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1 /*-------------------------------------------------------------------- |
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2 *© Portions copyright (c) 2006 Nokia Corporation. All rights reserved. |
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3 *-------------------------------------------------------------------- |
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4 */ |
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5 /* @(#)s_erf.c 5.1 93/09/24 */ |
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6 /* |
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7 * ==================================================== |
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8 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. |
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9 * |
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10 * Developed at SunPro, a Sun Microsystems, Inc. business. |
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11 * Permission to use, copy, modify, and distribute this |
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12 * software is freely granted, provided that this notice |
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13 * is preserved. |
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14 * ==================================================== |
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15 */ |
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16 #ifndef __SYMBIAN32__ |
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17 #ifndef lint |
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18 static char rcsid[] = "$FreeBSD: src/lib/msun/src/s_erf.c,v 1.7 2002/05/28 18:15:04 alfred Exp $"; |
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19 #endif |
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20 #endif //__SYMBIAN32__ |
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21 |
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22 /* double erf(double x) |
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23 * double erfc(double x) |
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24 * x |
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25 * 2 |\ |
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26 * erf(x) = --------- | exp(-t*t)dt |
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27 * sqrt(pi) \| |
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28 * 0 |
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29 * |
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30 * erfc(x) = 1-erf(x) |
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31 * Note that |
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32 * erf(-x) = -erf(x) |
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33 * erfc(-x) = 2 - erfc(x) |
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34 * |
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35 * Method: |
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36 * 1. For |x| in [0, 0.84375] |
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37 * erf(x) = x + x*R(x^2) |
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38 * erfc(x) = 1 - erf(x) if x in [-.84375,0.25] |
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39 * = 0.5 + ((0.5-x)-x*R) if x in [0.25,0.84375] |
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40 * where R = P/Q where P is an odd poly of degree 8 and |
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41 * Q is an odd poly of degree 10. |
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42 * -57.90 |
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43 * | R - (erf(x)-x)/x | <= 2 |
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44 * |
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45 * |
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46 * Remark. The formula is derived by noting |
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47 * erf(x) = (2/sqrt(pi))*(x - x^3/3 + x^5/10 - x^7/42 + ....) |
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48 * and that |
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49 * 2/sqrt(pi) = 1.128379167095512573896158903121545171688 |
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50 * is close to one. The interval is chosen because the fix |
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51 * point of erf(x) is near 0.6174 (i.e., erf(x)=x when x is |
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52 * near 0.6174), and by some experiment, 0.84375 is chosen to |
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53 * guarantee the error is less than one ulp for erf. |
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54 * |
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55 * 2. For |x| in [0.84375,1.25], let s = |x| - 1, and |
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56 * c = 0.84506291151 rounded to single (24 bits) |
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57 * erf(x) = sign(x) * (c + P1(s)/Q1(s)) |
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58 * erfc(x) = (1-c) - P1(s)/Q1(s) if x > 0 |
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59 * 1+(c+P1(s)/Q1(s)) if x < 0 |
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60 * |P1/Q1 - (erf(|x|)-c)| <= 2**-59.06 |
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61 * Remark: here we use the taylor series expansion at x=1. |
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62 * erf(1+s) = erf(1) + s*Poly(s) |
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63 * = 0.845.. + P1(s)/Q1(s) |
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64 * That is, we use rational approximation to approximate |
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65 * erf(1+s) - (c = (single)0.84506291151) |
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66 * Note that |P1/Q1|< 0.078 for x in [0.84375,1.25] |
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67 * where |
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68 * P1(s) = degree 6 poly in s |
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69 * Q1(s) = degree 6 poly in s |
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70 * |
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71 * 3. For x in [1.25,1/0.35(~2.857143)], |
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72 * erfc(x) = (1/x)*exp(-x*x-0.5625+R1/S1) |
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73 * erf(x) = 1 - erfc(x) |
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74 * where |
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75 * R1(z) = degree 7 poly in z, (z=1/x^2) |
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76 * S1(z) = degree 8 poly in z |
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77 * |
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78 * 4. For x in [1/0.35,28] |
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79 * erfc(x) = (1/x)*exp(-x*x-0.5625+R2/S2) if x > 0 |
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80 * = 2.0 - (1/x)*exp(-x*x-0.5625+R2/S2) if -6<x<0 |
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81 * = 2.0 - tiny (if x <= -6) |
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82 * erf(x) = sign(x)*(1.0 - erfc(x)) if x < 6, else |
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83 * erf(x) = sign(x)*(1.0 - tiny) |
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84 * where |
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85 * R2(z) = degree 6 poly in z, (z=1/x^2) |
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86 * S2(z) = degree 7 poly in z |
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87 * |
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88 * Note1: |
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89 * To compute exp(-x*x-0.5625+R/S), let s be a single |
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90 * precision number and s := x; then |
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91 * -x*x = -s*s + (s-x)*(s+x) |
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92 * exp(-x*x-0.5626+R/S) = |
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93 * exp(-s*s-0.5625)*exp((s-x)*(s+x)+R/S); |
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94 * Note2: |
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95 * Here 4 and 5 make use of the asymptotic series |
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96 * exp(-x*x) |
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97 * erfc(x) ~ ---------- * ( 1 + Poly(1/x^2) ) |
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98 * x*sqrt(pi) |
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99 * We use rational approximation to approximate |
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100 * g(s)=f(1/x^2) = log(erfc(x)*x) - x*x + 0.5625 |
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101 * Here is the error bound for R1/S1 and R2/S2 |
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102 * |R1/S1 - f(x)| < 2**(-62.57) |
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103 * |R2/S2 - f(x)| < 2**(-61.52) |
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104 * |
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105 * 5. For inf > x >= 28 |
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106 * erf(x) = sign(x) *(1 - tiny) (raise inexact) |
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107 * erfc(x) = tiny*tiny (raise underflow) if x > 0 |
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108 * = 2 - tiny if x<0 |
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109 * |
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110 * 7. Special case: |
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111 * erf(0) = 0, erf(inf) = 1, erf(-inf) = -1, |
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112 * erfc(0) = 1, erfc(inf) = 0, erfc(-inf) = 2, |
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113 * erfc/erf(NaN) is NaN |
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114 */ |
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115 |
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116 |
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117 #include <math.h> |
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118 #include "math_private.h" |
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119 |
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120 static const double |
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121 tiny = 1e-300, |
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122 half= 5.00000000000000000000e-01, /* 0x3FE00000, 0x00000000 */ |
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123 one = 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */ |
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124 two = 2.00000000000000000000e+00, /* 0x40000000, 0x00000000 */ |
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125 /* c = (float)0.84506291151 */ |
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126 erx = 8.45062911510467529297e-01, /* 0x3FEB0AC1, 0x60000000 */ |
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127 /* |
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128 * Coefficients for approximation to erf on [0,0.84375] |
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129 */ |
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130 efx = 1.28379167095512586316e-01, /* 0x3FC06EBA, 0x8214DB69 */ |
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131 efx8= 1.02703333676410069053e+00, /* 0x3FF06EBA, 0x8214DB69 */ |
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132 pp0 = 1.28379167095512558561e-01, /* 0x3FC06EBA, 0x8214DB68 */ |
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133 pp1 = -3.25042107247001499370e-01, /* 0xBFD4CD7D, 0x691CB913 */ |
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134 pp2 = -2.84817495755985104766e-02, /* 0xBF9D2A51, 0xDBD7194F */ |
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135 pp3 = -5.77027029648944159157e-03, /* 0xBF77A291, 0x236668E4 */ |
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136 pp4 = -2.37630166566501626084e-05, /* 0xBEF8EAD6, 0x120016AC */ |
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137 qq1 = 3.97917223959155352819e-01, /* 0x3FD97779, 0xCDDADC09 */ |
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138 qq2 = 6.50222499887672944485e-02, /* 0x3FB0A54C, 0x5536CEBA */ |
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139 qq3 = 5.08130628187576562776e-03, /* 0x3F74D022, 0xC4D36B0F */ |
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140 qq4 = 1.32494738004321644526e-04, /* 0x3F215DC9, 0x221C1A10 */ |
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141 qq5 = -3.96022827877536812320e-06, /* 0xBED09C43, 0x42A26120 */ |
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142 /* |
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143 * Coefficients for approximation to erf in [0.84375,1.25] |
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144 */ |
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145 pa0 = -2.36211856075265944077e-03, /* 0xBF6359B8, 0xBEF77538 */ |
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146 pa1 = 4.14856118683748331666e-01, /* 0x3FDA8D00, 0xAD92B34D */ |
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147 pa2 = -3.72207876035701323847e-01, /* 0xBFD7D240, 0xFBB8C3F1 */ |
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148 pa3 = 3.18346619901161753674e-01, /* 0x3FD45FCA, 0x805120E4 */ |
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149 pa4 = -1.10894694282396677476e-01, /* 0xBFBC6398, 0x3D3E28EC */ |
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150 pa5 = 3.54783043256182359371e-02, /* 0x3FA22A36, 0x599795EB */ |
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151 pa6 = -2.16637559486879084300e-03, /* 0xBF61BF38, 0x0A96073F */ |
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152 qa1 = 1.06420880400844228286e-01, /* 0x3FBB3E66, 0x18EEE323 */ |
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153 qa2 = 5.40397917702171048937e-01, /* 0x3FE14AF0, 0x92EB6F33 */ |
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154 qa3 = 7.18286544141962662868e-02, /* 0x3FB2635C, 0xD99FE9A7 */ |
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155 qa4 = 1.26171219808761642112e-01, /* 0x3FC02660, 0xE763351F */ |
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156 qa5 = 1.36370839120290507362e-02, /* 0x3F8BEDC2, 0x6B51DD1C */ |
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157 qa6 = 1.19844998467991074170e-02, /* 0x3F888B54, 0x5735151D */ |
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158 /* |
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159 * Coefficients for approximation to erfc in [1.25,1/0.35] |
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160 */ |
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161 ra0 = -9.86494403484714822705e-03, /* 0xBF843412, 0x600D6435 */ |
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162 ra1 = -6.93858572707181764372e-01, /* 0xBFE63416, 0xE4BA7360 */ |
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163 ra2 = -1.05586262253232909814e+01, /* 0xC0251E04, 0x41B0E726 */ |
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164 ra3 = -6.23753324503260060396e+01, /* 0xC04F300A, 0xE4CBA38D */ |
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165 ra4 = -1.62396669462573470355e+02, /* 0xC0644CB1, 0x84282266 */ |
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166 ra5 = -1.84605092906711035994e+02, /* 0xC067135C, 0xEBCCABB2 */ |
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167 ra6 = -8.12874355063065934246e+01, /* 0xC0545265, 0x57E4D2F2 */ |
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168 ra7 = -9.81432934416914548592e+00, /* 0xC023A0EF, 0xC69AC25C */ |
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169 sa1 = 1.96512716674392571292e+01, /* 0x4033A6B9, 0xBD707687 */ |
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170 sa2 = 1.37657754143519042600e+02, /* 0x4061350C, 0x526AE721 */ |
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171 sa3 = 4.34565877475229228821e+02, /* 0x407B290D, 0xD58A1A71 */ |
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172 sa4 = 6.45387271733267880336e+02, /* 0x40842B19, 0x21EC2868 */ |
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173 sa5 = 4.29008140027567833386e+02, /* 0x407AD021, 0x57700314 */ |
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174 sa6 = 1.08635005541779435134e+02, /* 0x405B28A3, 0xEE48AE2C */ |
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175 sa7 = 6.57024977031928170135e+00, /* 0x401A47EF, 0x8E484A93 */ |
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176 sa8 = -6.04244152148580987438e-02, /* 0xBFAEEFF2, 0xEE749A62 */ |
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177 /* |
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178 * Coefficients for approximation to erfc in [1/.35,28] |
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179 */ |
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180 rb0 = -9.86494292470009928597e-03, /* 0xBF843412, 0x39E86F4A */ |
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181 rb1 = -7.99283237680523006574e-01, /* 0xBFE993BA, 0x70C285DE */ |
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182 rb2 = -1.77579549177547519889e+01, /* 0xC031C209, 0x555F995A */ |
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183 rb3 = -1.60636384855821916062e+02, /* 0xC064145D, 0x43C5ED98 */ |
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184 rb4 = -6.37566443368389627722e+02, /* 0xC083EC88, 0x1375F228 */ |
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185 rb5 = -1.02509513161107724954e+03, /* 0xC0900461, 0x6A2E5992 */ |
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186 rb6 = -4.83519191608651397019e+02, /* 0xC07E384E, 0x9BDC383F */ |
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187 sb1 = 3.03380607434824582924e+01, /* 0x403E568B, 0x261D5190 */ |
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188 sb2 = 3.25792512996573918826e+02, /* 0x40745CAE, 0x221B9F0A */ |
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189 sb3 = 1.53672958608443695994e+03, /* 0x409802EB, 0x189D5118 */ |
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190 sb4 = 3.19985821950859553908e+03, /* 0x40A8FFB7, 0x688C246A */ |
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191 sb5 = 2.55305040643316442583e+03, /* 0x40A3F219, 0xCEDF3BE6 */ |
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192 sb6 = 4.74528541206955367215e+02, /* 0x407DA874, 0xE79FE763 */ |
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193 sb7 = -2.24409524465858183362e+01; /* 0xC03670E2, 0x42712D62 */ |
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194 |
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195 EXPORT_C double |
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196 erf(double x) |
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197 { |
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198 int32_t hx,ix,i; |
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199 double R,S,P,Q,s,y,z,r; |
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200 GET_HIGH_WORD(hx,x); |
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201 ix = hx&0x7fffffff; |
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202 if(ix>=0x7ff00000) { /* erf(nan)=nan */ |
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203 i = ((u_int32_t)hx>>31)<<1; |
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204 return (double)(1-i)+one/x; /* erf(+-inf)=+-1 */ |
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205 } |
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206 |
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207 if(ix < 0x3feb0000) { /* |x|<0.84375 */ |
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208 if(ix < 0x3e300000) { /* |x|<2**-28 */ |
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209 if (ix < 0x00800000) |
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210 return 0.125*(8.0*x+efx8*x); /*avoid underflow */ |
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211 return x + efx*x; |
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212 } |
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213 z = x*x; |
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214 r = pp0+z*(pp1+z*(pp2+z*(pp3+z*pp4))); |
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215 s = one+z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*qq5)))); |
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216 y = r/s; |
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217 return x + x*y; |
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218 } |
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219 if(ix < 0x3ff40000) { /* 0.84375 <= |x| < 1.25 */ |
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220 s = fabs(x)-one; |
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221 P = pa0+s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6))))); |
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222 Q = one+s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6))))); |
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223 if(hx>=0) return erx + P/Q; else return -erx - P/Q; |
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224 } |
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225 if (ix >= 0x40180000) { /* inf>|x|>=6 */ |
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226 if(hx>=0) return one-tiny; else return tiny-one; |
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227 } |
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228 x = fabs(x); |
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229 s = one/(x*x); |
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230 if(ix< 0x4006DB6E) { /* |x| < 1/0.35 */ |
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231 R=ra0+s*(ra1+s*(ra2+s*(ra3+s*(ra4+s*( |
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232 ra5+s*(ra6+s*ra7)))))); |
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233 S=one+s*(sa1+s*(sa2+s*(sa3+s*(sa4+s*( |
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234 sa5+s*(sa6+s*(sa7+s*sa8))))))); |
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235 } else { /* |x| >= 1/0.35 */ |
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236 R=rb0+s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*( |
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237 rb5+s*rb6))))); |
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238 S=one+s*(sb1+s*(sb2+s*(sb3+s*(sb4+s*( |
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239 sb5+s*(sb6+s*sb7)))))); |
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240 } |
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241 z = x; |
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242 SET_LOW_WORD(z,0); |
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243 r = __ieee754_exp(-z*z-0.5625)*__ieee754_exp((z-x)*(z+x)+R/S); |
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244 if(hx>=0) return one-r/x; else return r/x-one; |
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245 } |
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246 |
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247 EXPORT_C double |
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248 erfc(double x) |
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249 { |
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250 int32_t hx,ix; |
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251 double R,S,P,Q,s,y,z,r; |
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252 GET_HIGH_WORD(hx,x); |
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253 ix = hx&0x7fffffff; |
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254 if(ix>=0x7ff00000) { /* erfc(nan)=nan */ |
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255 /* erfc(+-inf)=0,2 */ |
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256 return (double)(((u_int32_t)hx>>31)<<1)+one/x; |
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257 } |
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258 |
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259 if(ix < 0x3feb0000) { /* |x|<0.84375 */ |
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260 if(ix < 0x3c700000) /* |x|<2**-56 */ |
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261 return one-x; |
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262 z = x*x; |
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263 r = pp0+z*(pp1+z*(pp2+z*(pp3+z*pp4))); |
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264 s = one+z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*qq5)))); |
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265 y = r/s; |
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266 if(hx < 0x3fd00000) { /* x<1/4 */ |
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267 return one-(x+x*y); |
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268 } else { |
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269 r = x*y; |
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270 r += (x-half); |
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271 return half - r ; |
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272 } |
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273 } |
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274 if(ix < 0x3ff40000) { /* 0.84375 <= |x| < 1.25 */ |
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275 s = fabs(x)-one; |
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276 P = pa0+s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6))))); |
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277 Q = one+s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6))))); |
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278 if(hx>=0) { |
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279 z = one-erx; return z - P/Q; |
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280 } else { |
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281 z = erx+P/Q; return one+z; |
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282 } |
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283 } |
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284 if (ix < 0x403c0000) { /* |x|<28 */ |
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285 x = fabs(x); |
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286 s = one/(x*x); |
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287 if(ix< 0x4006DB6D) { /* |x| < 1/.35 ~ 2.857143*/ |
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288 R=ra0+s*(ra1+s*(ra2+s*(ra3+s*(ra4+s*( |
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289 ra5+s*(ra6+s*ra7)))))); |
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290 S=one+s*(sa1+s*(sa2+s*(sa3+s*(sa4+s*( |
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291 sa5+s*(sa6+s*(sa7+s*sa8))))))); |
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292 } else { /* |x| >= 1/.35 ~ 2.857143 */ |
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293 if(hx<0&&ix>=0x40180000) return two-tiny;/* x < -6 */ |
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294 R=rb0+s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*( |
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295 rb5+s*rb6))))); |
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296 S=one+s*(sb1+s*(sb2+s*(sb3+s*(sb4+s*( |
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297 sb5+s*(sb6+s*sb7)))))); |
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298 } |
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299 z = x; |
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300 SET_LOW_WORD(z,0); |
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301 r = __ieee754_exp(-z*z-0.5625)* |
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302 __ieee754_exp((z-x)*(z+x)+R/S); |
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303 if(hx>0) return r/x; else return two-r/x; |
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304 } else { |
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305 if(hx>0) return tiny*tiny; else return two-tiny; |
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306 } |
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307 } |