| 1 | /* igami() |
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| 2 | * |
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| 3 | * Inverse of complemented imcomplete gamma integral |
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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 a, x, p, igami(); |
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| 10 | * |
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| 11 | * x = igami( a, p ); |
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| 12 | * |
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| 13 | * DESCRIPTION: |
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| 14 | * |
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| 15 | * Given p, the function finds x such that |
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| 16 | * |
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| 17 | * igamc( a, x ) = p. |
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| 18 | * |
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| 19 | * Starting with the approximate value |
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| 20 | * |
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| 21 | * 3 |
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| 22 | * x = a t |
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| 23 | * |
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| 24 | * where |
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| 25 | * |
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| 26 | * t = 1 - d - ndtri(p) sqrt(d) |
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| 27 | * |
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| 28 | * and |
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| 29 | * |
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| 30 | * d = 1/9a, |
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| 31 | * |
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| 32 | * the routine performs up to 10 Newton iterations to find the |
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| 33 | * root of igamc(a,x) - p = 0. |
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| 34 | * |
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| 35 | * ACCURACY: |
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| 36 | * |
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| 37 | * Tested at random a, p in the intervals indicated. |
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| 38 | * |
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| 39 | * a p Relative error: |
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| 40 | * arithmetic domain domain # trials peak rms |
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| 41 | * IEEE 0.5,100 0,0.5 100000 1.0e-14 1.7e-15 |
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| 42 | * IEEE 0.01,0.5 0,0.5 100000 9.0e-14 3.4e-15 |
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| 43 | * IEEE 0.5,10000 0,0.5 20000 2.3e-13 3.8e-14 |
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| 44 | */ |
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| 45 | |
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| 46 | /* |
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| 47 | Cephes Math Library Release 2.8: June, 2000 |
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| 48 | Copyright 1984, 1987, 1995, 2000 by Stephen L. Moshier |
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| 49 | */ |
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| 50 | |
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| 51 | #include "mconf.h" |
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| 52 | |
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| 53 | extern double MACHEP, MAXNUM, MAXLOG, MINLOG; |
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| 54 | #ifdef ANSIPROT |
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| 55 | extern double igamc ( double, double ); |
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| 56 | extern double ndtri ( double ); |
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| 57 | extern double exp ( double ); |
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| 58 | extern double fabs ( double ); |
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| 59 | extern double log ( double ); |
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| 60 | extern double sqrt ( double ); |
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| 61 | extern double lgam ( double ); |
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| 62 | #else |
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| 63 | double igamc(), ndtri(), exp(), fabs(), log(), sqrt(), lgam(); |
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| 64 | #endif |
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| 65 | |
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| 66 | double igami( a, y0 ) |
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| 67 | double a, y0; |
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| 68 | { |
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| 69 | double x0, x1, x, yl, yh, y, d, lgm, dithresh; |
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| 70 | int i, dir; |
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| 71 | |
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| 72 | /* bound the solution */ |
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| 73 | x0 = MAXNUM; |
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| 74 | yl = 0; |
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| 75 | x1 = 0; |
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| 76 | yh = 1.0; |
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| 77 | dithresh = 5.0 * MACHEP; |
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| 78 | |
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| 79 | /* approximation to inverse function */ |
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| 80 | d = 1.0/(9.0*a); |
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| 81 | y = ( 1.0 - d - ndtri(y0) * sqrt(d) ); |
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| 82 | x = a * y * y * y; |
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| 83 | |
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| 84 | lgm = lgam(a); |
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| 85 | |
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| 86 | for( i=0; i<10; i++ ) |
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| 87 | { |
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| 88 | if( x > x0 || x < x1 ) |
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| 89 | goto ihalve; |
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| 90 | y = igamc(a,x); |
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| 91 | if( y < yl || y > yh ) |
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| 92 | goto ihalve; |
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| 93 | if( y < y0 ) |
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| 94 | { |
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| 95 | x0 = x; |
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| 96 | yl = y; |
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| 97 | } |
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| 98 | else |
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| 99 | { |
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| 100 | x1 = x; |
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| 101 | yh = y; |
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| 102 | } |
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| 103 | /* compute the derivative of the function at this point */ |
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| 104 | d = (a - 1.0) * log(x) - x - lgm; |
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| 105 | if( d < -MAXLOG ) |
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| 106 | goto ihalve; |
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| 107 | d = -exp(d); |
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| 108 | /* compute the step to the next approximation of x */ |
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| 109 | d = (y - y0)/d; |
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| 110 | if( fabs(d/x) < MACHEP ) |
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| 111 | goto done; |
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| 112 | x = x - d; |
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| 113 | } |
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| 114 | |
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| 115 | /* Resort to interval halving if Newton iteration did not converge. */ |
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| 116 | ihalve: |
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| 117 | |
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| 118 | d = 0.0625; |
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| 119 | if( x0 == MAXNUM ) |
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| 120 | { |
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| 121 | if( x <= 0.0 ) |
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| 122 | x = 1.0; |
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| 123 | while( x0 == MAXNUM ) |
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| 124 | { |
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| 125 | x = (1.0 + d) * x; |
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| 126 | y = igamc( a, x ); |
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| 127 | if( y < y0 ) |
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| 128 | { |
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| 129 | x0 = x; |
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| 130 | yl = y; |
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| 131 | break; |
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| 132 | } |
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| 133 | d = d + d; |
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| 134 | } |
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| 135 | } |
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| 136 | d = 0.5; |
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| 137 | dir = 0; |
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| 138 | |
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| 139 | for( i=0; i<400; i++ ) |
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| 140 | { |
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| 141 | x = x1 + d * (x0 - x1); |
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| 142 | y = igamc( a, x ); |
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| 143 | lgm = (x0 - x1)/(x1 + x0); |
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| 144 | if( fabs(lgm) < dithresh ) |
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| 145 | break; |
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| 146 | lgm = (y - y0)/y0; |
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| 147 | if( fabs(lgm) < dithresh ) |
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| 148 | break; |
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| 149 | if( x <= 0.0 ) |
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| 150 | break; |
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| 151 | if( y >= y0 ) |
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| 152 | { |
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| 153 | x1 = x; |
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| 154 | yh = y; |
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| 155 | if( dir < 0 ) |
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| 156 | { |
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| 157 | dir = 0; |
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| 158 | d = 0.5; |
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| 159 | } |
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| 160 | else if( dir > 1 ) |
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| 161 | d = 0.5 * d + 0.5; |
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| 162 | else |
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| 163 | d = (y0 - yl)/(yh - yl); |
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| 164 | dir += 1; |
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| 165 | } |
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| 166 | else |
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| 167 | { |
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| 168 | x0 = x; |
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| 169 | yl = y; |
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| 170 | if( dir > 0 ) |
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| 171 | { |
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| 172 | dir = 0; |
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| 173 | d = 0.5; |
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| 174 | } |
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| 175 | else if( dir < -1 ) |
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| 176 | d = 0.5 * d; |
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| 177 | else |
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| 178 | d = (y0 - yl)/(yh - yl); |
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| 179 | dir -= 1; |
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| 180 | } |
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| 181 | } |
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| 182 | if( x == 0.0 ) |
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| 183 | mtherr( "igami", UNDERFLOW ); |
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| 184 | |
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| 185 | done: |
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| 186 | return( x ); |
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| 187 | } |
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