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