1 | /* jn.c |
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2 | * |
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3 | * Bessel function of integer order |
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4 | * |
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5 | * |
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6 | * |
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7 | * SYNOPSIS: |
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8 | * |
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9 | * int n; |
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10 | * double x, y, jn(); |
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11 | * |
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12 | * y = jn( n, x ); |
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13 | * |
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14 | * |
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15 | * |
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16 | * DESCRIPTION: |
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17 | * |
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18 | * Returns Bessel function of order n, where n is a |
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19 | * (possibly negative) integer. |
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20 | * |
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21 | * The ratio of jn(x) to j0(x) is computed by backward |
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22 | * recurrence. First the ratio jn/jn-1 is found by a |
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23 | * continued fraction expansion. Then the recurrence |
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24 | * relating successive orders is applied until j0 or j1 is |
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25 | * reached. |
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26 | * |
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27 | * If n = 0 or 1 the routine for j0 or j1 is called |
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28 | * directly. |
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29 | * |
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30 | * |
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31 | * |
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32 | * ACCURACY: |
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33 | * |
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34 | * Absolute error: |
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35 | * arithmetic range # trials peak rms |
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36 | * DEC 0, 30 5500 6.9e-17 9.3e-18 |
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37 | * IEEE 0, 30 5000 4.4e-16 7.9e-17 |
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38 | * |
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39 | * |
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40 | * Not suitable for large n or x. Use jv() instead. |
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41 | * |
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42 | */ |
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43 | |
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44 | /* jn.c |
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45 | Cephes Math Library Release 2.8: June, 2000 |
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46 | Copyright 1984, 1987, 2000 by Stephen L. Moshier |
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47 | */ |
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48 | |
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49 | #if FLOAT_SIZE > 4 |
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50 | |
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51 | double cephes_jn( int n, double x ); |
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52 | double cephes_jn( int n, double x ) { |
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53 | |
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54 | // PAK: seems to be machine epsilon/2 |
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55 | const double MACHEP = 1.11022302462515654042E-16; |
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56 | double pkm2, pkm1, pk, xk, r, ans; |
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57 | int k, sign; |
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58 | |
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59 | if( n < 0 ) { |
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60 | n = -n; |
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61 | if( (n & 1) == 0 ) /* -1**n */ |
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62 | sign = 1; |
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63 | else |
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64 | sign = -1; |
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65 | } else { |
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66 | sign = 1; |
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67 | } |
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68 | |
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69 | if( x < 0.0 ) { |
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70 | if( n & 1 ) |
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71 | sign = -sign; |
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72 | x = -x; |
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73 | } |
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74 | |
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75 | if( n == 0 ) |
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76 | return( sign * cephes_j0(x) ); |
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77 | if( n == 1 ) |
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78 | return( sign * cephes_j1(x) ); |
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79 | if( n == 2 ) |
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80 | return( sign * (2.0 * cephes_j1(x) / x - cephes_j0(x)) ); |
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81 | |
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82 | if( x < MACHEP ) |
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83 | return( 0.0 ); |
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84 | |
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85 | k = 53; |
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86 | pk = 2 * (n + k); |
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87 | ans = pk; |
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88 | xk = x * x; |
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89 | |
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90 | do { |
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91 | pk -= 2.0; |
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92 | ans = pk - (xk/ans); |
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93 | } while( --k > 0 ); |
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94 | |
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95 | /* backward recurrence */ |
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96 | |
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97 | pk = 1.0; |
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98 | |
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99 | ans = x/ans; |
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100 | pkm1 = 1.0/ans; |
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101 | |
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102 | k = n-1; |
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103 | r = 2 * k; |
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104 | |
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105 | do { |
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106 | pkm2 = (pkm1 * r - pk * x) / x; |
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107 | pk = pkm1; |
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108 | pkm1 = pkm2; |
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109 | r -= 2.0; |
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110 | } while( --k > 0 ); |
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111 | |
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112 | if( fabs(pk) > fabs(pkm1) ) |
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113 | ans = cephes_j1(x)/pk; |
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114 | else |
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115 | ans = cephes_j0(x)/pkm1; |
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116 | |
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117 | return( sign * ans ); |
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118 | } |
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119 | |
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120 | #else |
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121 | |
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122 | float cephes_jnf(int n, float x); |
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123 | float cephes_jnf(int n, float x) |
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124 | { |
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125 | // PAK: seems to be machine epsilon/2 |
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126 | const double MACHEP = 5.9604645e-08; |
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127 | float pkm2, pkm1, pk, xk, r, ans; |
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128 | int k, sign; |
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129 | |
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130 | if( n < 0 ) { |
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131 | n = -n; |
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132 | if( (n & 1) == 0 ) /* -1**n */ |
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133 | sign = 1; |
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134 | else |
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135 | sign = -1; |
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136 | } else { |
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137 | sign = 1; |
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138 | } |
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139 | |
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140 | if( x < 0.0 ) { |
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141 | if( n & 1 ) |
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142 | sign = -sign; |
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143 | x = -x; |
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144 | } |
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145 | |
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146 | if( n == 0 ) |
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147 | return( sign * cephes_j0f(x) ); |
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148 | if( n == 1 ) |
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149 | return( sign * cephes_j1f(x) ); |
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150 | if( n == 2 ) |
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151 | return( sign * (2.0 * cephes_j1f(x) / x - cephes_j0f(x)) ); |
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152 | |
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153 | if( x < MACHEP ) |
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154 | return( 0.0 ); |
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155 | |
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156 | k = 24; |
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157 | pk = 2 * (n + k); |
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158 | ans = pk; |
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159 | xk = x * x; |
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160 | |
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161 | do { |
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162 | pk -= 2.0; |
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163 | ans = pk - (xk/ans); |
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164 | } while( --k > 0 ); |
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165 | |
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166 | /* backward recurrence */ |
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167 | |
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168 | pk = 1.0; |
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169 | |
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170 | const float xinv = 1.0/x; |
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171 | pkm1 = ans * xinv; |
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172 | k = n-1; |
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173 | r = (float )(2 * k); |
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174 | |
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175 | do { |
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176 | pkm2 = (pkm1 * r - pk * x) * xinv; |
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177 | pk = pkm1; |
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178 | pkm1 = pkm2; |
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179 | r -= 2.0; |
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180 | } while( --k > 0 ); |
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181 | |
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182 | r = pk; |
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183 | if( r < 0 ) |
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184 | r = -r; |
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185 | ans = pkm1; |
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186 | if( ans < 0 ) |
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187 | ans = -ans; |
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188 | |
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189 | if( r > ans ) /* if( fabs(pk) > fabs(pkm1) ) */ |
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190 | ans = sign * cephes_j1f(x)/pk; |
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191 | else |
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192 | ans = sign * cephes_j0f(x)/pkm1; |
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193 | return( ans ); |
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194 | } |
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195 | #endif |
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196 | |
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197 | #if FLOAT_SIZE>4 |
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198 | #define sas_JN cephes_jn |
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199 | #else |
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200 | #define sas_JN cephes_jnf |
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201 | #endif |
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