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1 | package Math::BigInt; |
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2 | require Exporter; |
3 | @ISA = qw(Exporter); |
4 | |
5 | $VERSION='0.02'; |
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6 | |
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7 | use overload |
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8 | '+' => sub {new Math::BigInt &badd}, |
9 | '-' => sub {new Math::BigInt |
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10 | $_[2]? bsub($_[1],${$_[0]}) : bsub(${$_[0]},$_[1])}, |
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11 | '<=>' => sub {$_[2]? bcmp($_[1],${$_[0]}) : bcmp(${$_[0]},$_[1])}, |
12 | 'cmp' => sub {$_[2]? ($_[1] cmp ${$_[0]}) : (${$_[0]} cmp $_[1])}, |
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13 | '*' => sub {new Math::BigInt &bmul}, |
14 | '/' => sub {new Math::BigInt |
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15 | $_[2]? scalar bdiv($_[1],${$_[0]}) : |
16 | scalar bdiv(${$_[0]},$_[1])}, |
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17 | '%' => sub {new Math::BigInt |
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18 | $_[2]? bmod($_[1],${$_[0]}) : bmod(${$_[0]},$_[1])}, |
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19 | '**' => sub {new Math::BigInt |
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20 | $_[2]? bpow($_[1],${$_[0]}) : bpow(${$_[0]},$_[1])}, |
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21 | 'neg' => sub {new Math::BigInt &bneg}, |
22 | 'abs' => sub {new Math::BigInt &babs}, |
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23 | '<<' => sub {new Math::BigInt |
24 | $_[2]? blsft($_[1],${$_[0]}) : blsft(${$_[0]},$_[1])}, |
25 | '>>' => sub {new Math::BigInt |
26 | $_[2]? brsft($_[1],${$_[0]}) : brsft(${$_[0]},$_[1])}, |
27 | '&' => sub {new Math::BigInt &band}, |
28 | '|' => sub {new Math::BigInt &bior}, |
29 | '^' => sub {new Math::BigInt &bxor}, |
30 | '~' => sub {new Math::BigInt &bnot}, |
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31 | 'int' => sub { shift }, |
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32 | |
33 | qw( |
34 | "" stringify |
35 | 0+ numify) # Order of arguments unsignificant |
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36 | ; |
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37 | |
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38 | $NaNOK=1; |
39 | |
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40 | sub new { |
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41 | my($class) = shift; |
42 | my($foo) = bnorm(shift); |
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43 | die "Not a number initialized to Math::BigInt" if !$NaNOK && $foo eq "NaN"; |
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44 | bless \$foo, $class; |
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45 | } |
46 | sub stringify { "${$_[0]}" } |
47 | sub numify { 0 + "${$_[0]}" } # Not needed, additional overhead |
48 | # comparing to direct compilation based on |
49 | # stringify |
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50 | sub import { |
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51 | my $self = shift; |
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52 | return unless @_; |
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53 | for ( my $i; $i < @_ ; $i++ ) { |
54 | if ( $_[$i] eq ':constant' ) { |
55 | overload::constant integer => sub {Math::BigInt->new(shift)}; |
56 | splice @_, $i, 1; |
57 | last; |
58 | } |
59 | } |
60 | $self->SUPER::import(@_); |
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61 | } |
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62 | |
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63 | $zero = 0; |
64 | |
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65 | # overcome a floating point problem on certain osnames (posix-bc, os390) |
66 | BEGIN { |
67 | my $x = 100000.0; |
68 | my $use_mult = int($x*1e-5)*1e5 == $x ? 1 : 0; |
69 | } |
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70 | |
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71 | # normalize string form of number. Strip leading zeros. Strip any |
72 | # white space and add a sign, if missing. |
73 | # Strings that are not numbers result the value 'NaN'. |
74 | |
75 | sub bnorm { #(num_str) return num_str |
76 | local($_) = @_; |
77 | s/\s+//g; # strip white space |
78 | if (s/^([+-]?)0*(\d+)$/$1$2/) { # test if number |
79 | substr($_,$[,0) = '+' unless $1; # Add missing sign |
80 | s/^-0/+0/; |
81 | $_; |
82 | } else { |
83 | 'NaN'; |
84 | } |
85 | } |
86 | |
87 | # Convert a number from string format to internal base 100000 format. |
88 | # Assumes normalized value as input. |
89 | sub internal { #(num_str) return int_num_array |
90 | local($d) = @_; |
91 | ($is,$il) = (substr($d,$[,1),length($d)-2); |
92 | substr($d,$[,1) = ''; |
93 | ($is, reverse(unpack("a" . ($il%5+1) . ("a5" x ($il/5)), $d))); |
94 | } |
95 | |
96 | # Convert a number from internal base 100000 format to string format. |
97 | # This routine scribbles all over input array. |
98 | sub external { #(int_num_array) return num_str |
99 | $es = shift; |
100 | grep($_ > 9999 || ($_ = substr('0000'.$_,-5)), @_); # zero pad |
101 | &bnorm(join('', $es, reverse(@_))); # reverse concat and normalize |
102 | } |
103 | |
104 | # Negate input value. |
105 | sub bneg { #(num_str) return num_str |
106 | local($_) = &bnorm(@_); |
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107 | return $_ if $_ eq '+0' or $_ eq 'NaN'; |
108 | vec($_,0,8) ^= ord('+') ^ ord('-'); |
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109 | $_; |
110 | } |
111 | |
112 | # Returns the absolute value of the input. |
113 | sub babs { #(num_str) return num_str |
114 | &abs(&bnorm(@_)); |
115 | } |
116 | |
117 | sub abs { # post-normalized abs for internal use |
118 | local($_) = @_; |
119 | s/^-/+/; |
120 | $_; |
121 | } |
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122 | |
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123 | # Compares 2 values. Returns one of undef, <0, =0, >0. (suitable for sort) |
124 | sub bcmp { #(num_str, num_str) return cond_code |
125 | local($x,$y) = (&bnorm($_[$[]),&bnorm($_[$[+1])); |
126 | if ($x eq 'NaN') { |
127 | undef; |
128 | } elsif ($y eq 'NaN') { |
129 | undef; |
130 | } else { |
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131 | &cmp($x,$y) <=> 0; |
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132 | } |
133 | } |
134 | |
135 | sub cmp { # post-normalized compare for internal use |
136 | local($cx, $cy) = @_; |
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137 | |
138 | return 0 if ($cx eq $cy); |
139 | |
140 | local($sx, $sy) = (substr($cx, 0, 1), substr($cy, 0, 1)); |
141 | local($ld); |
142 | |
143 | if ($sx eq '+') { |
144 | return 1 if ($sy eq '-' || $cy eq '+0'); |
145 | $ld = length($cx) - length($cy); |
146 | return $ld if ($ld); |
147 | return $cx cmp $cy; |
148 | } else { # $sx eq '-' |
149 | return -1 if ($sy eq '+'); |
150 | $ld = length($cy) - length($cx); |
151 | return $ld if ($ld); |
152 | return $cy cmp $cx; |
153 | } |
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154 | } |
155 | |
156 | sub badd { #(num_str, num_str) return num_str |
157 | local(*x, *y); ($x, $y) = (&bnorm($_[$[]),&bnorm($_[$[+1])); |
158 | if ($x eq 'NaN') { |
159 | 'NaN'; |
160 | } elsif ($y eq 'NaN') { |
161 | 'NaN'; |
162 | } else { |
163 | @x = &internal($x); # convert to internal form |
164 | @y = &internal($y); |
165 | local($sx, $sy) = (shift @x, shift @y); # get signs |
166 | if ($sx eq $sy) { |
167 | &external($sx, &add(*x, *y)); # if same sign add |
168 | } else { |
169 | ($x, $y) = (&abs($x),&abs($y)); # make abs |
170 | if (&cmp($y,$x) > 0) { |
171 | &external($sy, &sub(*y, *x)); |
172 | } else { |
173 | &external($sx, &sub(*x, *y)); |
174 | } |
175 | } |
176 | } |
177 | } |
178 | |
179 | sub bsub { #(num_str, num_str) return num_str |
180 | &badd($_[$[],&bneg($_[$[+1])); |
181 | } |
182 | |
183 | # GCD -- Euclids algorithm Knuth Vol 2 pg 296 |
184 | sub bgcd { #(num_str, num_str) return num_str |
185 | local($x,$y) = (&bnorm($_[$[]),&bnorm($_[$[+1])); |
186 | if ($x eq 'NaN' || $y eq 'NaN') { |
187 | 'NaN'; |
188 | } else { |
189 | ($x, $y) = ($y,&bmod($x,$y)) while $y ne '+0'; |
190 | $x; |
191 | } |
192 | } |
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193 | |
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194 | # routine to add two base 1e5 numbers |
195 | # stolen from Knuth Vol 2 Algorithm A pg 231 |
196 | # there are separate routines to add and sub as per Kunth pg 233 |
197 | sub add { #(int_num_array, int_num_array) return int_num_array |
198 | local(*x, *y) = @_; |
199 | $car = 0; |
200 | for $x (@x) { |
201 | last unless @y || $car; |
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202 | $x -= 1e5 if $car = (($x += (@y ? shift(@y) : 0) + $car) >= 1e5) ? 1 : 0; |
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203 | } |
204 | for $y (@y) { |
205 | last unless $car; |
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206 | $y -= 1e5 if $car = (($y += $car) >= 1e5) ? 1 : 0; |
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207 | } |
208 | (@x, @y, $car); |
209 | } |
210 | |
211 | # subtract base 1e5 numbers -- stolen from Knuth Vol 2 pg 232, $x > $y |
212 | sub sub { #(int_num_array, int_num_array) return int_num_array |
213 | local(*sx, *sy) = @_; |
214 | $bar = 0; |
215 | for $sx (@sx) { |
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216 | last unless @sy || $bar; |
217 | $sx += 1e5 if $bar = (($sx -= (@sy ? shift(@sy) : 0) + $bar) < 0); |
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218 | } |
219 | @sx; |
220 | } |
221 | |
222 | # multiply two numbers -- stolen from Knuth Vol 2 pg 233 |
223 | sub bmul { #(num_str, num_str) return num_str |
224 | local(*x, *y); ($x, $y) = (&bnorm($_[$[]), &bnorm($_[$[+1])); |
225 | if ($x eq 'NaN') { |
226 | 'NaN'; |
227 | } elsif ($y eq 'NaN') { |
228 | 'NaN'; |
229 | } else { |
230 | @x = &internal($x); |
231 | @y = &internal($y); |
232 | &external(&mul(*x,*y)); |
233 | } |
234 | } |
235 | |
236 | # multiply two numbers in internal representation |
237 | # destroys the arguments, supposes that two arguments are different |
238 | sub mul { #(*int_num_array, *int_num_array) return int_num_array |
239 | local(*x, *y) = (shift, shift); |
240 | local($signr) = (shift @x ne shift @y) ? '-' : '+'; |
241 | @prod = (); |
242 | for $x (@x) { |
243 | ($car, $cty) = (0, $[); |
244 | for $y (@y) { |
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245 | $prod = $x * $y + ($prod[$cty] || 0) + $car; |
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246 | if ($use_mult) { |
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247 | $prod[$cty++] = |
248 | $prod - ($car = int($prod * 1e-5)) * 1e5; |
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249 | } |
250 | else { |
251 | $prod[$cty++] = |
252 | $prod - ($car = int($prod / 1e5)) * 1e5; |
253 | } |
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254 | } |
255 | $prod[$cty] += $car if $car; |
256 | $x = shift @prod; |
257 | } |
258 | ($signr, @x, @prod); |
259 | } |
260 | |
261 | # modulus |
262 | sub bmod { #(num_str, num_str) return num_str |
263 | (&bdiv(@_))[$[+1]; |
264 | } |
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265 | |
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266 | sub bdiv { #(dividend: num_str, divisor: num_str) return num_str |
267 | local (*x, *y); ($x, $y) = (&bnorm($_[$[]), &bnorm($_[$[+1])); |
268 | return wantarray ? ('NaN','NaN') : 'NaN' |
269 | if ($x eq 'NaN' || $y eq 'NaN' || $y eq '+0'); |
270 | return wantarray ? ('+0',$x) : '+0' if (&cmp(&abs($x),&abs($y)) < 0); |
271 | @x = &internal($x); @y = &internal($y); |
272 | $srem = $y[$[]; |
273 | $sr = (shift @x ne shift @y) ? '-' : '+'; |
274 | $car = $bar = $prd = 0; |
275 | if (($dd = int(1e5/($y[$#y]+1))) != 1) { |
276 | for $x (@x) { |
277 | $x = $x * $dd + $car; |
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278 | if ($use_mult) { |
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279 | $x -= ($car = int($x * 1e-5)) * 1e5; |
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280 | } |
281 | else { |
282 | $x -= ($car = int($x / 1e5)) * 1e5; |
283 | } |
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284 | } |
285 | push(@x, $car); $car = 0; |
286 | for $y (@y) { |
287 | $y = $y * $dd + $car; |
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288 | if ($use_mult) { |
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289 | $y -= ($car = int($y * 1e-5)) * 1e5; |
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290 | } |
291 | else { |
292 | $y -= ($car = int($y / 1e5)) * 1e5; |
293 | } |
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294 | } |
295 | } |
296 | else { |
297 | push(@x, 0); |
298 | } |
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299 | @q = (); ($v2,$v1) = @y[-2,-1]; |
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300 | $v2 = 0 unless $v2; |
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301 | while ($#x > $#y) { |
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302 | ($u2,$u1,$u0) = @x[-3..-1]; |
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303 | $u2 = 0 unless $u2; |
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304 | $q = (($u0 == $v1) ? 99999 : int(($u0*1e5+$u1)/$v1)); |
305 | --$q while ($v2*$q > ($u0*1e5+$u1-$q*$v1)*1e5+$u2); |
306 | if ($q) { |
307 | ($car, $bar) = (0,0); |
308 | for ($y = $[, $x = $#x-$#y+$[-1; $y <= $#y; ++$y,++$x) { |
309 | $prd = $q * $y[$y] + $car; |
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310 | if ($use_mult) { |
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311 | $prd -= ($car = int($prd * 1e-5)) * 1e5; |
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312 | } |
313 | else { |
314 | $prd -= ($car = int($prd / 1e5)) * 1e5; |
315 | } |
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316 | $x[$x] += 1e5 if ($bar = (($x[$x] -= $prd + $bar) < 0)); |
317 | } |
318 | if ($x[$#x] < $car + $bar) { |
319 | $car = 0; --$q; |
320 | for ($y = $[, $x = $#x-$#y+$[-1; $y <= $#y; ++$y,++$x) { |
321 | $x[$x] -= 1e5 |
322 | if ($car = (($x[$x] += $y[$y] + $car) > 1e5)); |
323 | } |
324 | } |
325 | } |
326 | pop(@x); unshift(@q, $q); |
327 | } |
328 | if (wantarray) { |
329 | @d = (); |
330 | if ($dd != 1) { |
331 | $car = 0; |
332 | for $x (reverse @x) { |
333 | $prd = $car * 1e5 + $x; |
334 | $car = $prd - ($tmp = int($prd / $dd)) * $dd; |
335 | unshift(@d, $tmp); |
336 | } |
337 | } |
338 | else { |
339 | @d = @x; |
340 | } |
341 | (&external($sr, @q), &external($srem, @d, $zero)); |
342 | } else { |
343 | &external($sr, @q); |
344 | } |
345 | } |
346 | |
347 | # compute power of two numbers -- stolen from Knuth Vol 2 pg 233 |
348 | sub bpow { #(num_str, num_str) return num_str |
349 | local(*x, *y); ($x, $y) = (&bnorm($_[$[]), &bnorm($_[$[+1])); |
350 | if ($x eq 'NaN') { |
351 | 'NaN'; |
352 | } elsif ($y eq 'NaN') { |
353 | 'NaN'; |
354 | } elsif ($x eq '+1') { |
355 | '+1'; |
356 | } elsif ($x eq '-1') { |
357 | &bmod($x,2) ? '-1': '+1'; |
358 | } elsif ($y =~ /^-/) { |
359 | 'NaN'; |
360 | } elsif ($x eq '+0' && $y eq '+0') { |
361 | 'NaN'; |
362 | } else { |
363 | @x = &internal($x); |
364 | local(@pow2)=@x; |
365 | local(@pow)=&internal("+1"); |
366 | local($y1,$res,@tmp1,@tmp2)=(1); # need tmp to send to mul |
367 | while ($y ne '+0') { |
368 | ($y,$res)=&bdiv($y,2); |
369 | if ($res ne '+0') {@tmp=@pow2; @pow=&mul(*pow,*tmp);} |
370 | if ($y ne '+0') {@tmp=@pow2;@pow2=&mul(*pow2,*tmp);} |
371 | } |
372 | &external(@pow); |
373 | } |
374 | } |
375 | |
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376 | # compute x << y, y >= 0 |
377 | sub blsft { #(num_str, num_str) return num_str |
378 | &bmul($_[$[], &bpow(2, $_[$[+1])); |
379 | } |
380 | |
381 | # compute x >> y, y >= 0 |
382 | sub brsft { #(num_str, num_str) return num_str |
383 | &bdiv($_[$[], &bpow(2, $_[$[+1])); |
384 | } |
385 | |
386 | # compute x & y |
387 | sub band { #(num_str, num_str) return num_str |
388 | local($x,$y,$r,$m,$xr,$yr) = (&bnorm($_[$[]),&bnorm($_[$[+1]),0,1); |
389 | if ($x eq 'NaN' || $y eq 'NaN') { |
390 | 'NaN'; |
391 | } else { |
392 | while ($x ne '+0' && $y ne '+0') { |
393 | ($x, $xr) = &bdiv($x, 0x10000); |
394 | ($y, $yr) = &bdiv($y, 0x10000); |
395 | $r = &badd(&bmul(int $xr & $yr, $m), $r); |
396 | $m = &bmul($m, 0x10000); |
397 | } |
398 | $r; |
399 | } |
400 | } |
401 | |
402 | # compute x | y |
403 | sub bior { #(num_str, num_str) return num_str |
404 | local($x,$y,$r,$m,$xr,$yr) = (&bnorm($_[$[]),&bnorm($_[$[+1]),0,1); |
405 | if ($x eq 'NaN' || $y eq 'NaN') { |
406 | 'NaN'; |
407 | } else { |
408 | while ($x ne '+0' || $y ne '+0') { |
409 | ($x, $xr) = &bdiv($x, 0x10000); |
410 | ($y, $yr) = &bdiv($y, 0x10000); |
411 | $r = &badd(&bmul(int $xr | $yr, $m), $r); |
412 | $m = &bmul($m, 0x10000); |
413 | } |
414 | $r; |
415 | } |
416 | } |
417 | |
418 | # compute x ^ y |
419 | sub bxor { #(num_str, num_str) return num_str |
420 | local($x,$y,$r,$m,$xr,$yr) = (&bnorm($_[$[]),&bnorm($_[$[+1]),0,1); |
421 | if ($x eq 'NaN' || $y eq 'NaN') { |
422 | 'NaN'; |
423 | } else { |
424 | while ($x ne '+0' || $y ne '+0') { |
425 | ($x, $xr) = &bdiv($x, 0x10000); |
426 | ($y, $yr) = &bdiv($y, 0x10000); |
427 | $r = &badd(&bmul(int $xr ^ $yr, $m), $r); |
428 | $m = &bmul($m, 0x10000); |
429 | } |
430 | $r; |
431 | } |
432 | } |
433 | |
434 | # represent ~x as twos-complement number |
435 | sub bnot { #(num_str) return num_str |
436 | &bsub(-1,$_[$[]); |
437 | } |
438 | |
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439 | 1; |
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440 | __END__ |
441 | |
442 | =head1 NAME |
443 | |
444 | Math::BigInt - Arbitrary size integer math package |
445 | |
446 | =head1 SYNOPSIS |
447 | |
448 | use Math::BigInt; |
449 | $i = Math::BigInt->new($string); |
450 | |
451 | $i->bneg return BINT negation |
452 | $i->babs return BINT absolute value |
453 | $i->bcmp(BINT) return CODE compare numbers (undef,<0,=0,>0) |
454 | $i->badd(BINT) return BINT addition |
455 | $i->bsub(BINT) return BINT subtraction |
456 | $i->bmul(BINT) return BINT multiplication |
457 | $i->bdiv(BINT) return (BINT,BINT) division (quo,rem) just quo if scalar |
458 | $i->bmod(BINT) return BINT modulus |
459 | $i->bgcd(BINT) return BINT greatest common divisor |
460 | $i->bnorm return BINT normalization |
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461 | $i->blsft(BINT) return BINT left shift |
462 | $i->brsft(BINT) return (BINT,BINT) right shift (quo,rem) just quo if scalar |
463 | $i->band(BINT) return BINT bit-wise and |
464 | $i->bior(BINT) return BINT bit-wise inclusive or |
465 | $i->bxor(BINT) return BINT bit-wise exclusive or |
466 | $i->bnot return BINT bit-wise not |
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467 | |
468 | =head1 DESCRIPTION |
469 | |
470 | All basic math operations are overloaded if you declare your big |
471 | integers as |
472 | |
473 | $i = new Math::BigInt '123 456 789 123 456 789'; |
474 | |
475 | |
476 | =over 2 |
477 | |
478 | =item Canonical notation |
479 | |
480 | Big integer value are strings of the form C</^[+-]\d+$/> with leading |
481 | zeros suppressed. |
482 | |
483 | =item Input |
484 | |
485 | Input values to these routines may be strings of the form |
486 | C</^\s*[+-]?[\d\s]+$/>. |
487 | |
488 | =item Output |
489 | |
490 | Output values always always in canonical form |
491 | |
492 | =back |
493 | |
494 | Actual math is done in an internal format consisting of an array |
495 | whose first element is the sign (/^[+-]$/) and whose remaining |
496 | elements are base 100000 digits with the least significant digit first. |
497 | The string 'NaN' is used to represent the result when input arguments |
498 | are not numbers, as well as the result of dividing by zero. |
499 | |
500 | =head1 EXAMPLES |
501 | |
502 | '+0' canonical zero value |
503 | ' -123 123 123' canonical value '-123123123' |
504 | '1 23 456 7890' canonical value '+1234567890' |
505 | |
506 | |
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507 | =head1 Autocreating constants |
508 | |
509 | After C<use Math::BigInt ':constant'> all the integer decimal constants |
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510 | in the given scope are converted to C<Math::BigInt>. This conversion |
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511 | happens at compile time. |
512 | |
513 | In particular |
514 | |
515 | perl -MMath::BigInt=:constant -e 'print 2**100' |
516 | |
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517 | print the integer value of C<2**100>. Note that without conversion of |
518 | constants the expression 2**100 will be calculated as floating point number. |
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519 | |
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520 | =head1 BUGS |
521 | |
522 | The current version of this module is a preliminary version of the |
523 | real thing that is currently (as of perl5.002) under development. |
524 | |
525 | =head1 AUTHOR |
526 | |
527 | Mark Biggar, overloaded interface by Ilya Zakharevich. |
528 | |
529 | =cut |