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annotate src/libm/e_log.c @ 5216:25ad4a50d34f

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OpenGL ES 2.0 rendering LIVES!
author Sam Lantinga <slouken@libsdl.org>
date Sun, 06 Feb 2011 19:12:55 -0800
parents dc1eb82ffdaa
children
rev   line source
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a98604b691c8 Expanded the libm support and put it into a separate directory.
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1 /* @(#)e_log.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 const char rcsid[] =
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15 "$NetBSD: e_log.c,v 1.8 1995/05/10 20:45:49 jtc Exp $";
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16 #endif
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17
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18 /* __ieee754_log(x)
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19 * Return the logrithm of x
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20 *
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21 * Method :
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22 * 1. Argument Reduction: find k and f such that
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23 * x = 2^k * (1+f),
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24 * where sqrt(2)/2 < 1+f < sqrt(2) .
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25 *
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26 * 2. Approximation of log(1+f).
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27 * Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
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28 * = 2s + 2/3 s**3 + 2/5 s**5 + .....,
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29 * = 2s + s*R
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30 * We use a special Reme algorithm on [0,0.1716] to generate
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31 * a polynomial of degree 14 to approximate R The maximum error
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32 * of this polynomial approximation is bounded by 2**-58.45. In
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33 * other words,
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34 * 2 4 6 8 10 12 14
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35 * R(z) ~ Lg1*s +Lg2*s +Lg3*s +Lg4*s +Lg5*s +Lg6*s +Lg7*s
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36 * (the values of Lg1 to Lg7 are listed in the program)
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37 * and
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38 * | 2 14 | -58.45
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39 * | Lg1*s +...+Lg7*s - R(z) | <= 2
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40 * | |
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41 * Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2.
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42 * In order to guarantee error in log below 1ulp, we compute log
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43 * by
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44 * log(1+f) = f - s*(f - R) (if f is not too large)
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45 * log(1+f) = f - (hfsq - s*(hfsq+R)). (better accuracy)
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46 *
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47 * 3. Finally, log(x) = k*ln2 + log(1+f).
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48 * = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo)))
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49 * Here ln2 is split into two floating point number:
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50 * ln2_hi + ln2_lo,
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51 * where n*ln2_hi is always exact for |n| < 2000.
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52 *
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53 * Special cases:
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54 * log(x) is NaN with signal if x < 0 (including -INF) ;
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55 * log(+INF) is +INF; log(0) is -INF with signal;
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56 * log(NaN) is that NaN with no signal.
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57 *
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58 * Accuracy:
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59 * according to an error analysis, the error is always less than
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60 * 1 ulp (unit in the last place).
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61 *
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62 * Constants:
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63 * The hexadecimal values are the intended ones for the following
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64 * constants. The decimal values may be used, provided that the
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65 * compiler will convert from decimal to binary accurately enough
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66 * to produce the hexadecimal values shown.
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67 */
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68
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69 #include "math.h"
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70 #include "math_private.h"
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71
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72 #ifdef __STDC__
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73 static const double
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74 #else
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75 static double
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76 #endif
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77 ln2_hi = 6.93147180369123816490e-01, /* 3fe62e42 fee00000 */
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78 ln2_lo = 1.90821492927058770002e-10, /* 3dea39ef 35793c76 */
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79 two54 = 1.80143985094819840000e+16, /* 43500000 00000000 */
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80 Lg1 = 6.666666666666735130e-01, /* 3FE55555 55555593 */
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81 Lg2 = 3.999999999940941908e-01, /* 3FD99999 9997FA04 */
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82 Lg3 = 2.857142874366239149e-01, /* 3FD24924 94229359 */
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83 Lg4 = 2.222219843214978396e-01, /* 3FCC71C5 1D8E78AF */
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84 Lg5 = 1.818357216161805012e-01, /* 3FC74664 96CB03DE */
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85 Lg6 = 1.531383769920937332e-01, /* 3FC39A09 D078C69F */
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86 Lg7 = 1.479819860511658591e-01; /* 3FC2F112 DF3E5244 */
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87
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88 #ifdef __STDC__
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89 static const double zero = 0.0;
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90 #else
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91 static double zero = 0.0;
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92 #endif
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93
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94 #ifdef __STDC__
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95 double attribute_hidden
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96 __ieee754_log(double x)
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97 #else
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98 double attribute_hidden
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99 __ieee754_log(x)
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100 double x;
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101 #endif
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102 {
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103 double hfsq, f, s, z, R, w, t1, t2, dk;
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104 int32_t k, hx, i, j;
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105 u_int32_t lx;
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106
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107 EXTRACT_WORDS(hx, lx, x);
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108
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109 k = 0;
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110 if (hx < 0x00100000) { /* x < 2**-1022 */
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111 if (((hx & 0x7fffffff) | lx) == 0)
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112 return -two54 / zero; /* log(+-0)=-inf */
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113 if (hx < 0)
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114 return (x - x) / zero; /* log(-#) = NaN */
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115 k -= 54;
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116 x *= two54; /* subnormal number, scale up x */
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117 GET_HIGH_WORD(hx, x);
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118 }
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119 if (hx >= 0x7ff00000)
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120 return x + x;
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121 k += (hx >> 20) - 1023;
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122 hx &= 0x000fffff;
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123 i = (hx + 0x95f64) & 0x100000;
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124 SET_HIGH_WORD(x, hx | (i ^ 0x3ff00000)); /* normalize x or x/2 */
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125 k += (i >> 20);
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126 f = x - 1.0;
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127 if ((0x000fffff & (2 + hx)) < 3) { /* |f| < 2**-20 */
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128 if (f == zero) {
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129 if (k == 0)
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130 return zero;
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131 else {
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132 dk = (double) k;
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133 return dk * ln2_hi + dk * ln2_lo;
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134 }
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135 }
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136 R = f * f * (0.5 - 0.33333333333333333 * f);
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137 if (k == 0)
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138 return f - R;
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139 else {
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140 dk = (double) k;
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141 return dk * ln2_hi - ((R - dk * ln2_lo) - f);
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142 }
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143 }
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144 s = f / (2.0 + f);
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145 dk = (double) k;
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146 z = s * s;
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147 i = hx - 0x6147a;
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148 w = z * z;
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149 j = 0x6b851 - hx;
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150 t1 = w * (Lg2 + w * (Lg4 + w * Lg6));
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151 t2 = z * (Lg1 + w * (Lg3 + w * (Lg5 + w * Lg7)));
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152 i |= j;
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153 R = t2 + t1;
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154 if (i > 0) {
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155 hfsq = 0.5 * f * f;
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156 if (k == 0)
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157 return f - (hfsq - s * (hfsq + R));
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158 else
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159 return dk * ln2_hi - ((hfsq - (s * (hfsq + R) + dk * ln2_lo)) -
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160 f);
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161 } else {
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162 if (k == 0)
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163 return f - s * (f - R);
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164 else
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165 return dk * ln2_hi - ((s * (f - R) - dk * ln2_lo) - f);
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166 }
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167 }