Commit | Line | Data |
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 (/^[+-]$/) and whose remaining |
19 | # elements are base 100000 digits with the least significant digit first. |
20 | # The string 'NaN' is used to represent the result when input arguments |
21 | # are not numbers, as well as the result of dividing by zero |
22 | # |
23 | # routines provided are: |
24 | # |
25 | # bneg(BINT) return BINT negation |
26 | # babs(BINT) return BINT absolute value |
27 | # bcmp(BINT,BINT) return CODE compare numbers (undef,<0,=0,>0) |
28 | # badd(BINT,BINT) return BINT addition |
29 | # bsub(BINT,BINT) return BINT subtraction |
30 | # bmul(BINT,BINT) return BINT multiplication |
31 | # bdiv(BINT,BINT) return (BINT,BINT) division (quo,rem) just quo if scalar |
32 | # bmod(BINT,BINT) return BINT modulus |
33 | # bgcd(BINT,BINT) return BINT greatest common divisor |
34 | # bnorm(BINT) return BINT normalization |
35 | # |
36 | \f |
37 | # normalize string form of number. Strip leading zeros. Strip any |
38 | # white space and add a sign, if missing. |
39 | # Strings that are not numbers result the value 'NaN'. |
40 | sub main'bnorm { #(num_str) return num_str |
41 | local($_) = @_; |
42 | s/\s+//g; # strip white space |
43 | if (s/^([+-]?)0*(\d+)$/$1$2/) { # test if number |
44 | substr($_,0,0) = '+' unless $1; # Add missing sign |
45 | s/^-0/+0/; |
46 | $_; |
47 | } else { |
48 | 'NaN'; |
49 | } |
50 | } |
51 | |
52 | # Convert a number from string format to internal base 100000 format. |
53 | # Assumes normalized value as input. |
54 | sub internal { #(num_str) return int_num_array |
55 | local($d) = @_; |
56 | ($is,$il) = (substr($d,0,1),length($d)-2); |
57 | substr($d,0,1) = ''; |
58 | ($is, reverse(unpack("a" . ($il%5+1) . ("a5" x ($il/5)), $d))); |
59 | } |
60 | |
61 | # Convert a number from internal base 100000 format to string format. |
62 | # This routine scribbles all over input array. |
63 | sub external { #(int_num_array) return num_str |
64 | $es = shift; |
65 | grep($_ > 9999 || ($_ = substr('0000'.$_,-5)), @_); # zero pad |
66 | &'bnorm(join('', $es, reverse(@_))); # reverse concat and normalize |
67 | } |
68 | |
69 | # Negate input value. |
70 | sub main'bneg { #(num_str) return num_str |
71 | local($_) = &'bnorm(@_); |
72 | vec($_,0,8) ^= ord('+') ^ ord('-') unless $_ eq '+0'; |
73 | s/^H/N/; |
74 | $_; |
75 | } |
76 | |
77 | # Returns the absolute value of the input. |
78 | sub main'babs { #(num_str) return num_str |
79 | &abs(&'bnorm(@_)); |
80 | } |
81 | |
82 | sub abs { # post-normalized abs for internal use |
83 | local($_) = @_; |
84 | s/^-/+/; |
85 | $_; |
86 | } |
87 | \f |
88 | # Compares 2 values. Returns one of undef, <0, =0, >0. (suitable for sort) |
89 | sub main'bcmp { #(num_str, num_str) return cond_code |
90 | local($x,$y) = (&'bnorm($_[0]),&'bnorm($_[1])); |
91 | if ($x eq 'NaN') { |
92 | undef; |
93 | } elsif ($y eq 'NaN') { |
94 | undef; |
95 | } else { |
96 | &cmp($x,$y); |
97 | } |
98 | } |
99 | |
100 | sub cmp { # post-normalized compare for internal use |
101 | local($cx, $cy) = @_; |
102 | $cx cmp $cy |
103 | && |
104 | ( |
105 | ord($cy) <=> ord($cx) |
106 | || |
107 | ($cx cmp ',') * (length($cy) <=> length($cx) || $cy cmp $cx) |
108 | ); |
109 | } |
110 | |
111 | sub main'badd { #(num_str, num_str) return num_str |
112 | local(*x, *y); ($x, $y) = (&'bnorm($_[0]),&'bnorm($_[1])); |
113 | if ($x eq 'NaN') { |
114 | 'NaN'; |
115 | } elsif ($y eq 'NaN') { |
116 | 'NaN'; |
117 | } else { |
118 | @x = &internal($x); # convert to internal form |
119 | @y = &internal($y); |
120 | local($sx, $sy) = (shift @x, shift @y); # get signs |
121 | if ($sx eq $sy) { |
122 | &external($sx, &add(*x, *y)); # if same sign add |
123 | } else { |
124 | ($x, $y) = (&abs($x),&abs($y)); # make abs |
125 | if (&cmp($y,$x) > 0) { |
126 | &external($sy, &sub(*y, *x)); |
127 | } else { |
128 | &external($sx, &sub(*x, *y)); |
129 | } |
130 | } |
131 | } |
132 | } |
133 | |
134 | sub main'bsub { #(num_str, num_str) return num_str |
135 | &'badd($_[0],&'bneg($_[1])); |
136 | } |
137 | |
138 | # GCD -- Euclids algorithm Knuth Vol 2 pg 296 |
139 | sub main'bgcd { #(num_str, num_str) return num_str |
140 | local($x,$y) = (&'bnorm($_[0]),&'bnorm($_[1])); |
141 | if ($x eq 'NaN') { |
142 | 'NaN'; |
143 | } |
144 | elsif ($y eq 'NaN') { |
145 | 'NaN'; |
146 | } |
147 | else { |
148 | ($x, $y) = ($y,&'bmod($x,$y)) while $y ne '+0'; |
149 | $x; |
150 | } |
151 | } |
152 | \f |
153 | # routine to add two base 100000 numbers |
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; |
161 | $x -= 100000 if $car = (($x += shift @y + $car) >= 100000); |
162 | } |
163 | for $y (@y) { |
164 | last unless $car; |
165 | $y -= 100000 if $car = (($y += $car) >= 100000); |
166 | } |
167 | (@x, @y, $car); |
168 | } |
169 | |
170 | # subtract base 100000 numbers -- stolen from Knuth Vol 2 pg 232, $x > $y |
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; |
176 | $sx += 100000 if $bar = (($sx -= shift @sy + $bar) < 0); |
177 | } |
178 | @sx; |
179 | } |
180 | |
181 | # multiply two numbers -- stolen from Knuth Vol 2 pg 233 |
182 | sub main'bmul { #(num_str, num_str) return num_str |
183 | local(*x, *y); ($x, $y) = (&'bnorm($_[0]), &'bnorm($_[1])); |
184 | if ($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 $x (@x) { |
194 | ($car, $cty) = (0, 0); |
195 | for $y (@y) { |
196 | $prod = $x * $y + $prod[$cty] + $car; |
197 | $prod[$cty++] = |
198 | $prod - ($car = int($prod * (1/100000))) * 100000; |
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 |
209 | (&'bdiv(@_))[1]; |
210 | } |
211 | \f |
212 | sub main'bdiv { #(dividend: num_str, divisor: num_str) return num_str |
213 | local (*x, *y); ($x, $y) = (&'bnorm($_[0]), &'bnorm($_[1])); |
214 | return wantarray ? ('NaN','NaN') : 'NaN' |
215 | if ($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); |
218 | $srem = $y[0]; |
219 | $sr = (shift @x ne shift @y) ? '-' : '+'; |
220 | $car = $bar = $prd = 0; |
221 | if (($dd = int(100000/($y[$#y]+1))) != 1) { |
222 | for $x (@x) { |
223 | $x = $x * $dd + $car; |
224 | $x -= ($car = int($x * (1/100000))) * 100000; |
225 | } |
226 | push(@x, $car); $car = 0; |
227 | for $y (@y) { |
228 | $y = $y * $dd + $car; |
229 | $y -= ($car = int($y * (1/100000))) * 100000; |
230 | } |
231 | } |
232 | else { |
233 | push(@x, 0); |
234 | } |
235 | @q = (); ($v2,$v1) = @y[$#y-1,$#y]; |
236 | while ($#x > $#y) { |
237 | ($u2,$u1,$u0) = @x[($#x-2)..$#x]; |
238 | $q = (($u0 == $v1) ? 99999 : int(($u0*100000+$u1)/$v1)); |
239 | --$q while ($v2*$q > ($u0*100000+$u1-$q*$v1)*100000+$u2); |
240 | if ($q) { |
241 | ($car, $bar) = (0,0); |
242 | for ($y = 0, $x = $#x-$#y-1; $y <= $#y; ++$y,++$x) { |
243 | $prd = $q * $y[$y] + $car; |
244 | $prd -= ($car = int($prd * (1/100000))) * 100000; |
245 | $x[$x] += 100000 if ($bar = (($x[$x] -= $prd + $bar) < 0)); |
246 | } |
247 | if ($x[$#x] < $car + $bar) { |
248 | $car = 0; --$q; |
249 | for ($y = 0, $x = $#x-$#y-1; $y <= $#y; ++$y,++$x) { |
250 | $x[$x] -= 100000 |
251 | if ($car = (($x[$x] += $y[$y] + $car) > 100000)); |
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) { |
262 | $prd = $car * 100000 + $x; |
263 | $car = $prd - ($tmp = int($prd / $dd)) * $dd; |
264 | unshift(@d, $tmp); |
265 | } |
266 | } |
267 | else { |
268 | @d = @x; |
269 | } |
270 | (&external($sr, @q), &external($srem, @d, 0)); |
271 | } else { |
272 | &external($sr, @q); |
273 | } |
274 | } |
275 | 1; |