1 | // |
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2 | // Lol Engine |
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3 | // |
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4 | // Copyright: (c) 2010-2012 Sam Hocevar <sam@hocevar.net> |
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5 | // This program is free software; you can redistribute it and/or |
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6 | // modify it under the terms of the Do What The Fuck You Want To |
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7 | // Public License, Version 2, as published by Sam Hocevar. See |
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8 | // http://sam.zoy.org/projects/COPYING.WTFPL for more details. |
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9 | // |
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10 | |
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11 | #if defined HAVE_CONFIG_H |
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12 | # include "config.h" |
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13 | #endif |
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14 | |
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15 | #if defined _XBOX |
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16 | # define _USE_MATH_DEFINES /* for M_PI */ |
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17 | # include <xtl.h> |
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18 | # undef near /* Fuck Microsoft */ |
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19 | # undef far /* Fuck Microsoft again */ |
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20 | #elif defined WIN32 |
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21 | # define _USE_MATH_DEFINES /* for M_PI */ |
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22 | # define WIN32_LEAN_AND_MEAN |
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23 | # include <windows.h> |
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24 | # undef near /* Fuck Microsoft */ |
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25 | # undef far /* Fuck Microsoft again */ |
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26 | #endif |
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27 | |
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28 | #include <new> |
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29 | #include <cstring> |
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30 | #include <cstdio> |
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31 | #include <cstdlib> |
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32 | |
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33 | #include "core.h" |
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34 | |
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35 | using namespace std; |
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36 | |
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37 | namespace lol |
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38 | { |
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39 | |
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40 | template<> real::Real() |
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41 | { |
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42 | m_mantissa = new uint32_t[BIGITS]; |
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43 | m_signexp = 0; |
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44 | } |
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45 | |
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46 | template<> real::Real(real const &x) |
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47 | { |
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48 | m_mantissa = new uint32_t[BIGITS]; |
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49 | memcpy(m_mantissa, x.m_mantissa, BIGITS * sizeof(uint32_t)); |
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50 | m_signexp = x.m_signexp; |
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51 | } |
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52 | |
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53 | template<> real const &real::operator =(real const &x) |
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54 | { |
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55 | if (&x != this) |
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56 | { |
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57 | memcpy(m_mantissa, x.m_mantissa, BIGITS * sizeof(uint32_t)); |
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58 | m_signexp = x.m_signexp; |
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59 | } |
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60 | |
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61 | return *this; |
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62 | } |
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63 | |
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64 | template<> real::~Real() |
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65 | { |
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66 | delete[] m_mantissa; |
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67 | } |
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68 | |
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69 | template<> real::Real(float f) { new(this) real((double)f); } |
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70 | template<> real::Real(int i) { new(this) real((double)i); } |
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71 | template<> real::Real(unsigned int i) { new(this) real((double)i); } |
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72 | |
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73 | template<> real::Real(double d) |
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74 | { |
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75 | new(this) real(); |
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76 | |
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77 | union { double d; uint64_t x; } u = { d }; |
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78 | |
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79 | uint32_t sign = (u.x >> 63) << 31; |
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80 | uint32_t exponent = (u.x << 1) >> 53; |
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81 | |
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82 | switch (exponent) |
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83 | { |
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84 | case 0x00: |
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85 | m_signexp = sign; |
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86 | break; |
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87 | case 0x7ff: |
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88 | m_signexp = sign | 0x7fffffffu; |
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89 | break; |
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90 | default: |
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91 | m_signexp = sign | (exponent + (1 << 30) - (1 << 10)); |
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92 | break; |
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93 | } |
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94 | |
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95 | m_mantissa[0] = (uint32_t)(u.x >> 20); |
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96 | m_mantissa[1] = (uint32_t)(u.x << 12); |
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97 | memset(m_mantissa + 2, 0, (BIGITS - 2) * sizeof(m_mantissa[0])); |
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98 | } |
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99 | |
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100 | template<> real::operator float() const { return (float)(double)(*this); } |
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101 | template<> real::operator int() const { return (int)(double)(*this); } |
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102 | template<> real::operator unsigned() const { return (unsigned)(double)(*this); } |
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103 | |
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104 | template<> real::operator double() const |
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105 | { |
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106 | union { double d; uint64_t x; } u; |
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107 | |
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108 | /* Get sign */ |
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109 | u.x = m_signexp >> 31; |
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110 | u.x <<= 11; |
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111 | |
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112 | /* Compute new exponent */ |
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113 | uint32_t exponent = (m_signexp << 1) >> 1; |
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114 | int e = (int)exponent - (1 << 30) + (1 << 10); |
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115 | |
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116 | if (e < 0) |
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117 | u.x <<= 52; |
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118 | else if (e >= 0x7ff) |
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119 | { |
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120 | u.x |= 0x7ff; |
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121 | u.x <<= 52; |
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122 | } |
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123 | else |
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124 | { |
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125 | u.x |= e; |
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126 | |
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127 | /* Store mantissa if necessary */ |
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128 | u.x <<= 32; |
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129 | u.x |= m_mantissa[0]; |
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130 | u.x <<= 20; |
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131 | u.x |= m_mantissa[1] >> 12; |
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132 | /* Rounding */ |
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133 | u.x += (m_mantissa[1] >> 11) & 1; |
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134 | } |
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135 | |
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136 | return u.d; |
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137 | } |
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138 | |
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139 | /* |
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140 | * Create a real number from an ASCII representation |
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141 | */ |
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142 | template<> real::Real(char const *str) |
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143 | { |
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144 | real ret = 0; |
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145 | int exponent = 0; |
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146 | bool comma = false, nonzero = false, negative = false, finished = false; |
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147 | |
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148 | for (char const *p = str; *p && !finished; p++) |
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149 | { |
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150 | switch (*p) |
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151 | { |
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152 | case '-': |
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153 | case '+': |
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154 | if (p != str) |
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155 | break; |
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156 | negative = (*p == '-'); |
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157 | break; |
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158 | case '.': |
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159 | if (comma) |
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160 | finished = true; |
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161 | comma = true; |
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162 | break; |
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163 | case '0': case '1': case '2': case '3': case '4': |
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164 | case '5': case '6': case '7': case '8': case '9': |
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165 | if (nonzero) |
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166 | { |
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167 | real x = ret + ret; |
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168 | x = x + x + ret; |
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169 | ret = x + x; |
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170 | } |
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171 | if (*p != '0') |
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172 | { |
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173 | ret += (int)(*p - '0'); |
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174 | nonzero = true; |
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175 | } |
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176 | if (comma) |
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177 | exponent--; |
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178 | break; |
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179 | case 'e': |
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180 | case 'E': |
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181 | exponent += atoi(p + 1); |
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182 | finished = true; |
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183 | break; |
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184 | default: |
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185 | finished = true; |
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186 | break; |
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187 | } |
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188 | } |
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189 | |
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190 | if (exponent) |
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191 | ret *= pow(R_10, (real)exponent); |
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192 | |
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193 | if (negative) |
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194 | ret = -ret; |
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195 | |
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196 | new(this) real(ret); |
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197 | } |
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198 | |
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199 | template<> real real::operator +() const |
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200 | { |
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201 | return *this; |
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202 | } |
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203 | |
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204 | template<> real real::operator -() const |
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205 | { |
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206 | real ret = *this; |
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207 | ret.m_signexp ^= 0x80000000u; |
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208 | return ret; |
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209 | } |
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210 | |
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211 | template<> real real::operator +(real const &x) const |
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212 | { |
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213 | if (x.m_signexp << 1 == 0) |
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214 | return *this; |
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215 | |
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216 | /* Ensure both arguments are positive. Otherwise, switch signs, |
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217 | * or replace + with -. */ |
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218 | if (m_signexp >> 31) |
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219 | return -(-*this + -x); |
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220 | |
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221 | if (x.m_signexp >> 31) |
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222 | return *this - (-x); |
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223 | |
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224 | /* Ensure *this has the larger exponent (no need for the mantissa to |
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225 | * be larger, as in subtraction). Otherwise, switch. */ |
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226 | if ((m_signexp << 1) < (x.m_signexp << 1)) |
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227 | return x + *this; |
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228 | |
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229 | real ret; |
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230 | |
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231 | int e1 = m_signexp - (1 << 30) + 1; |
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232 | int e2 = x.m_signexp - (1 << 30) + 1; |
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233 | |
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234 | int bigoff = (e1 - e2) / BIGIT_BITS; |
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235 | int off = e1 - e2 - bigoff * BIGIT_BITS; |
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236 | |
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237 | if (bigoff > BIGITS) |
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238 | return *this; |
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239 | |
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240 | ret.m_signexp = m_signexp; |
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241 | |
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242 | uint64_t carry = 0; |
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243 | for (int i = BIGITS; i--; ) |
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244 | { |
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245 | carry += m_mantissa[i]; |
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246 | if (i - bigoff >= 0) |
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247 | carry += x.m_mantissa[i - bigoff] >> off; |
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248 | |
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249 | if (off && i - bigoff > 0) |
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250 | carry += (x.m_mantissa[i - bigoff - 1] << (BIGIT_BITS - off)) & 0xffffffffu; |
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251 | else if (i - bigoff == 0) |
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252 | carry += (uint64_t)1 << (BIGIT_BITS - off); |
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253 | |
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254 | ret.m_mantissa[i] = (uint32_t)carry; |
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255 | carry >>= BIGIT_BITS; |
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256 | } |
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257 | |
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258 | /* Renormalise in case we overflowed the mantissa */ |
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259 | if (carry) |
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260 | { |
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261 | carry--; |
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262 | for (int i = 0; i < BIGITS; i++) |
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263 | { |
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264 | uint32_t tmp = ret.m_mantissa[i]; |
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265 | ret.m_mantissa[i] = ((uint32_t)carry << (BIGIT_BITS - 1)) |
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266 | | (tmp >> 1); |
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267 | carry = tmp & 1u; |
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268 | } |
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269 | ret.m_signexp++; |
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270 | } |
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271 | |
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272 | return ret; |
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273 | } |
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274 | |
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275 | template<> real real::operator -(real const &x) const |
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276 | { |
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277 | if (x.m_signexp << 1 == 0) |
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278 | return *this; |
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279 | |
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280 | /* Ensure both arguments are positive. Otherwise, switch signs, |
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281 | * or replace - with +. */ |
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282 | if (m_signexp >> 31) |
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283 | return -(-*this + x); |
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284 | |
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285 | if (x.m_signexp >> 31) |
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286 | return (*this) + (-x); |
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287 | |
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288 | /* Ensure *this is larger than x */ |
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289 | if (*this < x) |
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290 | return -(x - *this); |
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291 | |
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292 | real ret; |
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293 | |
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294 | int e1 = m_signexp - (1 << 30) + 1; |
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295 | int e2 = x.m_signexp - (1 << 30) + 1; |
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296 | |
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297 | int bigoff = (e1 - e2) / BIGIT_BITS; |
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298 | int off = e1 - e2 - bigoff * BIGIT_BITS; |
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299 | |
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300 | if (bigoff > BIGITS) |
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301 | return *this; |
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302 | |
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303 | ret.m_signexp = m_signexp; |
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304 | |
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305 | /* int64_t instead of uint64_t to preserve sign through shifts */ |
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306 | int64_t carry = 0; |
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307 | for (int i = 0; i < bigoff; i++) |
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308 | { |
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309 | carry -= x.m_mantissa[BIGITS - 1 - i]; |
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310 | /* Emulates a signed shift */ |
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311 | carry >>= BIGIT_BITS; |
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312 | carry |= carry << BIGIT_BITS; |
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313 | } |
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314 | if (bigoff < BIGITS) |
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315 | carry -= x.m_mantissa[BIGITS - 1 - bigoff] & (((int64_t)1 << off) - 1); |
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316 | carry /= (int64_t)1 << off; |
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317 | |
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318 | for (int i = BIGITS; i--; ) |
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319 | { |
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320 | carry += m_mantissa[i]; |
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321 | if (i - bigoff >= 0) |
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322 | carry -= x.m_mantissa[i - bigoff] >> off; |
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323 | |
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324 | if (off && i - bigoff > 0) |
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325 | carry -= (x.m_mantissa[i - bigoff - 1] << (BIGIT_BITS - off)) & 0xffffffffu; |
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326 | else if (i - bigoff == 0) |
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327 | carry -= (uint64_t)1 << (BIGIT_BITS - off); |
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328 | |
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329 | ret.m_mantissa[i] = (uint32_t)carry; |
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330 | carry >>= BIGIT_BITS; |
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331 | carry |= carry << BIGIT_BITS; |
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332 | } |
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333 | |
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334 | carry += 1; |
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335 | |
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336 | /* Renormalise if we underflowed the mantissa */ |
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337 | if (carry == 0) |
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338 | { |
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339 | /* How much do we need to shift the mantissa? FIXME: this could |
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340 | * be computed above */ |
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341 | off = 0; |
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342 | for (int i = 0; i < BIGITS; i++) |
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343 | { |
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344 | if (!ret.m_mantissa[i]) |
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345 | { |
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346 | off += BIGIT_BITS; |
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347 | continue; |
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348 | } |
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349 | |
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350 | for (uint32_t tmp = ret.m_mantissa[i]; tmp < 0x80000000u; tmp <<= 1) |
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351 | off++; |
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352 | break; |
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353 | } |
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354 | if (off == BIGITS * BIGIT_BITS) |
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355 | ret.m_signexp &= 0x80000000u; |
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356 | else |
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357 | { |
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358 | off++; /* Shift one more to get rid of the leading one */ |
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359 | ret.m_signexp -= off; |
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360 | |
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361 | bigoff = off / BIGIT_BITS; |
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362 | off -= bigoff * BIGIT_BITS; |
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363 | |
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364 | for (int i = 0; i < BIGITS; i++) |
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365 | { |
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366 | uint32_t tmp = 0; |
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367 | if (i + bigoff < BIGITS) |
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368 | tmp |= ret.m_mantissa[i + bigoff] << off; |
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369 | if (off && i + bigoff + 1 < BIGITS) |
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370 | tmp |= ret.m_mantissa[i + bigoff + 1] >> (BIGIT_BITS - off); |
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371 | ret.m_mantissa[i] = tmp; |
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372 | } |
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373 | } |
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374 | } |
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375 | |
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376 | return ret; |
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377 | } |
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378 | |
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379 | template<> real real::operator *(real const &x) const |
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380 | { |
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381 | real ret; |
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382 | |
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383 | if (m_signexp << 1 == 0 || x.m_signexp << 1 == 0) |
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384 | { |
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385 | ret = (m_signexp << 1 == 0) ? *this : x; |
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386 | ret.m_signexp ^= x.m_signexp & 0x80000000u; |
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387 | return ret; |
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388 | } |
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389 | |
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390 | ret.m_signexp = (m_signexp ^ x.m_signexp) & 0x80000000u; |
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391 | int e = (m_signexp & 0x7fffffffu) - (1 << 30) + 1 |
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392 | + (x.m_signexp & 0x7fffffffu) - (1 << 30) + 1; |
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393 | |
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394 | /* Accumulate low order product; no need to store it, we just |
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395 | * want the carry value */ |
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396 | uint64_t carry = 0, hicarry = 0, prev; |
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397 | for (int i = 0; i < BIGITS; i++) |
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398 | { |
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399 | for (int j = 0; j < i + 1; j++) |
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400 | { |
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401 | prev = carry; |
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402 | carry += (uint64_t)m_mantissa[BIGITS - 1 - j] |
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403 | * (uint64_t)x.m_mantissa[BIGITS - 1 + j - i]; |
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404 | if (carry < prev) |
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405 | hicarry++; |
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406 | } |
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407 | carry >>= BIGIT_BITS; |
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408 | carry |= hicarry << BIGIT_BITS; |
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409 | hicarry >>= BIGIT_BITS; |
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410 | } |
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411 | |
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412 | for (int i = 0; i < BIGITS; i++) |
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413 | { |
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414 | for (int j = i + 1; j < BIGITS; j++) |
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415 | { |
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416 | prev = carry; |
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417 | carry += (uint64_t)m_mantissa[BIGITS - 1 - j] |
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418 | * (uint64_t)x.m_mantissa[j - 1 - i]; |
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419 | if (carry < prev) |
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420 | hicarry++; |
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421 | } |
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422 | prev = carry; |
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423 | carry += m_mantissa[BIGITS - 1 - i]; |
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424 | carry += x.m_mantissa[BIGITS - 1 - i]; |
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425 | if (carry < prev) |
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426 | hicarry++; |
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427 | ret.m_mantissa[BIGITS - 1 - i] = carry & 0xffffffffu; |
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428 | carry >>= BIGIT_BITS; |
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429 | carry |= hicarry << BIGIT_BITS; |
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430 | hicarry >>= BIGIT_BITS; |
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431 | } |
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432 | |
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433 | /* Renormalise in case we overflowed the mantissa */ |
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434 | if (carry) |
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435 | { |
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436 | carry--; |
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437 | for (int i = 0; i < BIGITS; i++) |
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438 | { |
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439 | uint32_t tmp = (uint32_t)ret.m_mantissa[i]; |
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440 | ret.m_mantissa[i] = ((uint32_t)carry << (BIGIT_BITS - 1)) |
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441 | | (tmp >> 1); |
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442 | carry = tmp & 1u; |
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443 | } |
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444 | e++; |
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445 | } |
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446 | |
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447 | ret.m_signexp |= e + (1 << 30) - 1; |
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448 | |
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449 | return ret; |
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450 | } |
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451 | |
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452 | template<> real real::operator /(real const &x) const |
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453 | { |
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454 | return *this * re(x); |
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455 | } |
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456 | |
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457 | template<> real const &real::operator +=(real const &x) |
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458 | { |
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459 | real tmp = *this; |
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460 | return *this = tmp + x; |
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461 | } |
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462 | |
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463 | template<> real const &real::operator -=(real const &x) |
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464 | { |
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465 | real tmp = *this; |
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466 | return *this = tmp - x; |
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467 | } |
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468 | |
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469 | template<> real const &real::operator *=(real const &x) |
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470 | { |
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471 | real tmp = *this; |
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472 | return *this = tmp * x; |
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473 | } |
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474 | |
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475 | template<> real const &real::operator /=(real const &x) |
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476 | { |
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477 | real tmp = *this; |
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478 | return *this = tmp / x; |
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479 | } |
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480 | |
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481 | template<> bool real::operator ==(real const &x) const |
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482 | { |
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483 | if ((m_signexp << 1) == 0 && (x.m_signexp << 1) == 0) |
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484 | return true; |
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485 | |
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486 | if (m_signexp != x.m_signexp) |
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487 | return false; |
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488 | |
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489 | return memcmp(m_mantissa, x.m_mantissa, BIGITS * sizeof(uint32_t)) == 0; |
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490 | } |
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491 | |
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492 | template<> bool real::operator !=(real const &x) const |
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493 | { |
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494 | return !(*this == x); |
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495 | } |
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496 | |
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497 | template<> bool real::operator <(real const &x) const |
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498 | { |
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499 | /* Ensure both numbers are positive */ |
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500 | if (m_signexp >> 31) |
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501 | return (x.m_signexp >> 31) ? -*this > -x : true; |
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502 | |
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503 | if (x.m_signexp >> 31) |
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504 | return false; |
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505 | |
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506 | /* Compare all relevant bits */ |
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507 | if (m_signexp != x.m_signexp) |
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508 | return m_signexp < x.m_signexp; |
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509 | |
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510 | for (int i = 0; i < BIGITS; i++) |
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511 | if (m_mantissa[i] != x.m_mantissa[i]) |
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512 | return m_mantissa[i] < x.m_mantissa[i]; |
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513 | |
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514 | return false; |
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515 | } |
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516 | |
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517 | template<> bool real::operator <=(real const &x) const |
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518 | { |
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519 | return !(*this > x); |
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520 | } |
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521 | |
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522 | template<> bool real::operator >(real const &x) const |
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523 | { |
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524 | /* Ensure both numbers are positive */ |
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525 | if (m_signexp >> 31) |
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526 | return (x.m_signexp >> 31) ? -*this < -x : false; |
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527 | |
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528 | if (x.m_signexp >> 31) |
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529 | return true; |
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530 | |
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531 | /* Compare all relevant bits */ |
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532 | if (m_signexp != x.m_signexp) |
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533 | return m_signexp > x.m_signexp; |
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534 | |
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535 | for (int i = 0; i < BIGITS; i++) |
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536 | if (m_mantissa[i] != x.m_mantissa[i]) |
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537 | return m_mantissa[i] > x.m_mantissa[i]; |
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538 | |
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539 | return false; |
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540 | } |
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541 | |
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542 | template<> bool real::operator >=(real const &x) const |
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543 | { |
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544 | return !(*this < x); |
---|
545 | } |
---|
546 | |
---|
547 | template<> bool real::operator !() const |
---|
548 | { |
---|
549 | return !(bool)*this; |
---|
550 | } |
---|
551 | |
---|
552 | template<> real::operator bool() const |
---|
553 | { |
---|
554 | /* A real is "true" if it is non-zero (exponent is non-zero) AND |
---|
555 | * not NaN (exponent is not full bits OR higher order mantissa is zero) */ |
---|
556 | uint32_t exponent = m_signexp << 1; |
---|
557 | return exponent && (~exponent || m_mantissa[0] == 0); |
---|
558 | } |
---|
559 | |
---|
560 | template<> real min(real const &a, real const &b) |
---|
561 | { |
---|
562 | return (a < b) ? a : b; |
---|
563 | } |
---|
564 | |
---|
565 | template<> real max(real const &a, real const &b) |
---|
566 | { |
---|
567 | return (a > b) ? a : b; |
---|
568 | } |
---|
569 | |
---|
570 | template<> real clamp(real const &x, real const &a, real const &b) |
---|
571 | { |
---|
572 | return (x < a) ? a : (x > b) ? b : x; |
---|
573 | } |
---|
574 | |
---|
575 | template<> real re(real const &x) |
---|
576 | { |
---|
577 | if (!(x.m_signexp << 1)) |
---|
578 | { |
---|
579 | real ret = x; |
---|
580 | ret.m_signexp = x.m_signexp | 0x7fffffffu; |
---|
581 | ret.m_mantissa[0] = 0; |
---|
582 | return ret; |
---|
583 | } |
---|
584 | |
---|
585 | /* Use the system's float inversion to approximate 1/x */ |
---|
586 | union { float f; uint32_t x; } u = { 1.0f }, v = { 1.0f }; |
---|
587 | v.x |= x.m_mantissa[0] >> 9; |
---|
588 | v.f = 1.0f / v.f; |
---|
589 | |
---|
590 | real ret; |
---|
591 | ret.m_mantissa[0] = v.x << 9; |
---|
592 | |
---|
593 | uint32_t sign = x.m_signexp & 0x80000000u; |
---|
594 | ret.m_signexp = sign; |
---|
595 | |
---|
596 | int exponent = (x.m_signexp & 0x7fffffffu) + 1; |
---|
597 | exponent = -exponent + (v.x >> 23) - (u.x >> 23); |
---|
598 | ret.m_signexp |= (exponent - 1) & 0x7fffffffu; |
---|
599 | |
---|
600 | /* FIXME: 1+log2(BIGITS) steps of Newton-Raphson seems to be enough for |
---|
601 | * convergence, but this hasn't been checked seriously. */ |
---|
602 | for (int i = 1; i <= real::BIGITS; i *= 2) |
---|
603 | ret = ret * (real::R_2 - ret * x); |
---|
604 | |
---|
605 | return ret; |
---|
606 | } |
---|
607 | |
---|
608 | template<> real sqrt(real const &x) |
---|
609 | { |
---|
610 | /* if zero, return x */ |
---|
611 | if (!(x.m_signexp << 1)) |
---|
612 | return x; |
---|
613 | |
---|
614 | /* if negative, return NaN */ |
---|
615 | if (x.m_signexp >> 31) |
---|
616 | { |
---|
617 | real ret; |
---|
618 | ret.m_signexp = 0x7fffffffu; |
---|
619 | ret.m_mantissa[0] = 0xffffu; |
---|
620 | return ret; |
---|
621 | } |
---|
622 | |
---|
623 | /* Use the system's float inversion to approximate 1/sqrt(x). First |
---|
624 | * we construct a float in the [1..4[ range that has roughly the same |
---|
625 | * mantissa as our real. Its exponent is 0 or 1, depending on the |
---|
626 | * partity of x. The final exponent is 0, -1 or -2. We use the final |
---|
627 | * exponent and final mantissa to pre-fill the result. */ |
---|
628 | union { float f; uint32_t x; } u = { 1.0f }, v = { 2.0f }; |
---|
629 | v.x -= ((x.m_signexp & 1) << 23); |
---|
630 | v.x |= x.m_mantissa[0] >> 9; |
---|
631 | v.f = 1.0f / sqrtf(v.f); |
---|
632 | |
---|
633 | real ret; |
---|
634 | ret.m_mantissa[0] = v.x << 9; |
---|
635 | |
---|
636 | uint32_t sign = x.m_signexp & 0x80000000u; |
---|
637 | ret.m_signexp = sign; |
---|
638 | |
---|
639 | uint32_t exponent = (x.m_signexp & 0x7fffffffu); |
---|
640 | exponent = ((1 << 30) + (1 << 29) - 1) - (exponent + 1) / 2; |
---|
641 | exponent = exponent + (v.x >> 23) - (u.x >> 23); |
---|
642 | ret.m_signexp |= exponent & 0x7fffffffu; |
---|
643 | |
---|
644 | /* FIXME: 1+log2(BIGITS) steps of Newton-Raphson seems to be enough for |
---|
645 | * convergence, but this hasn't been checked seriously. */ |
---|
646 | for (int i = 1; i <= real::BIGITS; i *= 2) |
---|
647 | { |
---|
648 | ret = ret * (real::R_3 - ret * ret * x); |
---|
649 | ret.m_signexp--; |
---|
650 | } |
---|
651 | |
---|
652 | return ret * x; |
---|
653 | } |
---|
654 | |
---|
655 | template<> real cbrt(real const &x) |
---|
656 | { |
---|
657 | /* if zero, return x */ |
---|
658 | if (!(x.m_signexp << 1)) |
---|
659 | return x; |
---|
660 | |
---|
661 | /* Use the system's float inversion to approximate cbrt(x). First |
---|
662 | * we construct a float in the [1..8[ range that has roughly the same |
---|
663 | * mantissa as our real. Its exponent is 0, 1 or 2, depending on the |
---|
664 | * value of x. The final exponent is 0 or 1 (special case). We use |
---|
665 | * the final exponent and final mantissa to pre-fill the result. */ |
---|
666 | union { float f; uint32_t x; } u = { 1.0f }, v = { 1.0f }; |
---|
667 | v.x += ((x.m_signexp % 3) << 23); |
---|
668 | v.x |= x.m_mantissa[0] >> 9; |
---|
669 | v.f = powf(v.f, 0.33333333333333333f); |
---|
670 | |
---|
671 | real ret; |
---|
672 | ret.m_mantissa[0] = v.x << 9; |
---|
673 | |
---|
674 | uint32_t sign = x.m_signexp & 0x80000000u; |
---|
675 | ret.m_signexp = sign; |
---|
676 | |
---|
677 | int exponent = (x.m_signexp & 0x7fffffffu) - (1 << 30) + 1; |
---|
678 | exponent = exponent / 3 + (v.x >> 23) - (u.x >> 23); |
---|
679 | ret.m_signexp |= (exponent + (1 << 30) - 1) & 0x7fffffffu; |
---|
680 | |
---|
681 | /* FIXME: 1+log2(BIGITS) steps of Newton-Raphson seems to be enough for |
---|
682 | * convergence, but this hasn't been checked seriously. */ |
---|
683 | for (int i = 1; i <= real::BIGITS; i *= 2) |
---|
684 | { |
---|
685 | static real third = re(real::R_3); |
---|
686 | ret = third * (x / (ret * ret) + (ret / 2)); |
---|
687 | } |
---|
688 | |
---|
689 | return ret; |
---|
690 | } |
---|
691 | |
---|
692 | template<> real pow(real const &x, real const &y) |
---|
693 | { |
---|
694 | if (!y) |
---|
695 | return real::R_1; |
---|
696 | if (!x) |
---|
697 | return real::R_0; |
---|
698 | if (x > real::R_0) |
---|
699 | return exp(y * log(x)); |
---|
700 | else /* x < 0 */ |
---|
701 | { |
---|
702 | /* Odd integer exponent */ |
---|
703 | if (y == (round(y / 2) * 2)) |
---|
704 | return exp(y * log(-x)); |
---|
705 | |
---|
706 | /* Even integer exponent */ |
---|
707 | if (y == round(y)) |
---|
708 | return -exp(y * log(-x)); |
---|
709 | |
---|
710 | /* FIXME: negative nth root */ |
---|
711 | return real::R_0; |
---|
712 | } |
---|
713 | } |
---|
714 | |
---|
715 | static real fast_fact(int x) |
---|
716 | { |
---|
717 | real ret = real::R_1; |
---|
718 | int i = 1, multiplier = 1, exponent = 0; |
---|
719 | |
---|
720 | for (;;) |
---|
721 | { |
---|
722 | if (i++ >= x) |
---|
723 | /* Multiplication is a no-op if multiplier == 1 */ |
---|
724 | return ldexp(ret * multiplier, exponent); |
---|
725 | |
---|
726 | int tmp = i; |
---|
727 | while ((tmp & 1) == 0) |
---|
728 | { |
---|
729 | tmp >>= 1; |
---|
730 | exponent++; |
---|
731 | } |
---|
732 | if (multiplier * tmp / tmp != multiplier) |
---|
733 | { |
---|
734 | ret *= multiplier; |
---|
735 | multiplier = 1; |
---|
736 | } |
---|
737 | multiplier *= tmp; |
---|
738 | } |
---|
739 | } |
---|
740 | |
---|
741 | template<> real gamma(real const &x) |
---|
742 | { |
---|
743 | /* We use Spouge's formula. FIXME: precision is far from acceptable, |
---|
744 | * especially with large values. We need to compute this with higher |
---|
745 | * precision values in order to attain the desired accuracy. It might |
---|
746 | * also be useful to sort the ck values by decreasing absolute value |
---|
747 | * and do the addition in this order. */ |
---|
748 | int a = ceilf(logf(2) / logf(2 * M_PI) * real::BIGITS * real::BIGIT_BITS); |
---|
749 | |
---|
750 | real ret = sqrt(real::R_PI * 2); |
---|
751 | real fact_k_1 = real::R_1; |
---|
752 | |
---|
753 | for (int k = 1; k < a; k++) |
---|
754 | { |
---|
755 | real a_k = (real)(a - k); |
---|
756 | real ck = pow(a_k, (real)((float)k - 0.5)) * exp(a_k) |
---|
757 | / (fact_k_1 * (x + (real)(k - 1))); |
---|
758 | ret += ck; |
---|
759 | fact_k_1 *= (real)-k; |
---|
760 | } |
---|
761 | |
---|
762 | ret *= pow(x + (real)(a - 1), x - (real::R_1 / 2)); |
---|
763 | ret *= exp(-x - (real)(a - 1)); |
---|
764 | |
---|
765 | return ret; |
---|
766 | } |
---|
767 | |
---|
768 | template<> real fabs(real const &x) |
---|
769 | { |
---|
770 | real ret = x; |
---|
771 | ret.m_signexp &= 0x7fffffffu; |
---|
772 | return ret; |
---|
773 | } |
---|
774 | |
---|
775 | static real fast_log(real const &x) |
---|
776 | { |
---|
777 | /* This fast log method is tuned to work on the [1..2] range and |
---|
778 | * no effort whatsoever was made to improve convergence outside this |
---|
779 | * domain of validity. It can converge pretty fast, provided we use |
---|
780 | * the following variable substitutions: |
---|
781 | * y = sqrt(x) |
---|
782 | * z = (y - 1) / (y + 1) |
---|
783 | * |
---|
784 | * And the following identities: |
---|
785 | * ln(x) = 2 ln(y) |
---|
786 | * = 2 ln((1 + z) / (1 - z)) |
---|
787 | * = 4 z (1 + z^2 / 3 + z^4 / 5 + z^6 / 7...) |
---|
788 | * |
---|
789 | * Any additional sqrt() call would halve the convergence time, but |
---|
790 | * would also impact the final precision. For now we stick with one |
---|
791 | * sqrt() call. */ |
---|
792 | real y = sqrt(x); |
---|
793 | real z = (y - real::R_1) / (y + real::R_1), z2 = z * z, zn = z2; |
---|
794 | real sum = real::R_1; |
---|
795 | |
---|
796 | for (int i = 3; ; i += 2) |
---|
797 | { |
---|
798 | real newsum = sum + zn / (real)i; |
---|
799 | if (newsum == sum) |
---|
800 | break; |
---|
801 | sum = newsum; |
---|
802 | zn *= z2; |
---|
803 | } |
---|
804 | |
---|
805 | return z * sum * 4; |
---|
806 | } |
---|
807 | |
---|
808 | template<> real log(real const &x) |
---|
809 | { |
---|
810 | /* Strategy for log(x): if x = 2^E*M then log(x) = E log(2) + log(M), |
---|
811 | * with the property that M is in [1..2[, so fast_log() applies here. */ |
---|
812 | real tmp = x; |
---|
813 | if (x.m_signexp >> 31 || x.m_signexp == 0) |
---|
814 | { |
---|
815 | tmp.m_signexp = 0xffffffffu; |
---|
816 | tmp.m_mantissa[0] = 0xffffffffu; |
---|
817 | return tmp; |
---|
818 | } |
---|
819 | tmp.m_signexp = (1 << 30) - 1; |
---|
820 | return (real)(int)(x.m_signexp - (1 << 30) + 1) * real::R_LN2 |
---|
821 | + fast_log(tmp); |
---|
822 | } |
---|
823 | |
---|
824 | template<> real log2(real const &x) |
---|
825 | { |
---|
826 | /* Strategy for log2(x): see log(x). */ |
---|
827 | real tmp = x; |
---|
828 | if (x.m_signexp >> 31 || x.m_signexp == 0) |
---|
829 | { |
---|
830 | tmp.m_signexp = 0xffffffffu; |
---|
831 | tmp.m_mantissa[0] = 0xffffffffu; |
---|
832 | return tmp; |
---|
833 | } |
---|
834 | tmp.m_signexp = (1 << 30) - 1; |
---|
835 | return (real)(int)(x.m_signexp - (1 << 30) + 1) |
---|
836 | + fast_log(tmp) * real::R_LOG2E; |
---|
837 | } |
---|
838 | |
---|
839 | template<> real log10(real const &x) |
---|
840 | { |
---|
841 | return log(x) * real::R_LOG10E; |
---|
842 | } |
---|
843 | |
---|
844 | static real fast_exp_sub(real const &x, real const &y) |
---|
845 | { |
---|
846 | /* This fast exp method is tuned to work on the [-1..1] range and |
---|
847 | * no effort whatsoever was made to improve convergence outside this |
---|
848 | * domain of validity. The argument y is used for cases where we |
---|
849 | * don't want the leading 1 in the Taylor series. */ |
---|
850 | real ret = real::R_1 - y, xn = x; |
---|
851 | int i = 1; |
---|
852 | |
---|
853 | for (;;) |
---|
854 | { |
---|
855 | real newret = ret + xn; |
---|
856 | if (newret == ret) |
---|
857 | break; |
---|
858 | ret = newret * ++i; |
---|
859 | xn *= x; |
---|
860 | } |
---|
861 | |
---|
862 | return ret / fast_fact(i); |
---|
863 | } |
---|
864 | |
---|
865 | template<> real exp(real const &x) |
---|
866 | { |
---|
867 | /* Strategy for exp(x): the Taylor series does not converge very fast |
---|
868 | * with large positive or negative values. |
---|
869 | * |
---|
870 | * However, we know that the result is going to be in the form M*2^E, |
---|
871 | * where M is the mantissa and E the exponent. We first try to predict |
---|
872 | * a value for E, which is approximately log2(exp(x)) = x / log(2). |
---|
873 | * |
---|
874 | * Let E0 be an integer close to x / log(2). We need to find a value x0 |
---|
875 | * such that exp(x) = 2^E0 * exp(x0). We get x0 = x - E0 log(2). |
---|
876 | * |
---|
877 | * Thus the final algorithm: |
---|
878 | * int E0 = x / log(2) |
---|
879 | * real x0 = x - E0 log(2) |
---|
880 | * real x1 = exp(x0) |
---|
881 | * return x1 * 2^E0 |
---|
882 | */ |
---|
883 | int e0 = x / real::R_LN2; |
---|
884 | real x0 = x - (real)e0 * real::R_LN2; |
---|
885 | real x1 = fast_exp_sub(x0, real::R_0); |
---|
886 | x1.m_signexp += e0; |
---|
887 | return x1; |
---|
888 | } |
---|
889 | |
---|
890 | template<> real exp2(real const &x) |
---|
891 | { |
---|
892 | /* Strategy for exp2(x): see strategy in exp(). */ |
---|
893 | int e0 = x; |
---|
894 | real x0 = x - (real)e0; |
---|
895 | real x1 = fast_exp_sub(x0 * real::R_LN2, real::R_0); |
---|
896 | x1.m_signexp += e0; |
---|
897 | return x1; |
---|
898 | } |
---|
899 | |
---|
900 | template<> real sinh(real const &x) |
---|
901 | { |
---|
902 | /* We cannot always use (exp(x)-exp(-x))/2 because we'll lose |
---|
903 | * accuracy near zero. We only use this identity for |x|>0.5. If |
---|
904 | * |x|<=0.5, we compute exp(x)-1 and exp(-x)-1 instead. */ |
---|
905 | bool near_zero = (fabs(x) < real::R_1 / 2); |
---|
906 | real x1 = near_zero ? fast_exp_sub(x, real::R_1) : exp(x); |
---|
907 | real x2 = near_zero ? fast_exp_sub(-x, real::R_1) : exp(-x); |
---|
908 | return (x1 - x2) / 2; |
---|
909 | } |
---|
910 | |
---|
911 | template<> real tanh(real const &x) |
---|
912 | { |
---|
913 | /* See sinh() for the strategy here */ |
---|
914 | bool near_zero = (fabs(x) < real::R_1 / 2); |
---|
915 | real x1 = near_zero ? fast_exp_sub(x, real::R_1) : exp(x); |
---|
916 | real x2 = near_zero ? fast_exp_sub(-x, real::R_1) : exp(-x); |
---|
917 | real x3 = near_zero ? x1 + x2 + real::R_2 : x1 + x2; |
---|
918 | return (x1 - x2) / x3; |
---|
919 | } |
---|
920 | |
---|
921 | template<> real cosh(real const &x) |
---|
922 | { |
---|
923 | /* No need to worry about accuracy here; maybe the last bit is slightly |
---|
924 | * off, but that's about it. */ |
---|
925 | return (exp(x) + exp(-x)) / 2; |
---|
926 | } |
---|
927 | |
---|
928 | template<> real frexp(real const &x, int *exp) |
---|
929 | { |
---|
930 | if (!x) |
---|
931 | { |
---|
932 | *exp = 0; |
---|
933 | return x; |
---|
934 | } |
---|
935 | |
---|
936 | real ret = x; |
---|
937 | int exponent = (ret.m_signexp & 0x7fffffffu) - (1 << 30) + 1; |
---|
938 | *exp = exponent + 1; |
---|
939 | ret.m_signexp -= exponent + 1; |
---|
940 | return ret; |
---|
941 | } |
---|
942 | |
---|
943 | template<> real ldexp(real const &x, int exp) |
---|
944 | { |
---|
945 | real ret = x; |
---|
946 | if (ret) |
---|
947 | ret.m_signexp += exp; |
---|
948 | return ret; |
---|
949 | } |
---|
950 | |
---|
951 | template<> real modf(real const &x, real *iptr) |
---|
952 | { |
---|
953 | real absx = fabs(x); |
---|
954 | real tmp = floor(absx); |
---|
955 | |
---|
956 | *iptr = copysign(tmp, x); |
---|
957 | return copysign(absx - tmp, x); |
---|
958 | } |
---|
959 | |
---|
960 | template<> real ulp(real const &x) |
---|
961 | { |
---|
962 | real ret = real::R_1; |
---|
963 | if (x) |
---|
964 | ret.m_signexp = x.m_signexp + 1 - real::BIGITS * real::BIGIT_BITS; |
---|
965 | else |
---|
966 | ret.m_signexp = 0; |
---|
967 | return ret; |
---|
968 | } |
---|
969 | |
---|
970 | template<> real nextafter(real const &x, real const &y) |
---|
971 | { |
---|
972 | if (x == y) |
---|
973 | return x; |
---|
974 | else if (x < y) |
---|
975 | return x + ulp(x); |
---|
976 | else |
---|
977 | return x - ulp(x); |
---|
978 | } |
---|
979 | |
---|
980 | template<> real copysign(real const &x, real const &y) |
---|
981 | { |
---|
982 | real ret = x; |
---|
983 | ret.m_signexp &= 0x7fffffffu; |
---|
984 | ret.m_signexp |= y.m_signexp & 0x80000000u; |
---|
985 | return ret; |
---|
986 | } |
---|
987 | |
---|
988 | template<> real floor(real const &x) |
---|
989 | { |
---|
990 | /* Strategy for floor(x): |
---|
991 | * - if negative, return -ceil(-x) |
---|
992 | * - if zero or negative zero, return x |
---|
993 | * - if less than one, return zero |
---|
994 | * - otherwise, if e is the exponent, clear all bits except the |
---|
995 | * first e. */ |
---|
996 | if (x < -real::R_0) |
---|
997 | return -ceil(-x); |
---|
998 | if (!x) |
---|
999 | return x; |
---|
1000 | if (x < real::R_1) |
---|
1001 | return real::R_0; |
---|
1002 | |
---|
1003 | real ret = x; |
---|
1004 | int exponent = x.m_signexp - (1 << 30) + 1; |
---|
1005 | |
---|
1006 | for (int i = 0; i < real::BIGITS; i++) |
---|
1007 | { |
---|
1008 | if (exponent <= 0) |
---|
1009 | ret.m_mantissa[i] = 0; |
---|
1010 | else if (exponent < real::BIGIT_BITS) |
---|
1011 | ret.m_mantissa[i] &= ~((1 << (real::BIGIT_BITS - exponent)) - 1); |
---|
1012 | |
---|
1013 | exponent -= real::BIGIT_BITS; |
---|
1014 | } |
---|
1015 | |
---|
1016 | return ret; |
---|
1017 | } |
---|
1018 | |
---|
1019 | template<> real ceil(real const &x) |
---|
1020 | { |
---|
1021 | /* Strategy for ceil(x): |
---|
1022 | * - if negative, return -floor(-x) |
---|
1023 | * - if x == floor(x), return x |
---|
1024 | * - otherwise, return floor(x) + 1 */ |
---|
1025 | if (x < -real::R_0) |
---|
1026 | return -floor(-x); |
---|
1027 | real ret = floor(x); |
---|
1028 | if (x == ret) |
---|
1029 | return ret; |
---|
1030 | else |
---|
1031 | return ret + real::R_1; |
---|
1032 | } |
---|
1033 | |
---|
1034 | template<> real round(real const &x) |
---|
1035 | { |
---|
1036 | if (x < real::R_0) |
---|
1037 | return -round(-x); |
---|
1038 | |
---|
1039 | return floor(x + (real::R_1 / 2)); |
---|
1040 | } |
---|
1041 | |
---|
1042 | template<> real fmod(real const &x, real const &y) |
---|
1043 | { |
---|
1044 | if (!y) |
---|
1045 | return real::R_0; /* FIXME: return NaN */ |
---|
1046 | |
---|
1047 | if (!x) |
---|
1048 | return x; |
---|
1049 | |
---|
1050 | real tmp = round(x / y); |
---|
1051 | return x - tmp * y; |
---|
1052 | } |
---|
1053 | |
---|
1054 | template<> real sin(real const &x) |
---|
1055 | { |
---|
1056 | int switch_sign = x.m_signexp & 0x80000000u; |
---|
1057 | |
---|
1058 | real absx = fmod(fabs(x), real::R_PI * 2); |
---|
1059 | if (absx > real::R_PI) |
---|
1060 | { |
---|
1061 | absx -= real::R_PI; |
---|
1062 | switch_sign = !switch_sign; |
---|
1063 | } |
---|
1064 | |
---|
1065 | if (absx > real::R_PI_2) |
---|
1066 | absx = real::R_PI - absx; |
---|
1067 | |
---|
1068 | real ret = real::R_0, fact = real::R_1, xn = absx, mx2 = -absx * absx; |
---|
1069 | int i = 1; |
---|
1070 | for (;;) |
---|
1071 | { |
---|
1072 | real newret = ret + xn; |
---|
1073 | if (newret == ret) |
---|
1074 | break; |
---|
1075 | ret = newret * ((i + 1) * (i + 2)); |
---|
1076 | xn *= mx2; |
---|
1077 | i += 2; |
---|
1078 | } |
---|
1079 | ret /= fast_fact(i); |
---|
1080 | |
---|
1081 | /* Propagate sign */ |
---|
1082 | if (switch_sign) |
---|
1083 | ret.m_signexp ^= 0x80000000u; |
---|
1084 | return ret; |
---|
1085 | } |
---|
1086 | |
---|
1087 | template<> real cos(real const &x) |
---|
1088 | { |
---|
1089 | return sin(real::R_PI_2 - x); |
---|
1090 | } |
---|
1091 | |
---|
1092 | template<> real tan(real const &x) |
---|
1093 | { |
---|
1094 | /* Constrain input to [-π,π] */ |
---|
1095 | real y = fmod(x, real::R_PI); |
---|
1096 | |
---|
1097 | /* Constrain input to [-π/2,π/2] */ |
---|
1098 | if (y < -real::R_PI_2) |
---|
1099 | y += real::R_PI; |
---|
1100 | else if (y > real::R_PI_2) |
---|
1101 | y -= real::R_PI; |
---|
1102 | |
---|
1103 | /* In [-π/4,π/4] return sin/cos */ |
---|
1104 | if (fabs(y) <= real::R_PI_4) |
---|
1105 | return sin(y) / cos(y); |
---|
1106 | |
---|
1107 | /* Otherwise, return cos/sin */ |
---|
1108 | if (y > real::R_0) |
---|
1109 | y = real::R_PI_2 - y; |
---|
1110 | else |
---|
1111 | y = -real::R_PI_2 - y; |
---|
1112 | |
---|
1113 | return cos(y) / sin(y); |
---|
1114 | } |
---|
1115 | |
---|
1116 | static inline real asinacos(real const &x, int is_asin, int is_negative) |
---|
1117 | { |
---|
1118 | /* Strategy for asin(): in [-0.5..0.5], use a Taylor series around |
---|
1119 | * zero. In [0.5..1], use asin(x) = π/2 - 2*asin(sqrt((1-x)/2)), and |
---|
1120 | * in [-1..-0.5] just revert the sign. |
---|
1121 | * Strategy for acos(): use acos(x) = π/2 - asin(x) and try not to |
---|
1122 | * lose the precision around x=1. */ |
---|
1123 | real absx = fabs(x); |
---|
1124 | int around_zero = (absx < (real::R_1 / 2)); |
---|
1125 | |
---|
1126 | if (!around_zero) |
---|
1127 | absx = sqrt((real::R_1 - absx) / 2); |
---|
1128 | |
---|
1129 | real ret = absx, xn = absx, x2 = absx * absx, fact1 = 2, fact2 = 1; |
---|
1130 | for (int i = 1; ; i++) |
---|
1131 | { |
---|
1132 | xn *= x2; |
---|
1133 | real mul = (real)(2 * i + 1); |
---|
1134 | real newret = ret + ldexp(fact1 * xn / (mul * fact2), -2 * i); |
---|
1135 | if (newret == ret) |
---|
1136 | break; |
---|
1137 | ret = newret; |
---|
1138 | fact1 *= (real)((2 * i + 1) * (2 * i + 2)); |
---|
1139 | fact2 *= (real)((i + 1) * (i + 1)); |
---|
1140 | } |
---|
1141 | |
---|
1142 | if (is_negative) |
---|
1143 | ret = -ret; |
---|
1144 | |
---|
1145 | if (around_zero) |
---|
1146 | ret = is_asin ? ret : real::R_PI_2 - ret; |
---|
1147 | else |
---|
1148 | { |
---|
1149 | real adjust = is_negative ? real::R_PI : real::R_0; |
---|
1150 | if (is_asin) |
---|
1151 | ret = real::R_PI_2 - adjust - ret * 2; |
---|
1152 | else |
---|
1153 | ret = adjust + ret * 2; |
---|
1154 | } |
---|
1155 | |
---|
1156 | return ret; |
---|
1157 | } |
---|
1158 | |
---|
1159 | template<> real asin(real const &x) |
---|
1160 | { |
---|
1161 | return asinacos(x, 1, x.m_signexp >> 31); |
---|
1162 | } |
---|
1163 | |
---|
1164 | template<> real acos(real const &x) |
---|
1165 | { |
---|
1166 | return asinacos(x, 0, x.m_signexp >> 31); |
---|
1167 | } |
---|
1168 | |
---|
1169 | template<> real atan(real const &x) |
---|
1170 | { |
---|
1171 | /* Computing atan(x): we choose a different Taylor series depending on |
---|
1172 | * the value of x to help with convergence. |
---|
1173 | * |
---|
1174 | * If |x| < 0.5 we evaluate atan(y) near 0: |
---|
1175 | * atan(y) = y - y^3/3 + y^5/5 - y^7/7 + y^9/9 ... |
---|
1176 | * |
---|
1177 | * If 0.5 <= |x| < 1.5 we evaluate atan(1+y) near 0: |
---|
1178 | * atan(1+y) = π/4 + y/(1*2^1) - y^2/(2*2^1) + y^3/(3*2^2) |
---|
1179 | * - y^5/(5*2^3) + y^6/(6*2^3) - y^7/(7*2^4) |
---|
1180 | * + y^9/(9*2^5) - y^10/(10*2^5) + y^11/(11*2^6) ... |
---|
1181 | * |
---|
1182 | * If 1.5 <= |x| < 2 we evaluate atan(sqrt(3)+2y) near 0: |
---|
1183 | * atan(sqrt(3)+2y) = π/3 + 1/2 y - sqrt(3)/2 y^2/2 |
---|
1184 | * + y^3/3 - sqrt(3)/2 y^4/4 + 1/2 y^5/5 |
---|
1185 | * - 1/2 y^7/7 + sqrt(3)/2 y^8/8 |
---|
1186 | * - y^9/9 + sqrt(3)/2 y^10/10 - 1/2 y^11/11 |
---|
1187 | * + 1/2 y^13/13 - sqrt(3)/2 y^14/14 |
---|
1188 | * + y^15/15 - sqrt(3)/2 y^16/16 + 1/2 y^17/17 ... |
---|
1189 | * |
---|
1190 | * If |x| >= 2 we evaluate atan(y) near +∞: |
---|
1191 | * atan(y) = π/2 - y^-1 + y^-3/3 - y^-5/5 + y^-7/7 - y^-9/9 ... |
---|
1192 | */ |
---|
1193 | real absx = fabs(x); |
---|
1194 | |
---|
1195 | if (absx < (real::R_1 / 2)) |
---|
1196 | { |
---|
1197 | real ret = x, xn = x, mx2 = -x * x; |
---|
1198 | for (int i = 3; ; i += 2) |
---|
1199 | { |
---|
1200 | xn *= mx2; |
---|
1201 | real newret = ret + xn / (real)i; |
---|
1202 | if (newret == ret) |
---|
1203 | break; |
---|
1204 | ret = newret; |
---|
1205 | } |
---|
1206 | return ret; |
---|
1207 | } |
---|
1208 | |
---|
1209 | real ret = 0; |
---|
1210 | |
---|
1211 | if (absx < (real::R_3 / 2)) |
---|
1212 | { |
---|
1213 | real y = real::R_1 - absx; |
---|
1214 | real yn = y, my2 = -y * y; |
---|
1215 | for (int i = 0; ; i += 2) |
---|
1216 | { |
---|
1217 | real newret = ret + ldexp(yn / (real)(2 * i + 1), -i - 1); |
---|
1218 | yn *= y; |
---|
1219 | newret += ldexp(yn / (real)(2 * i + 2), -i - 1); |
---|
1220 | yn *= y; |
---|
1221 | newret += ldexp(yn / (real)(2 * i + 3), -i - 2); |
---|
1222 | if (newret == ret) |
---|
1223 | break; |
---|
1224 | ret = newret; |
---|
1225 | yn *= my2; |
---|
1226 | } |
---|
1227 | ret = real::R_PI_4 - ret; |
---|
1228 | } |
---|
1229 | else if (absx < real::R_2) |
---|
1230 | { |
---|
1231 | real y = (absx - real::R_SQRT3) / 2; |
---|
1232 | real yn = y, my2 = -y * y; |
---|
1233 | for (int i = 1; ; i += 6) |
---|
1234 | { |
---|
1235 | real newret = ret + ((yn / (real)i) / 2); |
---|
1236 | yn *= y; |
---|
1237 | newret -= (real::R_SQRT3 / 2) * yn / (real)(i + 1); |
---|
1238 | yn *= y; |
---|
1239 | newret += yn / (real)(i + 2); |
---|
1240 | yn *= y; |
---|
1241 | newret -= (real::R_SQRT3 / 2) * yn / (real)(i + 3); |
---|
1242 | yn *= y; |
---|
1243 | newret += (yn / (real)(i + 4)) / 2; |
---|
1244 | if (newret == ret) |
---|
1245 | break; |
---|
1246 | ret = newret; |
---|
1247 | yn *= my2; |
---|
1248 | } |
---|
1249 | ret = real::R_PI_3 + ret; |
---|
1250 | } |
---|
1251 | else |
---|
1252 | { |
---|
1253 | real y = re(absx); |
---|
1254 | real yn = y, my2 = -y * y; |
---|
1255 | ret = y; |
---|
1256 | for (int i = 3; ; i += 2) |
---|
1257 | { |
---|
1258 | yn *= my2; |
---|
1259 | real newret = ret + yn / (real)i; |
---|
1260 | if (newret == ret) |
---|
1261 | break; |
---|
1262 | ret = newret; |
---|
1263 | } |
---|
1264 | ret = real::R_PI_2 - ret; |
---|
1265 | } |
---|
1266 | |
---|
1267 | /* Propagate sign */ |
---|
1268 | ret.m_signexp |= (x.m_signexp & 0x80000000u); |
---|
1269 | return ret; |
---|
1270 | } |
---|
1271 | |
---|
1272 | template<> real atan2(real const &y, real const &x) |
---|
1273 | { |
---|
1274 | if (!y) |
---|
1275 | { |
---|
1276 | if ((x.m_signexp >> 31) == 0) |
---|
1277 | return y; |
---|
1278 | if (y.m_signexp >> 31) |
---|
1279 | return -real::R_PI; |
---|
1280 | return real::R_PI; |
---|
1281 | } |
---|
1282 | |
---|
1283 | if (!x) |
---|
1284 | { |
---|
1285 | if (y.m_signexp >> 31) |
---|
1286 | return -real::R_PI; |
---|
1287 | return real::R_PI; |
---|
1288 | } |
---|
1289 | |
---|
1290 | /* FIXME: handle the Inf and NaN cases */ |
---|
1291 | real z = y / x; |
---|
1292 | real ret = atan(z); |
---|
1293 | if (x < real::R_0) |
---|
1294 | ret += (y > real::R_0) ? real::R_PI : -real::R_PI; |
---|
1295 | return ret; |
---|
1296 | } |
---|
1297 | |
---|
1298 | template<> void real::hexprint() const |
---|
1299 | { |
---|
1300 | printf("%08x", m_signexp); |
---|
1301 | for (int i = 0; i < BIGITS; i++) |
---|
1302 | printf(" %08x", m_mantissa[i]); |
---|
1303 | printf("\n"); |
---|
1304 | } |
---|
1305 | |
---|
1306 | template<> void real::print(int ndigits) const |
---|
1307 | { |
---|
1308 | real x = *this; |
---|
1309 | |
---|
1310 | if (x.m_signexp >> 31) |
---|
1311 | { |
---|
1312 | printf("-"); |
---|
1313 | x = -x; |
---|
1314 | } |
---|
1315 | |
---|
1316 | if (!x) |
---|
1317 | { |
---|
1318 | printf("0.0\n"); |
---|
1319 | return; |
---|
1320 | } |
---|
1321 | |
---|
1322 | /* Normalise x so that mantissa is in [1..9.999] */ |
---|
1323 | /* FIXME: better use int64_t when the cast is implemented */ |
---|
1324 | int exponent = ceil(log10(x)); |
---|
1325 | x /= pow(R_10, (real)exponent); |
---|
1326 | |
---|
1327 | if (x < R_1) |
---|
1328 | { |
---|
1329 | x *= R_10; |
---|
1330 | exponent--; |
---|
1331 | } |
---|
1332 | |
---|
1333 | /* Print digits */ |
---|
1334 | for (int i = 0; i < ndigits; i++) |
---|
1335 | { |
---|
1336 | int digit = (int)floor(x); |
---|
1337 | printf("%i", digit); |
---|
1338 | if (i == 0) |
---|
1339 | printf("."); |
---|
1340 | x -= real(digit); |
---|
1341 | x *= R_10; |
---|
1342 | } |
---|
1343 | |
---|
1344 | /* Print exponent information */ |
---|
1345 | if (exponent < 0) |
---|
1346 | printf("e-%i", -exponent); |
---|
1347 | else if (exponent > 0) |
---|
1348 | printf("e+%i", exponent); |
---|
1349 | |
---|
1350 | printf("\n"); |
---|
1351 | } |
---|
1352 | |
---|
1353 | static real fast_pi() |
---|
1354 | { |
---|
1355 | /* Approximate Pi using Machin's formula: 16*atan(1/5)-4*atan(1/239) */ |
---|
1356 | real ret = 0, x0 = 5, x1 = 239; |
---|
1357 | real const m0 = -x0 * x0, m1 = -x1 * x1, r16 = 16, r4 = 4; |
---|
1358 | |
---|
1359 | for (int i = 1; ; i += 2) |
---|
1360 | { |
---|
1361 | real newret = ret + r16 / (x0 * (real)i) - r4 / (x1 * (real)i); |
---|
1362 | if (newret == ret) |
---|
1363 | break; |
---|
1364 | ret = newret; |
---|
1365 | x0 *= m0; |
---|
1366 | x1 *= m1; |
---|
1367 | } |
---|
1368 | |
---|
1369 | return ret; |
---|
1370 | } |
---|
1371 | |
---|
1372 | template<> real const real::R_0 = (real)0.0; |
---|
1373 | template<> real const real::R_1 = (real)1.0; |
---|
1374 | template<> real const real::R_2 = (real)2.0; |
---|
1375 | template<> real const real::R_3 = (real)3.0; |
---|
1376 | template<> real const real::R_10 = (real)10.0; |
---|
1377 | |
---|
1378 | /* |
---|
1379 | * Initialisation order is important here: |
---|
1380 | * - fast_log() requires R_1 |
---|
1381 | * - log() requires R_LN2 |
---|
1382 | * - re() require R_2 |
---|
1383 | * - exp() requires R_0, R_1, R_LN2 |
---|
1384 | * - sqrt() requires R_3 |
---|
1385 | */ |
---|
1386 | template<> real const real::R_LN2 = fast_log(R_2); |
---|
1387 | template<> real const real::R_LN10 = log(R_10); |
---|
1388 | template<> real const real::R_LOG2E = re(R_LN2); |
---|
1389 | template<> real const real::R_LOG10E = re(R_LN10); |
---|
1390 | template<> real const real::R_E = exp(R_1); |
---|
1391 | template<> real const real::R_PI = fast_pi(); |
---|
1392 | template<> real const real::R_PI_2 = R_PI / 2; |
---|
1393 | template<> real const real::R_PI_3 = R_PI / R_3; |
---|
1394 | template<> real const real::R_PI_4 = R_PI / 4; |
---|
1395 | template<> real const real::R_1_PI = re(R_PI); |
---|
1396 | template<> real const real::R_2_PI = R_1_PI * 2; |
---|
1397 | template<> real const real::R_2_SQRTPI = re(sqrt(R_PI)) * 2; |
---|
1398 | template<> real const real::R_SQRT2 = sqrt(R_2); |
---|
1399 | template<> real const real::R_SQRT3 = sqrt(R_3); |
---|
1400 | template<> real const real::R_SQRT1_2 = R_SQRT2 / 2; |
---|
1401 | |
---|
1402 | } /* namespace lol */ |
---|
1403 | |
---|