1 | /* ef_j1.c -- float version of e_j1.c. |
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2 | * Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com. |
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3 | */ |
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4 | |
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5 | /* |
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6 | * ==================================================== |
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7 | * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. |
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8 | * |
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9 | * Developed at SunPro, a Sun Microsystems, Inc. business. |
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10 | * Permission to use, copy, modify, and distribute this |
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11 | * software is freely granted, provided that this notice |
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12 | * is preserved. |
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13 | * ==================================================== |
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14 | */ |
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15 | |
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16 | #include "fdlibm.h" |
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17 | |
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18 | #ifdef __STDC__ |
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19 | static float ponef(float), qonef(float); |
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20 | #else |
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21 | static float ponef(), qonef(); |
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22 | #endif |
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23 | |
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24 | #ifdef __STDC__ |
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25 | static const float |
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26 | #else |
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27 | static float |
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28 | #endif |
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29 | huge = 1e30, |
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30 | one = 1.0, |
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31 | invsqrtpi= 5.6418961287e-01, /* 0x3f106ebb */ |
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32 | tpi = 6.3661974669e-01, /* 0x3f22f983 */ |
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33 | /* R0/S0 on [0,2] */ |
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34 | r00 = -6.2500000000e-02, /* 0xbd800000 */ |
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35 | r01 = 1.4070566976e-03, /* 0x3ab86cfd */ |
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36 | r02 = -1.5995563444e-05, /* 0xb7862e36 */ |
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37 | r03 = 4.9672799207e-08, /* 0x335557d2 */ |
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38 | s01 = 1.9153760746e-02, /* 0x3c9ce859 */ |
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39 | s02 = 1.8594678841e-04, /* 0x3942fab6 */ |
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40 | s03 = 1.1771846857e-06, /* 0x359dffc2 */ |
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41 | s04 = 5.0463624390e-09, /* 0x31ad6446 */ |
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42 | s05 = 1.2354227016e-11; /* 0x2d59567e */ |
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43 | |
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44 | #ifdef __STDC__ |
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45 | static const float zero = 0.0; |
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46 | #else |
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47 | static float zero = 0.0; |
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48 | #endif |
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49 | |
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50 | #ifdef __STDC__ |
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51 | float j1f(float x) |
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52 | #else |
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53 | float j1f(x) |
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54 | float x; |
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55 | #endif |
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56 | { |
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57 | float z, s,c,ss,cc,r,u,v,y; |
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58 | __int32_t hx,ix; |
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59 | |
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60 | GET_FLOAT_WORD(hx,x); |
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61 | ix = hx&0x7fffffff; |
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62 | if(ix>=0x7f800000) return one/x; |
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63 | y = fabsf(x); |
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64 | if(ix >= 0x40000000) { /* |x| >= 2.0 */ |
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65 | s = sinf(y); |
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66 | c = cosf(y); |
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67 | ss = -s-c; |
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68 | cc = s-c; |
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69 | if(ix<0x7f000000) { /* make sure y+y not overflow */ |
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70 | z = cosf(y+y); |
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71 | if ((s*c)>zero) cc = z/ss; |
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72 | else ss = z/cc; |
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73 | } |
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74 | /* |
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75 | * j1(x) = 1/sqrt(pi) * (P(1,x)*cc - Q(1,x)*ss) / sqrt(x) |
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76 | * y1(x) = 1/sqrt(pi) * (P(1,x)*ss + Q(1,x)*cc) / sqrt(x) |
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77 | */ |
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78 | if(ix>0x80000000) z = (invsqrtpi*cc)/sqrtf(y); |
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79 | else { |
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80 | u = ponef(y); v = qonef(y); |
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81 | z = invsqrtpi*(u*cc-v*ss)/sqrtf(y); |
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82 | } |
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83 | if(hx<0) return -z; |
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84 | else return z; |
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85 | } |
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86 | if(ix<0x32000000) { /* |x|<2**-27 */ |
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87 | if(huge+x>one) return (float)0.5*x;/* inexact if x!=0 necessary */ |
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88 | } |
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89 | z = x*x; |
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90 | r = z*(r00+z*(r01+z*(r02+z*r03))); |
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91 | s = one+z*(s01+z*(s02+z*(s03+z*(s04+z*s05)))); |
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92 | r *= x; |
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93 | return(x*(float)0.5+r/s); |
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94 | } |
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95 | |
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96 | #ifdef __STDC__ |
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97 | static const float U0[5] = { |
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98 | #else |
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99 | static float U0[5] = { |
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100 | #endif |
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101 | -1.9605709612e-01, /* 0xbe48c331 */ |
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102 | 5.0443872809e-02, /* 0x3d4e9e3c */ |
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103 | -1.9125689287e-03, /* 0xbafaaf2a */ |
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104 | 2.3525259166e-05, /* 0x37c5581c */ |
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105 | -9.1909917899e-08, /* 0xb3c56003 */ |
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106 | }; |
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107 | #ifdef __STDC__ |
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108 | static const float V0[5] = { |
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109 | #else |
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110 | static float V0[5] = { |
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111 | #endif |
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112 | 1.9916731864e-02, /* 0x3ca3286a */ |
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113 | 2.0255257550e-04, /* 0x3954644b */ |
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114 | 1.3560879779e-06, /* 0x35b602d4 */ |
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115 | 6.2274145840e-09, /* 0x31d5f8eb */ |
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116 | 1.6655924903e-11, /* 0x2d9281cf */ |
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117 | }; |
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118 | |
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119 | #ifdef __STDC__ |
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120 | float y1f(float x) |
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121 | #else |
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122 | float y1f(x) |
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123 | float x; |
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124 | #endif |
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125 | { |
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126 | float z, s,c,ss,cc,u,v; |
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127 | __int32_t hx,ix; |
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128 | |
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129 | GET_FLOAT_WORD(hx,x); |
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130 | ix = 0x7fffffff&hx; |
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131 | /* if Y1(NaN) is NaN, Y1(-inf) is NaN, Y1(inf) is 0 */ |
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132 | if(ix>=0x7f800000) return one/(x+x*x); |
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133 | if(ix==0) return -one/zero; |
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134 | if(hx<0) return zero/zero; |
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135 | if(ix >= 0x40000000) { /* |x| >= 2.0 */ |
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136 | s = sinf(x); |
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137 | c = cosf(x); |
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138 | ss = -s-c; |
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139 | cc = s-c; |
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140 | if(ix<0x7f000000) { /* make sure x+x not overflow */ |
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141 | z = cosf(x+x); |
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142 | if ((s*c)>zero) cc = z/ss; |
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143 | else ss = z/cc; |
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144 | } |
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145 | /* y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x0)+q1(x)*cos(x0)) |
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146 | * where x0 = x-3pi/4 |
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147 | * Better formula: |
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148 | * cos(x0) = cos(x)cos(3pi/4)+sin(x)sin(3pi/4) |
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149 | * = 1/sqrt(2) * (sin(x) - cos(x)) |
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150 | * sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4) |
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151 | * = -1/sqrt(2) * (cos(x) + sin(x)) |
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152 | * To avoid cancellation, use |
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153 | * sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x)) |
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154 | * to compute the worse one. |
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155 | */ |
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156 | if(ix>0x48000000) z = (invsqrtpi*ss)/sqrtf(x); |
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157 | else { |
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158 | u = ponef(x); v = qonef(x); |
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159 | z = invsqrtpi*(u*ss+v*cc)/sqrtf(x); |
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160 | } |
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161 | return z; |
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162 | } |
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163 | if(ix<=0x24800000) { /* x < 2**-54 */ |
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164 | return(-tpi/x); |
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165 | } |
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166 | z = x*x; |
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167 | u = U0[0]+z*(U0[1]+z*(U0[2]+z*(U0[3]+z*U0[4]))); |
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168 | v = one+z*(V0[0]+z*(V0[1]+z*(V0[2]+z*(V0[3]+z*V0[4])))); |
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169 | return(x*(u/v) + tpi*(j1f(x)*logf(x)-one/x)); |
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170 | } |
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171 | |
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172 | /* For x >= 8, the asymptotic expansions of pone is |
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173 | * 1 + 15/128 s^2 - 4725/2^15 s^4 - ..., where s = 1/x. |
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174 | * We approximate pone by |
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175 | * pone(x) = 1 + (R/S) |
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176 | * where R = pr0 + pr1*s^2 + pr2*s^4 + ... + pr5*s^10 |
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177 | * S = 1 + ps0*s^2 + ... + ps4*s^10 |
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178 | * and |
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179 | * | pone(x)-1-R/S | <= 2 ** ( -60.06) |
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180 | */ |
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181 | |
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182 | #ifdef __STDC__ |
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183 | static const float pr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */ |
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184 | #else |
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185 | static float pr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */ |
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186 | #endif |
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187 | 0.0000000000e+00, /* 0x00000000 */ |
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188 | 1.1718750000e-01, /* 0x3df00000 */ |
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189 | 1.3239480972e+01, /* 0x4153d4ea */ |
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190 | 4.1205184937e+02, /* 0x43ce06a3 */ |
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191 | 3.8747453613e+03, /* 0x45722bed */ |
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192 | 7.9144794922e+03, /* 0x45f753d6 */ |
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193 | }; |
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194 | #ifdef __STDC__ |
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195 | static const float ps8[5] = { |
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196 | #else |
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197 | static float ps8[5] = { |
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198 | #endif |
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199 | 1.1420736694e+02, /* 0x42e46a2c */ |
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200 | 3.6509309082e+03, /* 0x45642ee5 */ |
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201 | 3.6956207031e+04, /* 0x47105c35 */ |
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202 | 9.7602796875e+04, /* 0x47bea166 */ |
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203 | 3.0804271484e+04, /* 0x46f0a88b */ |
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204 | }; |
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205 | |
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206 | #ifdef __STDC__ |
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207 | static const float pr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */ |
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208 | #else |
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209 | static float pr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */ |
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210 | #endif |
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211 | 1.3199052094e-11, /* 0x2d68333f */ |
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212 | 1.1718749255e-01, /* 0x3defffff */ |
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213 | 6.8027510643e+00, /* 0x40d9b023 */ |
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214 | 1.0830818176e+02, /* 0x42d89dca */ |
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215 | 5.1763616943e+02, /* 0x440168b7 */ |
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216 | 5.2871520996e+02, /* 0x44042dc6 */ |
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217 | }; |
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218 | #ifdef __STDC__ |
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219 | static const float ps5[5] = { |
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220 | #else |
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221 | static float ps5[5] = { |
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222 | #endif |
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223 | 5.9280597687e+01, /* 0x426d1f55 */ |
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224 | 9.9140142822e+02, /* 0x4477d9b1 */ |
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225 | 5.3532670898e+03, /* 0x45a74a23 */ |
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226 | 7.8446904297e+03, /* 0x45f52586 */ |
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227 | 1.5040468750e+03, /* 0x44bc0180 */ |
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228 | }; |
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229 | |
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230 | #ifdef __STDC__ |
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231 | static const float pr3[6] = { |
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232 | #else |
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233 | static float pr3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */ |
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234 | #endif |
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235 | 3.0250391081e-09, /* 0x314fe10d */ |
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236 | 1.1718686670e-01, /* 0x3defffab */ |
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237 | 3.9329774380e+00, /* 0x407bb5e7 */ |
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238 | 3.5119403839e+01, /* 0x420c7a45 */ |
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239 | 9.1055007935e+01, /* 0x42b61c2a */ |
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240 | 4.8559066772e+01, /* 0x42423c7c */ |
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241 | }; |
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242 | #ifdef __STDC__ |
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243 | static const float ps3[5] = { |
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244 | #else |
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245 | static float ps3[5] = { |
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246 | #endif |
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247 | 3.4791309357e+01, /* 0x420b2a4d */ |
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248 | 3.3676245117e+02, /* 0x43a86198 */ |
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249 | 1.0468714600e+03, /* 0x4482dbe3 */ |
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250 | 8.9081134033e+02, /* 0x445eb3ed */ |
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251 | 1.0378793335e+02, /* 0x42cf936c */ |
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252 | }; |
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253 | |
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254 | #ifdef __STDC__ |
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255 | static const float pr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */ |
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256 | #else |
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257 | static float pr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */ |
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258 | #endif |
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259 | 1.0771083225e-07, /* 0x33e74ea8 */ |
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260 | 1.1717621982e-01, /* 0x3deffa16 */ |
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261 | 2.3685150146e+00, /* 0x401795c0 */ |
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262 | 1.2242610931e+01, /* 0x4143e1bc */ |
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263 | 1.7693971634e+01, /* 0x418d8d41 */ |
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264 | 5.0735230446e+00, /* 0x40a25a4d */ |
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265 | }; |
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266 | #ifdef __STDC__ |
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267 | static const float ps2[5] = { |
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268 | #else |
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269 | static float ps2[5] = { |
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270 | #endif |
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271 | 2.1436485291e+01, /* 0x41ab7dec */ |
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272 | 1.2529022980e+02, /* 0x42fa9499 */ |
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273 | 2.3227647400e+02, /* 0x436846c7 */ |
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274 | 1.1767937469e+02, /* 0x42eb5bd7 */ |
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275 | 8.3646392822e+00, /* 0x4105d590 */ |
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276 | }; |
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277 | |
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278 | #ifdef __STDC__ |
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279 | static float ponef(float x) |
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280 | #else |
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281 | static float ponef(x) |
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282 | float x; |
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283 | #endif |
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284 | { |
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285 | #ifdef __STDC__ |
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286 | const float *p,*q; |
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287 | #else |
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288 | float *p,*q; |
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289 | #endif |
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290 | float z,r,s; |
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291 | __int32_t ix; |
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292 | GET_FLOAT_WORD(ix,x); |
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293 | ix &= 0x7fffffff; |
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294 | if(ix>=0x41000000) {p = pr8; q= ps8;} |
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295 | else if(ix>=0x40f71c58){p = pr5; q= ps5;} |
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296 | else if(ix>=0x4036db68){p = pr3; q= ps3;} |
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297 | else {p = pr2; q= ps2;} |
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298 | z = one/(x*x); |
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299 | r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5])))); |
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300 | s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*q[4])))); |
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301 | return one+ r/s; |
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302 | } |
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303 | |
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304 | |
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305 | /* For x >= 8, the asymptotic expansions of qone is |
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306 | * 3/8 s - 105/1024 s^3 - ..., where s = 1/x. |
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307 | * We approximate qone by |
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308 | * qone(x) = s*(0.375 + (R/S)) |
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309 | * where R = qr1*s^2 + qr2*s^4 + ... + qr5*s^10 |
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310 | * S = 1 + qs1*s^2 + ... + qs6*s^12 |
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311 | * and |
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312 | * | qone(x)/s -0.375-R/S | <= 2 ** ( -61.13) |
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313 | */ |
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314 | |
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315 | #ifdef __STDC__ |
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316 | static const float qr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */ |
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317 | #else |
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318 | static float qr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */ |
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319 | #endif |
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320 | 0.0000000000e+00, /* 0x00000000 */ |
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321 | -1.0253906250e-01, /* 0xbdd20000 */ |
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322 | -1.6271753311e+01, /* 0xc1822c8d */ |
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323 | -7.5960174561e+02, /* 0xc43de683 */ |
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324 | -1.1849806641e+04, /* 0xc639273a */ |
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325 | -4.8438511719e+04, /* 0xc73d3683 */ |
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326 | }; |
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327 | #ifdef __STDC__ |
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328 | static const float qs8[6] = { |
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329 | #else |
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330 | static float qs8[6] = { |
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331 | #endif |
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332 | 1.6139537048e+02, /* 0x43216537 */ |
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333 | 7.8253862305e+03, /* 0x45f48b17 */ |
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334 | 1.3387534375e+05, /* 0x4802bcd6 */ |
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335 | 7.1965775000e+05, /* 0x492fb29c */ |
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336 | 6.6660125000e+05, /* 0x4922be94 */ |
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337 | -2.9449025000e+05, /* 0xc88fcb48 */ |
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338 | }; |
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339 | |
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340 | #ifdef __STDC__ |
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341 | static const float qr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */ |
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342 | #else |
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343 | static float qr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */ |
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344 | #endif |
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345 | -2.0897993405e-11, /* 0xadb7d219 */ |
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346 | -1.0253904760e-01, /* 0xbdd1fffe */ |
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347 | -8.0564479828e+00, /* 0xc100e736 */ |
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348 | -1.8366960144e+02, /* 0xc337ab6b */ |
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349 | -1.3731937256e+03, /* 0xc4aba633 */ |
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350 | -2.6124443359e+03, /* 0xc523471c */ |
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351 | }; |
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352 | #ifdef __STDC__ |
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353 | static const float qs5[6] = { |
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354 | #else |
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355 | static float qs5[6] = { |
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356 | #endif |
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357 | 8.1276550293e+01, /* 0x42a28d98 */ |
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358 | 1.9917987061e+03, /* 0x44f8f98f */ |
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359 | 1.7468484375e+04, /* 0x468878f8 */ |
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360 | 4.9851425781e+04, /* 0x4742bb6d */ |
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361 | 2.7948074219e+04, /* 0x46da5826 */ |
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362 | -4.7191835938e+03, /* 0xc5937978 */ |
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363 | }; |
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364 | |
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365 | #ifdef __STDC__ |
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366 | static const float qr3[6] = { |
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367 | #else |
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368 | static float qr3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */ |
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369 | #endif |
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370 | -5.0783124372e-09, /* 0xb1ae7d4f */ |
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371 | -1.0253783315e-01, /* 0xbdd1ff5b */ |
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372 | -4.6101160049e+00, /* 0xc0938612 */ |
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373 | -5.7847221375e+01, /* 0xc267638e */ |
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374 | -2.2824453735e+02, /* 0xc3643e9a */ |
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375 | -2.1921012878e+02, /* 0xc35b35cb */ |
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376 | }; |
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377 | #ifdef __STDC__ |
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378 | static const float qs3[6] = { |
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379 | #else |
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380 | static float qs3[6] = { |
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381 | #endif |
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382 | 4.7665153503e+01, /* 0x423ea91e */ |
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383 | 6.7386511230e+02, /* 0x4428775e */ |
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384 | 3.3801528320e+03, /* 0x45534272 */ |
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385 | 5.5477290039e+03, /* 0x45ad5dd5 */ |
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386 | 1.9031191406e+03, /* 0x44ede3d0 */ |
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387 | -1.3520118713e+02, /* 0xc3073381 */ |
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388 | }; |
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389 | |
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390 | #ifdef __STDC__ |
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391 | static const float qr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */ |
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392 | #else |
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393 | static float qr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */ |
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394 | #endif |
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395 | -1.7838172539e-07, /* 0xb43f8932 */ |
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396 | -1.0251704603e-01, /* 0xbdd1f475 */ |
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397 | -2.7522056103e+00, /* 0xc0302423 */ |
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398 | -1.9663616180e+01, /* 0xc19d4f16 */ |
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399 | -4.2325313568e+01, /* 0xc2294d1f */ |
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400 | -2.1371921539e+01, /* 0xc1aaf9b2 */ |
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401 | }; |
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402 | #ifdef __STDC__ |
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403 | static const float qs2[6] = { |
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404 | #else |
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405 | static float qs2[6] = { |
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406 | #endif |
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407 | 2.9533363342e+01, /* 0x41ec4454 */ |
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408 | 2.5298155212e+02, /* 0x437cfb47 */ |
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409 | 7.5750280762e+02, /* 0x443d602e */ |
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410 | 7.3939318848e+02, /* 0x4438d92a */ |
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411 | 1.5594900513e+02, /* 0x431bf2f2 */ |
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412 | -4.9594988823e+00, /* 0xc09eb437 */ |
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413 | }; |
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414 | |
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415 | #ifdef __STDC__ |
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416 | static float qonef(float x) |
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417 | #else |
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418 | static float qonef(x) |
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419 | float x; |
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420 | #endif |
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421 | { |
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422 | #ifdef __STDC__ |
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423 | const float *p,*q; |
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424 | #else |
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425 | float *p,*q; |
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426 | #endif |
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427 | float s,r,z; |
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428 | __int32_t ix; |
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429 | GET_FLOAT_WORD(ix,x); |
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430 | ix &= 0x7fffffff; |
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431 | if(ix>=0x40200000) {p = qr8; q= qs8;} |
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432 | else if(ix>=0x40f71c58){p = qr5; q= qs5;} |
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433 | else if(ix>=0x4036db68){p = qr3; q= qs3;} |
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434 | else {p = qr2; q= qs2;} |
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435 | z = one/(x*x); |
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436 | r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5])))); |
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437 | s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*(q[4]+z*q[5]))))); |
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438 | return ((float).375 + r/s)/x; |
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439 | } |
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