1 (* Title: HOL/Real/Float.ML |
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2 ID: $Id$ |
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3 Author: Steven Obua |
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4 *) |
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5 |
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6 structure ExactFloatingPoint : |
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7 sig |
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8 exception Destruct_floatstr of string |
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9 val destruct_floatstr : (char -> bool) -> (char -> bool) -> string -> bool * string * string * bool * string |
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10 |
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11 exception Floating_point of string |
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12 |
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13 type floatrep = IntInf.int * IntInf.int |
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14 val approx_dec_by_bin : IntInf.int -> floatrep -> floatrep * floatrep |
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15 val approx_decstr_by_bin : int -> string -> floatrep * floatrep |
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16 end |
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17 = |
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18 struct |
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19 |
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20 exception Destruct_floatstr of string; |
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21 |
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22 fun destruct_floatstr isDigit isExp number = |
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23 let |
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24 val numlist = filter (not o Char.isSpace) (String.explode number) |
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25 |
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26 fun countsigns ((#"+")::cs) = countsigns cs |
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27 | countsigns ((#"-")::cs) = |
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28 let |
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29 val (positive, rest) = countsigns cs |
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30 in |
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31 (not positive, rest) |
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32 end |
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33 | countsigns cs = (true, cs) |
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34 |
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35 fun readdigits [] = ([], []) |
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36 | readdigits (q as c::cs) = |
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37 if (isDigit c) then |
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38 let |
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39 val (digits, rest) = readdigits cs |
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40 in |
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41 (c::digits, rest) |
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42 end |
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43 else |
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44 ([], q) |
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45 |
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46 fun readfromexp_helper cs = |
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47 let |
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48 val (positive, rest) = countsigns cs |
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49 val (digits, rest') = readdigits rest |
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50 in |
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51 case rest' of |
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52 [] => (positive, digits) |
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53 | _ => raise (Destruct_floatstr number) |
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54 end |
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55 |
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56 fun readfromexp [] = (true, []) |
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57 | readfromexp (c::cs) = |
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58 if isExp c then |
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59 readfromexp_helper cs |
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60 else |
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61 raise (Destruct_floatstr number) |
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62 |
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63 fun readfromdot [] = ([], readfromexp []) |
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64 | readfromdot ((#".")::cs) = |
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65 let |
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66 val (digits, rest) = readdigits cs |
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67 val exp = readfromexp rest |
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68 in |
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69 (digits, exp) |
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70 end |
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71 | readfromdot cs = readfromdot ((#".")::cs) |
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72 |
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73 val (positive, numlist) = countsigns numlist |
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74 val (digits1, numlist) = readdigits numlist |
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75 val (digits2, exp) = readfromdot numlist |
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76 in |
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77 (positive, String.implode digits1, String.implode digits2, fst exp, String.implode (snd exp)) |
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78 end |
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79 |
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80 type floatrep = IntInf.int * IntInf.int |
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81 |
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82 exception Floating_point of string; |
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83 |
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84 val ln2_10 = (Math.ln 10.0)/(Math.ln 2.0) |
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85 |
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86 fun intmul a b = IntInf.* (a,b) |
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87 fun intsub a b = IntInf.- (a,b) |
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88 fun intadd a b = IntInf.+ (a,b) |
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89 fun intpow a b = IntInf.pow (a, IntInf.toInt b); |
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90 fun intle a b = IntInf.<= (a, b); |
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91 fun intless a b = IntInf.< (a, b); |
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92 fun intneg a = IntInf.~ a; |
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93 val zero = IntInf.fromInt 0; |
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94 val one = IntInf.fromInt 1; |
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95 val two = IntInf.fromInt 2; |
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96 val ten = IntInf.fromInt 10; |
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97 val five = IntInf.fromInt 5; |
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98 |
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99 fun find_most_significant q r = |
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100 let |
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101 fun int2real i = |
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102 case Real.fromString (IntInf.toString i) of |
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103 SOME r => r |
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104 | NONE => raise (Floating_point "int2real") |
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105 fun subtract (q, r) (q', r') = |
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106 if intle r r' then |
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107 (intsub q (intmul q' (intpow ten (intsub r' r))), r) |
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108 else |
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109 (intsub (intmul q (intpow ten (intsub r r'))) q', r') |
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110 fun bin2dec d = |
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111 if intle zero d then |
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112 (intpow two d, zero) |
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113 else |
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114 (intpow five (intneg d), d) |
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115 |
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116 val L = IntInf.fromInt (Real.floor (int2real (IntInf.fromInt (IntInf.log2 q)) + (int2real r) * ln2_10)) |
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117 val L1 = intadd L one |
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118 |
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119 val (q1, r1) = subtract (q, r) (bin2dec L1) |
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120 in |
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121 if intle zero q1 then |
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122 let |
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123 val (q2, r2) = subtract (q, r) (bin2dec (intadd L1 one)) |
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124 in |
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125 if intle zero q2 then |
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126 raise (Floating_point "find_most_significant") |
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127 else |
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128 (L1, (q1, r1)) |
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129 end |
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130 else |
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131 let |
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132 val (q0, r0) = subtract (q, r) (bin2dec L) |
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133 in |
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134 if intle zero q0 then |
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135 (L, (q0, r0)) |
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136 else |
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137 raise (Floating_point "find_most_significant") |
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138 end |
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139 end |
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140 |
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141 fun approx_dec_by_bin n (q,r) = |
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142 let |
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143 fun addseq acc d' [] = acc |
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144 | addseq acc d' (d::ds) = addseq (intadd acc (intpow two (intsub d d'))) d' ds |
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145 |
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146 fun seq2bin [] = (zero, zero) |
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147 | seq2bin (d::ds) = (intadd (addseq zero d ds) one, d) |
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148 |
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149 fun approx d_seq d0 precision (q,r) = |
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150 if q = zero then |
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151 let val x = seq2bin d_seq in |
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152 (x, x) |
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153 end |
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154 else |
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155 let |
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156 val (d, (q', r')) = find_most_significant q r |
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157 in |
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158 if intless precision (intsub d0 d) then |
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159 let |
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160 val d' = intsub d0 precision |
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161 val x1 = seq2bin (d_seq) |
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162 val x2 = (intadd (intmul (fst x1) (intpow two (intsub (snd x1) d'))) one, d') (* = seq2bin (d'::d_seq) *) |
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163 in |
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164 (x1, x2) |
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165 end |
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166 else |
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167 approx (d::d_seq) d0 precision (q', r') |
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168 end |
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169 |
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170 fun approx_start precision (q, r) = |
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171 if q = zero then |
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172 ((zero, zero), (zero, zero)) |
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173 else |
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174 let |
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175 val (d, (q', r')) = find_most_significant q r |
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176 in |
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177 if intle precision zero then |
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178 let |
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179 val x1 = seq2bin [d] |
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180 in |
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181 if q' = zero then |
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182 (x1, x1) |
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183 else |
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184 (x1, seq2bin [intadd d one]) |
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185 end |
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186 else |
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187 approx [d] d precision (q', r') |
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188 end |
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189 in |
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190 if intle zero q then |
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191 approx_start n (q,r) |
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192 else |
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193 let |
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194 val ((a1,b1), (a2, b2)) = approx_start n (intneg q, r) |
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195 in |
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196 ((intneg a2, b2), (intneg a1, b1)) |
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197 end |
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198 end |
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199 |
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200 fun approx_decstr_by_bin n decstr = |
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201 let |
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202 fun str2int s = case IntInf.fromString s of SOME x => x | NONE => zero |
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203 fun signint p x = if p then x else intneg x |
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204 |
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205 val (p, d1, d2, ep, e) = destruct_floatstr Char.isDigit (fn e => e = #"e" orelse e = #"E") decstr |
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206 val s = IntInf.fromInt (size d2) |
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207 |
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208 val q = signint p (intadd (intmul (str2int d1) (intpow ten s)) (str2int d2)) |
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209 val r = intsub (signint ep (str2int e)) s |
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210 in |
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211 approx_dec_by_bin (IntInf.fromInt n) (q,r) |
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212 end |
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213 |
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214 end; |
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215 |
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216 structure FloatArith = |
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217 struct |
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218 |
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219 type float = IntInf.int * IntInf.int |
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220 |
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221 val izero = IntInf.fromInt 0 |
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222 val ione = IntInf.fromInt 1 |
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223 val imone = IntInf.fromInt ~1 |
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224 val itwo = IntInf.fromInt 2 |
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225 fun imul a b = IntInf.* (a,b) |
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226 fun isub a b = IntInf.- (a,b) |
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227 fun iadd a b = IntInf.+ (a,b) |
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228 |
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229 val floatzero = (izero, izero) |
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230 |
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231 fun positive_part (a,b) = |
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232 (if IntInf.< (a,izero) then izero else a, b) |
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233 |
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234 fun negative_part (a,b) = |
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235 (if IntInf.< (a,izero) then a else izero, b) |
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236 |
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237 fun is_negative (a,b) = |
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238 if IntInf.< (a, izero) then true else false |
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239 |
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240 fun is_positive (a,b) = |
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241 if IntInf.< (izero, a) then true else false |
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242 |
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243 fun is_zero (a,b) = |
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244 if a = izero then true else false |
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245 |
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246 fun ipow2 a = IntInf.pow ((IntInf.fromInt 2), IntInf.toInt a) |
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247 |
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248 fun add (a1, b1) (a2, b2) = |
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249 if IntInf.< (b1, b2) then |
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250 (iadd a1 (imul a2 (ipow2 (isub b2 b1))), b1) |
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251 else |
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252 (iadd (imul a1 (ipow2 (isub b1 b2))) a2, b2) |
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253 |
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254 fun sub (a1, b1) (a2, b2) = |
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255 if IntInf.< (b1, b2) then |
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256 (isub a1 (imul a2 (ipow2 (isub b2 b1))), b1) |
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257 else |
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258 (isub (imul a1 (ipow2 (isub b1 b2))) a2, b2) |
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259 |
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260 fun neg (a, b) = (IntInf.~ a, b) |
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261 |
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262 fun is_equal a b = is_zero (sub a b) |
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263 |
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264 fun is_less a b = is_negative (sub a b) |
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265 |
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266 fun max a b = if is_less a b then b else a |
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267 |
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268 fun min a b = if is_less a b then a else b |
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269 |
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270 fun abs a = if is_negative a then neg a else a |
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271 |
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272 fun mul (a1, b1) (a2, b2) = (imul a1 a2, iadd b1 b2) |
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273 |
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274 end; |
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275 |
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276 |
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277 structure Float: |
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278 sig |
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279 type float = FloatArith.float |
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280 type floatfunc = float * float -> float * float |
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281 |
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282 val mk_intinf : typ -> IntInf.int -> term |
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283 val mk_float : float -> term |
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284 |
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285 exception Dest_intinf; |
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286 val dest_intinf : term -> IntInf.int |
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287 val dest_nat : term -> IntInf.int |
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288 |
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289 exception Dest_float; |
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290 val dest_float : term -> float |
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291 |
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292 val float_const : term |
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293 |
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294 val float_add_const : term |
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295 val float_diff_const : term |
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296 val float_uminus_const : term |
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297 val float_pprt_const : term |
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298 val float_nprt_const : term |
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299 val float_abs_const : term |
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300 val float_mult_const : term |
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301 val float_le_const : term |
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302 |
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303 val nat_le_const : term |
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304 val nat_less_const : term |
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305 val nat_eq_const : term |
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306 |
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307 val approx_float : int -> floatfunc -> string -> term * term |
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308 |
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309 end |
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310 = |
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311 struct |
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312 |
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313 structure Inttab = TableFun(type key = int val ord = (rev_order o int_ord)); |
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314 |
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315 type float = IntInf.int*IntInf.int |
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316 type floatfunc = float*float -> float*float |
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317 |
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318 val float_const = Const ("Float.float", HOLogic.mk_prodT (HOLogic.intT, HOLogic.intT) --> HOLogic.realT) |
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319 |
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320 val float_add_const = Const ("HOL.plus", HOLogic.realT --> HOLogic.realT --> HOLogic.realT) |
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321 val float_diff_const = Const ("HOL.minus", HOLogic.realT --> HOLogic.realT --> HOLogic.realT) |
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322 val float_mult_const = Const ("HOL.times", HOLogic.realT --> HOLogic.realT --> HOLogic.realT) |
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323 val float_uminus_const = Const ("HOL.uminus", HOLogic.realT --> HOLogic.realT) |
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324 val float_abs_const = Const ("HOL.abs", HOLogic.realT --> HOLogic.realT) |
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325 val float_le_const = Const ("Orderings.less_eq", HOLogic.realT --> HOLogic.realT --> HOLogic.boolT) |
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326 val float_pprt_const = Const ("OrderedGroup.pprt", HOLogic.realT --> HOLogic.realT) |
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327 val float_nprt_const = Const ("OrderedGroup.nprt", HOLogic.realT --> HOLogic.realT) |
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328 |
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329 val nat_le_const = Const ("Orderings.less_eq", HOLogic.natT --> HOLogic.natT --> HOLogic.boolT) |
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330 val nat_less_const = Const ("Orderings.less", HOLogic.natT --> HOLogic.natT --> HOLogic.boolT) |
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331 val nat_eq_const = Const ("op =", HOLogic.natT --> HOLogic.natT --> HOLogic.boolT) |
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332 |
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333 val zero = FloatArith.izero |
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334 val minus_one = FloatArith.imone |
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335 val two = FloatArith.itwo |
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336 |
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337 exception Dest_intinf; |
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338 exception Dest_float; |
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339 |
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340 fun mk_intinf ty n = HOLogic.number_of_const ty $ HOLogic.mk_numeral n; |
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341 |
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342 val dest_intinf = snd o HOLogic.dest_number |
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343 |
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344 fun mk_float (a,b) = |
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345 float_const $ (HOLogic.mk_prod ((mk_intinf HOLogic.intT a), (mk_intinf HOLogic.intT b))) |
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346 |
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347 fun dest_float f = |
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348 case f of |
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349 (Const ("Float.float", _) $ (Const ("Pair", _) $ a $ b)) => (dest_intinf a, dest_intinf b) |
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350 | Const ("Numeral.number_of",_) $ a => (dest_intinf f, 0) |
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351 | Const ("Numeral0", _) => (FloatArith.izero, FloatArith.izero) |
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352 | Const ("Numeral1", _) => (FloatArith.ione, FloatArith.izero) |
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353 | _ => raise Dest_float |
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354 |
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355 fun dest_nat n = |
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356 let |
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357 val v = dest_intinf n |
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358 in |
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359 if IntInf.< (v, FloatArith.izero) then |
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360 FloatArith.izero |
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361 else |
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362 v |
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363 end |
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364 |
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365 fun approx_float prec f value = |
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366 let |
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367 val interval = ExactFloatingPoint.approx_decstr_by_bin prec value |
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368 val (flower, fupper) = f interval |
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369 in |
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370 (mk_float flower, mk_float fupper) |
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371 end |
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372 |
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373 end; |
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