1 | /* origin: FreeBSD /usr/src/lib/msun/src/e_pow.c */
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2 | /*
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3 | * ====================================================
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4 | * Copyright (C) 2004 by Sun Microsystems, Inc. All rights reserved.
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5 | *
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6 | * Permission to use, copy, modify, and distribute this
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7 | * software is freely granted, provided that this notice
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8 | * is preserved.
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9 | * ====================================================
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10 | */
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11 | /* pow(x,y) return x**y
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12 | *
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13 | * n
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14 | * Method: Let x = 2 * (1+f)
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15 | * 1. Compute and return log2(x) in two pieces:
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16 | * log2(x) = w1 + w2,
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17 | * where w1 has 53-24 = 29 bit trailing zeros.
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18 | * 2. Perform y*log2(x) = n+y' by simulating muti-precision
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19 | * arithmetic, where |y'|<=0.5.
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20 | * 3. Return x**y = 2**n*exp(y'*log2)
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21 | *
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22 | * Special cases:
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23 | * 1. (anything) ** 0 is 1
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24 | * 2. 1 ** (anything) is 1
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25 | * 3. (anything except 1) ** NAN is NAN
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26 | * 4. NAN ** (anything except 0) is NAN
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27 | * 5. +-(|x| > 1) ** +INF is +INF
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28 | * 6. +-(|x| > 1) ** -INF is +0
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29 | * 7. +-(|x| < 1) ** +INF is +0
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30 | * 8. +-(|x| < 1) ** -INF is +INF
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31 | * 9. -1 ** +-INF is 1
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32 | * 10. +0 ** (+anything except 0, NAN) is +0
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33 | * 11. -0 ** (+anything except 0, NAN, odd integer) is +0
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34 | * 12. +0 ** (-anything except 0, NAN) is +INF, raise divbyzero
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35 | * 13. -0 ** (-anything except 0, NAN, odd integer) is +INF, raise divbyzero
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36 | * 14. -0 ** (+odd integer) is -0
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37 | * 15. -0 ** (-odd integer) is -INF, raise divbyzero
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38 | * 16. +INF ** (+anything except 0,NAN) is +INF
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39 | * 17. +INF ** (-anything except 0,NAN) is +0
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40 | * 18. -INF ** (+odd integer) is -INF
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41 | * 19. -INF ** (anything) = -0 ** (-anything), (anything except odd integer)
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42 | * 20. (anything) ** 1 is (anything)
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43 | * 21. (anything) ** -1 is 1/(anything)
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44 | * 22. (-anything) ** (integer) is (-1)**(integer)*(+anything**integer)
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45 | * 23. (-anything except 0 and inf) ** (non-integer) is NAN
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46 | *
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47 | * Accuracy:
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48 | * pow(x,y) returns x**y nearly rounded. In particular
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49 | * pow(integer,integer)
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50 | * always returns the correct integer provided it is
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51 | * representable.
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52 | *
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53 | * Constants :
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54 | * The hexadecimal values are the intended ones for the following
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55 | * constants. The decimal values may be used, provided that the
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56 | * compiler will convert from decimal to binary accurately enough
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57 | * to produce the hexadecimal values shown.
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58 | */
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59 |
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60 | #include "libm.h"
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61 |
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62 | static const double
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63 | bp[] = {1.0, 1.5,},
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64 | dp_h[] = { 0.0, 5.84962487220764160156e-01,}, /* 0x3FE2B803, 0x40000000 */
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65 | dp_l[] = { 0.0, 1.35003920212974897128e-08,}, /* 0x3E4CFDEB, 0x43CFD006 */
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66 | two53 = 9007199254740992.0, /* 0x43400000, 0x00000000 */
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67 | huge = 1.0e300,
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68 | tiny = 1.0e-300,
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69 | /* poly coefs for (3/2)*(log(x)-2s-2/3*s**3 */
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70 | L1 = 5.99999999999994648725e-01, /* 0x3FE33333, 0x33333303 */
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71 | L2 = 4.28571428578550184252e-01, /* 0x3FDB6DB6, 0xDB6FABFF */
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72 | L3 = 3.33333329818377432918e-01, /* 0x3FD55555, 0x518F264D */
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73 | L4 = 2.72728123808534006489e-01, /* 0x3FD17460, 0xA91D4101 */
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74 | L5 = 2.30660745775561754067e-01, /* 0x3FCD864A, 0x93C9DB65 */
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75 | L6 = 2.06975017800338417784e-01, /* 0x3FCA7E28, 0x4A454EEF */
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76 | P1 = 1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */
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77 | P2 = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */
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78 | P3 = 6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */
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79 | P4 = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */
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80 | P5 = 4.13813679705723846039e-08, /* 0x3E663769, 0x72BEA4D0 */
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81 | lg2 = 6.93147180559945286227e-01, /* 0x3FE62E42, 0xFEFA39EF */
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82 | lg2_h = 6.93147182464599609375e-01, /* 0x3FE62E43, 0x00000000 */
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83 | lg2_l = -1.90465429995776804525e-09, /* 0xBE205C61, 0x0CA86C39 */
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84 | ovt = 8.0085662595372944372e-017, /* -(1024-log2(ovfl+.5ulp)) */
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85 | cp = 9.61796693925975554329e-01, /* 0x3FEEC709, 0xDC3A03FD =2/(3ln2) */
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86 | cp_h = 9.61796700954437255859e-01, /* 0x3FEEC709, 0xE0000000 =(float)cp */
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87 | cp_l = -7.02846165095275826516e-09, /* 0xBE3E2FE0, 0x145B01F5 =tail of cp_h*/
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88 | ivln2 = 1.44269504088896338700e+00, /* 0x3FF71547, 0x652B82FE =1/ln2 */
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89 | ivln2_h = 1.44269502162933349609e+00, /* 0x3FF71547, 0x60000000 =24b 1/ln2*/
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90 | ivln2_l = 1.92596299112661746887e-08; /* 0x3E54AE0B, 0xF85DDF44 =1/ln2 tail*/
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91 |
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92 | double pow(double x, double y)
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93 | {
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94 | double z,ax,z_h,z_l,p_h,p_l;
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95 | double y1,t1,t2,r,s,t,u,v,w;
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96 | int32_t i,j,k,yisint,n;
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97 | int32_t hx,hy,ix,iy;
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98 | uint32_t lx,ly;
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99 |
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100 | EXTRACT_WORDS(hx, lx, x);
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101 | EXTRACT_WORDS(hy, ly, y);
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102 | ix = hx & 0x7fffffff;
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103 | iy = hy & 0x7fffffff;
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104 |
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105 | /* x**0 = 1, even if x is NaN */
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106 | if ((iy|ly) == 0)
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107 | return 1.0;
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108 | /* 1**y = 1, even if y is NaN */
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109 | if (hx == 0x3ff00000 && lx == 0)
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110 | return 1.0;
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111 | /* NaN if either arg is NaN */
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112 | if (ix > 0x7ff00000 || (ix == 0x7ff00000 && lx != 0) ||
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113 | iy > 0x7ff00000 || (iy == 0x7ff00000 && ly != 0))
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114 | return x + y;
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115 |
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116 | /* determine if y is an odd int when x < 0
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117 | * yisint = 0 ... y is not an integer
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118 | * yisint = 1 ... y is an odd int
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119 | * yisint = 2 ... y is an even int
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120 | */
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121 | yisint = 0;
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122 | if (hx < 0) {
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123 | if (iy >= 0x43400000)
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124 | yisint = 2; /* even integer y */
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125 | else if (iy >= 0x3ff00000) {
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126 | k = (iy>>20) - 0x3ff; /* exponent */
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127 | if (k > 20) {
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128 | uint32_t j = ly>>(52-k);
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129 | if ((j<<(52-k)) == ly)
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130 | yisint = 2 - (j&1);
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131 | } else if (ly == 0) {
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132 | uint32_t j = iy>>(20-k);
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133 | if ((j<<(20-k)) == iy)
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134 | yisint = 2 - (j&1);
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135 | }
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136 | }
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137 | }
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138 |
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139 | /* special value of y */
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140 | if (ly == 0) {
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141 | if (iy == 0x7ff00000) { /* y is +-inf */
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142 | if (((ix-0x3ff00000)|lx) == 0) /* (-1)**+-inf is 1 */
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143 | return 1.0;
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144 | else if (ix >= 0x3ff00000) /* (|x|>1)**+-inf = inf,0 */
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145 | return hy >= 0 ? y : 0.0;
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146 | else /* (|x|<1)**+-inf = 0,inf */
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147 | return hy >= 0 ? 0.0 : -y;
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148 | }
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149 | if (iy == 0x3ff00000) { /* y is +-1 */
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150 | if (hy >= 0)
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151 | return x;
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152 | y = 1/x;
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153 | #if FLT_EVAL_METHOD!=0
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154 | {
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155 | union {double f; uint64_t i;} u = {y};
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156 | uint64_t i = u.i & -1ULL/2;
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157 | if (i>>52 == 0 && (i&(i-1)))
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158 | FORCE_EVAL((float)y);
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159 | }
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160 | #endif
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161 | return y;
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162 | }
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163 | if (hy == 0x40000000) /* y is 2 */
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164 | return x*x;
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165 | if (hy == 0x3fe00000) { /* y is 0.5 */
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166 | if (hx >= 0) /* x >= +0 */
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167 | return sqrt(x);
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168 | }
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169 | }
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170 |
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171 | ax = fabs(x);
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172 | /* special value of x */
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173 | if (lx == 0) {
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174 | if (ix == 0x7ff00000 || ix == 0 || ix == 0x3ff00000) { /* x is +-0,+-inf,+-1 */
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175 | z = ax;
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176 | if (hy < 0) /* z = (1/|x|) */
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177 | z = 1.0/z;
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178 | if (hx < 0) {
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179 | if (((ix-0x3ff00000)|yisint) == 0) {
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180 | z = (z-z)/(z-z); /* (-1)**non-int is NaN */
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181 | } else if (yisint == 1)
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182 | z = -z; /* (x<0)**odd = -(|x|**odd) */
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183 | }
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184 | return z;
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185 | }
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186 | }
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187 |
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188 | s = 1.0; /* sign of result */
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189 | if (hx < 0) {
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190 | if (yisint == 0) /* (x<0)**(non-int) is NaN */
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191 | return (x-x)/(x-x);
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192 | if (yisint == 1) /* (x<0)**(odd int) */
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193 | s = -1.0;
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194 | }
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195 |
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196 | /* |y| is huge */
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197 | if (iy > 0x41e00000) { /* if |y| > 2**31 */
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198 | if (iy > 0x43f00000) { /* if |y| > 2**64, must o/uflow */
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199 | if (ix <= 0x3fefffff)
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200 | return hy < 0 ? huge*huge : tiny*tiny;
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201 | if (ix >= 0x3ff00000)
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202 | return hy > 0 ? huge*huge : tiny*tiny;
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203 | }
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204 | /* over/underflow if x is not close to one */
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205 | if (ix < 0x3fefffff)
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206 | return hy < 0 ? s*huge*huge : s*tiny*tiny;
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207 | if (ix > 0x3ff00000)
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208 | return hy > 0 ? s*huge*huge : s*tiny*tiny;
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209 | /* now |1-x| is tiny <= 2**-20, suffice to compute
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210 | log(x) by x-x^2/2+x^3/3-x^4/4 */
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211 | t = ax - 1.0; /* t has 20 trailing zeros */
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212 | w = (t*t)*(0.5 - t*(0.3333333333333333333333-t*0.25));
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213 | u = ivln2_h*t; /* ivln2_h has 21 sig. bits */
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214 | v = t*ivln2_l - w*ivln2;
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215 | t1 = u + v;
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216 | SET_LOW_WORD(t1, 0);
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217 | t2 = v - (t1-u);
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218 | } else {
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219 | double ss,s2,s_h,s_l,t_h,t_l;
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220 | n = 0;
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221 | /* take care subnormal number */
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222 | if (ix < 0x00100000) {
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223 | ax *= two53;
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224 | n -= 53;
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225 | GET_HIGH_WORD(ix,ax);
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226 | }
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227 | n += ((ix)>>20) - 0x3ff;
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228 | j = ix & 0x000fffff;
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229 | /* determine interval */
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230 | ix = j | 0x3ff00000; /* normalize ix */
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231 | if (j <= 0x3988E) /* |x|<sqrt(3/2) */
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232 | k = 0;
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233 | else if (j < 0xBB67A) /* |x|<sqrt(3) */
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234 | k = 1;
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235 | else {
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236 | k = 0;
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237 | n += 1;
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238 | ix -= 0x00100000;
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239 | }
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240 | SET_HIGH_WORD(ax, ix);
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241 |
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242 | /* compute ss = s_h+s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5) */
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243 | u = ax - bp[k]; /* bp[0]=1.0, bp[1]=1.5 */
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244 | v = 1.0/(ax+bp[k]);
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245 | ss = u*v;
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246 | s_h = ss;
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247 | SET_LOW_WORD(s_h, 0);
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248 | /* t_h=ax+bp[k] High */
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249 | t_h = 0.0;
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250 | SET_HIGH_WORD(t_h, ((ix>>1)|0x20000000) + 0x00080000 + (k<<18));
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251 | t_l = ax - (t_h-bp[k]);
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252 | s_l = v*((u-s_h*t_h)-s_h*t_l);
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253 | /* compute log(ax) */
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254 | s2 = ss*ss;
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255 | r = s2*s2*(L1+s2*(L2+s2*(L3+s2*(L4+s2*(L5+s2*L6)))));
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256 | r += s_l*(s_h+ss);
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257 | s2 = s_h*s_h;
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258 | t_h = 3.0 + s2 + r;
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259 | SET_LOW_WORD(t_h, 0);
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260 | t_l = r - ((t_h-3.0)-s2);
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261 | /* u+v = ss*(1+...) */
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262 | u = s_h*t_h;
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263 | v = s_l*t_h + t_l*ss;
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264 | /* 2/(3log2)*(ss+...) */
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265 | p_h = u + v;
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266 | SET_LOW_WORD(p_h, 0);
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267 | p_l = v - (p_h-u);
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268 | z_h = cp_h*p_h; /* cp_h+cp_l = 2/(3*log2) */
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269 | z_l = cp_l*p_h+p_l*cp + dp_l[k];
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270 | /* log2(ax) = (ss+..)*2/(3*log2) = n + dp_h + z_h + z_l */
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271 | t = (double)n;
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272 | t1 = ((z_h + z_l) + dp_h[k]) + t;
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273 | SET_LOW_WORD(t1, 0);
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274 | t2 = z_l - (((t1 - t) - dp_h[k]) - z_h);
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275 | }
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276 |
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277 | /* split up y into y1+y2 and compute (y1+y2)*(t1+t2) */
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278 | y1 = y;
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279 | SET_LOW_WORD(y1, 0);
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280 | p_l = (y-y1)*t1 + y*t2;
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281 | p_h = y1*t1;
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282 | z = p_l + p_h;
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283 | EXTRACT_WORDS(j, i, z);
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284 | if (j >= 0x40900000) { /* z >= 1024 */
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285 | if (((j-0x40900000)|i) != 0) /* if z > 1024 */
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286 | return s*huge*huge; /* overflow */
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287 | if (p_l + ovt > z - p_h)
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288 | return s*huge*huge; /* overflow */
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289 | } else if ((j&0x7fffffff) >= 0x4090cc00) { /* z <= -1075 */ // FIXME: instead of abs(j) use unsigned j
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290 | if (((j-0xc090cc00)|i) != 0) /* z < -1075 */
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291 | return s*tiny*tiny; /* underflow */
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292 | if (p_l <= z - p_h)
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293 | return s*tiny*tiny; /* underflow */
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294 | }
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295 | /*
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296 | * compute 2**(p_h+p_l)
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297 | */
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298 | i = j & 0x7fffffff;
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299 | k = (i>>20) - 0x3ff;
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300 | n = 0;
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301 | if (i > 0x3fe00000) { /* if |z| > 0.5, set n = [z+0.5] */
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302 | n = j + (0x00100000>>(k+1));
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303 | k = ((n&0x7fffffff)>>20) - 0x3ff; /* new k for n */
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304 | t = 0.0;
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305 | SET_HIGH_WORD(t, n & ~(0x000fffff>>k));
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306 | n = ((n&0x000fffff)|0x00100000)>>(20-k);
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307 | if (j < 0)
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308 | n = -n;
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309 | p_h -= t;
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310 | }
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311 | t = p_l + p_h;
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312 | SET_LOW_WORD(t, 0);
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313 | u = t*lg2_h;
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314 | v = (p_l-(t-p_h))*lg2 + t*lg2_l;
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315 | z = u + v;
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316 | w = v - (z-u);
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317 | t = z*z;
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318 | t1 = z - t*(P1+t*(P2+t*(P3+t*(P4+t*P5))));
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319 | r = (z*t1)/(t1-2.0) - (w + z*w);
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320 | z = 1.0 - (r-z);
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321 | GET_HIGH_WORD(j, z);
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322 | j += n<<20;
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323 | if ((j>>20) <= 0) /* subnormal output */
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324 | z = scalbn(z,n);
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325 | else
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326 | SET_HIGH_WORD(z, j);
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327 | return s*z;
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328 | }
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