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1 | package bigrat; |
2 | require "bigint.pl"; |
3 | |
4 | # Arbitrary size rational math package |
5 | # |
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6 | # by Mark Biggar |
7 | # |
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8 | # Input values to these routines consist of strings of the form |
9 | # m|^\s*[+-]?[\d\s]+(/[\d\s]+)?$|. |
10 | # Examples: |
11 | # "+0/1" canonical zero value |
12 | # "3" canonical value "+3/1" |
13 | # " -123/123 123" canonical value "-1/1001" |
14 | # "123 456/7890" canonical value "+20576/1315" |
15 | # Output values always include a sign and no leading zeros or |
16 | # white space. |
17 | # This package makes use of the bigint package. |
18 | # The string 'NaN' is used to represent the result when input arguments |
19 | # that are not numbers, as well as the result of dividing by zero and |
20 | # the sqrt of a negative number. |
21 | # Extreamly naive algorthims are used. |
22 | # |
23 | # Routines provided are: |
24 | # |
25 | # rneg(RAT) return RAT negation |
26 | # rabs(RAT) return RAT absolute value |
27 | # rcmp(RAT,RAT) return CODE compare numbers (undef,<0,=0,>0) |
28 | # radd(RAT,RAT) return RAT addition |
29 | # rsub(RAT,RAT) return RAT subtraction |
30 | # rmul(RAT,RAT) return RAT multiplication |
31 | # rdiv(RAT,RAT) return RAT division |
32 | # rmod(RAT) return (RAT,RAT) integer and fractional parts |
33 | # rnorm(RAT) return RAT normalization |
34 | # rsqrt(RAT, cycles) return RAT square root |
35 | \f |
36 | # Convert a number to the canonical string form m|^[+-]\d+/\d+|. |
37 | sub main'rnorm { #(string) return rat_num |
38 | local($_) = @_; |
39 | s/\s+//g; |
40 | if (m#^([+-]?\d+)(/(\d*[1-9]0*))?$#) { |
41 | &norm($1, $3 ? $3 : '+1'); |
42 | } else { |
43 | 'NaN'; |
44 | } |
45 | } |
46 | |
47 | # Normalize by reducing to lowest terms |
48 | sub norm { #(bint, bint) return rat_num |
49 | local($num,$dom) = @_; |
50 | if ($num eq 'NaN') { |
51 | 'NaN'; |
52 | } elsif ($dom eq 'NaN') { |
53 | 'NaN'; |
54 | } elsif ($dom =~ /^[+-]?0+$/) { |
55 | 'NaN'; |
56 | } else { |
57 | local($gcd) = &'bgcd($num,$dom); |
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58 | $gcd =~ s/^-/+/; |
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59 | if ($gcd ne '+1') { |
60 | $num = &'bdiv($num,$gcd); |
61 | $dom = &'bdiv($dom,$gcd); |
62 | } else { |
63 | $num = &'bnorm($num); |
64 | $dom = &'bnorm($dom); |
65 | } |
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66 | substr($dom,$[,1) = ''; |
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67 | "$num/$dom"; |
68 | } |
69 | } |
70 | |
71 | # negation |
72 | sub main'rneg { #(rat_num) return rat_num |
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73 | local($_) = &'rnorm(@_); |
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74 | tr/-+/+-/ if ($_ ne '+0/1'); |
75 | $_; |
76 | } |
77 | |
78 | # absolute value |
79 | sub main'rabs { #(rat_num) return $rat_num |
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80 | local($_) = &'rnorm(@_); |
81 | substr($_,$[,1) = '+' unless $_ eq 'NaN'; |
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82 | $_; |
83 | } |
84 | |
85 | # multipication |
86 | sub main'rmul { #(rat_num, rat_num) return rat_num |
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87 | local($xn,$xd) = split('/',&'rnorm($_[$[])); |
88 | local($yn,$yd) = split('/',&'rnorm($_[$[+1])); |
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89 | &norm(&'bmul($xn,$yn),&'bmul($xd,$yd)); |
90 | } |
91 | |
92 | # division |
93 | sub main'rdiv { #(rat_num, rat_num) return rat_num |
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94 | local($xn,$xd) = split('/',&'rnorm($_[$[])); |
95 | local($yn,$yd) = split('/',&'rnorm($_[$[+1])); |
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96 | &norm(&'bmul($xn,$yd),&'bmul($xd,$yn)); |
97 | } |
98 | \f |
99 | # addition |
100 | sub main'radd { #(rat_num, rat_num) return rat_num |
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101 | local($xn,$xd) = split('/',&'rnorm($_[$[])); |
102 | local($yn,$yd) = split('/',&'rnorm($_[$[+1])); |
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103 | &norm(&'badd(&'bmul($xn,$yd),&'bmul($yn,$xd)),&'bmul($xd,$yd)); |
104 | } |
105 | |
106 | # subtraction |
107 | sub main'rsub { #(rat_num, rat_num) return rat_num |
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108 | local($xn,$xd) = split('/',&'rnorm($_[$[])); |
109 | local($yn,$yd) = split('/',&'rnorm($_[$[+1])); |
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110 | &norm(&'bsub(&'bmul($xn,$yd),&'bmul($yn,$xd)),&'bmul($xd,$yd)); |
111 | } |
112 | |
113 | # comparison |
114 | sub main'rcmp { #(rat_num, rat_num) return cond_code |
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115 | local($xn,$xd) = split('/',&'rnorm($_[$[])); |
116 | local($yn,$yd) = split('/',&'rnorm($_[$[+1])); |
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117 | &bigint'cmp(&'bmul($xn,$yd),&'bmul($yn,$xd)); |
118 | } |
119 | |
120 | # int and frac parts |
121 | sub main'rmod { #(rat_num) return (rat_num,rat_num) |
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122 | local($xn,$xd) = split('/',&'rnorm(@_)); |
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123 | local($i,$f) = &'bdiv($xn,$xd); |
124 | if (wantarray) { |
125 | ("$i/1", "$f/$xd"); |
126 | } else { |
127 | "$i/1"; |
128 | } |
129 | } |
130 | |
131 | # square root by Newtons method. |
132 | # cycles specifies the number of iterations default: 5 |
133 | sub main'rsqrt { #(fnum_str[, cycles]) return fnum_str |
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134 | local($x, $scale) = (&'rnorm($_[$[]), $_[$[+1]); |
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135 | if ($x eq 'NaN') { |
136 | 'NaN'; |
137 | } elsif ($x =~ /^-/) { |
138 | 'NaN'; |
139 | } else { |
140 | local($gscale, $guess) = (0, '+1/1'); |
141 | $scale = 5 if (!$scale); |
142 | while ($gscale++ < $scale) { |
143 | $guess = &'rmul(&'radd($guess,&'rdiv($x,$guess)),"+1/2"); |
144 | } |
145 | "$guess"; # quotes necessary due to perl bug |
146 | } |
147 | } |
148 | |
149 | 1; |