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5303340c |
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 | # |
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36 | |
37 | $zero = 0; |
38 | |
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39 | \f |
40 | # normalize string form of number. Strip leading zeros. Strip any |
41 | # white space and add a sign, if missing. |
42 | # Strings that are not numbers result the value 'NaN'. |
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43 | |
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44 | sub main'bnorm { #(num_str) return num_str |
45 | local($_) = @_; |
46 | s/\s+//g; # strip white space |
47 | if (s/^([+-]?)0*(\d+)$/$1$2/) { # test if number |
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48 | substr($_,$[,0) = '+' unless $1; # Add missing sign |
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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) = @_; |
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60 | ($is,$il) = (substr($d,$[,1),length($d)-2); |
61 | substr($d,$[,1) = ''; |
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62 | ($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 | $_; |
90 | } |
91 | \f |
92 | # Compares 2 values. Returns one of undef, <0, =0, >0. (suitable for sort) |
93 | sub main'bcmp { #(num_str, num_str) return cond_code |
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94 | local($x,$y) = (&'bnorm($_[$[]),&'bnorm($_[$[+1])); |
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95 | if ($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) = @_; |
106 | $cx cmp $cy |
107 | && |
108 | ( |
109 | ord($cy) <=> ord($cx) |
110 | || |
111 | ($cx cmp ',') * (length($cy) <=> length($cx) || $cy cmp $cx) |
112 | ); |
113 | } |
114 | |
115 | sub main'badd { #(num_str, num_str) return num_str |
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116 | local(*x, *y); ($x, $y) = (&'bnorm($_[$[]),&'bnorm($_[$[+1])); |
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117 | if ($x eq 'NaN') { |
118 | 'NaN'; |
119 | } elsif ($y eq 'NaN') { |
120 | 'NaN'; |
121 | } else { |
122 | @x = &internal($x); # convert to internal form |
123 | @y = &internal($y); |
124 | local($sx, $sy) = (shift @x, shift @y); # get signs |
125 | if ($sx eq $sy) { |
126 | &external($sx, &add(*x, *y)); # if same sign add |
127 | } else { |
128 | ($x, $y) = (&abs($x),&abs($y)); # make abs |
129 | if (&cmp($y,$x) > 0) { |
130 | &external($sy, &sub(*y, *x)); |
131 | } else { |
132 | &external($sx, &sub(*x, *y)); |
133 | } |
134 | } |
135 | } |
136 | } |
137 | |
138 | sub main'bsub { #(num_str, num_str) return num_str |
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139 | &'badd($_[$[],&'bneg($_[$[+1])); |
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140 | } |
141 | |
142 | # GCD -- Euclids algorithm Knuth Vol 2 pg 296 |
143 | sub main'bgcd { #(num_str, num_str) return num_str |
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144 | local($x,$y) = (&'bnorm($_[$[]),&'bnorm($_[$[+1])); |
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145 | if ($x eq 'NaN' || $y eq 'NaN') { |
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146 | 'NaN'; |
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147 | } else { |
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148 | ($x, $y) = ($y,&'bmod($x,$y)) while $y ne '+0'; |
149 | $x; |
150 | } |
151 | } |
152 | \f |
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153 | # routine to add two base 1e5 numbers |
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154 | # stolen from Knuth Vol 2 Algorithm A pg 231 |
155 | # there are separate routines to add and sub as per Kunth pg 233 |
156 | sub add { #(int_num_array, int_num_array) return int_num_array |
157 | local(*x, *y) = @_; |
158 | $car = 0; |
159 | for $x (@x) { |
160 | last unless @y || $car; |
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161 | $x -= 1e5 if $car = (($x += shift(@y) + $car) >= 1e5); |
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162 | } |
163 | for $y (@y) { |
164 | last unless $car; |
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165 | $y -= 1e5 if $car = (($y += $car) >= 1e5); |
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166 | } |
167 | (@x, @y, $car); |
168 | } |
169 | |
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170 | # subtract base 1e5 numbers -- stolen from Knuth Vol 2 pg 232, $x > $y |
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171 | sub sub { #(int_num_array, int_num_array) return int_num_array |
172 | local(*sx, *sy) = @_; |
173 | $bar = 0; |
174 | for $sx (@sx) { |
175 | last unless @y || $bar; |
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176 | $sx += 1e5 if $bar = (($sx -= shift(@sy) + $bar) < 0); |
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177 | } |
178 | @sx; |
179 | } |
180 | |
181 | # multiply two numbers -- stolen from Knuth Vol 2 pg 233 |
182 | sub main'bmul { #(num_str, num_str) return num_str |
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183 | local(*x, *y); ($x, $y) = (&'bnorm($_[$[]), &'bnorm($_[$[+1])); |
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184 | if ($x eq 'NaN') { |
185 | 'NaN'; |
186 | } elsif ($y eq 'NaN') { |
187 | 'NaN'; |
188 | } else { |
189 | @x = &internal($x); |
190 | @y = &internal($y); |
191 | local($signr) = (shift @x ne shift @y) ? '-' : '+'; |
192 | @prod = (); |
193 | for $x (@x) { |
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194 | ($car, $cty) = (0, $[); |
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195 | for $y (@y) { |
196 | $prod = $x * $y + $prod[$cty] + $car; |
197 | $prod[$cty++] = |
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198 | $prod - ($car = int($prod * 1e-5)) * 1e5; |
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199 | } |
200 | $prod[$cty] += $car if $car; |
201 | $x = shift @prod; |
202 | } |
203 | &external($signr, @x, @prod); |
204 | } |
205 | } |
206 | |
207 | # modulus |
208 | sub main'bmod { #(num_str, num_str) return num_str |
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209 | (&'bdiv(@_))[$[+1]; |
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210 | } |
211 | \f |
212 | sub main'bdiv { #(dividend: num_str, divisor: num_str) return num_str |
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213 | local (*x, *y); ($x, $y) = (&'bnorm($_[$[]), &'bnorm($_[$[+1])); |
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214 | return wantarray ? ('NaN','NaN') : 'NaN' |
215 | if ($x eq 'NaN' || $y eq 'NaN' || $y eq '+0'); |
216 | return wantarray ? ('+0',$x) : '+0' if (&cmp(&abs($x),&abs($y)) < 0); |
217 | @x = &internal($x); @y = &internal($y); |
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218 | $srem = $y[$[]; |
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219 | $sr = (shift @x ne shift @y) ? '-' : '+'; |
220 | $car = $bar = $prd = 0; |
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221 | if (($dd = int(1e5/($y[$#y]+1))) != 1) { |
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222 | for $x (@x) { |
223 | $x = $x * $dd + $car; |
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224 | $x -= ($car = int($x * 1e-5)) * 1e5; |
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225 | } |
226 | push(@x, $car); $car = 0; |
227 | for $y (@y) { |
228 | $y = $y * $dd + $car; |
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229 | $y -= ($car = int($y * 1e-5)) * 1e5; |
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230 | } |
231 | } |
232 | else { |
233 | push(@x, 0); |
234 | } |
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235 | @q = (); ($v2,$v1) = @y[-2,-1]; |
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236 | while ($#x > $#y) { |
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237 | ($u2,$u1,$u0) = @x[-3..-1]; |
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238 | $q = (($u0 == $v1) ? 99999 : int(($u0*1e5+$u1)/$v1)); |
239 | --$q while ($v2*$q > ($u0*1e5+$u1-$q*$v1)*1e5+$u2); |
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240 | if ($q) { |
241 | ($car, $bar) = (0,0); |
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242 | for ($y = $[, $x = $#x-$#y+$[-1; $y <= $#y; ++$y,++$x) { |
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243 | $prd = $q * $y[$y] + $car; |
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244 | $prd -= ($car = int($prd * 1e-5)) * 1e5; |
245 | $x[$x] += 1e5 if ($bar = (($x[$x] -= $prd + $bar) < 0)); |
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246 | } |
247 | if ($x[$#x] < $car + $bar) { |
248 | $car = 0; --$q; |
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249 | for ($y = $[, $x = $#x-$#y+$[-1; $y <= $#y; ++$y,++$x) { |
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250 | $x[$x] -= 1e5 |
251 | if ($car = (($x[$x] += $y[$y] + $car) > 1e5)); |
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252 | } |
253 | } |
254 | } |
255 | pop(@x); unshift(@q, $q); |
256 | } |
257 | if (wantarray) { |
258 | @d = (); |
259 | if ($dd != 1) { |
260 | $car = 0; |
261 | for $x (reverse @x) { |
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262 | $prd = $car * 1e5 + $x; |
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263 | $car = $prd - ($tmp = int($prd / $dd)) * $dd; |
264 | unshift(@d, $tmp); |
265 | } |
266 | } |
267 | else { |
268 | @d = @x; |
269 | } |
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270 | (&external($sr, @q), &external($srem, @d, $zero)); |
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271 | } else { |
272 | &external($sr, @q); |
273 | } |
274 | } |
275 | 1; |