[1557a1e] | 1 | |
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| 2 | /* |
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| 3 | * Cephes Math Library Release 2.2: June, 1992 |
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| 4 | * Copyright 1984, 1987, 1988, 1992 by Stephen L. Moshier |
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| 5 | * Direct inquiries to 30 Frost Street, Cambridge, MA 02140 |
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| 6 | */ |
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| 7 | /* |
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| 8 | * |
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| 9 | * Error function |
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| 10 | * |
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| 11 | * |
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| 12 | * |
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| 13 | * SYNOPSIS: |
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| 14 | * |
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| 15 | * double x, y, erf(); |
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| 16 | * |
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| 17 | * y = erf( x ); |
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| 18 | * |
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| 19 | * |
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| 20 | * |
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| 21 | * DESCRIPTION: |
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| 22 | * |
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| 23 | * The integral is |
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| 24 | * |
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| 25 | * x |
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| 26 | * - |
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| 27 | * 2 | | 2 |
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| 28 | * erf(x) = -------- | exp( - t ) dt. |
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| 29 | * sqrt(pi) | | |
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| 30 | * - |
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| 31 | * 0 |
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| 32 | * |
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| 33 | * For 0 <= |x| < 1, erf(x) = x * P4(x**2)/Q5(x**2); otherwise |
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| 34 | * erf(x) = 1 - erfc(x). |
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| 35 | * |
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| 36 | * |
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| 37 | * |
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| 38 | * ACCURACY: |
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| 39 | * |
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| 40 | * Relative error: |
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| 41 | * arithmetic domain # trials peak rms |
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| 42 | * IEEE 0,1 30000 3.7e-16 1.0e-16 |
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| 43 | * |
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| 44 | */ |
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| 45 | /* |
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| 46 | * |
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| 47 | * Complementary error function |
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| 48 | * |
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| 49 | * |
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| 50 | * |
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| 51 | * SYNOPSIS: |
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| 52 | * |
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| 53 | * double x, y, erfc(); |
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| 54 | * |
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| 55 | * y = erfc( x ); |
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| 56 | * |
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| 57 | * |
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| 58 | * |
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| 59 | * DESCRIPTION: |
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| 60 | * |
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| 61 | * |
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| 62 | * 1 - erf(x) = |
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| 63 | * |
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| 64 | * inf. |
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| 65 | * - |
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| 66 | * 2 | | 2 |
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| 67 | * erfc(x) = -------- | exp( - t ) dt |
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| 68 | * sqrt(pi) | | |
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| 69 | * - |
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| 70 | * x |
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| 71 | * |
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| 72 | * |
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| 73 | * For small x, erfc(x) = 1 - erf(x); otherwise rational |
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| 74 | * approximations are computed. |
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| 75 | * |
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| 76 | * |
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| 77 | * |
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| 78 | * ACCURACY: |
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| 79 | * |
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| 80 | * Relative error: |
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| 81 | * arithmetic domain # trials peak rms |
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| 82 | * IEEE 0,26.6417 30000 5.7e-14 1.5e-14 |
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| 83 | */ |
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| 84 | |
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| 85 | #ifdef NEED_ERF |
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| 86 | |
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| 87 | #if FLOAT_SIZE>4 // DOUBLE_PRECISION |
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| 88 | double erf(double x); |
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| 89 | double erfc(double a); |
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| 90 | |
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| 91 | constant double PD[] = { |
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| 92 | 2.46196981473530512524E-10, |
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| 93 | 5.64189564831068821977E-1, |
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| 94 | 7.46321056442269912687E0, |
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| 95 | 4.86371970985681366614E1, |
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| 96 | 1.96520832956077098242E2, |
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| 97 | 5.26445194995477358631E2, |
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| 98 | 9.34528527171957607540E2, |
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| 99 | 1.02755188689515710272E3, |
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| 100 | 5.57535335369399327526E2 |
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| 101 | }; |
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| 102 | |
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| 103 | constant double QD[] = { |
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| 104 | /* 1.00000000000000000000E0, */ |
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| 105 | 1.32281951154744992508E1, |
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| 106 | 8.67072140885989742329E1, |
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| 107 | 3.54937778887819891062E2, |
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| 108 | 9.75708501743205489753E2, |
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| 109 | 1.82390916687909736289E3, |
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| 110 | 2.24633760818710981792E3, |
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| 111 | 1.65666309194161350182E3, |
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| 112 | 5.57535340817727675546E2 |
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| 113 | }; |
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| 114 | |
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| 115 | constant double RD[] = { |
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| 116 | 5.64189583547755073984E-1, |
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| 117 | 1.27536670759978104416E0, |
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| 118 | 5.01905042251180477414E0, |
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| 119 | 6.16021097993053585195E0, |
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| 120 | 7.40974269950448939160E0, |
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| 121 | 2.97886665372100240670E0 |
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| 122 | }; |
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| 123 | |
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| 124 | constant double SD[] = { |
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| 125 | /* 1.00000000000000000000E0, */ |
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| 126 | 2.26052863220117276590E0, |
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| 127 | 9.39603524938001434673E0, |
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| 128 | 1.20489539808096656605E1, |
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| 129 | 1.70814450747565897222E1, |
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| 130 | 9.60896809063285878198E0, |
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| 131 | 3.36907645100081516050E0 |
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| 132 | }; |
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| 133 | |
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| 134 | constant double TD[] = { |
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| 135 | 9.60497373987051638749E0, |
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| 136 | 9.00260197203842689217E1, |
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| 137 | 2.23200534594684319226E3, |
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| 138 | 7.00332514112805075473E3, |
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| 139 | 5.55923013010394962768E4 |
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| 140 | }; |
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| 141 | |
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| 142 | constant double UD[] = { |
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| 143 | /* 1.00000000000000000000E0, */ |
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| 144 | 3.35617141647503099647E1, |
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| 145 | 5.21357949780152679795E2, |
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| 146 | 4.59432382970980127987E3, |
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| 147 | 2.26290000613890934246E4, |
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| 148 | 4.92673942608635921086E4 |
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| 149 | }; |
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| 150 | |
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| 151 | double erfc(double a) |
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| 152 | { |
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| 153 | double MAXLOG = 88.72283905206835; |
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| 154 | double p, q, x, y, z; |
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| 155 | |
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| 156 | |
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| 157 | /*if (a < 0.0) |
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| 158 | x = -a; |
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| 159 | else |
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| 160 | x = a;*/ |
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| 161 | |
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| 162 | x = fabs(a); |
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| 163 | |
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| 164 | |
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| 165 | if (x < 1.0) { |
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| 166 | //The line bellow is a troublemaker for GPU, so sas_erf function |
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| 167 | //is explicit here for the case < 1.0 |
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| 168 | //return (1.0 - sas_erf(a)); |
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| 169 | z = x * x; |
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| 170 | y = x * polevl(z, TD, 4) / p1evl(z, UD, 5); |
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| 171 | |
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| 172 | return y; |
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| 173 | } |
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| 174 | |
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| 175 | z = -a * a; |
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| 176 | |
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| 177 | if (z < -MAXLOG) { |
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| 178 | if (a < 0) |
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| 179 | return (2.0); |
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| 180 | else |
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| 181 | return (0.0); |
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| 182 | } |
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| 183 | |
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| 184 | z = exp(z); |
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| 185 | |
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| 186 | |
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| 187 | if (x < 8.0) { |
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| 188 | p = polevl(x, PD, 8); |
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| 189 | q = p1evl(x, QD, 8); |
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| 190 | } |
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| 191 | else { |
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| 192 | p = polevl(x, RD, 5); |
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| 193 | q = p1evl(x, SD, 6); |
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| 194 | } |
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| 195 | y = (z * p) / q; |
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| 196 | |
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| 197 | if (a < 0) |
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| 198 | y = 2.0 - y; |
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| 199 | |
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| 200 | if (y == 0.0) { |
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| 201 | if (a < 0) |
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| 202 | return (2.0); |
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| 203 | else |
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| 204 | return (0.0); |
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| 205 | } |
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| 206 | return y; |
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| 207 | } |
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| 208 | |
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| 209 | |
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| 210 | double erf(double x) |
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| 211 | { |
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| 212 | double y, z; |
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| 213 | |
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| 214 | if (fabs(x) > 1.0) |
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| 215 | return (1.0 - erfc(x)); |
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| 216 | |
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| 217 | z = x * x; |
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| 218 | #if FLOAT_SIZE>4 |
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| 219 | y = x * polevl(z, TD, 4) / p1evl(z, UD, 5); |
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| 220 | #else |
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| 221 | y = x * polevl( z, TF, 6 ); |
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| 222 | #endif |
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| 223 | |
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| 224 | return y; |
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| 225 | } |
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| 226 | |
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| 227 | #else // SINGLE PRECISION |
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| 228 | |
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| 229 | double erff(double x); |
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| 230 | double erfcf(double a); |
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| 231 | |
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| 232 | /* erfc(x) = exp(-x^2) P(1/x), 1 < x < 2 */ |
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| 233 | constant double PF[] = { |
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| 234 | 2.326819970068386E-002, |
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| 235 | -1.387039388740657E-001, |
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| 236 | 3.687424674597105E-001, |
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| 237 | -5.824733027278666E-001, |
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| 238 | 6.210004621745983E-001, |
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| 239 | -4.944515323274145E-001, |
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| 240 | 3.404879937665872E-001, |
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| 241 | -2.741127028184656E-001, |
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| 242 | 5.638259427386472E-001 |
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| 243 | }; |
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| 244 | |
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| 245 | /* erfc(x) = exp(-x^2) 1/x P(1/x^2), 2 < x < 14 */ |
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| 246 | constant double RF[] = { |
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| 247 | -1.047766399936249E+001, |
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| 248 | 1.297719955372516E+001, |
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| 249 | -7.495518717768503E+000, |
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| 250 | 2.921019019210786E+000, |
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| 251 | -1.015265279202700E+000, |
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| 252 | 4.218463358204948E-001, |
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| 253 | -2.820767439740514E-001, |
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| 254 | 5.641895067754075E-001 |
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| 255 | }; |
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| 256 | |
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| 257 | /* erf(x) = x P(x^2), 0 < x < 1 */ |
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| 258 | constant double TF[] = { |
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| 259 | 7.853861353153693E-005, |
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| 260 | -8.010193625184903E-004, |
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| 261 | 5.188327685732524E-003, |
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| 262 | -2.685381193529856E-002, |
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| 263 | 1.128358514861418E-001, |
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| 264 | -3.761262582423300E-001, |
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| 265 | 1.128379165726710E+000 |
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| 266 | }; |
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| 267 | |
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| 268 | |
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| 269 | float erfcf(float a) |
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| 270 | { |
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| 271 | float MAXLOG = 88.72283905206835; |
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| 272 | float p, q, x, y, z; |
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| 273 | |
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| 274 | |
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| 275 | /*if (a < 0.0) |
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| 276 | x = -a; |
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| 277 | else |
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| 278 | x = a;*/ |
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| 279 | |
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| 280 | x = fabsf(a); |
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| 281 | |
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| 282 | |
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| 283 | if (x < 1.0) { |
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| 284 | //The line below is a troublemaker for GPU, so sas_erf function |
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| 285 | //is explicit here for the case < 1.0 |
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| 286 | //return (1.0 - sas_erf(a)); |
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| 287 | z = x * x; |
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| 288 | y = x * polevl( z, TF, 6 ); |
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| 289 | |
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| 290 | return y; |
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| 291 | } |
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| 292 | |
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| 293 | z = -a * a; |
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| 294 | |
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| 295 | if (z < -MAXLOG) { |
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| 296 | if (a < 0) |
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| 297 | return (2.0); |
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| 298 | else |
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| 299 | return (0.0); |
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| 300 | } |
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| 301 | |
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| 302 | z = expf(z); |
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| 303 | |
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| 304 | |
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| 305 | q=1.0/x; |
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| 306 | y=q*q; |
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| 307 | if( x < 2.0 ) { |
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| 308 | p = polevl( y, PF, 8 ); |
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| 309 | } else { |
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| 310 | p = polevl( y, RF, 7 ); |
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| 311 | } |
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| 312 | y = z * q * p; |
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| 313 | |
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| 314 | if (a < 0) |
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| 315 | y = 2.0 - y; |
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| 316 | |
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| 317 | if (y == 0.0) { |
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| 318 | if (a < 0) |
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| 319 | return (2.0); |
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| 320 | else |
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| 321 | return (0.0); |
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| 322 | } |
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| 323 | return y; |
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| 324 | } |
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| 325 | |
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| 326 | |
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| 327 | float erff(float x) |
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| 328 | { |
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| 329 | float y, z; |
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| 330 | |
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| 331 | if (fabsf(x) > 1.0) |
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| 332 | return (1.0 - erfcf(x)); |
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| 333 | |
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| 334 | z = x * x; |
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| 335 | y = x * polevl( z, TF, 6 ); |
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| 336 | |
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| 337 | return y; |
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| 338 | } |
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| 339 | |
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| 340 | #endif // SINGLE_PRECISION |
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| 341 | #endif // NEED_ERF |
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| 342 | |
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| 343 | #if FLOAT_SIZE>4 |
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| 344 | #define sas_erf erf |
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| 345 | #define sas_erfc erfc |
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| 346 | #else |
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| 347 | #define sas_erf erff |
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| 348 | #define sas_erfc erfcf |
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| 349 | #endif |
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