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1 | package Math::BigInt; |
2 | |
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3 | use overload |
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4 | '+' => sub {new Math::BigInt &badd}, |
5 | '-' => sub {new Math::BigInt |
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6 | $_[2]? bsub($_[1],${$_[0]}) : bsub(${$_[0]},$_[1])}, |
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7 | '<=>' => sub {$_[2]? bcmp($_[1],${$_[0]}) : bcmp(${$_[0]},$_[1])}, |
8 | 'cmp' => sub {$_[2]? ($_[1] cmp ${$_[0]}) : (${$_[0]} cmp $_[1])}, |
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9 | '*' => sub {new Math::BigInt &bmul}, |
10 | '/' => sub {new Math::BigInt |
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11 | $_[2]? scalar bdiv($_[1],${$_[0]}) : |
12 | scalar bdiv(${$_[0]},$_[1])}, |
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13 | '%' => sub {new Math::BigInt |
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14 | $_[2]? bmod($_[1],${$_[0]}) : bmod(${$_[0]},$_[1])}, |
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15 | '**' => sub {new Math::BigInt |
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16 | $_[2]? bpow($_[1],${$_[0]}) : bpow(${$_[0]},$_[1])}, |
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17 | 'neg' => sub {new Math::BigInt &bneg}, |
18 | 'abs' => sub {new Math::BigInt &babs}, |
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19 | '<<' => sub {new Math::BigInt |
20 | $_[2]? blsft($_[1],${$_[0]}) : blsft(${$_[0]},$_[1])}, |
21 | '>>' => sub {new Math::BigInt |
22 | $_[2]? brsft($_[1],${$_[0]}) : brsft(${$_[0]},$_[1])}, |
23 | '&' => sub {new Math::BigInt &band}, |
24 | '|' => sub {new Math::BigInt &bior}, |
25 | '^' => sub {new Math::BigInt &bxor}, |
26 | '~' => sub {new Math::BigInt &bnot}, |
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27 | |
28 | qw( |
29 | "" stringify |
30 | 0+ numify) # Order of arguments unsignificant |
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31 | ; |
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32 | |
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33 | $NaNOK=1; |
34 | |
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35 | sub new { |
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36 | my($class) = shift; |
37 | my($foo) = bnorm(shift); |
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38 | die "Not a number initialized to Math::BigInt" if !$NaNOK && $foo eq "NaN"; |
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39 | bless \$foo, $class; |
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40 | } |
41 | sub stringify { "${$_[0]}" } |
42 | sub numify { 0 + "${$_[0]}" } # Not needed, additional overhead |
43 | # comparing to direct compilation based on |
44 | # stringify |
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45 | sub import { |
46 | shift; |
47 | return unless @_; |
48 | die "unknown import: @_" unless @_ == 1 and $_[0] eq ':constant'; |
49 | overload::constant integer => sub {Math::BigInt->new(shift)}; |
50 | } |
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51 | |
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52 | $zero = 0; |
53 | |
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54 | |
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55 | # normalize string form of number. Strip leading zeros. Strip any |
56 | # white space and add a sign, if missing. |
57 | # Strings that are not numbers result the value 'NaN'. |
58 | |
59 | sub bnorm { #(num_str) return num_str |
60 | local($_) = @_; |
61 | s/\s+//g; # strip white space |
62 | if (s/^([+-]?)0*(\d+)$/$1$2/) { # test if number |
63 | substr($_,$[,0) = '+' unless $1; # Add missing sign |
64 | s/^-0/+0/; |
65 | $_; |
66 | } else { |
67 | 'NaN'; |
68 | } |
69 | } |
70 | |
71 | # Convert a number from string format to internal base 100000 format. |
72 | # Assumes normalized value as input. |
73 | sub internal { #(num_str) return int_num_array |
74 | local($d) = @_; |
75 | ($is,$il) = (substr($d,$[,1),length($d)-2); |
76 | substr($d,$[,1) = ''; |
77 | ($is, reverse(unpack("a" . ($il%5+1) . ("a5" x ($il/5)), $d))); |
78 | } |
79 | |
80 | # Convert a number from internal base 100000 format to string format. |
81 | # This routine scribbles all over input array. |
82 | sub external { #(int_num_array) return num_str |
83 | $es = shift; |
84 | grep($_ > 9999 || ($_ = substr('0000'.$_,-5)), @_); # zero pad |
85 | &bnorm(join('', $es, reverse(@_))); # reverse concat and normalize |
86 | } |
87 | |
88 | # Negate input value. |
89 | sub bneg { #(num_str) return num_str |
90 | local($_) = &bnorm(@_); |
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91 | return $_ if $_ eq '+0' or $_ eq 'NaN'; |
92 | vec($_,0,8) ^= ord('+') ^ ord('-'); |
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93 | $_; |
94 | } |
95 | |
96 | # Returns the absolute value of the input. |
97 | sub babs { #(num_str) return num_str |
98 | &abs(&bnorm(@_)); |
99 | } |
100 | |
101 | sub abs { # post-normalized abs for internal use |
102 | local($_) = @_; |
103 | s/^-/+/; |
104 | $_; |
105 | } |
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106 | |
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107 | # Compares 2 values. Returns one of undef, <0, =0, >0. (suitable for sort) |
108 | sub bcmp { #(num_str, num_str) return cond_code |
109 | local($x,$y) = (&bnorm($_[$[]),&bnorm($_[$[+1])); |
110 | if ($x eq 'NaN') { |
111 | undef; |
112 | } elsif ($y eq 'NaN') { |
113 | undef; |
114 | } else { |
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115 | &cmp($x,$y) <=> 0; |
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116 | } |
117 | } |
118 | |
119 | sub cmp { # post-normalized compare for internal use |
120 | local($cx, $cy) = @_; |
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121 | |
122 | return 0 if ($cx eq $cy); |
123 | |
124 | local($sx, $sy) = (substr($cx, 0, 1), substr($cy, 0, 1)); |
125 | local($ld); |
126 | |
127 | if ($sx eq '+') { |
128 | return 1 if ($sy eq '-' || $cy eq '+0'); |
129 | $ld = length($cx) - length($cy); |
130 | return $ld if ($ld); |
131 | return $cx cmp $cy; |
132 | } else { # $sx eq '-' |
133 | return -1 if ($sy eq '+'); |
134 | $ld = length($cy) - length($cx); |
135 | return $ld if ($ld); |
136 | return $cy cmp $cx; |
137 | } |
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138 | } |
139 | |
140 | sub badd { #(num_str, num_str) return num_str |
141 | local(*x, *y); ($x, $y) = (&bnorm($_[$[]),&bnorm($_[$[+1])); |
142 | if ($x eq 'NaN') { |
143 | 'NaN'; |
144 | } elsif ($y eq 'NaN') { |
145 | 'NaN'; |
146 | } else { |
147 | @x = &internal($x); # convert to internal form |
148 | @y = &internal($y); |
149 | local($sx, $sy) = (shift @x, shift @y); # get signs |
150 | if ($sx eq $sy) { |
151 | &external($sx, &add(*x, *y)); # if same sign add |
152 | } else { |
153 | ($x, $y) = (&abs($x),&abs($y)); # make abs |
154 | if (&cmp($y,$x) > 0) { |
155 | &external($sy, &sub(*y, *x)); |
156 | } else { |
157 | &external($sx, &sub(*x, *y)); |
158 | } |
159 | } |
160 | } |
161 | } |
162 | |
163 | sub bsub { #(num_str, num_str) return num_str |
164 | &badd($_[$[],&bneg($_[$[+1])); |
165 | } |
166 | |
167 | # GCD -- Euclids algorithm Knuth Vol 2 pg 296 |
168 | sub bgcd { #(num_str, num_str) return num_str |
169 | local($x,$y) = (&bnorm($_[$[]),&bnorm($_[$[+1])); |
170 | if ($x eq 'NaN' || $y eq 'NaN') { |
171 | 'NaN'; |
172 | } else { |
173 | ($x, $y) = ($y,&bmod($x,$y)) while $y ne '+0'; |
174 | $x; |
175 | } |
176 | } |
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177 | |
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178 | # routine to add two base 1e5 numbers |
179 | # stolen from Knuth Vol 2 Algorithm A pg 231 |
180 | # there are separate routines to add and sub as per Kunth pg 233 |
181 | sub add { #(int_num_array, int_num_array) return int_num_array |
182 | local(*x, *y) = @_; |
183 | $car = 0; |
184 | for $x (@x) { |
185 | last unless @y || $car; |
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186 | $x -= 1e5 if $car = (($x += (@y ? shift(@y) : 0) + $car) >= 1e5) ? 1 : 0; |
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187 | } |
188 | for $y (@y) { |
189 | last unless $car; |
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190 | $y -= 1e5 if $car = (($y += $car) >= 1e5) ? 1 : 0; |
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191 | } |
192 | (@x, @y, $car); |
193 | } |
194 | |
195 | # subtract base 1e5 numbers -- stolen from Knuth Vol 2 pg 232, $x > $y |
196 | sub sub { #(int_num_array, int_num_array) return int_num_array |
197 | local(*sx, *sy) = @_; |
198 | $bar = 0; |
199 | for $sx (@sx) { |
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200 | last unless @sy || $bar; |
201 | $sx += 1e5 if $bar = (($sx -= (@sy ? shift(@sy) : 0) + $bar) < 0); |
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202 | } |
203 | @sx; |
204 | } |
205 | |
206 | # multiply two numbers -- stolen from Knuth Vol 2 pg 233 |
207 | sub bmul { #(num_str, num_str) return num_str |
208 | local(*x, *y); ($x, $y) = (&bnorm($_[$[]), &bnorm($_[$[+1])); |
209 | if ($x eq 'NaN') { |
210 | 'NaN'; |
211 | } elsif ($y eq 'NaN') { |
212 | 'NaN'; |
213 | } else { |
214 | @x = &internal($x); |
215 | @y = &internal($y); |
216 | &external(&mul(*x,*y)); |
217 | } |
218 | } |
219 | |
220 | # multiply two numbers in internal representation |
221 | # destroys the arguments, supposes that two arguments are different |
222 | sub mul { #(*int_num_array, *int_num_array) return int_num_array |
223 | local(*x, *y) = (shift, shift); |
224 | local($signr) = (shift @x ne shift @y) ? '-' : '+'; |
225 | @prod = (); |
226 | for $x (@x) { |
227 | ($car, $cty) = (0, $[); |
228 | for $y (@y) { |
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229 | $prod = $x * $y + ($prod[$cty] || 0) + $car; |
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230 | $prod[$cty++] = |
231 | $prod - ($car = int($prod * 1e-5)) * 1e5; |
232 | } |
233 | $prod[$cty] += $car if $car; |
234 | $x = shift @prod; |
235 | } |
236 | ($signr, @x, @prod); |
237 | } |
238 | |
239 | # modulus |
240 | sub bmod { #(num_str, num_str) return num_str |
241 | (&bdiv(@_))[$[+1]; |
242 | } |
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243 | |
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244 | sub bdiv { #(dividend: num_str, divisor: num_str) return num_str |
245 | local (*x, *y); ($x, $y) = (&bnorm($_[$[]), &bnorm($_[$[+1])); |
246 | return wantarray ? ('NaN','NaN') : 'NaN' |
247 | if ($x eq 'NaN' || $y eq 'NaN' || $y eq '+0'); |
248 | return wantarray ? ('+0',$x) : '+0' if (&cmp(&abs($x),&abs($y)) < 0); |
249 | @x = &internal($x); @y = &internal($y); |
250 | $srem = $y[$[]; |
251 | $sr = (shift @x ne shift @y) ? '-' : '+'; |
252 | $car = $bar = $prd = 0; |
253 | if (($dd = int(1e5/($y[$#y]+1))) != 1) { |
254 | for $x (@x) { |
255 | $x = $x * $dd + $car; |
256 | $x -= ($car = int($x * 1e-5)) * 1e5; |
257 | } |
258 | push(@x, $car); $car = 0; |
259 | for $y (@y) { |
260 | $y = $y * $dd + $car; |
261 | $y -= ($car = int($y * 1e-5)) * 1e5; |
262 | } |
263 | } |
264 | else { |
265 | push(@x, 0); |
266 | } |
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267 | @q = (); ($v2,$v1) = @y[-2,-1]; |
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268 | $v2 = 0 unless $v2; |
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269 | while ($#x > $#y) { |
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270 | ($u2,$u1,$u0) = @x[-3..-1]; |
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271 | $u2 = 0 unless $u2; |
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272 | $q = (($u0 == $v1) ? 99999 : int(($u0*1e5+$u1)/$v1)); |
273 | --$q while ($v2*$q > ($u0*1e5+$u1-$q*$v1)*1e5+$u2); |
274 | if ($q) { |
275 | ($car, $bar) = (0,0); |
276 | for ($y = $[, $x = $#x-$#y+$[-1; $y <= $#y; ++$y,++$x) { |
277 | $prd = $q * $y[$y] + $car; |
278 | $prd -= ($car = int($prd * 1e-5)) * 1e5; |
279 | $x[$x] += 1e5 if ($bar = (($x[$x] -= $prd + $bar) < 0)); |
280 | } |
281 | if ($x[$#x] < $car + $bar) { |
282 | $car = 0; --$q; |
283 | for ($y = $[, $x = $#x-$#y+$[-1; $y <= $#y; ++$y,++$x) { |
284 | $x[$x] -= 1e5 |
285 | if ($car = (($x[$x] += $y[$y] + $car) > 1e5)); |
286 | } |
287 | } |
288 | } |
289 | pop(@x); unshift(@q, $q); |
290 | } |
291 | if (wantarray) { |
292 | @d = (); |
293 | if ($dd != 1) { |
294 | $car = 0; |
295 | for $x (reverse @x) { |
296 | $prd = $car * 1e5 + $x; |
297 | $car = $prd - ($tmp = int($prd / $dd)) * $dd; |
298 | unshift(@d, $tmp); |
299 | } |
300 | } |
301 | else { |
302 | @d = @x; |
303 | } |
304 | (&external($sr, @q), &external($srem, @d, $zero)); |
305 | } else { |
306 | &external($sr, @q); |
307 | } |
308 | } |
309 | |
310 | # compute power of two numbers -- stolen from Knuth Vol 2 pg 233 |
311 | sub bpow { #(num_str, num_str) return num_str |
312 | local(*x, *y); ($x, $y) = (&bnorm($_[$[]), &bnorm($_[$[+1])); |
313 | if ($x eq 'NaN') { |
314 | 'NaN'; |
315 | } elsif ($y eq 'NaN') { |
316 | 'NaN'; |
317 | } elsif ($x eq '+1') { |
318 | '+1'; |
319 | } elsif ($x eq '-1') { |
320 | &bmod($x,2) ? '-1': '+1'; |
321 | } elsif ($y =~ /^-/) { |
322 | 'NaN'; |
323 | } elsif ($x eq '+0' && $y eq '+0') { |
324 | 'NaN'; |
325 | } else { |
326 | @x = &internal($x); |
327 | local(@pow2)=@x; |
328 | local(@pow)=&internal("+1"); |
329 | local($y1,$res,@tmp1,@tmp2)=(1); # need tmp to send to mul |
330 | while ($y ne '+0') { |
331 | ($y,$res)=&bdiv($y,2); |
332 | if ($res ne '+0') {@tmp=@pow2; @pow=&mul(*pow,*tmp);} |
333 | if ($y ne '+0') {@tmp=@pow2;@pow2=&mul(*pow2,*tmp);} |
334 | } |
335 | &external(@pow); |
336 | } |
337 | } |
338 | |
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339 | # compute x << y, y >= 0 |
340 | sub blsft { #(num_str, num_str) return num_str |
341 | &bmul($_[$[], &bpow(2, $_[$[+1])); |
342 | } |
343 | |
344 | # compute x >> y, y >= 0 |
345 | sub brsft { #(num_str, num_str) return num_str |
346 | &bdiv($_[$[], &bpow(2, $_[$[+1])); |
347 | } |
348 | |
349 | # compute x & y |
350 | sub band { #(num_str, num_str) return num_str |
351 | local($x,$y,$r,$m,$xr,$yr) = (&bnorm($_[$[]),&bnorm($_[$[+1]),0,1); |
352 | if ($x eq 'NaN' || $y eq 'NaN') { |
353 | 'NaN'; |
354 | } else { |
355 | while ($x ne '+0' && $y ne '+0') { |
356 | ($x, $xr) = &bdiv($x, 0x10000); |
357 | ($y, $yr) = &bdiv($y, 0x10000); |
358 | $r = &badd(&bmul(int $xr & $yr, $m), $r); |
359 | $m = &bmul($m, 0x10000); |
360 | } |
361 | $r; |
362 | } |
363 | } |
364 | |
365 | # compute x | y |
366 | sub bior { #(num_str, num_str) return num_str |
367 | local($x,$y,$r,$m,$xr,$yr) = (&bnorm($_[$[]),&bnorm($_[$[+1]),0,1); |
368 | if ($x eq 'NaN' || $y eq 'NaN') { |
369 | 'NaN'; |
370 | } else { |
371 | while ($x ne '+0' || $y ne '+0') { |
372 | ($x, $xr) = &bdiv($x, 0x10000); |
373 | ($y, $yr) = &bdiv($y, 0x10000); |
374 | $r = &badd(&bmul(int $xr | $yr, $m), $r); |
375 | $m = &bmul($m, 0x10000); |
376 | } |
377 | $r; |
378 | } |
379 | } |
380 | |
381 | # compute x ^ y |
382 | sub bxor { #(num_str, num_str) return num_str |
383 | local($x,$y,$r,$m,$xr,$yr) = (&bnorm($_[$[]),&bnorm($_[$[+1]),0,1); |
384 | if ($x eq 'NaN' || $y eq 'NaN') { |
385 | 'NaN'; |
386 | } else { |
387 | while ($x ne '+0' || $y ne '+0') { |
388 | ($x, $xr) = &bdiv($x, 0x10000); |
389 | ($y, $yr) = &bdiv($y, 0x10000); |
390 | $r = &badd(&bmul(int $xr ^ $yr, $m), $r); |
391 | $m = &bmul($m, 0x10000); |
392 | } |
393 | $r; |
394 | } |
395 | } |
396 | |
397 | # represent ~x as twos-complement number |
398 | sub bnot { #(num_str) return num_str |
399 | &bsub(-1,$_[$[]); |
400 | } |
401 | |
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402 | 1; |
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403 | __END__ |
404 | |
405 | =head1 NAME |
406 | |
407 | Math::BigInt - Arbitrary size integer math package |
408 | |
409 | =head1 SYNOPSIS |
410 | |
411 | use Math::BigInt; |
412 | $i = Math::BigInt->new($string); |
413 | |
414 | $i->bneg return BINT negation |
415 | $i->babs return BINT absolute value |
416 | $i->bcmp(BINT) return CODE compare numbers (undef,<0,=0,>0) |
417 | $i->badd(BINT) return BINT addition |
418 | $i->bsub(BINT) return BINT subtraction |
419 | $i->bmul(BINT) return BINT multiplication |
420 | $i->bdiv(BINT) return (BINT,BINT) division (quo,rem) just quo if scalar |
421 | $i->bmod(BINT) return BINT modulus |
422 | $i->bgcd(BINT) return BINT greatest common divisor |
423 | $i->bnorm return BINT normalization |
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424 | $i->blsft(BINT) return BINT left shift |
425 | $i->brsft(BINT) return (BINT,BINT) right shift (quo,rem) just quo if scalar |
426 | $i->band(BINT) return BINT bit-wise and |
427 | $i->bior(BINT) return BINT bit-wise inclusive or |
428 | $i->bxor(BINT) return BINT bit-wise exclusive or |
429 | $i->bnot return BINT bit-wise not |
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430 | |
431 | =head1 DESCRIPTION |
432 | |
433 | All basic math operations are overloaded if you declare your big |
434 | integers as |
435 | |
436 | $i = new Math::BigInt '123 456 789 123 456 789'; |
437 | |
438 | |
439 | =over 2 |
440 | |
441 | =item Canonical notation |
442 | |
443 | Big integer value are strings of the form C</^[+-]\d+$/> with leading |
444 | zeros suppressed. |
445 | |
446 | =item Input |
447 | |
448 | Input values to these routines may be strings of the form |
449 | C</^\s*[+-]?[\d\s]+$/>. |
450 | |
451 | =item Output |
452 | |
453 | Output values always always in canonical form |
454 | |
455 | =back |
456 | |
457 | Actual math is done in an internal format consisting of an array |
458 | whose first element is the sign (/^[+-]$/) and whose remaining |
459 | elements are base 100000 digits with the least significant digit first. |
460 | The string 'NaN' is used to represent the result when input arguments |
461 | are not numbers, as well as the result of dividing by zero. |
462 | |
463 | =head1 EXAMPLES |
464 | |
465 | '+0' canonical zero value |
466 | ' -123 123 123' canonical value '-123123123' |
467 | '1 23 456 7890' canonical value '+1234567890' |
468 | |
469 | |
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470 | =head1 Autocreating constants |
471 | |
472 | After C<use Math::BigInt ':constant'> all the integer decimal constants |
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473 | in the given scope are converted to C<Math::BigInt>. This conversion |
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474 | happens at compile time. |
475 | |
476 | In particular |
477 | |
478 | perl -MMath::BigInt=:constant -e 'print 2**100' |
479 | |
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480 | print the integer value of C<2**100>. Note that without conversion of |
481 | constants the expression 2**100 will be calculated as floating point number. |
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482 | |
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483 | =head1 BUGS |
484 | |
485 | The current version of this module is a preliminary version of the |
486 | real thing that is currently (as of perl5.002) under development. |
487 | |
488 | =head1 AUTHOR |
489 | |
490 | Mark Biggar, overloaded interface by Ilya Zakharevich. |
491 | |
492 | =cut |