1 | /* ef_jn.c -- float version of e_jn.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 const float |
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20 | #else |
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21 | static float |
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22 | #endif |
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23 | invsqrtpi= 5.6418961287e-01, /* 0x3f106ebb */ |
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24 | two = 2.0000000000e+00, /* 0x40000000 */ |
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25 | one = 1.0000000000e+00; /* 0x3F800000 */ |
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26 | |
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27 | #ifdef __STDC__ |
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28 | static const float zero = 0.0000000000e+00; |
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29 | #else |
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30 | static float zero = 0.0000000000e+00; |
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31 | #endif |
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32 | |
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33 | #ifdef __STDC__ |
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34 | float __ieee754_jnf(int n, float x) |
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35 | #else |
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36 | float __ieee754_jnf(n,x) |
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37 | int n; float x; |
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38 | #endif |
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39 | { |
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40 | __int32_t i,hx,ix, sgn; |
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41 | float a, b, temp, di; |
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42 | float z, w; |
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43 | |
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44 | /* J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x) |
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45 | * Thus, J(-n,x) = J(n,-x) |
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46 | */ |
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47 | GET_FLOAT_WORD(hx,x); |
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48 | ix = 0x7fffffff&hx; |
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49 | /* if J(n,NaN) is NaN */ |
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50 | if(FLT_UWORD_IS_NAN(ix)) return x+x; |
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51 | if(n<0){ |
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52 | n = -n; |
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53 | x = -x; |
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54 | hx ^= 0x80000000; |
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55 | } |
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56 | if(n==0) return(__ieee754_j0f(x)); |
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57 | if(n==1) return(__ieee754_j1f(x)); |
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58 | sgn = (n&1)&(hx>>31); /* even n -- 0, odd n -- sign(x) */ |
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59 | x = fabsf(x); |
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60 | if(FLT_UWORD_IS_ZERO(ix)||FLT_UWORD_IS_INFINITE(ix)) |
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61 | b = zero; |
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62 | else if((float)n<=x) { |
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63 | /* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */ |
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64 | a = __ieee754_j0f(x); |
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65 | b = __ieee754_j1f(x); |
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66 | for(i=1;i<n;i++){ |
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67 | temp = b; |
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68 | b = b*((float)(i+i)/x) - a; /* avoid underflow */ |
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69 | a = temp; |
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70 | } |
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71 | } else { |
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72 | if(ix<0x30800000) { /* x < 2**-29 */ |
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73 | /* x is tiny, return the first Taylor expansion of J(n,x) |
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74 | * J(n,x) = 1/n!*(x/2)^n - ... |
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75 | */ |
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76 | if(n>33) /* underflow */ |
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77 | b = zero; |
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78 | else { |
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79 | temp = x*(float)0.5; b = temp; |
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80 | for (a=one,i=2;i<=n;i++) { |
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81 | a *= (float)i; /* a = n! */ |
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82 | b *= temp; /* b = (x/2)^n */ |
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83 | } |
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84 | b = b/a; |
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85 | } |
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86 | } else { |
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87 | /* use backward recurrence */ |
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88 | /* x x^2 x^2 |
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89 | * J(n,x)/J(n-1,x) = ---- ------ ------ ..... |
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90 | * 2n - 2(n+1) - 2(n+2) |
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91 | * |
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92 | * 1 1 1 |
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93 | * (for large x) = ---- ------ ------ ..... |
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94 | * 2n 2(n+1) 2(n+2) |
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95 | * -- - ------ - ------ - |
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96 | * x x x |
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97 | * |
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98 | * Let w = 2n/x and h=2/x, then the above quotient |
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99 | * is equal to the continued fraction: |
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100 | * 1 |
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101 | * = ----------------------- |
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102 | * 1 |
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103 | * w - ----------------- |
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104 | * 1 |
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105 | * w+h - --------- |
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106 | * w+2h - ... |
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107 | * |
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108 | * To determine how many terms needed, let |
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109 | * Q(0) = w, Q(1) = w(w+h) - 1, |
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110 | * Q(k) = (w+k*h)*Q(k-1) - Q(k-2), |
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111 | * When Q(k) > 1e4 good for single |
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112 | * When Q(k) > 1e9 good for double |
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113 | * When Q(k) > 1e17 good for quadruple |
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114 | */ |
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115 | /* determine k */ |
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116 | float t,v; |
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117 | float q0,q1,h,tmp; __int32_t k,m; |
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118 | w = (n+n)/(float)x; h = (float)2.0/(float)x; |
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119 | q0 = w; z = w+h; q1 = w*z - (float)1.0; k=1; |
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120 | while(q1<(float)1.0e9) { |
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121 | k += 1; z += h; |
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122 | tmp = z*q1 - q0; |
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123 | q0 = q1; |
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124 | q1 = tmp; |
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125 | } |
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126 | m = n+n; |
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127 | for(t=zero, i = 2*(n+k); i>=m; i -= 2) t = one/(i/x-t); |
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128 | a = t; |
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129 | b = one; |
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130 | /* estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n) |
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131 | * Hence, if n*(log(2n/x)) > ... |
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132 | * single 8.8722839355e+01 |
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133 | * double 7.09782712893383973096e+02 |
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134 | * long double 1.1356523406294143949491931077970765006170e+04 |
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135 | * then recurrent value may overflow and the result is |
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136 | * likely underflow to zero |
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137 | */ |
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138 | tmp = n; |
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139 | v = two/x; |
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140 | tmp = tmp*__ieee754_logf(fabsf(v*tmp)); |
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141 | if(tmp<(float)8.8721679688e+01) { |
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142 | for(i=n-1,di=(float)(i+i);i>0;i--){ |
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143 | temp = b; |
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144 | b *= di; |
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145 | b = b/x - a; |
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146 | a = temp; |
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147 | di -= two; |
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148 | } |
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149 | } else { |
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150 | for(i=n-1,di=(float)(i+i);i>0;i--){ |
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151 | temp = b; |
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152 | b *= di; |
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153 | b = b/x - a; |
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154 | a = temp; |
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155 | di -= two; |
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156 | /* scale b to avoid spurious overflow */ |
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157 | if(b>(float)1e10) { |
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158 | a /= b; |
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159 | t /= b; |
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160 | b = one; |
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161 | } |
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162 | } |
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163 | } |
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164 | b = (t*__ieee754_j0f(x)/b); |
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165 | } |
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166 | } |
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167 | if(sgn==1) return -b; else return b; |
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168 | } |
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169 | |
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170 | #ifdef __STDC__ |
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171 | float __ieee754_ynf(int n, float x) |
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172 | #else |
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173 | float __ieee754_ynf(n,x) |
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174 | int n; float x; |
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175 | #endif |
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176 | { |
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177 | __int32_t i,hx,ix,ib; |
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178 | __int32_t sign; |
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179 | float a, b, temp; |
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180 | |
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181 | GET_FLOAT_WORD(hx,x); |
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182 | ix = 0x7fffffff&hx; |
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183 | /* if Y(n,NaN) is NaN */ |
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184 | if(FLT_UWORD_IS_NAN(ix)) return x+x; |
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185 | if(FLT_UWORD_IS_ZERO(ix)) return -one/zero; |
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186 | if(hx<0) return zero/zero; |
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187 | sign = 1; |
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188 | if(n<0){ |
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189 | n = -n; |
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190 | sign = 1 - ((n&1)<<1); |
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191 | } |
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192 | if(n==0) return(__ieee754_y0f(x)); |
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193 | if(n==1) return(sign*__ieee754_y1f(x)); |
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194 | if(FLT_UWORD_IS_INFINITE(ix)) return zero; |
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195 | |
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196 | a = __ieee754_y0f(x); |
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197 | b = __ieee754_y1f(x); |
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198 | /* quit if b is -inf */ |
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199 | GET_FLOAT_WORD(ib,b); |
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200 | for(i=1;i<n&&ib!=0xff800000;i++){ |
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201 | temp = b; |
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202 | b = ((float)(i+i)/x)*b - a; |
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203 | GET_FLOAT_WORD(ib,b); |
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204 | a = temp; |
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205 | } |
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206 | if(sign>0) return b; else return -b; |
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207 | } |
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