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1 | package Math::BigInt; |
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2 | $VERSION='0.01'; |
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3 | |
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4 | use overload |
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5 | '+' => sub {new Math::BigInt &badd}, |
6 | '-' => sub {new Math::BigInt |
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7 | $_[2]? bsub($_[1],${$_[0]}) : bsub(${$_[0]},$_[1])}, |
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8 | '<=>' => sub {$_[2]? bcmp($_[1],${$_[0]}) : bcmp(${$_[0]},$_[1])}, |
9 | 'cmp' => sub {$_[2]? ($_[1] cmp ${$_[0]}) : (${$_[0]} cmp $_[1])}, |
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10 | '*' => sub {new Math::BigInt &bmul}, |
11 | '/' => sub {new Math::BigInt |
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12 | $_[2]? scalar bdiv($_[1],${$_[0]}) : |
13 | scalar bdiv(${$_[0]},$_[1])}, |
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14 | '%' => sub {new Math::BigInt |
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15 | $_[2]? bmod($_[1],${$_[0]}) : bmod(${$_[0]},$_[1])}, |
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16 | '**' => sub {new Math::BigInt |
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17 | $_[2]? bpow($_[1],${$_[0]}) : bpow(${$_[0]},$_[1])}, |
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18 | 'neg' => sub {new Math::BigInt &bneg}, |
19 | 'abs' => sub {new Math::BigInt &babs}, |
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20 | '<<' => sub {new Math::BigInt |
21 | $_[2]? blsft($_[1],${$_[0]}) : blsft(${$_[0]},$_[1])}, |
22 | '>>' => sub {new Math::BigInt |
23 | $_[2]? brsft($_[1],${$_[0]}) : brsft(${$_[0]},$_[1])}, |
24 | '&' => sub {new Math::BigInt &band}, |
25 | '|' => sub {new Math::BigInt &bior}, |
26 | '^' => sub {new Math::BigInt &bxor}, |
27 | '~' => sub {new Math::BigInt &bnot}, |
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28 | |
29 | qw( |
30 | "" stringify |
31 | 0+ numify) # Order of arguments unsignificant |
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32 | ; |
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33 | |
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34 | $NaNOK=1; |
35 | |
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36 | sub new { |
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37 | my($class) = shift; |
38 | my($foo) = bnorm(shift); |
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39 | die "Not a number initialized to Math::BigInt" if !$NaNOK && $foo eq "NaN"; |
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40 | bless \$foo, $class; |
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41 | } |
42 | sub stringify { "${$_[0]}" } |
43 | sub numify { 0 + "${$_[0]}" } # Not needed, additional overhead |
44 | # comparing to direct compilation based on |
45 | # stringify |
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46 | sub import { |
47 | shift; |
48 | return unless @_; |
49 | die "unknown import: @_" unless @_ == 1 and $_[0] eq ':constant'; |
50 | overload::constant integer => sub {Math::BigInt->new(shift)}; |
51 | } |
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52 | |
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53 | $zero = 0; |
54 | |
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55 | |
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56 | # normalize string form of number. Strip leading zeros. Strip any |
57 | # white space and add a sign, if missing. |
58 | # Strings that are not numbers result the value 'NaN'. |
59 | |
60 | sub bnorm { #(num_str) return num_str |
61 | local($_) = @_; |
62 | s/\s+//g; # strip white space |
63 | if (s/^([+-]?)0*(\d+)$/$1$2/) { # test if number |
64 | substr($_,$[,0) = '+' unless $1; # Add missing sign |
65 | s/^-0/+0/; |
66 | $_; |
67 | } else { |
68 | 'NaN'; |
69 | } |
70 | } |
71 | |
72 | # Convert a number from string format to internal base 100000 format. |
73 | # Assumes normalized value as input. |
74 | sub internal { #(num_str) return int_num_array |
75 | local($d) = @_; |
76 | ($is,$il) = (substr($d,$[,1),length($d)-2); |
77 | substr($d,$[,1) = ''; |
78 | ($is, reverse(unpack("a" . ($il%5+1) . ("a5" x ($il/5)), $d))); |
79 | } |
80 | |
81 | # Convert a number from internal base 100000 format to string format. |
82 | # This routine scribbles all over input array. |
83 | sub external { #(int_num_array) return num_str |
84 | $es = shift; |
85 | grep($_ > 9999 || ($_ = substr('0000'.$_,-5)), @_); # zero pad |
86 | &bnorm(join('', $es, reverse(@_))); # reverse concat and normalize |
87 | } |
88 | |
89 | # Negate input value. |
90 | sub bneg { #(num_str) return num_str |
91 | local($_) = &bnorm(@_); |
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92 | return $_ if $_ eq '+0' or $_ eq 'NaN'; |
93 | vec($_,0,8) ^= ord('+') ^ ord('-'); |
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94 | $_; |
95 | } |
96 | |
97 | # Returns the absolute value of the input. |
98 | sub babs { #(num_str) return num_str |
99 | &abs(&bnorm(@_)); |
100 | } |
101 | |
102 | sub abs { # post-normalized abs for internal use |
103 | local($_) = @_; |
104 | s/^-/+/; |
105 | $_; |
106 | } |
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107 | |
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108 | # Compares 2 values. Returns one of undef, <0, =0, >0. (suitable for sort) |
109 | sub bcmp { #(num_str, num_str) return cond_code |
110 | local($x,$y) = (&bnorm($_[$[]),&bnorm($_[$[+1])); |
111 | if ($x eq 'NaN') { |
112 | undef; |
113 | } elsif ($y eq 'NaN') { |
114 | undef; |
115 | } else { |
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116 | &cmp($x,$y) <=> 0; |
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117 | } |
118 | } |
119 | |
120 | sub cmp { # post-normalized compare for internal use |
121 | local($cx, $cy) = @_; |
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122 | |
123 | return 0 if ($cx eq $cy); |
124 | |
125 | local($sx, $sy) = (substr($cx, 0, 1), substr($cy, 0, 1)); |
126 | local($ld); |
127 | |
128 | if ($sx eq '+') { |
129 | return 1 if ($sy eq '-' || $cy eq '+0'); |
130 | $ld = length($cx) - length($cy); |
131 | return $ld if ($ld); |
132 | return $cx cmp $cy; |
133 | } else { # $sx eq '-' |
134 | return -1 if ($sy eq '+'); |
135 | $ld = length($cy) - length($cx); |
136 | return $ld if ($ld); |
137 | return $cy cmp $cx; |
138 | } |
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139 | } |
140 | |
141 | sub badd { #(num_str, num_str) return num_str |
142 | local(*x, *y); ($x, $y) = (&bnorm($_[$[]),&bnorm($_[$[+1])); |
143 | if ($x eq 'NaN') { |
144 | 'NaN'; |
145 | } elsif ($y eq 'NaN') { |
146 | 'NaN'; |
147 | } else { |
148 | @x = &internal($x); # convert to internal form |
149 | @y = &internal($y); |
150 | local($sx, $sy) = (shift @x, shift @y); # get signs |
151 | if ($sx eq $sy) { |
152 | &external($sx, &add(*x, *y)); # if same sign add |
153 | } else { |
154 | ($x, $y) = (&abs($x),&abs($y)); # make abs |
155 | if (&cmp($y,$x) > 0) { |
156 | &external($sy, &sub(*y, *x)); |
157 | } else { |
158 | &external($sx, &sub(*x, *y)); |
159 | } |
160 | } |
161 | } |
162 | } |
163 | |
164 | sub bsub { #(num_str, num_str) return num_str |
165 | &badd($_[$[],&bneg($_[$[+1])); |
166 | } |
167 | |
168 | # GCD -- Euclids algorithm Knuth Vol 2 pg 296 |
169 | sub bgcd { #(num_str, num_str) return num_str |
170 | local($x,$y) = (&bnorm($_[$[]),&bnorm($_[$[+1])); |
171 | if ($x eq 'NaN' || $y eq 'NaN') { |
172 | 'NaN'; |
173 | } else { |
174 | ($x, $y) = ($y,&bmod($x,$y)) while $y ne '+0'; |
175 | $x; |
176 | } |
177 | } |
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178 | |
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179 | # routine to add two base 1e5 numbers |
180 | # stolen from Knuth Vol 2 Algorithm A pg 231 |
181 | # there are separate routines to add and sub as per Kunth pg 233 |
182 | sub add { #(int_num_array, int_num_array) return int_num_array |
183 | local(*x, *y) = @_; |
184 | $car = 0; |
185 | for $x (@x) { |
186 | last unless @y || $car; |
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187 | $x -= 1e5 if $car = (($x += (@y ? shift(@y) : 0) + $car) >= 1e5) ? 1 : 0; |
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188 | } |
189 | for $y (@y) { |
190 | last unless $car; |
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191 | $y -= 1e5 if $car = (($y += $car) >= 1e5) ? 1 : 0; |
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192 | } |
193 | (@x, @y, $car); |
194 | } |
195 | |
196 | # subtract base 1e5 numbers -- stolen from Knuth Vol 2 pg 232, $x > $y |
197 | sub sub { #(int_num_array, int_num_array) return int_num_array |
198 | local(*sx, *sy) = @_; |
199 | $bar = 0; |
200 | for $sx (@sx) { |
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201 | last unless @sy || $bar; |
202 | $sx += 1e5 if $bar = (($sx -= (@sy ? shift(@sy) : 0) + $bar) < 0); |
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203 | } |
204 | @sx; |
205 | } |
206 | |
207 | # multiply two numbers -- stolen from Knuth Vol 2 pg 233 |
208 | sub bmul { #(num_str, num_str) return num_str |
209 | local(*x, *y); ($x, $y) = (&bnorm($_[$[]), &bnorm($_[$[+1])); |
210 | if ($x eq 'NaN') { |
211 | 'NaN'; |
212 | } elsif ($y eq 'NaN') { |
213 | 'NaN'; |
214 | } else { |
215 | @x = &internal($x); |
216 | @y = &internal($y); |
217 | &external(&mul(*x,*y)); |
218 | } |
219 | } |
220 | |
221 | # multiply two numbers in internal representation |
222 | # destroys the arguments, supposes that two arguments are different |
223 | sub mul { #(*int_num_array, *int_num_array) return int_num_array |
224 | local(*x, *y) = (shift, shift); |
225 | local($signr) = (shift @x ne shift @y) ? '-' : '+'; |
226 | @prod = (); |
227 | for $x (@x) { |
228 | ($car, $cty) = (0, $[); |
229 | for $y (@y) { |
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230 | $prod = $x * $y + ($prod[$cty] || 0) + $car; |
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231 | $prod[$cty++] = |
232 | $prod - ($car = int($prod * 1e-5)) * 1e5; |
233 | } |
234 | $prod[$cty] += $car if $car; |
235 | $x = shift @prod; |
236 | } |
237 | ($signr, @x, @prod); |
238 | } |
239 | |
240 | # modulus |
241 | sub bmod { #(num_str, num_str) return num_str |
242 | (&bdiv(@_))[$[+1]; |
243 | } |
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244 | |
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245 | sub bdiv { #(dividend: num_str, divisor: num_str) return num_str |
246 | local (*x, *y); ($x, $y) = (&bnorm($_[$[]), &bnorm($_[$[+1])); |
247 | return wantarray ? ('NaN','NaN') : 'NaN' |
248 | if ($x eq 'NaN' || $y eq 'NaN' || $y eq '+0'); |
249 | return wantarray ? ('+0',$x) : '+0' if (&cmp(&abs($x),&abs($y)) < 0); |
250 | @x = &internal($x); @y = &internal($y); |
251 | $srem = $y[$[]; |
252 | $sr = (shift @x ne shift @y) ? '-' : '+'; |
253 | $car = $bar = $prd = 0; |
254 | if (($dd = int(1e5/($y[$#y]+1))) != 1) { |
255 | for $x (@x) { |
256 | $x = $x * $dd + $car; |
257 | $x -= ($car = int($x * 1e-5)) * 1e5; |
258 | } |
259 | push(@x, $car); $car = 0; |
260 | for $y (@y) { |
261 | $y = $y * $dd + $car; |
262 | $y -= ($car = int($y * 1e-5)) * 1e5; |
263 | } |
264 | } |
265 | else { |
266 | push(@x, 0); |
267 | } |
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268 | @q = (); ($v2,$v1) = @y[-2,-1]; |
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269 | $v2 = 0 unless $v2; |
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270 | while ($#x > $#y) { |
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271 | ($u2,$u1,$u0) = @x[-3..-1]; |
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272 | $u2 = 0 unless $u2; |
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273 | $q = (($u0 == $v1) ? 99999 : int(($u0*1e5+$u1)/$v1)); |
274 | --$q while ($v2*$q > ($u0*1e5+$u1-$q*$v1)*1e5+$u2); |
275 | if ($q) { |
276 | ($car, $bar) = (0,0); |
277 | for ($y = $[, $x = $#x-$#y+$[-1; $y <= $#y; ++$y,++$x) { |
278 | $prd = $q * $y[$y] + $car; |
279 | $prd -= ($car = int($prd * 1e-5)) * 1e5; |
280 | $x[$x] += 1e5 if ($bar = (($x[$x] -= $prd + $bar) < 0)); |
281 | } |
282 | if ($x[$#x] < $car + $bar) { |
283 | $car = 0; --$q; |
284 | for ($y = $[, $x = $#x-$#y+$[-1; $y <= $#y; ++$y,++$x) { |
285 | $x[$x] -= 1e5 |
286 | if ($car = (($x[$x] += $y[$y] + $car) > 1e5)); |
287 | } |
288 | } |
289 | } |
290 | pop(@x); unshift(@q, $q); |
291 | } |
292 | if (wantarray) { |
293 | @d = (); |
294 | if ($dd != 1) { |
295 | $car = 0; |
296 | for $x (reverse @x) { |
297 | $prd = $car * 1e5 + $x; |
298 | $car = $prd - ($tmp = int($prd / $dd)) * $dd; |
299 | unshift(@d, $tmp); |
300 | } |
301 | } |
302 | else { |
303 | @d = @x; |
304 | } |
305 | (&external($sr, @q), &external($srem, @d, $zero)); |
306 | } else { |
307 | &external($sr, @q); |
308 | } |
309 | } |
310 | |
311 | # compute power of two numbers -- stolen from Knuth Vol 2 pg 233 |
312 | sub bpow { #(num_str, num_str) return num_str |
313 | local(*x, *y); ($x, $y) = (&bnorm($_[$[]), &bnorm($_[$[+1])); |
314 | if ($x eq 'NaN') { |
315 | 'NaN'; |
316 | } elsif ($y eq 'NaN') { |
317 | 'NaN'; |
318 | } elsif ($x eq '+1') { |
319 | '+1'; |
320 | } elsif ($x eq '-1') { |
321 | &bmod($x,2) ? '-1': '+1'; |
322 | } elsif ($y =~ /^-/) { |
323 | 'NaN'; |
324 | } elsif ($x eq '+0' && $y eq '+0') { |
325 | 'NaN'; |
326 | } else { |
327 | @x = &internal($x); |
328 | local(@pow2)=@x; |
329 | local(@pow)=&internal("+1"); |
330 | local($y1,$res,@tmp1,@tmp2)=(1); # need tmp to send to mul |
331 | while ($y ne '+0') { |
332 | ($y,$res)=&bdiv($y,2); |
333 | if ($res ne '+0') {@tmp=@pow2; @pow=&mul(*pow,*tmp);} |
334 | if ($y ne '+0') {@tmp=@pow2;@pow2=&mul(*pow2,*tmp);} |
335 | } |
336 | &external(@pow); |
337 | } |
338 | } |
339 | |
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340 | # compute x << y, y >= 0 |
341 | sub blsft { #(num_str, num_str) return num_str |
342 | &bmul($_[$[], &bpow(2, $_[$[+1])); |
343 | } |
344 | |
345 | # compute x >> y, y >= 0 |
346 | sub brsft { #(num_str, num_str) return num_str |
347 | &bdiv($_[$[], &bpow(2, $_[$[+1])); |
348 | } |
349 | |
350 | # compute x & y |
351 | sub band { #(num_str, num_str) return num_str |
352 | local($x,$y,$r,$m,$xr,$yr) = (&bnorm($_[$[]),&bnorm($_[$[+1]),0,1); |
353 | if ($x eq 'NaN' || $y eq 'NaN') { |
354 | 'NaN'; |
355 | } else { |
356 | while ($x ne '+0' && $y ne '+0') { |
357 | ($x, $xr) = &bdiv($x, 0x10000); |
358 | ($y, $yr) = &bdiv($y, 0x10000); |
359 | $r = &badd(&bmul(int $xr & $yr, $m), $r); |
360 | $m = &bmul($m, 0x10000); |
361 | } |
362 | $r; |
363 | } |
364 | } |
365 | |
366 | # compute x | y |
367 | sub bior { #(num_str, num_str) return num_str |
368 | local($x,$y,$r,$m,$xr,$yr) = (&bnorm($_[$[]),&bnorm($_[$[+1]),0,1); |
369 | if ($x eq 'NaN' || $y eq 'NaN') { |
370 | 'NaN'; |
371 | } else { |
372 | while ($x ne '+0' || $y ne '+0') { |
373 | ($x, $xr) = &bdiv($x, 0x10000); |
374 | ($y, $yr) = &bdiv($y, 0x10000); |
375 | $r = &badd(&bmul(int $xr | $yr, $m), $r); |
376 | $m = &bmul($m, 0x10000); |
377 | } |
378 | $r; |
379 | } |
380 | } |
381 | |
382 | # compute x ^ y |
383 | sub bxor { #(num_str, num_str) return num_str |
384 | local($x,$y,$r,$m,$xr,$yr) = (&bnorm($_[$[]),&bnorm($_[$[+1]),0,1); |
385 | if ($x eq 'NaN' || $y eq 'NaN') { |
386 | 'NaN'; |
387 | } else { |
388 | while ($x ne '+0' || $y ne '+0') { |
389 | ($x, $xr) = &bdiv($x, 0x10000); |
390 | ($y, $yr) = &bdiv($y, 0x10000); |
391 | $r = &badd(&bmul(int $xr ^ $yr, $m), $r); |
392 | $m = &bmul($m, 0x10000); |
393 | } |
394 | $r; |
395 | } |
396 | } |
397 | |
398 | # represent ~x as twos-complement number |
399 | sub bnot { #(num_str) return num_str |
400 | &bsub(-1,$_[$[]); |
401 | } |
402 | |
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403 | 1; |
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404 | __END__ |
405 | |
406 | =head1 NAME |
407 | |
408 | Math::BigInt - Arbitrary size integer math package |
409 | |
410 | =head1 SYNOPSIS |
411 | |
412 | use Math::BigInt; |
413 | $i = Math::BigInt->new($string); |
414 | |
415 | $i->bneg return BINT negation |
416 | $i->babs return BINT absolute value |
417 | $i->bcmp(BINT) return CODE compare numbers (undef,<0,=0,>0) |
418 | $i->badd(BINT) return BINT addition |
419 | $i->bsub(BINT) return BINT subtraction |
420 | $i->bmul(BINT) return BINT multiplication |
421 | $i->bdiv(BINT) return (BINT,BINT) division (quo,rem) just quo if scalar |
422 | $i->bmod(BINT) return BINT modulus |
423 | $i->bgcd(BINT) return BINT greatest common divisor |
424 | $i->bnorm return BINT normalization |
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425 | $i->blsft(BINT) return BINT left shift |
426 | $i->brsft(BINT) return (BINT,BINT) right shift (quo,rem) just quo if scalar |
427 | $i->band(BINT) return BINT bit-wise and |
428 | $i->bior(BINT) return BINT bit-wise inclusive or |
429 | $i->bxor(BINT) return BINT bit-wise exclusive or |
430 | $i->bnot return BINT bit-wise not |
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431 | |
432 | =head1 DESCRIPTION |
433 | |
434 | All basic math operations are overloaded if you declare your big |
435 | integers as |
436 | |
437 | $i = new Math::BigInt '123 456 789 123 456 789'; |
438 | |
439 | |
440 | =over 2 |
441 | |
442 | =item Canonical notation |
443 | |
444 | Big integer value are strings of the form C</^[+-]\d+$/> with leading |
445 | zeros suppressed. |
446 | |
447 | =item Input |
448 | |
449 | Input values to these routines may be strings of the form |
450 | C</^\s*[+-]?[\d\s]+$/>. |
451 | |
452 | =item Output |
453 | |
454 | Output values always always in canonical form |
455 | |
456 | =back |
457 | |
458 | Actual math is done in an internal format consisting of an array |
459 | whose first element is the sign (/^[+-]$/) and whose remaining |
460 | elements are base 100000 digits with the least significant digit first. |
461 | The string 'NaN' is used to represent the result when input arguments |
462 | are not numbers, as well as the result of dividing by zero. |
463 | |
464 | =head1 EXAMPLES |
465 | |
466 | '+0' canonical zero value |
467 | ' -123 123 123' canonical value '-123123123' |
468 | '1 23 456 7890' canonical value '+1234567890' |
469 | |
470 | |
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471 | =head1 Autocreating constants |
472 | |
473 | After C<use Math::BigInt ':constant'> all the integer decimal constants |
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474 | in the given scope are converted to C<Math::BigInt>. This conversion |
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475 | happens at compile time. |
476 | |
477 | In particular |
478 | |
479 | perl -MMath::BigInt=:constant -e 'print 2**100' |
480 | |
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481 | print the integer value of C<2**100>. Note that without conversion of |
482 | constants the expression 2**100 will be calculated as floating point number. |
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483 | |
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484 | =head1 BUGS |
485 | |
486 | The current version of this module is a preliminary version of the |
487 | real thing that is currently (as of perl5.002) under development. |
488 | |
489 | =head1 AUTHOR |
490 | |
491 | Mark Biggar, overloaded interface by Ilya Zakharevich. |
492 | |
493 | =cut |