1 | /* j1.c |
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2 | * |
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3 | * Bessel function of order one |
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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 | * double x, y, j1(); |
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10 | * |
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11 | * y = j1( x ); |
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12 | * |
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13 | * |
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14 | * |
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15 | * DESCRIPTION: |
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16 | * |
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17 | * Returns Bessel function of order one of the argument. |
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18 | * |
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19 | * The domain is divided into the intervals [0, 8] and |
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20 | * (8, infinity). In the first interval a 24 term Chebyshev |
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21 | * expansion is used. In the second, the asymptotic |
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22 | * trigonometric representation is employed using two |
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23 | * rational functions of degree 5/5. |
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24 | * |
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25 | * |
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26 | * |
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27 | * ACCURACY: |
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28 | * |
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29 | * Absolute error: |
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30 | * arithmetic domain # trials peak rms |
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31 | * DEC 0, 30 10000 4.0e-17 1.1e-17 |
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32 | * IEEE 0, 30 30000 2.6e-16 1.1e-16 |
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33 | * |
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34 | * |
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35 | */ |
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36 | |
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37 | /* |
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38 | Cephes Math Library Release 2.8: June, 2000 |
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39 | Copyright 1984, 1987, 1989, 2000 by Stephen L. Moshier |
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40 | */ |
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41 | |
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42 | #if FLOAT_SIZE>4 |
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43 | //Cephes double pression function |
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44 | double cephes_j1(double x); |
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45 | |
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46 | constant double RPJ1[8] = { |
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47 | -8.99971225705559398224E8, |
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48 | 4.52228297998194034323E11, |
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49 | -7.27494245221818276015E13, |
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50 | 3.68295732863852883286E15, |
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51 | 0.0, |
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52 | 0.0, |
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53 | 0.0, |
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54 | 0.0 }; |
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55 | |
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56 | constant double RQJ1[8] = { |
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57 | 6.20836478118054335476E2, |
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58 | 2.56987256757748830383E5, |
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59 | 8.35146791431949253037E7, |
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60 | 2.21511595479792499675E10, |
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61 | 4.74914122079991414898E12, |
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62 | 7.84369607876235854894E14, |
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63 | 8.95222336184627338078E16, |
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64 | 5.32278620332680085395E18 |
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65 | }; |
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66 | |
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67 | constant double PPJ1[8] = { |
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68 | 7.62125616208173112003E-4, |
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69 | 7.31397056940917570436E-2, |
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70 | 1.12719608129684925192E0, |
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71 | 5.11207951146807644818E0, |
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72 | 8.42404590141772420927E0, |
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73 | 5.21451598682361504063E0, |
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74 | 1.00000000000000000254E0, |
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75 | 0.0} ; |
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76 | |
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77 | |
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78 | constant double PQJ1[8] = { |
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79 | 5.71323128072548699714E-4, |
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80 | 6.88455908754495404082E-2, |
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81 | 1.10514232634061696926E0, |
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82 | 5.07386386128601488557E0, |
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83 | 8.39985554327604159757E0, |
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84 | 5.20982848682361821619E0, |
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85 | 9.99999999999999997461E-1, |
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86 | 0.0 }; |
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87 | |
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88 | constant double QPJ1[8] = { |
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89 | 5.10862594750176621635E-2, |
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90 | 4.98213872951233449420E0, |
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91 | 7.58238284132545283818E1, |
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92 | 3.66779609360150777800E2, |
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93 | 7.10856304998926107277E2, |
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94 | 5.97489612400613639965E2, |
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95 | 2.11688757100572135698E2, |
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96 | 2.52070205858023719784E1 }; |
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97 | |
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98 | constant double QQJ1[8] = { |
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99 | 7.42373277035675149943E1, |
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100 | 1.05644886038262816351E3, |
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101 | 4.98641058337653607651E3, |
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102 | 9.56231892404756170795E3, |
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103 | 7.99704160447350683650E3, |
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104 | 2.82619278517639096600E3, |
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105 | 3.36093607810698293419E2, |
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106 | 0.0 }; |
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107 | |
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108 | double cephes_j1(double x) |
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109 | { |
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110 | |
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111 | double w, z, p, q, abs_x, sign_x; |
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112 | |
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113 | const double Z1 = 1.46819706421238932572E1; |
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114 | const double Z2 = 4.92184563216946036703E1; |
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115 | |
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116 | // 2017-05-18 PAK - mathematica and mpmath use J1(-x) = -J1(x) |
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117 | if (x < 0) { |
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118 | abs_x = -x; |
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119 | sign_x = -1.0; |
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120 | } else { |
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121 | abs_x = x; |
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122 | sign_x = 1.0; |
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123 | } |
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124 | |
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125 | if( abs_x <= 5.0 ) { |
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126 | z = abs_x * abs_x; |
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127 | w = polevl( z, RPJ1, 3 ) / p1evl( z, RQJ1, 8 ); |
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128 | w = w * abs_x * (z - Z1) * (z - Z2); |
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129 | return( sign_x * w ); |
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130 | } |
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131 | |
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132 | w = 5.0/abs_x; |
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133 | z = w * w; |
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134 | p = polevl( z, PPJ1, 6)/polevl( z, PQJ1, 6 ); |
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135 | q = polevl( z, QPJ1, 7)/p1evl( z, QQJ1, 7 ); |
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136 | |
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137 | // 2017-05-19 PAK improve accuracy using trig identies |
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138 | // original: |
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139 | // const double THPIO4 = 2.35619449019234492885; |
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140 | // const double SQ2OPI = 0.79788456080286535588; |
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141 | // double sin_xn, cos_xn; |
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142 | // SINCOS(abs_x - THPIO4, sin_xn, cos_xn); |
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143 | // p = p * cos_xn - w * q * sin_xn; |
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144 | // return( sign_x * p * SQ2OPI / sqrt(abs_x) ); |
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145 | // expanding p*cos(a - 3 pi/4) - wq sin(a - 3 pi/4) |
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146 | // [ p(sin(a) - cos(a)) + wq(sin(a) + cos(a)) / sqrt(2) |
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147 | // note that sqrt(1/2) * sqrt(2/pi) = sqrt(1/pi) |
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148 | const double SQRT1_PI = 0.56418958354775628; |
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149 | double sin_x, cos_x; |
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150 | SINCOS(abs_x, sin_x, cos_x); |
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151 | p = p*(sin_x - cos_x) + w*q*(sin_x + cos_x); |
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152 | return( sign_x * p * SQRT1_PI / sqrt(abs_x) ); |
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153 | } |
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154 | |
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155 | #else |
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156 | //Single precission version of cephes |
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157 | float cephes_j1f(float x); |
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158 | |
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159 | constant float JPJ1[8] = { |
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160 | -4.878788132172128E-009, |
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161 | 6.009061827883699E-007, |
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162 | -4.541343896997497E-005, |
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163 | 1.937383947804541E-003, |
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164 | -3.405537384615824E-002, |
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165 | 0.0, |
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166 | 0.0, |
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167 | 0.0 |
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168 | }; |
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169 | |
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170 | constant float MO1J1[8] = { |
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171 | 6.913942741265801E-002, |
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172 | -2.284801500053359E-001, |
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173 | 3.138238455499697E-001, |
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174 | -2.102302420403875E-001, |
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175 | 5.435364690523026E-003, |
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176 | 1.493389585089498E-001, |
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177 | 4.976029650847191E-006, |
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178 | 7.978845453073848E-001 |
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179 | }; |
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180 | |
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181 | constant float PH1J1[8] = { |
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182 | -4.497014141919556E+001, |
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183 | 5.073465654089319E+001, |
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184 | -2.485774108720340E+001, |
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185 | 7.222973196770240E+000, |
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186 | -1.544842782180211E+000, |
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187 | 3.503787691653334E-001, |
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188 | -1.637986776941202E-001, |
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189 | 3.749989509080821E-001 |
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190 | }; |
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191 | |
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192 | float cephes_j1f(float xx) |
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193 | { |
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194 | |
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195 | float x, w, z, p, q, xn; |
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196 | |
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197 | const float Z1 = 1.46819706421238932572E1; |
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198 | |
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199 | |
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200 | // 2017-05-18 PAK - mathematica and mpmath use J1(-x) = -J1(x) |
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201 | x = xx; |
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202 | if( x < 0 ) |
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203 | x = -xx; |
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204 | |
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205 | if( x <= 2.0 ) { |
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206 | z = x * x; |
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207 | p = (z-Z1) * x * polevl( z, JPJ1, 4 ); |
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208 | return( xx < 0. ? -p : p ); |
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209 | } |
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210 | |
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211 | q = 1.0/x; |
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212 | w = sqrt(q); |
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213 | |
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214 | p = w * polevl( q, MO1J1, 7); |
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215 | w = q*q; |
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216 | // 2017-05-19 PAK improve accuracy using trig identies |
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217 | // original: |
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218 | // const float THPIO4F = 2.35619449019234492885; /* 3*pi/4 */ |
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219 | // xn = q * polevl( w, PH1J1, 7) - THPIO4F; |
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220 | // p = p * cos(xn + x); |
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221 | // return( xx < 0. ? -p : p ); |
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222 | // expanding cos(a + b - 3 pi/4) is |
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223 | // [sin(a)sin(b) + sin(a)cos(b) + cos(a)sin(b)-cos(a)cos(b)] / sqrt(2) |
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224 | xn = q * polevl( w, PH1J1, 7); |
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225 | float cos_xn, sin_xn; |
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226 | float cos_x, sin_x; |
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227 | SINCOS(xn, sin_xn, cos_xn); // about xn and 1 |
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228 | SINCOS(x, sin_x, cos_x); |
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229 | p *= M_SQRT1_2*(sin_xn*(sin_x+cos_x) + cos_xn*(sin_x-cos_x)); |
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230 | |
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231 | return( xx < 0. ? -p : p ); |
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232 | } |
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233 | #endif |
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234 | |
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235 | #if FLOAT_SIZE>4 |
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236 | #define sas_J1 cephes_j1 |
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237 | #else |
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238 | #define sas_J1 cephes_j1f |
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239 | #endif |
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240 | |
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241 | //Finally J1c function that equals 2*J1(x)/x |
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242 | double sas_2J1x_x(double x); |
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243 | double sas_2J1x_x(double x) |
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244 | { |
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245 | return (x != 0.0 ) ? 2.0*sas_J1(x)/x : 1.0; |
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246 | } |
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