1 | // |
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2 | // Lol Engine |
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3 | // |
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4 | // Copyright: (c) 2010-2011 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 | #include <cstring> |
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16 | #include <cstdio> |
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17 | |
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18 | #include "core.h" |
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19 | |
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20 | using namespace std; |
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21 | |
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22 | namespace lol |
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23 | { |
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24 | |
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25 | real::real(float f) { *this = (double)f; } |
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26 | real::real(int i) { *this = (double)i; } |
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27 | real::real(unsigned int i) { *this = (double)i; } |
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28 | |
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29 | real::real(double d) |
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30 | { |
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31 | union { double d; uint64_t x; } u = { d }; |
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32 | |
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33 | uint32_t sign = (u.x >> 63) << 31; |
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34 | uint32_t exponent = (u.x << 1) >> 53; |
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35 | |
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36 | switch (exponent) |
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37 | { |
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38 | case 0x00: |
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39 | m_signexp = sign; |
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40 | break; |
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41 | case 0x7ff: |
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42 | m_signexp = sign | 0x7fffffffu; |
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43 | break; |
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44 | default: |
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45 | m_signexp = sign | (exponent + (1 << 30) - (1 << 10)); |
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46 | break; |
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47 | } |
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48 | |
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49 | m_mantissa[0] = u.x >> 36; |
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50 | m_mantissa[1] = u.x >> 20; |
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51 | m_mantissa[2] = u.x >> 4; |
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52 | m_mantissa[3] = u.x << 12; |
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53 | memset(m_mantissa + 4, 0, sizeof(m_mantissa) - 4 * sizeof(m_mantissa[0])); |
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54 | } |
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55 | |
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56 | real::operator float() const { return (float)(double)(*this); } |
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57 | real::operator int() const { return (int)(double)(*this); } |
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58 | real::operator unsigned int() const { return (unsigned int)(double)(*this); } |
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59 | |
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60 | real::operator double() const |
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61 | { |
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62 | union { double d; uint64_t x; } u; |
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63 | |
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64 | /* Get sign */ |
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65 | u.x = m_signexp >> 31; |
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66 | u.x <<= 11; |
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67 | |
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68 | /* Compute new exponent */ |
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69 | uint32_t exponent = (m_signexp << 1) >> 1; |
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70 | int e = (int)exponent - (1 << 30) + (1 << 10); |
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71 | |
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72 | if (e < 0) |
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73 | u.x <<= 52; |
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74 | else if (e >= 0x7ff) |
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75 | { |
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76 | u.x |= 0x7ff; |
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77 | u.x <<= 52; |
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78 | } |
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79 | else |
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80 | { |
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81 | u.x |= e; |
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82 | |
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83 | /* Store mantissa if necessary */ |
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84 | u.x <<= 16; |
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85 | u.x |= m_mantissa[0]; |
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86 | u.x <<= 16; |
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87 | u.x |= m_mantissa[1]; |
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88 | u.x <<= 16; |
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89 | u.x |= m_mantissa[2]; |
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90 | u.x <<= 4; |
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91 | u.x |= m_mantissa[3] >> 12; |
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92 | /* Rounding */ |
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93 | u.x += (m_mantissa[3] >> 11) & 1; |
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94 | } |
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95 | |
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96 | return u.d; |
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97 | } |
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98 | |
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99 | real real::operator -() const |
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100 | { |
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101 | real ret = *this; |
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102 | ret.m_signexp ^= 0x80000000u; |
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103 | return ret; |
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104 | } |
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105 | |
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106 | real real::operator +(real const &x) const |
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107 | { |
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108 | if (x.m_signexp << 1 == 0) |
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109 | return *this; |
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110 | |
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111 | /* Ensure both arguments are positive. Otherwise, switch signs, |
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112 | * or replace + with -. */ |
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113 | if (m_signexp >> 31) |
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114 | return -(-*this + -x); |
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115 | |
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116 | if (x.m_signexp >> 31) |
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117 | return *this - (-x); |
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118 | |
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119 | /* Ensure *this has the larger exponent (no need for the mantissa to |
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120 | * be larger, as in subtraction). Otherwise, switch. */ |
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121 | if ((m_signexp << 1) < (x.m_signexp << 1)) |
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122 | return x + *this; |
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123 | |
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124 | real ret; |
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125 | |
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126 | int e1 = m_signexp - (1 << 30) + 1; |
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127 | int e2 = x.m_signexp - (1 << 30) + 1; |
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128 | |
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129 | int bigoff = (e1 - e2) / (sizeof(uint16_t) * 8); |
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130 | int off = e1 - e2 - bigoff * (sizeof(uint16_t) * 8); |
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131 | |
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132 | if (bigoff > BIGITS) |
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133 | return *this; |
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134 | |
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135 | ret.m_signexp = m_signexp; |
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136 | |
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137 | uint32_t carry = 0; |
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138 | for (int i = BIGITS; i--; ) |
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139 | { |
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140 | carry += m_mantissa[i]; |
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141 | if (i - bigoff >= 0) |
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142 | carry += x.m_mantissa[i - bigoff] >> off; |
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143 | |
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144 | if (i - bigoff > 0) |
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145 | carry += (x.m_mantissa[i - bigoff - 1] << (16 - off)) & 0xffffu; |
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146 | else if (i - bigoff == 0) |
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147 | carry += 0x0001u << (16 - off); |
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148 | |
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149 | ret.m_mantissa[i] = carry; |
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150 | carry >>= 16; |
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151 | } |
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152 | |
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153 | /* Renormalise in case we overflowed the mantissa */ |
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154 | if (carry) |
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155 | { |
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156 | carry--; |
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157 | for (int i = 0; i < BIGITS; i++) |
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158 | { |
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159 | uint16_t tmp = ret.m_mantissa[i]; |
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160 | ret.m_mantissa[i] = (carry << 15) | (tmp >> 1); |
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161 | carry = tmp & 0x0001u; |
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162 | } |
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163 | ret.m_signexp++; |
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164 | } |
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165 | |
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166 | return ret; |
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167 | } |
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168 | |
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169 | real real::operator -(real const &x) const |
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170 | { |
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171 | if (x.m_signexp << 1 == 0) |
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172 | return *this; |
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173 | |
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174 | /* Ensure both arguments are positive. Otherwise, switch signs, |
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175 | * or replace - with +. */ |
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176 | if (m_signexp >> 31) |
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177 | return -(-*this + x); |
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178 | |
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179 | if (x.m_signexp >> 31) |
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180 | return (*this) + (-x); |
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181 | |
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182 | /* Ensure *this is larger than x */ |
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183 | if (*this < x) |
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184 | return -(x - *this); |
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185 | |
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186 | real ret; |
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187 | |
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188 | int e1 = m_signexp - (1 << 30) + 1; |
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189 | int e2 = x.m_signexp - (1 << 30) + 1; |
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190 | |
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191 | int bigoff = (e1 - e2) / (sizeof(uint16_t) * 8); |
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192 | int off = e1 - e2 - bigoff * (sizeof(uint16_t) * 8); |
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193 | |
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194 | if (bigoff > BIGITS) |
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195 | return *this; |
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196 | |
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197 | ret.m_signexp = m_signexp; |
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198 | |
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199 | int32_t carry = 0; |
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200 | for (int i = 0; i < bigoff; i++) |
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201 | { |
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202 | carry -= x.m_mantissa[BIGITS - i]; |
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203 | carry = (carry & 0xffff0000u) | (carry >> 16); |
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204 | } |
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205 | carry -= x.m_mantissa[BIGITS - 1 - bigoff] & ((1 << off) - 1); |
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206 | carry /= (1 << off); |
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207 | |
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208 | for (int i = BIGITS; i--; ) |
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209 | { |
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210 | carry += m_mantissa[i]; |
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211 | if (i - bigoff >= 0) |
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212 | carry -= x.m_mantissa[i - bigoff] >> off; |
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213 | |
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214 | if (i - bigoff > 0) |
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215 | carry -= (x.m_mantissa[i - bigoff - 1] << (16 - off)) & 0xffffu; |
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216 | else if (i - bigoff == 0) |
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217 | carry -= 0x0001u << (16 - off); |
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218 | |
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219 | ret.m_mantissa[i] = carry; |
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220 | carry = (carry & 0xffff0000u) | (carry >> 16); |
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221 | } |
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222 | |
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223 | carry += 1; |
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224 | |
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225 | /* Renormalise if we underflowed the mantissa */ |
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226 | if (carry == 0) |
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227 | { |
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228 | /* How much do we need to shift the mantissa? FIXME: this could |
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229 | * be computed above */ |
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230 | off = 0; |
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231 | for (int i = 0; i < BIGITS; i++) |
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232 | { |
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233 | if (!ret.m_mantissa[i]) |
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234 | { |
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235 | off += sizeof(uint16_t) * 8; |
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236 | continue; |
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237 | } |
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238 | |
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239 | for (uint16_t tmp = ret.m_mantissa[i]; tmp < 0x8000u; tmp <<= 1) |
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240 | off++; |
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241 | break; |
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242 | } |
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243 | if (off == BIGITS * sizeof(uint16_t) * 8) |
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244 | ret.m_signexp &= 0x80000000u; |
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245 | else |
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246 | { |
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247 | off++; /* Shift one more to get rid of the leading one */ |
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248 | ret.m_signexp -= off; |
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249 | |
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250 | bigoff = off / (sizeof(uint16_t) * 8); |
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251 | off -= bigoff * sizeof(uint16_t) * 8; |
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252 | |
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253 | for (int i = 0; i < BIGITS; i++) |
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254 | { |
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255 | uint16_t tmp = 0; |
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256 | if (i + bigoff < BIGITS) |
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257 | tmp |= ret.m_mantissa[i + bigoff] << off; |
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258 | if (i + bigoff + 1 < BIGITS) |
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259 | tmp |= ret.m_mantissa[i + bigoff + 1] >> (16 - off); |
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260 | ret.m_mantissa[i] = tmp; |
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261 | } |
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262 | } |
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263 | } |
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264 | |
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265 | return ret; |
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266 | } |
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267 | |
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268 | real real::operator *(real const &x) const |
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269 | { |
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270 | real ret; |
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271 | |
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272 | if (m_signexp << 1 == 0 || x.m_signexp << 1 == 0) |
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273 | { |
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274 | ret = (m_signexp << 1 == 0) ? *this : x; |
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275 | ret.m_signexp ^= x.m_signexp & 0x80000000u; |
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276 | return ret; |
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277 | } |
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278 | |
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279 | ret.m_signexp = (m_signexp ^ x.m_signexp) & 0x80000000u; |
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280 | int e = (m_signexp & 0x7fffffffu) - (1 << 30) + 1 |
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281 | + (x.m_signexp & 0x7fffffffu) - (1 << 30) + 1; |
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282 | |
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283 | /* Accumulate low order product; no need to store it, we just |
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284 | * want the carry value */ |
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285 | uint64_t carry = 0; |
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286 | for (int i = 0; i < BIGITS; i++) |
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287 | { |
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288 | for (int j = 0; j < i + 1; j++) |
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289 | carry += (uint32_t)m_mantissa[BIGITS - 1 - j] |
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290 | * (uint32_t)x.m_mantissa[BIGITS - 1 + j - i]; |
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291 | carry >>= 16; |
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292 | } |
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293 | |
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294 | for (int i = 0; i < BIGITS; i++) |
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295 | { |
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296 | for (int j = i + 1; j < BIGITS; j++) |
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297 | carry += (uint32_t)m_mantissa[BIGITS - 1 - j] |
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298 | * (uint32_t)x.m_mantissa[j - 1 - i]; |
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299 | |
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300 | carry += m_mantissa[BIGITS - 1 - i]; |
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301 | carry += x.m_mantissa[BIGITS - 1 - i]; |
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302 | ret.m_mantissa[BIGITS - 1 - i] = carry & 0xffffu; |
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303 | carry >>= 16; |
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304 | } |
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305 | |
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306 | /* Renormalise in case we overflowed the mantissa */ |
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307 | if (carry) |
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308 | { |
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309 | carry--; |
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310 | for (int i = 0; i < BIGITS; i++) |
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311 | { |
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312 | uint16_t tmp = ret.m_mantissa[i]; |
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313 | ret.m_mantissa[i] = (carry << 15) | (tmp >> 1); |
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314 | carry = tmp & 0x0001u; |
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315 | } |
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316 | e++; |
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317 | } |
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318 | |
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319 | ret.m_signexp |= e + (1 << 30) - 1; |
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320 | |
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321 | return ret; |
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322 | } |
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323 | |
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324 | real real::operator /(real const &x) const |
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325 | { |
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326 | return *this * re(x); |
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327 | } |
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328 | |
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329 | real &real::operator +=(real const &x) |
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330 | { |
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331 | real tmp = *this; |
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332 | return *this = tmp + x; |
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333 | } |
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334 | |
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335 | real &real::operator -=(real const &x) |
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336 | { |
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337 | real tmp = *this; |
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338 | return *this = tmp - x; |
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339 | } |
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340 | |
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341 | real &real::operator *=(real const &x) |
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342 | { |
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343 | real tmp = *this; |
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344 | return *this = tmp * x; |
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345 | } |
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346 | |
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347 | real &real::operator /=(real const &x) |
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348 | { |
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349 | real tmp = *this; |
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350 | return *this = tmp / x; |
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351 | } |
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352 | |
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353 | real real::operator <<(int x) const |
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354 | { |
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355 | real tmp = *this; |
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356 | return tmp <<= x; |
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357 | } |
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358 | |
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359 | real real::operator >>(int x) const |
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360 | { |
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361 | real tmp = *this; |
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362 | return tmp >>= x; |
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363 | } |
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364 | |
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365 | real &real::operator <<=(int x) |
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366 | { |
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367 | if (m_signexp << 1) |
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368 | m_signexp += x; |
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369 | return *this; |
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370 | } |
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371 | |
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372 | real &real::operator >>=(int x) |
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373 | { |
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374 | if (m_signexp << 1) |
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375 | m_signexp -= x; |
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376 | return *this; |
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377 | } |
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378 | |
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379 | bool real::operator ==(real const &x) const |
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380 | { |
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381 | if ((m_signexp << 1) == 0 && (x.m_signexp << 1) == 0) |
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382 | return true; |
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383 | |
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384 | if (m_signexp != x.m_signexp) |
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385 | return false; |
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386 | |
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387 | return memcmp(m_mantissa, x.m_mantissa, sizeof(m_mantissa)) == 0; |
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388 | } |
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389 | |
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390 | bool real::operator !=(real const &x) const |
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391 | { |
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392 | return !(*this == x); |
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393 | } |
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394 | |
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395 | bool real::operator <(real const &x) const |
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396 | { |
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397 | /* Ensure both numbers are positive */ |
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398 | if (m_signexp >> 31) |
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399 | return (x.m_signexp >> 31) ? -*this > -x : true; |
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400 | |
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401 | if (x.m_signexp >> 31) |
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402 | return false; |
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403 | |
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404 | /* Compare all relevant bits */ |
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405 | if (m_signexp != x.m_signexp) |
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406 | return m_signexp < x.m_signexp; |
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407 | |
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408 | for (int i = 0; i < BIGITS; i++) |
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409 | if (m_mantissa[i] != x.m_mantissa[i]) |
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410 | return m_mantissa[i] < x.m_mantissa[i]; |
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411 | |
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412 | return false; |
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413 | } |
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414 | |
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415 | bool real::operator <=(real const &x) const |
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416 | { |
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417 | return !(*this > x); |
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418 | } |
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419 | |
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420 | bool real::operator >(real const &x) const |
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421 | { |
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422 | /* Ensure both numbers are positive */ |
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423 | if (m_signexp >> 31) |
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424 | return (x.m_signexp >> 31) ? -*this < -x : false; |
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425 | |
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426 | if (x.m_signexp >> 31) |
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427 | return true; |
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428 | |
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429 | /* Compare all relevant bits */ |
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430 | if (m_signexp != x.m_signexp) |
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431 | return m_signexp > x.m_signexp; |
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432 | |
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433 | for (int i = 0; i < BIGITS; i++) |
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434 | if (m_mantissa[i] != x.m_mantissa[i]) |
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435 | return m_mantissa[i] > x.m_mantissa[i]; |
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436 | |
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437 | return false; |
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438 | } |
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439 | |
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440 | bool real::operator >=(real const &x) const |
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441 | { |
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442 | return !(*this < x); |
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443 | } |
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444 | |
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445 | bool real::operator !() const |
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446 | { |
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447 | return !(bool)*this; |
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448 | } |
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449 | |
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450 | real::operator bool() const |
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451 | { |
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452 | /* A real is "true" if it is non-zero (exponent is non-zero) AND |
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453 | * not NaN (exponent is not full bits OR higher order mantissa is zero) */ |
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454 | uint32_t exponent = m_signexp << 1; |
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455 | return exponent && (~exponent || m_mantissa[0] == 0); |
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456 | } |
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457 | |
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458 | real re(real const &x) |
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459 | { |
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460 | if (!(x.m_signexp << 1)) |
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461 | { |
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462 | real ret = x; |
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463 | ret.m_signexp = x.m_signexp | 0x7fffffffu; |
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464 | ret.m_mantissa[0] = 0; |
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465 | return ret; |
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466 | } |
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467 | |
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468 | /* Use the system's float inversion to approximate 1/x */ |
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469 | union { float f; uint32_t x; } u = { 1.0f }, v = { 1.0f }; |
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470 | v.x |= (uint32_t)x.m_mantissa[0] << 7; |
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471 | v.x |= (uint32_t)x.m_mantissa[1] >> 9; |
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472 | v.f = 1.0 / v.f; |
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473 | |
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474 | real ret; |
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475 | ret.m_mantissa[0] = (v.x >> 7) & 0xffffu; |
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476 | ret.m_mantissa[1] = (v.x << 9) & 0xffffu; |
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477 | |
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478 | uint32_t sign = x.m_signexp & 0x80000000u; |
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479 | ret.m_signexp = sign; |
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480 | |
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481 | int exponent = (x.m_signexp & 0x7fffffffu) + 1; |
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482 | exponent = -exponent + (v.x >> 23) - (u.x >> 23); |
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483 | ret.m_signexp |= (exponent - 1) & 0x7fffffffu; |
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484 | |
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485 | /* Five steps of Newton-Raphson seems enough for 32-bigit reals. */ |
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486 | real two = 2; |
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487 | ret = ret * (two - ret * x); |
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488 | ret = ret * (two - ret * x); |
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489 | ret = ret * (two - ret * x); |
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490 | ret = ret * (two - ret * x); |
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491 | ret = ret * (two - ret * x); |
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492 | |
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493 | return ret; |
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494 | } |
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495 | |
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496 | real sqrt(real const &x) |
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497 | { |
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498 | /* if zero, return x */ |
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499 | if (!(x.m_signexp << 1)) |
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500 | return x; |
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501 | |
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502 | /* if negative, return NaN */ |
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503 | if (x.m_signexp >> 31) |
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504 | { |
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505 | real ret; |
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506 | ret.m_signexp = 0x7fffffffu; |
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507 | ret.m_mantissa[0] = 0xffffu; |
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508 | return ret; |
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509 | } |
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510 | |
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511 | /* Use the system's float inversion to approximate 1/sqrt(x). First |
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512 | * we construct a float in the [1..4[ range that has roughly the same |
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513 | * mantissa as our real. Its exponent is 0 or 1, depending on the |
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514 | * partity of x. The final exponent is 0, -1 or -2. We use the final |
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515 | * exponent and final mantissa to pre-fill the result. */ |
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516 | union { float f; uint32_t x; } u = { 1.0f }, v = { 2.0f }; |
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517 | v.x -= ((x.m_signexp & 1) << 23); |
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518 | v.x |= (uint32_t)x.m_mantissa[0] << 7; |
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519 | v.x |= (uint32_t)x.m_mantissa[1] >> 9; |
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520 | v.f = 1.0 / sqrtf(v.f); |
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521 | |
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522 | real ret; |
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523 | ret.m_mantissa[0] = (v.x >> 7) & 0xffffu; |
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524 | ret.m_mantissa[1] = (v.x << 9) & 0xffffu; |
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525 | |
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526 | uint32_t sign = x.m_signexp & 0x80000000u; |
---|
527 | ret.m_signexp = sign; |
---|
528 | |
---|
529 | int exponent = (x.m_signexp & 0x7fffffffu) - ((1 << 30) - 1); |
---|
530 | exponent = - (exponent / 2) + (v.x >> 23) - (u.x >> 23); |
---|
531 | ret.m_signexp |= (exponent + ((1 << 30) - 1)) & 0x7fffffffu; |
---|
532 | |
---|
533 | /* Five steps of Newton-Raphson seems enough for 32-bigit reals. */ |
---|
534 | real three = 3; |
---|
535 | ret = ret * (three - ret * ret * x); |
---|
536 | ret.m_signexp--; |
---|
537 | ret = ret * (three - ret * ret * x); |
---|
538 | ret.m_signexp--; |
---|
539 | ret = ret * (three - ret * ret * x); |
---|
540 | ret.m_signexp--; |
---|
541 | ret = ret * (three - ret * ret * x); |
---|
542 | ret.m_signexp--; |
---|
543 | ret = ret * (three - ret * ret * x); |
---|
544 | ret.m_signexp--; |
---|
545 | |
---|
546 | return ret * x; |
---|
547 | } |
---|
548 | |
---|
549 | real fabs(real const &x) |
---|
550 | { |
---|
551 | real ret = x; |
---|
552 | ret.m_signexp &= 0x7fffffffu; |
---|
553 | return ret; |
---|
554 | } |
---|
555 | |
---|
556 | static real fastlog(real const &x) |
---|
557 | { |
---|
558 | /* This fast log method is tuned to work on the [1..2] range and |
---|
559 | * no effort whatsoever was made to improve convergence outside this |
---|
560 | * domain of validity. It can converge pretty fast, provided we use |
---|
561 | * the following variable substitutions: |
---|
562 | * y = sqrt(x) |
---|
563 | * z = (y - 1) / (y + 1) |
---|
564 | * |
---|
565 | * And the following identities: |
---|
566 | * ln(x) = 2 ln(y) |
---|
567 | * = 2 ln((1 + z) / (1 - z)) |
---|
568 | * = 4 z (1 + z^2 / 3 + z^4 / 5 + z^6 / 7...) |
---|
569 | * |
---|
570 | * Any additional sqrt() call would halve the convergence time, but |
---|
571 | * would also impact the final precision. For now we stick with one |
---|
572 | * sqrt() call. */ |
---|
573 | real y = sqrt(x); |
---|
574 | real z = (y - (real)1) / (y + (real)1), z2 = z * z, zn = z2; |
---|
575 | real sum = 1.0; |
---|
576 | |
---|
577 | for (int i = 3; i < 200; i += 2) |
---|
578 | { |
---|
579 | sum += zn / (real)i; |
---|
580 | zn *= z2; |
---|
581 | } |
---|
582 | |
---|
583 | return z * (sum << 2); |
---|
584 | } |
---|
585 | |
---|
586 | static real LOG_2 = fastlog((real)2); |
---|
587 | |
---|
588 | real log(real const &x) |
---|
589 | { |
---|
590 | /* Strategy for log(x): if x = 2^E*M then log(x) = E log(2) + log(M), |
---|
591 | * with the property that M is in [1..2[, so fastlog() applies here. */ |
---|
592 | real tmp = x; |
---|
593 | if (x.m_signexp >> 31 || x.m_signexp == 0) |
---|
594 | { |
---|
595 | tmp.m_signexp = 0xffffffffu; |
---|
596 | tmp.m_mantissa[0] = 0xffffu; |
---|
597 | return tmp; |
---|
598 | } |
---|
599 | tmp.m_signexp = (1 << 30) - 1; |
---|
600 | return (real)(x.m_signexp - (1 << 30) + 1) * LOG_2 + fastlog(tmp); |
---|
601 | } |
---|
602 | |
---|
603 | real exp(real const &x) |
---|
604 | { |
---|
605 | /* Strategy for exp(x): the Taylor series does not converge very fast |
---|
606 | * with large positive or negative values. |
---|
607 | * |
---|
608 | * However, we know that the result is going to be in the form M*2^E, |
---|
609 | * where M is the mantissa and E the exponent. We first try to predict |
---|
610 | * a value for E, which is approximately log2(exp(x)) = x / log(2). |
---|
611 | * |
---|
612 | * Let E0 be an integer close to x / log(2). We need to find a value x0 |
---|
613 | * such that exp(x) = 2^E0 * exp(x0). We get x0 = x - E0 log(2). |
---|
614 | * |
---|
615 | * Thus the final algorithm: |
---|
616 | * int E0 = x / log(2) |
---|
617 | * real x0 = x - E0 log(2) |
---|
618 | * real x1 = exp(x0) |
---|
619 | * return x1 * 2^E0 |
---|
620 | */ |
---|
621 | int e0 = x / LOG_2; |
---|
622 | real x0 = x - (real)e0 * LOG_2; |
---|
623 | real x1 = 1.0, fact = 1.0, xn = x0; |
---|
624 | |
---|
625 | for (int i = 1; i < 100; i++) |
---|
626 | { |
---|
627 | fact *= (real)i; |
---|
628 | x1 += xn / fact; |
---|
629 | xn *= x0; |
---|
630 | } |
---|
631 | |
---|
632 | x1.m_signexp += e0; |
---|
633 | return x1; |
---|
634 | } |
---|
635 | |
---|
636 | real sin(real const &x) |
---|
637 | { |
---|
638 | real ret = 0.0, fact = 1.0, xn = x, x2 = x * x; |
---|
639 | |
---|
640 | for (int i = 1; ; i += 2) |
---|
641 | { |
---|
642 | real newret = ret + xn / fact; |
---|
643 | if (ret == newret) |
---|
644 | break; |
---|
645 | ret = newret; |
---|
646 | xn *= x2; |
---|
647 | fact *= (real)(-(i + 1) * (i + 2)); |
---|
648 | } |
---|
649 | |
---|
650 | return ret; |
---|
651 | } |
---|
652 | |
---|
653 | real cos(real const &x) |
---|
654 | { |
---|
655 | real ret = 0.0, fact = 1.0, xn = 1.0, x2 = x * x; |
---|
656 | |
---|
657 | for (int i = 1; ; i += 2) |
---|
658 | { |
---|
659 | real newret = ret + xn / fact; |
---|
660 | if (ret == newret) |
---|
661 | break; |
---|
662 | ret = newret; |
---|
663 | xn *= x2; |
---|
664 | fact *= (real)(-i * (i + 1)); |
---|
665 | } |
---|
666 | |
---|
667 | return ret; |
---|
668 | } |
---|
669 | |
---|
670 | void real::print(int ndigits) const |
---|
671 | { |
---|
672 | real const r1 = 1, r10 = 10; |
---|
673 | real x = *this; |
---|
674 | |
---|
675 | if (x.m_signexp >> 31) |
---|
676 | { |
---|
677 | printf("-"); |
---|
678 | x = -x; |
---|
679 | } |
---|
680 | |
---|
681 | /* Normalise x so that mantissa is in [1..9.999] */ |
---|
682 | int exponent = 0; |
---|
683 | if (x.m_signexp) |
---|
684 | { |
---|
685 | for (real div = r1, newdiv; true; div = newdiv) |
---|
686 | { |
---|
687 | newdiv = div * r10; |
---|
688 | if (x < newdiv) |
---|
689 | { |
---|
690 | x /= div; |
---|
691 | break; |
---|
692 | } |
---|
693 | exponent++; |
---|
694 | } |
---|
695 | for (real mul = 1, newx; true; mul *= r10) |
---|
696 | { |
---|
697 | newx = x * mul; |
---|
698 | if (newx >= r1) |
---|
699 | { |
---|
700 | x = newx; |
---|
701 | break; |
---|
702 | } |
---|
703 | exponent--; |
---|
704 | } |
---|
705 | } |
---|
706 | |
---|
707 | /* Print digits */ |
---|
708 | for (int i = 0; i < ndigits; i++) |
---|
709 | { |
---|
710 | int digit = (int)x; |
---|
711 | printf("%i", digit); |
---|
712 | if (i == 0) |
---|
713 | printf("."); |
---|
714 | x -= real(digit); |
---|
715 | x *= r10; |
---|
716 | } |
---|
717 | |
---|
718 | /* Print exponent information */ |
---|
719 | if (exponent < 0) |
---|
720 | printf("e-%i", -exponent); |
---|
721 | else if (exponent > 0) |
---|
722 | printf("e+%i", exponent); |
---|
723 | |
---|
724 | printf("\n"); |
---|
725 | } |
---|
726 | |
---|
727 | static real fast_pi() |
---|
728 | { |
---|
729 | /* Approximate Pi using Machin's formula: 16*atan(1/5)-4*atan(1/239) */ |
---|
730 | real ret = 0.0, x0 = 5.0, x1 = 239.0; |
---|
731 | real const m0 = -x0 * x0, m1 = -x1 * x1, r16 = 16.0, r4 = 4.0; |
---|
732 | |
---|
733 | /* Degree 240 is required for 512-bit mantissa precision */ |
---|
734 | for (int i = 1; i < 240; i += 2) |
---|
735 | { |
---|
736 | ret += r16 / (x0 * (real)i) - r4 / (x1 * (real)i); |
---|
737 | x0 *= m0; |
---|
738 | x1 *= m1; |
---|
739 | } |
---|
740 | |
---|
741 | return ret; |
---|
742 | } |
---|
743 | |
---|
744 | real const real::R_0 = (real)0.0; |
---|
745 | real const real::R_1 = (real)1.0; |
---|
746 | real const real::R_2 = (real)2.0; |
---|
747 | real const real::R_10 = (real)10.0; |
---|
748 | |
---|
749 | real const real::R_E = exp(R_1); |
---|
750 | real const real::R_LN2 = log(R_2); |
---|
751 | real const real::R_LN10 = log(R_10); |
---|
752 | real const real::R_LOG2E = re(R_LN2); |
---|
753 | real const real::R_LOG10E = re(R_LN10); |
---|
754 | real const real::R_PI = fast_pi(); |
---|
755 | real const real::R_PI_2 = R_PI >> 1; |
---|
756 | real const real::R_PI_4 = R_PI >> 2; |
---|
757 | real const real::R_1_PI = re(R_PI); |
---|
758 | real const real::R_2_PI = R_1_PI << 1; |
---|
759 | real const real::R_2_SQRTPI = re(sqrt(R_PI)) << 1; |
---|
760 | real const real::R_SQRT2 = sqrt(R_2); |
---|
761 | real const real::R_SQRT1_2 = R_SQRT2 >> 1; |
---|
762 | |
---|
763 | } /* namespace lol */ |
---|
764 | |
---|