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annotate src/video/e_sqrt.h @ 1670:eef792d31de8 SDL-1.3

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Work in progress. :)
author Sam Lantinga <slouken@libsdl.org>
date Wed, 07 Jun 2006 16:10:28 +0000
parents 4da1ee79c9af
children
rev   line source
1330
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
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1 /* @(#)e_sqrt.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_sqrt.c,v 1.8 1995/05/10 20:46:17 jtc Exp $";
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15 #endif
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16
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17 /* __ieee754_sqrt(x)
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18 * Return correctly rounded sqrt.
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19 * ------------------------------------------
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20 * | Use the hardware sqrt if you have one |
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21 * ------------------------------------------
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22 * Method:
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23 * Bit by bit method using integer arithmetic. (Slow, but portable)
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24 * 1. Normalization
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25 * Scale x to y in [1,4) with even powers of 2:
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26 * find an integer k such that 1 <= (y=x*2^(2k)) < 4, then
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27 * sqrt(x) = 2^k * sqrt(y)
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28 * 2. Bit by bit computation
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29 * Let q = sqrt(y) truncated to i bit after binary point (q = 1),
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30 * i 0
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31 * i+1 2
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32 * s = 2*q , and y = 2 * ( y - q ). (1)
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33 * i i i i
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34 *
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35 * To compute q from q , one checks whether
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36 * i+1 i
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37 *
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38 * -(i+1) 2
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39 * (q + 2 ) <= y. (2)
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40 * i
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41 * -(i+1)
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42 * If (2) is false, then q = q ; otherwise q = q + 2 .
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43 * i+1 i i+1 i
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44 *
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45 * With some algebric manipulation, it is not difficult to see
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46 * that (2) is equivalent to
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47 * -(i+1)
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48 * s + 2 <= y (3)
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49 * i i
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50 *
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51 * The advantage of (3) is that s and y can be computed by
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52 * i i
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53 * the following recurrence formula:
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54 * if (3) is false
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55 *
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56 * s = s , y = y ; (4)
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57 * i+1 i i+1 i
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58 *
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59 * otherwise,
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60 * -i -(i+1)
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61 * s = s + 2 , y = y - s - 2 (5)
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62 * i+1 i i+1 i i
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63 *
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64 * One may easily use induction to prove (4) and (5).
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65 * Note. Since the left hand side of (3) contain only i+2 bits,
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66 * it does not necessary to do a full (53-bit) comparison
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67 * in (3).
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68 * 3. Final rounding
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69 * After generating the 53 bits result, we compute one more bit.
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70 * Together with the remainder, we can decide whether the
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71 * result is exact, bigger than 1/2ulp, or less than 1/2ulp
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72 * (it will never equal to 1/2ulp).
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73 * The rounding mode can be detected by checking whether
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74 * huge + tiny is equal to huge, and whether huge - tiny is
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75 * equal to huge for some floating point number "huge" and "tiny".
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76 *
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77 * Special cases:
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78 * sqrt(+-0) = +-0 ... exact
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79 * sqrt(inf) = inf
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80 * sqrt(-ve) = NaN ... with invalid signal
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81 * sqrt(NaN) = NaN ... with invalid signal for signaling NaN
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82 *
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83 * Other methods : see the appended file at the end of the program below.
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84 *---------------
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85 */
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86
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87 /*#include "math.h"*/
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88 #include "math_private.h"
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89
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90 #ifdef __STDC__
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91 double SDL_NAME(copysign) (double x, double y)
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92 #else
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93 double SDL_NAME(copysign) (x, y)
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94 double
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95 x,
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96 y;
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97 #endif
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98 {
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99 u_int32_t hx, hy;
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100 GET_HIGH_WORD(hx, x);
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101 GET_HIGH_WORD(hy, y);
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102 SET_HIGH_WORD(x, (hx & 0x7fffffff) | (hy & 0x80000000));
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103 return x;
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104 }
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105
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106 #ifdef __STDC__
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107 double SDL_NAME(scalbn) (double x, int n)
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108 #else
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109 double SDL_NAME(scalbn) (x, n)
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110 double
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111 x;
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112 int
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113 n;
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114 #endif
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115 {
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116 int32_t k, hx, lx;
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117 EXTRACT_WORDS(hx, lx, x);
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118 k = (hx & 0x7ff00000) >> 20; /* extract exponent */
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119 if (k == 0) { /* 0 or subnormal x */
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120 if ((lx | (hx & 0x7fffffff)) == 0)
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121 return x; /* +-0 */
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122 x *= two54;
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123 GET_HIGH_WORD(hx, x);
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124 k = ((hx & 0x7ff00000) >> 20) - 54;
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125 if (n < -50000)
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126 return tiny * x; /*underflow */
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127 }
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128 if (k == 0x7ff)
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129 return x + x; /* NaN or Inf */
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130 k = k + n;
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131 if (k > 0x7fe)
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132 return huge * SDL_NAME(copysign) (huge, x); /* overflow */
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133 if (k > 0) { /* normal result */
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134 SET_HIGH_WORD(x, (hx & 0x800fffff) | (k << 20));
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135 return x;
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136 }
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137 if (k <= -54) {
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138 if (n > 50000) /* in case integer overflow in n+k */
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139 return huge * SDL_NAME(copysign) (huge, x); /*overflow */
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140 else
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141 return tiny * SDL_NAME(copysign) (tiny, x); /*underflow */
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142 }
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143 k += 54; /* subnormal result */
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144 SET_HIGH_WORD(x, (hx & 0x800fffff) | (k << 20));
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145 return x * twom54;
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146 }
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147
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148 #ifdef __STDC__
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149 double
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150 __ieee754_sqrt(double x)
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151 #else
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152 double
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153 __ieee754_sqrt(x)
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154 double x;
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155 #endif
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156 {
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157 double z;
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158 int32_t sign = (int) 0x80000000;
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159 int32_t ix0, s0, q, m, t, i;
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160 u_int32_t r, t1, s1, ix1, q1;
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161
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162 EXTRACT_WORDS(ix0, ix1, x);
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163
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164 /* take care of Inf and NaN */
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165 if ((ix0 & 0x7ff00000) == 0x7ff00000) {
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166 return x * x + x; /* sqrt(NaN)=NaN, sqrt(+inf)=+inf
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167 sqrt(-inf)=sNaN */
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168 }
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169 /* take care of zero */
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170 if (ix0 <= 0) {
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171 if (((ix0 & (~sign)) | ix1) == 0)
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172 return x; /* sqrt(+-0) = +-0 */
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173 else if (ix0 < 0)
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174 return (x - x) / (x - x); /* sqrt(-ve) = sNaN */
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175 }
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176 /* normalize x */
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177 m = (ix0 >> 20);
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178 if (m == 0) { /* subnormal x */
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179 while (ix0 == 0) {
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180 m -= 21;
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181 ix0 |= (ix1 >> 11);
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182 ix1 <<= 21;
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183 }
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184 for (i = 0; (ix0 & 0x00100000) == 0; i++)
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185 ix0 <<= 1;
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186 m -= i - 1;
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187 ix0 |= (ix1 >> (32 - i));
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188 ix1 <<= i;
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189 }
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190 m -= 1023; /* unbias exponent */
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191 ix0 = (ix0 & 0x000fffff) | 0x00100000;
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192 if (m & 1) { /* odd m, double x to make it even */
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193 ix0 += ix0 + ((ix1 & sign) >> 31);
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194 ix1 += ix1;
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195 }
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196 m >>= 1; /* m = [m/2] */
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197
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198 /* generate sqrt(x) bit by bit */
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199 ix0 += ix0 + ((ix1 & sign) >> 31);
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200 ix1 += ix1;
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201 q = q1 = s0 = s1 = 0; /* [q,q1] = sqrt(x) */
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202 r = 0x00200000; /* r = moving bit from right to left */
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203
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204 while (r != 0) {
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205 t = s0 + r;
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206 if (t <= ix0) {
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207 s0 = t + r;
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208 ix0 -= t;
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209 q += r;
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210 }
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211 ix0 += ix0 + ((ix1 & sign) >> 31);
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212 ix1 += ix1;
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213 r >>= 1;
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214 }
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215
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216 r = sign;
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217 while (r != 0) {
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218 t1 = s1 + r;
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219 t = s0;
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220 if ((t < ix0) || ((t == ix0) && (t1 <= ix1))) {
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221 s1 = t1 + r;
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222 if (((int32_t) (t1 & sign) == sign) && (s1 & sign) == 0)
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223 s0 += 1;
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224 ix0 -= t;
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225 if (ix1 < t1)
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226 ix0 -= 1;
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227 ix1 -= t1;
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228 q1 += r;
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229 }
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230 ix0 += ix0 + ((ix1 & sign) >> 31);
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231 ix1 += ix1;
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232 r >>= 1;
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233 }
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234
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235 /* use floating add to find out rounding direction */
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236 if ((ix0 | ix1) != 0) {
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237 z = one - tiny; /* trigger inexact flag */
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238 if (z >= one) {
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239 z = one + tiny;
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240 if (q1 == (u_int32_t) 0xffffffff) {
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241 q1 = 0;
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242 q += 1;
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243 } else if (z > one) {
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244 if (q1 == (u_int32_t) 0xfffffffe)
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245 q += 1;
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246 q1 += 2;
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247 } else
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248 q1 += (q1 & 1);
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249 }
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250 }
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251 ix0 = (q >> 1) + 0x3fe00000;
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252 ix1 = q1 >> 1;
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253 if ((q & 1) == 1)
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254 ix1 |= sign;
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255 ix0 += (m << 20);
1668
4da1ee79c9af more tweaking indent options
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256 INSERT_WORDS(z, ix0, ix1);
1662
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257 return z;
1330
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258 }
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259
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260 /*
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261 Other methods (use floating-point arithmetic)
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262 -------------
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263 (This is a copy of a drafted paper by Prof W. Kahan
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264 and K.C. Ng, written in May, 1986)
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265
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266 Two algorithms are given here to implement sqrt(x)
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267 (IEEE double precision arithmetic) in software.
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268 Both supply sqrt(x) correctly rounded. The first algorithm (in
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269 Section A) uses newton iterations and involves four divisions.
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270 The second one uses reciproot iterations to avoid division, but
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271 requires more multiplications. Both algorithms need the ability
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272 to chop results of arithmetic operations instead of round them,
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273 and the INEXACT flag to indicate when an arithmetic operation
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274 is executed exactly with no roundoff error, all part of the
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275 standard (IEEE 754-1985). The ability to perform shift, add,
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276 subtract and logical AND operations upon 32-bit words is needed
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277 too, though not part of the standard.
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278
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279 A. sqrt(x) by Newton Iteration
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280
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281 (1) Initial approximation
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282
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283 Let x0 and x1 be the leading and the trailing 32-bit words of
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284 a floating point number x (in IEEE double format) respectively
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285
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286 1 11 52 ...widths
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287 ------------------------------------------------------
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288 x: |s| e | f |
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289 ------------------------------------------------------
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290 msb lsb msb lsb ...order
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291
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292
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293 ------------------------ ------------------------
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294 x0: |s| e | f1 | x1: | f2 |
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295 ------------------------ ------------------------
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296
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297 By performing shifts and subtracts on x0 and x1 (both regarded
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298 as integers), we obtain an 8-bit approximation of sqrt(x) as
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299 follows.
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300
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301 k := (x0>>1) + 0x1ff80000;
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302 y0 := k - T1[31&(k>>15)]. ... y ~ sqrt(x) to 8 bits
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303 Here k is a 32-bit integer and T1[] is an integer array containing
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304 correction terms. Now magically the floating value of y (y's
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305 leading 32-bit word is y0, the value of its trailing word is 0)
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306 approximates sqrt(x) to almost 8-bit.
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307
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308 Value of T1:
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309 static int T1[32]= {
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310 0, 1024, 3062, 5746, 9193, 13348, 18162, 23592,
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311 29598, 36145, 43202, 50740, 58733, 67158, 75992, 85215,
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312 83599, 71378, 60428, 50647, 41945, 34246, 27478, 21581,
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313 16499, 12183, 8588, 5674, 3403, 1742, 661, 130,};
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314
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315 (2) Iterative refinement
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316
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317 Apply Heron's rule three times to y, we have y approximates
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318 sqrt(x) to within 1 ulp (Unit in the Last Place):
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319
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320 y := (y+x/y)/2 ... almost 17 sig. bits
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321 y := (y+x/y)/2 ... almost 35 sig. bits
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322 y := y-(y-x/y)/2 ... within 1 ulp
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323
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324
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325 Remark 1.
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326 Another way to improve y to within 1 ulp is:
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327
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328 y := (y+x/y) ... almost 17 sig. bits to 2*sqrt(x)
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329 y := y - 0x00100006 ... almost 18 sig. bits to sqrt(x)
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330
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331 2
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332 (x-y )*y
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333 y := y + 2* ---------- ...within 1 ulp
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334 2
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335 3y + x
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336
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337
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338 This formula has one division fewer than the one above; however,
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339 it requires more multiplications and additions. Also x must be
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340 scaled in advance to avoid spurious overflow in evaluating the
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341 expression 3y*y+x. Hence it is not recommended uless division
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342 is slow. If division is very slow, then one should use the
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343 reciproot algorithm given in section B.
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344
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345 (3) Final adjustment
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346
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347 By twiddling y's last bit it is possible to force y to be
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348 correctly rounded according to the prevailing rounding mode
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349 as follows. Let r and i be copies of the rounding mode and
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350 inexact flag before entering the square root program. Also we
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351 use the expression y+-ulp for the next representable floating
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352 numbers (up and down) of y. Note that y+-ulp = either fixed
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353 point y+-1, or multiply y by nextafter(1,+-inf) in chopped
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354 mode.
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355
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356 I := FALSE; ... reset INEXACT flag I
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357 R := RZ; ... set rounding mode to round-toward-zero
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358 z := x/y; ... chopped quotient, possibly inexact
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359 If(not I) then { ... if the quotient is exact
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360 if(z=y) {
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361 I := i; ... restore inexact flag
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362 R := r; ... restore rounded mode
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363 return sqrt(x):=y.
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364 } else {
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365 z := z - ulp; ... special rounding
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366 }
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367 }
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368 i := TRUE; ... sqrt(x) is inexact
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369 If (r=RN) then z=z+ulp ... rounded-to-nearest
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370 If (r=RP) then { ... round-toward-+inf
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371 y = y+ulp; z=z+ulp;
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372 }
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373 y := y+z; ... chopped sum
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374 y0:=y0-0x00100000; ... y := y/2 is correctly rounded.
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375 I := i; ... restore inexact flag
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376 R := r; ... restore rounded mode
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377 return sqrt(x):=y.
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378
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
379 (4) Special cases
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
380
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
381 Square root of +inf, +-0, or NaN is itself;
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
382 Square root of a negative number is NaN with invalid signal.
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
383
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
384
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
385 B. sqrt(x) by Reciproot Iteration
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
386
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
387 (1) Initial approximation
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
388
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
389 Let x0 and x1 be the leading and the trailing 32-bit words of
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
390 a floating point number x (in IEEE double format) respectively
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
391 (see section A). By performing shifs and subtracts on x0 and y0,
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
392 we obtain a 7.8-bit approximation of 1/sqrt(x) as follows.
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
393
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
394 k := 0x5fe80000 - (x0>>1);
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
395 y0:= k - T2[63&(k>>14)]. ... y ~ 1/sqrt(x) to 7.8 bits
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
396
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
397 Here k is a 32-bit integer and T2[] is an integer array
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
398 containing correction terms. Now magically the floating
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
399 value of y (y's leading 32-bit word is y0, the value of
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
400 its trailing word y1 is set to zero) approximates 1/sqrt(x)
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
401 to almost 7.8-bit.
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
402
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
403 Value of T2:
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
404 static int T2[64]= {
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
405 0x1500, 0x2ef8, 0x4d67, 0x6b02, 0x87be, 0xa395, 0xbe7a, 0xd866,
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
406 0xf14a, 0x1091b,0x11fcd,0x13552,0x14999,0x15c98,0x16e34,0x17e5f,
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
407 0x18d03,0x19a01,0x1a545,0x1ae8a,0x1b5c4,0x1bb01,0x1bfde,0x1c28d,
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
408 0x1c2de,0x1c0db,0x1ba73,0x1b11c,0x1a4b5,0x1953d,0x18266,0x16be0,
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
409 0x1683e,0x179d8,0x18a4d,0x19992,0x1a789,0x1b445,0x1bf61,0x1c989,
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
410 0x1d16d,0x1d77b,0x1dddf,0x1e2ad,0x1e5bf,0x1e6e8,0x1e654,0x1e3cd,
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
411 0x1df2a,0x1d635,0x1cb16,0x1be2c,0x1ae4e,0x19bde,0x1868e,0x16e2e,
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
412 0x1527f,0x1334a,0x11051,0xe951, 0xbe01, 0x8e0d, 0x5924, 0x1edd,};
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
413
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
414 (2) Iterative refinement
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
415
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
416 Apply Reciproot iteration three times to y and multiply the
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
417 result by x to get an approximation z that matches sqrt(x)
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
418 to about 1 ulp. To be exact, we will have
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
419 -1ulp < sqrt(x)-z<1.0625ulp.
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
420
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
421 ... set rounding mode to Round-to-nearest
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
422 y := y*(1.5-0.5*x*y*y) ... almost 15 sig. bits to 1/sqrt(x)
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
423 y := y*((1.5-2^-30)+0.5*x*y*y)... about 29 sig. bits to 1/sqrt(x)
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
424 ... special arrangement for better accuracy
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
425 z := x*y ... 29 bits to sqrt(x), with z*y<1
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
426 z := z + 0.5*z*(1-z*y) ... about 1 ulp to sqrt(x)
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
427
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
428 Remark 2. The constant 1.5-2^-30 is chosen to bias the error so that
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
429 (a) the term z*y in the final iteration is always less than 1;
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
430 (b) the error in the final result is biased upward so that
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
431 -1 ulp < sqrt(x) - z < 1.0625 ulp
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
432 instead of |sqrt(x)-z|<1.03125ulp.
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
433
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
434 (3) Final adjustment
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
435
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
436 By twiddling y's last bit it is possible to force y to be
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
437 correctly rounded according to the prevailing rounding mode
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
438 as follows. Let r and i be copies of the rounding mode and
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
439 inexact flag before entering the square root program. Also we
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
440 use the expression y+-ulp for the next representable floating
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
441 numbers (up and down) of y. Note that y+-ulp = either fixed
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
442 point y+-1, or multiply y by nextafter(1,+-inf) in chopped
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
443 mode.
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
444
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
445 R := RZ; ... set rounding mode to round-toward-zero
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
446 switch(r) {
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
447 case RN: ... round-to-nearest
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
448 if(x<= z*(z-ulp)...chopped) z = z - ulp; else
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
449 if(x<= z*(z+ulp)...chopped) z = z; else z = z+ulp;
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
450 break;
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
451 case RZ:case RM: ... round-to-zero or round-to--inf
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
452 R:=RP; ... reset rounding mod to round-to-+inf
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
453 if(x<z*z ... rounded up) z = z - ulp; else
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
454 if(x>=(z+ulp)*(z+ulp) ...rounded up) z = z+ulp;
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
455 break;
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
456 case RP: ... round-to-+inf
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
457 if(x>(z+ulp)*(z+ulp)...chopped) z = z+2*ulp; else
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
458 if(x>z*z ...chopped) z = z+ulp;
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
459 break;
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
460 }
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
461
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
462 Remark 3. The above comparisons can be done in fixed point. For
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
463 example, to compare x and w=z*z chopped, it suffices to compare
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
464 x1 and w1 (the trailing parts of x and w), regarding them as
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
465 two's complement integers.
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
466
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
467 ...Is z an exact square root?
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
468 To determine whether z is an exact square root of x, let z1 be the
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
469 trailing part of z, and also let x0 and x1 be the leading and
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
470 trailing parts of x.
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
471
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
472 If ((z1&0x03ffffff)!=0) ... not exact if trailing 26 bits of z!=0
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
473 I := 1; ... Raise Inexact flag: z is not exact
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
474 else {
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
475 j := 1 - [(x0>>20)&1] ... j = logb(x) mod 2
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
476 k := z1 >> 26; ... get z's 25-th and 26-th
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
477 fraction bits
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
478 I := i or (k&j) or ((k&(j+j+1))!=(x1&3));
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
479 }
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
480 R:= r ... restore rounded mode
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
481 return sqrt(x):=z.
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
482
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
483 If multiplication is cheaper then the foregoing red tape, the
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
484 Inexact flag can be evaluated by
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
485
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
486 I := i;
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
487 I := (z*z!=x) or I.
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
488
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
489 Note that z*z can overwrite I; this value must be sensed if it is
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
490 True.
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
491
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
492 Remark 4. If z*z = x exactly, then bit 25 to bit 0 of z1 must be
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
493 zero.
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
494
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
495 --------------------
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
496 z1: | f2 |
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
497 --------------------
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
498 bit 31 bit 0
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
499
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
500 Further more, bit 27 and 26 of z1, bit 0 and 1 of x1, and the odd
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
501 or even of logb(x) have the following relations:
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
502
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
503 -------------------------------------------------
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
504 bit 27,26 of z1 bit 1,0 of x1 logb(x)
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
505 -------------------------------------------------
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
506 00 00 odd and even
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
507 01 01 even
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
508 10 10 odd
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
509 10 00 even
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
510 11 01 even
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
511 -------------------------------------------------
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
512
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
513 (4) Special cases (see (4) of Section A).
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
514
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
515 */
1662
782fd950bd46 Revamp of the video system in progress - adding support for multiple displays, multiple windows, and a full video mode selection API.
Sam Lantinga <slouken@libsdl.org>
parents: 1424
diff changeset
516 /* vi: set ts=4 sw=4 expandtab: */