bround(@r); |
e745a66c JH |
721 | } |
722 | # inf, nan handling etc | |
b282a552 | 723 | $x->badd($self->bone(),@r); # badd() does round |
58cde26e JH |
724 | } |
725 | ||
726 | sub bdec | |
727 | { | |
728 | # decrement arg by one | |
b282a552 | 729 | my ($self,$x,@r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_); |
e745a66c | 730 | |
50109ad0 RGS |
731 | return $x if $x->modify('bdec'); |
732 | ||
9b924220 | 733 | if ($x->{_es} eq '-') |
e745a66c | 734 | { |
b282a552 | 735 | return $x->badd($self->bone('-'),@r); # digits after dot |
e745a66c JH |
736 | } |
737 | ||
9b924220 | 738 | if (!$MBI->_is_zero($x->{_e})) |
e745a66c | 739 | { |
9b924220 RGS |
740 | $x->{_m} = $MBI->_lsft($x->{_m}, $x->{_e},10); # 1e2 => 100 |
741 | $x->{_e} = $MBI->_zero(); # normalize | |
742 | $x->{_es} = '+'; | |
e745a66c JH |
743 | } |
744 | # now $x->{_e} == 0 | |
745 | my $zero = $x->is_zero(); | |
746 | # <= 0 | |
747 | if (($x->{sign} eq '-') || $zero) | |
748 | { | |
9b924220 RGS |
749 | $MBI->_inc($x->{_m}); |
750 | $x->{sign} = '-' if $zero; # 0 => 1 => -1 | |
751 | $x->{sign} = '+' if $MBI->_is_zero($x->{_m}); # -1 +1 => -0 => +0 | |
b282a552 | 752 | return $x->bnorm()->round(@r); |
e745a66c JH |
753 | } |
754 | # > 0 | |
755 | elsif ($x->{sign} eq '+') | |
756 | { | |
9b924220 | 757 | $MBI->_dec($x->{_m}); |
b282a552 | 758 | return $x->bnorm()->round(@r); |
e745a66c JH |
759 | } |
760 | # inf, nan handling etc | |
9b924220 | 761 | $x->badd($self->bone('-'),@r); # does round |
58cde26e JH |
762 | } |
763 | ||
990fb837 RGS |
764 | sub DEBUG () { 0; } |
765 | ||
61f5c3f5 T |
766 | sub blog |
767 | { | |
990fb837 | 768 | my ($self,$x,$base,$a,$p,$r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_); |
61f5c3f5 | 769 | |
50109ad0 RGS |
770 | return $x if $x->modify('blog'); |
771 | ||
990fb837 RGS |
772 | # $base > 0, $base != 1; if $base == undef default to $base == e |
773 | # $x >= 0 | |
9393ace2 | 774 | |
b3abae2a JH |
775 | # we need to limit the accuracy to protect against overflow |
776 | my $fallback = 0; | |
990fb837 RGS |
777 | my ($scale,@params); |
778 | ($x,@params) = $x->_find_round_parameters($a,$p,$r); | |
61f5c3f5 | 779 | |
990fb837 RGS |
780 | # also takes care of the "error in _find_round_parameters?" case |
781 | return $x->bnan() if $x->{sign} ne '+' || $x->is_zero(); | |
091c87b1 | 782 | |
b3abae2a | 783 | # no rounding at all, so must use fallback |
990fb837 | 784 | if (scalar @params == 0) |
b3abae2a JH |
785 | { |
786 | # simulate old behaviour | |
990fb837 RGS |
787 | $params[0] = $self->div_scale(); # and round to it as accuracy |
788 | $params[1] = undef; # P = undef | |
789 | $scale = $params[0]+4; # at least four more for proper round | |
790 | $params[2] = $r; # round mode by caller or undef | |
b3abae2a JH |
791 | $fallback = 1; # to clear a/p afterwards |
792 | } | |
793 | else | |
794 | { | |
795 | # the 4 below is empirical, and there might be cases where it is not | |
796 | # enough... | |
990fb837 | 797 | $scale = abs($params[0] || $params[1]) + 4; # take whatever is defined |
b3abae2a | 798 | } |
61f5c3f5 | 799 | |
b3abae2a | 800 | return $x->bzero(@params) if $x->is_one(); |
50109ad0 | 801 | # base not defined => base == Euler's number e |
990fb837 RGS |
802 | if (defined $base) |
803 | { | |
091c87b1 | 804 | # make object, since we don't feed it through objectify() to still get the |
990fb837 RGS |
805 | # case of $base == undef |
806 | $base = $self->new($base) unless ref($base); | |
807 | # $base > 0; $base != 1 | |
808 | return $x->bnan() if $base->is_zero() || $base->is_one() || | |
809 | $base->{sign} ne '+'; | |
091c87b1 | 810 | # if $x == $base, we know the result must be 1.0 |
90d1b129 T |
811 | if ($x->bcmp($base) == 0) |
812 | { | |
813 | $x->bone('+',@params); | |
814 | if ($fallback) | |
815 | { | |
816 | # clear a/p after round, since user did not request it | |
817 | delete $x->{_a}; delete $x->{_p}; | |
818 | } | |
819 | return $x; | |
820 | } | |
990fb837 | 821 | } |
61f5c3f5 | 822 | |
b3abae2a | 823 | # when user set globals, they would interfere with our calculation, so |
56d9de68 | 824 | # disable them and later re-enable them |
b3abae2a JH |
825 | no strict 'refs'; |
826 | my $abr = "$self\::accuracy"; my $ab = $$abr; $$abr = undef; | |
827 | my $pbr = "$self\::precision"; my $pb = $$pbr; $$pbr = undef; | |
828 | # we also need to disable any set A or P on $x (_find_round_parameters took | |
829 | # them already into account), since these would interfere, too | |
830 | delete $x->{_a}; delete $x->{_p}; | |
9393ace2 | 831 | # need to disable $upgrade in BigInt, to avoid deep recursion |
b3abae2a | 832 | local $Math::BigInt::upgrade = undef; |
93c87d9d | 833 | local $Math::BigFloat::downgrade = undef; |
990fb837 RGS |
834 | |
835 | # upgrade $x if $x is not a BigFloat (handle BigInt input) | |
7d193e39 | 836 | # XXX TODO: rebless! |
990fb837 RGS |
837 | if (!$x->isa('Math::BigFloat')) |
838 | { | |
839 | $x = Math::BigFloat->new($x); | |
840 | $self = ref($x); | |
841 | } | |
b282a552 T |
842 | |
843 | my $done = 0; | |
844 | ||
845 | # If the base is defined and an integer, try to calculate integer result | |
846 | # first. This is very fast, and in case the real result was found, we can | |
847 | # stop right here. | |
848 | if (defined $base && $base->is_int() && $x->is_int()) | |
849 | { | |
9b924220 RGS |
850 | my $i = $MBI->_copy( $x->{_m} ); |
851 | $MBI->_lsft( $i, $x->{_e}, 10 ) unless $MBI->_is_zero($x->{_e}); | |
852 | my $int = Math::BigInt->bzero(); | |
853 | $int->{value} = $i; | |
b282a552 T |
854 | $int->blog($base->as_number()); |
855 | # if ($exact) | |
9b924220 | 856 | if ($base->as_number()->bpow($int) == $x) |
b282a552 T |
857 | { |
858 | # found result, return it | |
9b924220 RGS |
859 | $x->{_m} = $int->{value}; |
860 | $x->{_e} = $MBI->_zero(); | |
861 | $x->{_es} = '+'; | |
b282a552 T |
862 | $x->bnorm(); |
863 | $done = 1; | |
864 | } | |
865 | } | |
866 | ||
867 | if ($done == 0) | |
9393ace2 | 868 | { |
7d193e39 T |
869 | # base is undef, so base should be e (Euler's number), so first calculate the |
870 | # log to base e (using reduction by 10 (and probably 2)): | |
b282a552 | 871 | $self->_log_10($x,$scale); |
9b924220 | 872 | |
b282a552 T |
873 | # and if a different base was requested, convert it |
874 | if (defined $base) | |
875 | { | |
876 | $base = Math::BigFloat->new($base) unless $base->isa('Math::BigFloat'); | |
877 | # not ln, but some other base (don't modify $base) | |
878 | $x->bdiv( $base->copy()->blog(undef,$scale), $scale ); | |
879 | } | |
9393ace2 | 880 | } |
990fb837 | 881 | |
091c87b1 | 882 | # shortcut to not run through _find_round_parameters again |
990fb837 | 883 | if (defined $params[0]) |
b3abae2a | 884 | { |
990fb837 | 885 | $x->bround($params[0],$params[2]); # then round accordingly |
b3abae2a JH |
886 | } |
887 | else | |
888 | { | |
990fb837 | 889 | $x->bfround($params[1],$params[2]); # then round accordingly |
b3abae2a JH |
890 | } |
891 | if ($fallback) | |
892 | { | |
893 | # clear a/p after round, since user did not request it | |
ef9466ea | 894 | delete $x->{_a}; delete $x->{_p}; |
b3abae2a JH |
895 | } |
896 | # restore globals | |
897 | $$abr = $ab; $$pbr = $pb; | |
898 | ||
899 | $x; | |
61f5c3f5 T |
900 | } |
901 | ||
50109ad0 RGS |
902 | sub _len_to_steps |
903 | { | |
904 | # Given D (digits in decimal), compute N so that N! (N factorial) is | |
905 | # at least D digits long. D should be at least 50. | |
906 | my $d = shift; | |
907 | ||
908 | # two constants for the Ramanujan estimate of ln(N!) | |
909 | my $lg2 = log(2 * 3.14159265) / 2; | |
910 | my $lg10 = log(10); | |
911 | ||
912 | # D = 50 => N => 42, so L = 40 and R = 50 | |
913 | my $l = 40; my $r = $d; | |
914 | ||
915 | # Otherwise this does not work under -Mbignum and we do not yet have "no bignum;" :( | |
916 | $l = $l->numify if ref($l); | |
917 | $r = $r->numify if ref($r); | |
918 | $lg2 = $lg2->numify if ref($lg2); | |
919 | $lg10 = $lg10->numify if ref($lg10); | |
920 | ||
921 | # binary search for the right value (could this be written as the reverse of lg(n!)?) | |
922 | while ($r - $l > 1) | |
923 | { | |
924 | my $n = int(($r - $l) / 2) + $l; | |
925 | my $ramanujan = | |
926 | int(($n * log($n) - $n + log( $n * (1 + 4*$n*(1+2*$n)) ) / 6 + $lg2) / $lg10); | |
927 | $ramanujan > $d ? $r = $n : $l = $n; | |
928 | } | |
929 | $l; | |
930 | } | |
931 | ||
932 | sub bnok | |
933 | { | |
934 | # Calculate n over k (binomial coefficient or "choose" function) as integer. | |
935 | # set up parameters | |
936 | my ($self,$x,$y,@r) = (ref($_[0]),@_); | |
937 | ||
938 | # objectify is costly, so avoid it | |
939 | if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1]))) | |
940 | { | |
941 | ($self,$x,$y,@r) = objectify(2,@_); | |
942 | } | |
943 | ||
944 | return $x if $x->modify('bnok'); | |
945 | ||
946 | return $x->bnan() if $x->is_nan() || $y->is_nan(); | |
947 | return $x->binf() if $x->is_inf(); | |
948 | ||
949 | my $u = $x->as_int(); | |
950 | $u->bnok($y->as_int()); | |
951 | ||
952 | $x->{_m} = $u->{value}; | |
953 | $x->{_e} = $MBI->_zero(); | |
954 | $x->{_es} = '+'; | |
955 | $x->{sign} = '+'; | |
956 | $x->bnorm(@r); | |
957 | } | |
958 | ||
7d193e39 T |
959 | sub bexp |
960 | { | |
50109ad0 | 961 | # Calculate e ** X (Euler's number to the power of X) |
7d193e39 T |
962 | my ($self,$x,$a,$p,$r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_); |
963 | ||
50109ad0 RGS |
964 | return $x if $x->modify('bexp'); |
965 | ||
7d193e39 T |
966 | return $x->binf() if $x->{sign} eq '+inf'; |
967 | return $x->bzero() if $x->{sign} eq '-inf'; | |
968 | ||
969 | # we need to limit the accuracy to protect against overflow | |
970 | my $fallback = 0; | |
971 | my ($scale,@params); | |
972 | ($x,@params) = $x->_find_round_parameters($a,$p,$r); | |
973 | ||
974 | # also takes care of the "error in _find_round_parameters?" case | |
975 | return $x if $x->{sign} eq 'NaN'; | |
976 | ||
977 | # no rounding at all, so must use fallback | |
978 | if (scalar @params == 0) | |
979 | { | |
980 | # simulate old behaviour | |
981 | $params[0] = $self->div_scale(); # and round to it as accuracy | |
982 | $params[1] = undef; # P = undef | |
983 | $scale = $params[0]+4; # at least four more for proper round | |
984 | $params[2] = $r; # round mode by caller or undef | |
985 | $fallback = 1; # to clear a/p afterwards | |
986 | } | |
987 | else | |
988 | { | |
989 | # the 4 below is empirical, and there might be cases where it's not enough... | |
990 | $scale = abs($params[0] || $params[1]) + 4; # take whatever is defined | |
991 | } | |
992 | ||
993 | return $x->bone(@params) if $x->is_zero(); | |
994 | ||
995 | if (!$x->isa('Math::BigFloat')) | |
996 | { | |
997 | $x = Math::BigFloat->new($x); | |
998 | $self = ref($x); | |
999 | } | |
1000 | ||
1001 | # when user set globals, they would interfere with our calculation, so | |
1002 | # disable them and later re-enable them | |
1003 | no strict 'refs'; | |
1004 | my $abr = "$self\::accuracy"; my $ab = $$abr; $$abr = undef; | |
1005 | my $pbr = "$self\::precision"; my $pb = $$pbr; $$pbr = undef; | |
1006 | # we also need to disable any set A or P on $x (_find_round_parameters took | |
1007 | # them already into account), since these would interfere, too | |
1008 | delete $x->{_a}; delete $x->{_p}; | |
1009 | # need to disable $upgrade in BigInt, to avoid deep recursion | |
1010 | local $Math::BigInt::upgrade = undef; | |
1011 | local $Math::BigFloat::downgrade = undef; | |
1012 | ||
1013 | my $x_org = $x->copy(); | |
7d193e39 T |
1014 | |
1015 | # We use the following Taylor series: | |
1016 | ||
1017 | # x x^2 x^3 x^4 | |
1018 | # e = 1 + --- + --- + --- + --- ... | |
1019 | # 1! 2! 3! 4! | |
1020 | ||
1021 | # The difference for each term is X and N, which would result in: | |
1022 | # 2 copy, 2 mul, 2 add, 1 inc, 1 div operations per term | |
1023 | ||
1024 | # But it is faster to compute exp(1) and then raising it to the | |
1025 | # given power, esp. if $x is really big and an integer because: | |
1026 | ||
1027 | # * The numerator is always 1, making the computation faster | |
1028 | # * the series converges faster in the case of x == 1 | |
1029 | # * We can also easily check when we have reached our limit: when the | |
1030 | # term to be added is smaller than "1E$scale", we can stop - f.i. | |
1031 | # scale == 5, and we have 1/40320, then we stop since 1/40320 < 1E-5. | |
1032 | # * we can compute the *exact* result by simulating bigrat math: | |
1033 | ||
1034 | # 1 1 gcd(3,4) = 1 1*24 + 1*6 5 | |
1035 | # - + - = ---------- = -- | |
1036 | # 6 24 6*24 24 | |
1037 | ||
1038 | # We do not compute the gcd() here, but simple do: | |
1039 | # 1 1 1*24 + 1*6 30 | |
1040 | # - + - = --------- = -- | |
1041 | # 6 24 6*24 144 | |
1042 | ||
1043 | # In general: | |
1044 | # a c a*d + c*b and note that c is always 1 and d = (b*f) | |
1045 | # - + - = --------- | |
1046 | # b d b*d | |
1047 | ||
1048 | # This leads to: which can be reduced by b to: | |
1049 | # a 1 a*b*f + b a*f + 1 | |
1050 | # - + - = --------- = ------- | |
1051 | # b b*f b*b*f b*f | |
1052 | ||
1053 | # The first terms in the series are: | |
1054 | ||
1055 | # 1 1 1 1 1 1 1 1 13700 | |
1056 | # -- + -- + -- + -- + -- + --- + --- + ---- = ----- | |
1057 | # 1 1 2 6 24 120 720 5040 5040 | |
1058 | ||
50109ad0 | 1059 | # Note that we cannot simple reduce 13700/5040 to 685/252, but must keep A and B! |
7d193e39 | 1060 | |
50109ad0 | 1061 | if ($scale <= 75) |
7d193e39 | 1062 | { |
50109ad0 RGS |
1063 | # set $x directly from a cached string form |
1064 | $x->{_m} = $MBI->_new( | |
1065 | "27182818284590452353602874713526624977572470936999595749669676277240766303535476"); | |
1066 | $x->{sign} = '+'; | |
1067 | $x->{_es} = '-'; | |
1068 | $x->{_e} = $MBI->_new(79); | |
7d193e39 | 1069 | } |
50109ad0 RGS |
1070 | else |
1071 | { | |
1072 | # compute A and B so that e = A / B. | |
1073 | ||
1074 | # After some terms we end up with this, so we use it as a starting point: | |
1075 | my $A = $MBI->_new("90933395208605785401971970164779391644753259799242"); | |
1076 | my $F = $MBI->_new(42); my $step = 42; | |
1077 | ||
1078 | # Compute how many steps we need to take to get $A and $B sufficiently big | |
1079 | my $steps = _len_to_steps($scale - 4); | |
1080 | # print STDERR "# Doing $steps steps for ", $scale-4, " digits\n"; | |
1081 | while ($step++ <= $steps) | |
1082 | { | |
1083 | # calculate $a * $f + 1 | |
1084 | $A = $MBI->_mul($A, $F); | |
1085 | $A = $MBI->_inc($A); | |
1086 | # increment f | |
1087 | $F = $MBI->_inc($F); | |
1088 | } | |
1089 | # compute $B as factorial of $steps (this is faster than doing it manually) | |
1090 | my $B = $MBI->_fac($MBI->_new($steps)); | |
1091 | ||
1092 | # print "A ", $MBI->_str($A), "\nB ", $MBI->_str($B), "\n"; | |
7d193e39 | 1093 | |
50109ad0 RGS |
1094 | # compute A/B with $scale digits in the result (truncate, not round) |
1095 | $A = $MBI->_lsft( $A, $MBI->_new($scale), 10); | |
1096 | $A = $MBI->_div( $A, $B ); | |
7d193e39 | 1097 | |
50109ad0 RGS |
1098 | $x->{_m} = $A; |
1099 | $x->{sign} = '+'; | |
1100 | $x->{_es} = '-'; | |
1101 | $x->{_e} = $MBI->_new($scale); | |
1102 | } | |
7d193e39 | 1103 | |
50109ad0 RGS |
1104 | # $x contains now an estimate of e, with some surplus digits, so we can round |
1105 | if (!$x_org->is_one()) | |
7d193e39 | 1106 | { |
50109ad0 RGS |
1107 | # raise $x to the wanted power and round it in one step: |
1108 | $x->bpow($x_org, @params); | |
7d193e39 T |
1109 | } |
1110 | else | |
1111 | { | |
50109ad0 RGS |
1112 | # else just round the already computed result |
1113 | delete $x->{_a}; delete $x->{_p}; | |
1114 | # shortcut to not run through _find_round_parameters again | |
1115 | if (defined $params[0]) | |
1116 | { | |
1117 | $x->bround($params[0],$params[2]); # then round accordingly | |
1118 | } | |
1119 | else | |
1120 | { | |
1121 | $x->bfround($params[1],$params[2]); # then round accordingly | |
1122 | } | |
7d193e39 T |
1123 | } |
1124 | if ($fallback) | |
1125 | { | |
1126 | # clear a/p after round, since user did not request it | |
1127 | delete $x->{_a}; delete $x->{_p}; | |
1128 | } | |
1129 | # restore globals | |
1130 | $$abr = $ab; $$pbr = $pb; | |
1131 | ||
1132 | $x; # return modified $x | |
1133 | } | |
1134 | ||
990fb837 RGS |
1135 | sub _log |
1136 | { | |
091c87b1 | 1137 | # internal log function to calculate ln() based on Taylor series. |
990fb837 RGS |
1138 | # Modifies $x in place. |
1139 | my ($self,$x,$scale) = @_; | |
1140 | ||
091c87b1 T |
1141 | # in case of $x == 1, result is 0 |
1142 | return $x->bzero() if $x->is_one(); | |
1143 | ||
9681bfa6 | 1144 | # XXX TODO: rewrite this in a similar manner to bexp() |
7d193e39 | 1145 | |
990fb837 RGS |
1146 | # http://www.efunda.com/math/taylor_series/logarithmic.cfm?search_string=log |
1147 | ||
1148 | # u = x-1, v = x+1 | |
1149 | # _ _ | |
1150 | # Taylor: | u 1 u^3 1 u^5 | | |
1151 | # ln (x) = 2 | --- + - * --- + - * --- + ... | x > 0 | |
1152 | # |_ v 3 v^3 5 v^5 _| | |
1153 | ||
1154 | # This takes much more steps to calculate the result and is thus not used | |
1155 | # u = x-1 | |
1156 | # _ _ | |
1157 | # Taylor: | u 1 u^2 1 u^3 | | |
1158 | # ln (x) = 2 | --- + - * --- + - * --- + ... | x > 1/2 | |
1159 | # |_ x 2 x^2 3 x^3 _| | |
1160 | ||
990fb837 RGS |
1161 | my ($limit,$v,$u,$below,$factor,$two,$next,$over,$f); |
1162 | ||
1163 | $v = $x->copy(); $v->binc(); # v = x+1 | |
1164 | $x->bdec(); $u = $x->copy(); # u = x-1; x = x-1 | |
1165 | $x->bdiv($v,$scale); # first term: u/v | |
1166 | $below = $v->copy(); | |
1167 | $over = $u->copy(); | |
1168 | $u *= $u; $v *= $v; # u^2, v^2 | |
1169 | $below->bmul($v); # u^3, v^3 | |
1170 | $over->bmul($u); | |
1171 | $factor = $self->new(3); $f = $self->new(2); | |
1172 | ||
1173 | my $steps = 0 if DEBUG; | |
1174 | $limit = $self->new("1E-". ($scale-1)); | |
1175 | while (3 < 5) | |
1176 | { | |
1177 | # we calculate the next term, and add it to the last | |
1178 | # when the next term is below our limit, it won't affect the outcome | |
1179 | # anymore, so we stop | |
1180 | ||
1181 | # calculating the next term simple from over/below will result in quite | |
1182 | # a time hog if the input has many digits, since over and below will | |
1183 | # accumulate more and more digits, and the result will also have many | |
1184 | # digits, but in the end it is rounded to $scale digits anyway. So if we | |
1185 | # round $over and $below first, we save a lot of time for the division | |
1186 | # (not with log(1.2345), but try log (123**123) to see what I mean. This | |
1187 | # can introduce a rounding error if the division result would be f.i. | |
1188 | # 0.1234500000001 and we round it to 5 digits it would become 0.12346, but | |
091c87b1 T |
1189 | # if we truncated $over and $below we might get 0.12345. Does this matter |
1190 | # for the end result? So we give $over and $below 4 more digits to be | |
1191 | # on the safe side (unscientific error handling as usual... :+D | |
7d193e39 | 1192 | |
990fb837 RGS |
1193 | $next = $over->copy->bround($scale+4)->bdiv( |
1194 | $below->copy->bmul($factor)->bround($scale+4), | |
1195 | $scale); | |
1196 | ||
1197 | ## old version: | |
1198 | ## $next = $over->copy()->bdiv($below->copy()->bmul($factor),$scale); | |
1199 | ||
1200 | last if $next->bacmp($limit) <= 0; | |
1201 | ||
1202 | delete $next->{_a}; delete $next->{_p}; | |
1203 | $x->badd($next); | |
990fb837 RGS |
1204 | # calculate things for the next term |
1205 | $over *= $u; $below *= $v; $factor->badd($f); | |
1206 | if (DEBUG) | |
1207 | { | |
1208 | $steps++; print "step $steps = $x\n" if $steps % 10 == 0; | |
1209 | } | |
1210 | } | |
990fb837 | 1211 | print "took $steps steps\n" if DEBUG; |
7d193e39 | 1212 | $x->bmul($f); # $x *= 2 |
990fb837 RGS |
1213 | } |
1214 | ||
1215 | sub _log_10 | |
1216 | { | |
091c87b1 T |
1217 | # Internal log function based on reducing input to the range of 0.1 .. 9.99 |
1218 | # and then "correcting" the result to the proper one. Modifies $x in place. | |
990fb837 RGS |
1219 | my ($self,$x,$scale) = @_; |
1220 | ||
7d193e39 | 1221 | # Taking blog() from numbers greater than 10 takes a *very long* time, so we |
990fb837 | 1222 | # break the computation down into parts based on the observation that: |
7d193e39 T |
1223 | # blog(X*Y) = blog(X) + blog(Y) |
1224 | # We set Y here to multiples of 10 so that $x becomes below 1 - the smaller | |
1225 | # $x is the faster it gets. Since 2*$x takes about 10 times as | |
1226 | # long, we make it faster by about a factor of 100 by dividing $x by 10. | |
1227 | ||
1228 | # The same observation is valid for numbers smaller than 0.1, e.g. computing | |
1229 | # log(1) is fastest, and the further away we get from 1, the longer it takes. | |
1230 | # So we also 'break' this down by multiplying $x with 10 and subtract the | |
990fb837 RGS |
1231 | # log(10) afterwards to get the correct result. |
1232 | ||
7d193e39 T |
1233 | # To get $x even closer to 1, we also divide by 2 and then use log(2) to |
1234 | # correct for this. For instance if $x is 2.4, we use the formula: | |
1235 | # blog(2.4 * 2) == blog (1.2) + blog(2) | |
1236 | # and thus calculate only blog(1.2) and blog(2), which is faster in total | |
1237 | # than calculating blog(2.4). | |
1238 | ||
1239 | # In addition, the values for blog(2) and blog(10) are cached. | |
1240 | ||
1241 | # Calculate nr of digits before dot: | |
9b924220 RGS |
1242 | my $dbd = $MBI->_num($x->{_e}); |
1243 | $dbd = -$dbd if $x->{_es} eq '-'; | |
1244 | $dbd += $MBI->_len($x->{_m}); | |
990fb837 RGS |
1245 | |
1246 | # more than one digit (e.g. at least 10), but *not* exactly 10 to avoid | |
1247 | # infinite recursion | |
1248 | ||
1249 | my $calc = 1; # do some calculation? | |
1250 | ||
1251 | # disable the shortcut for 10, since we need log(10) and this would recurse | |
1252 | # infinitely deep | |
9b924220 | 1253 | if ($x->{_es} eq '+' && $MBI->_is_one($x->{_e}) && $MBI->_is_one($x->{_m})) |
990fb837 RGS |
1254 | { |
1255 | $dbd = 0; # disable shortcut | |
1256 | # we can use the cached value in these cases | |
1257 | if ($scale <= $LOG_10_A) | |
1258 | { | |
7d193e39 | 1259 | $x->bzero(); $x->badd($LOG_10); # modify $x in place |
990fb837 RGS |
1260 | $calc = 0; # no need to calc, but round |
1261 | } | |
7d193e39 | 1262 | # if we can't use the shortcut, we continue normally |
990fb837 | 1263 | } |
091c87b1 | 1264 | else |
990fb837 | 1265 | { |
091c87b1 | 1266 | # disable the shortcut for 2, since we maybe have it cached |
9b924220 | 1267 | if (($MBI->_is_zero($x->{_e}) && $MBI->_is_two($x->{_m}))) |
990fb837 | 1268 | { |
091c87b1 T |
1269 | $dbd = 0; # disable shortcut |
1270 | # we can use the cached value in these cases | |
1271 | if ($scale <= $LOG_2_A) | |
1272 | { | |
7d193e39 | 1273 | $x->bzero(); $x->badd($LOG_2); # modify $x in place |
091c87b1 T |
1274 | $calc = 0; # no need to calc, but round |
1275 | } | |
7d193e39 | 1276 | # if we can't use the shortcut, we continue normally |
990fb837 RGS |
1277 | } |
1278 | } | |
1279 | ||
1280 | # if $x = 0.1, we know the result must be 0-log(10) | |
9b924220 RGS |
1281 | if ($calc != 0 && $x->{_es} eq '-' && $MBI->_is_one($x->{_e}) && |
1282 | $MBI->_is_one($x->{_m})) | |
990fb837 RGS |
1283 | { |
1284 | $dbd = 0; # disable shortcut | |
1285 | # we can use the cached value in these cases | |
1286 | if ($scale <= $LOG_10_A) | |
1287 | { | |
1288 | $x->bzero(); $x->bsub($LOG_10); | |
1289 | $calc = 0; # no need to calc, but round | |
1290 | } | |
1291 | } | |
1292 | ||
091c87b1 T |
1293 | return if $calc == 0; # already have the result |
1294 | ||
990fb837 RGS |
1295 | # default: these correction factors are undef and thus not used |
1296 | my $l_10; # value of ln(10) to A of $scale | |
1297 | my $l_2; # value of ln(2) to A of $scale | |
1298 | ||
7d193e39 T |
1299 | my $two = $self->new(2); |
1300 | ||
990fb837 RGS |
1301 | # $x == 2 => 1, $x == 13 => 2, $x == 0.1 => 0, $x == 0.01 => -1 |
1302 | # so don't do this shortcut for 1 or 0 | |
1303 | if (($dbd > 1) || ($dbd < 0)) | |
1304 | { | |
1305 | # convert our cached value to an object if not already (avoid doing this | |
1306 | # at import() time, since not everybody needs this) | |
1307 | $LOG_10 = $self->new($LOG_10,undef,undef) unless ref $LOG_10; | |
1308 | ||
1309 | #print "x = $x, dbd = $dbd, calc = $calc\n"; | |
1310 | # got more than one digit before the dot, or more than one zero after the | |
1311 | # dot, so do: | |
1312 | # log(123) == log(1.23) + log(10) * 2 | |
1313 | # log(0.0123) == log(1.23) - log(10) * 2 | |
1314 | ||
1315 | if ($scale <= $LOG_10_A) | |
1316 | { | |
1317 | # use cached value | |
990fb837 RGS |
1318 | $l_10 = $LOG_10->copy(); # copy for mul |
1319 | } | |
1320 | else | |
1321 | { | |
7d193e39 | 1322 | # else: slower, compute and cache result |
990fb837 RGS |
1323 | # also disable downgrade for this code path |
1324 | local $Math::BigFloat::downgrade = undef; | |
7d193e39 T |
1325 | |
1326 | # shorten the time to calculate log(10) based on the following: | |
1327 | # log(1.25 * 8) = log(1.25) + log(8) | |
1328 | # = log(1.25) + log(2) + log(2) + log(2) | |
1329 | ||
1330 | # first get $l_2 (and possible compute and cache log(2)) | |
1331 | $LOG_2 = $self->new($LOG_2,undef,undef) unless ref $LOG_2; | |
1332 | if ($scale <= $LOG_2_A) | |
1333 | { | |
1334 | # use cached value | |
1335 | $l_2 = $LOG_2->copy(); # copy() for the mul below | |
1336 | } | |
1337 | else | |
1338 | { | |
1339 | # else: slower, compute and cache result | |
1340 | $l_2 = $two->copy(); $self->_log($l_2, $scale); # scale+4, actually | |
1341 | $LOG_2 = $l_2->copy(); # cache the result for later | |
1342 | # the copy() is for mul below | |
1343 | $LOG_2_A = $scale; | |
1344 | } | |
1345 | ||
1346 | # now calculate log(1.25): | |
1347 | $l_10 = $self->new('1.25'); $self->_log($l_10, $scale); # scale+4, actually | |
1348 | ||
1349 | # log(1.25) + log(2) + log(2) + log(2): | |
1350 | $l_10->badd($l_2); | |
1351 | $l_10->badd($l_2); | |
1352 | $l_10->badd($l_2); | |
1353 | $LOG_10 = $l_10->copy(); # cache the result for later | |
1354 | # the copy() is for mul below | |
1355 | $LOG_10_A = $scale; | |
990fb837 RGS |
1356 | } |
1357 | $dbd-- if ($dbd > 1); # 20 => dbd=2, so make it dbd=1 | |
9b924220 RGS |
1358 | $l_10->bmul( $self->new($dbd)); # log(10) * (digits_before_dot-1) |
1359 | my $dbd_sign = '+'; | |
1360 | if ($dbd < 0) | |
1361 | { | |
1362 | $dbd = -$dbd; | |
1363 | $dbd_sign = '-'; | |
1364 | } | |
1365 | ($x->{_e}, $x->{_es}) = | |
1366 | _e_sub( $x->{_e}, $MBI->_new($dbd), $x->{_es}, $dbd_sign); # 123 => 1.23 | |
990fb837 RGS |
1367 | |
1368 | } | |
1369 | ||
1370 | # Now: 0.1 <= $x < 10 (and possible correction in l_10) | |
1371 | ||
1372 | ### Since $x in the range 0.5 .. 1.5 is MUCH faster, we do a repeated div | |
1373 | ### or mul by 2 (maximum times 3, since x < 10 and x > 0.1) | |
1374 | ||
27e7b8bb T |
1375 | $HALF = $self->new($HALF) unless ref($HALF); |
1376 | ||
091c87b1 | 1377 | my $twos = 0; # default: none (0 times) |
7d193e39 | 1378 | while ($x->bacmp($HALF) <= 0) # X <= 0.5 |
990fb837 | 1379 | { |
091c87b1 T |
1380 | $twos--; $x->bmul($two); |
1381 | } | |
7d193e39 | 1382 | while ($x->bacmp($two) >= 0) # X >= 2 |
091c87b1 T |
1383 | { |
1384 | $twos++; $x->bdiv($two,$scale+4); # keep all digits | |
1385 | } | |
7d193e39 T |
1386 | # $twos > 0 => did mul 2, < 0 => did div 2 (but we never did both) |
1387 | # So calculate correction factor based on ln(2): | |
091c87b1 T |
1388 | if ($twos != 0) |
1389 | { | |
1390 | $LOG_2 = $self->new($LOG_2,undef,undef) unless ref $LOG_2; | |
1391 | if ($scale <= $LOG_2_A) | |
990fb837 | 1392 | { |
091c87b1 | 1393 | # use cached value |
7d193e39 | 1394 | $l_2 = $LOG_2->copy(); # copy() for the mul below |
990fb837 | 1395 | } |
091c87b1 | 1396 | else |
990fb837 | 1397 | { |
7d193e39 | 1398 | # else: slower, compute and cache result |
091c87b1 T |
1399 | # also disable downgrade for this code path |
1400 | local $Math::BigFloat::downgrade = undef; | |
7d193e39 T |
1401 | $l_2 = $two->copy(); $self->_log($l_2, $scale); # scale+4, actually |
1402 | $LOG_2 = $l_2->copy(); # cache the result for later | |
1403 | # the copy() is for mul below | |
1404 | $LOG_2_A = $scale; | |
990fb837 | 1405 | } |
091c87b1 | 1406 | $l_2->bmul($twos); # * -2 => subtract, * 2 => add |
990fb837 RGS |
1407 | } |
1408 | ||
091c87b1 T |
1409 | $self->_log($x,$scale); # need to do the "normal" way |
1410 | $x->badd($l_10) if defined $l_10; # correct it by ln(10) | |
1411 | $x->badd($l_2) if defined $l_2; # and maybe by ln(2) | |
7d193e39 | 1412 | |
990fb837 | 1413 | # all done, $x contains now the result |
7d193e39 | 1414 | $x; |
990fb837 RGS |
1415 | } |
1416 | ||
58cde26e JH |
1417 | sub blcm |
1418 | { | |
ee15d750 | 1419 | # (BFLOAT or num_str, BFLOAT or num_str) return BFLOAT |
58cde26e JH |
1420 | # does not modify arguments, but returns new object |
1421 | # Lowest Common Multiplicator | |
58cde26e JH |
1422 | |
1423 | my ($self,@arg) = objectify(0,@_); | |
1424 | my $x = $self->new(shift @arg); | |
b68b7ab1 | 1425 | while (@arg) { $x = Math::BigInt::__lcm($x,shift @arg); } |
58cde26e JH |
1426 | $x; |
1427 | } | |
1428 | ||
b68b7ab1 T |
1429 | sub bgcd |
1430 | { | |
1431 | # (BINT or num_str, BINT or num_str) return BINT | |
58cde26e | 1432 | # does not modify arguments, but returns new object |
b68b7ab1 T |
1433 | |
1434 | my $y = shift; | |
1435 | $y = __PACKAGE__->new($y) if !ref($y); | |
1436 | my $self = ref($y); | |
1437 | my $x = $y->copy()->babs(); # keep arguments | |
1438 | ||
1439 | return $x->bnan() if $x->{sign} !~ /^[+-]$/ # x NaN? | |
1440 | || !$x->is_int(); # only for integers now | |
1441 | ||
1442 | while (@_) | |
1443 | { | |
1444 | my $t = shift; $t = $self->new($t) if !ref($t); | |
1445 | $y = $t->copy()->babs(); | |
1446 | ||
1447 | return $x->bnan() if $y->{sign} !~ /^[+-]$/ # y NaN? | |
1448 | || !$y->is_int(); # only for integers now | |
1449 | ||
1450 | # greatest common divisor | |
1451 | while (! $y->is_zero()) | |
1452 | { | |
1453 | ($x,$y) = ($y->copy(), $x->copy()->bmod($y)); | |
1454 | } | |
1455 | ||
1456 | last if $x->is_one(); | |
1457 | } | |
58cde26e JH |
1458 | $x; |
1459 | } | |
1460 | ||
9b924220 | 1461 | ############################################################################## |
b3abae2a | 1462 | |
9b924220 | 1463 | sub _e_add |
091c87b1 | 1464 | { |
9b924220 RGS |
1465 | # Internal helper sub to take two positive integers and their signs and |
1466 | # then add them. Input ($CALC,$CALC,('+'|'-'),('+'|'-')), | |
1467 | # output ($CALC,('+'|'-')) | |
1468 | my ($x,$y,$xs,$ys) = @_; | |
1469 | ||
1470 | # if the signs are equal we can add them (-5 + -3 => -(5 + 3) => -8) | |
1471 | if ($xs eq $ys) | |
1472 | { | |
1473 | $x = $MBI->_add ($x, $y ); # a+b | |
1474 | # the sign follows $xs | |
1475 | return ($x, $xs); | |
1476 | } | |
091c87b1 | 1477 | |
9b924220 RGS |
1478 | my $a = $MBI->_acmp($x,$y); |
1479 | if ($a > 0) | |
1480 | { | |
1481 | $x = $MBI->_sub ($x , $y); # abs sub | |
1482 | } | |
1483 | elsif ($a == 0) | |
1484 | { | |
1485 | $x = $MBI->_zero(); # result is 0 | |
1486 | $xs = '+'; | |
1487 | } | |
1488 | else # a < 0 | |
1489 | { | |
1490 | $x = $MBI->_sub ( $y, $x, 1 ); # abs sub | |
1491 | $xs = $ys; | |
1492 | } | |
1493 | ($x,$xs); | |
091c87b1 T |
1494 | } |
1495 | ||
9b924220 RGS |
1496 | sub _e_sub |
1497 | { | |
1498 | # Internal helper sub to take two positive integers and their signs and | |
1499 | # then subtract them. Input ($CALC,$CALC,('+'|'-'),('+'|'-')), | |
1500 | # output ($CALC,('+'|'-')) | |
1501 | my ($x,$y,$xs,$ys) = @_; | |
1502 | ||
1503 | # flip sign | |
1504 | $ys =~ tr/+-/-+/; | |
1505 | _e_add($x,$y,$xs,$ys); # call add (does subtract now) | |
1506 | } | |
1507 | ||
1508 | ############################################################################### | |
1509 | # is_foo methods (is_negative, is_positive are inherited from BigInt) | |
1510 | ||
b3abae2a JH |
1511 | sub is_int |
1512 | { | |
1513 | # return true if arg (BFLOAT or num_str) is an integer | |
091c87b1 | 1514 | my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_); |
b3abae2a | 1515 | |
80365507 T |
1516 | (($x->{sign} =~ /^[+-]$/) && # NaN and +-inf aren't |
1517 | ($x->{_es} eq '+')) ? 1 : 0; # 1e-1 => no integer | |
b3abae2a JH |
1518 | } |
1519 | ||
58cde26e JH |
1520 | sub is_zero |
1521 | { | |
b3abae2a | 1522 | # return true if arg (BFLOAT or num_str) is zero |
091c87b1 | 1523 | my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_); |
574bacfe | 1524 | |
80365507 | 1525 | ($x->{sign} eq '+' && $MBI->_is_zero($x->{_m})) ? 1 : 0; |
58cde26e JH |
1526 | } |
1527 | ||
1528 | sub is_one | |
1529 | { | |
b3abae2a | 1530 | # return true if arg (BFLOAT or num_str) is +1 or -1 if signis given |
091c87b1 | 1531 | my ($self,$x,$sign) = ref($_[0]) ? (undef,@_) : objectify(1,@_); |
ee15d750 | 1532 | |
990fb837 | 1533 | $sign = '+' if !defined $sign || $sign ne '-'; |
80365507 T |
1534 | |
1535 | ($x->{sign} eq $sign && | |
1536 | $MBI->_is_zero($x->{_e}) && | |
1537 | $MBI->_is_one($x->{_m}) ) ? 1 : 0; | |
58cde26e JH |
1538 | } |
1539 | ||
1540 | sub is_odd | |
1541 | { | |
ee15d750 | 1542 | # return true if arg (BFLOAT or num_str) is odd or false if even |
091c87b1 | 1543 | my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_); |
0716bf9b | 1544 | |
80365507 T |
1545 | (($x->{sign} =~ /^[+-]$/) && # NaN & +-inf aren't |
1546 | ($MBI->_is_zero($x->{_e})) && | |
1547 | ($MBI->_is_odd($x->{_m}))) ? 1 : 0; | |
58cde26e JH |
1548 | } |
1549 | ||
1550 | sub is_even | |
1551 | { | |
b22b3e31 | 1552 | # return true if arg (BINT or num_str) is even or false if odd |
091c87b1 | 1553 | my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_); |
0716bf9b | 1554 | |
80365507 T |
1555 | (($x->{sign} =~ /^[+-]$/) && # NaN & +-inf aren't |
1556 | ($x->{_es} eq '+') && # 123.45 isn't | |
1557 | ($MBI->_is_even($x->{_m}))) ? 1 : 0; # but 1200 is | |
58cde26e JH |
1558 | } |
1559 | ||
80365507 | 1560 | sub bmul |
58cde26e | 1561 | { |
80365507 | 1562 | # multiply two numbers |
f9a08e12 JH |
1563 | |
1564 | # set up parameters | |
80365507 | 1565 | my ($self,$x,$y,@r) = (ref($_[0]),@_); |
f9a08e12 JH |
1566 | # objectify is costly, so avoid it |
1567 | if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1]))) | |
1568 | { | |
80365507 | 1569 | ($self,$x,$y,@r) = objectify(2,@_); |
f9a08e12 | 1570 | } |
58cde26e | 1571 | |
50109ad0 RGS |
1572 | return $x if $x->modify('bmul'); |
1573 | ||
58cde26e JH |
1574 | return $x->bnan() if (($x->{sign} eq $nan) || ($y->{sign} eq $nan)); |
1575 | ||
574bacfe JH |
1576 | # inf handling |
1577 | if (($x->{sign} =~ /^[+-]inf$/) || ($y->{sign} =~ /^[+-]inf$/)) | |
1578 | { | |
13a12e00 | 1579 | return $x->bnan() if $x->is_zero() || $y->is_zero(); |
574bacfe JH |
1580 | # result will always be +-inf: |
1581 | # +inf * +/+inf => +inf, -inf * -/-inf => +inf | |
1582 | # +inf * -/-inf => -inf, -inf * +/+inf => -inf | |
1583 | return $x->binf() if ($x->{sign} =~ /^\+/ && $y->{sign} =~ /^\+/); | |
1584 | return $x->binf() if ($x->{sign} =~ /^-/ && $y->{sign} =~ /^-/); | |
1585 | return $x->binf('-'); | |
1586 | } | |
8f675a64 | 1587 | |
80365507 T |
1588 | return $upgrade->bmul($x,$y,@r) if defined $upgrade && |
1589 | ((!$x->isa($self)) || (!$y->isa($self))); | |
1590 | ||
1591 | # aEb * cEd = (a*c)E(b+d) | |
1592 | $MBI->_mul($x->{_m},$y->{_m}); | |
1593 | ($x->{_e}, $x->{_es}) = _e_add($x->{_e}, $y->{_e}, $x->{_es}, $y->{_es}); | |
1594 | ||
1595 | $r[3] = $y; # no push! | |
1596 | ||
1597 | # adjust sign: | |
1598 | $x->{sign} = $x->{sign} ne $y->{sign} ? '-' : '+'; | |
1599 | $x->bnorm->round(@r); | |
1600 | } | |
1601 | ||
1602 | sub bmuladd | |
1603 | { | |
1604 | # multiply two numbers and add the third to the result | |
1605 | ||
1606 | # set up parameters | |
1607 | my ($self,$x,$y,$z,@r) = (ref($_[0]),@_); | |
1608 | # objectify is costly, so avoid it | |
1609 | if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1]))) | |
1610 | { | |
1611 | ($self,$x,$y,$z,@r) = objectify(3,@_); | |
1612 | } | |
1613 | ||
1614 | return $x if $x->modify('bmuladd'); | |
1615 | ||
1616 | return $x->bnan() if (($x->{sign} eq $nan) || | |
1617 | ($y->{sign} eq $nan) || | |
1618 | ($z->{sign} eq $nan)); | |
1619 | ||
1620 | # inf handling | |
1621 | if (($x->{sign} =~ /^[+-]inf$/) || ($y->{sign} =~ /^[+-]inf$/)) | |
1622 | { | |
1623 | return $x->bnan() if $x->is_zero() || $y->is_zero(); | |
1624 | # result will always be +-inf: | |
1625 | # +inf * +/+inf => +inf, -inf * -/-inf => +inf | |
1626 | # +inf * -/-inf => -inf, -inf * +/+inf => -inf | |
1627 | return $x->binf() if ($x->{sign} =~ /^\+/ && $y->{sign} =~ /^\+/); | |
1628 | return $x->binf() if ($x->{sign} =~ /^-/ && $y->{sign} =~ /^-/); | |
1629 | return $x->binf('-'); | |
1630 | } | |
1631 | ||
1632 | return $upgrade->bmul($x,$y,@r) if defined $upgrade && | |
8f675a64 | 1633 | ((!$x->isa($self)) || (!$y->isa($self))); |
574bacfe | 1634 | |
58cde26e | 1635 | # aEb * cEd = (a*c)E(b+d) |
9b924220 RGS |
1636 | $MBI->_mul($x->{_m},$y->{_m}); |
1637 | ($x->{_e}, $x->{_es}) = _e_add($x->{_e}, $y->{_e}, $x->{_es}, $y->{_es}); | |
1638 | ||
80365507 T |
1639 | $r[3] = $y; # no push! |
1640 | ||
58cde26e JH |
1641 | # adjust sign: |
1642 | $x->{sign} = $x->{sign} ne $y->{sign} ? '-' : '+'; | |
80365507 T |
1643 | |
1644 | # z=inf handling (z=NaN handled above) | |
1645 | $x->{sign} = $z->{sign}, return $x if $z->{sign} =~ /^[+-]inf$/; | |
1646 | ||
1647 | # take lower of the two e's and adapt m1 to it to match m2 | |
1648 | my $e = $z->{_e}; | |
1649 | $e = $MBI->_zero() if !defined $e; # if no BFLOAT? | |
1650 | $e = $MBI->_copy($e); # make copy (didn't do it yet) | |
1651 | ||
1652 | my $es; | |
1653 | ||
1654 | ($e,$es) = _e_sub($e, $x->{_e}, $z->{_es} || '+', $x->{_es}); | |
1655 | ||
1656 | my $add = $MBI->_copy($z->{_m}); | |
1657 | ||
1658 | if ($es eq '-') # < 0 | |
1659 | { | |
1660 | $MBI->_lsft( $x->{_m}, $e, 10); | |
1661 | ($x->{_e},$x->{_es}) = _e_add($x->{_e}, $e, $x->{_es}, $es); | |
1662 | } | |
1663 | elsif (!$MBI->_is_zero($e)) # > 0 | |
1664 | { | |
1665 | $MBI->_lsft($add, $e, 10); | |
1666 | } | |
1667 | # else: both e are the same, so just leave them | |
1668 | ||
1669 | if ($x->{sign} eq $z->{sign}) | |
1670 | { | |
1671 | # add | |
1672 | $x->{_m} = $MBI->_add($x->{_m}, $add); | |
1673 | } | |
1674 | else | |
1675 | { | |
1676 | ($x->{_m}, $x->{sign}) = | |
1677 | _e_add($x->{_m}, $add, $x->{sign}, $z->{sign}); | |
1678 | } | |
1679 | ||
1680 | # delete trailing zeros, then round | |
1681 | $x->bnorm()->round(@r); | |
58cde26e JH |
1682 | } |
1683 | ||
1684 | sub bdiv | |
1685 | { | |
1686 | # (dividend: BFLOAT or num_str, divisor: BFLOAT or num_str) return | |
9393ace2 | 1687 | # (BFLOAT,BFLOAT) (quo,rem) or BFLOAT (only rem) |
f9a08e12 JH |
1688 | |
1689 | # set up parameters | |
1690 | my ($self,$x,$y,$a,$p,$r) = (ref($_[0]),@_); | |
1691 | # objectify is costly, so avoid it | |
1692 | if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1]))) | |
1693 | { | |
1694 | ($self,$x,$y,$a,$p,$r) = objectify(2,@_); | |
1695 | } | |
58cde26e | 1696 | |
50109ad0 RGS |
1697 | return $x if $x->modify('bdiv'); |
1698 | ||
13a12e00 JH |
1699 | return $self->_div_inf($x,$y) |
1700 | if (($x->{sign} !~ /^[+-]$/) || ($y->{sign} !~ /^[+-]$/) || $y->is_zero()); | |
574bacfe | 1701 | |
13a12e00 | 1702 | # x== 0 # also: or y == 1 or y == -1 |
394e6ffb | 1703 | return wantarray ? ($x,$self->bzero()) : $x if $x->is_zero(); |
0716bf9b | 1704 | |
9393ace2 JH |
1705 | # upgrade ? |
1706 | return $upgrade->bdiv($upgrade->new($x),$y,$a,$p,$r) if defined $upgrade; | |
13a12e00 | 1707 | |
58cde26e | 1708 | # we need to limit the accuracy to protect against overflow |
574bacfe | 1709 | my $fallback = 0; |
990fb837 RGS |
1710 | my (@params,$scale); |
1711 | ($x,@params) = $x->_find_round_parameters($a,$p,$r,$y); | |
1712 | ||
1713 | return $x if $x->is_nan(); # error in _find_round_parameters? | |
ee15d750 JH |
1714 | |
1715 | # no rounding at all, so must use fallback | |
990fb837 | 1716 | if (scalar @params == 0) |
58cde26e | 1717 | { |
0716bf9b | 1718 | # simulate old behaviour |
990fb837 RGS |
1719 | $params[0] = $self->div_scale(); # and round to it as accuracy |
1720 | $scale = $params[0]+4; # at least four more for proper round | |
1721 | $params[2] = $r; # round mode by caller or undef | |
ee15d750 JH |
1722 | $fallback = 1; # to clear a/p afterwards |
1723 | } | |
1724 | else | |
1725 | { | |
1726 | # the 4 below is empirical, and there might be cases where it is not | |
1727 | # enough... | |
990fb837 | 1728 | $scale = abs($params[0] || $params[1]) + 4; # take whatever is defined |
a0d0e21e | 1729 | } |
03874afe T |
1730 | |
1731 | my $rem; $rem = $self->bzero() if wantarray; | |
1732 | ||
1733 | $y = $self->new($y) unless $y->isa('Math::BigFloat'); | |
1734 | ||
9b924220 | 1735 | my $lx = $MBI->_len($x->{_m}); my $ly = $MBI->_len($y->{_m}); |
58cde26e | 1736 | $scale = $lx if $lx > $scale; |
58cde26e | 1737 | $scale = $ly if $ly > $scale; |
0716bf9b JH |
1738 | my $diff = $ly - $lx; |
1739 | $scale += $diff if $diff > 0; # if lx << ly, but not if ly << lx! | |
a87115f0 RGS |
1740 | |
1741 | # already handled inf/NaN/-inf above: | |
1742 | ||
233f7bc0 T |
1743 | # check that $y is not 1 nor -1 and cache the result: |
1744 | my $y_not_one = !($MBI->_is_zero($y->{_e}) && $MBI->_is_one($y->{_m})); | |
1745 | ||
7596a890 RGS |
1746 | # flipping the sign of $y will also flip the sign of $x for the special |
1747 | # case of $x->bsub($x); so we can catch it below: | |
1748 | my $xsign = $x->{sign}; | |
1749 | $y->{sign} =~ tr/+-/-+/; | |
1750 | ||
a87115f0 | 1751 | if ($xsign ne $x->{sign}) |
b3abae2a | 1752 | { |
a87115f0 | 1753 | # special case of $x /= $x results in 1 |
233f7bc0 | 1754 | $x->bone(); # "fixes" also sign of $y, since $x is $y |
b3abae2a | 1755 | } |
03874afe | 1756 | else |
58cde26e | 1757 | { |
a87115f0 RGS |
1758 | # correct $y's sign again |
1759 | $y->{sign} =~ tr/+-/-+/; | |
1760 | # continue with normal div code: | |
1761 | ||
03874afe | 1762 | # make copy of $x in case of list context for later reminder calculation |
a87115f0 | 1763 | if (wantarray && $y_not_one) |
03874afe T |
1764 | { |
1765 | $rem = $x->copy(); | |
1766 | } | |
394e6ffb | 1767 | |
03874afe | 1768 | $x->{sign} = $x->{sign} ne $y->sign() ? '-' : '+'; |
9b924220 | 1769 | |
03874afe | 1770 | # check for / +-1 ( +/- 1E0) |
a87115f0 | 1771 | if ($y_not_one) |
03874afe T |
1772 | { |
1773 | # promote BigInts and it's subclasses (except when already a BigFloat) | |
1774 | $y = $self->new($y) unless $y->isa('Math::BigFloat'); | |
1775 | ||
1776 | # calculate the result to $scale digits and then round it | |
1777 | # a * 10 ** b / c * 10 ** d => a/c * 10 ** (b-d) | |
1778 | $MBI->_lsft($x->{_m},$MBI->_new($scale),10); | |
1779 | $MBI->_div ($x->{_m},$y->{_m}); # a/c | |
1780 | ||
1781 | # correct exponent of $x | |
1782 | ($x->{_e},$x->{_es}) = _e_sub($x->{_e}, $y->{_e}, $x->{_es}, $y->{_es}); | |
1783 | # correct for 10**scale | |
1784 | ($x->{_e},$x->{_es}) = _e_sub($x->{_e}, $MBI->_new($scale), $x->{_es}, '+'); | |
1785 | $x->bnorm(); # remove trailing 0's | |
1786 | } | |
1787 | } # ende else $x != $y | |
a5f75d66 | 1788 | |
091c87b1 | 1789 | # shortcut to not run through _find_round_parameters again |
990fb837 | 1790 | if (defined $params[0]) |
ee15d750 | 1791 | { |
ef9466ea | 1792 | delete $x->{_a}; # clear before round |
990fb837 | 1793 | $x->bround($params[0],$params[2]); # then round accordingly |
ee15d750 JH |
1794 | } |
1795 | else | |
1796 | { | |
ef9466ea | 1797 | delete $x->{_p}; # clear before round |
990fb837 | 1798 | $x->bfround($params[1],$params[2]); # then round accordingly |
ee15d750 | 1799 | } |
574bacfe JH |
1800 | if ($fallback) |
1801 | { | |
1802 | # clear a/p after round, since user did not request it | |
ef9466ea | 1803 | delete $x->{_a}; delete $x->{_p}; |
574bacfe | 1804 | } |
03874afe | 1805 | |
58cde26e JH |
1806 | if (wantarray) |
1807 | { | |
a87115f0 | 1808 | if ($y_not_one) |
394e6ffb | 1809 | { |
990fb837 | 1810 | $rem->bmod($y,@params); # copy already done |
394e6ffb | 1811 | } |
574bacfe JH |
1812 | if ($fallback) |
1813 | { | |
1814 | # clear a/p after round, since user did not request it | |
ef9466ea | 1815 | delete $rem->{_a}; delete $rem->{_p}; |
574bacfe | 1816 | } |
0716bf9b | 1817 | return ($x,$rem); |
58cde26e | 1818 | } |
9393ace2 | 1819 | $x; |
58cde26e | 1820 | } |
a0d0e21e | 1821 | |
58cde26e JH |
1822 | sub bmod |
1823 | { | |
1824 | # (dividend: BFLOAT or num_str, divisor: BFLOAT or num_str) return reminder | |
f9a08e12 JH |
1825 | |
1826 | # set up parameters | |
1827 | my ($self,$x,$y,$a,$p,$r) = (ref($_[0]),@_); | |
1828 | # objectify is costly, so avoid it | |
1829 | if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1]))) | |
1830 | { | |
1831 | ($self,$x,$y,$a,$p,$r) = objectify(2,@_); | |
1832 | } | |
a0d0e21e | 1833 | |
50109ad0 RGS |
1834 | return $x if $x->modify('bmod'); |
1835 | ||
9b924220 | 1836 | # handle NaN, inf, -inf |
61f5c3f5 T |
1837 | if (($x->{sign} !~ /^[+-]$/) || ($y->{sign} !~ /^[+-]$/)) |
1838 | { | |
1839 | my ($d,$re) = $self->SUPER::_div_inf($x,$y); | |
f9a08e12 JH |
1840 | $x->{sign} = $re->{sign}; |
1841 | $x->{_e} = $re->{_e}; | |
1842 | $x->{_m} = $re->{_m}; | |
1843 | return $x->round($a,$p,$r,$y); | |
61f5c3f5 | 1844 | } |
9b924220 RGS |
1845 | if ($y->is_zero()) |
1846 | { | |
1847 | return $x->bnan() if $x->is_zero(); | |
1848 | return $x; | |
1849 | } | |
233f7bc0 | 1850 | |
7596a890 RGS |
1851 | return $x->bzero() if $x->is_zero() |
1852 | || ($x->is_int() && | |
1853 | # check that $y == +1 or $y == -1: | |
1854 | ($MBI->_is_zero($y->{_e}) && $MBI->_is_one($y->{_m}))); | |
58cde26e | 1855 | |
61f5c3f5 T |
1856 | my $cmp = $x->bacmp($y); # equal or $x < $y? |
1857 | return $x->bzero($a,$p) if $cmp == 0; # $x == $y => result 0 | |
1858 | ||
1859 | # only $y of the operands negative? | |
1860 | my $neg = 0; $neg = 1 if $x->{sign} ne $y->{sign}; | |
1861 | ||
1862 | $x->{sign} = $y->{sign}; # calc sign first | |
1863 | return $x->round($a,$p,$r) if $cmp < 0 && $neg == 0; # $x < $y => result $x | |
1864 | ||
9b924220 | 1865 | my $ym = $MBI->_copy($y->{_m}); |
61f5c3f5 T |
1866 | |
1867 | # 2e1 => 20 | |
9b924220 RGS |
1868 | $MBI->_lsft( $ym, $y->{_e}, 10) |
1869 | if $y->{_es} eq '+' && !$MBI->_is_zero($y->{_e}); | |
61f5c3f5 T |
1870 | |
1871 | # if $y has digits after dot | |
1872 | my $shifty = 0; # correct _e of $x by this | |
9b924220 | 1873 | if ($y->{_es} eq '-') # has digits after dot |
61f5c3f5 T |
1874 | { |
1875 | # 123 % 2.5 => 1230 % 25 => 5 => 0.5 | |
9b924220 RGS |
1876 | $shifty = $MBI->_num($y->{_e}); # no more digits after dot |
1877 | $MBI->_lsft($x->{_m}, $y->{_e}, 10);# 123 => 1230, $y->{_m} is already 25 | |
61f5c3f5 T |
1878 | } |
1879 | # $ym is now mantissa of $y based on exponent 0 | |
b3abae2a | 1880 | |
61f5c3f5 | 1881 | my $shiftx = 0; # correct _e of $x by this |
9b924220 | 1882 | if ($x->{_es} eq '-') # has digits after dot |
61f5c3f5 T |
1883 | { |
1884 | # 123.4 % 20 => 1234 % 200 | |
9b924220 RGS |
1885 | $shiftx = $MBI->_num($x->{_e}); # no more digits after dot |
1886 | $MBI->_lsft($ym, $x->{_e}, 10); # 123 => 1230 | |
61f5c3f5 T |
1887 | } |
1888 | # 123e1 % 20 => 1230 % 20 | |
9b924220 | 1889 | if ($x->{_es} eq '+' && !$MBI->_is_zero($x->{_e})) |
61f5c3f5 | 1890 | { |
9b924220 | 1891 | $MBI->_lsft( $x->{_m}, $x->{_e},10); # es => '+' here |
61f5c3f5 | 1892 | } |
9b924220 RGS |
1893 | |
1894 | $x->{_e} = $MBI->_new($shiftx); | |
1895 | $x->{_es} = '+'; | |
1896 | $x->{_es} = '-' if $shiftx != 0 || $shifty != 0; | |
1897 | $MBI->_add( $x->{_e}, $MBI->_new($shifty)) if $shifty != 0; | |
61f5c3f5 T |
1898 | |
1899 | # now mantissas are equalized, exponent of $x is adjusted, so calc result | |
b3abae2a | 1900 | |
9b924220 | 1901 | $x->{_m} = $MBI->_mod( $x->{_m}, $ym); |
61f5c3f5 | 1902 | |
9b924220 | 1903 | $x->{sign} = '+' if $MBI->_is_zero($x->{_m}); # fix sign for -0 |
61f5c3f5 T |
1904 | $x->bnorm(); |
1905 | ||
1906 | if ($neg != 0) # one of them negative => correct in place | |
1907 | { | |
1908 | my $r = $y - $x; | |
1909 | $x->{_m} = $r->{_m}; | |
1910 | $x->{_e} = $r->{_e}; | |
9b924220 RGS |
1911 | $x->{_es} = $r->{_es}; |
1912 | $x->{sign} = '+' if $MBI->_is_zero($x->{_m}); # fix sign for -0 | |
61f5c3f5 T |
1913 | $x->bnorm(); |
1914 | } | |
1915 | ||
1916 | $x->round($a,$p,$r,$y); # round and return | |
58cde26e JH |
1917 | } |
1918 | ||
990fb837 RGS |
1919 | sub broot |
1920 | { | |
1921 | # calculate $y'th root of $x | |
3a427a11 RGS |
1922 | |
1923 | # set up parameters | |
1924 | my ($self,$x,$y,$a,$p,$r) = (ref($_[0]),@_); | |
1925 | # objectify is costly, so avoid it | |
1926 | if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1]))) | |
1927 | { | |
1928 | ($self,$x,$y,$a,$p,$r) = objectify(2,@_); | |
1929 | } | |
990fb837 | 1930 | |
50109ad0 RGS |
1931 | return $x if $x->modify('broot'); |
1932 | ||
990fb837 RGS |
1933 | # NaN handling: $x ** 1/0, x or y NaN, or y inf/-inf or y == 0 |
1934 | return $x->bnan() if $x->{sign} !~ /^\+/ || $y->is_zero() || | |
1935 | $y->{sign} !~ /^\+$/; | |
1936 | ||
1937 | return $x if $x->is_zero() || $x->is_one() || $x->is_inf() || $y->is_one(); | |
1938 | ||
1939 | # we need to limit the accuracy to protect against overflow | |
1940 | my $fallback = 0; | |
1941 | my (@params,$scale); | |
1942 | ($x,@params) = $x->_find_round_parameters($a,$p,$r); | |
1943 | ||
1944 | return $x if $x->is_nan(); # error in _find_round_parameters? | |
1945 | ||
1946 | # no rounding at all, so must use fallback | |
1947 | if (scalar @params == 0) | |
1948 | { | |
1949 | # simulate old behaviour | |
1950 | $params[0] = $self->div_scale(); # and round to it as accuracy | |
1951 | $scale = $params[0]+4; # at least four more for proper round | |
9b924220 | 1952 | $params[2] = $r; # iound mode by caller or undef |
990fb837 RGS |
1953 | $fallback = 1; # to clear a/p afterwards |
1954 | } | |
1955 | else | |
1956 | { | |
1957 | # the 4 below is empirical, and there might be cases where it is not | |
1958 | # enough... | |
1959 | $scale = abs($params[0] || $params[1]) + 4; # take whatever is defined | |
1960 | } | |
1961 | ||
1962 | # when user set globals, they would interfere with our calculation, so | |
1963 | # disable them and later re-enable them | |
1964 | no strict 'refs'; | |
1965 | my $abr = "$self\::accuracy"; my $ab = $$abr; $$abr = undef; | |
1966 | my $pbr = "$self\::precision"; my $pb = $$pbr; $$pbr = undef; | |
1967 | # we also need to disable any set A or P on $x (_find_round_parameters took | |
1968 | # them already into account), since these would interfere, too | |
1969 | delete $x->{_a}; delete $x->{_p}; | |
1970 | # need to disable $upgrade in BigInt, to avoid deep recursion | |
1971 | local $Math::BigInt::upgrade = undef; # should be really parent class vs MBI | |
1972 | ||
1973 | # remember sign and make $x positive, since -4 ** (1/2) => -2 | |
27e7b8bb T |
1974 | my $sign = 0; $sign = 1 if $x->{sign} eq '-'; $x->{sign} = '+'; |
1975 | ||
1976 | my $is_two = 0; | |
1977 | if ($y->isa('Math::BigFloat')) | |
1978 | { | |
1979 | $is_two = ($y->{sign} eq '+' && $MBI->_is_two($y->{_m}) && $MBI->_is_zero($y->{_e})); | |
1980 | } | |
1981 | else | |
1982 | { | |
1983 | $is_two = ($y == 2); | |
1984 | } | |
990fb837 | 1985 | |
27e7b8bb T |
1986 | # normal square root if $y == 2: |
1987 | if ($is_two) | |
990fb837 RGS |
1988 | { |
1989 | $x->bsqrt($scale+4); | |
1990 | } | |
1991 | elsif ($y->is_one('-')) | |
1992 | { | |
1993 | # $x ** -1 => 1/$x | |
1994 | my $u = $self->bone()->bdiv($x,$scale); | |
1995 | # copy private parts over | |
1996 | $x->{_m} = $u->{_m}; | |
1997 | $x->{_e} = $u->{_e}; | |
9b924220 | 1998 | $x->{_es} = $u->{_es}; |
990fb837 RGS |
1999 | } |
2000 | else | |
2001 | { | |
3a427a11 RGS |
2002 | # calculate the broot() as integer result first, and if it fits, return |
2003 | # it rightaway (but only if $x and $y are integer): | |
2004 | ||
2005 | my $done = 0; # not yet | |
2006 | if ($y->is_int() && $x->is_int()) | |
2007 | { | |
9b924220 RGS |
2008 | my $i = $MBI->_copy( $x->{_m} ); |
2009 | $MBI->_lsft( $i, $x->{_e}, 10 ) unless $MBI->_is_zero($x->{_e}); | |
2010 | my $int = Math::BigInt->bzero(); | |
2011 | $int->{value} = $i; | |
3a427a11 RGS |
2012 | $int->broot($y->as_number()); |
2013 | # if ($exact) | |
2014 | if ($int->copy()->bpow($y) == $x) | |
2015 | { | |
2016 | # found result, return it | |
9b924220 RGS |
2017 | $x->{_m} = $int->{value}; |
2018 | $x->{_e} = $MBI->_zero(); | |
2019 | $x->{_es} = '+'; | |
3a427a11 RGS |
2020 | $x->bnorm(); |
2021 | $done = 1; | |
2022 | } | |
2023 | } | |
2024 | if ($done == 0) | |
2025 | { | |
2026 | my $u = $self->bone()->bdiv($y,$scale+4); | |
2027 | delete $u->{_a}; delete $u->{_p}; # otherwise it conflicts | |
2028 | $x->bpow($u,$scale+4); # el cheapo | |
2029 | } | |
990fb837 RGS |
2030 | } |
2031 | $x->bneg() if $sign == 1; | |
2032 | ||
091c87b1 | 2033 | # shortcut to not run through _find_round_parameters again |
990fb837 RGS |
2034 | if (defined $params[0]) |
2035 | { | |
2036 | $x->bround($params[0],$params[2]); # then round accordingly | |
2037 | } | |
2038 | else | |
2039 | { | |
2040 | $x->bfround($params[1],$params[2]); # then round accordingly | |
2041 | } | |
2042 | if ($fallback) | |
2043 | { | |
2044 | # clear a/p after round, since user did not request it | |
ef9466ea | 2045 | delete $x->{_a}; delete $x->{_p}; |
990fb837 RGS |
2046 | } |
2047 | # restore globals | |
2048 | $$abr = $ab; $$pbr = $pb; | |
2049 | $x; | |
2050 | } | |
2051 | ||
58cde26e JH |
2052 | sub bsqrt |
2053 | { | |
990fb837 | 2054 | # calculate square root |
ee15d750 | 2055 | my ($self,$x,$a,$p,$r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_); |
58cde26e | 2056 | |
50109ad0 RGS |
2057 | return $x if $x->modify('bsqrt'); |
2058 | ||
990fb837 RGS |
2059 | return $x->bnan() if $x->{sign} !~ /^[+]/; # NaN, -inf or < 0 |
2060 | return $x if $x->{sign} eq '+inf'; # sqrt(inf) == inf | |
2061 | return $x->round($a,$p,$r) if $x->is_zero() || $x->is_one(); | |
58cde26e | 2062 | |
61f5c3f5 | 2063 | # we need to limit the accuracy to protect against overflow |
574bacfe | 2064 | my $fallback = 0; |
990fb837 RGS |
2065 | my (@params,$scale); |
2066 | ($x,@params) = $x->_find_round_parameters($a,$p,$r); | |
2067 | ||
2068 | return $x if $x->is_nan(); # error in _find_round_parameters? | |
61f5c3f5 T |
2069 | |
2070 | # no rounding at all, so must use fallback | |
990fb837 | 2071 | if (scalar @params == 0) |
0716bf9b JH |
2072 | { |
2073 | # simulate old behaviour | |
990fb837 RGS |
2074 | $params[0] = $self->div_scale(); # and round to it as accuracy |
2075 | $scale = $params[0]+4; # at least four more for proper round | |
2076 | $params[2] = $r; # round mode by caller or undef | |
ee15d750 | 2077 | $fallback = 1; # to clear a/p afterwards |
0716bf9b | 2078 | } |
61f5c3f5 T |
2079 | else |
2080 | { | |
2081 | # the 4 below is empirical, and there might be cases where it is not | |
2082 | # enough... | |
990fb837 | 2083 | $scale = abs($params[0] || $params[1]) + 4; # take whatever is defined |
61f5c3f5 T |
2084 | } |
2085 | ||
2086 | # when user set globals, they would interfere with our calculation, so | |
9393ace2 | 2087 | # disable them and later re-enable them |
61f5c3f5 T |
2088 | no strict 'refs'; |
2089 | my $abr = "$self\::accuracy"; my $ab = $$abr; $$abr = undef; | |
b3abae2a | 2090 | my $pbr = "$self\::precision"; my $pb = $$pbr; $$pbr = undef; |
61f5c3f5 T |
2091 | # we also need to disable any set A or P on $x (_find_round_parameters took |
2092 | # them already into account), since these would interfere, too | |
2093 | delete $x->{_a}; delete $x->{_p}; | |
9393ace2 JH |
2094 | # need to disable $upgrade in BigInt, to avoid deep recursion |
2095 | local $Math::BigInt::upgrade = undef; # should be really parent class vs MBI | |
61f5c3f5 | 2096 | |
9b924220 RGS |
2097 | my $i = $MBI->_copy( $x->{_m} ); |
2098 | $MBI->_lsft( $i, $x->{_e}, 10 ) unless $MBI->_is_zero($x->{_e}); | |
2099 | my $xas = Math::BigInt->bzero(); | |
2100 | $xas->{value} = $i; | |
2101 | ||
394e6ffb | 2102 | my $gs = $xas->copy()->bsqrt(); # some guess |
b3abae2a | 2103 | |
9b924220 | 2104 | if (($x->{_es} ne '-') # guess can't be accurate if there are |
394e6ffb | 2105 | # digits after the dot |
b3abae2a | 2106 | && ($xas->bacmp($gs * $gs) == 0)) # guess hit the nail on the head? |
394e6ffb | 2107 | { |
9b924220 RGS |
2108 | # exact result, copy result over to keep $x |
2109 | $x->{_m} = $gs->{value}; $x->{_e} = $MBI->_zero(); $x->{_es} = '+'; | |
2110 | $x->bnorm(); | |
091c87b1 | 2111 | # shortcut to not run through _find_round_parameters again |
990fb837 | 2112 | if (defined $params[0]) |
61f5c3f5 | 2113 | { |
990fb837 | 2114 | $x->bround($params[0],$params[2]); # then round accordingly |
61f5c3f5 T |
2115 | } |
2116 | else | |
2117 | { | |
990fb837 | 2118 | $x->bfround($params[1],$params[2]); # then round accordingly |
61f5c3f5 T |
2119 | } |
2120 | if ($fallback) | |
2121 | { | |
2122 | # clear a/p after round, since user did not request it | |
ef9466ea | 2123 | delete $x->{_a}; delete $x->{_p}; |
61f5c3f5 | 2124 | } |
9393ace2 | 2125 | # re-enable A and P, upgrade is taken care of by "local" |
b3abae2a | 2126 | ${"$self\::accuracy"} = $ab; ${"$self\::precision"} = $pb; |
61f5c3f5 | 2127 | return $x; |
394e6ffb | 2128 | } |
2ab5f49d T |
2129 | |
2130 | # sqrt(2) = 1.4 because sqrt(2*100) = 1.4*10; so we can increase the accuracy | |
9681bfa6 | 2131 | # of the result by multiplying the input by 100 and then divide the integer |
2ab5f49d | 2132 | # result of sqrt(input) by 10. Rounding afterwards returns the real result. |
9b924220 RGS |
2133 | |
2134 | # The following steps will transform 123.456 (in $x) into 123456 (in $y1) | |
2135 | my $y1 = $MBI->_copy($x->{_m}); | |
2136 | ||
2137 | my $length = $MBI->_len($y1); | |
2138 | ||
2139 | # Now calculate how many digits the result of sqrt(y1) would have | |
2140 | my $digits = int($length / 2); | |
2141 | ||
2142 | # But we need at least $scale digits, so calculate how many are missing | |
2143 | my $shift = $scale - $digits; | |
2144 | ||
0dceeee6 RGS |
2145 | # This happens if the input had enough digits |
2146 | # (we take care of integer guesses above) | |
2147 | $shift = 0 if $shift < 0; | |
9b924220 RGS |
2148 | |
2149 | # Multiply in steps of 100, by shifting left two times the "missing" digits | |
2150 | my $s2 = $shift * 2; | |
2151 | ||
2ab5f49d T |
2152 | # We now make sure that $y1 has the same odd or even number of digits than |
2153 | # $x had. So when _e of $x is odd, we must shift $y1 by one digit left, | |
2154 | # because we always must multiply by steps of 100 (sqrt(100) is 10) and not | |
2155 | # steps of 10. The length of $x does not count, since an even or odd number | |
2156 | # of digits before the dot is not changed by adding an even number of digits | |
2157 | # after the dot (the result is still odd or even digits long). | |
9b924220 RGS |
2158 | $s2++ if $MBI->_is_odd($x->{_e}); |
2159 | ||
2160 | $MBI->_lsft( $y1, $MBI->_new($s2), 10); | |
2161 | ||
2ab5f49d | 2162 | # now take the square root and truncate to integer |
9b924220 RGS |
2163 | $y1 = $MBI->_sqrt($y1); |
2164 | ||
2ab5f49d T |
2165 | # By "shifting" $y1 right (by creating a negative _e) we calculate the final |
2166 | # result, which is than later rounded to the desired scale. | |
990fb837 RGS |
2167 | |
2168 | # calculate how many zeros $x had after the '.' (or before it, depending | |
9b924220 RGS |
2169 | # on sign of $dat, the result should have half as many: |
2170 | my $dat = $MBI->_num($x->{_e}); | |
2171 | $dat = -$dat if $x->{_es} eq '-'; | |
2172 | $dat += $length; | |
990fb837 RGS |
2173 | |
2174 | if ($dat > 0) | |
2175 | { | |
2176 | # no zeros after the dot (e.g. 1.23, 0.49 etc) | |
2177 | # preserve half as many digits before the dot than the input had | |
2178 | # (but round this "up") | |
2179 | $dat = int(($dat+1)/2); | |
2180 | } | |
2181 | else | |
2182 | { | |
2183 | $dat = int(($dat)/2); | |
2184 | } | |
9b924220 RGS |
2185 | $dat -= $MBI->_len($y1); |
2186 | if ($dat < 0) | |
2187 | { | |
2188 | $dat = abs($dat); | |
2189 | $x->{_e} = $MBI->_new( $dat ); | |
2190 | $x->{_es} = '-'; | |
2191 | } | |
2192 | else | |
2193 | { | |
2194 | $x->{_e} = $MBI->_new( $dat ); | |
2195 | $x->{_es} = '+'; | |
2196 | } | |
2ab5f49d | 2197 | $x->{_m} = $y1; |
9b924220 | 2198 | $x->bnorm(); |
61f5c3f5 | 2199 | |
091c87b1 | 2200 | # shortcut to not run through _find_round_parameters again |
990fb837 | 2201 | if (defined $params[0]) |
61f5c3f5 | 2202 | { |
990fb837 | 2203 | $x->bround($params[0],$params[2]); # then round accordingly |
61f5c3f5 T |
2204 | } |
2205 | else | |
2206 | { | |
990fb837 | 2207 | $x->bfround($params[1],$params[2]); # then round accordingly |
61f5c3f5 | 2208 | } |
574bacfe JH |
2209 | if ($fallback) |
2210 | { | |
2211 | # clear a/p after round, since user did not request it | |
ef9466ea | 2212 | delete $x->{_a}; delete $x->{_p}; |
574bacfe | 2213 | } |
61f5c3f5 | 2214 | # restore globals |
b3abae2a | 2215 | $$abr = $ab; $$pbr = $pb; |
574bacfe | 2216 | $x; |
58cde26e JH |
2217 | } |
2218 | ||
b3abae2a JH |
2219 | sub bfac |
2220 | { | |
28df3e88 | 2221 | # (BFLOAT or num_str, BFLOAT or num_str) return BFLOAT |
091c87b1 | 2222 | # compute factorial number, modifies first argument |
b3abae2a | 2223 | |
b282a552 T |
2224 | # set up parameters |
2225 | my ($self,$x,@r) = (ref($_[0]),@_); | |
2226 | # objectify is costly, so avoid it | |
2227 | ($self,$x,@r) = objectify(1,@_) if !ref($x); | |
2228 | ||
50109ad0 RGS |
2229 | # inf => inf |
2230 | return $x if $x->modify('bfac') || $x->{sign} eq '+inf'; | |
2231 | ||
28df3e88 JH |
2232 | return $x->bnan() |
2233 | if (($x->{sign} ne '+') || # inf, NaN, <0 etc => NaN | |
9b924220 | 2234 | ($x->{_es} ne '+')); # digits after dot? |
b3abae2a | 2235 | |
b3abae2a | 2236 | # use BigInt's bfac() for faster calc |
9b924220 | 2237 | if (! $MBI->_is_zero($x->{_e})) |
091c87b1 | 2238 | { |
9b924220 RGS |
2239 | $MBI->_lsft($x->{_m}, $x->{_e},10); # change 12e1 to 120e0 |
2240 | $x->{_e} = $MBI->_zero(); # normalize | |
2241 | $x->{_es} = '+'; | |
091c87b1 | 2242 | } |
9b924220 | 2243 | $MBI->_fac($x->{_m}); # calculate factorial |
091c87b1 | 2244 | $x->bnorm()->round(@r); # norm again and round result |
b3abae2a JH |
2245 | } |
2246 | ||
9393ace2 JH |
2247 | sub _pow |
2248 | { | |
60a1aa19 T |
2249 | # Calculate a power where $y is a non-integer, like 2 ** 0.3 |
2250 | my ($x,$y,@r) = @_; | |
9393ace2 JH |
2251 | my $self = ref($x); |
2252 | ||
2253 | # if $y == 0.5, it is sqrt($x) | |
27e7b8bb | 2254 | $HALF = $self->new($HALF) unless ref($HALF); |
60a1aa19 | 2255 | return $x->bsqrt(@r,$y) if $y->bcmp($HALF) == 0; |
9393ace2 | 2256 | |
990fb837 RGS |
2257 | # Using: |
2258 | # a ** x == e ** (x * ln a) | |
2259 | ||
9393ace2 | 2260 | # u = y * ln x |
990fb837 RGS |
2261 | # _ _ |
2262 | # Taylor: | u u^2 u^3 | | |
2263 | # x ** y = 1 + | --- + --- + ----- + ... | | |
2264 | # |_ 1 1*2 1*2*3 _| | |
9393ace2 JH |
2265 | |
2266 | # we need to limit the accuracy to protect against overflow | |
2267 | my $fallback = 0; | |
990fb837 | 2268 | my ($scale,@params); |
60a1aa19 | 2269 | ($x,@params) = $x->_find_round_parameters(@r); |
990fb837 RGS |
2270 | |
2271 | return $x if $x->is_nan(); # error in _find_round_parameters? | |
9393ace2 JH |
2272 | |
2273 | # no rounding at all, so must use fallback | |
990fb837 | 2274 | if (scalar @params == 0) |
9393ace2 JH |
2275 | { |
2276 | # simulate old behaviour | |
990fb837 RGS |
2277 | $params[0] = $self->div_scale(); # and round to it as accuracy |
2278 | $params[1] = undef; # disable P | |
2279 | $scale = $params[0]+4; # at least four more for proper round | |
60a1aa19 | 2280 | $params[2] = $r[2]; # round mode by caller or undef |
9393ace2 JH |
2281 | $fallback = 1; # to clear a/p afterwards |
2282 | } | |
2283 | else | |
2284 | { | |
2285 | # the 4 below is empirical, and there might be cases where it is not | |
2286 | # enough... | |
990fb837 | 2287 | $scale = abs($params[0] || $params[1]) + 4; # take whatever is defined |
9393ace2 JH |
2288 | } |
2289 | ||
2290 | # when user set globals, they would interfere with our calculation, so | |
56d9de68 | 2291 | # disable them and later re-enable them |
9393ace2 JH |
2292 | no strict 'refs'; |
2293 | my $abr = "$self\::accuracy"; my $ab = $$abr; $$abr = undef; | |
2294 | my $pbr = "$self\::precision"; my $pb = $$pbr; $$pbr = undef; | |
2295 | # we also need to disable any set A or P on $x (_find_round_parameters took | |
2296 | # them already into account), since these would interfere, too | |
2297 | delete $x->{_a}; delete $x->{_p}; | |
2298 | # need to disable $upgrade in BigInt, to avoid deep recursion | |
2299 | local $Math::BigInt::upgrade = undef; | |
2300 | ||
2301 | my ($limit,$v,$u,$below,$factor,$next,$over); | |
2302 | ||
990fb837 | 2303 | $u = $x->copy()->blog(undef,$scale)->bmul($y); |
9393ace2 JH |
2304 | $v = $self->bone(); # 1 |
2305 | $factor = $self->new(2); # 2 | |
2306 | $x->bone(); # first term: 1 | |
2307 | ||
2308 | $below = $v->copy(); | |
2309 | $over = $u->copy(); | |
ae161977 | 2310 | |
9393ace2 JH |
2311 | $limit = $self->new("1E-". ($scale-1)); |
2312 | #my $steps = 0; | |
2313 | while (3 < 5) | |
2314 | { | |
2315 | # we calculate the next term, and add it to the last | |
2316 | # when the next term is below our limit, it won't affect the outcome | |
7d193e39 | 2317 | # anymore, so we stop: |
9393ace2 | 2318 | $next = $over->copy()->bdiv($below,$scale); |
990fb837 | 2319 | last if $next->bacmp($limit) <= 0; |
9393ace2 | 2320 | $x->badd($next); |
9393ace2 JH |
2321 | # calculate things for the next term |
2322 | $over *= $u; $below *= $factor; $factor->binc(); | |
9b924220 RGS |
2323 | |
2324 | last if $x->{sign} !~ /^[-+]$/; | |
2325 | ||
9393ace2 JH |
2326 | #$steps++; |
2327 | } | |
2328 | ||
091c87b1 | 2329 | # shortcut to not run through _find_round_parameters again |
990fb837 | 2330 | if (defined $params[0]) |
9393ace2 | 2331 | { |
990fb837 | 2332 | $x->bround($params[0],$params[2]); # then round accordingly |
9393ace2 JH |
2333 | } |
2334 | else | |
2335 | { | |
990fb837 | 2336 | $x->bfround($params[1],$params[2]); # then round accordingly |
9393ace2 JH |
2337 | } |
2338 | if ($fallback) | |
2339 | { | |
2340 | # clear a/p after round, since user did not request it | |
ef9466ea | 2341 | delete $x->{_a}; delete $x->{_p}; |
9393ace2 JH |
2342 | } |
2343 | # restore globals | |
2344 | $$abr = $ab; $$pbr = $pb; | |
2345 | $x; | |
2346 | } | |
2347 | ||
58cde26e JH |
2348 | sub bpow |
2349 | { | |
2350 | # (BFLOAT or num_str, BFLOAT or num_str) return BFLOAT | |
2351 | # compute power of two numbers, second arg is used as integer | |
2352 | # modifies first argument | |
2353 | ||
f9a08e12 JH |
2354 | # set up parameters |
2355 | my ($self,$x,$y,$a,$p,$r) = (ref($_[0]),@_); | |
2356 | # objectify is costly, so avoid it | |
2357 | if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1]))) | |
2358 | { | |
2359 | ($self,$x,$y,$a,$p,$r) = objectify(2,@_); | |
2360 | } | |
58cde26e | 2361 | |
50109ad0 RGS |
2362 | return $x if $x->modify('bpow'); |
2363 | ||
58cde26e | 2364 | return $x->bnan() if $x->{sign} eq $nan || $y->{sign} eq $nan; |
2d2b2744 T |
2365 | return $x if $x->{sign} =~ /^[+-]inf$/; |
2366 | ||
ae161977 RGS |
2367 | # cache the result of is_zero |
2368 | my $y_is_zero = $y->is_zero(); | |
2369 | return $x->bone() if $y_is_zero; | |
58cde26e | 2370 | return $x if $x->is_one() || $y->is_one(); |
9393ace2 | 2371 | |
ae161977 | 2372 | my $x_is_zero = $x->is_zero(); |
2d2b2744 | 2373 | return $x->_pow($y,$a,$p,$r) if !$x_is_zero && !$y->is_int(); # non-integer power |
9393ace2 | 2374 | |
ae161977 | 2375 | my $y1 = $y->as_number()->{value}; # make MBI part |
9b924220 | 2376 | |
394e6ffb | 2377 | # if ($x == -1) |
9b924220 | 2378 | if ($x->{sign} eq '-' && $MBI->_is_one($x->{_m}) && $MBI->_is_zero($x->{_e})) |
58cde26e JH |
2379 | { |
2380 | # if $x == -1 and odd/even y => +1/-1 because +-1 ^ (+-1) => +-1 | |
9b924220 | 2381 | return $MBI->_is_odd($y1) ? $x : $x->babs(1); |
288d023a | 2382 | } |
ae161977 | 2383 | if ($x_is_zero) |
28df3e88 JH |
2384 | { |
2385 | return $x if $y->{sign} eq '+'; # 0**y => 0 (if not y <= 0) | |
9b924220 RGS |
2386 | # 0 ** -y => 1 / (0 ** y) => 1 / 0! (1 / 0 => +inf) |
2387 | return $x->binf(); | |
28df3e88 | 2388 | } |
58cde26e | 2389 | |
9b924220 | 2390 | my $new_sign = '+'; |
ae161977 | 2391 | $new_sign = $MBI->_is_odd($y1) ? '-' : '+' if $x->{sign} ne '+'; |
9b924220 | 2392 | |
58cde26e | 2393 | # calculate $x->{_m} ** $y and $x->{_e} * $y separately (faster) |
9b924220 | 2394 | $x->{_m} = $MBI->_pow( $x->{_m}, $y1); |
ae161977 | 2395 | $x->{_e} = $MBI->_mul ($x->{_e}, $y1); |
9b924220 RGS |
2396 | |
2397 | $x->{sign} = $new_sign; | |
58cde26e JH |
2398 | $x->bnorm(); |
2399 | if ($y->{sign} eq '-') | |
2400 | { | |
2401 | # modify $x in place! | |
ae161977 | 2402 | my $z = $x->copy(); $x->bone(); |
7b29e1e6 | 2403 | return scalar $x->bdiv($z,$a,$p,$r); # round in one go (might ignore y's A!) |
a0d0e21e | 2404 | } |
28df3e88 | 2405 | $x->round($a,$p,$r,$y); |
58cde26e JH |
2406 | } |
2407 | ||
80365507 T |
2408 | sub bmodpow |
2409 | { | |
2410 | # takes a very large number to a very large exponent in a given very | |
2411 | # large modulus, quickly, thanks to binary exponentation. Supports | |
2412 | # negative exponents. | |
2413 | my ($self,$num,$exp,$mod,@r) = objectify(3,@_); | |
2414 | ||
2415 | return $num if $num->modify('bmodpow'); | |
2416 | ||
2417 | # check modulus for valid values | |
2418 | return $num->bnan() if ($mod->{sign} ne '+' # NaN, - , -inf, +inf | |
2419 | || $mod->is_zero()); | |
2420 | ||
2421 | # check exponent for valid values | |
2422 | if ($exp->{sign} =~ /\w/) | |
2423 | { | |
2424 | # i.e., if it's NaN, +inf, or -inf... | |
2425 | return $num->bnan(); | |
2426 | } | |
2427 | ||
2428 | $num->bmodinv ($mod) if ($exp->{sign} eq '-'); | |
2429 | ||
2430 | # check num for valid values (also NaN if there was no inverse but $exp < 0) | |
2431 | return $num->bnan() if $num->{sign} !~ /^[+-]$/; | |
2432 | ||
2433 | # $mod is positive, sign on $exp is ignored, result also positive | |
2434 | ||
2435 | # XXX TODO: speed it up when all three numbers are integers | |
2436 | $num->bpow($exp)->bmod($mod); | |
2437 | } | |
2438 | ||
58cde26e | 2439 | ############################################################################### |
fdb4b05f T |
2440 | # trigonometric functions |
2441 | ||
20e2035c | 2442 | # helper function for bpi() and batan2(), calculates arcus tanges (1/x) |
fdb4b05f | 2443 | |
20e2035c | 2444 | sub _atan_inv |
fdb4b05f | 2445 | { |
20e2035c T |
2446 | # return a/b so that a/b approximates atan(1/x) to at least limit digits |
2447 | my ($self, $x, $limit) = @_; | |
fdb4b05f | 2448 | |
20e2035c T |
2449 | # Taylor: x^3 x^5 x^7 x^9 |
2450 | # atan = x - --- + --- - --- + --- - ... | |
2451 | # 3 5 7 9 | |
fdb4b05f | 2452 | |
20e2035c T |
2453 | # 1 1 1 1 |
2454 | # atan 1/x = - - ------- + ------- - ------- + ... | |
2455 | # x x^3 * 3 x^5 * 5 x^7 * 7 | |
2456 | ||
2457 | # 1 1 1 1 | |
2458 | # atan 1/x = - - --------- + ---------- - ----------- + ... | |
2459 | # 5 3 * 125 5 * 3125 7 * 78125 | |
2460 | ||
2461 | # Subtraction/addition of a rational: | |
2462 | ||
2463 | # 5 7 5*3 +- 7*4 | |
2464 | # - +- - = ---------- | |
2465 | # 4 3 4*3 | |
2466 | ||
2467 | # Term: N N+1 | |
2468 | # | |
2469 | # a 1 a * d * c +- b | |
2470 | # ----- +- ------------------ = ---------------- | |
2471 | # b d * c b * d * c | |
2472 | ||
2473 | # since b1 = b0 * (d-2) * c | |
2474 | ||
2475 | # a 1 a * d +- b / c | |
2476 | # ----- +- ------------------ = ---------------- | |
2477 | # b d * c b * d | |
2478 | ||
2479 | # and d = d + 2 | |
2480 | # and c = c * x * x | |
2481 | ||
2482 | # u = d * c | |
2483 | # stop if length($u) > limit | |
2484 | # a = a * u +- b | |
2485 | # b = b * u | |
2486 | # d = d + 2 | |
2487 | # c = c * x * x | |
2488 | # sign = 1 - sign | |
2489 | ||
2490 | my $a = $MBI->_one(); | |
30afc38d | 2491 | my $b = $MBI->_copy($x); |
fdb4b05f | 2492 | |
30afc38d | 2493 | my $x2 = $MBI->_mul( $MBI->_copy($x), $b); # x2 = x * x |
20e2035c | 2494 | my $d = $MBI->_new( 3 ); # d = 3 |
30afc38d | 2495 | my $c = $MBI->_mul( $MBI->_copy($x), $x2); # c = x ^ 3 |
20e2035c T |
2496 | my $two = $MBI->_new( 2 ); |
2497 | ||
2498 | # run the first step unconditionally | |
2499 | my $u = $MBI->_mul( $MBI->_copy($d), $c); | |
2500 | $a = $MBI->_mul($a, $u); | |
2501 | $a = $MBI->_sub($a, $b); | |
2502 | $b = $MBI->_mul($b, $u); | |
2503 | $d = $MBI->_add($d, $two); | |
2504 | $c = $MBI->_mul($c, $x2); | |
2505 | ||
2506 | # a is now a * (d-3) * c | |
2507 | # b is now b * (d-2) * c | |
2508 | ||
2509 | # run the second step unconditionally | |
2510 | $u = $MBI->_mul( $MBI->_copy($d), $c); | |
2511 | $a = $MBI->_mul($a, $u); | |
2512 | $a = $MBI->_add($a, $b); | |
2513 | $b = $MBI->_mul($b, $u); | |
2514 | $d = $MBI->_add($d, $two); | |
2515 | $c = $MBI->_mul($c, $x2); | |
2516 | ||
2517 | # a is now a * (d-3) * (d-5) * c * c | |
2518 | # b is now b * (d-2) * (d-4) * c * c | |
2519 | ||
2520 | # so we can remove c * c from both a and b to shorten the numbers involved: | |
2521 | $a = $MBI->_div($a, $x2); | |
2522 | $b = $MBI->_div($b, $x2); | |
2523 | $a = $MBI->_div($a, $x2); | |
2524 | $b = $MBI->_div($b, $x2); | |
2525 | ||
2526 | # my $step = 0; | |
2527 | my $sign = 0; # 0 => -, 1 => + | |
2528 | while (3 < 5) | |
fdb4b05f | 2529 | { |
20e2035c T |
2530 | # $step++; |
2531 | # if (($i++ % 100) == 0) | |
2532 | # { | |
2533 | # print "a=",$MBI->_str($a),"\n"; | |
2534 | # print "b=",$MBI->_str($b),"\n"; | |
2535 | # } | |
2536 | # print "d=",$MBI->_str($d),"\n"; | |
2537 | # print "x2=",$MBI->_str($x2),"\n"; | |
2538 | # print "c=",$MBI->_str($c),"\n"; | |
2539 | ||
2540 | my $u = $MBI->_mul( $MBI->_copy($d), $c); | |
2541 | # use _alen() for libs like GMP where _len() would be O(N^2) | |
2542 | last if $MBI->_alen($u) > $limit; | |
2543 | my ($bc,$r) = $MBI->_div( $MBI->_copy($b), $c); | |
2544 | if ($MBI->_is_zero($r)) | |
fdb4b05f | 2545 | { |
20e2035c T |
2546 | # b / c is an integer, so we can remove c from all terms |
2547 | # this happens almost every time: | |
2548 | $a = $MBI->_mul($a, $d); | |
2549 | $a = $MBI->_sub($a, $bc) if $sign == 0; | |
2550 | $a = $MBI->_add($a, $bc) if $sign == 1; | |
2551 | $b = $MBI->_mul($b, $d); | |
fdb4b05f T |
2552 | } |
2553 | else | |
2554 | { | |
20e2035c T |
2555 | # b / c is not an integer, so we keep c in the terms |
2556 | # this happens very rarely, for instance for x = 5, this happens only | |
2557 | # at the following steps: | |
2558 | # 1, 5, 14, 32, 72, 157, 340, ... | |
2559 | $a = $MBI->_mul($a, $u); | |
2560 | $a = $MBI->_sub($a, $b) if $sign == 0; | |
2561 | $a = $MBI->_add($a, $b) if $sign == 1; | |
2562 | $b = $MBI->_mul($b, $u); | |
fdb4b05f | 2563 | } |
20e2035c T |
2564 | $d = $MBI->_add($d, $two); |
2565 | $c = $MBI->_mul($c, $x2); | |
2566 | $sign = 1 - $sign; | |
2567 | ||
fdb4b05f T |
2568 | } |
2569 | ||
30afc38d | 2570 | # print "Took $step steps for ", $MBI->_str($x),"\n"; |
20e2035c T |
2571 | # print "a=",$MBI->_str($a),"\n"; print "b=",$MBI->_str($b),"\n"; |
2572 | # return a/b so that a/b approximates atan(1/x) | |
2573 | ($a,$b); | |
fdb4b05f T |
2574 | } |
2575 | ||
2576 | sub bpi | |
2577 | { | |
fdb4b05f | 2578 | my ($self,$n) = @_; |
80365507 T |
2579 | if (@_ == 0) |
2580 | { | |
2581 | $self = $class; | |
2582 | } | |
fdb4b05f T |
2583 | if (@_ == 1) |
2584 | { | |
2585 | # called like Math::BigFloat::bpi(10); | |
2586 | $n = $self; $self = $class; | |
60a1aa19 T |
2587 | # called like Math::BigFloat->bpi(); |
2588 | $n = undef if $n eq 'Math::BigFloat'; | |
fdb4b05f T |
2589 | } |
2590 | $self = ref($self) if ref($self); | |
36ec1dfe | 2591 | my $fallback = defined $n ? 0 : 1; |
fdb4b05f T |
2592 | $n = 40 if !defined $n || $n < 1; |
2593 | ||
20e2035c T |
2594 | # after 黃見利 (Hwang Chien-Lih) (1997) |
2595 | # pi/4 = 183 * atan(1/239) + 32 * atan(1/1023) – 68 * atan(1/5832) | |
2596 | # + 12 * atan(1/110443) - 12 * atan(1/4841182) - 100 * atan(1/6826318) | |
2597 | ||
2598 | # a few more to prevent rounding errors | |
2599 | $n += 4; | |
2600 | ||
30afc38d T |
2601 | my ($a,$b) = $self->_atan_inv( $MBI->_new(239),$n); |
2602 | my ($c,$d) = $self->_atan_inv( $MBI->_new(1023),$n); | |
2603 | my ($e,$f) = $self->_atan_inv( $MBI->_new(5832),$n); | |
2604 | my ($g,$h) = $self->_atan_inv( $MBI->_new(110443),$n); | |
2605 | my ($i,$j) = $self->_atan_inv( $MBI->_new(4841182),$n); | |
2606 | my ($k,$l) = $self->_atan_inv( $MBI->_new(6826318),$n); | |
20e2035c T |
2607 | |
2608 | $MBI->_mul($a, $MBI->_new(732)); | |
2609 | $MBI->_mul($c, $MBI->_new(128)); | |
2610 | $MBI->_mul($e, $MBI->_new(272)); | |
2611 | $MBI->_mul($g, $MBI->_new(48)); | |
2612 | $MBI->_mul($i, $MBI->_new(48)); | |
2613 | $MBI->_mul($k, $MBI->_new(400)); | |
2614 | ||
2615 | my $x = $self->bone(); $x->{_m} = $a; my $x_d = $self->bone(); $x_d->{_m} = $b; | |
2616 | my $y = $self->bone(); $y->{_m} = $c; my $y_d = $self->bone(); $y_d->{_m} = $d; | |
2617 | my $z = $self->bone(); $z->{_m} = $e; my $z_d = $self->bone(); $z_d->{_m} = $f; | |
2618 | my $u = $self->bone(); $u->{_m} = $g; my $u_d = $self->bone(); $u_d->{_m} = $h; | |
2619 | my $v = $self->bone(); $v->{_m} = $i; my $v_d = $self->bone(); $v_d->{_m} = $j; | |
2620 | my $w = $self->bone(); $w->{_m} = $k; my $w_d = $self->bone(); $w_d->{_m} = $l; | |
2621 | $x->bdiv($x_d, $n); | |
2622 | $y->bdiv($y_d, $n); | |
2623 | $z->bdiv($z_d, $n); | |
2624 | $u->bdiv($u_d, $n); | |
2625 | $v->bdiv($v_d, $n); | |
2626 | $w->bdiv($w_d, $n); | |
2627 | ||
36ec1dfe T |
2628 | delete $x->{_a}; delete $y->{_a}; delete $z->{_a}; |
2629 | delete $u->{_a}; delete $v->{_a}; delete $w->{_a}; | |
20e2035c T |
2630 | $x->badd($y)->bsub($z)->badd($u)->bsub($v)->bsub($w); |
2631 | ||
36ec1dfe T |
2632 | $x->bround($n-4); |
2633 | delete $x->{_a} if $fallback == 1; | |
2634 | $x; | |
fdb4b05f T |
2635 | } |
2636 | ||
60a1aa19 T |
2637 | sub bcos |
2638 | { | |
2639 | # Calculate a cosinus of x. | |
2640 | my ($self,$x,@r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_); | |
2641 | ||
2642 | # Taylor: x^2 x^4 x^6 x^8 | |
2643 | # cos = 1 - --- + --- - --- + --- ... | |
2644 | # 2! 4! 6! 8! | |
2645 | ||
2646 | # we need to limit the accuracy to protect against overflow | |
2647 | my $fallback = 0; | |
2648 | my ($scale,@params); | |
2649 | ($x,@params) = $x->_find_round_parameters(@r); | |
2650 | ||
2651 | # constant object or error in _find_round_parameters? | |
2652 | return $x if $x->modify('bcos') || $x->is_nan(); | |
2653 | ||
2654 | return $x->bone(@r) if $x->is_zero(); | |
2655 | ||
2656 | # no rounding at all, so must use fallback | |
2657 | if (scalar @params == 0) | |
2658 | { | |
2659 | # simulate old behaviour | |
2660 | $params[0] = $self->div_scale(); # and round to it as accuracy | |
2661 | $params[1] = undef; # disable P | |
2662 | $scale = $params[0]+4; # at least four more for proper round | |
2663 | $params[2] = $r[2]; # round mode by caller or undef | |
2664 | $fallback = 1; # to clear a/p afterwards | |
2665 | } | |
2666 | else | |
2667 | { | |
2668 | # the 4 below is empirical, and there might be cases where it is not | |
2669 | # enough... | |
2670 | $scale = abs($params[0] || $params[1]) + 4; # take whatever is defined | |
2671 | } | |
2672 | ||
2673 | # when user set globals, they would interfere with our calculation, so | |
2674 | # disable them and later re-enable them | |
2675 | no strict 'refs'; | |
2676 | my $abr = "$self\::accuracy"; my $ab = $$abr; $$abr = undef; | |
2677 | my $pbr = "$self\::precision"; my $pb = $$pbr; $$pbr = undef; | |
2678 | # we also need to disable any set A or P on $x (_find_round_parameters took | |
2679 | # them already into account), since these would interfere, too | |
2680 | delete $x->{_a}; delete $x->{_p}; | |
2681 | # need to disable $upgrade in BigInt, to avoid deep recursion | |
2682 | local $Math::BigInt::upgrade = undef; | |
2683 | ||
2684 | my $last = 0; | |
2685 | my $over = $x * $x; # X ^ 2 | |
2686 | my $x2 = $over->copy(); # X ^ 2; difference between terms | |
2687 | my $sign = 1; # start with -= | |
2688 | my $below = $self->new(2); my $factorial = $self->new(3); | |
36ec1dfe | 2689 | $x->bone(); delete $x->{_a}; delete $x->{_p}; |
60a1aa19 T |
2690 | |
2691 | my $limit = $self->new("1E-". ($scale-1)); | |
2692 | #my $steps = 0; | |
2693 | while (3 < 5) | |
2694 | { | |
2695 | # we calculate the next term, and add it to the last | |
2696 | # when the next term is below our limit, it won't affect the outcome | |
2697 | # anymore, so we stop: | |
2698 | my $next = $over->copy()->bdiv($below,$scale); | |
2699 | last if $next->bacmp($limit) <= 0; | |
2700 | ||
2701 | if ($sign == 0) | |
2702 | { | |
2703 | $x->badd($next); | |
2704 | } | |
2705 | else | |
2706 | { | |
2707 | $x->bsub($next); | |
2708 | } | |
2709 | $sign = 1-$sign; # alternate | |
2710 | # calculate things for the next term | |
2711 | $over->bmul($x2); # $x*$x | |
2712 | $below->bmul($factorial); $factorial->binc(); # n*(n+1) | |
2713 | $below->bmul($factorial); $factorial->binc(); # n*(n+1) | |
2714 | } | |
2715 | ||
2716 | # shortcut to not run through _find_round_parameters again | |
2717 | if (defined $params[0]) | |
2718 | { | |
2719 | $x->bround($params[0],$params[2]); # then round accordingly | |
2720 | } | |
2721 | else | |
2722 | { | |
2723 | $x->bfround($params[1],$params[2]); # then round accordingly | |
2724 | } | |
2725 | if ($fallback) | |
2726 | { | |
2727 | # clear a/p after round, since user did not request it | |
2728 | delete $x->{_a}; delete $x->{_p}; | |
2729 | } | |
2730 | # restore globals | |
2731 | $$abr = $ab; $$pbr = $pb; | |
2732 | $x; | |
2733 | } | |
2734 | ||
2735 | sub bsin | |
2736 | { | |
2737 | # Calculate a sinus of x. | |
2738 | my ($self,$x,@r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_); | |
2739 | ||
2740 | # taylor: x^3 x^5 x^7 x^9 | |
2741 | # sin = x - --- + --- - --- + --- ... | |
2742 | # 3! 5! 7! 9! | |
2743 | ||
2744 | # we need to limit the accuracy to protect against overflow | |
2745 | my $fallback = 0; | |
2746 | my ($scale,@params); | |
2747 | ($x,@params) = $x->_find_round_parameters(@r); | |
2748 | ||
2749 | # constant object or error in _find_round_parameters? | |
2750 | return $x if $x->modify('bsin') || $x->is_nan(); | |
2751 | ||
2752 | return $x->bzero(@r) if $x->is_zero(); | |
2753 | ||
2754 | # no rounding at all, so must use fallback | |
2755 | if (scalar @params == 0) | |
2756 | { | |
2757 | # simulate old behaviour | |
2758 | $params[0] = $self->div_scale(); # and round to it as accuracy | |
2759 | $params[1] = undef; # disable P | |
2760 | $scale = $params[0]+4; # at least four more for proper round | |
2761 | $params[2] = $r[2]; # round mode by caller or undef | |
2762 | $fallback = 1; # to clear a/p afterwards | |
2763 | } | |
2764 | else | |
2765 | { | |
2766 | # the 4 below is empirical, and there might be cases where it is not | |
2767 | # enough... | |
2768 | $scale = abs($params[0] || $params[1]) + 4; # take whatever is defined | |
2769 | } | |
2770 | ||
2771 | # when user set globals, they would interfere with our calculation, so | |
2772 | # disable them and later re-enable them | |
2773 | no strict 'refs'; | |
2774 | my $abr = "$self\::accuracy"; my $ab = $$abr; $$abr = undef; | |
2775 | my $pbr = "$self\::precision"; my $pb = $$pbr; $$pbr = undef; | |
2776 | # we also need to disable any set A or P on $x (_find_round_parameters took | |
2777 | # them already into account), since these would interfere, too | |
2778 | delete $x->{_a}; delete $x->{_p}; | |
2779 | # need to disable $upgrade in BigInt, to avoid deep recursion | |
2780 | local $Math::BigInt::upgrade = undef; | |
2781 | ||
2782 | my $last = 0; | |
2783 | my $over = $x * $x; # X ^ 2 | |
2784 | my $x2 = $over->copy(); # X ^ 2; difference between terms | |
2785 | $over->bmul($x); # X ^ 3 as starting value | |
2786 | my $sign = 1; # start with -= | |
2787 | my $below = $self->new(6); my $factorial = $self->new(4); | |
36ec1dfe | 2788 | delete $x->{_a}; delete $x->{_p}; |
60a1aa19 T |
2789 | |
2790 | my $limit = $self->new("1E-". ($scale-1)); | |
2791 | #my $steps = 0; | |
2792 | while (3 < 5) | |
2793 | { | |
2794 | # we calculate the next term, and add it to the last | |
2795 | # when the next term is below our limit, it won't affect the outcome | |
2796 | # anymore, so we stop: | |
2797 | my $next = $over->copy()->bdiv($below,$scale); | |
2798 | last if $next->bacmp($limit) <= 0; | |
2799 | ||
2800 | if ($sign == 0) | |
2801 | { | |
2802 | $x->badd($next); | |
2803 | } | |
2804 | else | |
2805 | { | |
2806 | $x->bsub($next); | |
2807 | } | |
2808 | $sign = 1-$sign; # alternate | |
2809 | # calculate things for the next term | |
2810 | $over->bmul($x2); # $x*$x | |
2811 | $below->bmul($factorial); $factorial->binc(); # n*(n+1) | |
2812 | $below->bmul($factorial); $factorial->binc(); # n*(n+1) | |
2813 | } | |
2814 | ||
2815 | # shortcut to not run through _find_round_parameters again | |
2816 | if (defined $params[0]) | |
2817 | { | |
2818 | $x->bround($params[0],$params[2]); # then round accordingly | |
2819 | } | |
2820 | else | |
2821 | { | |
2822 | $x->bfround($params[1],$params[2]); # then round accordingly | |
2823 | } | |
2824 | if ($fallback) | |
2825 | { | |
2826 | # clear a/p after round, since user did not request it | |
2827 | delete $x->{_a}; delete $x->{_p}; | |
2828 | } | |
2829 | # restore globals | |
2830 | $$abr = $ab; $$pbr = $pb; | |
2831 | $x; | |
2832 | } | |
2833 | ||
20e2035c T |
2834 | sub batan2 |
2835 | { | |
30afc38d | 2836 | # calculate arcus tangens of ($y/$x) |
20e2035c T |
2837 | |
2838 | # set up parameters | |
30afc38d | 2839 | my ($self,$y,$x,@r) = (ref($_[0]),@_); |
20e2035c T |
2840 | # objectify is costly, so avoid it |
2841 | if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1]))) | |
2842 | { | |
30afc38d | 2843 | ($self,$y,$x,@r) = objectify(2,@_); |
20e2035c T |
2844 | } |
2845 | ||
30afc38d | 2846 | return $y if $y->modify('batan2'); |
20e2035c | 2847 | |
30afc38d | 2848 | return $y->bnan() if ($y->{sign} eq $nan) || ($x->{sign} eq $nan); |
20e2035c | 2849 | |
30afc38d T |
2850 | # Y X |
2851 | # 0 0 result is 0 | |
2852 | # 0 +x result is 0 | |
0dceeee6 RGS |
2853 | # ? inf result is 0 |
2854 | return $y->bzero(@r) if ($x->is_inf('+') && !$y->is_inf()) || ($y->is_zero() && $x->{sign} eq '+'); | |
2855 | ||
2856 | # Y X | |
2857 | # != 0 -inf result is +- pi | |
2858 | if ($x->is_inf() || $y->is_inf()) | |
2859 | { | |
2860 | # calculate PI | |
2861 | my $pi = $self->bpi(@r); | |
2862 | if ($y->is_inf()) | |
2863 | { | |
2864 | # upgrade to BigRat etc. | |
2865 | return $upgrade->new($y)->batan2($upgrade->new($x),@r) if defined $upgrade; | |
2866 | if ($x->{sign} eq '-inf') | |
2867 | { | |
2868 | # calculate 3 pi/4 | |
2869 | $MBI->_mul($pi->{_m}, $MBI->_new(3)); | |
2870 | $MBI->_div($pi->{_m}, $MBI->_new(4)); | |
2871 | } | |
2872 | elsif ($x->{sign} eq '+inf') | |
2873 | { | |
2874 | # calculate pi/4 | |
2875 | $MBI->_div($pi->{_m}, $MBI->_new(4)); | |
2876 | } | |
2877 | else | |
2878 | { | |
2879 | # calculate pi/2 | |
2880 | $MBI->_div($pi->{_m}, $MBI->_new(2)); | |
2881 | } | |
2882 | $y->{sign} = substr($y->{sign},0,1); # keep +/- | |
2883 | } | |
2884 | # modify $y in place | |
2885 | $y->{_m} = $pi->{_m}; | |
2886 | $y->{_e} = $pi->{_e}; | |
2887 | $y->{_es} = $pi->{_es}; | |
2888 | # keep the sign of $y | |
2889 | return $y; | |
2890 | } | |
2891 | ||
2892 | return $upgrade->new($y)->batan2($upgrade->new($x),@r) if defined $upgrade; | |
30afc38d T |
2893 | |
2894 | # Y X | |
2895 | # 0 -x result is PI | |
2896 | if ($y->is_zero()) | |
20e2035c | 2897 | { |
30afc38d T |
2898 | # calculate PI |
2899 | my $pi = $self->bpi(@r); | |
0dceeee6 | 2900 | # modify $y in place |
30afc38d T |
2901 | $y->{_m} = $pi->{_m}; |
2902 | $y->{_e} = $pi->{_e}; | |
2903 | $y->{_es} = $pi->{_es}; | |
2904 | $y->{sign} = '+'; | |
2905 | return $y; | |
2906 | } | |
2907 | ||
2908 | # Y X | |
2909 | # +y 0 result is PI/2 | |
2910 | # -y 0 result is -PI/2 | |
0dceeee6 | 2911 | if ($x->is_zero()) |
30afc38d T |
2912 | { |
2913 | # calculate PI/2 | |
2914 | my $pi = $self->bpi(@r); | |
0dceeee6 | 2915 | # modify $y in place |
30afc38d T |
2916 | $y->{_m} = $pi->{_m}; |
2917 | $y->{_e} = $pi->{_e}; | |
2918 | $y->{_es} = $pi->{_es}; | |
2919 | # -y => -PI/2, +y => PI/2 | |
30afc38d T |
2920 | $MBI->_div($y->{_m}, $MBI->_new(2)); |
2921 | return $y; | |
20e2035c T |
2922 | } |
2923 | ||
30afc38d T |
2924 | # we need to limit the accuracy to protect against overflow |
2925 | my $fallback = 0; | |
2926 | my ($scale,@params); | |
2927 | ($y,@params) = $y->_find_round_parameters(@r); | |
2928 | ||
2929 | # error in _find_round_parameters? | |
2930 | return $y if $y->is_nan(); | |
2931 | ||
2932 | # no rounding at all, so must use fallback | |
2933 | if (scalar @params == 0) | |
2934 | { | |
2935 | # simulate old behaviour | |
2936 | $params[0] = $self->div_scale(); # and round to it as accuracy | |
2937 | $params[1] = undef; # disable P | |
2938 | $scale = $params[0]+4; # at least four more for proper round | |
2939 | $params[2] = $r[2]; # round mode by caller or undef | |
2940 | $fallback = 1; # to clear a/p afterwards | |
2941 | } | |
2942 | else | |
2943 | { | |
2944 | # the 4 below is empirical, and there might be cases where it is not | |
2945 | # enough... | |
2946 | $scale = abs($params[0] || $params[1]) + 4; # take whatever is defined | |
2947 | } | |
2948 | ||
2949 | # inlined is_one() && is_one('-') | |
2950 | if ($MBI->_is_one($y->{_m}) && $MBI->_is_zero($y->{_e})) | |
2951 | { | |
2952 | # shortcut: 1 1 result is PI/4 | |
2953 | # inlined is_one() && is_one('-') | |
2954 | if ($MBI->_is_one($x->{_m}) && $MBI->_is_zero($x->{_e})) | |
2955 | { | |
2956 | # 1,1 => PI/4 | |
2957 | my $pi_4 = $self->bpi( $scale - 3); | |
0dceeee6 | 2958 | # modify $y in place |
30afc38d T |
2959 | $y->{_m} = $pi_4->{_m}; |
2960 | $y->{_e} = $pi_4->{_e}; | |
2961 | $y->{_es} = $pi_4->{_es}; | |
2962 | # 1 1 => + | |
2963 | # -1 1 => - | |
2964 | # 1 -1 => - | |
2965 | # -1 -1 => + | |
2966 | $y->{sign} = $x->{sign} eq $y->{sign} ? '+' : '-'; | |
2967 | $MBI->_div($y->{_m}, $MBI->_new(4)); | |
2968 | return $y; | |
2969 | } | |
2970 | # shortcut: 1 int(X) result is _atan_inv(X) | |
2971 | ||
2972 | # is integer | |
2973 | if ($x->{_es} eq '+') | |
2974 | { | |
2975 | my $x1 = $MBI->_copy($x->{_m}); | |
2976 | $MBI->_lsft($x1, $x->{_e},10) unless $MBI->_is_zero($x->{_e}); | |
2977 | ||
2978 | my ($a,$b) = $self->_atan_inv($x1, $scale); | |
2979 | my $y_sign = $y->{sign}; | |
2980 | # calculate A/B | |
2981 | $y->bone(); $y->{_m} = $a; my $y_d = $self->bone(); $y_d->{_m} = $b; | |
2982 | $y->bdiv($y_d, @r); | |
2983 | $y->{sign} = $y_sign; | |
2984 | return $y; | |
2985 | } | |
2986 | } | |
2987 | ||
2988 | # handle all other cases | |
2989 | # X Y | |
2990 | # +x +y 0 to PI/2 | |
2991 | # -x +y PI/2 to PI | |
2992 | # +x -y 0 to -PI/2 | |
2993 | # -x -y -PI/2 to -PI | |
2994 | ||
2995 | my $y_sign = $y->{sign}; | |
20e2035c T |
2996 | |
2997 | # divide $x by $y | |
30afc38d T |
2998 | $y->bdiv($x, $scale) unless $x->is_one(); |
2999 | $y->batan(@r); | |
20e2035c | 3000 | |
30afc38d T |
3001 | # restore sign |
3002 | $y->{sign} = $y_sign; | |
20e2035c | 3003 | |
30afc38d | 3004 | $y; |
20e2035c T |
3005 | } |
3006 | ||
60a1aa19 T |
3007 | sub batan |
3008 | { | |
3009 | # Calculate a arcus tangens of x. | |
3010 | my ($x,@r) = @_; | |
3011 | my $self = ref($x); | |
3012 | ||
3013 | # taylor: x^3 x^5 x^7 x^9 | |
3014 | # atan = x - --- + --- - --- + --- ... | |
3015 | # 3 5 7 9 | |
3016 | ||
60a1aa19 T |
3017 | # we need to limit the accuracy to protect against overflow |
3018 | my $fallback = 0; | |
3019 | my ($scale,@params); | |
3020 | ($x,@params) = $x->_find_round_parameters(@r); | |
3021 | ||
3022 | # constant object or error in _find_round_parameters? | |
3023 | return $x if $x->modify('batan') || $x->is_nan(); | |
3024 | ||
30afc38d T |
3025 | if ($x->{sign} =~ /^[+-]inf\z/) |
3026 | { | |
3027 | # +inf result is PI/2 | |
3028 | # -inf result is -PI/2 | |
3029 | # calculate PI/2 | |
3030 | my $pi = $self->bpi(@r); | |
3031 | # modify $x in place | |
3032 | $x->{_m} = $pi->{_m}; | |
3033 | $x->{_e} = $pi->{_e}; | |
3034 | $x->{_es} = $pi->{_es}; | |
3035 | # -y => -PI/2, +y => PI/2 | |
3036 | $x->{sign} = substr($x->{sign},0,1); # +inf => + | |
3037 | $MBI->_div($x->{_m}, $MBI->_new(2)); | |
3038 | return $x; | |
3039 | } | |
3040 | ||
3041 | return $x->bzero(@r) if $x->is_zero(); | |
3042 | ||
60a1aa19 T |
3043 | # no rounding at all, so must use fallback |
3044 | if (scalar @params == 0) | |
3045 | { | |
3046 | # simulate old behaviour | |
3047 | $params[0] = $self->div_scale(); # and round to it as accuracy | |
3048 | $params[1] = undef; # disable P | |
3049 | $scale = $params[0]+4; # at least four more for proper round | |
3050 | $params[2] = $r[2]; # round mode by caller or undef | |
3051 | $fallback = 1; # to clear a/p afterwards | |
3052 | } | |
3053 | else | |
3054 | { | |
3055 | # the 4 below is empirical, and there might be cases where it is not | |
3056 | # enough... | |
3057 | $scale = abs($params[0] || $params[1]) + 4; # take whatever is defined | |
3058 | } | |
3059 | ||
30afc38d T |
3060 | # 1 or -1 => PI/4 |
3061 | # inlined is_one() && is_one('-') | |
3062 | if ($MBI->_is_one($x->{_m}) && $MBI->_is_zero($x->{_e})) | |
3063 | { | |
3064 | my $pi = $self->bpi($scale - 3); | |
3065 | # modify $x in place | |
3066 | $x->{_m} = $pi->{_m}; | |
3067 | $x->{_e} = $pi->{_e}; | |
3068 | $x->{_es} = $pi->{_es}; | |
3069 | # leave the sign of $x alone (+1 => +PI/4, -1 => -PI/4) | |
3070 | $MBI->_div($x->{_m}, $MBI->_new(4)); | |
3071 | return $x; | |
3072 | } | |
3073 | ||
3074 | # This series is only valid if -1 < x < 1, so for other x we need to | |
c97ef841 | 3075 | # to calculate PI/2 - atan(1/x): |
30afc38d T |
3076 | my $one = $MBI->_new(1); |
3077 | my $pi = undef; | |
3078 | if ($x->{_es} eq '+' && ($MBI->_acmp($x->{_m},$one) >= 0)) | |
3079 | { | |
3080 | # calculate PI/2 | |
3081 | $pi = $self->bpi($scale - 3); | |
3082 | $MBI->_div($pi->{_m}, $MBI->_new(2)); | |
3083 | # calculate 1/$x: | |
3084 | my $x_copy = $x->copy(); | |
3085 | # modify $x in place | |
3086 | $x->bone(); $x->bdiv($x_copy,$scale); | |
3087 | } | |
3088 | ||
60a1aa19 T |
3089 | # when user set globals, they would interfere with our calculation, so |
3090 | # disable them and later re-enable them | |
3091 | no strict 'refs'; | |
3092 | my $abr = "$self\::accuracy"; my $ab = $$abr; $$abr = undef; | |
3093 | my $pbr = "$self\::precision"; my $pb = $$pbr; $$pbr = undef; | |
3094 | # we also need to disable any set A or P on $x (_find_round_parameters took | |
3095 | # them already into account), since these would interfere, too | |
3096 | delete $x->{_a}; delete $x->{_p}; | |
3097 | # need to disable $upgrade in BigInt, to avoid deep recursion | |
3098 | local $Math::BigInt::upgrade = undef; | |
3099 | ||
3100 | my $last = 0; | |
3101 | my $over = $x * $x; # X ^ 2 | |
3102 | my $x2 = $over->copy(); # X ^ 2; difference between terms | |
3103 | $over->bmul($x); # X ^ 3 as starting value | |
3104 | my $sign = 1; # start with -= | |
3105 | my $below = $self->new(3); | |
3106 | my $two = $self->new(2); | |
36ec1dfe | 3107 | delete $x->{_a}; delete $x->{_p}; |
60a1aa19 T |
3108 | |
3109 | my $limit = $self->new("1E-". ($scale-1)); | |
3110 | #my $steps = 0; | |
3111 | while (3 < 5) | |
3112 | { | |
3113 | # we calculate the next term, and add it to the last | |
3114 | # when the next term is below our limit, it won't affect the outcome | |
3115 | # anymore, so we stop: | |
3116 | my $next = $over->copy()->bdiv($below,$scale); | |
3117 | last if $next->bacmp($limit) <= 0; | |
3118 | ||
3119 | if ($sign == 0) | |
3120 | { | |
3121 | $x->badd($next); | |
3122 | } | |
3123 | else | |
3124 | { | |
3125 | $x->bsub($next); | |
3126 | } | |
3127 | $sign = 1-$sign; # alternate | |
3128 | # calculate things for the next term | |
3129 | $over->bmul($x2); # $x*$x | |
3130 | $below->badd($two); # n += 2 | |
3131 | } | |
3132 | ||
30afc38d T |
3133 | if (defined $pi) |
3134 | { | |
3135 | my $x_copy = $x->copy(); | |
3136 | # modify $x in place | |
3137 | $x->{_m} = $pi->{_m}; | |
3138 | $x->{_e} = $pi->{_e}; | |
3139 | $x->{_es} = $pi->{_es}; | |
3140 | # PI/2 - $x | |
3141 | $x->bsub($x_copy); | |
3142 | } | |
3143 | ||
60a1aa19 T |
3144 | # shortcut to not run through _find_round_parameters again |
3145 | if (defined $params[0]) | |
3146 | { | |
3147 | $x->bround($params[0],$params[2]); # then round accordingly | |
3148 | } | |
3149 | else | |
3150 | { | |
3151 | $x->bfround($params[1],$params[2]); # then round accordingly | |
3152 | } | |
3153 | if ($fallback) | |
3154 | { | |
3155 | # clear a/p after round, since user did not request it | |
3156 | delete $x->{_a}; delete $x->{_p}; | |
3157 | } | |
3158 | # restore globals | |
3159 | $$abr = $ab; $$pbr = $pb; | |
3160 | $x; | |
3161 | } | |
3162 | ||
fdb4b05f | 3163 | ############################################################################### |
58cde26e JH |
3164 | # rounding functions |
3165 | ||
3166 | sub bfround | |
3167 | { | |
3168 | # precision: round to the $Nth digit left (+$n) or right (-$n) from the '.' | |
3169 | # $n == 0 means round to integer | |
3170 | # expects and returns normalized numbers! | |
ee15d750 | 3171 | my $x = shift; my $self = ref($x) || $x; $x = $self->new(shift) if !ref($x); |
a0d0e21e | 3172 | |
b68b7ab1 T |
3173 | my ($scale,$mode) = $x->_scale_p(@_); |
3174 | return $x if !defined $scale || $x->modify('bfround'); # no-op | |
58cde26e | 3175 | |
574bacfe | 3176 | # never round a 0, +-inf, NaN |
61f5c3f5 T |
3177 | if ($x->is_zero()) |
3178 | { | |
3179 | $x->{_p} = $scale if !defined $x->{_p} || $x->{_p} < $scale; # -3 < -2 | |
3180 | return $x; | |
3181 | } | |
3182 | return $x if $x->{sign} !~ /^[+-]$/; | |
58cde26e | 3183 | |
ee15d750 JH |
3184 | # don't round if x already has lower precision |
3185 | return $x if (defined $x->{_p} && $x->{_p} < 0 && $scale < $x->{_p}); | |
3186 | ||
3187 | $x->{_p} = $scale; # remember round in any case | |
ef9466ea | 3188 | delete $x->{_a}; # and clear A |
58cde26e JH |
3189 | if ($scale < 0) |
3190 | { | |
58cde26e | 3191 | # round right from the '.' |
f9a08e12 | 3192 | |
9b924220 | 3193 | return $x if $x->{_es} eq '+'; # e >= 0 => nothing to round |
f9a08e12 | 3194 | |
58cde26e | 3195 | $scale = -$scale; # positive for simplicity |
9b924220 | 3196 | my $len = $MBI->_len($x->{_m}); # length of mantissa |
f9a08e12 JH |
3197 | |
3198 | # the following poses a restriction on _e, but if _e is bigger than a | |
3199 | # scalar, you got other problems (memory etc) anyway | |
9b924220 | 3200 | my $dad = -(0+ ($x->{_es}.$MBI->_num($x->{_e}))); # digits after dot |
58cde26e | 3201 | my $zad = 0; # zeros after dot |
f9a08e12 | 3202 | $zad = $dad - $len if (-$dad < -$len); # for 0.00..00xxx style |
9b924220 RGS |
3203 | |
3204 | # p rint "scale $scale dad $dad zad $zad len $len\n"; | |
58cde26e JH |
3205 | # number bsstr len zad dad |
3206 | # 0.123 123e-3 3 0 3 | |
3207 | # 0.0123 123e-4 3 1 4 | |
3208 | # 0.001 1e-3 1 2 3 | |
3209 | # 1.23 123e-2 3 0 2 | |
3210 | # 1.2345 12345e-4 5 0 4 | |
3211 | ||
3212 | # do not round after/right of the $dad | |
3213 | return $x if $scale > $dad; # 0.123, scale >= 3 => exit | |
3214 | ||
ee15d750 JH |
3215 | # round to zero if rounding inside the $zad, but not for last zero like: |
3216 | # 0.0065, scale -2, round last '0' with following '65' (scale == zad case) | |
3217 | return $x->bzero() if $scale < $zad; | |
3218 | if ($scale == $zad) # for 0.006, scale -3 and trunc | |
58cde26e | 3219 | { |
b3abae2a | 3220 | $scale = -$len; |
58cde26e JH |
3221 | } |
3222 | else | |
3223 | { | |
3224 | # adjust round-point to be inside mantissa | |
3225 | if ($zad != 0) | |
3226 | { | |
3227 | $scale = $scale-$zad; | |
3228 | } | |
3229 | else | |
3230 | { | |
3231 | my $dbd = $len - $dad; $dbd = 0 if $dbd < 0; # digits before dot | |
3232 | $scale = $dbd+$scale; | |
3233 | } | |
3234 | } | |
a0d0e21e | 3235 | } |
58cde26e JH |
3236 | else |
3237 | { | |
f9a08e12 JH |
3238 | # round left from the '.' |
3239 | ||
58cde26e | 3240 | # 123 => 100 means length(123) = 3 - $scale (2) => 1 |
a5f75d66 | 3241 | |
9b924220 | 3242 | my $dbt = $MBI->_len($x->{_m}); |
b3abae2a | 3243 | # digits before dot |
9b924220 | 3244 | my $dbd = $dbt + ($x->{_es} . $MBI->_num($x->{_e})); |
b3abae2a JH |
3245 | # should be the same, so treat it as this |
3246 | $scale = 1 if $scale == 0; | |
3247 | # shortcut if already integer | |
3248 | return $x if $scale == 1 && $dbt <= $dbd; | |
3249 | # maximum digits before dot | |
3250 | ++$dbd; | |
3251 | ||
3252 | if ($scale > $dbd) | |
3253 | { | |
3254 | # not enough digits before dot, so round to zero | |
3255 | return $x->bzero; | |
3256 | } | |
3257 | elsif ( $scale == $dbd ) | |
3258 | { | |
3259 | # maximum | |
3260 | $scale = -$dbt; | |
3261 | } | |
58cde26e | 3262 | else |
b3abae2a JH |
3263 | { |
3264 | $scale = $dbd - $scale; | |
3265 | } | |
a0d0e21e | 3266 | } |
574bacfe | 3267 | # pass sign to bround for rounding modes '+inf' and '-inf' |
ae161977 | 3268 | my $m = bless { sign => $x->{sign}, value => $x->{_m} }, 'Math::BigInt'; |
9b924220 RGS |
3269 | $m->bround($scale,$mode); |
3270 | $x->{_m} = $m->{value}; # get our mantissa back | |
58cde26e JH |
3271 | $x->bnorm(); |
3272 | } | |
3273 | ||
3274 | sub bround | |
3275 | { | |
3276 | # accuracy: preserve $N digits, and overwrite the rest with 0's | |
ee15d750 | 3277 | my $x = shift; my $self = ref($x) || $x; $x = $self->new(shift) if !ref($x); |
9b924220 | 3278 | |
990fb837 RGS |
3279 | if (($_[0] || 0) < 0) |
3280 | { | |
3281 | require Carp; Carp::croak ('bround() needs positive accuracy'); | |
3282 | } | |
58cde26e | 3283 | |
b68b7ab1 T |
3284 | my ($scale,$mode) = $x->_scale_a(@_); |
3285 | return $x if !defined $scale || $x->modify('bround'); # no-op | |
61f5c3f5 | 3286 | |
ee15d750 JH |
3287 | # scale is now either $x->{_a}, $accuracy, or the user parameter |
3288 | # test whether $x already has lower accuracy, do nothing in this case | |
3289 | # but do round if the accuracy is the same, since a math operation might | |
3290 | # want to round a number with A=5 to 5 digits afterwards again | |
b68b7ab1 | 3291 | return $x if defined $x->{_a} && $x->{_a} < $scale; |
58cde26e | 3292 | |
61f5c3f5 | 3293 | # scale < 0 makes no sense |
b68b7ab1 | 3294 | # scale == 0 => keep all digits |
61f5c3f5 | 3295 | # never round a +-inf, NaN |
b68b7ab1 | 3296 | return $x if ($scale <= 0) || $x->{sign} !~ /^[+-]$/; |
58cde26e | 3297 | |
b68b7ab1 T |
3298 | # 1: never round a 0 |
3299 | # 2: if we should keep more digits than the mantissa has, do nothing | |
3300 | if ($x->is_zero() || $MBI->_len($x->{_m}) <= $scale) | |
61f5c3f5 T |
3301 | { |
3302 | $x->{_a} = $scale if !defined $x->{_a} || $x->{_a} > $scale; | |
3303 | return $x; | |
3304 | } | |
f216259d | 3305 | |
58cde26e | 3306 | # pass sign to bround for '+inf' and '-inf' rounding modes |
ae161977 | 3307 | my $m = bless { sign => $x->{sign}, value => $x->{_m} }, 'Math::BigInt'; |
9b924220 RGS |
3308 | |
3309 | $m->bround($scale,$mode); # round mantissa | |
3310 | $x->{_m} = $m->{value}; # get our mantissa back | |
ee15d750 | 3311 | $x->{_a} = $scale; # remember rounding |
ef9466ea | 3312 | delete $x->{_p}; # and clear P |
574bacfe | 3313 | $x->bnorm(); # del trailing zeros gen. by bround() |
58cde26e JH |
3314 | } |
3315 | ||
3316 | sub bfloor | |
3317 | { | |
3318 | # return integer less or equal then $x | |
ee15d750 | 3319 | my ($self,$x,$a,$p,$r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_); |
58cde26e JH |
3320 | |
3321 | return $x if $x->modify('bfloor'); | |
3322 | ||
3323 | return $x if $x->{sign} !~ /^[+-]$/; # nan, +inf, -inf | |
3324 | ||
3325 | # if $x has digits after dot | |
9b924220 | 3326 | if ($x->{_es} eq '-') |
58cde26e | 3327 | { |
9b924220 RGS |
3328 | $x->{_m} = $MBI->_rsft($x->{_m},$x->{_e},10); # cut off digits after dot |
3329 | $x->{_e} = $MBI->_zero(); # trunc/norm | |
3330 | $x->{_es} = '+'; # abs e | |
3331 | $MBI->_inc($x->{_m}) if $x->{sign} eq '-'; # increment if negative | |
f216259d | 3332 | } |
61f5c3f5 | 3333 | $x->round($a,$p,$r); |
58cde26e | 3334 | } |
288d023a | 3335 | |
58cde26e JH |
3336 | sub bceil |
3337 | { | |
3338 | # return integer greater or equal then $x | |
ee15d750 | 3339 | my ($self,$x,$a,$p,$r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_); |
58cde26e JH |
3340 | |
3341 | return $x if $x->modify('bceil'); | |
3342 | return $x if $x->{sign} !~ /^[+-]$/; # nan, +inf, -inf | |
3343 | ||
3344 | # if $x has digits after dot | |
9b924220 | 3345 | if ($x->{_es} eq '-') |
58cde26e | 3346 | { |
9b924220 RGS |
3347 | $x->{_m} = $MBI->_rsft($x->{_m},$x->{_e},10); # cut off digits after dot |
3348 | $x->{_e} = $MBI->_zero(); # trunc/norm | |
3349 | $x->{_es} = '+'; # abs e | |
3350 | $MBI->_inc($x->{_m}) if $x->{sign} eq '+'; # increment if positive | |
a0d0e21e | 3351 | } |
61f5c3f5 | 3352 | $x->round($a,$p,$r); |
58cde26e JH |
3353 | } |
3354 | ||
394e6ffb JH |
3355 | sub brsft |
3356 | { | |
f9a08e12 JH |
3357 | # shift right by $y (divide by power of $n) |
3358 | ||
3359 | # set up parameters | |
3360 | my ($self,$x,$y,$n,$a,$p,$r) = (ref($_[0]),@_); | |
3361 | # objectify is costly, so avoid it | |
3362 | if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1]))) | |
3363 | { | |
3364 | ($self,$x,$y,$n,$a,$p,$r) = objectify(2,@_); | |
3365 | } | |
394e6ffb JH |
3366 | |
3367 | return $x if $x->modify('brsft'); | |
3368 | return $x if $x->{sign} !~ /^[+-]$/; # nan, +inf, -inf | |
3369 | ||
f9a08e12 | 3370 | $n = 2 if !defined $n; $n = $self->new($n); |
7b29e1e6 T |
3371 | |
3372 | # negative amount? | |
3373 | return $x->blsft($y->copy()->babs(),$n) if $y->{sign} =~ /^-/; | |
3374 | ||
3375 | # the following call to bdiv() will return either quo or (quo,reminder): | |
f9a08e12 | 3376 | $x->bdiv($n->bpow($y),$a,$p,$r,$y); |
394e6ffb JH |
3377 | } |
3378 | ||
3379 | sub blsft | |
3380 | { | |
f9a08e12 JH |
3381 | # shift left by $y (multiply by power of $n) |
3382 | ||
3383 | # set up parameters | |
3384 | my ($self,$x,$y,$n,$a,$p,$r) = (ref($_[0]),@_); | |
3385 | # objectify is costly, so avoid it | |
3386 | if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1]))) | |
3387 | { | |
3388 | ($self,$x,$y,$n,$a,$p,$r) = objectify(2,@_); | |
3389 | } | |
394e6ffb | 3390 | |
f9a08e12 | 3391 | return $x if $x->modify('blsft'); |
394e6ffb JH |
3392 | return $x if $x->{sign} !~ /^[+-]$/; # nan, +inf, -inf |
3393 | ||
f9a08e12 | 3394 | $n = 2 if !defined $n; $n = $self->new($n); |
7b29e1e6 T |
3395 | |
3396 | # negative amount? | |
3397 | return $x->brsft($y->copy()->babs(),$n) if $y->{sign} =~ /^-/; | |
3398 | ||
f9a08e12 | 3399 | $x->bmul($n->bpow($y),$a,$p,$r,$y); |
394e6ffb JH |
3400 | } |
3401 | ||
58cde26e | 3402 | ############################################################################### |
a5f75d66 | 3403 | |
58cde26e JH |
3404 | sub DESTROY |
3405 | { | |
b282a552 | 3406 | # going through AUTOLOAD for every DESTROY is costly, avoid it by empty sub |
58cde26e JH |
3407 | } |
3408 | ||
3409 | sub AUTOLOAD | |
3410 | { | |
b3abae2a JH |
3411 | # make fxxx and bxxx both work by selectively mapping fxxx() to MBF::bxxx() |
3412 | # or falling back to MBI::bxxx() | |
58cde26e JH |
3413 | my $name = $AUTOLOAD; |
3414 | ||
ef9466ea T |
3415 | $name =~ s/(.*):://; # split package |
3416 | my $c = $1 || $class; | |
ee15d750 | 3417 | no strict 'refs'; |
ef9466ea | 3418 | $c->import() if $IMPORT == 0; |
7b29e1e6 | 3419 | if (!_method_alias($name)) |
58cde26e | 3420 | { |
ee15d750 JH |
3421 | if (!defined $name) |
3422 | { | |
3423 | # delayed load of Carp and avoid recursion | |
3424 | require Carp; | |
ef9466ea | 3425 | Carp::croak ("$c: Can't call a method without name"); |
ee15d750 | 3426 | } |
7b29e1e6 | 3427 | if (!_method_hand_up($name)) |
ee15d750 JH |
3428 | { |
3429 | # delayed load of Carp and avoid recursion | |
3430 | require Carp; | |
ef9466ea | 3431 | Carp::croak ("Can't call $c\-\>$name, not a valid method"); |
ee15d750 JH |
3432 | } |
3433 | # try one level up, but subst. bxxx() for fxxx() since MBI only got bxxx() | |
3434 | $name =~ s/^f/b/; | |
9b924220 | 3435 | return &{"Math::BigInt"."::$name"}(@_); |
a0d0e21e | 3436 | } |
58cde26e | 3437 | my $bname = $name; $bname =~ s/^f/b/; |
ef9466ea T |
3438 | $c .= "::$name"; |
3439 | *{$c} = \&{$bname}; | |
3440 | &{$c}; # uses @_ | |
58cde26e JH |
3441 | } |
3442 | ||
3443 | sub exponent | |
3444 | { | |
3445 | # return a copy of the exponent | |
ee15d750 | 3446 | my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_); |
58cde26e | 3447 | |
ee15d750 JH |
3448 | if ($x->{sign} !~ /^[+-]$/) |
3449 | { | |
3450 | my $s = $x->{sign}; $s =~ s/^[+-]//; | |
9b924220 | 3451 | return Math::BigInt->new($s); # -inf, +inf => +inf |
ee15d750 | 3452 | } |
9b924220 | 3453 | Math::BigInt->new( $x->{_es} . $MBI->_str($x->{_e})); |
58cde26e JH |
3454 | } |
3455 | ||
3456 | sub mantissa | |
3457 | { | |
3458 | # return a copy of the mantissa | |
ee15d750 | 3459 | my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_); |
58cde26e | 3460 | |
ee15d750 JH |
3461 | if ($x->{sign} !~ /^[+-]$/) |
3462 | { | |
3463 | my $s = $x->{sign}; $s =~ s/^[+]//; | |
9b924220 | 3464 | return Math::BigInt->new($s); # -inf, +inf => +inf |
ee15d750 | 3465 | } |
9b924220 | 3466 | my $m = Math::BigInt->new( $MBI->_str($x->{_m})); |
ee15d750 | 3467 | $m->bneg() if $x->{sign} eq '-'; |
58cde26e | 3468 | |
61f5c3f5 | 3469 | $m; |
58cde26e JH |
3470 | } |
3471 | ||
3472 | sub parts | |
3473 | { | |
3474 | # return a copy of both the exponent and the mantissa | |
ee15d750 | 3475 | my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_); |
58cde26e | 3476 | |
ee15d750 JH |
3477 | if ($x->{sign} !~ /^[+-]$/) |
3478 | { | |
3479 | my $s = $x->{sign}; $s =~ s/^[+]//; my $se = $s; $se =~ s/^[-]//; | |
3480 | return ($self->new($s),$self->new($se)); # +inf => inf and -inf,+inf => inf | |
3481 | } | |
9b924220 RGS |
3482 | my $m = Math::BigInt->bzero(); |
3483 | $m->{value} = $MBI->_copy($x->{_m}); | |
ee15d750 | 3484 | $m->bneg() if $x->{sign} eq '-'; |
9b924220 | 3485 | ($m, Math::BigInt->new( $x->{_es} . $MBI->_num($x->{_e}) )); |
58cde26e JH |
3486 | } |
3487 | ||
3488 | ############################################################################## | |
3489 | # private stuff (internal use only) | |
3490 | ||
58cde26e JH |
3491 | sub import |
3492 | { | |
3493 | my $self = shift; | |
8f675a64 JH |
3494 | my $l = scalar @_; |
3495 | my $lib = ''; my @a; | |
50109ad0 | 3496 | my $lib_kind = 'try'; |
990fb837 | 3497 | $IMPORT=1; |
8f675a64 | 3498 | for ( my $i = 0; $i < $l ; $i++) |
58cde26e JH |
3499 | { |
3500 | if ( $_[$i] eq ':constant' ) | |
3501 | { | |
091c87b1 T |
3502 | # This causes overlord er load to step in. 'binary' and 'integer' |
3503 | # are handled by BigInt. | |
58cde26e | 3504 | overload::constant float => sub { $self->new(shift); }; |
b3abae2a JH |
3505 | } |
3506 | elsif ($_[$i] eq 'upgrade') | |
3507 | { | |
3508 | # this causes upgrading | |
28df3e88 | 3509 | $upgrade = $_[$i+1]; # or undef to disable |
8f675a64 | 3510 | $i++; |
28df3e88 JH |
3511 | } |
3512 | elsif ($_[$i] eq 'downgrade') | |
3513 | { | |
3514 | # this causes downgrading | |
3515 | $downgrade = $_[$i+1]; # or undef to disable | |
8f675a64 | 3516 | $i++; |
58cde26e | 3517 | } |
50109ad0 | 3518 | elsif ($_[$i] =~ /^(lib|try|only)\z/) |
56b9c951 | 3519 | { |
990fb837 | 3520 | # alternative library |
56b9c951 | 3521 | $lib = $_[$i+1] || ''; # default Calc |
50109ad0 | 3522 | $lib_kind = $1; # lib, try or only |
8f675a64 | 3523 | $i++; |
56b9c951 JH |
3524 | } |
3525 | elsif ($_[$i] eq 'with') | |
3526 | { | |
990fb837 | 3527 | # alternative class for our private parts() |
9b924220 RGS |
3528 | # XXX: no longer supported |
3529 | # $MBI = $_[$i+1] || 'Math::BigInt'; | |
8f675a64 JH |
3530 | $i++; |
3531 | } | |
3532 | else | |
3533 | { | |
3534 | push @a, $_[$i]; | |
56b9c951 | 3535 | } |
58cde26e | 3536 | } |
8f675a64 | 3537 | |
b68b7ab1 | 3538 | $lib =~ tr/a-zA-Z0-9,://cd; # restrict to sane characters |
56b9c951 JH |
3539 | # let use Math::BigInt lib => 'GMP'; use Math::BigFloat; still work |
3540 | my $mbilib = eval { Math::BigInt->config()->{lib} }; | |
9b924220 | 3541 | if ((defined $mbilib) && ($MBI eq 'Math::BigInt::Calc')) |
8f675a64 JH |
3542 | { |
3543 | # MBI already loaded | |
50109ad0 | 3544 | Math::BigInt->import( $lib_kind, "$lib,$mbilib", 'objectify'); |
8f675a64 JH |
3545 | } |
3546 | else | |
3547 | { | |
9b924220 | 3548 | # MBI not loaded, or with ne "Math::BigInt::Calc" |
8f675a64 | 3549 | $lib .= ",$mbilib" if defined $mbilib; |
07d34614 | 3550 | $lib =~ s/^,//; # don't leave empty |
12fc2493 | 3551 | |
990fb837 | 3552 | # replacement library can handle lib statement, but also could ignore it |
12fc2493 AMS |
3553 | |
3554 | # Perl < 5.6.0 dies with "out of memory!" when eval() and ':constant' is | |
3555 | # used in the same script, or eval inside import(). So we require MBI: | |
3556 | require Math::BigInt; | |
50109ad0 | 3557 | Math::BigInt->import( $lib_kind => $lib, 'objectify' ); |
8f675a64 | 3558 | } |
990fb837 RGS |
3559 | if ($@) |
3560 | { | |
9b924220 | 3561 | require Carp; Carp::croak ("Couldn't load $lib: $! $@"); |
990fb837 | 3562 | } |
b68b7ab1 | 3563 | # find out which one was actually loaded |
9b924220 | 3564 | $MBI = Math::BigInt->config()->{lib}; |
56b9c951 | 3565 | |
b68b7ab1 T |
3566 | # register us with MBI to get notified of future lib changes |
3567 | Math::BigInt::_register_callback( $self, sub { $MBI = $_[0]; } ); | |
fdb4b05f T |
3568 | |
3569 | $self->export_to_level(1,$self,@a); # export wanted functions | |
58cde26e JH |
3570 | } |
3571 | ||
3572 | sub bnorm | |
3573 | { | |
3574 | # adjust m and e so that m is smallest possible | |
9b924220 | 3575 | my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_); |
58cde26e | 3576 | |
0716bf9b | 3577 | return $x if $x->{sign} !~ /^[+-]$/; # inf, nan etc |
58cde26e | 3578 | |
9b924220 | 3579 | my $zeros = $MBI->_zeros($x->{_m}); # correct for trailing zeros |
b282a552 T |
3580 | if ($zeros != 0) |
3581 | { | |
9b924220 RGS |
3582 | my $z = $MBI->_new($zeros); |
3583 | $x->{_m} = $MBI->_rsft ($x->{_m}, $z, 10); | |
3584 | if ($x->{_es} eq '-') | |
3585 | { | |
3586 | if ($MBI->_acmp($x->{_e},$z) >= 0) | |
3587 | { | |
80365507 | 3588 | $x->{_e} = $MBI->_sub ($x->{_e}, $z); |
27e7b8bb | 3589 | $x->{_es} = '+' if $MBI->_is_zero($x->{_e}); |
9b924220 RGS |
3590 | } |
3591 | else | |
3592 | { | |
80365507 | 3593 | $x->{_e} = $MBI->_sub ( $MBI->_copy($z), $x->{_e}); |
9b924220 RGS |
3594 | $x->{_es} = '+'; |
3595 | } | |
3596 | } | |
3597 | else | |
3598 | { | |
80365507 | 3599 | $x->{_e} = $MBI->_add ($x->{_e}, $z); |
9b924220 | 3600 | } |
b282a552 T |
3601 | } |
3602 | else | |
3603 | { | |
3604 | # $x can only be 0Ey if there are no trailing zeros ('0' has 0 trailing | |
3605 | # zeros). So, for something like 0Ey, set y to 1, and -0 => +0 | |
9b924220 RGS |
3606 | $x->{sign} = '+', $x->{_es} = '+', $x->{_e} = $MBI->_one() |
3607 | if $MBI->_is_zero($x->{_m}); | |
b282a552 T |
3608 | } |
3609 | ||
61f5c3f5 T |
3610 | $x; # MBI bnorm is no-op, so dont call it |
3611 | } | |
58cde26e JH |
3612 | |
3613 | ############################################################################## | |
56d9de68 T |
3614 | |
3615 | sub as_hex | |
3616 | { | |
3617 | # return number as hexadecimal string (only for integers defined) | |
3618 | my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_); | |
3619 | ||
3620 | return $x->bstr() if $x->{sign} !~ /^[+-]$/; # inf, nan etc | |
3621 | return '0x0' if $x->is_zero(); | |
3622 | ||
9b924220 | 3623 | return $nan if $x->{_es} ne '+'; # how to do 1e-1 in hex!? |
56d9de68 | 3624 | |
9b924220 RGS |
3625 | my $z = $MBI->_copy($x->{_m}); |
3626 | if (! $MBI->_is_zero($x->{_e})) # > 0 | |
56d9de68 | 3627 | { |
9b924220 | 3628 | $MBI->_lsft( $z, $x->{_e},10); |
56d9de68 | 3629 | } |
9b924220 | 3630 | $z = Math::BigInt->new( $x->{sign} . $MBI->_num($z)); |
56d9de68 T |
3631 | $z->as_hex(); |
3632 | } | |
3633 | ||
3634 | sub as_bin | |
3635 | { | |
3636 | # return number as binary digit string (only for integers defined) | |
3637 | my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_); | |
3638 | ||
3639 | return $x->bstr() if $x->{sign} !~ /^[+-]$/; # inf, nan etc | |
3640 | return '0b0' if $x->is_zero(); | |
3641 | ||
9b924220 | 3642 | return $nan if $x->{_es} ne '+'; # how to do 1e-1 in hex!? |
56d9de68 | 3643 | |
9b924220 RGS |
3644 | my $z = $MBI->_copy($x->{_m}); |
3645 | if (! $MBI->_is_zero($x->{_e})) # > 0 | |
56d9de68 | 3646 | { |
9b924220 | 3647 | $MBI->_lsft( $z, $x->{_e},10); |
56d9de68 | 3648 | } |
9b924220 | 3649 | $z = Math::BigInt->new( $x->{sign} . $MBI->_num($z)); |
56d9de68 T |
3650 | $z->as_bin(); |
3651 | } | |
58cde26e | 3652 | |
7b29e1e6 T |
3653 | sub as_oct |
3654 | { | |
3655 | # return number as octal digit string (only for integers defined) | |
3656 | my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_); | |
3657 | ||
3658 | return $x->bstr() if $x->{sign} !~ /^[+-]$/; # inf, nan etc | |
3659 | return '0' if $x->is_zero(); | |
3660 | ||
3661 | return $nan if $x->{_es} ne '+'; # how to do 1e-1 in hex!? | |
3662 | ||
3663 | my $z = $MBI->_copy($x->{_m}); | |
3664 | if (! $MBI->_is_zero($x->{_e})) # > 0 | |
3665 | { | |
3666 | $MBI->_lsft( $z, $x->{_e},10); | |
3667 | } | |
3668 | $z = Math::BigInt->new( $x->{sign} . $MBI->_num($z)); | |
3669 | $z->as_oct(); | |
3670 | } | |
3671 | ||
58cde26e JH |
3672 | sub as_number |
3673 | { | |
394e6ffb JH |
3674 | # return copy as a bigint representation of this BigFloat number |
3675 | my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_); | |
58cde26e | 3676 | |
50109ad0 RGS |
3677 | return $x if $x->modify('as_number'); |
3678 | ||
0dceeee6 RGS |
3679 | if (!$x->isa('Math::BigFloat')) |
3680 | { | |
3681 | # if the object can as_number(), use it | |
3682 | return $x->as_number() if $x->can('as_number'); | |
3683 | # otherwise, get us a float and then a number | |
3684 | $x = $x->can('as_float') ? $x->as_float() : $self->new(0+"$x"); | |
3685 | } | |
3686 | ||
3d6017f5 PJA |
3687 | return Math::BigInt->binf($x->sign()) if $x->is_inf(); |
3688 | return Math::BigInt->bnan() if $x->is_nan(); | |
3689 | ||
9b924220 RGS |
3690 | my $z = $MBI->_copy($x->{_m}); |
3691 | if ($x->{_es} eq '-') # < 0 | |
58cde26e | 3692 | { |
9b924220 | 3693 | $MBI->_rsft( $z, $x->{_e},10); |
0716bf9b | 3694 | } |
9b924220 | 3695 | elsif (! $MBI->_is_zero($x->{_e})) # > 0 |
0716bf9b | 3696 | { |
9b924220 | 3697 | $MBI->_lsft( $z, $x->{_e},10); |
58cde26e | 3698 | } |
26c6d654 | 3699 | $z = Math::BigInt->new( $x->{sign} . $MBI->_str($z)); |
61f5c3f5 | 3700 | $z; |
58cde26e JH |
3701 | } |
3702 | ||
3703 | sub length | |
3704 | { | |
ee15d750 JH |
3705 | my $x = shift; |
3706 | my $class = ref($x) || $x; | |
3707 | $x = $class->new(shift) unless ref($x); | |
58cde26e | 3708 | |
9b924220 RGS |
3709 | return 1 if $MBI->_is_zero($x->{_m}); |
3710 | ||
3711 | my $len = $MBI->_len($x->{_m}); | |
3712 | $len += $MBI->_num($x->{_e}) if $x->{_es} eq '+'; | |
58cde26e JH |
3713 | if (wantarray()) |
3714 | { | |
9b924220 RGS |
3715 | my $t = 0; |
3716 | $t = $MBI->_num($x->{_e}) if $x->{_es} eq '-'; | |
3717 | return ($len, $t); | |
58cde26e | 3718 | } |
61f5c3f5 | 3719 | $len; |
58cde26e | 3720 | } |
a0d0e21e LW |
3721 | |
3722 | 1; | |
a5f75d66 AD |
3723 | __END__ |
3724 | ||
3725 | =head1 NAME | |
3726 | ||
58cde26e | 3727 | Math::BigFloat - Arbitrary size floating point math package |
a5f75d66 AD |
3728 | |
3729 | =head1 SYNOPSIS | |
3730 | ||
a2008d6d | 3731 | use Math::BigFloat; |
58cde26e | 3732 | |
b3abae2a | 3733 | # Number creation |
fdb4b05f T |
3734 | my $x = Math::BigFloat->new($str); # defaults to 0 |
3735 | my $y = $x->copy(); # make a true copy | |
3736 | my $nan = Math::BigFloat->bnan(); # create a NotANumber | |
3737 | my $zero = Math::BigFloat->bzero(); # create a +0 | |
3738 | my $inf = Math::BigFloat->binf(); # create a +inf | |
3739 | my $inf = Math::BigFloat->binf('-'); # create a -inf | |
3740 | my $one = Math::BigFloat->bone(); # create a +1 | |
3741 | my $mone = Math::BigFloat->bone('-'); # create a -1 | |
3742 | ||
3743 | my $pi = Math::BigFloat->bpi(100); # PI to 100 digits | |
58cde26e | 3744 | |
60a1aa19 T |
3745 | # the following examples compute their result to 100 digits accuracy: |
3746 | my $cos = Math::BigFloat->new(1)->bcos(100); # cosinus(1) | |
3747 | my $sin = Math::BigFloat->new(1)->bsin(100); # sinus(1) | |
3748 | my $atan = Math::BigFloat->new(1)->batan(100); # arcus tangens(1) | |
3749 | ||
30afc38d T |
3750 | my $atan2 = Math::BigFloat->new( 1 )->batan2( 1 ,100); # batan(1) |
3751 | my $atan2 = Math::BigFloat->new( 1 )->batan2( 8 ,100); # batan(1/8) | |
3752 | my $atan2 = Math::BigFloat->new( -2 )->batan2( 1 ,100); # batan(-2) | |
3753 | ||
58cde26e | 3754 | # Testing |
b3abae2a JH |
3755 | $x->is_zero(); # true if arg is +0 |
3756 | $x->is_nan(); # true if arg is NaN | |
0716bf9b JH |
3757 | $x->is_one(); # true if arg is +1 |
3758 | $x->is_one('-'); # true if arg is -1 | |
3759 | $x->is_odd(); # true if odd, false for even | |
3760 | $x->is_even(); # true if even, false for odd | |
ef9466ea T |
3761 | $x->is_pos(); # true if >= 0 |
3762 | $x->is_neg(); # true if < 0 | |
b3abae2a JH |
3763 | $x->is_inf(sign); # true if +inf, or -inf (default is '+') |
3764 | ||
58cde26e JH |
3765 | $x->bcmp($y); # compare numbers (undef,<0,=0,>0) |
3766 | $x->bacmp($y); # compare absolutely (undef,<0,=0,>0) | |
3767 | $x->sign(); # return the sign, either +,- or NaN | |
b3abae2a JH |
3768 | $x->digit($n); # return the nth digit, counting from right |
3769 | $x->digit(-$n); # return the nth digit, counting from left | |
58cde26e | 3770 | |
990fb837 RGS |
3771 | # The following all modify their first argument. If you want to preserve |
3772 | # $x, use $z = $x->copy()->bXXX($y); See under L<CAVEATS> for why this is | |
3c4b39be | 3773 | # necessary when mixing $a = $b assignments with non-overloaded math. |
990fb837 | 3774 | |
58cde26e JH |
3775 | # set |
3776 | $x->bzero(); # set $i to 0 | |
3777 | $x->bnan(); # set $i to NaN | |
b3abae2a JH |
3778 | $x->bone(); # set $x to +1 |
3779 | $x->bone('-'); # set $x to -1 | |
3780 | $x->binf(); # set $x to inf | |
3781 | $x->binf('-'); # set $x to -inf | |
58cde26e JH |
3782 | |
3783 | $x->bneg(); # negation | |
3784 | $x->babs(); # absolute value | |
3785 | $x->bnorm(); # normalize (no-op) | |
3786 | $x->bnot(); # two's complement (bit wise not) | |
3787 | $x->binc(); # increment x by 1 | |
3788 | $x->bdec(); # decrement x by 1 | |
3789 | ||
3790 | $x->badd($y); # addition (add $y to $x) | |
3791 | $x->bsub($y); # subtraction (subtract $y from $x) | |
3792 | $x->bmul($y); # multiplication (multiply $x by $y) | |
990fb837 | 3793 | $x->bdiv($y); # divide, set $x to quotient |
58cde26e JH |
3794 | # return (quo,rem) or quo if scalar |
3795 | ||
990fb837 RGS |
3796 | $x->bmod($y); # modulus ($x % $y) |
3797 | $x->bpow($y); # power of arguments ($x ** $y) | |
80365507 | 3798 | $x->bmodpow($exp,$mod); # modular exponentation (($num**$exp) % $mod)) |
7b29e1e6 T |
3799 | $x->blsft($y, $n); # left shift by $y places in base $n |
3800 | $x->brsft($y, $n); # right shift by $y places in base $n | |
3801 | # returns (quo,rem) or quo if in scalar context | |
58cde26e | 3802 | |
990fb837 RGS |
3803 | $x->blog(); # logarithm of $x to base e (Euler's number) |
3804 | $x->blog($base); # logarithm of $x to base $base (f.i. 2) | |
7d193e39 | 3805 | $x->bexp(); # calculate e ** $x where e is Euler's number |
61f5c3f5 | 3806 | |
58cde26e JH |
3807 | $x->band($y); # bit-wise and |
3808 | $x->bior($y); # bit-wise inclusive or | |
3809 | $x->bxor($y); # bit-wise exclusive or | |
3810 | $x->bnot(); # bit-wise not (two's complement) | |
b3abae2a JH |
3811 | |
3812 | $x->bsqrt(); # calculate square-root | |
990fb837 | 3813 | $x->broot($y); # $y'th root of $x (e.g. $y == 3 => cubic root) |
b3abae2a JH |
3814 | $x->bfac(); # factorial of $x (1*2*3*4*..$x) |
3815 | ||
990fb837 | 3816 | $x->bround($N); # accuracy: preserve $N digits |
58cde26e JH |
3817 | $x->bfround($N); # precision: round to the $Nth digit |
3818 | ||
990fb837 RGS |
3819 | $x->bfloor(); # return integer less or equal than $x |
3820 | $x->bceil(); # return integer greater or equal than $x | |
3821 | ||
58cde26e | 3822 | # The following do not modify their arguments: |
990fb837 | 3823 | |
58cde26e JH |
3824 | bgcd(@values); # greatest common divisor |
3825 | blcm(@values); # lowest common multiplicator | |
3826 | ||
3827 | $x->bstr(); # return string | |
3828 | $x->bsstr(); # return string in scientific notation | |
ef9466ea T |
3829 | |
3830 | $x->as_int(); # return $x as BigInt | |
58cde26e JH |
3831 | $x->exponent(); # return exponent as BigInt |
3832 | $x->mantissa(); # return mantissa as BigInt | |
3833 | $x->parts(); # return (mantissa,exponent) as BigInt | |
3834 | ||
3835 | $x->length(); # number of digits (w/o sign and '.') | |
3836 | ($l,$f) = $x->length(); # number of digits, and length of fraction | |
a5f75d66 | 3837 | |
f9a08e12 JH |
3838 | $x->precision(); # return P of $x (or global, if P of $x undef) |
3839 | $x->precision($n); # set P of $x to $n | |
3840 | $x->accuracy(); # return A of $x (or global, if A of $x undef) | |
723d369b | 3841 | $x->accuracy($n); # set A $x to $n |
f9a08e12 | 3842 | |
990fb837 RGS |
3843 | # these get/set the appropriate global value for all BigFloat objects |
3844 | Math::BigFloat->precision(); # Precision | |
3845 | Math::BigFloat->accuracy(); # Accuracy | |
3846 | Math::BigFloat->round_mode(); # rounding mode | |
f9a08e12 | 3847 | |
a5f75d66 AD |
3848 | =head1 DESCRIPTION |
3849 | ||
3c4b39be | 3850 | All operators (including basic math operations) are overloaded if you |
58cde26e | 3851 | declare your big floating point numbers as |
a5f75d66 | 3852 | |
58cde26e JH |
3853 | $i = new Math::BigFloat '12_3.456_789_123_456_789E-2'; |
3854 | ||
3855 | Operations with overloaded operators preserve the arguments, which is | |
3856 | exactly what you expect. | |
3857 | ||
3858 | =head2 Canonical notation | |
3859 | ||
3860 | Input to these routines are either BigFloat objects, or strings of the | |
3861 | following four forms: | |
a5f75d66 AD |
3862 | |
3863 | =over 2 | |
3864 | ||
58cde26e JH |
3865 | =item * |
3866 | ||
3867 | C</^[+-]\d+$/> | |
a5f75d66 | 3868 | |
58cde26e | 3869 | =item * |
a5f75d66 | 3870 | |
58cde26e | 3871 | C</^[+-]\d+\.\d*$/> |
a5f75d66 | 3872 | |
58cde26e | 3873 | =item * |
a5f75d66 | 3874 | |
58cde26e | 3875 | C</^[+-]\d+E[+-]?\d+$/> |
a5f75d66 | 3876 | |
58cde26e | 3877 | =item * |
a5f75d66 | 3878 | |
58cde26e | 3879 | C</^[+-]\d*\.\d+E[+-]?\d+$/> |
5d7098d5 | 3880 | |
58cde26e JH |
3881 | =back |
3882 | ||
3c4b39be | 3883 | all with optional leading and trailing zeros and/or spaces. Additionally, |
58cde26e JH |
3884 | numbers are allowed to have an underscore between any two digits. |
3885 | ||
3886 | Empty strings as well as other illegal numbers results in 'NaN'. | |
3887 | ||
3888 | bnorm() on a BigFloat object is now effectively a no-op, since the numbers | |
3889 | are always stored in normalized form. On a string, it creates a BigFloat | |
3890 | object. | |
3891 | ||
3892 | =head2 Output | |
3893 | ||
3894 | Output values are BigFloat objects (normalized), except for bstr() and bsstr(). | |
3895 | ||
3896 | The string output will always have leading and trailing zeros stripped and drop | |
3897 | a plus sign. C<bstr()> will give you always the form with a decimal point, | |
990fb837 | 3898 | while C<bsstr()> (s for scientific) gives you the scientific notation. |
58cde26e JH |
3899 | |
3900 | Input bstr() bsstr() | |
3901 | '-0' '0' '0E1' | |
3902 | ' -123 123 123' '-123123123' '-123123123E0' | |
3903 | '00.0123' '0.0123' '123E-4' | |
3904 | '123.45E-2' '1.2345' '12345E-4' | |
3905 | '10E+3' '10000' '1E4' | |
3906 | ||
3907 | Some routines (C<is_odd()>, C<is_even()>, C<is_zero()>, C<is_one()>, | |
3908 | C<is_nan()>) return true or false, while others (C<bcmp()>, C<bacmp()>) | |
3909 | return either undef, <0, 0 or >0 and are suited for sort. | |
3910 | ||
fd9ea5b0 | 3911 | Actual math is done by using the class defined with C<< with => Class; >> (which |
990fb837 RGS |
3912 | defaults to BigInts) to represent the mantissa and exponent. |
3913 | ||
58cde26e JH |
3914 | The sign C</^[+-]$/> is stored separately. The string 'NaN' is used to |
3915 | represent the result when input arguments are not numbers, as well as | |
3916 | the result of dividing by zero. | |
3917 | ||
3918 | =head2 C<mantissa()>, C<exponent()> and C<parts()> | |
3919 | ||
3920 | C<mantissa()> and C<exponent()> return the said parts of the BigFloat | |
3921 | as BigInts such that: | |
3922 | ||
3923 | $m = $x->mantissa(); | |
3924 | $e = $x->exponent(); | |
3925 | $y = $m * ( 10 ** $e ); | |
3926 | print "ok\n" if $x == $y; | |
3927 | ||
3928 | C<< ($m,$e) = $x->parts(); >> is just a shortcut giving you both of them. | |
3929 | ||
3930 | A zero is represented and returned as C<0E1>, B<not> C<0E0> (after Knuth). | |
3931 | ||
3932 | Currently the mantissa is reduced as much as possible, favouring higher | |
3933 | exponents over lower ones (e.g. returning 1e7 instead of 10e6 or 10000000e0). | |
3934 | This might change in the future, so do not depend on it. | |
3935 | ||
3936 | =head2 Accuracy vs. Precision | |
3937 | ||
3938 | See also: L<Rounding|Rounding>. | |
3939 | ||
233f7bc0 T |
3940 | Math::BigFloat supports both precision (rounding to a certain place before or |
3941 | after the dot) and accuracy (rounding to a certain number of digits). For a | |
3942 | full documentation, examples and tips on these topics please see the large | |
3943 | section about rounding in L<Math::BigInt>. | |
5d7098d5 | 3944 | |
233f7bc0 T |
3945 | Since things like C<sqrt(2)> or C<1 / 3> must presented with a limited |
3946 | accuracy lest a operation consumes all resources, each operation produces | |
3947 | no more than the requested number of digits. | |
990fb837 | 3948 | |
9681bfa6 | 3949 | If there is no global precision or accuracy set, B<and> the operation in |
233f7bc0 T |
3950 | question was not called with a requested precision or accuracy, B<and> the |
3951 | input $x has no accuracy or precision set, then a fallback parameter will | |
3952 | be used. For historical reasons, it is called C<div_scale> and can be accessed | |
3953 | via: | |
990fb837 RGS |
3954 | |
3955 | $d = Math::BigFloat->div_scale(); # query | |
3956 | Math::BigFloat->div_scale($n); # set to $n digits | |
3957 | ||
233f7bc0 | 3958 | The default value for C<div_scale> is 40. |
58cde26e | 3959 | |
233f7bc0 | 3960 | In case the result of one operation has more digits than specified, |
58cde26e JH |
3961 | it is rounded. The rounding mode taken is either the default mode, or the one |
3962 | supplied to the operation after the I<scale>: | |
3963 | ||
3964 | $x = Math::BigFloat->new(2); | |
a87115f0 RGS |
3965 | Math::BigFloat->accuracy(5); # 5 digits max |
3966 | $y = $x->copy()->bdiv(3); # will give 0.66667 | |
3967 | $y = $x->copy()->bdiv(3,6); # will give 0.666667 | |
3968 | $y = $x->copy()->bdiv(3,6,undef,'odd'); # will give 0.666667 | |
990fb837 | 3969 | Math::BigFloat->round_mode('zero'); |
a87115f0 RGS |
3970 | $y = $x->copy()->bdiv(3,6); # will also give 0.666667 |
3971 | ||
3972 | Note that C<< Math::BigFloat->accuracy() >> and C<< Math::BigFloat->precision() >> | |
3973 | set the global variables, and thus B<any> newly created number will be subject | |
3c4b39be | 3974 | to the global rounding B<immediately>. This means that in the examples above, the |
233f7bc0 | 3975 | C<3> as argument to C<bdiv()> will also get an accuracy of B<5>. |
a87115f0 RGS |
3976 | |
3977 | It is less confusing to either calculate the result fully, and afterwards | |
3c4b39be | 3978 | round it explicitly, or use the additional parameters to the math |
a87115f0 RGS |
3979 | functions like so: |
3980 | ||
3981 | use Math::BigFloat; | |
3982 | $x = Math::BigFloat->new(2); | |
3983 | $y = $x->copy()->bdiv(3); | |
3984 | print $y->bround(5),"\n"; # will give 0.66667 | |
3985 | ||
3986 | or | |
3987 | ||
3988 | use Math::BigFloat; | |
3989 | $x = Math::BigFloat->new(2); | |
3990 | $y = $x->copy()->bdiv(3,5); # will give 0.66667 | |
3991 | print "$y\n"; | |
58cde26e JH |
3992 | |
3993 | =head2 Rounding | |
3994 | ||
3995 | =over 2 | |
3996 | ||
5dc6f178 | 3997 | =item ffround ( +$scale ) |
58cde26e | 3998 | |
0716bf9b JH |
3999 | Rounds to the $scale'th place left from the '.', counting from the dot. |
4000 | The first digit is numbered 1. | |
58cde26e | 4001 | |
5dc6f178 | 4002 | =item ffround ( -$scale ) |
58cde26e | 4003 | |
0716bf9b | 4004 | Rounds to the $scale'th place right from the '.', counting from the dot. |
58cde26e | 4005 | |
5dc6f178 JH |
4006 | =item ffround ( 0 ) |
4007 | ||
0716bf9b | 4008 | Rounds to an integer. |
5dc6f178 JH |
4009 | |
4010 | =item fround ( +$scale ) | |
4011 | ||
0716bf9b JH |
4012 | Preserves accuracy to $scale digits from the left (aka significant digits) |
4013 | and pads the rest with zeros. If the number is between 1 and -1, the | |
4014 | significant digits count from the first non-zero after the '.' | |
5dc6f178 JH |
4015 | |
4016 | =item fround ( -$scale ) and fround ( 0 ) | |
4017 | ||
990fb837 | 4018 | These are effectively no-ops. |
5d7098d5 | 4019 | |
a5f75d66 AD |
4020 | =back |
4021 | ||
0716bf9b | 4022 | All rounding functions take as a second parameter a rounding mode from one of |
86b76201 | 4023 | the following: 'even', 'odd', '+inf', '-inf', 'zero', 'trunc' or 'common'. |
58cde26e JH |
4024 | |
4025 | The default rounding mode is 'even'. By using | |
990fb837 | 4026 | C<< Math::BigFloat->round_mode($round_mode); >> you can get and set the default |
ee15d750 | 4027 | mode for subsequent rounding. The usage of C<$Math::BigFloat::$round_mode> is |
0716bf9b | 4028 | no longer supported. |
b22b3e31 | 4029 | The second parameter to the round functions then overrides the default |
0716bf9b | 4030 | temporarily. |
58cde26e | 4031 | |
990fb837 | 4032 | The C<as_number()> function returns a BigInt from a Math::BigFloat. It uses |
58cde26e JH |
4033 | 'trunc' as rounding mode to make it equivalent to: |
4034 | ||
4035 | $x = 2.5; | |
4036 | $y = int($x) + 2; | |
4037 | ||
4038 | You can override this by passing the desired rounding mode as parameter to | |
4039 | C<as_number()>: | |
4040 | ||
4041 | $x = Math::BigFloat->new(2.5); | |
4042 | $y = $x->as_number('odd'); # $y = 3 | |
4043 | ||
233f7bc0 T |
4044 | =head1 METHODS |
4045 | ||
86b76201 T |
4046 | Math::BigFloat supports all methods that Math::BigInt supports, except it |
4047 | calculates non-integer results when possible. Please see L<Math::BigInt> | |
4048 | for a full description of each method. Below are just the most important | |
4049 | differences: | |
4050 | ||
233f7bc0 T |
4051 | =head2 accuracy |
4052 | ||
4053 | $x->accuracy(5); # local for $x | |
4054 | CLASS->accuracy(5); # global for all members of CLASS | |
4055 | # Note: This also applies to new()! | |
4056 | ||
4057 | $A = $x->accuracy(); # read out accuracy that affects $x | |
4058 | $A = CLASS->accuracy(); # read out global accuracy | |
4059 | ||
4060 | Set or get the global or local accuracy, aka how many significant digits the | |
4061 | results have. If you set a global accuracy, then this also applies to new()! | |
4062 | ||
4063 | Warning! The accuracy I<sticks>, e.g. once you created a number under the | |
4064 | influence of C<< CLASS->accuracy($A) >>, all results from math operations with | |
4065 | that number will also be rounded. | |
4066 | ||
3c4b39be | 4067 | In most cases, you should probably round the results explicitly using one of |
233f7bc0 T |
4068 | L<round()>, L<bround()> or L<bfround()> or by passing the desired accuracy |
4069 | to the math operation as additional parameter: | |
4070 | ||
4071 | my $x = Math::BigInt->new(30000); | |
4072 | my $y = Math::BigInt->new(7); | |
4073 | print scalar $x->copy()->bdiv($y, 2); # print 4300 | |
4074 | print scalar $x->copy()->bdiv($y)->bround(2); # print 4300 | |
4075 | ||
4076 | =head2 precision() | |
4077 | ||
4078 | $x->precision(-2); # local for $x, round at the second digit right of the dot | |
4079 | $x->precision(2); # ditto, round at the second digit left of the dot | |
4080 | ||
4081 | CLASS->precision(5); # Global for all members of CLASS | |
4082 | # This also applies to new()! | |
4083 | CLASS->precision(-5); # ditto | |
4084 | ||
4085 | $P = CLASS->precision(); # read out global precision | |
4086 | $P = $x->precision(); # read out precision that affects $x | |
4087 | ||
4088 | Note: You probably want to use L<accuracy()> instead. With L<accuracy> you | |
4089 | set the number of digits each result should have, with L<precision> you | |
4090 | set the place where to round! | |
58cde26e | 4091 | |
86b76201 T |
4092 | =head2 bexp() |
4093 | ||
4094 | $x->bexp($accuracy); # calculate e ** X | |
4095 | ||
4096 | Calculates the expression C<e ** $x> where C<e> is Euler's number. | |
4097 | ||
4098 | This method was added in v1.82 of Math::BigInt (April 2007). | |
4099 | ||
4100 | =head2 bnok() | |
4101 | ||
4102 | $x->bnok($y); # x over y (binomial coefficient n over k) | |
4103 | ||
4104 | Calculates the binomial coefficient n over k, also called the "choose" | |
4105 | function. The result is equivalent to: | |
4106 | ||
4107 | ( n ) n! | |
4108 | | - | = ------- | |
4109 | ( k ) k!(n-k)! | |
4110 | ||
4111 | This method was added in v1.84 of Math::BigInt (April 2007). | |
4112 | ||
fdb4b05f T |
4113 | =head2 bpi() |
4114 | ||
4115 | print Math::BigFloat->bpi(100), "\n"; | |
4116 | ||
20e2035c T |
4117 | Calculate PI to N digits (including the 3 before the dot). The result is |
4118 | rounded according to the current rounding mode, which defaults to "even". | |
fdb4b05f T |
4119 | |
4120 | This method was added in v1.87 of Math::BigInt (June 2007). | |
4121 | ||
60a1aa19 T |
4122 | =head2 bcos() |
4123 | ||
4124 | my $x = Math::BigFloat->new(1); | |
4125 | print $x->bcos(100), "\n"; | |
4126 | ||
4127 | Calculate the cosinus of $x, modifying $x in place. | |
4128 | ||
4129 | This method was added in v1.87 of Math::BigInt (June 2007). | |
4130 | ||
4131 | =head2 bsin() | |
4132 | ||
4133 | my $x = Math::BigFloat->new(1); | |
4134 | print $x->bsin(100), "\n"; | |
4135 | ||
4136 | Calculate the sinus of $x, modifying $x in place. | |
4137 | ||
4138 | This method was added in v1.87 of Math::BigInt (June 2007). | |
4139 | ||
30afc38d | 4140 | =head2 batan2() |
60a1aa19 | 4141 | |
30afc38d T |
4142 | my $y = Math::BigFloat->new(2); |
4143 | my $x = Math::BigFloat->new(3); | |
4144 | print $y->batan2($x), "\n"; | |
60a1aa19 | 4145 | |
30afc38d T |
4146 | Calculate the arcus tanges of C<$y> divided by C<$x>, modifying $y in place. |
4147 | See also L<batan()>. | |
20e2035c T |
4148 | |
4149 | This method was added in v1.87 of Math::BigInt (June 2007). | |
4150 | ||
30afc38d | 4151 | =head2 batan() |
20e2035c | 4152 | |
30afc38d T |
4153 | my $x = Math::BigFloat->new(1); |
4154 | print $x->batan(100), "\n"; | |
20e2035c | 4155 | |
30afc38d | 4156 | Calculate the arcus tanges of $x, modifying $x in place. See also L<batan2()>. |
60a1aa19 T |
4157 | |
4158 | This method was added in v1.87 of Math::BigInt (June 2007). | |
4159 | ||
80365507 T |
4160 | =head2 bmuladd() |
4161 | ||
4162 | $x->bmuladd($y,$z); | |
4163 | ||
4164 | Multiply $x by $y, and then add $z to the result. | |
4165 | ||
4166 | This method was added in v1.87 of Math::BigInt (June 2007). | |
4167 | ||
58cde26e JH |
4168 | =head1 Autocreating constants |
4169 | ||
4170 | After C<use Math::BigFloat ':constant'> all the floating point constants | |
4171 | in the given scope are converted to C<Math::BigFloat>. This conversion | |
4172 | happens at compile time. | |
4173 | ||
4174 | In particular | |
4175 | ||
4176 | perl -MMath::BigFloat=:constant -e 'print 2E-100,"\n"' | |
4177 | ||
56b9c951 | 4178 | prints the value of C<2E-100>. Note that without conversion of |
58cde26e JH |
4179 | constants the expression 2E-100 will be calculated as normal floating point |
4180 | number. | |
4181 | ||
56b9c951 JH |
4182 | Please note that ':constant' does not affect integer constants, nor binary |
4183 | nor hexadecimal constants. Use L<bignum> or L<Math::BigInt> to get this to | |
4184 | work. | |
4185 | ||
4186 | =head2 Math library | |
4187 | ||
4188 | Math with the numbers is done (by default) by a module called | |
4189 | Math::BigInt::Calc. This is equivalent to saying: | |
4190 | ||
4191 | use Math::BigFloat lib => 'Calc'; | |
4192 | ||
4193 | You can change this by using: | |
4194 | ||
86b76201 | 4195 | use Math::BigFloat lib => 'GMP'; |
56b9c951 | 4196 | |
0dceeee6 RGS |
4197 | B<Note>: General purpose packages should not be explicit about the library |
4198 | to use; let the script author decide which is best. | |
4199 | ||
86f0d17a T |
4200 | Note: The keyword 'lib' will warn when the requested library could not be |
4201 | loaded. To suppress the warning use 'try' instead: | |
4202 | ||
4203 | use Math::BigFloat try => 'GMP'; | |
4204 | ||
0dceeee6 RGS |
4205 | If your script works with huge numbers and Calc is too slow for them, |
4206 | you can also for the loading of one of these libraries and if none | |
4207 | of them can be used, the code will die: | |
86f0d17a | 4208 | |
0dceeee6 | 4209 | use Math::BigFloat only => 'GMP,Pari'; |
86f0d17a | 4210 | |
56b9c951 JH |
4211 | The following would first try to find Math::BigInt::Foo, then |
4212 | Math::BigInt::Bar, and when this also fails, revert to Math::BigInt::Calc: | |
4213 | ||
4214 | use Math::BigFloat lib => 'Foo,Math::BigInt::Bar'; | |
4215 | ||
86b76201 | 4216 | See the respective low-level library documentation for further details. |
56b9c951 JH |
4217 | |
4218 | Please note that Math::BigFloat does B<not> use the denoted library itself, | |
4219 | but it merely passes the lib argument to Math::BigInt. So, instead of the need | |
4220 | to do: | |
4221 | ||
4222 | use Math::BigInt lib => 'GMP'; | |
4223 | use Math::BigFloat; | |
4224 | ||
4225 | you can roll it all into one line: | |
4226 | ||
4227 | use Math::BigFloat lib => 'GMP'; | |
4228 | ||
990fb837 RGS |
4229 | It is also possible to just require Math::BigFloat: |
4230 | ||
4231 | require Math::BigFloat; | |
4232 | ||
3c4b39be | 4233 | This will load the necessary things (like BigInt) when they are needed, and |
990fb837 RGS |
4234 | automatically. |
4235 | ||
86b76201 T |
4236 | See L<Math::BigInt> for more details than you ever wanted to know about using |
4237 | a different low-level library. | |
56b9c951 JH |
4238 | |
4239 | =head2 Using Math::BigInt::Lite | |
4240 | ||
86b76201 T |
4241 | For backwards compatibility reasons it is still possible to |
4242 | request a different storage class for use with Math::BigFloat: | |
56b9c951 | 4243 | |
56b9c951 JH |
4244 | use Math::BigFloat with => 'Math::BigInt::Lite'; |
4245 | ||
86b76201 | 4246 | However, this request is ignored, as the current code now uses the low-level |
df0693ed | 4247 | math library for directly storing the number parts. |
56b9c951 | 4248 | |
fdb4b05f T |
4249 | =head1 EXPORTS |
4250 | ||
4251 | C<Math::BigFloat> exports nothing by default, but can export the C<bpi()> method: | |
4252 | ||
4253 | use Math::BigFloat qw/bpi/; | |
4254 | ||
4255 | print bpi(10), "\n"; | |
4256 | ||
86b76201 | 4257 | =head1 BUGS |
58cde26e | 4258 | |
86b76201 | 4259 | Please see the file BUGS in the CPAN distribution Math::BigInt for known bugs. |
58cde26e | 4260 | |
86b76201 | 4261 | =head1 CAVEATS |
990fb837 | 4262 | |
86b76201 T |
4263 | Do not try to be clever to insert some operations in between switching |
4264 | libraries: | |
990fb837 RGS |
4265 | |
4266 | require Math::BigFloat; | |
86b76201 | 4267 | my $matter = Math::BigFloat->bone() + 4; # load BigInt and Calc |
990fb837 | 4268 | Math::BigFloat->import( lib => 'Pari' ); # load Pari, too |
86f0d17a | 4269 | my $anti_matter = Math::BigFloat->bone()+4; # now use Pari |
990fb837 | 4270 | |
86b76201 T |
4271 | This will create objects with numbers stored in two different backend libraries, |
4272 | and B<VERY BAD THINGS> will happen when you use these together: | |
990fb837 | 4273 | |
86b76201 | 4274 | my $flash_and_bang = $matter + $anti_matter; # Don't do this! |
58cde26e JH |
4275 | |
4276 | =over 1 | |
4277 | ||
4278 | =item stringify, bstr() | |
4279 | ||
4280 | Both stringify and bstr() now drop the leading '+'. The old code would return | |
4281 | '+1.23', the new returns '1.23'. See the documentation in L<Math::BigInt> for | |
4282 | reasoning and details. | |
4283 | ||
4284 | =item bdiv | |
4285 | ||
7b29e1e6 | 4286 | The following will probably not print what you expect: |
58cde26e JH |
4287 | |
4288 | print $c->bdiv(123.456),"\n"; | |
4289 | ||
4290 | It prints both quotient and reminder since print works in list context. Also, | |
3c4b39be | 4291 | bdiv() will modify $c, so be careful. You probably want to use |
58cde26e JH |
4292 | |
4293 | print $c / 123.456,"\n"; | |
4294 | print scalar $c->bdiv(123.456),"\n"; # or if you want to modify $c | |
4295 | ||
4296 | instead. | |
4297 | ||
7b29e1e6 T |
4298 | =item brsft |
4299 | ||
4300 | The following will probably not print what you expect: | |
4301 | ||
4302 | my $c = Math::BigFloat->new('3.14159'); | |
4303 | print $c->brsft(3,10),"\n"; # prints 0.00314153.1415 | |
4304 | ||
4305 | It prints both quotient and remainder, since print calls C<brsft()> in list | |
4306 | context. Also, C<< $c->brsft() >> will modify $c, so be careful. | |
4307 | You probably want to use | |
4308 | ||
4309 | print scalar $c->copy()->brsft(3,10),"\n"; | |
4310 | # or if you really want to modify $c | |
4311 | print scalar $c->brsft(3,10),"\n"; | |
4312 | ||
4313 | instead. | |
4314 | ||
58cde26e JH |
4315 | =item Modifying and = |
4316 | ||
4317 | Beware of: | |
4318 | ||
4319 | $x = Math::BigFloat->new(5); | |
4320 | $y = $x; | |
4321 | ||
4322 | It will not do what you think, e.g. making a copy of $x. Instead it just makes | |
4323 | a second reference to the B<same> object and stores it in $y. Thus anything | |
990fb837 RGS |
4324 | that modifies $x will modify $y (except overloaded math operators), and vice |
4325 | versa. See L<Math::BigInt> for details and how to avoid that. | |
58cde26e JH |
4326 | |
4327 | =item bpow | |
4328 | ||
4329 | C<bpow()> now modifies the first argument, unlike the old code which left | |
4330 | it alone and only returned the result. This is to be consistent with | |
4331 | C<badd()> etc. The first will modify $x, the second one won't: | |
4332 | ||
4333 | print bpow($x,$i),"\n"; # modify $x | |
4334 | print $x->bpow($i),"\n"; # ditto | |
4335 | print $x ** $i,"\n"; # leave $x alone | |
4336 | ||
233f7bc0 T |
4337 | =item precision() vs. accuracy() |
4338 | ||
4339 | A common pitfall is to use L<precision()> when you want to round a result to | |
4340 | a certain number of digits: | |
4341 | ||
4342 | use Math::BigFloat; | |
4343 | ||
4344 | Math::BigFloat->precision(4); # does not do what you think it does | |
4345 | my $x = Math::BigFloat->new(12345); # rounds $x to "12000"! | |
4346 | print "$x\n"; # print "12000" | |
4347 | my $y = Math::BigFloat->new(3); # rounds $y to "0"! | |
4348 | print "$y\n"; # print "0" | |
4349 | $z = $x / $y; # 12000 / 0 => NaN! | |
4350 | print "$z\n"; | |
4351 | print $z->precision(),"\n"; # 4 | |
4352 | ||
4353 | Replacing L<precision> with L<accuracy> is probably not what you want, either: | |
4354 | ||
4355 | use Math::BigFloat; | |
4356 | ||
4357 | Math::BigFloat->accuracy(4); # enables global rounding: | |
3c4b39be | 4358 | my $x = Math::BigFloat->new(123456); # rounded immediately to "12350" |
233f7bc0 T |
4359 | print "$x\n"; # print "123500" |
4360 | my $y = Math::BigFloat->new(3); # rounded to "3 | |
4361 | print "$y\n"; # print "3" | |
4362 | print $z = $x->copy()->bdiv($y),"\n"; # 41170 | |
4363 | print $z->accuracy(),"\n"; # 4 | |
4364 | ||
4365 | What you want to use instead is: | |
4366 | ||
4367 | use Math::BigFloat; | |
4368 | ||
4369 | my $x = Math::BigFloat->new(123456); # no rounding | |
4370 | print "$x\n"; # print "123456" | |
4371 | my $y = Math::BigFloat->new(3); # no rounding | |
4372 | print "$y\n"; # print "3" | |
4373 | print $z = $x->copy()->bdiv($y,4),"\n"; # 41150 | |
4374 | print $z->accuracy(),"\n"; # undef | |
4375 | ||
4376 | In addition to computing what you expected, the last example also does B<not> | |
4377 | "taint" the result with an accuracy or precision setting, which would | |
4378 | influence any further operation. | |
4379 | ||
58cde26e JH |
4380 | =back |
4381 | ||
990fb837 RGS |
4382 | =head1 SEE ALSO |
4383 | ||
4384 | L<Math::BigInt>, L<Math::BigRat> and L<Math::Big> as well as | |
4385 | L<Math::BigInt::BitVect>, L<Math::BigInt::Pari> and L<Math::BigInt::GMP>. | |
4386 | ||
4387 | The pragmas L<bignum>, L<bigint> and L<bigrat> might also be of interest | |
4388 | because they solve the autoupgrading/downgrading issue, at least partly. | |
4389 | ||
7b29e1e6 | 4390 | The package at L<http://search.cpan.org/~tels/Math-BigInt> contains |
990fb837 RGS |
4391 | more documentation including a full version history, testcases, empty |
4392 | subclass files and benchmarks. | |
4393 | ||
58cde26e | 4394 | =head1 LICENSE |
a5f75d66 | 4395 | |
58cde26e JH |
4396 | This program is free software; you may redistribute it and/or modify it under |
4397 | the same terms as Perl itself. | |
5d7098d5 | 4398 | |
58cde26e | 4399 | =head1 AUTHORS |
5d7098d5 | 4400 | |
58cde26e | 4401 | Mark Biggar, overloaded interface by Ilya Zakharevich. |
7b29e1e6 T |
4402 | Completely rewritten by Tels L<http://bloodgate.com> in 2001 - 2006, and still |
4403 | at it in 2007. | |
a5f75d66 | 4404 | |
a5f75d66 | 4405 | =cut |