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 | return *this; |
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102 | } |
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103 | |
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104 | real real::operator -() const |
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105 | { |
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106 | real ret = *this; |
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107 | ret.m_signexp ^= 0x80000000u; |
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108 | return ret; |
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109 | } |
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110 | |
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111 | real real::operator +(real const &x) const |
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112 | { |
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113 | if (x.m_signexp << 1 == 0) |
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114 | return *this; |
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115 | |
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116 | /* Ensure both arguments are positive. Otherwise, switch signs, |
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117 | * or replace + with -. */ |
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118 | if (m_signexp >> 31) |
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119 | return -(-*this + -x); |
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120 | |
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121 | if (x.m_signexp >> 31) |
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122 | return *this - (-x); |
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123 | |
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124 | /* Ensure *this has the larger exponent (no need for the mantissa to |
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125 | * be larger, as in subtraction). Otherwise, switch. */ |
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126 | if ((m_signexp << 1) < (x.m_signexp << 1)) |
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127 | return x + *this; |
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128 | |
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129 | real ret; |
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130 | |
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131 | int e1 = m_signexp - (1 << 30) + 1; |
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132 | int e2 = x.m_signexp - (1 << 30) + 1; |
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133 | |
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134 | int bigoff = (e1 - e2) / (sizeof(uint16_t) * 8); |
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135 | int off = e1 - e2 - bigoff * (sizeof(uint16_t) * 8); |
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136 | |
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137 | if (bigoff > BIGITS) |
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138 | return *this; |
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139 | |
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140 | ret.m_signexp = m_signexp; |
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141 | |
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142 | uint32_t carry = 0; |
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143 | for (int i = BIGITS; i--; ) |
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144 | { |
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145 | carry += m_mantissa[i]; |
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146 | if (i - bigoff >= 0) |
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147 | carry += x.m_mantissa[i - bigoff] >> off; |
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148 | |
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149 | if (i - bigoff > 0) |
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150 | carry += (x.m_mantissa[i - bigoff - 1] << (16 - off)) & 0xffffu; |
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151 | else if (i - bigoff == 0) |
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152 | carry += 0x0001u << (16 - off); |
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153 | |
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154 | ret.m_mantissa[i] = carry; |
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155 | carry >>= 16; |
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156 | } |
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157 | |
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158 | /* Renormalise in case we overflowed the mantissa */ |
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159 | if (carry) |
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160 | { |
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161 | carry--; |
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162 | for (int i = 0; i < BIGITS; i++) |
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163 | { |
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164 | uint16_t tmp = ret.m_mantissa[i]; |
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165 | ret.m_mantissa[i] = (carry << 15) | (tmp >> 1); |
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166 | carry = tmp & 0x0001u; |
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167 | } |
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168 | ret.m_signexp++; |
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169 | } |
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170 | |
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171 | return ret; |
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172 | } |
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173 | |
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174 | real real::operator -(real const &x) const |
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175 | { |
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176 | if (x.m_signexp << 1 == 0) |
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177 | return *this; |
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178 | |
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179 | /* Ensure both arguments are positive. Otherwise, switch signs, |
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180 | * or replace - with +. */ |
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181 | if (m_signexp >> 31) |
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182 | return -(-*this + x); |
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183 | |
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184 | if (x.m_signexp >> 31) |
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185 | return (*this) + (-x); |
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186 | |
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187 | /* Ensure *this is larger than x */ |
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188 | if (*this < x) |
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189 | return -(x - *this); |
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190 | |
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191 | real ret; |
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192 | |
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193 | int e1 = m_signexp - (1 << 30) + 1; |
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194 | int e2 = x.m_signexp - (1 << 30) + 1; |
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195 | |
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196 | int bigoff = (e1 - e2) / (sizeof(uint16_t) * 8); |
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197 | int off = e1 - e2 - bigoff * (sizeof(uint16_t) * 8); |
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198 | |
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199 | if (bigoff > BIGITS) |
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200 | return *this; |
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201 | |
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202 | ret.m_signexp = m_signexp; |
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203 | |
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204 | int32_t carry = 0; |
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205 | for (int i = 0; i < bigoff; i++) |
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206 | { |
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207 | carry -= x.m_mantissa[BIGITS - i]; |
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208 | carry = (carry & 0xffff0000u) | (carry >> 16); |
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209 | } |
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210 | carry -= x.m_mantissa[BIGITS - 1 - bigoff] & ((1 << off) - 1); |
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211 | carry /= (1 << off); |
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212 | |
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213 | for (int i = BIGITS; i--; ) |
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214 | { |
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215 | carry += m_mantissa[i]; |
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216 | if (i - bigoff >= 0) |
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217 | carry -= x.m_mantissa[i - bigoff] >> off; |
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218 | |
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219 | if (i - bigoff > 0) |
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220 | carry -= (x.m_mantissa[i - bigoff - 1] << (16 - off)) & 0xffffu; |
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221 | else if (i - bigoff == 0) |
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222 | carry -= 0x0001u << (16 - off); |
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223 | |
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224 | ret.m_mantissa[i] = carry; |
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225 | carry = (carry & 0xffff0000u) | (carry >> 16); |
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226 | } |
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227 | |
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228 | carry += 1; |
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229 | |
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230 | /* Renormalise if we underflowed the mantissa */ |
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231 | if (carry == 0) |
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232 | { |
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233 | /* How much do we need to shift the mantissa? FIXME: this could |
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234 | * be computed above */ |
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235 | off = 0; |
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236 | for (int i = 0; i < BIGITS; i++) |
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237 | { |
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238 | if (!ret.m_mantissa[i]) |
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239 | { |
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240 | off += sizeof(uint16_t) * 8; |
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241 | continue; |
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242 | } |
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243 | |
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244 | for (uint16_t tmp = ret.m_mantissa[i]; tmp < 0x8000u; tmp <<= 1) |
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245 | off++; |
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246 | break; |
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247 | } |
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248 | if (off == BIGITS * sizeof(uint16_t) * 8) |
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249 | ret.m_signexp &= 0x80000000u; |
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250 | else |
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251 | { |
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252 | off++; /* Shift one more to get rid of the leading one */ |
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253 | ret.m_signexp -= off; |
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254 | |
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255 | bigoff = off / (sizeof(uint16_t) * 8); |
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256 | off -= bigoff * sizeof(uint16_t) * 8; |
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257 | |
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258 | for (int i = 0; i < BIGITS; i++) |
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259 | { |
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260 | uint16_t tmp = 0; |
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261 | if (i + bigoff < BIGITS) |
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262 | tmp |= ret.m_mantissa[i + bigoff] << off; |
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263 | if (i + bigoff + 1 < BIGITS) |
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264 | tmp |= ret.m_mantissa[i + bigoff + 1] >> (16 - off); |
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265 | ret.m_mantissa[i] = tmp; |
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266 | } |
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267 | } |
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268 | } |
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269 | |
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270 | return ret; |
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271 | } |
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272 | |
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273 | real real::operator *(real const &x) const |
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274 | { |
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275 | real ret; |
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276 | |
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277 | if (m_signexp << 1 == 0 || x.m_signexp << 1 == 0) |
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278 | { |
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279 | ret = (m_signexp << 1 == 0) ? *this : x; |
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280 | ret.m_signexp ^= x.m_signexp & 0x80000000u; |
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281 | return ret; |
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282 | } |
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283 | |
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284 | ret.m_signexp = (m_signexp ^ x.m_signexp) & 0x80000000u; |
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285 | int e = (m_signexp & 0x7fffffffu) - (1 << 30) + 1 |
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286 | + (x.m_signexp & 0x7fffffffu) - (1 << 30) + 1; |
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287 | |
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288 | /* Accumulate low order product; no need to store it, we just |
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289 | * want the carry value */ |
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290 | uint64_t carry = 0; |
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291 | for (int i = 0; i < BIGITS; i++) |
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292 | { |
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293 | for (int j = 0; j < i + 1; j++) |
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294 | carry += (uint32_t)m_mantissa[BIGITS - 1 - j] |
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295 | * (uint32_t)x.m_mantissa[BIGITS - 1 + j - i]; |
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296 | carry >>= 16; |
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297 | } |
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298 | |
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299 | for (int i = 0; i < BIGITS; i++) |
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300 | { |
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301 | for (int j = i + 1; j < BIGITS; j++) |
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302 | carry += (uint32_t)m_mantissa[BIGITS - 1 - j] |
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303 | * (uint32_t)x.m_mantissa[j - 1 - i]; |
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304 | |
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305 | carry += m_mantissa[BIGITS - 1 - i]; |
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306 | carry += x.m_mantissa[BIGITS - 1 - i]; |
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307 | ret.m_mantissa[BIGITS - 1 - i] = carry & 0xffffu; |
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308 | carry >>= 16; |
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309 | } |
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310 | |
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311 | /* Renormalise in case we overflowed the mantissa */ |
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312 | if (carry) |
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313 | { |
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314 | carry--; |
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315 | for (int i = 0; i < BIGITS; i++) |
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316 | { |
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317 | uint16_t tmp = ret.m_mantissa[i]; |
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318 | ret.m_mantissa[i] = (carry << 15) | (tmp >> 1); |
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319 | carry = tmp & 0x0001u; |
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320 | } |
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321 | e++; |
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322 | } |
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323 | |
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324 | ret.m_signexp |= e + (1 << 30) - 1; |
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325 | |
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326 | return ret; |
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327 | } |
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328 | |
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329 | real real::operator /(real const &x) const |
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330 | { |
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331 | return *this * re(x); |
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332 | } |
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333 | |
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334 | real &real::operator +=(real const &x) |
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335 | { |
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336 | real tmp = *this; |
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337 | return *this = tmp + x; |
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338 | } |
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339 | |
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340 | real &real::operator -=(real const &x) |
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341 | { |
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342 | real tmp = *this; |
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343 | return *this = tmp - x; |
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344 | } |
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345 | |
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346 | real &real::operator *=(real const &x) |
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347 | { |
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348 | real tmp = *this; |
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349 | return *this = tmp * x; |
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350 | } |
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351 | |
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352 | real &real::operator /=(real const &x) |
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353 | { |
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354 | real tmp = *this; |
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355 | return *this = tmp / x; |
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356 | } |
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357 | |
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358 | real real::operator <<(int x) const |
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359 | { |
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360 | real tmp = *this; |
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361 | return tmp <<= x; |
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362 | } |
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363 | |
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364 | real real::operator >>(int x) const |
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365 | { |
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366 | real tmp = *this; |
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367 | return tmp >>= x; |
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368 | } |
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369 | |
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370 | real &real::operator <<=(int x) |
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371 | { |
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372 | if (m_signexp << 1) |
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373 | m_signexp += x; |
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374 | return *this; |
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375 | } |
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376 | |
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377 | real &real::operator >>=(int x) |
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378 | { |
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379 | if (m_signexp << 1) |
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380 | m_signexp -= x; |
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381 | return *this; |
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382 | } |
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383 | |
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384 | bool real::operator ==(real const &x) const |
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385 | { |
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386 | if ((m_signexp << 1) == 0 && (x.m_signexp << 1) == 0) |
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387 | return true; |
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388 | |
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389 | if (m_signexp != x.m_signexp) |
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390 | return false; |
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391 | |
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392 | return memcmp(m_mantissa, x.m_mantissa, sizeof(m_mantissa)) == 0; |
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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 | return !(*this == x); |
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398 | } |
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399 | |
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400 | bool real::operator <(real const &x) const |
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401 | { |
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402 | /* Ensure both numbers are positive */ |
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403 | if (m_signexp >> 31) |
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404 | return (x.m_signexp >> 31) ? -*this > -x : true; |
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405 | |
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406 | if (x.m_signexp >> 31) |
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407 | return false; |
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408 | |
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409 | /* Compare all relevant bits */ |
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410 | if (m_signexp != x.m_signexp) |
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411 | return m_signexp < x.m_signexp; |
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412 | |
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413 | for (int i = 0; i < BIGITS; i++) |
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414 | if (m_mantissa[i] != x.m_mantissa[i]) |
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415 | return m_mantissa[i] < x.m_mantissa[i]; |
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416 | |
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417 | return false; |
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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 | return !(*this > x); |
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423 | } |
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424 | |
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425 | bool real::operator >(real const &x) const |
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426 | { |
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427 | /* Ensure both numbers are positive */ |
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428 | if (m_signexp >> 31) |
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429 | return (x.m_signexp >> 31) ? -*this < -x : false; |
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430 | |
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431 | if (x.m_signexp >> 31) |
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432 | return true; |
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433 | |
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434 | /* Compare all relevant bits */ |
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435 | if (m_signexp != x.m_signexp) |
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436 | return m_signexp > x.m_signexp; |
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437 | |
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438 | for (int i = 0; i < BIGITS; i++) |
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439 | if (m_mantissa[i] != x.m_mantissa[i]) |
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440 | return m_mantissa[i] > x.m_mantissa[i]; |
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441 | |
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442 | return false; |
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443 | } |
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444 | |
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445 | bool real::operator >=(real const &x) const |
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446 | { |
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447 | return !(*this < x); |
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448 | } |
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449 | |
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450 | bool real::operator !() const |
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451 | { |
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452 | return !(bool)*this; |
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453 | } |
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454 | |
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455 | real::operator bool() const |
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456 | { |
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457 | /* A real is "true" if it is non-zero (exponent is non-zero) AND |
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458 | * not NaN (exponent is not full bits OR higher order mantissa is zero) */ |
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459 | uint32_t exponent = m_signexp << 1; |
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460 | return exponent && (~exponent || m_mantissa[0] == 0); |
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461 | } |
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462 | |
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463 | real re(real const &x) |
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464 | { |
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465 | if (!(x.m_signexp << 1)) |
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466 | { |
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467 | real ret = x; |
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468 | ret.m_signexp = x.m_signexp | 0x7fffffffu; |
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469 | ret.m_mantissa[0] = 0; |
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470 | return ret; |
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471 | } |
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472 | |
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473 | /* Use the system's float inversion to approximate 1/x */ |
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474 | union { float f; uint32_t x; } u = { 1.0f }, v = { 1.0f }; |
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475 | v.x |= (uint32_t)x.m_mantissa[0] << 7; |
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476 | v.x |= (uint32_t)x.m_mantissa[1] >> 9; |
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477 | v.f = 1.0 / v.f; |
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478 | |
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479 | real ret; |
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480 | ret.m_mantissa[0] = (v.x >> 7) & 0xffffu; |
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481 | ret.m_mantissa[1] = (v.x << 9) & 0xffffu; |
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482 | |
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483 | uint32_t sign = x.m_signexp & 0x80000000u; |
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484 | ret.m_signexp = sign; |
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485 | |
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486 | int exponent = (x.m_signexp & 0x7fffffffu) + 1; |
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487 | exponent = -exponent + (v.x >> 23) - (u.x >> 23); |
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488 | ret.m_signexp |= (exponent - 1) & 0x7fffffffu; |
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489 | |
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490 | /* Five steps of Newton-Raphson seems enough for 32-bigit reals. */ |
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491 | real two = 2; |
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492 | ret = ret * (two - ret * x); |
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493 | ret = ret * (two - ret * x); |
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494 | ret = ret * (two - ret * x); |
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495 | ret = ret * (two - ret * x); |
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496 | ret = ret * (two - ret * x); |
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497 | |
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498 | return ret; |
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499 | } |
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500 | |
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501 | real sqrt(real const &x) |
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502 | { |
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503 | /* if zero, return x */ |
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504 | if (!(x.m_signexp << 1)) |
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505 | return x; |
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506 | |
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507 | /* if negative, return NaN */ |
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508 | if (x.m_signexp >> 31) |
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509 | { |
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510 | real ret; |
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511 | ret.m_signexp = 0x7fffffffu; |
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512 | ret.m_mantissa[0] = 0xffffu; |
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513 | return ret; |
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514 | } |
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515 | |
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516 | /* Use the system's float inversion to approximate 1/sqrt(x). First |
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517 | * we construct a float in the [1..4[ range that has roughly the same |
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518 | * mantissa as our real. Its exponent is 0 or 1, depending on the |
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519 | * partity of x. The final exponent is 0, -1 or -2. We use the final |
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520 | * exponent and final mantissa to pre-fill the result. */ |
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521 | union { float f; uint32_t x; } u = { 1.0f }, v = { 2.0f }; |
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522 | v.x -= ((x.m_signexp & 1) << 23); |
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523 | v.x |= (uint32_t)x.m_mantissa[0] << 7; |
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524 | v.x |= (uint32_t)x.m_mantissa[1] >> 9; |
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525 | v.f = 1.0 / sqrtf(v.f); |
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526 | |
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527 | real ret; |
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528 | ret.m_mantissa[0] = (v.x >> 7) & 0xffffu; |
---|
529 | ret.m_mantissa[1] = (v.x << 9) & 0xffffu; |
---|
530 | |
---|
531 | uint32_t sign = x.m_signexp & 0x80000000u; |
---|
532 | ret.m_signexp = sign; |
---|
533 | |
---|
534 | uint32_t exponent = (x.m_signexp & 0x7fffffffu); |
---|
535 | exponent = ((1 << 30) + (1 << 29) -1) - (exponent + 1) / 2; |
---|
536 | exponent = exponent + (v.x >> 23) - (u.x >> 23); |
---|
537 | ret.m_signexp |= exponent & 0x7fffffffu; |
---|
538 | |
---|
539 | /* Five steps of Newton-Raphson seems enough for 32-bigit reals. */ |
---|
540 | real three = 3; |
---|
541 | ret = ret * (three - ret * ret * x); |
---|
542 | ret.m_signexp--; |
---|
543 | ret = ret * (three - ret * ret * x); |
---|
544 | ret.m_signexp--; |
---|
545 | ret = ret * (three - ret * ret * x); |
---|
546 | ret.m_signexp--; |
---|
547 | ret = ret * (three - ret * ret * x); |
---|
548 | ret.m_signexp--; |
---|
549 | ret = ret * (three - ret * ret * x); |
---|
550 | ret.m_signexp--; |
---|
551 | |
---|
552 | return ret * x; |
---|
553 | } |
---|
554 | |
---|
555 | real fabs(real const &x) |
---|
556 | { |
---|
557 | real ret = x; |
---|
558 | ret.m_signexp &= 0x7fffffffu; |
---|
559 | return ret; |
---|
560 | } |
---|
561 | |
---|
562 | static real fastlog(real const &x) |
---|
563 | { |
---|
564 | /* This fast log method is tuned to work on the [1..2] range and |
---|
565 | * no effort whatsoever was made to improve convergence outside this |
---|
566 | * domain of validity. It can converge pretty fast, provided we use |
---|
567 | * the following variable substitutions: |
---|
568 | * y = sqrt(x) |
---|
569 | * z = (y - 1) / (y + 1) |
---|
570 | * |
---|
571 | * And the following identities: |
---|
572 | * ln(x) = 2 ln(y) |
---|
573 | * = 2 ln((1 + z) / (1 - z)) |
---|
574 | * = 4 z (1 + z^2 / 3 + z^4 / 5 + z^6 / 7...) |
---|
575 | * |
---|
576 | * Any additional sqrt() call would halve the convergence time, but |
---|
577 | * would also impact the final precision. For now we stick with one |
---|
578 | * sqrt() call. */ |
---|
579 | real y = sqrt(x); |
---|
580 | real z = (y - (real)1) / (y + (real)1), z2 = z * z, zn = z2; |
---|
581 | real sum = 1.0; |
---|
582 | |
---|
583 | for (int i = 3; i < 200; i += 2) |
---|
584 | { |
---|
585 | sum += zn / (real)i; |
---|
586 | zn *= z2; |
---|
587 | } |
---|
588 | |
---|
589 | return z * (sum << 2); |
---|
590 | } |
---|
591 | |
---|
592 | static real LOG_2 = fastlog((real)2); |
---|
593 | |
---|
594 | real log(real const &x) |
---|
595 | { |
---|
596 | /* Strategy for log(x): if x = 2^E*M then log(x) = E log(2) + log(M), |
---|
597 | * with the property that M is in [1..2[, so fastlog() applies here. */ |
---|
598 | real tmp = x; |
---|
599 | if (x.m_signexp >> 31 || x.m_signexp == 0) |
---|
600 | { |
---|
601 | tmp.m_signexp = 0xffffffffu; |
---|
602 | tmp.m_mantissa[0] = 0xffffu; |
---|
603 | return tmp; |
---|
604 | } |
---|
605 | tmp.m_signexp = (1 << 30) - 1; |
---|
606 | return (real)(x.m_signexp - (1 << 30) + 1) * LOG_2 + fastlog(tmp); |
---|
607 | } |
---|
608 | |
---|
609 | real exp(real const &x) |
---|
610 | { |
---|
611 | /* Strategy for exp(x): the Taylor series does not converge very fast |
---|
612 | * with large positive or negative values. |
---|
613 | * |
---|
614 | * However, we know that the result is going to be in the form M*2^E, |
---|
615 | * where M is the mantissa and E the exponent. We first try to predict |
---|
616 | * a value for E, which is approximately log2(exp(x)) = x / log(2). |
---|
617 | * |
---|
618 | * Let E0 be an integer close to x / log(2). We need to find a value x0 |
---|
619 | * such that exp(x) = 2^E0 * exp(x0). We get x0 = x - E0 log(2). |
---|
620 | * |
---|
621 | * Thus the final algorithm: |
---|
622 | * int E0 = x / log(2) |
---|
623 | * real x0 = x - E0 log(2) |
---|
624 | * real x1 = exp(x0) |
---|
625 | * return x1 * 2^E0 |
---|
626 | */ |
---|
627 | int e0 = x / LOG_2; |
---|
628 | real x0 = x - (real)e0 * LOG_2; |
---|
629 | real x1 = 1.0, fact = 1.0, xn = x0; |
---|
630 | |
---|
631 | for (int i = 1; i < 100; i++) |
---|
632 | { |
---|
633 | fact *= (real)i; |
---|
634 | x1 += xn / fact; |
---|
635 | xn *= x0; |
---|
636 | } |
---|
637 | |
---|
638 | x1.m_signexp += e0; |
---|
639 | return x1; |
---|
640 | } |
---|
641 | |
---|
642 | real floor(real const &x) |
---|
643 | { |
---|
644 | /* Strategy for floor(x): |
---|
645 | * - if negative, return -ceil(-x) |
---|
646 | * - if zero or negative zero, return x |
---|
647 | * - if less than one, return zero |
---|
648 | * - otherwise, if e is the exponent, clear all bits except the |
---|
649 | * first e. */ |
---|
650 | if (x < -real::R_0) |
---|
651 | return -ceil(-x); |
---|
652 | if (!x) |
---|
653 | return x; |
---|
654 | if (x < real::R_1) |
---|
655 | return real::R_0; |
---|
656 | |
---|
657 | real ret = x; |
---|
658 | int exponent = x.m_signexp - (1 << 30) + 1; |
---|
659 | |
---|
660 | for (int i = 0; i < real::BIGITS; i++) |
---|
661 | { |
---|
662 | if (exponent <= 0) |
---|
663 | ret.m_mantissa[i] = 0; |
---|
664 | else if (exponent < 8 * (int)sizeof(uint16_t)) |
---|
665 | ret.m_mantissa[i] &= ~((1 << (16 - exponent)) - 1); |
---|
666 | |
---|
667 | exponent -= 8 * sizeof(uint16_t); |
---|
668 | } |
---|
669 | |
---|
670 | return ret; |
---|
671 | } |
---|
672 | |
---|
673 | real ceil(real const &x) |
---|
674 | { |
---|
675 | /* Strategy for ceil(x): |
---|
676 | * - if negative, return -floor(-x) |
---|
677 | * - if x == floor(x), return x |
---|
678 | * - otherwise, return floor(x) + 1 */ |
---|
679 | if (x < -real::R_0) |
---|
680 | return -floor(-x); |
---|
681 | real ret = floor(x); |
---|
682 | if (x == ret) |
---|
683 | return ret; |
---|
684 | else |
---|
685 | return ret + real::R_1; |
---|
686 | } |
---|
687 | |
---|
688 | real round(real const &x) |
---|
689 | { |
---|
690 | if (x < real::R_0) |
---|
691 | return -round(-x); |
---|
692 | |
---|
693 | return floor(x + (real::R_1 >> 1)); |
---|
694 | } |
---|
695 | |
---|
696 | real fmod(real const &x, real const &y) |
---|
697 | { |
---|
698 | if (!y) |
---|
699 | return real::R_0; /* FIXME: return NaN */ |
---|
700 | |
---|
701 | if (!x) |
---|
702 | return x; |
---|
703 | |
---|
704 | real tmp = round(x / y); |
---|
705 | return x - tmp * y; |
---|
706 | } |
---|
707 | |
---|
708 | real sin(real const &x) |
---|
709 | { |
---|
710 | bool switch_sign = x.m_signexp & 0x80000000u; |
---|
711 | |
---|
712 | real absx = fmod(fabs(x), real::R_PI << 1); |
---|
713 | if (absx > real::R_PI) |
---|
714 | { |
---|
715 | absx -= real::R_PI; |
---|
716 | switch_sign = !switch_sign; |
---|
717 | } |
---|
718 | |
---|
719 | if (absx > real::R_PI_2) |
---|
720 | absx = real::R_PI - absx; |
---|
721 | |
---|
722 | real ret = 0.0, fact = 1.0, xn = absx, x2 = absx * absx; |
---|
723 | for (int i = 1; ; i += 2) |
---|
724 | { |
---|
725 | real newret = ret + xn / fact; |
---|
726 | if (ret == newret) |
---|
727 | break; |
---|
728 | ret = newret; |
---|
729 | xn *= x2; |
---|
730 | fact *= (real)(-(i + 1) * (i + 2)); |
---|
731 | } |
---|
732 | |
---|
733 | /* Propagate sign */ |
---|
734 | if (switch_sign) |
---|
735 | ret.m_signexp ^= 0x80000000u; |
---|
736 | return ret; |
---|
737 | } |
---|
738 | |
---|
739 | real cos(real const &x) |
---|
740 | { |
---|
741 | bool switch_sign = false; |
---|
742 | real absx = fmod(fabs(x), real::R_PI << 1); |
---|
743 | |
---|
744 | if (absx > real::R_PI) |
---|
745 | absx = (real::R_PI << 1) - absx; |
---|
746 | |
---|
747 | if (absx > real::R_PI_2) |
---|
748 | { |
---|
749 | absx = real::R_PI - absx; |
---|
750 | switch_sign = true; |
---|
751 | } |
---|
752 | |
---|
753 | real ret = 0.0, fact = 1.0, xn = 1.0, x2 = absx * absx; |
---|
754 | for (int i = 1; ; i += 2) |
---|
755 | { |
---|
756 | real newret = ret + xn / fact; |
---|
757 | if (ret == newret) |
---|
758 | break; |
---|
759 | ret = newret; |
---|
760 | xn *= x2; |
---|
761 | fact *= (real)(-i * (i + 1)); |
---|
762 | } |
---|
763 | |
---|
764 | /* Propagate sign */ |
---|
765 | if (switch_sign) |
---|
766 | ret.m_signexp ^= 0x80000000u; |
---|
767 | return ret; |
---|
768 | } |
---|
769 | |
---|
770 | static real asinacos(real const &x, bool is_asin, bool is_negative) |
---|
771 | { |
---|
772 | /* Strategy for asin(): in [-0.5..0.5], use a Taylor series around |
---|
773 | * zero. In [0.5..1], use asin(x) = π/2 - 2*asin(sqrt((1-x)/2)), and |
---|
774 | * in [-1..-0.5] just revert the sign. |
---|
775 | * Strategy for acos(): use acos(x) = π/2 - asin(x) and try not to |
---|
776 | * lose the precision around x=1. */ |
---|
777 | real absx = fabs(x); |
---|
778 | bool around_zero = (absx < (real::R_1 >> 1)); |
---|
779 | |
---|
780 | if (!around_zero) |
---|
781 | absx = sqrt((real::R_1 - absx) >> 1); |
---|
782 | |
---|
783 | real ret = absx, xn = absx, x2 = absx * absx, fact1 = 2, fact2 = 1; |
---|
784 | for (int i = 1; i < 280; i++) |
---|
785 | { |
---|
786 | xn *= x2; |
---|
787 | ret += (fact1 * xn / ((real)(2 * i + 1) * fact2)) >> (i * 2); |
---|
788 | fact1 *= (real)((2 * i + 1) * (2 * i + 2)); |
---|
789 | fact2 *= (real)((i + 1) * (i + 1)); |
---|
790 | } |
---|
791 | |
---|
792 | if (is_negative) |
---|
793 | ret = -ret; |
---|
794 | |
---|
795 | if (around_zero) |
---|
796 | ret = is_asin ? ret : real::R_PI_2 - ret; |
---|
797 | else |
---|
798 | { |
---|
799 | real adjust = is_negative ? real::R_PI : real::R_0; |
---|
800 | if (is_asin) |
---|
801 | ret = real::R_PI_2 - adjust - (ret << 1); |
---|
802 | else |
---|
803 | ret = adjust + (ret << 1); |
---|
804 | } |
---|
805 | |
---|
806 | return ret; |
---|
807 | } |
---|
808 | |
---|
809 | real asin(real const &x) |
---|
810 | { |
---|
811 | return asinacos(x, true, x.m_signexp >> 31); |
---|
812 | } |
---|
813 | |
---|
814 | real acos(real const &x) |
---|
815 | { |
---|
816 | return asinacos(x, false, x.m_signexp >> 31); |
---|
817 | } |
---|
818 | |
---|
819 | real atan(real const &x) |
---|
820 | { |
---|
821 | /* Computing atan(x): we choose a different Taylor series depending on |
---|
822 | * the value of x to help with convergence. |
---|
823 | * |
---|
824 | * If |x| < 0.5 we evaluate atan(y) near 0: |
---|
825 | * atan(y) = y - y^3/3 + y^5/5 - y^7/7 + y^9/9 ... |
---|
826 | * |
---|
827 | * If 0.5 <= |x| < 1.5 we evaluate atan(1+y) near 0: |
---|
828 | * atan(1+y) = π/4 + y/(1*2^1) - y^2/(2*2^1) + y^3/(3*2^2) |
---|
829 | * - y^5/(5*2^3) + y^6/(6*2^3) - y^7/(7*2^4) |
---|
830 | * + y^9/(9*2^5) - y^10/(10*2^5) + y^11/(11*2^6) ... |
---|
831 | * |
---|
832 | * If 1.5 <= |x| < 2 we evaluate atan(sqrt(3)+2y) near 0: |
---|
833 | * atan(sqrt(3)+2y) = π/3 + 1/2 y - sqrt(3)/2 y^2/2 |
---|
834 | * + y^3/3 - sqrt(3)/2 y^4/4 + 1/2 y^5/5 |
---|
835 | * - 1/2 y^7/7 + sqrt(3)/2 y^8/8 |
---|
836 | * - y^9/9 + sqrt(3)/2 y^10/10 - 1/2 y^11/11 |
---|
837 | * + 1/2 y^13/13 - sqrt(3)/2 y^14/14 |
---|
838 | * + y^15/15 - sqrt(3)/2 y^16/16 + 1/2 y^17/17 ... |
---|
839 | * |
---|
840 | * If |x| >= 2 we evaluate atan(y) near +∞: |
---|
841 | * atan(y) = π/2 - y^-1 + y^-3/3 - y^-5/5 + y^-7/7 - y^-9/9 ... |
---|
842 | */ |
---|
843 | real absx = fabs(x); |
---|
844 | |
---|
845 | if (absx < (real::R_1 >> 1)) |
---|
846 | { |
---|
847 | real ret = x, xn = x, mx2 = -x * x; |
---|
848 | for (int i = 3; i < 100; i += 2) |
---|
849 | { |
---|
850 | xn *= mx2; |
---|
851 | ret += xn / (real)i; |
---|
852 | } |
---|
853 | return ret; |
---|
854 | } |
---|
855 | |
---|
856 | real ret = 0; |
---|
857 | |
---|
858 | if (absx < (real::R_3 >> 1)) |
---|
859 | { |
---|
860 | real y = real::R_1 - absx; |
---|
861 | real yn = y, my2 = -y * y; |
---|
862 | for (int i = 0; i < 200; i += 2) |
---|
863 | { |
---|
864 | ret += (yn / (real)(2 * i + 1)) >> (i + 1); |
---|
865 | yn *= y; |
---|
866 | ret += (yn / (real)(2 * i + 2)) >> (i + 1); |
---|
867 | yn *= y; |
---|
868 | ret += (yn / (real)(2 * i + 3)) >> (i + 2); |
---|
869 | yn *= my2; |
---|
870 | } |
---|
871 | ret = real::R_PI_4 - ret; |
---|
872 | } |
---|
873 | else if (absx < real::R_2) |
---|
874 | { |
---|
875 | real y = (absx - real::R_SQRT3) >> 1; |
---|
876 | real yn = y, my2 = -y * y; |
---|
877 | for (int i = 1; i < 200; i += 6) |
---|
878 | { |
---|
879 | ret += (yn / (real)i) >> 1; |
---|
880 | yn *= y; |
---|
881 | ret -= (real::R_SQRT3 >> 1) * yn / (real)(i + 1); |
---|
882 | yn *= y; |
---|
883 | ret += yn / (real)(i + 2); |
---|
884 | yn *= y; |
---|
885 | ret -= (real::R_SQRT3 >> 1) * yn / (real)(i + 3); |
---|
886 | yn *= y; |
---|
887 | ret += (yn / (real)(i + 4)) >> 1; |
---|
888 | yn *= my2; |
---|
889 | } |
---|
890 | ret = real::R_PI_3 + ret; |
---|
891 | } |
---|
892 | else |
---|
893 | { |
---|
894 | real y = re(absx); |
---|
895 | real yn = y, my2 = -y * y; |
---|
896 | ret = y; |
---|
897 | for (int i = 3; i < 120; i += 2) |
---|
898 | { |
---|
899 | yn *= my2; |
---|
900 | ret += yn / (real)i; |
---|
901 | } |
---|
902 | ret = real::R_PI_2 - ret; |
---|
903 | } |
---|
904 | |
---|
905 | /* Propagate sign */ |
---|
906 | ret.m_signexp |= (x.m_signexp & 0x80000000u); |
---|
907 | return ret; |
---|
908 | } |
---|
909 | |
---|
910 | void real::print(int ndigits) const |
---|
911 | { |
---|
912 | real const r1 = 1, r10 = 10; |
---|
913 | real x = *this; |
---|
914 | |
---|
915 | if (x.m_signexp >> 31) |
---|
916 | { |
---|
917 | printf("-"); |
---|
918 | x = -x; |
---|
919 | } |
---|
920 | |
---|
921 | /* Normalise x so that mantissa is in [1..9.999] */ |
---|
922 | int exponent = 0; |
---|
923 | if (x.m_signexp) |
---|
924 | { |
---|
925 | for (real div = r1, newdiv; true; div = newdiv) |
---|
926 | { |
---|
927 | newdiv = div * r10; |
---|
928 | if (x < newdiv) |
---|
929 | { |
---|
930 | x /= div; |
---|
931 | break; |
---|
932 | } |
---|
933 | exponent++; |
---|
934 | } |
---|
935 | for (real mul = 1, newx; true; mul *= r10) |
---|
936 | { |
---|
937 | newx = x * mul; |
---|
938 | if (newx >= r1) |
---|
939 | { |
---|
940 | x = newx; |
---|
941 | break; |
---|
942 | } |
---|
943 | exponent--; |
---|
944 | } |
---|
945 | } |
---|
946 | |
---|
947 | /* Print digits */ |
---|
948 | for (int i = 0; i < ndigits; i++) |
---|
949 | { |
---|
950 | int digit = (int)x; |
---|
951 | printf("%i", digit); |
---|
952 | if (i == 0) |
---|
953 | printf("."); |
---|
954 | x -= real(digit); |
---|
955 | x *= r10; |
---|
956 | } |
---|
957 | |
---|
958 | /* Print exponent information */ |
---|
959 | if (exponent < 0) |
---|
960 | printf("e-%i", -exponent); |
---|
961 | else if (exponent > 0) |
---|
962 | printf("e+%i", exponent); |
---|
963 | |
---|
964 | printf("\n"); |
---|
965 | } |
---|
966 | |
---|
967 | static real fast_pi() |
---|
968 | { |
---|
969 | /* Approximate Pi using Machin's formula: 16*atan(1/5)-4*atan(1/239) */ |
---|
970 | real ret = 0.0, x0 = 5.0, x1 = 239.0; |
---|
971 | real const m0 = -x0 * x0, m1 = -x1 * x1, r16 = 16.0, r4 = 4.0; |
---|
972 | |
---|
973 | /* Degree 240 is required for 512-bit mantissa precision */ |
---|
974 | for (int i = 1; i < 240; i += 2) |
---|
975 | { |
---|
976 | ret += r16 / (x0 * (real)i) - r4 / (x1 * (real)i); |
---|
977 | x0 *= m0; |
---|
978 | x1 *= m1; |
---|
979 | } |
---|
980 | |
---|
981 | return ret; |
---|
982 | } |
---|
983 | |
---|
984 | real const real::R_0 = (real)0.0; |
---|
985 | real const real::R_1 = (real)1.0; |
---|
986 | real const real::R_2 = (real)2.0; |
---|
987 | real const real::R_3 = (real)3.0; |
---|
988 | real const real::R_10 = (real)10.0; |
---|
989 | |
---|
990 | real const real::R_E = exp(R_1); |
---|
991 | real const real::R_LN2 = log(R_2); |
---|
992 | real const real::R_LN10 = log(R_10); |
---|
993 | real const real::R_LOG2E = re(R_LN2); |
---|
994 | real const real::R_LOG10E = re(R_LN10); |
---|
995 | real const real::R_PI = fast_pi(); |
---|
996 | real const real::R_PI_2 = R_PI >> 1; |
---|
997 | real const real::R_PI_3 = R_PI / R_3; |
---|
998 | real const real::R_PI_4 = R_PI >> 2; |
---|
999 | real const real::R_1_PI = re(R_PI); |
---|
1000 | real const real::R_2_PI = R_1_PI << 1; |
---|
1001 | real const real::R_2_SQRTPI = re(sqrt(R_PI)) << 1; |
---|
1002 | real const real::R_SQRT2 = sqrt(R_2); |
---|
1003 | real const real::R_SQRT3 = sqrt(R_3); |
---|
1004 | real const real::R_SQRT1_2 = R_SQRT2 >> 1; |
---|
1005 | |
---|
1006 | } /* namespace lol */ |
---|
1007 | |
---|