Commit | Line | Data |
a0d0e21e |
1 | package Math::BigInt; |
2 | |
3 | %OVERLOAD = ( |
4 | # Anonymous subroutines: |
5 | '+' => sub {new BigInt &badd}, |
6 | '-' => sub {new BigInt |
7 | $_[2]? bsub($_[1],${$_[0]}) : bsub(${$_[0]},$_[1])}, |
8 | '<=>' => sub {new BigInt |
9 | $_[2]? bcmp($_[1],${$_[0]}) : bcmp(${$_[0]},$_[1])}, |
10 | 'cmp' => sub {new BigInt |
11 | $_[2]? ($_[1] cmp ${$_[0]}) : (${$_[0]} cmp $_[1])}, |
12 | '*' => sub {new BigInt &bmul}, |
13 | '/' => sub {new BigInt |
14 | $_[2]? scalar bdiv($_[1],${$_[0]}) : |
15 | scalar bdiv(${$_[0]},$_[1])}, |
16 | '%' => sub {new BigInt |
17 | $_[2]? bmod($_[1],${$_[0]}) : bmod(${$_[0]},$_[1])}, |
18 | '**' => sub {new BigInt |
19 | $_[2]? bpow($_[1],${$_[0]}) : bpow(${$_[0]},$_[1])}, |
20 | 'neg' => sub {new BigInt &bneg}, |
21 | 'abs' => sub {new BigInt &babs}, |
22 | |
23 | qw( |
24 | "" stringify |
25 | 0+ numify) # Order of arguments unsignificant |
26 | ); |
27 | |
28 | sub new { |
29 | my $foo = bnorm($_[1]); |
30 | die "Not a number initialized to BigInt" if $foo eq "NaN"; |
31 | bless \$foo; |
32 | } |
33 | sub stringify { "${$_[0]}" } |
34 | sub numify { 0 + "${$_[0]}" } # Not needed, additional overhead |
35 | # comparing to direct compilation based on |
36 | # stringify |
37 | |
38 | # arbitrary size integer math package |
39 | # |
40 | # by Mark Biggar |
41 | # |
42 | # Canonical Big integer value are strings of the form |
43 | # /^[+-]\d+$/ with leading zeros suppressed |
44 | # Input values to these routines may be strings of the form |
45 | # /^\s*[+-]?[\d\s]+$/. |
46 | # Examples: |
47 | # '+0' canonical zero value |
48 | # ' -123 123 123' canonical value '-123123123' |
49 | # '1 23 456 7890' canonical value '+1234567890' |
50 | # Output values always always in canonical form |
51 | # |
52 | # Actual math is done in an internal format consisting of an array |
53 | # whose first element is the sign (/^[+-]$/) and whose remaining |
54 | # elements are base 100000 digits with the least significant digit first. |
55 | # The string 'NaN' is used to represent the result when input arguments |
56 | # are not numbers, as well as the result of dividing by zero |
57 | # |
58 | # routines provided are: |
59 | # |
60 | # bneg(BINT) return BINT negation |
61 | # babs(BINT) return BINT absolute value |
62 | # bcmp(BINT,BINT) return CODE compare numbers (undef,<0,=0,>0) |
63 | # badd(BINT,BINT) return BINT addition |
64 | # bsub(BINT,BINT) return BINT subtraction |
65 | # bmul(BINT,BINT) return BINT multiplication |
66 | # bdiv(BINT,BINT) return (BINT,BINT) division (quo,rem) just quo if scalar |
67 | # bmod(BINT,BINT) return BINT modulus |
68 | # bgcd(BINT,BINT) return BINT greatest common divisor |
69 | # bnorm(BINT) return BINT normalization |
70 | # |
71 | |
72 | $zero = 0; |
73 | |
74 | \f |
75 | # normalize string form of number. Strip leading zeros. Strip any |
76 | # white space and add a sign, if missing. |
77 | # Strings that are not numbers result the value 'NaN'. |
78 | |
79 | sub bnorm { #(num_str) return num_str |
80 | local($_) = @_; |
81 | s/\s+//g; # strip white space |
82 | if (s/^([+-]?)0*(\d+)$/$1$2/) { # test if number |
83 | substr($_,$[,0) = '+' unless $1; # Add missing sign |
84 | s/^-0/+0/; |
85 | $_; |
86 | } else { |
87 | 'NaN'; |
88 | } |
89 | } |
90 | |
91 | # Convert a number from string format to internal base 100000 format. |
92 | # Assumes normalized value as input. |
93 | sub internal { #(num_str) return int_num_array |
94 | local($d) = @_; |
95 | ($is,$il) = (substr($d,$[,1),length($d)-2); |
96 | substr($d,$[,1) = ''; |
97 | ($is, reverse(unpack("a" . ($il%5+1) . ("a5" x ($il/5)), $d))); |
98 | } |
99 | |
100 | # Convert a number from internal base 100000 format to string format. |
101 | # This routine scribbles all over input array. |
102 | sub external { #(int_num_array) return num_str |
103 | $es = shift; |
104 | grep($_ > 9999 || ($_ = substr('0000'.$_,-5)), @_); # zero pad |
105 | &bnorm(join('', $es, reverse(@_))); # reverse concat and normalize |
106 | } |
107 | |
108 | # Negate input value. |
109 | sub bneg { #(num_str) return num_str |
110 | local($_) = &bnorm(@_); |
111 | vec($_,0,8) ^= ord('+') ^ ord('-') unless $_ eq '+0'; |
112 | s/^H/N/; |
113 | $_; |
114 | } |
115 | |
116 | # Returns the absolute value of the input. |
117 | sub babs { #(num_str) return num_str |
118 | &abs(&bnorm(@_)); |
119 | } |
120 | |
121 | sub abs { # post-normalized abs for internal use |
122 | local($_) = @_; |
123 | s/^-/+/; |
124 | $_; |
125 | } |
126 | \f |
127 | # Compares 2 values. Returns one of undef, <0, =0, >0. (suitable for sort) |
128 | sub bcmp { #(num_str, num_str) return cond_code |
129 | local($x,$y) = (&bnorm($_[$[]),&bnorm($_[$[+1])); |
130 | if ($x eq 'NaN') { |
131 | undef; |
132 | } elsif ($y eq 'NaN') { |
133 | undef; |
134 | } else { |
135 | &cmp($x,$y); |
136 | } |
137 | } |
138 | |
139 | sub cmp { # post-normalized compare for internal use |
140 | local($cx, $cy) = @_; |
141 | $cx cmp $cy |
142 | && |
143 | ( |
144 | ord($cy) <=> ord($cx) |
145 | || |
146 | ($cx cmp ',') * (length($cy) <=> length($cx) || $cy cmp $cx) |
147 | ); |
148 | } |
149 | |
150 | sub badd { #(num_str, num_str) return num_str |
151 | local(*x, *y); ($x, $y) = (&bnorm($_[$[]),&bnorm($_[$[+1])); |
152 | if ($x eq 'NaN') { |
153 | 'NaN'; |
154 | } elsif ($y eq 'NaN') { |
155 | 'NaN'; |
156 | } else { |
157 | @x = &internal($x); # convert to internal form |
158 | @y = &internal($y); |
159 | local($sx, $sy) = (shift @x, shift @y); # get signs |
160 | if ($sx eq $sy) { |
161 | &external($sx, &add(*x, *y)); # if same sign add |
162 | } else { |
163 | ($x, $y) = (&abs($x),&abs($y)); # make abs |
164 | if (&cmp($y,$x) > 0) { |
165 | &external($sy, &sub(*y, *x)); |
166 | } else { |
167 | &external($sx, &sub(*x, *y)); |
168 | } |
169 | } |
170 | } |
171 | } |
172 | |
173 | sub bsub { #(num_str, num_str) return num_str |
174 | &badd($_[$[],&bneg($_[$[+1])); |
175 | } |
176 | |
177 | # GCD -- Euclids algorithm Knuth Vol 2 pg 296 |
178 | sub bgcd { #(num_str, num_str) return num_str |
179 | local($x,$y) = (&bnorm($_[$[]),&bnorm($_[$[+1])); |
180 | if ($x eq 'NaN' || $y eq 'NaN') { |
181 | 'NaN'; |
182 | } else { |
183 | ($x, $y) = ($y,&bmod($x,$y)) while $y ne '+0'; |
184 | $x; |
185 | } |
186 | } |
187 | \f |
188 | # routine to add two base 1e5 numbers |
189 | # stolen from Knuth Vol 2 Algorithm A pg 231 |
190 | # there are separate routines to add and sub as per Kunth pg 233 |
191 | sub add { #(int_num_array, int_num_array) return int_num_array |
192 | local(*x, *y) = @_; |
193 | $car = 0; |
194 | for $x (@x) { |
195 | last unless @y || $car; |
196 | $x -= 1e5 if $car = (($x += shift(@y) + $car) >= 1e5); |
197 | } |
198 | for $y (@y) { |
199 | last unless $car; |
200 | $y -= 1e5 if $car = (($y += $car) >= 1e5); |
201 | } |
202 | (@x, @y, $car); |
203 | } |
204 | |
205 | # subtract base 1e5 numbers -- stolen from Knuth Vol 2 pg 232, $x > $y |
206 | sub sub { #(int_num_array, int_num_array) return int_num_array |
207 | local(*sx, *sy) = @_; |
208 | $bar = 0; |
209 | for $sx (@sx) { |
210 | last unless @y || $bar; |
211 | $sx += 1e5 if $bar = (($sx -= shift(@sy) + $bar) < 0); |
212 | } |
213 | @sx; |
214 | } |
215 | |
216 | # multiply two numbers -- stolen from Knuth Vol 2 pg 233 |
217 | sub bmul { #(num_str, num_str) return num_str |
218 | local(*x, *y); ($x, $y) = (&bnorm($_[$[]), &bnorm($_[$[+1])); |
219 | if ($x eq 'NaN') { |
220 | 'NaN'; |
221 | } elsif ($y eq 'NaN') { |
222 | 'NaN'; |
223 | } else { |
224 | @x = &internal($x); |
225 | @y = &internal($y); |
226 | &external(&mul(*x,*y)); |
227 | } |
228 | } |
229 | |
230 | # multiply two numbers in internal representation |
231 | # destroys the arguments, supposes that two arguments are different |
232 | sub mul { #(*int_num_array, *int_num_array) return int_num_array |
233 | local(*x, *y) = (shift, shift); |
234 | local($signr) = (shift @x ne shift @y) ? '-' : '+'; |
235 | @prod = (); |
236 | for $x (@x) { |
237 | ($car, $cty) = (0, $[); |
238 | for $y (@y) { |
239 | $prod = $x * $y + $prod[$cty] + $car; |
240 | $prod[$cty++] = |
241 | $prod - ($car = int($prod * 1e-5)) * 1e5; |
242 | } |
243 | $prod[$cty] += $car if $car; |
244 | $x = shift @prod; |
245 | } |
246 | ($signr, @x, @prod); |
247 | } |
248 | |
249 | # modulus |
250 | sub bmod { #(num_str, num_str) return num_str |
251 | (&bdiv(@_))[$[+1]; |
252 | } |
253 | \f |
254 | sub bdiv { #(dividend: num_str, divisor: num_str) return num_str |
255 | local (*x, *y); ($x, $y) = (&bnorm($_[$[]), &bnorm($_[$[+1])); |
256 | return wantarray ? ('NaN','NaN') : 'NaN' |
257 | if ($x eq 'NaN' || $y eq 'NaN' || $y eq '+0'); |
258 | return wantarray ? ('+0',$x) : '+0' if (&cmp(&abs($x),&abs($y)) < 0); |
259 | @x = &internal($x); @y = &internal($y); |
260 | $srem = $y[$[]; |
261 | $sr = (shift @x ne shift @y) ? '-' : '+'; |
262 | $car = $bar = $prd = 0; |
263 | if (($dd = int(1e5/($y[$#y]+1))) != 1) { |
264 | for $x (@x) { |
265 | $x = $x * $dd + $car; |
266 | $x -= ($car = int($x * 1e-5)) * 1e5; |
267 | } |
268 | push(@x, $car); $car = 0; |
269 | for $y (@y) { |
270 | $y = $y * $dd + $car; |
271 | $y -= ($car = int($y * 1e-5)) * 1e5; |
272 | } |
273 | } |
274 | else { |
275 | push(@x, 0); |
276 | } |
277 | @q = (); ($v2,$v1) = @y[-2,-1]; |
278 | while ($#x > $#y) { |
279 | ($u2,$u1,$u0) = @x[-3..-1]; |
280 | $q = (($u0 == $v1) ? 99999 : int(($u0*1e5+$u1)/$v1)); |
281 | --$q while ($v2*$q > ($u0*1e5+$u1-$q*$v1)*1e5+$u2); |
282 | if ($q) { |
283 | ($car, $bar) = (0,0); |
284 | for ($y = $[, $x = $#x-$#y+$[-1; $y <= $#y; ++$y,++$x) { |
285 | $prd = $q * $y[$y] + $car; |
286 | $prd -= ($car = int($prd * 1e-5)) * 1e5; |
287 | $x[$x] += 1e5 if ($bar = (($x[$x] -= $prd + $bar) < 0)); |
288 | } |
289 | if ($x[$#x] < $car + $bar) { |
290 | $car = 0; --$q; |
291 | for ($y = $[, $x = $#x-$#y+$[-1; $y <= $#y; ++$y,++$x) { |
292 | $x[$x] -= 1e5 |
293 | if ($car = (($x[$x] += $y[$y] + $car) > 1e5)); |
294 | } |
295 | } |
296 | } |
297 | pop(@x); unshift(@q, $q); |
298 | } |
299 | if (wantarray) { |
300 | @d = (); |
301 | if ($dd != 1) { |
302 | $car = 0; |
303 | for $x (reverse @x) { |
304 | $prd = $car * 1e5 + $x; |
305 | $car = $prd - ($tmp = int($prd / $dd)) * $dd; |
306 | unshift(@d, $tmp); |
307 | } |
308 | } |
309 | else { |
310 | @d = @x; |
311 | } |
312 | (&external($sr, @q), &external($srem, @d, $zero)); |
313 | } else { |
314 | &external($sr, @q); |
315 | } |
316 | } |
317 | |
318 | # compute power of two numbers -- stolen from Knuth Vol 2 pg 233 |
319 | sub bpow { #(num_str, num_str) return num_str |
320 | local(*x, *y); ($x, $y) = (&bnorm($_[$[]), &bnorm($_[$[+1])); |
321 | if ($x eq 'NaN') { |
322 | 'NaN'; |
323 | } elsif ($y eq 'NaN') { |
324 | 'NaN'; |
325 | } elsif ($x eq '+1') { |
326 | '+1'; |
327 | } elsif ($x eq '-1') { |
328 | &bmod($x,2) ? '-1': '+1'; |
329 | } elsif ($y =~ /^-/) { |
330 | 'NaN'; |
331 | } elsif ($x eq '+0' && $y eq '+0') { |
332 | 'NaN'; |
333 | } else { |
334 | @x = &internal($x); |
335 | local(@pow2)=@x; |
336 | local(@pow)=&internal("+1"); |
337 | local($y1,$res,@tmp1,@tmp2)=(1); # need tmp to send to mul |
338 | while ($y ne '+0') { |
339 | ($y,$res)=&bdiv($y,2); |
340 | if ($res ne '+0') {@tmp=@pow2; @pow=&mul(*pow,*tmp);} |
341 | if ($y ne '+0') {@tmp=@pow2;@pow2=&mul(*pow2,*tmp);} |
342 | } |
343 | &external(@pow); |
344 | } |
345 | } |
346 | |
347 | 1; |