| 1 | package Math::BigFloat; |
| 2 | |
| 3 | # |
| 4 | # Mike grinned. 'Two down, infinity to go' - Mike Nostrus in 'Before and After' |
| 5 | # |
| 6 | |
| 7 | # The following hash values are internally used: |
| 8 | # _e : exponent (ref to $CALC object) |
| 9 | # _m : mantissa (ref to $CALC object) |
| 10 | # _es : sign of _e |
| 11 | # sign : +,-,+inf,-inf, or "NaN" if not a number |
| 12 | # _a : accuracy |
| 13 | # _p : precision |
| 14 | |
| 15 | $VERSION = '1.99_01'; |
| 16 | require 5.006002; |
| 17 | |
| 18 | require Exporter; |
| 19 | @ISA = qw/Math::BigInt/; |
| 20 | @EXPORT_OK = qw/bpi/; |
| 21 | |
| 22 | use strict; |
| 23 | # $_trap_inf/$_trap_nan are internal and should never be accessed from outside |
| 24 | use vars qw/$AUTOLOAD $accuracy $precision $div_scale $round_mode $rnd_mode |
| 25 | $upgrade $downgrade $_trap_nan $_trap_inf/; |
| 26 | my $class = "Math::BigFloat"; |
| 27 | |
| 28 | use overload |
| 29 | '<=>' => sub { my $rc = $_[2] ? |
| 30 | ref($_[0])->bcmp($_[1],$_[0]) : |
| 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 | }, |
| 43 | 'int' => sub { $_[0]->as_number() }, # 'trunc' to bigint |
| 44 | ; |
| 45 | |
| 46 | ############################################################################## |
| 47 | # global constants, flags and assorted stuff |
| 48 | |
| 49 | # the following are public, but their usage is not recommended. Use the |
| 50 | # accessor methods instead. |
| 51 | |
| 52 | # class constants, use Class->constant_name() to access |
| 53 | # one of 'even', 'odd', '+inf', '-inf', 'zero', 'trunc' or 'common' |
| 54 | $round_mode = 'even'; |
| 55 | $accuracy = undef; |
| 56 | $precision = undef; |
| 57 | $div_scale = 40; |
| 58 | |
| 59 | $upgrade = undef; |
| 60 | $downgrade = undef; |
| 61 | # the package we are using for our private parts, defaults to: |
| 62 | # Math::BigInt->config()->{lib} |
| 63 | my $MBI = 'Math::BigInt::FastCalc'; |
| 64 | |
| 65 | # are NaNs ok? (otherwise it dies when encountering an NaN) set w/ config() |
| 66 | $_trap_nan = 0; |
| 67 | # the same for infinity |
| 68 | $_trap_inf = 0; |
| 69 | |
| 70 | # constant for easier life |
| 71 | my $nan = 'NaN'; |
| 72 | |
| 73 | my $IMPORT = 0; # was import() called yet? used to make require work |
| 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; |
| 83 | my $HALF = '0.5'; # made into an object if nec. |
| 84 | |
| 85 | ############################################################################## |
| 86 | # the old code had $rnd_mode, so we need to support it, too |
| 87 | |
| 88 | sub TIESCALAR { my ($class) = @_; bless \$round_mode, $class; } |
| 89 | sub FETCH { return $round_mode; } |
| 90 | sub STORE { $rnd_mode = $_[0]->round_mode($_[1]); } |
| 91 | |
| 92 | BEGIN |
| 93 | { |
| 94 | # when someone sets $rnd_mode, we catch this and check the value to see |
| 95 | # whether it is valid or not. |
| 96 | $rnd_mode = 'even'; tie $rnd_mode, 'Math::BigFloat'; |
| 97 | |
| 98 | # we need both of them in this package: |
| 99 | *as_int = \&as_number; |
| 100 | } |
| 101 | |
| 102 | ############################################################################## |
| 103 | |
| 104 | { |
| 105 | # valid method aliases for AUTOLOAD |
| 106 | my %methods = map { $_ => 1 } |
| 107 | qw / fadd fsub fmul fdiv fround ffround fsqrt fmod fstr fsstr fpow fnorm |
| 108 | fint facmp fcmp fzero fnan finf finc fdec ffac fneg |
| 109 | fceil ffloor frsft flsft fone flog froot fexp |
| 110 | /; |
| 111 | # valid methods that can be handed up (for AUTOLOAD) |
| 112 | my %hand_ups = map { $_ => 1 } |
| 113 | qw / is_nan is_inf is_negative is_positive is_pos is_neg |
| 114 | accuracy precision div_scale round_mode fabs fnot |
| 115 | objectify upgrade downgrade |
| 116 | bone binf bnan bzero |
| 117 | bsub |
| 118 | /; |
| 119 | |
| 120 | sub _method_alias { exists $methods{$_[0]||''}; } |
| 121 | sub _method_hand_up { exists $hand_ups{$_[0]||''}; } |
| 122 | } |
| 123 | |
| 124 | ############################################################################## |
| 125 | # constructors |
| 126 | |
| 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" |
| 133 | |
| 134 | my ($class,$wanted,@r) = @_; |
| 135 | |
| 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'); |
| 139 | |
| 140 | $class->import() if $IMPORT == 0; # make require work |
| 141 | |
| 142 | my $self = {}; bless $self, $class; |
| 143 | # shortcut for bigints and its subclasses |
| 144 | if ((ref($wanted)) && UNIVERSAL::can( $wanted, "as_number")) |
| 145 | { |
| 146 | $self->{_m} = $wanted->as_number()->{value}; # get us a bigint copy |
| 147 | $self->{_e} = $MBI->_zero(); |
| 148 | $self->{_es} = '+'; |
| 149 | $self->{sign} = $wanted->sign(); |
| 150 | return $self->bnorm(); |
| 151 | } |
| 152 | # else: got a string or something masquerading as number (with overload) |
| 153 | |
| 154 | # handle '+inf', '-inf' first |
| 155 | if ($wanted =~ /^[+-]?inf\z/) |
| 156 | { |
| 157 | return $downgrade->new($wanted) if $downgrade; |
| 158 | |
| 159 | $self->{sign} = $wanted; # set a default sign for bstr() |
| 160 | return $self->binf($wanted); |
| 161 | } |
| 162 | |
| 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 | |
| 174 | my ($mis,$miv,$mfv,$es,$ev) = Math::BigInt::_split($wanted); |
| 175 | if (!ref $mis) |
| 176 | { |
| 177 | if ($_trap_nan) |
| 178 | { |
| 179 | require Carp; |
| 180 | Carp::croak ("$wanted is not a number initialized to $class"); |
| 181 | } |
| 182 | |
| 183 | return $downgrade->bnan() if $downgrade; |
| 184 | |
| 185 | $self->{_e} = $MBI->_zero(); |
| 186 | $self->{_es} = '+'; |
| 187 | $self->{_m} = $MBI->_zero(); |
| 188 | $self->{sign} = $nan; |
| 189 | } |
| 190 | else |
| 191 | { |
| 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. |
| 198 | |
| 199 | # 3.123E0 = 3123E-3, and 3.123E-2 => 3123E-5 |
| 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 | } |
| 206 | # we can only have trailing zeros on the mantissa if $$mfv eq '' |
| 207 | else |
| 208 | { |
| 209 | # Use a regexp to count the trailing zeros in $$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 $$miv =~ /[1-9](0*)$/; |
| 212 | if ($zeros != 0) |
| 213 | { |
| 214 | my $z = $MBI->_new($zeros); |
| 215 | # turn '120e2' into '12e3' |
| 216 | $MBI->_rsft ( $self->{_m}, $z, 10); |
| 217 | ($self->{_e}, $self->{_es}) = |
| 218 | _e_add ( $self->{_e}, $z, $self->{_es}, '+'); |
| 219 | } |
| 220 | } |
| 221 | $self->{sign} = $$mis; |
| 222 | |
| 223 | # for something like 0Ey, set y to 1, and -0 => +0 |
| 224 | # Check $$miv for being '0' and $$mfv eq '', because otherwise _m could not |
| 225 | # have become 0. That's faster than to call $MBI->_is_zero(). |
| 226 | $self->{sign} = '+', $self->{_e} = $MBI->_one() |
| 227 | if $$miv eq '0' and $$mfv eq ''; |
| 228 | |
| 229 | return $self->round(@r) if !$downgrade; |
| 230 | } |
| 231 | # if downgrade, inf, NaN or integers go down |
| 232 | |
| 233 | if ($downgrade && $self->{_es} eq '+') |
| 234 | { |
| 235 | if ($MBI->_is_zero( $self->{_e} )) |
| 236 | { |
| 237 | return $downgrade->new($$mis . $MBI->_str( $self->{_m} )); |
| 238 | } |
| 239 | return $downgrade->new($self->bsstr()); |
| 240 | } |
| 241 | $self->bnorm()->round(@r); # first normalize, then round |
| 242 | } |
| 243 | |
| 244 | sub copy |
| 245 | { |
| 246 | # if two arguments, the first one is the class to "swallow" subclasses |
| 247 | if (@_ > 1) |
| 248 | { |
| 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; |
| 259 | } |
| 260 | |
| 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]); |
| 267 | |
| 268 | $self->{_a} = $_[0]->{_a} if defined $_[0]->{_a}; |
| 269 | $self->{_p} = $_[0]->{_p} if defined $_[0]->{_p}; |
| 270 | $self; |
| 271 | } |
| 272 | |
| 273 | sub _bnan |
| 274 | { |
| 275 | # used by parent class bone() to initialize number to NaN |
| 276 | my $self = shift; |
| 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 |
| 286 | $self->{_m} = $MBI->_zero(); |
| 287 | $self->{_e} = $MBI->_zero(); |
| 288 | $self->{_es} = '+'; |
| 289 | } |
| 290 | |
| 291 | sub _binf |
| 292 | { |
| 293 | # used by parent class bone() to initialize number to +-inf |
| 294 | my $self = shift; |
| 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 |
| 304 | $self->{_m} = $MBI->_zero(); |
| 305 | $self->{_e} = $MBI->_zero(); |
| 306 | $self->{_es} = '+'; |
| 307 | } |
| 308 | |
| 309 | sub _bone |
| 310 | { |
| 311 | # used by parent class bone() to initialize number to 1 |
| 312 | my $self = shift; |
| 313 | $IMPORT=1; # call our import only once |
| 314 | $self->{_m} = $MBI->_one(); |
| 315 | $self->{_e} = $MBI->_zero(); |
| 316 | $self->{_es} = '+'; |
| 317 | } |
| 318 | |
| 319 | sub _bzero |
| 320 | { |
| 321 | # used by parent class bone() to initialize number to 0 |
| 322 | my $self = shift; |
| 323 | $IMPORT=1; # call our import only once |
| 324 | $self->{_m} = $MBI->_zero(); |
| 325 | $self->{_e} = $MBI->_one(); |
| 326 | $self->{_es} = '+'; |
| 327 | } |
| 328 | |
| 329 | sub isa |
| 330 | { |
| 331 | my ($self,$class) = @_; |
| 332 | return if $class =~ /^Math::BigInt/; # we aren't one of these |
| 333 | UNIVERSAL::isa($self,$class); |
| 334 | } |
| 335 | |
| 336 | sub config |
| 337 | { |
| 338 | # return (later set?) configuration data as hash ref |
| 339 | my $class = shift || 'Math::BigFloat'; |
| 340 | |
| 341 | if (@_ == 1 && ref($_[0]) ne 'HASH') |
| 342 | { |
| 343 | my $cfg = $class->SUPER::config(); |
| 344 | return $cfg->{$_[0]}; |
| 345 | } |
| 346 | |
| 347 | my $cfg = $class->SUPER::config(@_); |
| 348 | |
| 349 | # now we need only to override the ones that are different from our parent |
| 350 | $cfg->{class} = $class; |
| 351 | $cfg->{with} = $MBI; |
| 352 | $cfg; |
| 353 | } |
| 354 | |
| 355 | ############################################################################## |
| 356 | # string conversion |
| 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") |
| 363 | my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_); |
| 364 | |
| 365 | if ($x->{sign} !~ /^[+-]$/) |
| 366 | { |
| 367 | return $x->{sign} unless $x->{sign} eq '+inf'; # -inf, NaN |
| 368 | return 'inf'; # +inf |
| 369 | } |
| 370 | |
| 371 | my $es = '0'; my $len = 1; my $cad = 0; my $dot = '.'; |
| 372 | |
| 373 | # $x is zero? |
| 374 | my $not_zero = !($x->{sign} eq '+' && $MBI->_is_zero($x->{_m})); |
| 375 | if ($not_zero) |
| 376 | { |
| 377 | $es = $MBI->_str($x->{_m}); |
| 378 | $len = CORE::length($es); |
| 379 | my $e = $MBI->_num($x->{_e}); |
| 380 | $e = -$e if $x->{_es} eq '-'; |
| 381 | if ($e < 0) |
| 382 | { |
| 383 | $dot = ''; |
| 384 | # if _e is bigger than a scalar, the following will blow your memory |
| 385 | if ($e <= -$len) |
| 386 | { |
| 387 | my $r = abs($e) - $len; |
| 388 | $es = '0.'. ('0' x $r) . $es; $cad = -($len+$r); |
| 389 | } |
| 390 | else |
| 391 | { |
| 392 | substr($es,$e,0) = '.'; $cad = $MBI->_num($x->{_e}); |
| 393 | $cad = -$cad if $x->{_es} eq '-'; |
| 394 | } |
| 395 | } |
| 396 | elsif ($e > 0) |
| 397 | { |
| 398 | # expand with zeros |
| 399 | $es .= '0' x $e; $len += $e; $cad = 0; |
| 400 | } |
| 401 | } # if not zero |
| 402 | |
| 403 | $es = '-'.$es if $x->{sign} eq '-'; |
| 404 | # if set accuracy or precision, pad with zeros on the right side |
| 405 | if ((defined $x->{_a}) && ($not_zero)) |
| 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; |
| 410 | $es .= $dot.'0' x $zeros if $zeros > 0; |
| 411 | } |
| 412 | elsif ((($x->{_p} || 0) < 0)) |
| 413 | { |
| 414 | # 123400 => 6, 0.1234 => 4, 0.001234 => 6 |
| 415 | my $zeros = -$x->{_p} + $cad; |
| 416 | $es .= $dot.'0' x $zeros if $zeros > 0; |
| 417 | } |
| 418 | $es; |
| 419 | } |
| 420 | |
| 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") |
| 426 | my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_); |
| 427 | |
| 428 | if ($x->{sign} !~ /^[+-]$/) |
| 429 | { |
| 430 | return $x->{sign} unless $x->{sign} eq '+inf'; # -inf, NaN |
| 431 | return 'inf'; # +inf |
| 432 | } |
| 433 | my $sep = 'e'.$x->{_es}; |
| 434 | my $sign = $x->{sign}; $sign = '' if $sign eq '+'; |
| 435 | $sign . $MBI->_str($x->{_m}) . $sep . $MBI->_str($x->{_e}); |
| 436 | } |
| 437 | |
| 438 | sub numify |
| 439 | { |
| 440 | # Make a number from a BigFloat object |
| 441 | # simple return a string and let Perl's atoi()/atof() handle the rest |
| 442 | my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_); |
| 443 | $x->bsstr(); |
| 444 | } |
| 445 | |
| 446 | ############################################################################## |
| 447 | # public stuff (usually prefixed with "b") |
| 448 | |
| 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 | |
| 462 | # tels 2001-08-04 |
| 463 | # XXX TODO this must be overwritten and return NaN for non-integer values |
| 464 | # band(), bior(), bxor(), too |
| 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) |
| 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 | } |
| 481 | |
| 482 | return $upgrade->bcmp($x,$y) if defined $upgrade && |
| 483 | ((!$x->isa($self)) || (!$y->isa($self))); |
| 484 | |
| 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'; |
| 493 | return +1; |
| 494 | } |
| 495 | |
| 496 | # check sign for speed first |
| 497 | return 1 if $x->{sign} eq '+' && $y->{sign} eq '-'; # does also 0 <=> -y |
| 498 | return -1 if $x->{sign} eq '-' && $y->{sign} eq '+'; # does also -x <=> 0 |
| 499 | |
| 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 |
| 506 | |
| 507 | # adjust so that exponents are equal |
| 508 | my $lxm = $MBI->_len($x->{_m}); |
| 509 | my $lym = $MBI->_len($y->{_m}); |
| 510 | # the numify somewhat limits our length, but makes it much faster |
| 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}); |
| 516 | my $l = $lx - $ly; $l = -$l if $x->{sign} eq '-'; |
| 517 | return $l <=> 0 if $l != 0; |
| 518 | |
| 519 | # lengths (corrected by exponent) are equal |
| 520 | # so make mantissa equal length by padding with zero (shift left) |
| 521 | my $diff = $lxm - $lym; |
| 522 | my $xm = $x->{_m}; # not yet copy it |
| 523 | my $ym = $y->{_m}; |
| 524 | if ($diff > 0) |
| 525 | { |
| 526 | $ym = $MBI->_copy($y->{_m}); |
| 527 | $ym = $MBI->_lsft($ym, $MBI->_new($diff), 10); |
| 528 | } |
| 529 | elsif ($diff < 0) |
| 530 | { |
| 531 | $xm = $MBI->_copy($x->{_m}); |
| 532 | $xm = $MBI->_lsft($xm, $MBI->_new(-$diff), 10); |
| 533 | } |
| 534 | my $rc = $MBI->_acmp($xm,$ym); |
| 535 | $rc = -$rc if $x->{sign} eq '-'; # -124 < -123 |
| 536 | $rc <=> 0; |
| 537 | } |
| 538 | |
| 539 | sub bacmp |
| 540 | { |
| 541 | # Compares 2 values, ignoring their signs. |
| 542 | # Returns one of undef, <0, =0, >0. (suitable for sort) |
| 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 | } |
| 551 | |
| 552 | return $upgrade->bacmp($x,$y) if defined $upgrade && |
| 553 | ((!$x->isa($self)) || (!$y->isa($self))); |
| 554 | |
| 555 | # handle +-inf and NaN's |
| 556 | if ($x->{sign} !~ /^[+-]$/ || $y->{sign} !~ /^[+-]$/) |
| 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()); |
| 561 | return -1; |
| 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 |
| 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 '+'; |
| 577 | # the numify somewhat limits our length, but makes it much faster |
| 578 | my $lx = $lxm + $xes * $MBI->_num($x->{_e}); |
| 579 | my $ly = $lym + $yes * $MBI->_num($y->{_e}); |
| 580 | my $l = $lx - $ly; |
| 581 | return $l <=> 0 if $l != 0; |
| 582 | |
| 583 | # lengths (corrected by exponent) are equal |
| 584 | # so make mantissa equal-length by padding with zero (shift left) |
| 585 | my $diff = $lxm - $lym; |
| 586 | my $xm = $x->{_m}; # not yet copy it |
| 587 | my $ym = $y->{_m}; |
| 588 | if ($diff > 0) |
| 589 | { |
| 590 | $ym = $MBI->_copy($y->{_m}); |
| 591 | $ym = $MBI->_lsft($ym, $MBI->_new($diff), 10); |
| 592 | } |
| 593 | elsif ($diff < 0) |
| 594 | { |
| 595 | $xm = $MBI->_copy($x->{_m}); |
| 596 | $xm = $MBI->_lsft($xm, $MBI->_new(-$diff), 10); |
| 597 | } |
| 598 | $MBI->_acmp($xm,$ym); |
| 599 | } |
| 600 | |
| 601 | sub badd |
| 602 | { |
| 603 | # add second arg (BFLOAT or string) to first (BFLOAT) (modifies first) |
| 604 | # return result as BFLOAT |
| 605 | |
| 606 | # set up parameters |
| 607 | my ($self,$x,$y,@r) = (ref($_[0]),@_); |
| 608 | # objectify is costly, so avoid it |
| 609 | if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1]))) |
| 610 | { |
| 611 | ($self,$x,$y,@r) = objectify(2,@_); |
| 612 | } |
| 613 | |
| 614 | return $x if $x->modify('badd'); |
| 615 | |
| 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)); |
| 621 | # inf handling |
| 622 | if (($x->{sign} =~ /^[+-]inf$/) && ($y->{sign} =~ /^[+-]inf$/)) |
| 623 | { |
| 624 | # +inf++inf or -inf+-inf => same, rest is NaN |
| 625 | return $x if $x->{sign} eq $y->{sign}; |
| 626 | return $x->bnan(); |
| 627 | } |
| 628 | # +-inf + something => +inf; something +-inf => +-inf |
| 629 | $x->{sign} = $y->{sign}, return $x if $y->{sign} =~ /^[+-]inf$/; |
| 630 | return $x; |
| 631 | } |
| 632 | |
| 633 | return $upgrade->badd($x,$y,@r) if defined $upgrade && |
| 634 | ((!$x->isa($self)) || (!$y->isa($self))); |
| 635 | |
| 636 | $r[3] = $y; # no push! |
| 637 | |
| 638 | # speed: no add for 0+y or x+0 |
| 639 | return $x->bround(@r) if $y->is_zero(); # x+0 |
| 640 | if ($x->is_zero()) # 0+y |
| 641 | { |
| 642 | # make copy, clobbering up x (modify in place!) |
| 643 | $x->{_e} = $MBI->_copy($y->{_e}); |
| 644 | $x->{_es} = $y->{_es}; |
| 645 | $x->{_m} = $MBI->_copy($y->{_m}); |
| 646 | $x->{sign} = $y->{sign} || $nan; |
| 647 | return $x->round(@r); |
| 648 | } |
| 649 | |
| 650 | # take lower of the two e's and adapt m1 to it to match m2 |
| 651 | my $e = $y->{_e}; |
| 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 |
| 662 | { |
| 663 | $MBI->_lsft( $x->{_m}, $e, 10); |
| 664 | ($x->{_e},$x->{_es}) = _e_add($x->{_e}, $e, $x->{_es}, $es); |
| 665 | } |
| 666 | elsif (!$MBI->_is_zero($e)) # > 0 |
| 667 | { |
| 668 | $MBI->_lsft($add, $e, 10); |
| 669 | } |
| 670 | # else: both e are the same, so just leave them |
| 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 | |
| 683 | # delete trailing zeros, then round |
| 684 | $x->bnorm()->round(@r); |
| 685 | } |
| 686 | |
| 687 | # sub bsub is inherited from Math::BigInt! |
| 688 | |
| 689 | sub binc |
| 690 | { |
| 691 | # increment arg by one |
| 692 | my ($self,$x,@r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_); |
| 693 | |
| 694 | return $x if $x->modify('binc'); |
| 695 | |
| 696 | if ($x->{_es} eq '-') |
| 697 | { |
| 698 | return $x->badd($self->bone(),@r); # digits after dot |
| 699 | } |
| 700 | |
| 701 | if (!$MBI->_is_zero($x->{_e})) # _e == 0 for NaN, inf, -inf |
| 702 | { |
| 703 | # 1e2 => 100, so after the shift below _m has a '0' as last digit |
| 704 | $x->{_m} = $MBI->_lsft($x->{_m}, $x->{_e},10); # 1e2 => 100 |
| 705 | $x->{_e} = $MBI->_zero(); # normalize |
| 706 | $x->{_es} = '+'; |
| 707 | # we know that the last digit of $x will be '1' or '9', depending on the |
| 708 | # sign |
| 709 | } |
| 710 | # now $x->{_e} == 0 |
| 711 | if ($x->{sign} eq '+') |
| 712 | { |
| 713 | $MBI->_inc($x->{_m}); |
| 714 | return $x->bnorm()->bround(@r); |
| 715 | } |
| 716 | elsif ($x->{sign} eq '-') |
| 717 | { |
| 718 | $MBI->_dec($x->{_m}); |
| 719 | $x->{sign} = '+' if $MBI->_is_zero($x->{_m}); # -1 +1 => -0 => +0 |
| 720 | return $x->bnorm()->bround(@r); |
| 721 | } |
| 722 | # inf, nan handling etc |
| 723 | $x->badd($self->bone(),@r); # badd() does round |
| 724 | } |
| 725 | |
| 726 | sub bdec |
| 727 | { |
| 728 | # decrement arg by one |
| 729 | my ($self,$x,@r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_); |
| 730 | |
| 731 | return $x if $x->modify('bdec'); |
| 732 | |
| 733 | if ($x->{_es} eq '-') |
| 734 | { |
| 735 | return $x->badd($self->bone('-'),@r); # digits after dot |
| 736 | } |
| 737 | |
| 738 | if (!$MBI->_is_zero($x->{_e})) |
| 739 | { |
| 740 | $x->{_m} = $MBI->_lsft($x->{_m}, $x->{_e},10); # 1e2 => 100 |
| 741 | $x->{_e} = $MBI->_zero(); # normalize |
| 742 | $x->{_es} = '+'; |
| 743 | } |
| 744 | # now $x->{_e} == 0 |
| 745 | my $zero = $x->is_zero(); |
| 746 | # <= 0 |
| 747 | if (($x->{sign} eq '-') || $zero) |
| 748 | { |
| 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 |
| 752 | return $x->bnorm()->round(@r); |
| 753 | } |
| 754 | # > 0 |
| 755 | elsif ($x->{sign} eq '+') |
| 756 | { |
| 757 | $MBI->_dec($x->{_m}); |
| 758 | return $x->bnorm()->round(@r); |
| 759 | } |
| 760 | # inf, nan handling etc |
| 761 | $x->badd($self->bone('-'),@r); # does round |
| 762 | } |
| 763 | |
| 764 | sub DEBUG () { 0; } |
| 765 | |
| 766 | sub blog |
| 767 | { |
| 768 | my ($self,$x,$base,$a,$p,$r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_); |
| 769 | |
| 770 | return $x if $x->modify('blog'); |
| 771 | |
| 772 | # $base > 0, $base != 1; if $base == undef default to $base == e |
| 773 | # $x >= 0 |
| 774 | |
| 775 | # we need to limit the accuracy to protect against overflow |
| 776 | my $fallback = 0; |
| 777 | my ($scale,@params); |
| 778 | ($x,@params) = $x->_find_round_parameters($a,$p,$r); |
| 779 | |
| 780 | # also takes care of the "error in _find_round_parameters?" case |
| 781 | return $x->bnan() if $x->{sign} ne '+' || $x->is_zero(); |
| 782 | |
| 783 | # no rounding at all, so must use fallback |
| 784 | if (scalar @params == 0) |
| 785 | { |
| 786 | # simulate old behaviour |
| 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 |
| 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... |
| 797 | $scale = abs($params[0] || $params[1]) + 4; # take whatever is defined |
| 798 | } |
| 799 | |
| 800 | return $x->bzero(@params) if $x->is_one(); |
| 801 | # base not defined => base == Euler's number e |
| 802 | if (defined $base) |
| 803 | { |
| 804 | # make object, since we don't feed it through objectify() to still get the |
| 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 '+'; |
| 810 | # if $x == $base, we know the result must be 1.0 |
| 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 | } |
| 821 | } |
| 822 | |
| 823 | # when user set globals, they would interfere with our calculation, so |
| 824 | # disable them and later re-enable them |
| 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}; |
| 831 | # need to disable $upgrade in BigInt, to avoid deep recursion |
| 832 | local $Math::BigInt::upgrade = undef; |
| 833 | local $Math::BigFloat::downgrade = undef; |
| 834 | |
| 835 | # upgrade $x if $x is not a BigFloat (handle BigInt input) |
| 836 | # XXX TODO: rebless! |
| 837 | if (!$x->isa('Math::BigFloat')) |
| 838 | { |
| 839 | $x = Math::BigFloat->new($x); |
| 840 | $self = ref($x); |
| 841 | } |
| 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 | { |
| 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; |
| 854 | $int->blog($base->as_number()); |
| 855 | # if ($exact) |
| 856 | if ($base->as_number()->bpow($int) == $x) |
| 857 | { |
| 858 | # found result, return it |
| 859 | $x->{_m} = $int->{value}; |
| 860 | $x->{_e} = $MBI->_zero(); |
| 861 | $x->{_es} = '+'; |
| 862 | $x->bnorm(); |
| 863 | $done = 1; |
| 864 | } |
| 865 | } |
| 866 | |
| 867 | if ($done == 0) |
| 868 | { |
| 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)): |
| 871 | $self->_log_10($x,$scale); |
| 872 | |
| 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 | } |
| 880 | } |
| 881 | |
| 882 | # shortcut to not run through _find_round_parameters again |
| 883 | if (defined $params[0]) |
| 884 | { |
| 885 | $x->bround($params[0],$params[2]); # then round accordingly |
| 886 | } |
| 887 | else |
| 888 | { |
| 889 | $x->bfround($params[1],$params[2]); # then round accordingly |
| 890 | } |
| 891 | if ($fallback) |
| 892 | { |
| 893 | # clear a/p after round, since user did not request it |
| 894 | delete $x->{_a}; delete $x->{_p}; |
| 895 | } |
| 896 | # restore globals |
| 897 | $$abr = $ab; $$pbr = $pb; |
| 898 | |
| 899 | $x; |
| 900 | } |
| 901 | |
| 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 | |
| 959 | sub bexp |
| 960 | { |
| 961 | # Calculate e ** X (Euler's number to the power of X) |
| 962 | my ($self,$x,$a,$p,$r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_); |
| 963 | |
| 964 | return $x if $x->modify('bexp'); |
| 965 | |
| 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(); |
| 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 | |
| 1059 | # Note that we cannot simple reduce 13700/5040 to 685/252, but must keep A and B! |
| 1060 | |
| 1061 | if ($scale <= 75) |
| 1062 | { |
| 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); |
| 1069 | } |
| 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"; |
| 1093 | |
| 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 ); |
| 1097 | |
| 1098 | $x->{_m} = $A; |
| 1099 | $x->{sign} = '+'; |
| 1100 | $x->{_es} = '-'; |
| 1101 | $x->{_e} = $MBI->_new($scale); |
| 1102 | } |
| 1103 | |
| 1104 | # $x contains now an estimate of e, with some surplus digits, so we can round |
| 1105 | if (!$x_org->is_one()) |
| 1106 | { |
| 1107 | # raise $x to the wanted power and round it in one step: |
| 1108 | $x->bpow($x_org, @params); |
| 1109 | } |
| 1110 | else |
| 1111 | { |
| 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 | } |
| 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 | |
| 1135 | sub _log |
| 1136 | { |
| 1137 | # internal log function to calculate ln() based on Taylor series. |
| 1138 | # Modifies $x in place. |
| 1139 | my ($self,$x,$scale) = @_; |
| 1140 | |
| 1141 | # in case of $x == 1, result is 0 |
| 1142 | return $x->bzero() if $x->is_one(); |
| 1143 | |
| 1144 | # XXX TODO: rewrite this in a similar manner to bexp() |
| 1145 | |
| 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 | |
| 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 |
| 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 |
| 1192 | |
| 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); |
| 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 | } |
| 1211 | print "took $steps steps\n" if DEBUG; |
| 1212 | $x->bmul($f); # $x *= 2 |
| 1213 | } |
| 1214 | |
| 1215 | sub _log_10 |
| 1216 | { |
| 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. |
| 1219 | my ($self,$x,$scale) = @_; |
| 1220 | |
| 1221 | # Taking blog() from numbers greater than 10 takes a *very long* time, so we |
| 1222 | # break the computation down into parts based on the observation that: |
| 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 |
| 1231 | # log(10) afterwards to get the correct result. |
| 1232 | |
| 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: |
| 1242 | my $dbd = $MBI->_num($x->{_e}); |
| 1243 | $dbd = -$dbd if $x->{_es} eq '-'; |
| 1244 | $dbd += $MBI->_len($x->{_m}); |
| 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 |
| 1253 | if ($x->{_es} eq '+' && $MBI->_is_one($x->{_e}) && $MBI->_is_one($x->{_m})) |
| 1254 | { |
| 1255 | $dbd = 0; # disable shortcut |
| 1256 | # we can use the cached value in these cases |
| 1257 | if ($scale <= $LOG_10_A) |
| 1258 | { |
| 1259 | $x->bzero(); $x->badd($LOG_10); # modify $x in place |
| 1260 | $calc = 0; # no need to calc, but round |
| 1261 | } |
| 1262 | # if we can't use the shortcut, we continue normally |
| 1263 | } |
| 1264 | else |
| 1265 | { |
| 1266 | # disable the shortcut for 2, since we maybe have it cached |
| 1267 | if (($MBI->_is_zero($x->{_e}) && $MBI->_is_two($x->{_m}))) |
| 1268 | { |
| 1269 | $dbd = 0; # disable shortcut |
| 1270 | # we can use the cached value in these cases |
| 1271 | if ($scale <= $LOG_2_A) |
| 1272 | { |
| 1273 | $x->bzero(); $x->badd($LOG_2); # modify $x in place |
| 1274 | $calc = 0; # no need to calc, but round |
| 1275 | } |
| 1276 | # if we can't use the shortcut, we continue normally |
| 1277 | } |
| 1278 | } |
| 1279 | |
| 1280 | # if $x = 0.1, we know the result must be 0-log(10) |
| 1281 | if ($calc != 0 && $x->{_es} eq '-' && $MBI->_is_one($x->{_e}) && |
| 1282 | $MBI->_is_one($x->{_m})) |
| 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 | |
| 1293 | return if $calc == 0; # already have the result |
| 1294 | |
| 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 | |
| 1299 | my $two = $self->new(2); |
| 1300 | |
| 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 |
| 1318 | $l_10 = $LOG_10->copy(); # copy for mul |
| 1319 | } |
| 1320 | else |
| 1321 | { |
| 1322 | # else: slower, compute and cache result |
| 1323 | # also disable downgrade for this code path |
| 1324 | local $Math::BigFloat::downgrade = undef; |
| 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; |
| 1356 | } |
| 1357 | $dbd-- if ($dbd > 1); # 20 => dbd=2, so make it dbd=1 |
| 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 |
| 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 | |
| 1375 | $HALF = $self->new($HALF) unless ref($HALF); |
| 1376 | |
| 1377 | my $twos = 0; # default: none (0 times) |
| 1378 | while ($x->bacmp($HALF) <= 0) # X <= 0.5 |
| 1379 | { |
| 1380 | $twos--; $x->bmul($two); |
| 1381 | } |
| 1382 | while ($x->bacmp($two) >= 0) # X >= 2 |
| 1383 | { |
| 1384 | $twos++; $x->bdiv($two,$scale+4); # keep all digits |
| 1385 | } |
| 1386 | # $twos > 0 => did mul 2, < 0 => did div 2 (but we never did both) |
| 1387 | # So calculate correction factor based on ln(2): |
| 1388 | if ($twos != 0) |
| 1389 | { |
| 1390 | $LOG_2 = $self->new($LOG_2,undef,undef) unless ref $LOG_2; |
| 1391 | if ($scale <= $LOG_2_A) |
| 1392 | { |
| 1393 | # use cached value |
| 1394 | $l_2 = $LOG_2->copy(); # copy() for the mul below |
| 1395 | } |
| 1396 | else |
| 1397 | { |
| 1398 | # else: slower, compute and cache result |
| 1399 | # also disable downgrade for this code path |
| 1400 | local $Math::BigFloat::downgrade = undef; |
| 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; |
| 1405 | } |
| 1406 | $l_2->bmul($twos); # * -2 => subtract, * 2 => add |
| 1407 | } |
| 1408 | |
| 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) |
| 1412 | |
| 1413 | # all done, $x contains now the result |
| 1414 | $x; |
| 1415 | } |
| 1416 | |
| 1417 | sub blcm |
| 1418 | { |
| 1419 | # (BFLOAT or num_str, BFLOAT or num_str) return BFLOAT |
| 1420 | # does not modify arguments, but returns new object |
| 1421 | # Lowest Common Multiplicator |
| 1422 | |
| 1423 | my ($self,@arg) = objectify(0,@_); |
| 1424 | my $x = $self->new(shift @arg); |
| 1425 | while (@arg) { $x = Math::BigInt::__lcm($x,shift @arg); } |
| 1426 | $x; |
| 1427 | } |
| 1428 | |
| 1429 | sub bgcd |
| 1430 | { |
| 1431 | # (BINT or num_str, BINT or num_str) return BINT |
| 1432 | # does not modify arguments, but returns new object |
| 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 | } |
| 1458 | $x; |
| 1459 | } |
| 1460 | |
| 1461 | ############################################################################## |
| 1462 | |
| 1463 | sub _e_add |
| 1464 | { |
| 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 | } |
| 1477 | |
| 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); |
| 1494 | } |
| 1495 | |
| 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 | |
| 1511 | sub is_int |
| 1512 | { |
| 1513 | # return true if arg (BFLOAT or num_str) is an integer |
| 1514 | my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_); |
| 1515 | |
| 1516 | (($x->{sign} =~ /^[+-]$/) && # NaN and +-inf aren't |
| 1517 | ($x->{_es} eq '+')) ? 1 : 0; # 1e-1 => no integer |
| 1518 | } |
| 1519 | |
| 1520 | sub is_zero |
| 1521 | { |
| 1522 | # return true if arg (BFLOAT or num_str) is zero |
| 1523 | my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_); |
| 1524 | |
| 1525 | ($x->{sign} eq '+' && $MBI->_is_zero($x->{_m})) ? 1 : 0; |
| 1526 | } |
| 1527 | |
| 1528 | sub is_one |
| 1529 | { |
| 1530 | # return true if arg (BFLOAT or num_str) is +1 or -1 if signis given |
| 1531 | my ($self,$x,$sign) = ref($_[0]) ? (undef,@_) : objectify(1,@_); |
| 1532 | |
| 1533 | $sign = '+' if !defined $sign || $sign ne '-'; |
| 1534 | |
| 1535 | ($x->{sign} eq $sign && |
| 1536 | $MBI->_is_zero($x->{_e}) && |
| 1537 | $MBI->_is_one($x->{_m}) ) ? 1 : 0; |
| 1538 | } |
| 1539 | |
| 1540 | sub is_odd |
| 1541 | { |
| 1542 | # return true if arg (BFLOAT or num_str) is odd or false if even |
| 1543 | my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_); |
| 1544 | |
| 1545 | (($x->{sign} =~ /^[+-]$/) && # NaN & +-inf aren't |
| 1546 | ($MBI->_is_zero($x->{_e})) && |
| 1547 | ($MBI->_is_odd($x->{_m}))) ? 1 : 0; |
| 1548 | } |
| 1549 | |
| 1550 | sub is_even |
| 1551 | { |
| 1552 | # return true if arg (BINT or num_str) is even or false if odd |
| 1553 | my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_); |
| 1554 | |
| 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 |
| 1558 | } |
| 1559 | |
| 1560 | sub bmul |
| 1561 | { |
| 1562 | # multiply two numbers |
| 1563 | |
| 1564 | # set up parameters |
| 1565 | my ($self,$x,$y,@r) = (ref($_[0]),@_); |
| 1566 | # objectify is costly, so avoid it |
| 1567 | if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1]))) |
| 1568 | { |
| 1569 | ($self,$x,$y,@r) = objectify(2,@_); |
| 1570 | } |
| 1571 | |
| 1572 | return $x if $x->modify('bmul'); |
| 1573 | |
| 1574 | return $x->bnan() if (($x->{sign} eq $nan) || ($y->{sign} eq $nan)); |
| 1575 | |
| 1576 | # inf handling |
| 1577 | if (($x->{sign} =~ /^[+-]inf$/) || ($y->{sign} =~ /^[+-]inf$/)) |
| 1578 | { |
| 1579 | return $x->bnan() if $x->is_zero() || $y->is_zero(); |
| 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 | } |
| 1587 | |
| 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 && |
| 1633 | ((!$x->isa($self)) || (!$y->isa($self))); |
| 1634 | |
| 1635 | # aEb * cEd = (a*c)E(b+d) |
| 1636 | $MBI->_mul($x->{_m},$y->{_m}); |
| 1637 | ($x->{_e}, $x->{_es}) = _e_add($x->{_e}, $y->{_e}, $x->{_es}, $y->{_es}); |
| 1638 | |
| 1639 | $r[3] = $y; # no push! |
| 1640 | |
| 1641 | # adjust sign: |
| 1642 | $x->{sign} = $x->{sign} ne $y->{sign} ? '-' : '+'; |
| 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); |
| 1682 | } |
| 1683 | |
| 1684 | sub bdiv |
| 1685 | { |
| 1686 | # (dividend: BFLOAT or num_str, divisor: BFLOAT or num_str) return |
| 1687 | # (BFLOAT,BFLOAT) (quo,rem) or BFLOAT (only rem) |
| 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 | } |
| 1696 | |
| 1697 | return $x if $x->modify('bdiv'); |
| 1698 | |
| 1699 | return $self->_div_inf($x,$y) |
| 1700 | if (($x->{sign} !~ /^[+-]$/) || ($y->{sign} !~ /^[+-]$/) || $y->is_zero()); |
| 1701 | |
| 1702 | # x== 0 # also: or y == 1 or y == -1 |
| 1703 | return wantarray ? ($x,$self->bzero()) : $x if $x->is_zero(); |
| 1704 | |
| 1705 | # upgrade ? |
| 1706 | return $upgrade->bdiv($upgrade->new($x),$y,$a,$p,$r) if defined $upgrade; |
| 1707 | |
| 1708 | # we need to limit the accuracy to protect against overflow |
| 1709 | my $fallback = 0; |
| 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? |
| 1714 | |
| 1715 | # no rounding at all, so must use fallback |
| 1716 | if (scalar @params == 0) |
| 1717 | { |
| 1718 | # simulate old behaviour |
| 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 |
| 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... |
| 1728 | $scale = abs($params[0] || $params[1]) + 4; # take whatever is defined |
| 1729 | } |
| 1730 | |
| 1731 | my $rem; $rem = $self->bzero() if wantarray; |
| 1732 | |
| 1733 | $y = $self->new($y) unless $y->isa('Math::BigFloat'); |
| 1734 | |
| 1735 | my $lx = $MBI->_len($x->{_m}); my $ly = $MBI->_len($y->{_m}); |
| 1736 | $scale = $lx if $lx > $scale; |
| 1737 | $scale = $ly if $ly > $scale; |
| 1738 | my $diff = $ly - $lx; |
| 1739 | $scale += $diff if $diff > 0; # if lx << ly, but not if ly << lx! |
| 1740 | |
| 1741 | # already handled inf/NaN/-inf above: |
| 1742 | |
| 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 | |
| 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 | |
| 1751 | if ($xsign ne $x->{sign}) |
| 1752 | { |
| 1753 | # special case of $x /= $x results in 1 |
| 1754 | $x->bone(); # "fixes" also sign of $y, since $x is $y |
| 1755 | } |
| 1756 | else |
| 1757 | { |
| 1758 | # correct $y's sign again |
| 1759 | $y->{sign} =~ tr/+-/-+/; |
| 1760 | # continue with normal div code: |
| 1761 | |
| 1762 | # make copy of $x in case of list context for later reminder calculation |
| 1763 | if (wantarray && $y_not_one) |
| 1764 | { |
| 1765 | $rem = $x->copy(); |
| 1766 | } |
| 1767 | |
| 1768 | $x->{sign} = $x->{sign} ne $y->sign() ? '-' : '+'; |
| 1769 | |
| 1770 | # check for / +-1 ( +/- 1E0) |
| 1771 | if ($y_not_one) |
| 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 |
| 1788 | |
| 1789 | # shortcut to not run through _find_round_parameters again |
| 1790 | if (defined $params[0]) |
| 1791 | { |
| 1792 | delete $x->{_a}; # clear before round |
| 1793 | $x->bround($params[0],$params[2]); # then round accordingly |
| 1794 | } |
| 1795 | else |
| 1796 | { |
| 1797 | delete $x->{_p}; # clear before round |
| 1798 | $x->bfround($params[1],$params[2]); # then round accordingly |
| 1799 | } |
| 1800 | if ($fallback) |
| 1801 | { |
| 1802 | # clear a/p after round, since user did not request it |
| 1803 | delete $x->{_a}; delete $x->{_p}; |
| 1804 | } |
| 1805 | |
| 1806 | if (wantarray) |
| 1807 | { |
| 1808 | if ($y_not_one) |
| 1809 | { |
| 1810 | $rem->bmod($y,@params); # copy already done |
| 1811 | } |
| 1812 | if ($fallback) |
| 1813 | { |
| 1814 | # clear a/p after round, since user did not request it |
| 1815 | delete $rem->{_a}; delete $rem->{_p}; |
| 1816 | } |
| 1817 | return ($x,$rem); |
| 1818 | } |
| 1819 | $x; |
| 1820 | } |
| 1821 | |
| 1822 | sub bmod |
| 1823 | { |
| 1824 | # (dividend: BFLOAT or num_str, divisor: BFLOAT or num_str) return reminder |
| 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 | } |
| 1833 | |
| 1834 | return $x if $x->modify('bmod'); |
| 1835 | |
| 1836 | # handle NaN, inf, -inf |
| 1837 | if (($x->{sign} !~ /^[+-]$/) || ($y->{sign} !~ /^[+-]$/)) |
| 1838 | { |
| 1839 | my ($d,$re) = $self->SUPER::_div_inf($x,$y); |
| 1840 | $x->{sign} = $re->{sign}; |
| 1841 | $x->{_e} = $re->{_e}; |
| 1842 | $x->{_m} = $re->{_m}; |
| 1843 | return $x->round($a,$p,$r,$y); |
| 1844 | } |
| 1845 | if ($y->is_zero()) |
| 1846 | { |
| 1847 | return $x->bnan() if $x->is_zero(); |
| 1848 | return $x; |
| 1849 | } |
| 1850 | |
| 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}))); |
| 1855 | |
| 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 | |
| 1865 | my $ym = $MBI->_copy($y->{_m}); |
| 1866 | |
| 1867 | # 2e1 => 20 |
| 1868 | $MBI->_lsft( $ym, $y->{_e}, 10) |
| 1869 | if $y->{_es} eq '+' && !$MBI->_is_zero($y->{_e}); |
| 1870 | |
| 1871 | # if $y has digits after dot |
| 1872 | my $shifty = 0; # correct _e of $x by this |
| 1873 | if ($y->{_es} eq '-') # has digits after dot |
| 1874 | { |
| 1875 | # 123 % 2.5 => 1230 % 25 => 5 => 0.5 |
| 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 |
| 1878 | } |
| 1879 | # $ym is now mantissa of $y based on exponent 0 |
| 1880 | |
| 1881 | my $shiftx = 0; # correct _e of $x by this |
| 1882 | if ($x->{_es} eq '-') # has digits after dot |
| 1883 | { |
| 1884 | # 123.4 % 20 => 1234 % 200 |
| 1885 | $shiftx = $MBI->_num($x->{_e}); # no more digits after dot |
| 1886 | $MBI->_lsft($ym, $x->{_e}, 10); # 123 => 1230 |
| 1887 | } |
| 1888 | # 123e1 % 20 => 1230 % 20 |
| 1889 | if ($x->{_es} eq '+' && !$MBI->_is_zero($x->{_e})) |
| 1890 | { |
| 1891 | $MBI->_lsft( $x->{_m}, $x->{_e},10); # es => '+' here |
| 1892 | } |
| 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; |
| 1898 | |
| 1899 | # now mantissas are equalized, exponent of $x is adjusted, so calc result |
| 1900 | |
| 1901 | $x->{_m} = $MBI->_mod( $x->{_m}, $ym); |
| 1902 | |
| 1903 | $x->{sign} = '+' if $MBI->_is_zero($x->{_m}); # fix sign for -0 |
| 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}; |
| 1911 | $x->{_es} = $r->{_es}; |
| 1912 | $x->{sign} = '+' if $MBI->_is_zero($x->{_m}); # fix sign for -0 |
| 1913 | $x->bnorm(); |
| 1914 | } |
| 1915 | |
| 1916 | $x->round($a,$p,$r,$y); # round and return |
| 1917 | } |
| 1918 | |
| 1919 | sub broot |
| 1920 | { |
| 1921 | # calculate $y'th root of $x |
| 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 | } |
| 1930 | |
| 1931 | return $x if $x->modify('broot'); |
| 1932 | |
| 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 |
| 1952 | $params[2] = $r; # iound mode by caller or undef |
| 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 |
| 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 | } |
| 1985 | |
| 1986 | # normal square root if $y == 2: |
| 1987 | if ($is_two) |
| 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}; |
| 1998 | $x->{_es} = $u->{_es}; |
| 1999 | } |
| 2000 | else |
| 2001 | { |
| 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 | { |
| 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; |
| 2012 | $int->broot($y->as_number()); |
| 2013 | # if ($exact) |
| 2014 | if ($int->copy()->bpow($y) == $x) |
| 2015 | { |
| 2016 | # found result, return it |
| 2017 | $x->{_m} = $int->{value}; |
| 2018 | $x->{_e} = $MBI->_zero(); |
| 2019 | $x->{_es} = '+'; |
| 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 | } |
| 2030 | } |
| 2031 | $x->bneg() if $sign == 1; |
| 2032 | |
| 2033 | # shortcut to not run through _find_round_parameters again |
| 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 |
| 2045 | delete $x->{_a}; delete $x->{_p}; |
| 2046 | } |
| 2047 | # restore globals |
| 2048 | $$abr = $ab; $$pbr = $pb; |
| 2049 | $x; |
| 2050 | } |
| 2051 | |
| 2052 | sub bsqrt |
| 2053 | { |
| 2054 | # calculate square root |
| 2055 | my ($self,$x,$a,$p,$r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_); |
| 2056 | |
| 2057 | return $x if $x->modify('bsqrt'); |
| 2058 | |
| 2059 | return $x->bnan() if $x->{sign} !~ /^[+]/; # NaN, -inf or < 0 |
| 2060 | return $x if $x->{sign} eq '+inf'; # sqrt(inf) == inf |
| 2061 | return $x->round($a,$p,$r) if $x->is_zero() || $x->is_one(); |
| 2062 | |
| 2063 | # we need to limit the accuracy to protect against overflow |
| 2064 | my $fallback = 0; |
| 2065 | my (@params,$scale); |
| 2066 | ($x,@params) = $x->_find_round_parameters($a,$p,$r); |
| 2067 | |
| 2068 | return $x if $x->is_nan(); # error in _find_round_parameters? |
| 2069 | |
| 2070 | # no rounding at all, so must use fallback |
| 2071 | if (scalar @params == 0) |
| 2072 | { |
| 2073 | # simulate old behaviour |
| 2074 | $params[0] = $self->div_scale(); # and round to it as accuracy |
| 2075 | $scale = $params[0]+4; # at least four more for proper round |
| 2076 | $params[2] = $r; # round mode by caller or undef |
| 2077 | $fallback = 1; # to clear a/p afterwards |
| 2078 | } |
| 2079 | else |
| 2080 | { |
| 2081 | # the 4 below is empirical, and there might be cases where it is not |
| 2082 | # enough... |
| 2083 | $scale = abs($params[0] || $params[1]) + 4; # take whatever is defined |
| 2084 | } |
| 2085 | |
| 2086 | # when user set globals, they would interfere with our calculation, so |
| 2087 | # disable them and later re-enable them |
| 2088 | no strict 'refs'; |
| 2089 | my $abr = "$self\::accuracy"; my $ab = $$abr; $$abr = undef; |
| 2090 | my $pbr = "$self\::precision"; my $pb = $$pbr; $$pbr = undef; |
| 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}; |
| 2094 | # need to disable $upgrade in BigInt, to avoid deep recursion |
| 2095 | local $Math::BigInt::upgrade = undef; # should be really parent class vs MBI |
| 2096 | |
| 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 | |
| 2102 | my $gs = $xas->copy()->bsqrt(); # some guess |
| 2103 | |
| 2104 | if (($x->{_es} ne '-') # guess can't be accurate if there are |
| 2105 | # digits after the dot |
| 2106 | && ($xas->bacmp($gs * $gs) == 0)) # guess hit the nail on the head? |
| 2107 | { |
| 2108 | # exact result, copy result over to keep $x |
| 2109 | $x->{_m} = $gs->{value}; $x->{_e} = $MBI->_zero(); $x->{_es} = '+'; |
| 2110 | $x->bnorm(); |
| 2111 | # shortcut to not run through _find_round_parameters again |
| 2112 | if (defined $params[0]) |
| 2113 | { |
| 2114 | $x->bround($params[0],$params[2]); # then round accordingly |
| 2115 | } |
| 2116 | else |
| 2117 | { |
| 2118 | $x->bfround($params[1],$params[2]); # then round accordingly |
| 2119 | } |
| 2120 | if ($fallback) |
| 2121 | { |
| 2122 | # clear a/p after round, since user did not request it |
| 2123 | delete $x->{_a}; delete $x->{_p}; |
| 2124 | } |
| 2125 | # re-enable A and P, upgrade is taken care of by "local" |
| 2126 | ${"$self\::accuracy"} = $ab; ${"$self\::precision"} = $pb; |
| 2127 | return $x; |
| 2128 | } |
| 2129 | |
| 2130 | # sqrt(2) = 1.4 because sqrt(2*100) = 1.4*10; so we can increase the accuracy |
| 2131 | # of the result by multiplying the input by 100 and then divide the integer |
| 2132 | # result of sqrt(input) by 10. Rounding afterwards returns the real result. |
| 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 | |
| 2145 | # This happens if the input had enough digits |
| 2146 | # (we take care of integer guesses above) |
| 2147 | $shift = 0 if $shift < 0; |
| 2148 | |
| 2149 | # Multiply in steps of 100, by shifting left two times the "missing" digits |
| 2150 | my $s2 = $shift * 2; |
| 2151 | |
| 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). |
| 2158 | $s2++ if $MBI->_is_odd($x->{_e}); |
| 2159 | |
| 2160 | $MBI->_lsft( $y1, $MBI->_new($s2), 10); |
| 2161 | |
| 2162 | # now take the square root and truncate to integer |
| 2163 | $y1 = $MBI->_sqrt($y1); |
| 2164 | |
| 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. |
| 2167 | |
| 2168 | # calculate how many zeros $x had after the '.' (or before it, depending |
| 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; |
| 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 | } |
| 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 | } |
| 2197 | $x->{_m} = $y1; |
| 2198 | $x->bnorm(); |
| 2199 | |
| 2200 | # shortcut to not run through _find_round_parameters again |
| 2201 | if (defined $params[0]) |
| 2202 | { |
| 2203 | $x->bround($params[0],$params[2]); # then round accordingly |
| 2204 | } |
| 2205 | else |
| 2206 | { |
| 2207 | $x->bfround($params[1],$params[2]); # then round accordingly |
| 2208 | } |
| 2209 | if ($fallback) |
| 2210 | { |
| 2211 | # clear a/p after round, since user did not request it |
| 2212 | delete $x->{_a}; delete $x->{_p}; |
| 2213 | } |
| 2214 | # restore globals |
| 2215 | $$abr = $ab; $$pbr = $pb; |
| 2216 | $x; |
| 2217 | } |
| 2218 | |
| 2219 | sub bfac |
| 2220 | { |
| 2221 | # (BFLOAT or num_str, BFLOAT or num_str) return BFLOAT |
| 2222 | # compute factorial number, modifies first argument |
| 2223 | |
| 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 | |
| 2229 | # inf => inf |
| 2230 | return $x if $x->modify('bfac') || $x->{sign} eq '+inf'; |
| 2231 | |
| 2232 | return $x->bnan() |
| 2233 | if (($x->{sign} ne '+') || # inf, NaN, <0 etc => NaN |
| 2234 | ($x->{_es} ne '+')); # digits after dot? |
| 2235 | |
| 2236 | # use BigInt's bfac() for faster calc |
| 2237 | if (! $MBI->_is_zero($x->{_e})) |
| 2238 | { |
| 2239 | $MBI->_lsft($x->{_m}, $x->{_e},10); # change 12e1 to 120e0 |
| 2240 | $x->{_e} = $MBI->_zero(); # normalize |
| 2241 | $x->{_es} = '+'; |
| 2242 | } |
| 2243 | $MBI->_fac($x->{_m}); # calculate factorial |
| 2244 | $x->bnorm()->round(@r); # norm again and round result |
| 2245 | } |
| 2246 | |
| 2247 | sub _pow |
| 2248 | { |
| 2249 | # Calculate a power where $y is a non-integer, like 2 ** 0.3 |
| 2250 | my ($x,$y,@r) = @_; |
| 2251 | my $self = ref($x); |
| 2252 | |
| 2253 | # if $y == 0.5, it is sqrt($x) |
| 2254 | $HALF = $self->new($HALF) unless ref($HALF); |
| 2255 | return $x->bsqrt(@r,$y) if $y->bcmp($HALF) == 0; |
| 2256 | |
| 2257 | # Using: |
| 2258 | # a ** x == e ** (x * ln a) |
| 2259 | |
| 2260 | # u = y * ln x |
| 2261 | # _ _ |
| 2262 | # Taylor: | u u^2 u^3 | |
| 2263 | # x ** y = 1 + | --- + --- + ----- + ... | |
| 2264 | # |_ 1 1*2 1*2*3 _| |
| 2265 | |
| 2266 | # we need to limit the accuracy to protect against overflow |
| 2267 | my $fallback = 0; |
| 2268 | my ($scale,@params); |
| 2269 | ($x,@params) = $x->_find_round_parameters(@r); |
| 2270 | |
| 2271 | return $x if $x->is_nan(); # error in _find_round_parameters? |
| 2272 | |
| 2273 | # no rounding at all, so must use fallback |
| 2274 | if (scalar @params == 0) |
| 2275 | { |
| 2276 | # simulate old behaviour |
| 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 |
| 2280 | $params[2] = $r[2]; # round mode by caller or undef |
| 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... |
| 2287 | $scale = abs($params[0] || $params[1]) + 4; # take whatever is defined |
| 2288 | } |
| 2289 | |
| 2290 | # when user set globals, they would interfere with our calculation, so |
| 2291 | # disable them and later re-enable them |
| 2292 | no strict 'refs'; |
| 2293 | my $abr = "$self\::accuracy"; my $ab = $$abr; $$abr = undef; |
| 2294 | my $pbr = "$self\::precision"; my $pb = $$pbr; $$pbr = undef; |
| 2295 | # we also need to disable any set A or P on $x (_find_round_parameters took |
| 2296 | # them already into account), since these would interfere, too |
| 2297 | delete $x->{_a}; delete $x->{_p}; |
| 2298 | # need to disable $upgrade in BigInt, to avoid deep recursion |
| 2299 | local $Math::BigInt::upgrade = undef; |
| 2300 | |
| 2301 | my ($limit,$v,$u,$below,$factor,$next,$over); |
| 2302 | |
| 2303 | $u = $x->copy()->blog(undef,$scale)->bmul($y); |
| 2304 | $v = $self->bone(); # 1 |
| 2305 | $factor = $self->new(2); # 2 |
| 2306 | $x->bone(); # first term: 1 |
| 2307 | |
| 2308 | $below = $v->copy(); |
| 2309 | $over = $u->copy(); |
| 2310 | |
| 2311 | $limit = $self->new("1E-". ($scale-1)); |
| 2312 | #my $steps = 0; |
| 2313 | while (3 < 5) |
| 2314 | { |
| 2315 | # we calculate the next term, and add it to the last |
| 2316 | # when the next term is below our limit, it won't affect the outcome |
| 2317 | # anymore, so we stop: |
| 2318 | $next = $over->copy()->bdiv($below,$scale); |
| 2319 | last if $next->bacmp($limit) <= 0; |
| 2320 | $x->badd($next); |
| 2321 | # calculate things for the next term |
| 2322 | $over *= $u; $below *= $factor; $factor->binc(); |
| 2323 | |
| 2324 | last if $x->{sign} !~ /^[-+]$/; |
| 2325 | |
| 2326 | #$steps++; |
| 2327 | } |
| 2328 | |
| 2329 | # shortcut to not run through _find_round_parameters again |
| 2330 | if (defined $params[0]) |
| 2331 | { |
| 2332 | $x->bround($params[0],$params[2]); # then round accordingly |
| 2333 | } |
| 2334 | else |
| 2335 | { |
| 2336 | $x->bfround($params[1],$params[2]); # then round accordingly |
| 2337 | } |
| 2338 | if ($fallback) |
| 2339 | { |
| 2340 | # clear a/p after round, since user did not request it |
| 2341 | delete $x->{_a}; delete $x->{_p}; |
| 2342 | } |
| 2343 | # restore globals |
| 2344 | $$abr = $ab; $$pbr = $pb; |
| 2345 | $x; |
| 2346 | } |
| 2347 | |
| 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 | |
| 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 | } |
| 2361 | |
| 2362 | return $x if $x->modify('bpow'); |
| 2363 | |
| 2364 | return $x->bnan() if $x->{sign} eq $nan || $y->{sign} eq $nan; |
| 2365 | return $x if $x->{sign} =~ /^[+-]inf$/; |
| 2366 | |
| 2367 | # cache the result of is_zero |
| 2368 | my $y_is_zero = $y->is_zero(); |
| 2369 | return $x->bone() if $y_is_zero; |
| 2370 | return $x if $x->is_one() || $y->is_one(); |
| 2371 | |
| 2372 | my $x_is_zero = $x->is_zero(); |
| 2373 | return $x->_pow($y,$a,$p,$r) if !$x_is_zero && !$y->is_int(); # non-integer power |
| 2374 | |
| 2375 | my $y1 = $y->as_number()->{value}; # make MBI part |
| 2376 | |
| 2377 | # if ($x == -1) |
| 2378 | if ($x->{sign} eq '-' && $MBI->_is_one($x->{_m}) && $MBI->_is_zero($x->{_e})) |
| 2379 | { |
| 2380 | # if $x == -1 and odd/even y => +1/-1 because +-1 ^ (+-1) => +-1 |
| 2381 | return $MBI->_is_odd($y1) ? $x : $x->babs(1); |
| 2382 | } |
| 2383 | if ($x_is_zero) |
| 2384 | { |
| 2385 | return $x if $y->{sign} eq '+'; # 0**y => 0 (if not y <= 0) |
| 2386 | # 0 ** -y => 1 / (0 ** y) => 1 / 0! (1 / 0 => +inf) |
| 2387 | return $x->binf(); |
| 2388 | } |
| 2389 | |
| 2390 | my $new_sign = '+'; |
| 2391 | $new_sign = $MBI->_is_odd($y1) ? '-' : '+' if $x->{sign} ne '+'; |
| 2392 | |
| 2393 | # calculate $x->{_m} ** $y and $x->{_e} * $y separately (faster) |
| 2394 | $x->{_m} = $MBI->_pow( $x->{_m}, $y1); |
| 2395 | $x->{_e} = $MBI->_mul ($x->{_e}, $y1); |
| 2396 | |
| 2397 | $x->{sign} = $new_sign; |
| 2398 | $x->bnorm(); |
| 2399 | if ($y->{sign} eq '-') |
| 2400 | { |
| 2401 | # modify $x in place! |
| 2402 | my $z = $x->copy(); $x->bone(); |
| 2403 | return scalar $x->bdiv($z,$a,$p,$r); # round in one go (might ignore y's A!) |
| 2404 | } |
| 2405 | $x->round($a,$p,$r,$y); |
| 2406 | } |
| 2407 | |
| 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 | |
| 2439 | ############################################################################### |
| 2440 | # trigonometric functions |
| 2441 | |
| 2442 | # helper function for bpi() and batan2(), calculates arcus tanges (1/x) |
| 2443 | |
| 2444 | sub _atan_inv |
| 2445 | { |
| 2446 | # return a/b so that a/b approximates atan(1/x) to at least limit digits |
| 2447 | my ($self, $x, $limit) = @_; |
| 2448 | |
| 2449 | # Taylor: x^3 x^5 x^7 x^9 |
| 2450 | # atan = x - --- + --- - --- + --- - ... |
| 2451 | # 3 5 7 9 |
| 2452 | |
| 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(); |
| 2491 | my $b = $MBI->_copy($x); |
| 2492 | |
| 2493 | my $x2 = $MBI->_mul( $MBI->_copy($x), $b); # x2 = x * x |
| 2494 | my $d = $MBI->_new( 3 ); # d = 3 |
| 2495 | my $c = $MBI->_mul( $MBI->_copy($x), $x2); # c = x ^ 3 |
| 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) |
| 2529 | { |
| 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)) |
| 2545 | { |
| 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); |
| 2552 | } |
| 2553 | else |
| 2554 | { |
| 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); |
| 2563 | } |
| 2564 | $d = $MBI->_add($d, $two); |
| 2565 | $c = $MBI->_mul($c, $x2); |
| 2566 | $sign = 1 - $sign; |
| 2567 | |
| 2568 | } |
| 2569 | |
| 2570 | # print "Took $step steps for ", $MBI->_str($x),"\n"; |
| 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); |
| 2574 | } |
| 2575 | |
| 2576 | sub bpi |
| 2577 | { |
| 2578 | my ($self,$n) = @_; |
| 2579 | if (@_ == 0) |
| 2580 | { |
| 2581 | $self = $class; |
| 2582 | } |
| 2583 | if (@_ == 1) |
| 2584 | { |
| 2585 | # called like Math::BigFloat::bpi(10); |
| 2586 | $n = $self; $self = $class; |
| 2587 | # called like Math::BigFloat->bpi(); |
| 2588 | $n = undef if $n eq 'Math::BigFloat'; |
| 2589 | } |
| 2590 | $self = ref($self) if ref($self); |
| 2591 | my $fallback = defined $n ? 0 : 1; |
| 2592 | $n = 40 if !defined $n || $n < 1; |
| 2593 | |
| 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 | |
| 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); |
| 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 | |
| 2628 | delete $x->{_a}; delete $y->{_a}; delete $z->{_a}; |
| 2629 | delete $u->{_a}; delete $v->{_a}; delete $w->{_a}; |
| 2630 | $x->badd($y)->bsub($z)->badd($u)->bsub($v)->bsub($w); |
| 2631 | |
| 2632 | $x->bround($n-4); |
| 2633 | delete $x->{_a} if $fallback == 1; |
| 2634 | $x; |
| 2635 | } |
| 2636 | |
| 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); |
| 2689 | $x->bone(); delete $x->{_a}; delete $x->{_p}; |
| 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); |
| 2788 | delete $x->{_a}; delete $x->{_p}; |
| 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 | |
| 2834 | sub batan2 |
| 2835 | { |
| 2836 | # calculate arcus tangens of ($y/$x) |
| 2837 | |
| 2838 | # set up parameters |
| 2839 | my ($self,$y,$x,@r) = (ref($_[0]),@_); |
| 2840 | # objectify is costly, so avoid it |
| 2841 | if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1]))) |
| 2842 | { |
| 2843 | ($self,$y,$x,@r) = objectify(2,@_); |
| 2844 | } |
| 2845 | |
| 2846 | return $y if $y->modify('batan2'); |
| 2847 | |
| 2848 | return $y->bnan() if ($y->{sign} eq $nan) || ($x->{sign} eq $nan); |
| 2849 | |
| 2850 | # Y X |
| 2851 | # 0 0 result is 0 |
| 2852 | # 0 +x result is 0 |
| 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; |
| 2893 | |
| 2894 | # Y X |
| 2895 | # 0 -x result is PI |
| 2896 | if ($y->is_zero()) |
| 2897 | { |
| 2898 | # calculate PI |
| 2899 | my $pi = $self->bpi(@r); |
| 2900 | # modify $y in place |
| 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 |
| 2911 | if ($x->is_zero()) |
| 2912 | { |
| 2913 | # calculate PI/2 |
| 2914 | my $pi = $self->bpi(@r); |
| 2915 | # modify $y in place |
| 2916 | $y->{_m} = $pi->{_m}; |
| 2917 | $y->{_e} = $pi->{_e}; |
| 2918 | $y->{_es} = $pi->{_es}; |
| 2919 | # -y => -PI/2, +y => PI/2 |
| 2920 | $MBI->_div($y->{_m}, $MBI->_new(2)); |
| 2921 | return $y; |
| 2922 | } |
| 2923 | |
| 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); |
| 2958 | # modify $y in place |
| 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}; |
| 2996 | |
| 2997 | # divide $x by $y |
| 2998 | $y->bdiv($x, $scale) unless $x->is_one(); |
| 2999 | $y->batan(@r); |
| 3000 | |
| 3001 | # restore sign |
| 3002 | $y->{sign} = $y_sign; |
| 3003 | |
| 3004 | $y; |
| 3005 | } |
| 3006 | |
| 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 | |
| 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 | |
| 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 | |
| 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 | |
| 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 |
| 3075 | # to calculate PI/2 - atan(1/x): |
| 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 | |
| 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); |
| 3107 | delete $x->{_a}; delete $x->{_p}; |
| 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 | |
| 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 | |
| 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 | |
| 3163 | ############################################################################### |
| 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! |
| 3171 | my $x = shift; my $self = ref($x) || $x; $x = $self->new(shift) if !ref($x); |
| 3172 | |
| 3173 | my ($scale,$mode) = $x->_scale_p(@_); |
| 3174 | return $x if !defined $scale || $x->modify('bfround'); # no-op |
| 3175 | |
| 3176 | # never round a 0, +-inf, NaN |
| 3177 | if ($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 $x->{sign} !~ /^[+-]$/; |
| 3183 | |
| 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 |
| 3188 | delete $x->{_a}; # and clear A |
| 3189 | if ($scale < 0) |
| 3190 | { |
| 3191 | # round right from the '.' |
| 3192 | |
| 3193 | return $x if $x->{_es} eq '+'; # e >= 0 => nothing to round |
| 3194 | |
| 3195 | $scale = -$scale; # positive for simplicity |
| 3196 | my $len = $MBI->_len($x->{_m}); # length of mantissa |
| 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 |
| 3200 | my $dad = -(0+ ($x->{_es}.$MBI->_num($x->{_e}))); # digits after dot |
| 3201 | my $zad = 0; # zeros after dot |
| 3202 | $zad = $dad - $len if (-$dad < -$len); # for 0.00..00xxx style |
| 3203 | |
| 3204 | # p rint "scale $scale dad $dad zad $zad len $len\n"; |
| 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 | |
| 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 |
| 3219 | { |
| 3220 | $scale = -$len; |
| 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 | } |
| 3235 | } |
| 3236 | else |
| 3237 | { |
| 3238 | # round left from the '.' |
| 3239 | |
| 3240 | # 123 => 100 means length(123) = 3 - $scale (2) => 1 |
| 3241 | |
| 3242 | my $dbt = $MBI->_len($x->{_m}); |
| 3243 | # digits before dot |
| 3244 | my $dbd = $dbt + ($x->{_es} . $MBI->_num($x->{_e})); |
| 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 | } |
| 3262 | else |
| 3263 | { |
| 3264 | $scale = $dbd - $scale; |
| 3265 | } |
| 3266 | } |
| 3267 | # pass sign to bround for rounding modes '+inf' and '-inf' |
| 3268 | my $m = bless { sign => $x->{sign}, value => $x->{_m} }, 'Math::BigInt'; |
| 3269 | $m->bround($scale,$mode); |
| 3270 | $x->{_m} = $m->{value}; # get our mantissa back |
| 3271 | $x->bnorm(); |
| 3272 | } |
| 3273 | |
| 3274 | sub bround |
| 3275 | { |
| 3276 | # accuracy: preserve $N digits, and overwrite the rest with 0's |
| 3277 | my $x = shift; my $self = ref($x) || $x; $x = $self->new(shift) if !ref($x); |
| 3278 | |
| 3279 | if (($_[0] || 0) < 0) |
| 3280 | { |
| 3281 | require Carp; Carp::croak ('bround() needs positive accuracy'); |
| 3282 | } |
| 3283 | |
| 3284 | my ($scale,$mode) = $x->_scale_a(@_); |
| 3285 | return $x if !defined $scale || $x->modify('bround'); # no-op |
| 3286 | |
| 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 |
| 3291 | return $x if defined $x->{_a} && $x->{_a} < $scale; |
| 3292 | |
| 3293 | # scale < 0 makes no sense |
| 3294 | # scale == 0 => keep all digits |
| 3295 | # never round a +-inf, NaN |
| 3296 | return $x if ($scale <= 0) || $x->{sign} !~ /^[+-]$/; |
| 3297 | |
| 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) |
| 3301 | { |
| 3302 | $x->{_a} = $scale if !defined $x->{_a} || $x->{_a} > $scale; |
| 3303 | return $x; |
| 3304 | } |
| 3305 | |
| 3306 | # pass sign to bround for '+inf' and '-inf' rounding modes |
| 3307 | my $m = bless { sign => $x->{sign}, value => $x->{_m} }, 'Math::BigInt'; |
| 3308 | |
| 3309 | $m->bround($scale,$mode); # round mantissa |
| 3310 | $x->{_m} = $m->{value}; # get our mantissa back |
| 3311 | $x->{_a} = $scale; # remember rounding |
| 3312 | delete $x->{_p}; # and clear P |
| 3313 | $x->bnorm(); # del trailing zeros gen. by bround() |
| 3314 | } |
| 3315 | |
| 3316 | sub bfloor |
| 3317 | { |
| 3318 | # return integer less or equal then $x |
| 3319 | my ($self,$x,$a,$p,$r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_); |
| 3320 | |
| 3321 | return $x if $x->modify('bfloor'); |
| 3322 | |
| 3323 | return $x if $x->{sign} !~ /^[+-]$/; # nan, +inf, -inf |
| 3324 | |
| 3325 | # if $x has digits after dot |
| 3326 | if ($x->{_es} eq '-') |
| 3327 | { |
| 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 |
| 3332 | } |
| 3333 | $x->round($a,$p,$r); |
| 3334 | } |
| 3335 | |
| 3336 | sub bceil |
| 3337 | { |
| 3338 | # return integer greater or equal then $x |
| 3339 | my ($self,$x,$a,$p,$r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_); |
| 3340 | |
| 3341 | return $x if $x->modify('bceil'); |
| 3342 | return $x if $x->{sign} !~ /^[+-]$/; # nan, +inf, -inf |
| 3343 | |
| 3344 | # if $x has digits after dot |
| 3345 | if ($x->{_es} eq '-') |
| 3346 | { |
| 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 |
| 3351 | } |
| 3352 | $x->round($a,$p,$r); |
| 3353 | } |
| 3354 | |
| 3355 | sub brsft |
| 3356 | { |
| 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 | } |
| 3366 | |
| 3367 | return $x if $x->modify('brsft'); |
| 3368 | return $x if $x->{sign} !~ /^[+-]$/; # nan, +inf, -inf |
| 3369 | |
| 3370 | $n = 2 if !defined $n; $n = $self->new($n); |
| 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): |
| 3376 | $x->bdiv($n->bpow($y),$a,$p,$r,$y); |
| 3377 | } |
| 3378 | |
| 3379 | sub blsft |
| 3380 | { |
| 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 | } |
| 3390 | |
| 3391 | return $x if $x->modify('blsft'); |
| 3392 | return $x if $x->{sign} !~ /^[+-]$/; # nan, +inf, -inf |
| 3393 | |
| 3394 | $n = 2 if !defined $n; $n = $self->new($n); |
| 3395 | |
| 3396 | # negative amount? |
| 3397 | return $x->brsft($y->copy()->babs(),$n) if $y->{sign} =~ /^-/; |
| 3398 | |
| 3399 | $x->bmul($n->bpow($y),$a,$p,$r,$y); |
| 3400 | } |
| 3401 | |
| 3402 | ############################################################################### |
| 3403 | |
| 3404 | sub DESTROY |
| 3405 | { |
| 3406 | # going through AUTOLOAD for every DESTROY is costly, avoid it by empty sub |
| 3407 | } |
| 3408 | |
| 3409 | sub AUTOLOAD |
| 3410 | { |
| 3411 | # make fxxx and bxxx both work by selectively mapping fxxx() to MBF::bxxx() |
| 3412 | # or falling back to MBI::bxxx() |
| 3413 | my $name = $AUTOLOAD; |
| 3414 | |
| 3415 | $name =~ s/(.*):://; # split package |
| 3416 | my $c = $1 || $class; |
| 3417 | no strict 'refs'; |
| 3418 | $c->import() if $IMPORT == 0; |
| 3419 | if (!_method_alias($name)) |
| 3420 | { |
| 3421 | if (!defined $name) |
| 3422 | { |
| 3423 | # delayed load of Carp and avoid recursion |
| 3424 | require Carp; |
| 3425 | Carp::croak ("$c: Can't call a method without name"); |
| 3426 | } |
| 3427 | if (!_method_hand_up($name)) |
| 3428 | { |
| 3429 | # delayed load of Carp and avoid recursion |
| 3430 | require Carp; |
| 3431 | Carp::croak ("Can't call $c\-\>$name, not a valid method"); |
| 3432 | } |
| 3433 | # try one level up, but subst. bxxx() for fxxx() since MBI only got bxxx() |
| 3434 | $name =~ s/^f/b/; |
| 3435 | return &{"Math::BigInt"."::$name"}(@_); |
| 3436 | } |
| 3437 | my $bname = $name; $bname =~ s/^f/b/; |
| 3438 | $c .= "::$name"; |
| 3439 | *{$c} = \&{$bname}; |
| 3440 | &{$c}; # uses @_ |
| 3441 | } |
| 3442 | |
| 3443 | sub exponent |
| 3444 | { |
| 3445 | # return a copy of the exponent |
| 3446 | my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_); |
| 3447 | |
| 3448 | if ($x->{sign} !~ /^[+-]$/) |
| 3449 | { |
| 3450 | my $s = $x->{sign}; $s =~ s/^[+-]//; |
| 3451 | return Math::BigInt->new($s); # -inf, +inf => +inf |
| 3452 | } |
| 3453 | Math::BigInt->new( $x->{_es} . $MBI->_str($x->{_e})); |
| 3454 | } |
| 3455 | |
| 3456 | sub mantissa |
| 3457 | { |
| 3458 | # return a copy of the mantissa |
| 3459 | my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_); |
| 3460 | |
| 3461 | if ($x->{sign} !~ /^[+-]$/) |
| 3462 | { |
| 3463 | my $s = $x->{sign}; $s =~ s/^[+]//; |
| 3464 | return Math::BigInt->new($s); # -inf, +inf => +inf |
| 3465 | } |
| 3466 | my $m = Math::BigInt->new( $MBI->_str($x->{_m})); |
| 3467 | $m->bneg() if $x->{sign} eq '-'; |
| 3468 | |
| 3469 | $m; |
| 3470 | } |
| 3471 | |
| 3472 | sub parts |
| 3473 | { |
| 3474 | # return a copy of both the exponent and the mantissa |
| 3475 | my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_); |
| 3476 | |
| 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 | } |
| 3482 | my $m = Math::BigInt->bzero(); |
| 3483 | $m->{value} = $MBI->_copy($x->{_m}); |
| 3484 | $m->bneg() if $x->{sign} eq '-'; |
| 3485 | ($m, Math::BigInt->new( $x->{_es} . $MBI->_num($x->{_e}) )); |
| 3486 | } |
| 3487 | |
| 3488 | ############################################################################## |
| 3489 | # private stuff (internal use only) |
| 3490 | |
| 3491 | sub import |
| 3492 | { |
| 3493 | my $self = shift; |
| 3494 | my $l = scalar @_; |
| 3495 | my $lib = ''; my @a; |
| 3496 | my $lib_kind = 'try'; |
| 3497 | $IMPORT=1; |
| 3498 | for ( my $i = 0; $i < $l ; $i++) |
| 3499 | { |
| 3500 | if ( $_[$i] eq ':constant' ) |
| 3501 | { |
| 3502 | # This causes overlord er load to step in. 'binary' and 'integer' |
| 3503 | # are handled by BigInt. |
| 3504 | overload::constant float => sub { $self->new(shift); }; |
| 3505 | } |
| 3506 | elsif ($_[$i] eq 'upgrade') |
| 3507 | { |
| 3508 | # this causes upgrading |
| 3509 | $upgrade = $_[$i+1]; # or undef to disable |
| 3510 | $i++; |
| 3511 | } |
| 3512 | elsif ($_[$i] eq 'downgrade') |
| 3513 | { |
| 3514 | # this causes downgrading |
| 3515 | $downgrade = $_[$i+1]; # or undef to disable |
| 3516 | $i++; |
| 3517 | } |
| 3518 | elsif ($_[$i] =~ /^(lib|try|only)\z/) |
| 3519 | { |
| 3520 | # alternative library |
| 3521 | $lib = $_[$i+1] || ''; # default Calc |
| 3522 | $lib_kind = $1; # lib, try or only |
| 3523 | $i++; |
| 3524 | } |
| 3525 | elsif ($_[$i] eq 'with') |
| 3526 | { |
| 3527 | # alternative class for our private parts() |
| 3528 | # XXX: no longer supported |
| 3529 | # $MBI = $_[$i+1] || 'Math::BigInt'; |
| 3530 | $i++; |
| 3531 | } |
| 3532 | else |
| 3533 | { |
| 3534 | push @a, $_[$i]; |
| 3535 | } |
| 3536 | } |
| 3537 | |
| 3538 | $lib =~ tr/a-zA-Z0-9,://cd; # restrict to sane characters |
| 3539 | # let use Math::BigInt lib => 'GMP'; use Math::BigFloat; still work |
| 3540 | my $mbilib = eval { Math::BigInt->config()->{lib} }; |
| 3541 | if ((defined $mbilib) && ($MBI eq 'Math::BigInt::Calc')) |
| 3542 | { |
| 3543 | # MBI already loaded |
| 3544 | Math::BigInt->import( $lib_kind, "$lib,$mbilib", 'objectify'); |
| 3545 | } |
| 3546 | else |
| 3547 | { |
| 3548 | # MBI not loaded, or with ne "Math::BigInt::Calc" |
| 3549 | $lib .= ",$mbilib" if defined $mbilib; |
| 3550 | $lib =~ s/^,//; # don't leave empty |
| 3551 | |
| 3552 | # replacement library can handle lib statement, but also could ignore it |
| 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; |
| 3557 | Math::BigInt->import( $lib_kind => $lib, 'objectify' ); |
| 3558 | } |
| 3559 | if ($@) |
| 3560 | { |
| 3561 | require Carp; Carp::croak ("Couldn't load $lib: $! $@"); |
| 3562 | } |
| 3563 | # find out which one was actually loaded |
| 3564 | $MBI = Math::BigInt->config()->{lib}; |
| 3565 | |
| 3566 | # register us with MBI to get notified of future lib changes |
| 3567 | Math::BigInt::_register_callback( $self, sub { $MBI = $_[0]; } ); |
| 3568 | |
| 3569 | $self->export_to_level(1,$self,@a); # export wanted functions |
| 3570 | } |
| 3571 | |
| 3572 | sub bnorm |
| 3573 | { |
| 3574 | # adjust m and e so that m is smallest possible |
| 3575 | my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_); |
| 3576 | |
| 3577 | return $x if $x->{sign} !~ /^[+-]$/; # inf, nan etc |
| 3578 | |
| 3579 | my $zeros = $MBI->_zeros($x->{_m}); # correct for trailing zeros |
| 3580 | if ($zeros != 0) |
| 3581 | { |
| 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 | { |
| 3588 | $x->{_e} = $MBI->_sub ($x->{_e}, $z); |
| 3589 | $x->{_es} = '+' if $MBI->_is_zero($x->{_e}); |
| 3590 | } |
| 3591 | else |
| 3592 | { |
| 3593 | $x->{_e} = $MBI->_sub ( $MBI->_copy($z), $x->{_e}); |
| 3594 | $x->{_es} = '+'; |
| 3595 | } |
| 3596 | } |
| 3597 | else |
| 3598 | { |
| 3599 | $x->{_e} = $MBI->_add ($x->{_e}, $z); |
| 3600 | } |
| 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 |
| 3606 | $x->{sign} = '+', $x->{_es} = '+', $x->{_e} = $MBI->_one() |
| 3607 | if $MBI->_is_zero($x->{_m}); |
| 3608 | } |
| 3609 | |
| 3610 | $x; # MBI bnorm is no-op, so dont call it |
| 3611 | } |
| 3612 | |
| 3613 | ############################################################################## |
| 3614 | |
| 3615 | sub as_hex |
| 3616 | { |
| 3617 | # return number as hexadecimal string (only for integers defined) |
| 3618 | my ($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 | |
| 3623 | return $nan if $x->{_es} ne '+'; # how to do 1e-1 in hex!? |
| 3624 | |
| 3625 | my $z = $MBI->_copy($x->{_m}); |
| 3626 | if (! $MBI->_is_zero($x->{_e})) # > 0 |
| 3627 | { |
| 3628 | $MBI->_lsft( $z, $x->{_e},10); |
| 3629 | } |
| 3630 | $z = Math::BigInt->new( $x->{sign} . $MBI->_num($z)); |
| 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 | |
| 3642 | return $nan if $x->{_es} ne '+'; # how to do 1e-1 in hex!? |
| 3643 | |
| 3644 | my $z = $MBI->_copy($x->{_m}); |
| 3645 | if (! $MBI->_is_zero($x->{_e})) # > 0 |
| 3646 | { |
| 3647 | $MBI->_lsft( $z, $x->{_e},10); |
| 3648 | } |
| 3649 | $z = Math::BigInt->new( $x->{sign} . $MBI->_num($z)); |
| 3650 | $z->as_bin(); |
| 3651 | } |
| 3652 | |
| 3653 | sub as_oct |
| 3654 | { |
| 3655 | # return number as octal digit string (only for integers defined) |
| 3656 | my ($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 | $z->as_oct(); |
| 3670 | } |
| 3671 | |
| 3672 | sub as_number |
| 3673 | { |
| 3674 | # return copy as a bigint representation of this BigFloat number |
| 3675 | my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_); |
| 3676 | |
| 3677 | return $x if $x->modify('as_number'); |
| 3678 | |
| 3679 | if (!$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 | |
| 3687 | return Math::BigInt->binf($x->sign()) if $x->is_inf(); |
| 3688 | return Math::BigInt->bnan() if $x->is_nan(); |
| 3689 | |
| 3690 | my $z = $MBI->_copy($x->{_m}); |
| 3691 | if ($x->{_es} eq '-') # < 0 |
| 3692 | { |
| 3693 | $MBI->_rsft( $z, $x->{_e},10); |
| 3694 | } |
| 3695 | elsif (! $MBI->_is_zero($x->{_e})) # > 0 |
| 3696 | { |
| 3697 | $MBI->_lsft( $z, $x->{_e},10); |
| 3698 | } |
| 3699 | $z = Math::BigInt->new( $x->{sign} . $MBI->_str($z)); |
| 3700 | $z; |
| 3701 | } |
| 3702 | |
| 3703 | sub length |
| 3704 | { |
| 3705 | my $x = shift; |
| 3706 | my $class = ref($x) || $x; |
| 3707 | $x = $class->new(shift) unless ref($x); |
| 3708 | |
| 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 '+'; |
| 3713 | if (wantarray()) |
| 3714 | { |
| 3715 | my $t = 0; |
| 3716 | $t = $MBI->_num($x->{_e}) if $x->{_es} eq '-'; |
| 3717 | return ($len, $t); |
| 3718 | } |
| 3719 | $len; |
| 3720 | } |
| 3721 | |
| 3722 | 1; |
| 3723 | __END__ |
| 3724 | |
| 3725 | =head1 NAME |
| 3726 | |
| 3727 | Math::BigFloat - Arbitrary size floating point math package |
| 3728 | |
| 3729 | =head1 SYNOPSIS |
| 3730 | |
| 3731 | use Math::BigFloat; |
| 3732 | |
| 3733 | # Number creation |
| 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 |
| 3744 | |
| 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 | |
| 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 | |
| 3754 | # Testing |
| 3755 | $x->is_zero(); # true if arg is +0 |
| 3756 | $x->is_nan(); # true if arg is NaN |
| 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 |
| 3761 | $x->is_pos(); # true if >= 0 |
| 3762 | $x->is_neg(); # true if < 0 |
| 3763 | $x->is_inf(sign); # true if +inf, or -inf (default is '+') |
| 3764 | |
| 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 |
| 3768 | $x->digit($n); # return the nth digit, counting from right |
| 3769 | $x->digit(-$n); # return the nth digit, counting from left |
| 3770 | |
| 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 |
| 3773 | # necessary when mixing $a = $b assignments with non-overloaded math. |
| 3774 | |
| 3775 | # set |
| 3776 | $x->bzero(); # set $i to 0 |
| 3777 | $x->bnan(); # set $i to NaN |
| 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 |
| 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) |
| 3793 | $x->bdiv($y); # divide, set $x to quotient |
| 3794 | # return (quo,rem) or quo if scalar |
| 3795 | |
| 3796 | $x->bmod($y); # modulus ($x % $y) |
| 3797 | $x->bpow($y); # power of arguments ($x ** $y) |
| 3798 | $x->bmodpow($exp,$mod); # modular exponentation (($num**$exp) % $mod)) |
| 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 |
| 3802 | |
| 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) |
| 3805 | $x->bexp(); # calculate e ** $x where e is Euler's number |
| 3806 | |
| 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) |
| 3811 | |
| 3812 | $x->bsqrt(); # calculate square-root |
| 3813 | $x->broot($y); # $y'th root of $x (e.g. $y == 3 => cubic root) |
| 3814 | $x->bfac(); # factorial of $x (1*2*3*4*..$x) |
| 3815 | |
| 3816 | $x->bround($N); # accuracy: preserve $N digits |
| 3817 | $x->bfround($N); # precision: round to the $Nth digit |
| 3818 | |
| 3819 | $x->bfloor(); # return integer less or equal than $x |
| 3820 | $x->bceil(); # return integer greater or equal than $x |
| 3821 | |
| 3822 | # The following do not modify their arguments: |
| 3823 | |
| 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 |
| 3829 | |
| 3830 | $x->as_int(); # return $x as BigInt |
| 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 |
| 3837 | |
| 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) |
| 3841 | $x->accuracy($n); # set A $x to $n |
| 3842 | |
| 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 |
| 3847 | |
| 3848 | =head1 DESCRIPTION |
| 3849 | |
| 3850 | All operators (including basic math operations) are overloaded if you |
| 3851 | declare your big floating point numbers as |
| 3852 | |
| 3853 | $i = new Math::BigFloat '12_3.456_789_123_456_789E-2'; |
| 3854 | |
| 3855 | Operations with overloaded operators preserve the arguments, which is |
| 3856 | exactly what you expect. |
| 3857 | |
| 3858 | =head2 Canonical notation |
| 3859 | |
| 3860 | Input to these routines are either BigFloat objects, or strings of the |
| 3861 | following four forms: |
| 3862 | |
| 3863 | =over 2 |
| 3864 | |
| 3865 | =item * |
| 3866 | |
| 3867 | C</^[+-]\d+$/> |
| 3868 | |
| 3869 | =item * |
| 3870 | |
| 3871 | C</^[+-]\d+\.\d*$/> |
| 3872 | |
| 3873 | =item * |
| 3874 | |
| 3875 | C</^[+-]\d+E[+-]?\d+$/> |
| 3876 | |
| 3877 | =item * |
| 3878 | |
| 3879 | C</^[+-]\d*\.\d+E[+-]?\d+$/> |
| 3880 | |
| 3881 | =back |
| 3882 | |
| 3883 | all with optional leading and trailing zeros and/or spaces. Additionally, |
| 3884 | numbers are allowed to have an underscore between any two digits. |
| 3885 | |
| 3886 | Empty strings as well as other illegal numbers results in 'NaN'. |
| 3887 | |
| 3888 | bnorm() on a BigFloat object is now effectively a no-op, since the numbers |
| 3889 | are always stored in normalized form. On a string, it creates a BigFloat |
| 3890 | object. |
| 3891 | |
| 3892 | =head2 Output |
| 3893 | |
| 3894 | Output values are BigFloat objects (normalized), except for bstr() and bsstr(). |
| 3895 | |
| 3896 | The string output will always have leading and trailing zeros stripped and drop |
| 3897 | a plus sign. C<bstr()> will give you always the form with a decimal point, |
| 3898 | while C<bsstr()> (s for scientific) gives you the scientific notation. |
| 3899 | |
| 3900 | Input bstr() bsstr() |
| 3901 | '-0' '0' '0E1' |
| 3902 | ' -123 123 123' '-123123123' '-123123123E0' |
| 3903 | '00.0123' '0.0123' '123E-4' |
| 3904 | '123.45E-2' '1.2345' '12345E-4' |
| 3905 | '10E+3' '10000' '1E4' |
| 3906 | |
| 3907 | Some routines (C<is_odd()>, C<is_even()>, C<is_zero()>, C<is_one()>, |
| 3908 | C<is_nan()>) return true or false, while others (C<bcmp()>, C<bacmp()>) |
| 3909 | return either undef, <0, 0 or >0 and are suited for sort. |
| 3910 | |
| 3911 | Actual math is done by using the class defined with C<< with => Class; >> (which |
| 3912 | defaults to BigInts) to represent the mantissa and exponent. |
| 3913 | |
| 3914 | The sign C</^[+-]$/> 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 | |
| 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>. |
| 3944 | |
| 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. |
| 3948 | |
| 3949 | If there is no global precision or accuracy set, B<and> the operation in |
| 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: |
| 3954 | |
| 3955 | $d = Math::BigFloat->div_scale(); # query |
| 3956 | Math::BigFloat->div_scale($n); # set to $n digits |
| 3957 | |
| 3958 | The default value for C<div_scale> is 40. |
| 3959 | |
| 3960 | In case the result of one operation has more digits than specified, |
| 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); |
| 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 |
| 3969 | Math::BigFloat->round_mode('zero'); |
| 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 |
| 3974 | to the global rounding B<immediately>. This means that in the examples above, the |
| 3975 | C<3> as argument to C<bdiv()> will also get an accuracy of B<5>. |
| 3976 | |
| 3977 | It is less confusing to either calculate the result fully, and afterwards |
| 3978 | round it explicitly, or use the additional parameters to the math |
| 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"; |
| 3992 | |
| 3993 | =head2 Rounding |
| 3994 | |
| 3995 | =over 2 |
| 3996 | |
| 3997 | =item ffround ( +$scale ) |
| 3998 | |
| 3999 | Rounds to the $scale'th place left from the '.', counting from the dot. |
| 4000 | The first digit is numbered 1. |
| 4001 | |
| 4002 | =item ffround ( -$scale ) |
| 4003 | |
| 4004 | Rounds to the $scale'th place right from the '.', counting from the dot. |
| 4005 | |
| 4006 | =item ffround ( 0 ) |
| 4007 | |
| 4008 | Rounds to an integer. |
| 4009 | |
| 4010 | =item fround ( +$scale ) |
| 4011 | |
| 4012 | Preserves accuracy to $scale digits from the left (aka significant digits) |
| 4013 | and pads the rest with zeros. If the number is between 1 and -1, the |
| 4014 | significant digits count from the first non-zero after the '.' |
| 4015 | |
| 4016 | =item fround ( -$scale ) and fround ( 0 ) |
| 4017 | |
| 4018 | These are effectively no-ops. |
| 4019 | |
| 4020 | =back |
| 4021 | |
| 4022 | All rounding functions take as a second parameter a rounding mode from one of |
| 4023 | the following: 'even', 'odd', '+inf', '-inf', 'zero', 'trunc' or 'common'. |
| 4024 | |
| 4025 | The default rounding mode is 'even'. By using |
| 4026 | C<< Math::BigFloat->round_mode($round_mode); >> you can get and set the default |
| 4027 | mode for subsequent rounding. The usage of C<$Math::BigFloat::$round_mode> is |
| 4028 | no longer supported. |
| 4029 | The second parameter to the round functions then overrides the default |
| 4030 | temporarily. |
| 4031 | |
| 4032 | The C<as_number()> function returns a BigInt from a Math::BigFloat. It uses |
| 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 | |
| 4044 | =head1 METHODS |
| 4045 | |
| 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 | |
| 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 | |
| 4067 | In most cases, you should probably round the results explicitly using one of |
| 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! |
| 4091 | |
| 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 | |
| 4113 | =head2 bpi() |
| 4114 | |
| 4115 | print Math::BigFloat->bpi(100), "\n"; |
| 4116 | |
| 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". |
| 4119 | |
| 4120 | This method was added in v1.87 of Math::BigInt (June 2007). |
| 4121 | |
| 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 | |
| 4140 | =head2 batan2() |
| 4141 | |
| 4142 | my $y = Math::BigFloat->new(2); |
| 4143 | my $x = Math::BigFloat->new(3); |
| 4144 | print $y->batan2($x), "\n"; |
| 4145 | |
| 4146 | Calculate the arcus tanges of C<$y> divided by C<$x>, modifying $y in place. |
| 4147 | See also L<batan()>. |
| 4148 | |
| 4149 | This method was added in v1.87 of Math::BigInt (June 2007). |
| 4150 | |
| 4151 | =head2 batan() |
| 4152 | |
| 4153 | my $x = Math::BigFloat->new(1); |
| 4154 | print $x->batan(100), "\n"; |
| 4155 | |
| 4156 | Calculate the arcus tanges of $x, modifying $x in place. See also L<batan2()>. |
| 4157 | |
| 4158 | This method was added in v1.87 of Math::BigInt (June 2007). |
| 4159 | |
| 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 | |
| 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 | |
| 4178 | prints the value of C<2E-100>. Note that without conversion of |
| 4179 | constants the expression 2E-100 will be calculated as normal floating point |
| 4180 | number. |
| 4181 | |
| 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 | |
| 4195 | use Math::BigFloat lib => 'GMP'; |
| 4196 | |
| 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 | |
| 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 | |
| 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: |
| 4208 | |
| 4209 | use Math::BigFloat only => 'GMP,Pari'; |
| 4210 | |
| 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 | |
| 4216 | See the respective low-level library documentation for further details. |
| 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 | |
| 4229 | It is also possible to just require Math::BigFloat: |
| 4230 | |
| 4231 | require Math::BigFloat; |
| 4232 | |
| 4233 | This will load the necessary things (like BigInt) when they are needed, and |
| 4234 | automatically. |
| 4235 | |
| 4236 | See L<Math::BigInt> for more details than you ever wanted to know about using |
| 4237 | a different low-level library. |
| 4238 | |
| 4239 | =head2 Using Math::BigInt::Lite |
| 4240 | |
| 4241 | For backwards compatibility reasons it is still possible to |
| 4242 | request a different storage class for use with Math::BigFloat: |
| 4243 | |
| 4244 | use Math::BigFloat with => 'Math::BigInt::Lite'; |
| 4245 | |
| 4246 | However, this request is ignored, as the current code now uses the low-level |
| 4247 | math library for directly storing the number parts. |
| 4248 | |
| 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 | |
| 4257 | =head1 BUGS |
| 4258 | |
| 4259 | Please see the file BUGS in the CPAN distribution Math::BigInt for known bugs. |
| 4260 | |
| 4261 | =head1 CAVEATS |
| 4262 | |
| 4263 | Do not try to be clever to insert some operations in between switching |
| 4264 | libraries: |
| 4265 | |
| 4266 | require Math::BigFloat; |
| 4267 | my $matter = Math::BigFloat->bone() + 4; # load BigInt and Calc |
| 4268 | Math::BigFloat->import( lib => 'Pari' ); # load Pari, too |
| 4269 | my $anti_matter = Math::BigFloat->bone()+4; # now use Pari |
| 4270 | |
| 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: |
| 4273 | |
| 4274 | my $flash_and_bang = $matter + $anti_matter; # Don't do this! |
| 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 | |
| 4286 | The following will probably not print what you expect: |
| 4287 | |
| 4288 | print $c->bdiv(123.456),"\n"; |
| 4289 | |
| 4290 | It prints both quotient and reminder since print works in list context. Also, |
| 4291 | bdiv() will modify $c, so be careful. You probably want to use |
| 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 | |
| 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 | |
| 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 |
| 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. |
| 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 | |
| 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: |
| 4358 | my $x = Math::BigFloat->new(123456); # rounded immediately to "12350" |
| 4359 | print "$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 | |
| 4380 | =back |
| 4381 | |
| 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 | |
| 4390 | The package at L<http://search.cpan.org/~tels/Math-BigInt> contains |
| 4391 | more documentation including a full version history, testcases, empty |
| 4392 | subclass files and benchmarks. |
| 4393 | |
| 4394 | =head1 LICENSE |
| 4395 | |
| 4396 | This program is free software; you may redistribute it and/or modify it under |
| 4397 | the same terms as Perl itself. |
| 4398 | |
| 4399 | =head1 AUTHORS |
| 4400 | |
| 4401 | Mark Biggar, overloaded interface by Ilya Zakharevich. |
| 4402 | Completely rewritten by Tels L<http://bloodgate.com> in 2001 - 2006, and still |
| 4403 | at it in 2007. |
| 4404 | |
| 4405 | =cut |