Mercurial > sdl-ios-xcode
annotate src/video/e_sqrt.h @ 2153:760c1bd2d625
Reverted mousewheel support in 1.2, since it breaks binary compatibility.
author | Sam Lantinga <slouken@libsdl.org> |
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date | Fri, 06 Jul 2007 13:45:10 +0000 |
parents | c121d94672cb |
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rev | line source |
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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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SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head.
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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); |
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256 INSERT_WORDS(z, ix0, ix1); |
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257 return z; |
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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 |
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379 (4) Special cases |
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380 |
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381 Square root of +inf, +-0, or NaN is itself; |
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382 Square root of a negative number is NaN with invalid signal. |
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383 |
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384 |
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385 B. sqrt(x) by Reciproot Iteration |
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386 |
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387 (1) Initial approximation |
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388 |
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389 Let x0 and x1 be the leading and the trailing 32-bit words of |
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390 a floating point number x (in IEEE double format) respectively |
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391 (see section A). By performing shifs and subtracts on x0 and y0, |
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392 we obtain a 7.8-bit approximation of 1/sqrt(x) as follows. |
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393 |
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394 k := 0x5fe80000 - (x0>>1); |
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395 y0:= k - T2[63&(k>>14)]. ... y ~ 1/sqrt(x) to 7.8 bits |
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396 |
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397 Here k is a 32-bit integer and T2[] is an integer array |
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398 containing correction terms. Now magically the floating |
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399 value of y (y's leading 32-bit word is y0, the value of |
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400 its trailing word y1 is set to zero) approximates 1/sqrt(x) |
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401 to almost 7.8-bit. |
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402 |
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403 Value of T2: |
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404 static int T2[64]= { |
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405 0x1500, 0x2ef8, 0x4d67, 0x6b02, 0x87be, 0xa395, 0xbe7a, 0xd866, |
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406 0xf14a, 0x1091b,0x11fcd,0x13552,0x14999,0x15c98,0x16e34,0x17e5f, |
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407 0x18d03,0x19a01,0x1a545,0x1ae8a,0x1b5c4,0x1bb01,0x1bfde,0x1c28d, |
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408 0x1c2de,0x1c0db,0x1ba73,0x1b11c,0x1a4b5,0x1953d,0x18266,0x16be0, |
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409 0x1683e,0x179d8,0x18a4d,0x19992,0x1a789,0x1b445,0x1bf61,0x1c989, |
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410 0x1d16d,0x1d77b,0x1dddf,0x1e2ad,0x1e5bf,0x1e6e8,0x1e654,0x1e3cd, |
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411 0x1df2a,0x1d635,0x1cb16,0x1be2c,0x1ae4e,0x19bde,0x1868e,0x16e2e, |
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412 0x1527f,0x1334a,0x11051,0xe951, 0xbe01, 0x8e0d, 0x5924, 0x1edd,}; |
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413 |
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414 (2) Iterative refinement |
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415 |
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416 Apply Reciproot iteration three times to y and multiply the |
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417 result by x to get an approximation z that matches sqrt(x) |
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418 to about 1 ulp. To be exact, we will have |
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419 -1ulp < sqrt(x)-z<1.0625ulp. |
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420 |
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421 ... set rounding mode to Round-to-nearest |
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422 y := y*(1.5-0.5*x*y*y) ... almost 15 sig. bits to 1/sqrt(x) |
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423 y := y*((1.5-2^-30)+0.5*x*y*y)... about 29 sig. bits to 1/sqrt(x) |
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424 ... special arrangement for better accuracy |
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425 z := x*y ... 29 bits to sqrt(x), with z*y<1 |
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426 z := z + 0.5*z*(1-z*y) ... about 1 ulp to sqrt(x) |
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427 |
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428 Remark 2. The constant 1.5-2^-30 is chosen to bias the error so that |
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429 (a) the term z*y in the final iteration is always less than 1; |
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430 (b) the error in the final result is biased upward so that |
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431 -1 ulp < sqrt(x) - z < 1.0625 ulp |
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432 instead of |sqrt(x)-z|<1.03125ulp. |
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433 |
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434 (3) Final adjustment |
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435 |
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436 By twiddling y's last bit it is possible to force y to be |
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437 correctly rounded according to the prevailing rounding mode |
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438 as follows. Let r and i be copies of the rounding mode and |
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439 inexact flag before entering the square root program. Also we |
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440 use the expression y+-ulp for the next representable floating |
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441 numbers (up and down) of y. Note that y+-ulp = either fixed |
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442 point y+-1, or multiply y by nextafter(1,+-inf) in chopped |
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443 mode. |
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444 |
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445 R := RZ; ... set rounding mode to round-toward-zero |
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446 switch(r) { |
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447 case RN: ... round-to-nearest |
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448 if(x<= z*(z-ulp)...chopped) z = z - ulp; else |
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449 if(x<= z*(z+ulp)...chopped) z = z; else z = z+ulp; |
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450 break; |
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451 case RZ:case RM: ... round-to-zero or round-to--inf |
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452 R:=RP; ... reset rounding mod to round-to-+inf |
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453 if(x<z*z ... rounded up) z = z - ulp; else |
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454 if(x>=(z+ulp)*(z+ulp) ...rounded up) z = z+ulp; |
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455 break; |
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456 case RP: ... round-to-+inf |
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457 if(x>(z+ulp)*(z+ulp)...chopped) z = z+2*ulp; else |
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458 if(x>z*z ...chopped) z = z+ulp; |
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459 break; |
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460 } |
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461 |
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462 Remark 3. The above comparisons can be done in fixed point. For |
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463 example, to compare x and w=z*z chopped, it suffices to compare |
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464 x1 and w1 (the trailing parts of x and w), regarding them as |
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465 two's complement integers. |
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466 |
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467 ...Is z an exact square root? |
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468 To determine whether z is an exact square root of x, let z1 be the |
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469 trailing part of z, and also let x0 and x1 be the leading and |
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470 trailing parts of x. |
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471 |
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472 If ((z1&0x03ffffff)!=0) ... not exact if trailing 26 bits of z!=0 |
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473 I := 1; ... Raise Inexact flag: z is not exact |
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474 else { |
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475 j := 1 - [(x0>>20)&1] ... j = logb(x) mod 2 |
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476 k := z1 >> 26; ... get z's 25-th and 26-th |
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477 fraction bits |
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478 I := i or (k&j) or ((k&(j+j+1))!=(x1&3)); |
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479 } |
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480 R:= r ... restore rounded mode |
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481 return sqrt(x):=z. |
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482 |
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483 If multiplication is cheaper then the foregoing red tape, the |
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484 Inexact flag can be evaluated by |
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485 |
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486 I := i; |
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487 I := (z*z!=x) or I. |
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488 |
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489 Note that z*z can overwrite I; this value must be sensed if it is |
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490 True. |
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491 |
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492 Remark 4. If z*z = x exactly, then bit 25 to bit 0 of z1 must be |
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493 zero. |
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494 |
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495 -------------------- |
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496 z1: | f2 | |
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497 -------------------- |
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498 bit 31 bit 0 |
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499 |
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500 Further more, bit 27 and 26 of z1, bit 0 and 1 of x1, and the odd |
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501 or even of logb(x) have the following relations: |
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502 |
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503 ------------------------------------------------- |
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504 bit 27,26 of z1 bit 1,0 of x1 logb(x) |
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505 ------------------------------------------------- |
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506 00 00 odd and even |
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507 01 01 even |
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508 10 10 odd |
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509 10 00 even |
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510 11 01 even |
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511 ------------------------------------------------- |
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512 |
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513 (4) Special cases (see (4) of Section A). |
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514 |
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515 */ |
1895
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516 /* vi: set ts=4 sw=4 expandtab: */ |