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5303340c LW |
1 | package bigint; |
2 | ||
3 | # arbitrary size integer math package | |
4 | # | |
5 | # by Mark Biggar | |
6 | # | |
7 | # Canonical Big integer value are strings of the form | |
8 | # /^[+-]\d+$/ with leading zeros suppressed | |
9 | # Input values to these routines may be strings of the form | |
10 | # /^\s*[+-]?[\d\s]+$/. | |
11 | # Examples: | |
12 | # '+0' canonical zero value | |
13 | # ' -123 123 123' canonical value '-123123123' | |
14 | # '1 23 456 7890' canonical value '+1234567890' | |
15 | # Output values always always in canonical form | |
16 | # | |
17 | # Actual math is done in an internal format consisting of an array | |
18 | # whose first element is the sign (/^[+-]$/) and whose remaining | |
19 | # elements are base 100000 digits with the least significant digit first. | |
20 | # The string 'NaN' is used to represent the result when input arguments | |
21 | # are not numbers, as well as the result of dividing by zero | |
22 | # | |
23 | # routines provided are: | |
24 | # | |
25 | # bneg(BINT) return BINT negation | |
26 | # babs(BINT) return BINT absolute value | |
27 | # bcmp(BINT,BINT) return CODE compare numbers (undef,<0,=0,>0) | |
28 | # badd(BINT,BINT) return BINT addition | |
29 | # bsub(BINT,BINT) return BINT subtraction | |
30 | # bmul(BINT,BINT) return BINT multiplication | |
31 | # bdiv(BINT,BINT) return (BINT,BINT) division (quo,rem) just quo if scalar | |
32 | # bmod(BINT,BINT) return BINT modulus | |
33 | # bgcd(BINT,BINT) return BINT greatest common divisor | |
34 | # bnorm(BINT) return BINT normalization | |
35 | # | |
8990e307 LW |
36 | |
37 | $zero = 0; | |
38 | ||
5303340c LW |
39 | \f |
40 | # normalize string form of number. Strip leading zeros. Strip any | |
41 | # white space and add a sign, if missing. | |
42 | # Strings that are not numbers result the value 'NaN'. | |
8990e307 | 43 | |
5303340c LW |
44 | sub main'bnorm { #(num_str) return num_str |
45 | local($_) = @_; | |
46 | s/\s+//g; # strip white space | |
47 | if (s/^([+-]?)0*(\d+)$/$1$2/) { # test if number | |
79072805 | 48 | substr($_,$[,0) = '+' unless $1; # Add missing sign |
5303340c LW |
49 | s/^-0/+0/; |
50 | $_; | |
51 | } else { | |
52 | 'NaN'; | |
53 | } | |
54 | } | |
55 | ||
56 | # Convert a number from string format to internal base 100000 format. | |
57 | # Assumes normalized value as input. | |
58 | sub internal { #(num_str) return int_num_array | |
59 | local($d) = @_; | |
79072805 LW |
60 | ($is,$il) = (substr($d,$[,1),length($d)-2); |
61 | substr($d,$[,1) = ''; | |
5303340c LW |
62 | ($is, reverse(unpack("a" . ($il%5+1) . ("a5" x ($il/5)), $d))); |
63 | } | |
64 | ||
65 | # Convert a number from internal base 100000 format to string format. | |
66 | # This routine scribbles all over input array. | |
67 | sub external { #(int_num_array) return num_str | |
68 | $es = shift; | |
69 | grep($_ > 9999 || ($_ = substr('0000'.$_,-5)), @_); # zero pad | |
70 | &'bnorm(join('', $es, reverse(@_))); # reverse concat and normalize | |
71 | } | |
72 | ||
73 | # Negate input value. | |
74 | sub main'bneg { #(num_str) return num_str | |
75 | local($_) = &'bnorm(@_); | |
76 | vec($_,0,8) ^= ord('+') ^ ord('-') unless $_ eq '+0'; | |
77 | s/^H/N/; | |
78 | $_; | |
79 | } | |
80 | ||
81 | # Returns the absolute value of the input. | |
82 | sub main'babs { #(num_str) return num_str | |
83 | &abs(&'bnorm(@_)); | |
84 | } | |
85 | ||
86 | sub abs { # post-normalized abs for internal use | |
87 | local($_) = @_; | |
88 | s/^-/+/; | |
89 | $_; | |
90 | } | |
91 | \f | |
92 | # Compares 2 values. Returns one of undef, <0, =0, >0. (suitable for sort) | |
93 | sub main'bcmp { #(num_str, num_str) return cond_code | |
79072805 | 94 | local($x,$y) = (&'bnorm($_[$[]),&'bnorm($_[$[+1])); |
5303340c LW |
95 | if ($x eq 'NaN') { |
96 | undef; | |
97 | } elsif ($y eq 'NaN') { | |
98 | undef; | |
99 | } else { | |
100 | &cmp($x,$y); | |
101 | } | |
102 | } | |
103 | ||
104 | sub cmp { # post-normalized compare for internal use | |
105 | local($cx, $cy) = @_; | |
1e2e1ae8 JH |
106 | return 0 if ($cx eq $cy); |
107 | ||
108 | local($sx, $sy) = (substr($cx, 0, 1), substr($cy, 0, 1)); | |
109 | local($ld); | |
110 | ||
111 | if ($sx eq '+') { | |
112 | return 1 if ($sy eq '-' || $cy eq '+0'); | |
113 | $ld = length($cx) - length($cy); | |
114 | return $ld if ($ld); | |
115 | return $cx cmp $cy; | |
116 | } else { # $sx eq '-' | |
117 | return -1 if ($sy eq '+'); | |
118 | $ld = length($cy) - length($cx); | |
119 | return $ld if ($ld); | |
120 | return $cy cmp $cx; | |
121 | } | |
122 | ||
5303340c LW |
123 | } |
124 | ||
125 | sub main'badd { #(num_str, num_str) return num_str | |
79072805 | 126 | local(*x, *y); ($x, $y) = (&'bnorm($_[$[]),&'bnorm($_[$[+1])); |
5303340c LW |
127 | if ($x eq 'NaN') { |
128 | 'NaN'; | |
129 | } elsif ($y eq 'NaN') { | |
130 | 'NaN'; | |
131 | } else { | |
132 | @x = &internal($x); # convert to internal form | |
133 | @y = &internal($y); | |
134 | local($sx, $sy) = (shift @x, shift @y); # get signs | |
135 | if ($sx eq $sy) { | |
136 | &external($sx, &add(*x, *y)); # if same sign add | |
137 | } else { | |
138 | ($x, $y) = (&abs($x),&abs($y)); # make abs | |
139 | if (&cmp($y,$x) > 0) { | |
140 | &external($sy, &sub(*y, *x)); | |
141 | } else { | |
142 | &external($sx, &sub(*x, *y)); | |
143 | } | |
144 | } | |
145 | } | |
146 | } | |
147 | ||
148 | sub main'bsub { #(num_str, num_str) return num_str | |
79072805 | 149 | &'badd($_[$[],&'bneg($_[$[+1])); |
5303340c LW |
150 | } |
151 | ||
152 | # GCD -- Euclids algorithm Knuth Vol 2 pg 296 | |
153 | sub main'bgcd { #(num_str, num_str) return num_str | |
79072805 | 154 | local($x,$y) = (&'bnorm($_[$[]),&'bnorm($_[$[+1])); |
68decaef | 155 | if ($x eq 'NaN' || $y eq 'NaN') { |
5303340c | 156 | 'NaN'; |
68decaef | 157 | } else { |
5303340c LW |
158 | ($x, $y) = ($y,&'bmod($x,$y)) while $y ne '+0'; |
159 | $x; | |
160 | } | |
161 | } | |
162 | \f | |
68decaef | 163 | # routine to add two base 1e5 numbers |
5303340c LW |
164 | # stolen from Knuth Vol 2 Algorithm A pg 231 |
165 | # there are separate routines to add and sub as per Kunth pg 233 | |
166 | sub add { #(int_num_array, int_num_array) return int_num_array | |
167 | local(*x, *y) = @_; | |
168 | $car = 0; | |
169 | for $x (@x) { | |
170 | last unless @y || $car; | |
55497cff | 171 | $x -= 1e5 if $car = (($x += shift(@y) + $car) >= 1e5) ? 1 : 0; |
5303340c LW |
172 | } |
173 | for $y (@y) { | |
174 | last unless $car; | |
55497cff | 175 | $y -= 1e5 if $car = (($y += $car) >= 1e5) ? 1 : 0; |
5303340c LW |
176 | } |
177 | (@x, @y, $car); | |
178 | } | |
179 | ||
68decaef | 180 | # subtract base 1e5 numbers -- stolen from Knuth Vol 2 pg 232, $x > $y |
5303340c LW |
181 | sub sub { #(int_num_array, int_num_array) return int_num_array |
182 | local(*sx, *sy) = @_; | |
183 | $bar = 0; | |
184 | for $sx (@sx) { | |
185 | last unless @y || $bar; | |
e334a159 | 186 | $sx += 1e5 if $bar = (($sx -= shift(@sy) + $bar) < 0); |
5303340c LW |
187 | } |
188 | @sx; | |
189 | } | |
190 | ||
191 | # multiply two numbers -- stolen from Knuth Vol 2 pg 233 | |
192 | sub main'bmul { #(num_str, num_str) return num_str | |
79072805 | 193 | local(*x, *y); ($x, $y) = (&'bnorm($_[$[]), &'bnorm($_[$[+1])); |
5303340c LW |
194 | if ($x eq 'NaN') { |
195 | 'NaN'; | |
196 | } elsif ($y eq 'NaN') { | |
197 | 'NaN'; | |
198 | } else { | |
199 | @x = &internal($x); | |
200 | @y = &internal($y); | |
201 | local($signr) = (shift @x ne shift @y) ? '-' : '+'; | |
202 | @prod = (); | |
203 | for $x (@x) { | |
79072805 | 204 | ($car, $cty) = (0, $[); |
5303340c LW |
205 | for $y (@y) { |
206 | $prod = $x * $y + $prod[$cty] + $car; | |
207 | $prod[$cty++] = | |
68decaef | 208 | $prod - ($car = int($prod * 1e-5)) * 1e5; |
5303340c LW |
209 | } |
210 | $prod[$cty] += $car if $car; | |
211 | $x = shift @prod; | |
212 | } | |
213 | &external($signr, @x, @prod); | |
214 | } | |
215 | } | |
216 | ||
217 | # modulus | |
218 | sub main'bmod { #(num_str, num_str) return num_str | |
79072805 | 219 | (&'bdiv(@_))[$[+1]; |
5303340c LW |
220 | } |
221 | \f | |
222 | sub main'bdiv { #(dividend: num_str, divisor: num_str) return num_str | |
79072805 | 223 | local (*x, *y); ($x, $y) = (&'bnorm($_[$[]), &'bnorm($_[$[+1])); |
5303340c LW |
224 | return wantarray ? ('NaN','NaN') : 'NaN' |
225 | if ($x eq 'NaN' || $y eq 'NaN' || $y eq '+0'); | |
226 | return wantarray ? ('+0',$x) : '+0' if (&cmp(&abs($x),&abs($y)) < 0); | |
227 | @x = &internal($x); @y = &internal($y); | |
79072805 | 228 | $srem = $y[$[]; |
5303340c LW |
229 | $sr = (shift @x ne shift @y) ? '-' : '+'; |
230 | $car = $bar = $prd = 0; | |
68decaef | 231 | if (($dd = int(1e5/($y[$#y]+1))) != 1) { |
5303340c LW |
232 | for $x (@x) { |
233 | $x = $x * $dd + $car; | |
68decaef | 234 | $x -= ($car = int($x * 1e-5)) * 1e5; |
5303340c LW |
235 | } |
236 | push(@x, $car); $car = 0; | |
237 | for $y (@y) { | |
238 | $y = $y * $dd + $car; | |
68decaef | 239 | $y -= ($car = int($y * 1e-5)) * 1e5; |
5303340c LW |
240 | } |
241 | } | |
242 | else { | |
243 | push(@x, 0); | |
244 | } | |
ed6116ce | 245 | @q = (); ($v2,$v1) = @y[-2,-1]; |
5303340c | 246 | while ($#x > $#y) { |
ed6116ce | 247 | ($u2,$u1,$u0) = @x[-3..-1]; |
68decaef LW |
248 | $q = (($u0 == $v1) ? 99999 : int(($u0*1e5+$u1)/$v1)); |
249 | --$q while ($v2*$q > ($u0*1e5+$u1-$q*$v1)*1e5+$u2); | |
5303340c LW |
250 | if ($q) { |
251 | ($car, $bar) = (0,0); | |
79072805 | 252 | for ($y = $[, $x = $#x-$#y+$[-1; $y <= $#y; ++$y,++$x) { |
5303340c | 253 | $prd = $q * $y[$y] + $car; |
68decaef LW |
254 | $prd -= ($car = int($prd * 1e-5)) * 1e5; |
255 | $x[$x] += 1e5 if ($bar = (($x[$x] -= $prd + $bar) < 0)); | |
5303340c LW |
256 | } |
257 | if ($x[$#x] < $car + $bar) { | |
258 | $car = 0; --$q; | |
79072805 | 259 | for ($y = $[, $x = $#x-$#y+$[-1; $y <= $#y; ++$y,++$x) { |
68decaef LW |
260 | $x[$x] -= 1e5 |
261 | if ($car = (($x[$x] += $y[$y] + $car) > 1e5)); | |
5303340c LW |
262 | } |
263 | } | |
264 | } | |
265 | pop(@x); unshift(@q, $q); | |
266 | } | |
267 | if (wantarray) { | |
268 | @d = (); | |
269 | if ($dd != 1) { | |
270 | $car = 0; | |
271 | for $x (reverse @x) { | |
68decaef | 272 | $prd = $car * 1e5 + $x; |
5303340c LW |
273 | $car = $prd - ($tmp = int($prd / $dd)) * $dd; |
274 | unshift(@d, $tmp); | |
275 | } | |
276 | } | |
277 | else { | |
278 | @d = @x; | |
279 | } | |
8990e307 | 280 | (&external($sr, @q), &external($srem, @d, $zero)); |
5303340c LW |
281 | } else { |
282 | &external($sr, @q); | |
283 | } | |
284 | } | |
285 | 1; |