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1 /* @(#)e_pow.c 5.1 93/09/24 */ |
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2 /* |
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3 * ==================================================== |
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4 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. |
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5 * |
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6 * Developed at SunPro, a Sun Microsystems, Inc. business. |
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7 * Permission to use, copy, modify, and distribute this |
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8 * software is freely granted, provided that this notice |
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9 * is preserved. |
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10 * ==================================================== |
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11 */ |
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12 |
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13 #if defined(LIBM_SCCS) && !defined(lint) |
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14 static char rcsid[] = "$NetBSD: e_pow.c,v 1.9 1995/05/12 04:57:32 jtc Exp $"; |
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15 #endif |
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16 |
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17 /* __ieee754_pow(x,y) return x**y |
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18 * |
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19 * n |
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20 * Method: Let x = 2 * (1+f) |
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21 * 1. Compute and return log2(x) in two pieces: |
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22 * log2(x) = w1 + w2, |
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23 * where w1 has 53-24 = 29 bit trailing zeros. |
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24 * 2. Perform y*log2(x) = n+y' by simulating muti-precision |
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25 * arithmetic, where |y'|<=0.5. |
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26 * 3. Return x**y = 2**n*exp(y'*log2) |
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27 * |
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28 * Special cases: |
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29 * 1. (anything) ** 0 is 1 |
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30 * 2. (anything) ** 1 is itself |
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31 * 3. (anything) ** NAN is NAN |
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32 * 4. NAN ** (anything except 0) is NAN |
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33 * 5. +-(|x| > 1) ** +INF is +INF |
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34 * 6. +-(|x| > 1) ** -INF is +0 |
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35 * 7. +-(|x| < 1) ** +INF is +0 |
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36 * 8. +-(|x| < 1) ** -INF is +INF |
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37 * 9. +-1 ** +-INF is NAN |
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38 * 10. +0 ** (+anything except 0, NAN) is +0 |
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39 * 11. -0 ** (+anything except 0, NAN, odd integer) is +0 |
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40 * 12. +0 ** (-anything except 0, NAN) is +INF |
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41 * 13. -0 ** (-anything except 0, NAN, odd integer) is +INF |
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42 * 14. -0 ** (odd integer) = -( +0 ** (odd integer) ) |
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43 * 15. +INF ** (+anything except 0,NAN) is +INF |
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44 * 16. +INF ** (-anything except 0,NAN) is +0 |
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45 * 17. -INF ** (anything) = -0 ** (-anything) |
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46 * 18. (-anything) ** (integer) is (-1)**(integer)*(+anything**integer) |
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47 * 19. (-anything except 0 and inf) ** (non-integer) is NAN |
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48 * |
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49 * Accuracy: |
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50 * pow(x,y) returns x**y nearly rounded. In particular |
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51 * pow(integer,integer) |
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52 * always returns the correct integer provided it is |
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53 * representable. |
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54 * |
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55 * Constants : |
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56 * The hexadecimal values are the intended ones for the following |
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57 * constants. The decimal values may be used, provided that the |
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58 * compiler will convert from decimal to binary accurately enough |
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59 * to produce the hexadecimal values shown. |
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60 */ |
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61 |
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62 /*#include "math.h"*/ |
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63 #include "math_private.h" |
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64 |
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65 #ifdef __STDC__ |
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66 static const double |
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67 #else |
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68 static double |
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69 #endif |
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70 bp[] = {1.0, 1.5,}, |
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71 dp_h[] = { 0.0, 5.84962487220764160156e-01,}, /* 0x3FE2B803, 0x40000000 */ |
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72 dp_l[] = { 0.0, 1.35003920212974897128e-08,}, /* 0x3E4CFDEB, 0x43CFD006 */ |
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73 /* poly coefs for (3/2)*(log(x)-2s-2/3*s**3 */ |
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74 L1 = 5.99999999999994648725e-01, /* 0x3FE33333, 0x33333303 */ |
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75 L2 = 4.28571428578550184252e-01, /* 0x3FDB6DB6, 0xDB6FABFF */ |
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76 L3 = 3.33333329818377432918e-01, /* 0x3FD55555, 0x518F264D */ |
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77 L4 = 2.72728123808534006489e-01, /* 0x3FD17460, 0xA91D4101 */ |
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78 L5 = 2.30660745775561754067e-01, /* 0x3FCD864A, 0x93C9DB65 */ |
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79 L6 = 2.06975017800338417784e-01, /* 0x3FCA7E28, 0x4A454EEF */ |
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80 P1 = 1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */ |
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81 P2 = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */ |
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82 P3 = 6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */ |
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83 P4 = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */ |
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84 P5 = 4.13813679705723846039e-08, /* 0x3E663769, 0x72BEA4D0 */ |
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85 lg2 = 6.93147180559945286227e-01, /* 0x3FE62E42, 0xFEFA39EF */ |
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86 lg2_h = 6.93147182464599609375e-01, /* 0x3FE62E43, 0x00000000 */ |
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87 lg2_l = -1.90465429995776804525e-09, /* 0xBE205C61, 0x0CA86C39 */ |
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88 ovt = 8.0085662595372944372e-0017, /* -(1024-log2(ovfl+.5ulp)) */ |
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89 cp = 9.61796693925975554329e-01, /* 0x3FEEC709, 0xDC3A03FD =2/(3ln2) */ |
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90 cp_h = 9.61796700954437255859e-01, /* 0x3FEEC709, 0xE0000000 =(float)cp */ |
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91 cp_l = -7.02846165095275826516e-09, /* 0xBE3E2FE0, 0x145B01F5 =tail of cp_h*/ |
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92 ivln2 = 1.44269504088896338700e+00, /* 0x3FF71547, 0x652B82FE =1/ln2 */ |
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93 ivln2_h = 1.44269502162933349609e+00, /* 0x3FF71547, 0x60000000 =24b 1/ln2*/ |
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94 ivln2_l = 1.92596299112661746887e-08; /* 0x3E54AE0B, 0xF85DDF44 =1/ln2 tail*/ |
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95 |
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96 #ifdef __STDC__ |
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97 double __ieee754_pow(double x, double y) |
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98 #else |
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99 double __ieee754_pow(x,y) |
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100 double x, y; |
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101 #endif |
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102 { |
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103 double z,ax,z_h,z_l,p_h,p_l; |
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104 double y1,t1,t2,r,s,t,u,v,w; |
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105 int32_t i,j,k,yisint,n; |
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106 int32_t hx,hy,ix,iy; |
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107 u_int32_t lx,ly; |
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108 |
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109 EXTRACT_WORDS(hx,lx,x); |
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110 EXTRACT_WORDS(hy,ly,y); |
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111 ix = hx&0x7fffffff; iy = hy&0x7fffffff; |
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112 |
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113 /* y==zero: x**0 = 1 */ |
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114 if((iy|ly)==0) return one; |
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115 |
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116 /* +-NaN return x+y */ |
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117 if(ix > 0x7ff00000 || ((ix==0x7ff00000)&&(lx!=0)) || |
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118 iy > 0x7ff00000 || ((iy==0x7ff00000)&&(ly!=0))) |
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119 return x+y; |
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120 |
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121 /* determine if y is an odd int when x < 0 |
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122 * yisint = 0 ... y is not an integer |
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123 * yisint = 1 ... y is an odd int |
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124 * yisint = 2 ... y is an even int |
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125 */ |
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126 yisint = 0; |
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127 if(hx<0) { |
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128 if(iy>=0x43400000) yisint = 2; /* even integer y */ |
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129 else if(iy>=0x3ff00000) { |
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130 k = (iy>>20)-0x3ff; /* exponent */ |
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131 if(k>20) { |
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132 j = ly>>(52-k); |
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133 if((u_int32_t)(j<<(52-k))==ly) yisint = 2-(j&1); |
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134 } else if(ly==0) { |
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135 j = iy>>(20-k); |
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136 if((j<<(20-k))==iy) yisint = 2-(j&1); |
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137 } |
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138 } |
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139 } |
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140 |
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141 /* special value of y */ |
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142 if(ly==0) { |
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143 if (iy==0x7ff00000) { /* y is +-inf */ |
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144 if(((ix-0x3ff00000)|lx)==0) |
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145 return y - y; /* inf**+-1 is NaN */ |
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146 else if (ix >= 0x3ff00000)/* (|x|>1)**+-inf = inf,0 */ |
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147 return (hy>=0)? y: zero; |
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148 else /* (|x|<1)**-,+inf = inf,0 */ |
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149 return (hy<0)?-y: zero; |
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150 } |
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151 if(iy==0x3ff00000) { /* y is +-1 */ |
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152 if(hy<0) return one/x; else return x; |
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153 } |
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154 if(hy==0x40000000) return x*x; /* y is 2 */ |
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155 if(hy==0x3fe00000) { /* y is 0.5 */ |
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156 if(hx>=0) /* x >= +0 */ |
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157 return __ieee754_sqrt(x); |
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158 } |
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159 } |
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160 |
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161 ax = x < 0 ? -x : x; /*fabs(x);*/ |
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162 /* special value of x */ |
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163 if(lx==0) { |
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164 if(ix==0x7ff00000||ix==0||ix==0x3ff00000){ |
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165 z = ax; /*x is +-0,+-inf,+-1*/ |
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166 if(hy<0) z = one/z; /* z = (1/|x|) */ |
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167 if(hx<0) { |
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168 if(((ix-0x3ff00000)|yisint)==0) { |
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169 z = (z-z)/(z-z); /* (-1)**non-int is NaN */ |
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170 } else if(yisint==1) |
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171 z = -z; /* (x<0)**odd = -(|x|**odd) */ |
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172 } |
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173 return z; |
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174 } |
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175 } |
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176 |
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177 /* (x<0)**(non-int) is NaN */ |
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178 if(((((u_int32_t)hx>>31)-1)|yisint)==0) return (x-x)/(x-x); |
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179 |
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180 /* |y| is huge */ |
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181 if(iy>0x41e00000) { /* if |y| > 2**31 */ |
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182 if(iy>0x43f00000){ /* if |y| > 2**64, must o/uflow */ |
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183 if(ix<=0x3fefffff) return (hy<0)? huge*huge:tiny*tiny; |
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184 if(ix>=0x3ff00000) return (hy>0)? huge*huge:tiny*tiny; |
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185 } |
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186 /* over/underflow if x is not close to one */ |
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187 if(ix<0x3fefffff) return (hy<0)? huge*huge:tiny*tiny; |
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188 if(ix>0x3ff00000) return (hy>0)? huge*huge:tiny*tiny; |
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189 /* now |1-x| is tiny <= 2**-20, suffice to compute |
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190 log(x) by x-x^2/2+x^3/3-x^4/4 */ |
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191 t = x-1; /* t has 20 trailing zeros */ |
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192 w = (t*t)*(0.5-t*(0.3333333333333333333333-t*0.25)); |
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193 u = ivln2_h*t; /* ivln2_h has 21 sig. bits */ |
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194 v = t*ivln2_l-w*ivln2; |
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195 t1 = u+v; |
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196 SET_LOW_WORD(t1,0); |
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197 t2 = v-(t1-u); |
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198 } else { |
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199 double s2,s_h,s_l,t_h,t_l; |
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200 n = 0; |
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201 /* take care subnormal number */ |
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202 if(ix<0x00100000) |
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203 {ax *= two53; n -= 53; GET_HIGH_WORD(ix,ax); } |
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204 n += ((ix)>>20)-0x3ff; |
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205 j = ix&0x000fffff; |
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206 /* determine interval */ |
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207 ix = j|0x3ff00000; /* normalize ix */ |
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208 if(j<=0x3988E) k=0; /* |x|<sqrt(3/2) */ |
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209 else if(j<0xBB67A) k=1; /* |x|<sqrt(3) */ |
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210 else {k=0;n+=1;ix -= 0x00100000;} |
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211 SET_HIGH_WORD(ax,ix); |
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212 |
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213 /* compute s = s_h+s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5) */ |
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214 u = ax-bp[k]; /* bp[0]=1.0, bp[1]=1.5 */ |
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215 v = one/(ax+bp[k]); |
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216 s = u*v; |
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217 s_h = s; |
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218 SET_LOW_WORD(s_h,0); |
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219 /* t_h=ax+bp[k] High */ |
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220 t_h = zero; |
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221 SET_HIGH_WORD(t_h,((ix>>1)|0x20000000)+0x00080000+(k<<18)); |
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222 t_l = ax - (t_h-bp[k]); |
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223 s_l = v*((u-s_h*t_h)-s_h*t_l); |
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224 /* compute log(ax) */ |
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225 s2 = s*s; |
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226 r = s2*s2*(L1+s2*(L2+s2*(L3+s2*(L4+s2*(L5+s2*L6))))); |
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227 r += s_l*(s_h+s); |
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228 s2 = s_h*s_h; |
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229 t_h = 3.0+s2+r; |
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230 SET_LOW_WORD(t_h,0); |
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231 t_l = r-((t_h-3.0)-s2); |
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232 /* u+v = s*(1+...) */ |
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233 u = s_h*t_h; |
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234 v = s_l*t_h+t_l*s; |
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235 /* 2/(3log2)*(s+...) */ |
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236 p_h = u+v; |
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237 SET_LOW_WORD(p_h,0); |
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238 p_l = v-(p_h-u); |
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239 z_h = cp_h*p_h; /* cp_h+cp_l = 2/(3*log2) */ |
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240 z_l = cp_l*p_h+p_l*cp+dp_l[k]; |
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241 /* log2(ax) = (s+..)*2/(3*log2) = n + dp_h + z_h + z_l */ |
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242 t = (double)n; |
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243 t1 = (((z_h+z_l)+dp_h[k])+t); |
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244 SET_LOW_WORD(t1,0); |
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245 t2 = z_l-(((t1-t)-dp_h[k])-z_h); |
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246 } |
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247 |
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248 s = one; /* s (sign of result -ve**odd) = -1 else = 1 */ |
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249 if(((((u_int32_t)hx>>31)-1)|(yisint-1))==0) |
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250 s = -one;/* (-ve)**(odd int) */ |
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251 |
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252 /* split up y into y1+y2 and compute (y1+y2)*(t1+t2) */ |
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253 y1 = y; |
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254 SET_LOW_WORD(y1,0); |
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255 p_l = (y-y1)*t1+y*t2; |
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256 p_h = y1*t1; |
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257 z = p_l+p_h; |
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258 EXTRACT_WORDS(j,i,z); |
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259 if (j>=0x40900000) { /* z >= 1024 */ |
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260 if(((j-0x40900000)|i)!=0) /* if z > 1024 */ |
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261 return s*huge*huge; /* overflow */ |
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262 else { |
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263 if(p_l+ovt>z-p_h) return s*huge*huge; /* overflow */ |
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264 } |
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265 } else if((j&0x7fffffff)>=0x4090cc00 ) { /* z <= -1075 */ |
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266 if(((j-0xc090cc00)|i)!=0) /* z < -1075 */ |
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267 return s*tiny*tiny; /* underflow */ |
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268 else { |
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269 if(p_l<=z-p_h) return s*tiny*tiny; /* underflow */ |
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270 } |
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271 } |
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272 /* |
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273 * compute 2**(p_h+p_l) |
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274 */ |
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275 i = j&0x7fffffff; |
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276 k = (i>>20)-0x3ff; |
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277 n = 0; |
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278 if(i>0x3fe00000) { /* if |z| > 0.5, set n = [z+0.5] */ |
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279 n = j+(0x00100000>>(k+1)); |
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280 k = ((n&0x7fffffff)>>20)-0x3ff; /* new k for n */ |
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281 t = zero; |
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282 SET_HIGH_WORD(t,n&~(0x000fffff>>k)); |
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283 n = ((n&0x000fffff)|0x00100000)>>(20-k); |
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284 if(j<0) n = -n; |
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285 p_h -= t; |
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286 } |
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287 t = p_l+p_h; |
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288 SET_LOW_WORD(t,0); |
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289 u = t*lg2_h; |
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290 v = (p_l-(t-p_h))*lg2+t*lg2_l; |
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291 z = u+v; |
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292 w = v-(z-u); |
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293 t = z*z; |
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294 t1 = z - t*(P1+t*(P2+t*(P3+t*(P4+t*P5)))); |
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295 r = (z*t1)/(t1-two)-(w+z*w); |
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296 z = one-(r-z); |
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297 GET_HIGH_WORD(j,z); |
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298 j += (n<<20); |
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299 if((j>>20)<=0) z = SDL_NAME(scalbn)(z,n); /* subnormal output */ |
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300 else SET_HIGH_WORD(z,j); |
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301 return s*z; |
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302 } |