1 | /* origin: FreeBSD /usr/src/lib/msun/src/e_powf.c */
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2 | /*
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3 | * Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com.
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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 "libm.h"
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17 |
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18 | static const float
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19 | bp[] = {1.0, 1.5,},
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20 | dp_h[] = { 0.0, 5.84960938e-01,}, /* 0x3f15c000 */
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21 | dp_l[] = { 0.0, 1.56322085e-06,}, /* 0x35d1cfdc */
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22 | two24 = 16777216.0, /* 0x4b800000 */
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23 | huge = 1.0e30,
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24 | tiny = 1.0e-30,
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25 | /* poly coefs for (3/2)*(log(x)-2s-2/3*s**3 */
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26 | L1 = 6.0000002384e-01, /* 0x3f19999a */
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27 | L2 = 4.2857143283e-01, /* 0x3edb6db7 */
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28 | L3 = 3.3333334327e-01, /* 0x3eaaaaab */
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29 | L4 = 2.7272811532e-01, /* 0x3e8ba305 */
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30 | L5 = 2.3066075146e-01, /* 0x3e6c3255 */
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31 | L6 = 2.0697501302e-01, /* 0x3e53f142 */
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32 | P1 = 1.6666667163e-01, /* 0x3e2aaaab */
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33 | P2 = -2.7777778450e-03, /* 0xbb360b61 */
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34 | P3 = 6.6137559770e-05, /* 0x388ab355 */
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35 | P4 = -1.6533901999e-06, /* 0xb5ddea0e */
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36 | P5 = 4.1381369442e-08, /* 0x3331bb4c */
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37 | lg2 = 6.9314718246e-01, /* 0x3f317218 */
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38 | lg2_h = 6.93145752e-01, /* 0x3f317200 */
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39 | lg2_l = 1.42860654e-06, /* 0x35bfbe8c */
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40 | ovt = 4.2995665694e-08, /* -(128-log2(ovfl+.5ulp)) */
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41 | cp = 9.6179670095e-01, /* 0x3f76384f =2/(3ln2) */
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42 | cp_h = 9.6191406250e-01, /* 0x3f764000 =12b cp */
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43 | cp_l = -1.1736857402e-04, /* 0xb8f623c6 =tail of cp_h */
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44 | ivln2 = 1.4426950216e+00, /* 0x3fb8aa3b =1/ln2 */
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45 | ivln2_h = 1.4426879883e+00, /* 0x3fb8aa00 =16b 1/ln2*/
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46 | ivln2_l = 7.0526075433e-06; /* 0x36eca570 =1/ln2 tail*/
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47 |
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48 | float powf(float x, float y)
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49 | {
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50 | float z,ax,z_h,z_l,p_h,p_l;
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51 | float y1,t1,t2,r,s,sn,t,u,v,w;
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52 | int32_t i,j,k,yisint,n;
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53 | int32_t hx,hy,ix,iy,is;
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54 |
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55 | GET_FLOAT_WORD(hx, x);
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56 | GET_FLOAT_WORD(hy, y);
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57 | ix = hx & 0x7fffffff;
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58 | iy = hy & 0x7fffffff;
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59 |
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60 | /* x**0 = 1, even if x is NaN */
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61 | if (iy == 0)
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62 | return 1.0f;
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63 | /* 1**y = 1, even if y is NaN */
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64 | if (hx == 0x3f800000)
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65 | return 1.0f;
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66 | /* NaN if either arg is NaN */
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67 | if (ix > 0x7f800000 || iy > 0x7f800000)
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68 | return x + y;
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69 |
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70 | /* determine if y is an odd int when x < 0
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71 | * yisint = 0 ... y is not an integer
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72 | * yisint = 1 ... y is an odd int
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73 | * yisint = 2 ... y is an even int
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74 | */
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75 | yisint = 0;
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76 | if (hx < 0) {
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77 | if (iy >= 0x4b800000)
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78 | yisint = 2; /* even integer y */
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79 | else if (iy >= 0x3f800000) {
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80 | k = (iy>>23) - 0x7f; /* exponent */
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81 | j = iy>>(23-k);
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82 | if ((j<<(23-k)) == iy)
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83 | yisint = 2 - (j & 1);
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84 | }
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85 | }
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86 |
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87 | /* special value of y */
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88 | if (iy == 0x7f800000) { /* y is +-inf */
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89 | if (ix == 0x3f800000) /* (-1)**+-inf is 1 */
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90 | return 1.0f;
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91 | else if (ix > 0x3f800000) /* (|x|>1)**+-inf = inf,0 */
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92 | return hy >= 0 ? y : 0.0f;
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93 | else /* (|x|<1)**+-inf = 0,inf */
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94 | return hy >= 0 ? 0.0f: -y;
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95 | }
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96 | if (iy == 0x3f800000) /* y is +-1 */
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97 | return hy >= 0 ? x : 1.0f/x;
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98 | if (hy == 0x40000000) /* y is 2 */
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99 | return x*x;
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100 | if (hy == 0x3f000000) { /* y is 0.5 */
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101 | if (hx >= 0) /* x >= +0 */
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102 | return sqrtf(x);
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103 | }
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104 |
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105 | ax = fabsf(x);
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106 | /* special value of x */
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107 | if (ix == 0x7f800000 || ix == 0 || ix == 0x3f800000) { /* x is +-0,+-inf,+-1 */
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108 | z = ax;
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109 | if (hy < 0) /* z = (1/|x|) */
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110 | z = 1.0f/z;
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111 | if (hx < 0) {
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112 | if (((ix-0x3f800000)|yisint) == 0) {
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113 | z = (z-z)/(z-z); /* (-1)**non-int is NaN */
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114 | } else if (yisint == 1)
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115 | z = -z; /* (x<0)**odd = -(|x|**odd) */
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116 | }
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117 | return z;
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118 | }
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119 |
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120 | sn = 1.0f; /* sign of result */
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121 | if (hx < 0) {
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122 | if (yisint == 0) /* (x<0)**(non-int) is NaN */
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123 | return (x-x)/(x-x);
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124 | if (yisint == 1) /* (x<0)**(odd int) */
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125 | sn = -1.0f;
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126 | }
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127 |
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128 | /* |y| is huge */
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129 | if (iy > 0x4d000000) { /* if |y| > 2**27 */
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130 | /* over/underflow if x is not close to one */
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131 | if (ix < 0x3f7ffff8)
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132 | return hy < 0 ? sn*huge*huge : sn*tiny*tiny;
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133 | if (ix > 0x3f800007)
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134 | return hy > 0 ? sn*huge*huge : sn*tiny*tiny;
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135 | /* now |1-x| is tiny <= 2**-20, suffice to compute
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136 | log(x) by x-x^2/2+x^3/3-x^4/4 */
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137 | t = ax - 1; /* t has 20 trailing zeros */
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138 | w = (t*t)*(0.5f - t*(0.333333333333f - t*0.25f));
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139 | u = ivln2_h*t; /* ivln2_h has 16 sig. bits */
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140 | v = t*ivln2_l - w*ivln2;
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141 | t1 = u + v;
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142 | GET_FLOAT_WORD(is, t1);
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143 | SET_FLOAT_WORD(t1, is & 0xfffff000);
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144 | t2 = v - (t1-u);
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145 | } else {
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146 | float s2,s_h,s_l,t_h,t_l;
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147 | n = 0;
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148 | /* take care subnormal number */
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149 | if (ix < 0x00800000) {
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150 | ax *= two24;
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151 | n -= 24;
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152 | GET_FLOAT_WORD(ix, ax);
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153 | }
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154 | n += ((ix)>>23) - 0x7f;
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155 | j = ix & 0x007fffff;
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156 | /* determine interval */
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157 | ix = j | 0x3f800000; /* normalize ix */
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158 | if (j <= 0x1cc471) /* |x|<sqrt(3/2) */
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159 | k = 0;
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160 | else if (j < 0x5db3d7) /* |x|<sqrt(3) */
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161 | k = 1;
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162 | else {
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163 | k = 0;
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164 | n += 1;
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165 | ix -= 0x00800000;
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166 | }
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167 | SET_FLOAT_WORD(ax, ix);
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168 |
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169 | /* compute s = s_h+s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5) */
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170 | u = ax - bp[k]; /* bp[0]=1.0, bp[1]=1.5 */
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171 | v = 1.0f/(ax+bp[k]);
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172 | s = u*v;
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173 | s_h = s;
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174 | GET_FLOAT_WORD(is, s_h);
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175 | SET_FLOAT_WORD(s_h, is & 0xfffff000);
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176 | /* t_h=ax+bp[k] High */
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177 | is = ((ix>>1) & 0xfffff000) | 0x20000000;
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178 | SET_FLOAT_WORD(t_h, is + 0x00400000 + (k<<21));
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179 | t_l = ax - (t_h - bp[k]);
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180 | s_l = v*((u - s_h*t_h) - s_h*t_l);
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181 | /* compute log(ax) */
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182 | s2 = s*s;
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183 | r = s2*s2*(L1+s2*(L2+s2*(L3+s2*(L4+s2*(L5+s2*L6)))));
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184 | r += s_l*(s_h+s);
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185 | s2 = s_h*s_h;
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186 | t_h = 3.0f + s2 + r;
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187 | GET_FLOAT_WORD(is, t_h);
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188 | SET_FLOAT_WORD(t_h, is & 0xfffff000);
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189 | t_l = r - ((t_h - 3.0f) - s2);
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190 | /* u+v = s*(1+...) */
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191 | u = s_h*t_h;
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192 | v = s_l*t_h + t_l*s;
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193 | /* 2/(3log2)*(s+...) */
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194 | p_h = u + v;
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195 | GET_FLOAT_WORD(is, p_h);
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196 | SET_FLOAT_WORD(p_h, is & 0xfffff000);
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197 | p_l = v - (p_h - u);
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198 | z_h = cp_h*p_h; /* cp_h+cp_l = 2/(3*log2) */
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199 | z_l = cp_l*p_h + p_l*cp+dp_l[k];
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200 | /* log2(ax) = (s+..)*2/(3*log2) = n + dp_h + z_h + z_l */
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201 | t = (float)n;
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202 | t1 = (((z_h + z_l) + dp_h[k]) + t);
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203 | GET_FLOAT_WORD(is, t1);
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204 | SET_FLOAT_WORD(t1, is & 0xfffff000);
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205 | t2 = z_l - (((t1 - t) - dp_h[k]) - z_h);
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206 | }
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207 |
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208 | /* split up y into y1+y2 and compute (y1+y2)*(t1+t2) */
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209 | GET_FLOAT_WORD(is, y);
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210 | SET_FLOAT_WORD(y1, is & 0xfffff000);
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211 | p_l = (y-y1)*t1 + y*t2;
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212 | p_h = y1*t1;
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213 | z = p_l + p_h;
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214 | GET_FLOAT_WORD(j, z);
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215 | if (j > 0x43000000) /* if z > 128 */
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216 | return sn*huge*huge; /* overflow */
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217 | else if (j == 0x43000000) { /* if z == 128 */
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218 | if (p_l + ovt > z - p_h)
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219 | return sn*huge*huge; /* overflow */
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220 | } else if ((j&0x7fffffff) > 0x43160000) /* z < -150 */ // FIXME: check should be (uint32_t)j > 0xc3160000
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221 | return sn*tiny*tiny; /* underflow */
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222 | else if (j == 0xc3160000) { /* z == -150 */
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223 | if (p_l <= z-p_h)
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224 | return sn*tiny*tiny; /* underflow */
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225 | }
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226 | /*
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227 | * compute 2**(p_h+p_l)
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228 | */
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229 | i = j & 0x7fffffff;
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230 | k = (i>>23) - 0x7f;
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231 | n = 0;
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232 | if (i > 0x3f000000) { /* if |z| > 0.5, set n = [z+0.5] */
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233 | n = j + (0x00800000>>(k+1));
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234 | k = ((n&0x7fffffff)>>23) - 0x7f; /* new k for n */
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235 | SET_FLOAT_WORD(t, n & ~(0x007fffff>>k));
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236 | n = ((n&0x007fffff)|0x00800000)>>(23-k);
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237 | if (j < 0)
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238 | n = -n;
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239 | p_h -= t;
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240 | }
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241 | t = p_l + p_h;
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242 | GET_FLOAT_WORD(is, t);
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243 | SET_FLOAT_WORD(t, is & 0xffff8000);
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244 | u = t*lg2_h;
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245 | v = (p_l-(t-p_h))*lg2 + t*lg2_l;
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246 | z = u + v;
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247 | w = v - (z - u);
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248 | t = z*z;
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249 | t1 = z - t*(P1+t*(P2+t*(P3+t*(P4+t*P5))));
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250 | r = (z*t1)/(t1-2.0f) - (w+z*w);
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251 | z = 1.0f - (r - z);
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252 | GET_FLOAT_WORD(j, z);
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253 | j += n<<23;
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254 | if ((j>>23) <= 0) /* subnormal output */
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255 | z = scalbnf(z, n);
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256 | else
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257 | SET_FLOAT_WORD(z, j);
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258 | return sn*z;
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259 | }
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