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1 /* Copyright (C) 2007, 2009 Free Software Foundation, Inc.
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2
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3 This file is part of GCC.
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4
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5 GCC is free software; you can redistribute it and/or modify it under
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6 the terms of the GNU General Public License as published by the Free
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7 Software Foundation; either version 3, or (at your option) any later
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8 version.
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9
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10 GCC is distributed in the hope that it will be useful, but WITHOUT ANY
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11 WARRANTY; without even the implied warranty of MERCHANTABILITY or
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12 FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License
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13 for more details.
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14
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15 Under Section 7 of GPL version 3, you are granted additional
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16 permissions described in the GCC Runtime Library Exception, version
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17 3.1, as published by the Free Software Foundation.
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18
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19 You should have received a copy of the GNU General Public License and
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20 a copy of the GCC Runtime Library Exception along with this program;
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21 see the files COPYING3 and COPYING.RUNTIME respectively. If not, see
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22 <http://www.gnu.org/licenses/>. */
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23
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24 #include "bid_internal.h"
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25
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26
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27 #if DECIMAL_CALL_BY_REFERENCE
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28 void
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29 bid64dq_add (UINT64 * pres, UINT64 * px, UINT128 * py
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30 _RND_MODE_PARAM _EXC_FLAGS_PARAM _EXC_MASKS_PARAM
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31 _EXC_INFO_PARAM) {
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32 UINT64 x = *px;
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33 #if !DECIMAL_GLOBAL_ROUNDING
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34 unsigned int rnd_mode = *prnd_mode;
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35 #endif
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36 #else
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37 UINT64
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38 bid64dq_add (UINT64 x, UINT128 y
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39 _RND_MODE_PARAM _EXC_FLAGS_PARAM _EXC_MASKS_PARAM
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40 _EXC_INFO_PARAM) {
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41 #endif
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42 UINT64 res = 0xbaddbaddbaddbaddull;
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43 UINT128 x1;
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44
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45 #if DECIMAL_CALL_BY_REFERENCE
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46 bid64_to_bid128 (&x1, &x _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG);
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47 bid64qq_add (&res, &x1, py
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48 _RND_MODE_ARG _EXC_FLAGS_ARG _EXC_MASKS_ARG
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49 _EXC_INFO_ARG);
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50 #else
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51 x1 = bid64_to_bid128 (x _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG);
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52 res = bid64qq_add (x1, y
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53 _RND_MODE_ARG _EXC_FLAGS_ARG _EXC_MASKS_ARG
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54 _EXC_INFO_ARG);
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55 #endif
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56 BID_RETURN (res);
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57 }
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58
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59
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60 #if DECIMAL_CALL_BY_REFERENCE
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61 void
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62 bid64qd_add (UINT64 * pres, UINT128 * px, UINT64 * py
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63 _RND_MODE_PARAM _EXC_FLAGS_PARAM _EXC_MASKS_PARAM
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64 _EXC_INFO_PARAM) {
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65 UINT64 y = *py;
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66 #if !DECIMAL_GLOBAL_ROUNDING
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67 unsigned int rnd_mode = *prnd_mode;
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68 #endif
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69 #else
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70 UINT64
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71 bid64qd_add (UINT128 x, UINT64 y
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72 _RND_MODE_PARAM _EXC_FLAGS_PARAM _EXC_MASKS_PARAM
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73 _EXC_INFO_PARAM) {
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74 #endif
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75 UINT64 res = 0xbaddbaddbaddbaddull;
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76 UINT128 y1;
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77
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78 #if DECIMAL_CALL_BY_REFERENCE
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79 bid64_to_bid128 (&y1, &y _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG);
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80 bid64qq_add (&res, px, &y1
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81 _RND_MODE_ARG _EXC_FLAGS_ARG _EXC_MASKS_ARG
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82 _EXC_INFO_ARG);
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83 #else
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84 y1 = bid64_to_bid128 (y _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG);
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85 res = bid64qq_add (x, y1
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86 _RND_MODE_ARG _EXC_FLAGS_ARG _EXC_MASKS_ARG
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87 _EXC_INFO_ARG);
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88 #endif
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89 BID_RETURN (res);
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90 }
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91
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92
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93 #if DECIMAL_CALL_BY_REFERENCE
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94 void
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95 bid64qq_add (UINT64 * pres, UINT128 * px, UINT128 * py
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96 _RND_MODE_PARAM _EXC_FLAGS_PARAM _EXC_MASKS_PARAM
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97 _EXC_INFO_PARAM) {
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98 UINT128 x = *px, y = *py;
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99 #if !DECIMAL_GLOBAL_ROUNDING
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100 unsigned int rnd_mode = *prnd_mode;
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101 #endif
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102 #else
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103 UINT64
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104 bid64qq_add (UINT128 x, UINT128 y
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105 _RND_MODE_PARAM _EXC_FLAGS_PARAM _EXC_MASKS_PARAM
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106 _EXC_INFO_PARAM) {
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107 #endif
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108
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109 UINT128 one = { {0x0000000000000001ull, 0x3040000000000000ull}
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110 };
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111 UINT64 res = 0xbaddbaddbaddbaddull;
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112
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113 BID_SWAP128 (one);
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114 #if DECIMAL_CALL_BY_REFERENCE
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115 bid64qqq_fma (&res, &one, &x, &y
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116 _RND_MODE_ARG _EXC_FLAGS_ARG _EXC_MASKS_ARG
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117 _EXC_INFO_ARG);
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118 #else
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119 res = bid64qqq_fma (one, x, y
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120 _RND_MODE_ARG _EXC_FLAGS_ARG _EXC_MASKS_ARG
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121 _EXC_INFO_ARG);
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122 #endif
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123 BID_RETURN (res);
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124 }
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125
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126
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127 #if DECIMAL_CALL_BY_REFERENCE
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128 void
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129 bid128dd_add (UINT128 * pres, UINT64 * px, UINT64 * py
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130 _RND_MODE_PARAM _EXC_FLAGS_PARAM _EXC_MASKS_PARAM
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131 _EXC_INFO_PARAM) {
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132 UINT64 x = *px, y = *py;
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133 #if !DECIMAL_GLOBAL_ROUNDING
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134 unsigned int rnd_mode = *prnd_mode;
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135 #endif
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136 #else
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137 UINT128
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138 bid128dd_add (UINT64 x, UINT64 y
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139 _RND_MODE_PARAM _EXC_FLAGS_PARAM _EXC_MASKS_PARAM
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140 _EXC_INFO_PARAM) {
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141 #endif
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142 UINT128 res = { {0xbaddbaddbaddbaddull, 0xbaddbaddbaddbaddull}
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143 };
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144 UINT128 x1, y1;
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145
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146 #if DECIMAL_CALL_BY_REFERENCE
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147 bid64_to_bid128 (&x1, &x _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG);
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148 bid64_to_bid128 (&y1, &y _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG);
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149 bid128_add (&res, &x1, &y1
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150 _RND_MODE_ARG _EXC_FLAGS_ARG _EXC_MASKS_ARG
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151 _EXC_INFO_ARG);
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152 #else
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153 x1 = bid64_to_bid128 (x _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG);
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154 y1 = bid64_to_bid128 (y _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG);
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155 res = bid128_add (x1, y1
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156 _RND_MODE_ARG _EXC_FLAGS_ARG _EXC_MASKS_ARG
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157 _EXC_INFO_ARG);
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158 #endif
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159 BID_RETURN (res);
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160 }
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161
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162
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163 #if DECIMAL_CALL_BY_REFERENCE
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164 void
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165 bid128dq_add (UINT128 * pres, UINT64 * px, UINT128 * py
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166 _RND_MODE_PARAM _EXC_FLAGS_PARAM _EXC_MASKS_PARAM
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167 _EXC_INFO_PARAM) {
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168 UINT64 x = *px;
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169 #if !DECIMAL_GLOBAL_ROUNDING
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170 unsigned int rnd_mode = *prnd_mode;
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171 #endif
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172 #else
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173 UINT128
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174 bid128dq_add (UINT64 x, UINT128 y
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175 _RND_MODE_PARAM _EXC_FLAGS_PARAM _EXC_MASKS_PARAM
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176 _EXC_INFO_PARAM) {
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177 #endif
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178 UINT128 res = { {0xbaddbaddbaddbaddull, 0xbaddbaddbaddbaddull}
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179 };
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180 UINT128 x1;
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181
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182 #if DECIMAL_CALL_BY_REFERENCE
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183 bid64_to_bid128 (&x1, &x _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG);
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184 bid128_add (&res, &x1, py
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185 _RND_MODE_ARG _EXC_FLAGS_ARG _EXC_MASKS_ARG
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186 _EXC_INFO_ARG);
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187 #else
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188 x1 = bid64_to_bid128 (x _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG);
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189 res = bid128_add (x1, y
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190 _RND_MODE_ARG _EXC_FLAGS_ARG _EXC_MASKS_ARG
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191 _EXC_INFO_ARG);
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192 #endif
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193 BID_RETURN (res);
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194 }
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195
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196
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197 #if DECIMAL_CALL_BY_REFERENCE
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198 void
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199 bid128qd_add (UINT128 * pres, UINT128 * px, UINT64 * py
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200 _RND_MODE_PARAM _EXC_FLAGS_PARAM _EXC_MASKS_PARAM
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201 _EXC_INFO_PARAM) {
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202 UINT64 y = *py;
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203 #if !DECIMAL_GLOBAL_ROUNDING
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204 unsigned int rnd_mode = *prnd_mode;
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205 #endif
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206 #else
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207 UINT128
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208 bid128qd_add (UINT128 x, UINT64 y
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209 _RND_MODE_PARAM _EXC_FLAGS_PARAM _EXC_MASKS_PARAM
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210 _EXC_INFO_PARAM) {
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211 #endif
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212 UINT128 res = { {0xbaddbaddbaddbaddull, 0xbaddbaddbaddbaddull}
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213 };
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214 UINT128 y1;
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215
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216 #if DECIMAL_CALL_BY_REFERENCE
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217 bid64_to_bid128 (&y1, &y _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG);
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218 bid128_add (&res, px, &y1
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219 _RND_MODE_ARG _EXC_FLAGS_ARG _EXC_MASKS_ARG
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220 _EXC_INFO_ARG);
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221 #else
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222 y1 = bid64_to_bid128 (y _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG);
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223 res = bid128_add (x, y1
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224 _RND_MODE_ARG _EXC_FLAGS_ARG _EXC_MASKS_ARG
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225 _EXC_INFO_ARG);
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226 #endif
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227 BID_RETURN (res);
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228 }
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229
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230
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231 // bid128_add stands for bid128qq_add
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232
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233
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234 /*****************************************************************************
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235 * BID64/BID128 sub
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236 ****************************************************************************/
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237
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238 #if DECIMAL_CALL_BY_REFERENCE
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239 void
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240 bid64dq_sub (UINT64 * pres, UINT64 * px, UINT128 * py
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241 _RND_MODE_PARAM _EXC_FLAGS_PARAM _EXC_MASKS_PARAM
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242 _EXC_INFO_PARAM) {
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243 UINT64 x = *px;
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244 #if !DECIMAL_GLOBAL_ROUNDING
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245 unsigned int rnd_mode = *prnd_mode;
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246 #endif
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247 #else
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248 UINT64
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249 bid64dq_sub (UINT64 x, UINT128 y
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250 _RND_MODE_PARAM _EXC_FLAGS_PARAM _EXC_MASKS_PARAM
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251 _EXC_INFO_PARAM) {
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252 #endif
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253 UINT64 res = 0xbaddbaddbaddbaddull;
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254 UINT128 x1;
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255
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256 #if DECIMAL_CALL_BY_REFERENCE
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257 bid64_to_bid128 (&x1, &x _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG);
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258 bid64qq_sub (&res, &x1, py
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259 _RND_MODE_ARG _EXC_FLAGS_ARG _EXC_MASKS_ARG
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260 _EXC_INFO_ARG);
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261 #else
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262 x1 = bid64_to_bid128 (x _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG);
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263 res = bid64qq_sub (x1, y
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264 _RND_MODE_ARG _EXC_FLAGS_ARG _EXC_MASKS_ARG
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265 _EXC_INFO_ARG);
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266 #endif
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267 BID_RETURN (res);
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268 }
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269
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270
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271 #if DECIMAL_CALL_BY_REFERENCE
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272 void
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273 bid64qd_sub (UINT64 * pres, UINT128 * px, UINT64 * py
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274 _RND_MODE_PARAM _EXC_FLAGS_PARAM _EXC_MASKS_PARAM
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275 _EXC_INFO_PARAM) {
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276 UINT64 y = *py;
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277 #if !DECIMAL_GLOBAL_ROUNDING
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278 unsigned int rnd_mode = *prnd_mode;
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279 #endif
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280 #else
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281 UINT64
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282 bid64qd_sub (UINT128 x, UINT64 y
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283 _RND_MODE_PARAM _EXC_FLAGS_PARAM _EXC_MASKS_PARAM
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284 _EXC_INFO_PARAM) {
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285 #endif
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286 UINT64 res = 0xbaddbaddbaddbaddull;
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287 UINT128 y1;
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288
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289 #if DECIMAL_CALL_BY_REFERENCE
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290 bid64_to_bid128 (&y1, &y _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG);
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291 bid64qq_sub (&res, px, &y1
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292 _RND_MODE_ARG _EXC_FLAGS_ARG _EXC_MASKS_ARG
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293 _EXC_INFO_ARG);
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294 #else
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295 y1 = bid64_to_bid128 (y _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG);
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296 res = bid64qq_sub (x, y1
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297 _RND_MODE_ARG _EXC_FLAGS_ARG _EXC_MASKS_ARG
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298 _EXC_INFO_ARG);
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299 #endif
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300 BID_RETURN (res);
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301 }
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302
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303
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304 #if DECIMAL_CALL_BY_REFERENCE
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305 void
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306 bid64qq_sub (UINT64 * pres, UINT128 * px, UINT128 * py
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307 _RND_MODE_PARAM _EXC_FLAGS_PARAM _EXC_MASKS_PARAM
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308 _EXC_INFO_PARAM) {
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309 UINT128 x = *px, y = *py;
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310 #if !DECIMAL_GLOBAL_ROUNDING
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311 unsigned int rnd_mode = *prnd_mode;
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312 #endif
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313 #else
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314 UINT64
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315 bid64qq_sub (UINT128 x, UINT128 y
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316 _RND_MODE_PARAM _EXC_FLAGS_PARAM _EXC_MASKS_PARAM
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317 _EXC_INFO_PARAM) {
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318 #endif
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319
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320 UINT128 one = { {0x0000000000000001ull, 0x3040000000000000ull}
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321 };
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322 UINT64 res = 0xbaddbaddbaddbaddull;
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323 UINT64 y_sign;
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324
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325 BID_SWAP128 (one);
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326 if ((y.w[HIGH_128W] & MASK_NAN) != MASK_NAN) { // y is not NAN
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327 // change its sign
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328 y_sign = y.w[HIGH_128W] & MASK_SIGN; // 0 for positive, MASK_SIGN for negative
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329 if (y_sign)
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330 y.w[HIGH_128W] = y.w[HIGH_128W] & 0x7fffffffffffffffull;
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331 else
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332 y.w[HIGH_128W] = y.w[HIGH_128W] | 0x8000000000000000ull;
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333 }
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334 #if DECIMAL_CALL_BY_REFERENCE
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335 bid64qqq_fma (&res, &one, &x, &y
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336 _RND_MODE_ARG _EXC_FLAGS_ARG _EXC_MASKS_ARG
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337 _EXC_INFO_ARG);
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338 #else
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339 res = bid64qqq_fma (one, x, y
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340 _RND_MODE_ARG _EXC_FLAGS_ARG _EXC_MASKS_ARG
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341 _EXC_INFO_ARG);
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342 #endif
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343 BID_RETURN (res);
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344 }
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345
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346
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347 #if DECIMAL_CALL_BY_REFERENCE
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348 void
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349 bid128dd_sub (UINT128 * pres, UINT64 * px, UINT64 * py
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350 _RND_MODE_PARAM _EXC_FLAGS_PARAM _EXC_MASKS_PARAM
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351 _EXC_INFO_PARAM) {
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352 UINT64 x = *px, y = *py;
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353 #if !DECIMAL_GLOBAL_ROUNDING
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354 unsigned int rnd_mode = *prnd_mode;
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355 #endif
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356 #else
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357 UINT128
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358 bid128dd_sub (UINT64 x, UINT64 y
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359 _RND_MODE_PARAM _EXC_FLAGS_PARAM _EXC_MASKS_PARAM
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360 _EXC_INFO_PARAM) {
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361 #endif
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362 UINT128 res = { {0xbaddbaddbaddbaddull, 0xbaddbaddbaddbaddull}
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|
363 };
|
|
|
364 UINT128 x1, y1;
|
|
|
365
|
|
|
366 #if DECIMAL_CALL_BY_REFERENCE
|
|
|
367 bid64_to_bid128 (&x1, &x _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG);
|
|
|
368 bid64_to_bid128 (&y1, &y _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG);
|
|
|
369 bid128_sub (&res, &x1, &y1
|
|
|
370 _RND_MODE_ARG _EXC_FLAGS_ARG _EXC_MASKS_ARG
|
|
|
371 _EXC_INFO_ARG);
|
|
|
372 #else
|
|
|
373 x1 = bid64_to_bid128 (x _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG);
|
|
|
374 y1 = bid64_to_bid128 (y _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG);
|
|
|
375 res = bid128_sub (x1, y1
|
|
|
376 _RND_MODE_ARG _EXC_FLAGS_ARG _EXC_MASKS_ARG
|
|
|
377 _EXC_INFO_ARG);
|
|
|
378 #endif
|
|
|
379 BID_RETURN (res);
|
|
|
380 }
|
|
|
381
|
|
|
382
|
|
|
383 #if DECIMAL_CALL_BY_REFERENCE
|
|
|
384 void
|
|
|
385 bid128dq_sub (UINT128 * pres, UINT64 * px, UINT128 * py
|
|
|
386 _RND_MODE_PARAM _EXC_FLAGS_PARAM _EXC_MASKS_PARAM
|
|
|
387 _EXC_INFO_PARAM) {
|
|
|
388 UINT64 x = *px;
|
|
|
389 #if !DECIMAL_GLOBAL_ROUNDING
|
|
|
390 unsigned int rnd_mode = *prnd_mode;
|
|
|
391 #endif
|
|
|
392 #else
|
|
|
393 UINT128
|
|
|
394 bid128dq_sub (UINT64 x, UINT128 y
|
|
|
395 _RND_MODE_PARAM _EXC_FLAGS_PARAM _EXC_MASKS_PARAM
|
|
|
396 _EXC_INFO_PARAM) {
|
|
|
397 #endif
|
|
|
398 UINT128 res = { {0xbaddbaddbaddbaddull, 0xbaddbaddbaddbaddull}
|
|
|
399 };
|
|
|
400 UINT128 x1;
|
|
|
401
|
|
|
402 #if DECIMAL_CALL_BY_REFERENCE
|
|
|
403 bid64_to_bid128 (&x1, &x _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG);
|
|
|
404 bid128_sub (&res, &x1, py
|
|
|
405 _RND_MODE_ARG _EXC_FLAGS_ARG _EXC_MASKS_ARG
|
|
|
406 _EXC_INFO_ARG);
|
|
|
407 #else
|
|
|
408 x1 = bid64_to_bid128 (x _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG);
|
|
|
409 res = bid128_sub (x1, y
|
|
|
410 _RND_MODE_ARG _EXC_FLAGS_ARG _EXC_MASKS_ARG
|
|
|
411 _EXC_INFO_ARG);
|
|
|
412 #endif
|
|
|
413 BID_RETURN (res);
|
|
|
414 }
|
|
|
415
|
|
|
416
|
|
|
417 #if DECIMAL_CALL_BY_REFERENCE
|
|
|
418 void
|
|
|
419 bid128qd_sub (UINT128 * pres, UINT128 * px, UINT64 * py
|
|
|
420 _RND_MODE_PARAM _EXC_FLAGS_PARAM _EXC_MASKS_PARAM
|
|
|
421 _EXC_INFO_PARAM) {
|
|
|
422 UINT64 y = *py;
|
|
|
423 #if !DECIMAL_GLOBAL_ROUNDING
|
|
|
424 unsigned int rnd_mode = *prnd_mode;
|
|
|
425 #endif
|
|
|
426 #else
|
|
|
427 UINT128
|
|
|
428 bid128qd_sub (UINT128 x, UINT64 y
|
|
|
429 _RND_MODE_PARAM _EXC_FLAGS_PARAM _EXC_MASKS_PARAM
|
|
|
430 _EXC_INFO_PARAM) {
|
|
|
431 #endif
|
|
|
432 UINT128 res = { {0xbaddbaddbaddbaddull, 0xbaddbaddbaddbaddull}
|
|
|
433 };
|
|
|
434 UINT128 y1;
|
|
|
435
|
|
|
436 #if DECIMAL_CALL_BY_REFERENCE
|
|
|
437 bid64_to_bid128 (&y1, &y _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG);
|
|
|
438 bid128_sub (&res, px, &y1
|
|
|
439 _RND_MODE_ARG _EXC_FLAGS_ARG _EXC_MASKS_ARG
|
|
|
440 _EXC_INFO_ARG);
|
|
|
441 #else
|
|
|
442 y1 = bid64_to_bid128 (y _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG);
|
|
|
443 res = bid128_sub (x, y1
|
|
|
444 _RND_MODE_ARG _EXC_FLAGS_ARG _EXC_MASKS_ARG
|
|
|
445 _EXC_INFO_ARG);
|
|
|
446 #endif
|
|
|
447 BID_RETURN (res);
|
|
|
448 }
|
|
|
449
|
|
|
450 #if DECIMAL_CALL_BY_REFERENCE
|
|
|
451 void
|
|
|
452 bid128_add (UINT128 * pres, UINT128 * px, UINT128 * py
|
|
|
453 _RND_MODE_PARAM _EXC_FLAGS_PARAM _EXC_MASKS_PARAM
|
|
|
454 _EXC_INFO_PARAM) {
|
|
|
455 UINT128 x = *px, y = *py;
|
|
|
456 #if !DECIMAL_GLOBAL_ROUNDING
|
|
|
457 unsigned int rnd_mode = *prnd_mode;
|
|
|
458 #endif
|
|
|
459 #else
|
|
|
460 UINT128
|
|
|
461 bid128_add (UINT128 x, UINT128 y
|
|
|
462 _RND_MODE_PARAM _EXC_FLAGS_PARAM _EXC_MASKS_PARAM
|
|
|
463 _EXC_INFO_PARAM) {
|
|
|
464 #endif
|
|
|
465 UINT128 res = { {0xbaddbaddbaddbaddull, 0xbaddbaddbaddbaddull}
|
|
|
466 };
|
|
|
467 UINT64 x_sign, y_sign, tmp_sign;
|
|
|
468 UINT64 x_exp, y_exp, tmp_exp; // e1 = x_exp, e2 = y_exp
|
|
|
469 UINT64 C1_hi, C2_hi, tmp_signif_hi;
|
|
|
470 UINT64 C1_lo, C2_lo, tmp_signif_lo;
|
|
|
471 // Note: C1.w[1], C1.w[0] represent C1_hi, C1_lo (all UINT64)
|
|
|
472 // Note: C2.w[1], C2.w[0] represent C2_hi, C2_lo (all UINT64)
|
|
|
473 UINT64 tmp64, tmp64A, tmp64B;
|
|
|
474 BID_UI64DOUBLE tmp1, tmp2;
|
|
|
475 int x_nr_bits, y_nr_bits;
|
|
|
476 int q1, q2, delta, scale, x1, ind, shift, tmp_inexact = 0;
|
|
|
477 UINT64 halfulp64;
|
|
|
478 UINT128 halfulp128;
|
|
|
479 UINT128 C1, C2;
|
|
|
480 UINT128 ten2m1;
|
|
|
481 UINT128 highf2star; // top 128 bits in f2*; low 128 bits in R256[1], R256[0]
|
|
|
482 UINT256 P256, Q256, R256;
|
|
|
483 int is_inexact = 0, is_midpoint_lt_even = 0, is_midpoint_gt_even = 0;
|
|
|
484 int is_inexact_lt_midpoint = 0, is_inexact_gt_midpoint = 0;
|
|
|
485 int second_pass = 0;
|
|
|
486
|
|
|
487 BID_SWAP128 (x);
|
|
|
488 BID_SWAP128 (y);
|
|
|
489 x_sign = x.w[1] & MASK_SIGN; // 0 for positive, MASK_SIGN for negative
|
|
|
490 y_sign = y.w[1] & MASK_SIGN; // 0 for positive, MASK_SIGN for negative
|
|
|
491
|
|
|
492 // check for NaN or Infinity
|
|
|
493 if (((x.w[1] & MASK_SPECIAL) == MASK_SPECIAL)
|
|
|
494 || ((y.w[1] & MASK_SPECIAL) == MASK_SPECIAL)) {
|
|
|
495 // x is special or y is special
|
|
|
496 if ((x.w[1] & MASK_NAN) == MASK_NAN) { // x is NAN
|
|
|
497 // check first for non-canonical NaN payload
|
|
|
498 if (((x.w[1] & 0x00003fffffffffffull) > 0x0000314dc6448d93ull) ||
|
|
|
499 (((x.w[1] & 0x00003fffffffffffull) == 0x0000314dc6448d93ull)
|
|
|
500 && (x.w[0] > 0x38c15b09ffffffffull))) {
|
|
|
501 x.w[1] = x.w[1] & 0xffffc00000000000ull;
|
|
|
502 x.w[0] = 0x0ull;
|
|
|
503 }
|
|
|
504 if ((x.w[1] & MASK_SNAN) == MASK_SNAN) { // x is SNAN
|
|
|
505 // set invalid flag
|
|
|
506 *pfpsf |= INVALID_EXCEPTION;
|
|
|
507 // return quiet (x)
|
|
|
508 res.w[1] = x.w[1] & 0xfc003fffffffffffull;
|
|
|
509 // clear out also G[6]-G[16]
|
|
|
510 res.w[0] = x.w[0];
|
|
|
511 } else { // x is QNaN
|
|
|
512 // return x
|
|
|
513 res.w[1] = x.w[1] & 0xfc003fffffffffffull;
|
|
|
514 // clear out G[6]-G[16]
|
|
|
515 res.w[0] = x.w[0];
|
|
|
516 // if y = SNaN signal invalid exception
|
|
|
517 if ((y.w[1] & MASK_SNAN) == MASK_SNAN) {
|
|
|
518 // set invalid flag
|
|
|
519 *pfpsf |= INVALID_EXCEPTION;
|
|
|
520 }
|
|
|
521 }
|
|
|
522 BID_SWAP128 (res);
|
|
|
523 BID_RETURN (res);
|
|
|
524 } else if ((y.w[1] & MASK_NAN) == MASK_NAN) { // y is NAN
|
|
|
525 // check first for non-canonical NaN payload
|
|
|
526 if (((y.w[1] & 0x00003fffffffffffull) > 0x0000314dc6448d93ull) ||
|
|
|
527 (((y.w[1] & 0x00003fffffffffffull) == 0x0000314dc6448d93ull)
|
|
|
528 && (y.w[0] > 0x38c15b09ffffffffull))) {
|
|
|
529 y.w[1] = y.w[1] & 0xffffc00000000000ull;
|
|
|
530 y.w[0] = 0x0ull;
|
|
|
531 }
|
|
|
532 if ((y.w[1] & MASK_SNAN) == MASK_SNAN) { // y is SNAN
|
|
|
533 // set invalid flag
|
|
|
534 *pfpsf |= INVALID_EXCEPTION;
|
|
|
535 // return quiet (y)
|
|
|
536 res.w[1] = y.w[1] & 0xfc003fffffffffffull;
|
|
|
537 // clear out also G[6]-G[16]
|
|
|
538 res.w[0] = y.w[0];
|
|
|
539 } else { // y is QNaN
|
|
|
540 // return y
|
|
|
541 res.w[1] = y.w[1] & 0xfc003fffffffffffull;
|
|
|
542 // clear out G[6]-G[16]
|
|
|
543 res.w[0] = y.w[0];
|
|
|
544 }
|
|
|
545 BID_SWAP128 (res);
|
|
|
546 BID_RETURN (res);
|
|
|
547 } else { // neither x not y is NaN; at least one is infinity
|
|
|
548 if ((x.w[1] & MASK_ANY_INF) == MASK_INF) { // x is infinity
|
|
|
549 if ((y.w[1] & MASK_ANY_INF) == MASK_INF) { // y is infinity
|
|
|
550 // if same sign, return either of them
|
|
|
551 if ((x.w[1] & MASK_SIGN) == (y.w[1] & MASK_SIGN)) {
|
|
|
552 res.w[1] = x_sign | MASK_INF;
|
|
|
553 res.w[0] = 0x0ull;
|
|
|
554 } else { // x and y are infinities of opposite signs
|
|
|
555 // set invalid flag
|
|
|
556 *pfpsf |= INVALID_EXCEPTION;
|
|
|
557 // return QNaN Indefinite
|
|
|
558 res.w[1] = 0x7c00000000000000ull;
|
|
|
559 res.w[0] = 0x0000000000000000ull;
|
|
|
560 }
|
|
|
561 } else { // y is 0 or finite
|
|
|
562 // return x
|
|
|
563 res.w[1] = x_sign | MASK_INF;
|
|
|
564 res.w[0] = 0x0ull;
|
|
|
565 }
|
|
|
566 } else { // x is not NaN or infinity, so y must be infinity
|
|
|
567 res.w[1] = y_sign | MASK_INF;
|
|
|
568 res.w[0] = 0x0ull;
|
|
|
569 }
|
|
|
570 BID_SWAP128 (res);
|
|
|
571 BID_RETURN (res);
|
|
|
572 }
|
|
|
573 }
|
|
|
574 // unpack the arguments
|
|
|
575
|
|
|
576 // unpack x
|
|
|
577 C1_hi = x.w[1] & MASK_COEFF;
|
|
|
578 C1_lo = x.w[0];
|
|
|
579 // test for non-canonical values:
|
|
|
580 // - values whose encoding begins with x00, x01, or x10 and whose
|
|
|
581 // coefficient is larger than 10^34 -1, or
|
|
|
582 // - values whose encoding begins with x1100, x1101, x1110 (if NaNs
|
|
|
583 // and infinitis were eliminated already this test is reduced to
|
|
|
584 // checking for x10x)
|
|
|
585
|
|
|
586 // x is not infinity; check for non-canonical values - treated as zero
|
|
|
587 if ((x.w[1] & 0x6000000000000000ull) == 0x6000000000000000ull) {
|
|
|
588 // G0_G1=11; non-canonical
|
|
|
589 x_exp = (x.w[1] << 2) & MASK_EXP; // biased and shifted left 49 bits
|
|
|
590 C1_hi = 0; // significand high
|
|
|
591 C1_lo = 0; // significand low
|
|
|
592 } else { // G0_G1 != 11
|
|
|
593 x_exp = x.w[1] & MASK_EXP; // biased and shifted left 49 bits
|
|
|
594 if (C1_hi > 0x0001ed09bead87c0ull ||
|
|
|
595 (C1_hi == 0x0001ed09bead87c0ull
|
|
|
596 && C1_lo > 0x378d8e63ffffffffull)) {
|
|
|
597 // x is non-canonical if coefficient is larger than 10^34 -1
|
|
|
598 C1_hi = 0;
|
|
|
599 C1_lo = 0;
|
|
|
600 } else { // canonical
|
|
|
601 ;
|
|
|
602 }
|
|
|
603 }
|
|
|
604
|
|
|
605 // unpack y
|
|
|
606 C2_hi = y.w[1] & MASK_COEFF;
|
|
|
607 C2_lo = y.w[0];
|
|
|
608 // y is not infinity; check for non-canonical values - treated as zero
|
|
|
609 if ((y.w[1] & 0x6000000000000000ull) == 0x6000000000000000ull) {
|
|
|
610 // G0_G1=11; non-canonical
|
|
|
611 y_exp = (y.w[1] << 2) & MASK_EXP; // biased and shifted left 49 bits
|
|
|
612 C2_hi = 0; // significand high
|
|
|
613 C2_lo = 0; // significand low
|
|
|
614 } else { // G0_G1 != 11
|
|
|
615 y_exp = y.w[1] & MASK_EXP; // biased and shifted left 49 bits
|
|
|
616 if (C2_hi > 0x0001ed09bead87c0ull ||
|
|
|
617 (C2_hi == 0x0001ed09bead87c0ull
|
|
|
618 && C2_lo > 0x378d8e63ffffffffull)) {
|
|
|
619 // y is non-canonical if coefficient is larger than 10^34 -1
|
|
|
620 C2_hi = 0;
|
|
|
621 C2_lo = 0;
|
|
|
622 } else { // canonical
|
|
|
623 ;
|
|
|
624 }
|
|
|
625 }
|
|
|
626
|
|
|
627 if ((C1_hi == 0x0ull) && (C1_lo == 0x0ull)) {
|
|
|
628 // x is 0 and y is not special
|
|
|
629 // if y is 0 return 0 with the smaller exponent
|
|
|
630 if ((C2_hi == 0x0ull) && (C2_lo == 0x0ull)) {
|
|
|
631 if (x_exp < y_exp)
|
|
|
632 res.w[1] = x_exp;
|
|
|
633 else
|
|
|
634 res.w[1] = y_exp;
|
|
|
635 if (x_sign && y_sign)
|
|
|
636 res.w[1] = res.w[1] | x_sign; // both negative
|
|
|
637 else if (rnd_mode == ROUNDING_DOWN && x_sign != y_sign)
|
|
|
638 res.w[1] = res.w[1] | 0x8000000000000000ull; // -0
|
|
|
639 // else; // res = +0
|
|
|
640 res.w[0] = 0;
|
|
|
641 } else {
|
|
|
642 // for 0 + y return y, with the preferred exponent
|
|
|
643 if (y_exp <= x_exp) {
|
|
|
644 res.w[1] = y.w[1];
|
|
|
645 res.w[0] = y.w[0];
|
|
|
646 } else { // if y_exp > x_exp
|
|
|
647 // return (C2 * 10^scale) * 10^(y_exp - scale)
|
|
|
648 // where scale = min (P34-q2, y_exp-x_exp)
|
|
|
649 // determine q2 = nr. of decimal digits in y
|
|
|
650 // determine first the nr. of bits in y (y_nr_bits)
|
|
|
651
|
|
|
652 if (C2_hi == 0) { // y_bits is the nr. of bits in C2_lo
|
|
|
653 if (C2_lo >= 0x0020000000000000ull) { // y >= 2^53
|
|
|
654 // split the 64-bit value in two 32-bit halves to avoid
|
|
|
655 // rounding errors
|
|
|
656 if (C2_lo >= 0x0000000100000000ull) { // y >= 2^32
|
|
|
657 tmp2.d = (double) (C2_lo >> 32); // exact conversion
|
|
|
658 y_nr_bits =
|
|
|
659 32 +
|
|
|
660 ((((unsigned int) (tmp2.ui64 >> 52)) & 0x7ff) - 0x3ff);
|
|
|
661 } else { // y < 2^32
|
|
|
662 tmp2.d = (double) (C2_lo); // exact conversion
|
|
|
663 y_nr_bits =
|
|
|
664 ((((unsigned int) (tmp2.ui64 >> 52)) & 0x7ff) - 0x3ff);
|
|
|
665 }
|
|
|
666 } else { // if y < 2^53
|
|
|
667 tmp2.d = (double) C2_lo; // exact conversion
|
|
|
668 y_nr_bits =
|
|
|
669 ((((unsigned int) (tmp2.ui64 >> 52)) & 0x7ff) - 0x3ff);
|
|
|
670 }
|
|
|
671 } else { // C2_hi != 0 => nr. bits = 64 + nr_bits (C2_hi)
|
|
|
672 tmp2.d = (double) C2_hi; // exact conversion
|
|
|
673 y_nr_bits =
|
|
|
674 64 + ((((unsigned int) (tmp2.ui64 >> 52)) & 0x7ff) - 0x3ff);
|
|
|
675 }
|
|
|
676 q2 = nr_digits[y_nr_bits].digits;
|
|
|
677 if (q2 == 0) {
|
|
|
678 q2 = nr_digits[y_nr_bits].digits1;
|
|
|
679 if (C2_hi > nr_digits[y_nr_bits].threshold_hi ||
|
|
|
680 (C2_hi == nr_digits[y_nr_bits].threshold_hi &&
|
|
|
681 C2_lo >= nr_digits[y_nr_bits].threshold_lo))
|
|
|
682 q2++;
|
|
|
683 }
|
|
|
684 // return (C2 * 10^scale) * 10^(y_exp - scale)
|
|
|
685 // where scale = min (P34-q2, y_exp-x_exp)
|
|
|
686 scale = P34 - q2;
|
|
|
687 ind = (y_exp - x_exp) >> 49;
|
|
|
688 if (ind < scale)
|
|
|
689 scale = ind;
|
|
|
690 if (scale == 0) {
|
|
|
691 res.w[1] = y.w[1];
|
|
|
692 res.w[0] = y.w[0];
|
|
|
693 } else if (q2 <= 19) { // y fits in 64 bits
|
|
|
694 if (scale <= 19) { // 10^scale fits in 64 bits
|
|
|
695 // 64 x 64 C2_lo * ten2k64[scale]
|
|
|
696 __mul_64x64_to_128MACH (res, C2_lo, ten2k64[scale]);
|
|
|
697 } else { // 10^scale fits in 128 bits
|
|
|
698 // 64 x 128 C2_lo * ten2k128[scale - 20]
|
|
|
699 __mul_128x64_to_128 (res, C2_lo, ten2k128[scale - 20]);
|
|
|
700 }
|
|
|
701 } else { // y fits in 128 bits, but 10^scale must fit in 64 bits
|
|
|
702 // 64 x 128 ten2k64[scale] * C2
|
|
|
703 C2.w[1] = C2_hi;
|
|
|
704 C2.w[0] = C2_lo;
|
|
|
705 __mul_128x64_to_128 (res, ten2k64[scale], C2);
|
|
|
706 }
|
|
|
707 // subtract scale from the exponent
|
|
|
708 y_exp = y_exp - ((UINT64) scale << 49);
|
|
|
709 res.w[1] = res.w[1] | y_sign | y_exp;
|
|
|
710 }
|
|
|
711 }
|
|
|
712 BID_SWAP128 (res);
|
|
|
713 BID_RETURN (res);
|
|
|
714 } else if ((C2_hi == 0x0ull) && (C2_lo == 0x0ull)) {
|
|
|
715 // y is 0 and x is not special, and not zero
|
|
|
716 // for x + 0 return x, with the preferred exponent
|
|
|
717 if (x_exp <= y_exp) {
|
|
|
718 res.w[1] = x.w[1];
|
|
|
719 res.w[0] = x.w[0];
|
|
|
720 } else { // if x_exp > y_exp
|
|
|
721 // return (C1 * 10^scale) * 10^(x_exp - scale)
|
|
|
722 // where scale = min (P34-q1, x_exp-y_exp)
|
|
|
723 // determine q1 = nr. of decimal digits in x
|
|
|
724 // determine first the nr. of bits in x
|
|
|
725 if (C1_hi == 0) { // x_bits is the nr. of bits in C1_lo
|
|
|
726 if (C1_lo >= 0x0020000000000000ull) { // x >= 2^53
|
|
|
727 // split the 64-bit value in two 32-bit halves to avoid
|
|
|
728 // rounding errors
|
|
|
729 if (C1_lo >= 0x0000000100000000ull) { // x >= 2^32
|
|
|
730 tmp1.d = (double) (C1_lo >> 32); // exact conversion
|
|
|
731 x_nr_bits =
|
|
|
732 32 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) -
|
|
|
733 0x3ff);
|
|
|
734 } else { // x < 2^32
|
|
|
735 tmp1.d = (double) (C1_lo); // exact conversion
|
|
|
736 x_nr_bits =
|
|
|
737 ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff);
|
|
|
738 }
|
|
|
739 } else { // if x < 2^53
|
|
|
740 tmp1.d = (double) C1_lo; // exact conversion
|
|
|
741 x_nr_bits =
|
|
|
742 ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff);
|
|
|
743 }
|
|
|
744 } else { // C1_hi != 0 => nr. bits = 64 + nr_bits (C1_hi)
|
|
|
745 tmp1.d = (double) C1_hi; // exact conversion
|
|
|
746 x_nr_bits =
|
|
|
747 64 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff);
|
|
|
748 }
|
|
|
749 q1 = nr_digits[x_nr_bits].digits;
|
|
|
750 if (q1 == 0) {
|
|
|
751 q1 = nr_digits[x_nr_bits].digits1;
|
|
|
752 if (C1_hi > nr_digits[x_nr_bits].threshold_hi ||
|
|
|
753 (C1_hi == nr_digits[x_nr_bits].threshold_hi &&
|
|
|
754 C1_lo >= nr_digits[x_nr_bits].threshold_lo))
|
|
|
755 q1++;
|
|
|
756 }
|
|
|
757 // return (C1 * 10^scale) * 10^(x_exp - scale)
|
|
|
758 // where scale = min (P34-q1, x_exp-y_exp)
|
|
|
759 scale = P34 - q1;
|
|
|
760 ind = (x_exp - y_exp) >> 49;
|
|
|
761 if (ind < scale)
|
|
|
762 scale = ind;
|
|
|
763 if (scale == 0) {
|
|
|
764 res.w[1] = x.w[1];
|
|
|
765 res.w[0] = x.w[0];
|
|
|
766 } else if (q1 <= 19) { // x fits in 64 bits
|
|
|
767 if (scale <= 19) { // 10^scale fits in 64 bits
|
|
|
768 // 64 x 64 C1_lo * ten2k64[scale]
|
|
|
769 __mul_64x64_to_128MACH (res, C1_lo, ten2k64[scale]);
|
|
|
770 } else { // 10^scale fits in 128 bits
|
|
|
771 // 64 x 128 C1_lo * ten2k128[scale - 20]
|
|
|
772 __mul_128x64_to_128 (res, C1_lo, ten2k128[scale - 20]);
|
|
|
773 }
|
|
|
774 } else { // x fits in 128 bits, but 10^scale must fit in 64 bits
|
|
|
775 // 64 x 128 ten2k64[scale] * C1
|
|
|
776 C1.w[1] = C1_hi;
|
|
|
777 C1.w[0] = C1_lo;
|
|
|
778 __mul_128x64_to_128 (res, ten2k64[scale], C1);
|
|
|
779 }
|
|
|
780 // subtract scale from the exponent
|
|
|
781 x_exp = x_exp - ((UINT64) scale << 49);
|
|
|
782 res.w[1] = res.w[1] | x_sign | x_exp;
|
|
|
783 }
|
|
|
784 BID_SWAP128 (res);
|
|
|
785 BID_RETURN (res);
|
|
|
786 } else { // x and y are not canonical, not special, and are not zero
|
|
|
787 // note that the result may still be zero, and then it has to have the
|
|
|
788 // preferred exponent
|
|
|
789 if (x_exp < y_exp) { // if exp_x < exp_y then swap x and y
|
|
|
790 tmp_sign = x_sign;
|
|
|
791 tmp_exp = x_exp;
|
|
|
792 tmp_signif_hi = C1_hi;
|
|
|
793 tmp_signif_lo = C1_lo;
|
|
|
794 x_sign = y_sign;
|
|
|
795 x_exp = y_exp;
|
|
|
796 C1_hi = C2_hi;
|
|
|
797 C1_lo = C2_lo;
|
|
|
798 y_sign = tmp_sign;
|
|
|
799 y_exp = tmp_exp;
|
|
|
800 C2_hi = tmp_signif_hi;
|
|
|
801 C2_lo = tmp_signif_lo;
|
|
|
802 }
|
|
|
803 // q1 = nr. of decimal digits in x
|
|
|
804 // determine first the nr. of bits in x
|
|
|
805 if (C1_hi == 0) { // x_bits is the nr. of bits in C1_lo
|
|
|
806 if (C1_lo >= 0x0020000000000000ull) { // x >= 2^53
|
|
|
807 //split the 64-bit value in two 32-bit halves to avoid rounding errors
|
|
|
808 if (C1_lo >= 0x0000000100000000ull) { // x >= 2^32
|
|
|
809 tmp1.d = (double) (C1_lo >> 32); // exact conversion
|
|
|
810 x_nr_bits =
|
|
|
811 32 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff);
|
|
|
812 } else { // x < 2^32
|
|
|
813 tmp1.d = (double) (C1_lo); // exact conversion
|
|
|
814 x_nr_bits =
|
|
|
815 ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff);
|
|
|
816 }
|
|
|
817 } else { // if x < 2^53
|
|
|
818 tmp1.d = (double) C1_lo; // exact conversion
|
|
|
819 x_nr_bits =
|
|
|
820 ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff);
|
|
|
821 }
|
|
|
822 } else { // C1_hi != 0 => nr. bits = 64 + nr_bits (C1_hi)
|
|
|
823 tmp1.d = (double) C1_hi; // exact conversion
|
|
|
824 x_nr_bits =
|
|
|
825 64 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff);
|
|
|
826 }
|
|
|
827
|
|
|
828 q1 = nr_digits[x_nr_bits].digits;
|
|
|
829 if (q1 == 0) {
|
|
|
830 q1 = nr_digits[x_nr_bits].digits1;
|
|
|
831 if (C1_hi > nr_digits[x_nr_bits].threshold_hi ||
|
|
|
832 (C1_hi == nr_digits[x_nr_bits].threshold_hi &&
|
|
|
833 C1_lo >= nr_digits[x_nr_bits].threshold_lo))
|
|
|
834 q1++;
|
|
|
835 }
|
|
|
836 // q2 = nr. of decimal digits in y
|
|
|
837 // determine first the nr. of bits in y (y_nr_bits)
|
|
|
838 if (C2_hi == 0) { // y_bits is the nr. of bits in C2_lo
|
|
|
839 if (C2_lo >= 0x0020000000000000ull) { // y >= 2^53
|
|
|
840 //split the 64-bit value in two 32-bit halves to avoid rounding errors
|
|
|
841 if (C2_lo >= 0x0000000100000000ull) { // y >= 2^32
|
|
|
842 tmp2.d = (double) (C2_lo >> 32); // exact conversion
|
|
|
843 y_nr_bits =
|
|
|
844 32 + ((((unsigned int) (tmp2.ui64 >> 52)) & 0x7ff) - 0x3ff);
|
|
|
845 } else { // y < 2^32
|
|
|
846 tmp2.d = (double) (C2_lo); // exact conversion
|
|
|
847 y_nr_bits =
|
|
|
848 ((((unsigned int) (tmp2.ui64 >> 52)) & 0x7ff) - 0x3ff);
|
|
|
849 }
|
|
|
850 } else { // if y < 2^53
|
|
|
851 tmp2.d = (double) C2_lo; // exact conversion
|
|
|
852 y_nr_bits =
|
|
|
853 ((((unsigned int) (tmp2.ui64 >> 52)) & 0x7ff) - 0x3ff);
|
|
|
854 }
|
|
|
855 } else { // C2_hi != 0 => nr. bits = 64 + nr_bits (C2_hi)
|
|
|
856 tmp2.d = (double) C2_hi; // exact conversion
|
|
|
857 y_nr_bits =
|
|
|
858 64 + ((((unsigned int) (tmp2.ui64 >> 52)) & 0x7ff) - 0x3ff);
|
|
|
859 }
|
|
|
860
|
|
|
861 q2 = nr_digits[y_nr_bits].digits;
|
|
|
862 if (q2 == 0) {
|
|
|
863 q2 = nr_digits[y_nr_bits].digits1;
|
|
|
864 if (C2_hi > nr_digits[y_nr_bits].threshold_hi ||
|
|
|
865 (C2_hi == nr_digits[y_nr_bits].threshold_hi &&
|
|
|
866 C2_lo >= nr_digits[y_nr_bits].threshold_lo))
|
|
|
867 q2++;
|
|
|
868 }
|
|
|
869
|
|
|
870 delta = q1 + (int) (x_exp >> 49) - q2 - (int) (y_exp >> 49);
|
|
|
871
|
|
|
872 if (delta >= P34) {
|
|
|
873 // round the result directly because 0 < C2 < ulp (C1 * 10^(x_exp-e2))
|
|
|
874 // n = C1 * 10^e1 or n = C1 +/- 10^(q1-P34)) * 10^e1
|
|
|
875 // the result is inexact; the preferred exponent is the least possible
|
|
|
876
|
|
|
877 if (delta >= P34 + 1) {
|
|
|
878 // for RN the result is the operand with the larger magnitude,
|
|
|
879 // possibly scaled up by 10^(P34-q1)
|
|
|
880 // an overflow cannot occur in this case (rounding to nearest)
|
|
|
881 if (q1 < P34) { // scale C1 up by 10^(P34-q1)
|
|
|
882 // Note: because delta >= P34+1 it is certain that
|
|
|
883 // x_exp - ((UINT64)scale << 49) will stay above e_min
|
|
|
884 scale = P34 - q1;
|
|
|
885 if (q1 <= 19) { // C1 fits in 64 bits
|
|
|
886 // 1 <= q1 <= 19 => 15 <= scale <= 33
|
|
|
887 if (scale <= 19) { // 10^scale fits in 64 bits
|
|
|
888 __mul_64x64_to_128MACH (C1, ten2k64[scale], C1_lo);
|
|
|
889 } else { // if 20 <= scale <= 33
|
|
|
890 // C1 * 10^scale = (C1 * 10^(scale-19)) * 10^19 where
|
|
|
891 // (C1 * 10^(scale-19)) fits in 64 bits
|
|
|
892 C1_lo = C1_lo * ten2k64[scale - 19];
|
|
|
893 __mul_64x64_to_128MACH (C1, ten2k64[19], C1_lo);
|
|
|
894 }
|
|
|
895 } else { //if 20 <= q1 <= 33=P34-1 then C1 fits only in 128 bits
|
|
|
896 // => 1 <= P34 - q1 <= 14 so 10^(P34-q1) fits in 64 bits
|
|
|
897 C1.w[1] = C1_hi;
|
|
|
898 C1.w[0] = C1_lo;
|
|
|
899 // C1 = ten2k64[P34 - q1] * C1
|
|
|
900 __mul_128x64_to_128 (C1, ten2k64[P34 - q1], C1);
|
|
|
901 }
|
|
|
902 x_exp = x_exp - ((UINT64) scale << 49);
|
|
|
903 C1_hi = C1.w[1];
|
|
|
904 C1_lo = C1.w[0];
|
|
|
905 }
|
|
|
906 // some special cases arise: if delta = P34 + 1 and C1 = 10^(P34-1)
|
|
|
907 // (after scaling) and x_sign != y_sign and C2 > 5*10^(q2-1) =>
|
|
|
908 // subtract 1 ulp
|
|
|
909 // Note: do this only for rounding to nearest; for other rounding
|
|
|
910 // modes the correction will be applied next
|
|
|
911 if ((rnd_mode == ROUNDING_TO_NEAREST
|
|
|
912 || rnd_mode == ROUNDING_TIES_AWAY) && delta == (P34 + 1)
|
|
|
913 && C1_hi == 0x0000314dc6448d93ull
|
|
|
914 && C1_lo == 0x38c15b0a00000000ull && x_sign != y_sign
|
|
|
915 && ((q2 <= 19 && C2_lo > midpoint64[q2 - 1]) || (q2 >= 20
|
|
|
916 && (C2_hi >
|
|
|
917 midpoint128
|
|
|
918 [q2 -
|
|
|
919 20].
|
|
|
920 w[1]
|
|
|
921 ||
|
|
|
922 (C2_hi
|
|
|
923 ==
|
|
|
924 midpoint128
|
|
|
925 [q2 -
|
|
|
926 20].
|
|
|
927 w[1]
|
|
|
928 &&
|
|
|
929 C2_lo
|
|
|
930 >
|
|
|
931 midpoint128
|
|
|
932 [q2 -
|
|
|
933 20].
|
|
|
934 w
|
|
|
935 [0])))))
|
|
|
936 {
|
|
|
937 // C1 = 10^34 - 1 and decrement x_exp by 1 (no underflow possible)
|
|
|
938 C1_hi = 0x0001ed09bead87c0ull;
|
|
|
939 C1_lo = 0x378d8e63ffffffffull;
|
|
|
940 x_exp = x_exp - EXP_P1;
|
|
|
941 }
|
|
|
942 if (rnd_mode != ROUNDING_TO_NEAREST) {
|
|
|
943 if ((rnd_mode == ROUNDING_DOWN && x_sign && y_sign) ||
|
|
|
944 (rnd_mode == ROUNDING_UP && !x_sign && !y_sign)) {
|
|
|
945 // add 1 ulp and then check for overflow
|
|
|
946 C1_lo = C1_lo + 1;
|
|
|
947 if (C1_lo == 0) { // rounding overflow in the low 64 bits
|
|
|
948 C1_hi = C1_hi + 1;
|
|
|
949 }
|
|
|
950 if (C1_hi == 0x0001ed09bead87c0ull
|
|
|
951 && C1_lo == 0x378d8e6400000000ull) {
|
|
|
952 // C1 = 10^34 => rounding overflow
|
|
|
953 C1_hi = 0x0000314dc6448d93ull;
|
|
|
954 C1_lo = 0x38c15b0a00000000ull; // 10^33
|
|
|
955 x_exp = x_exp + EXP_P1;
|
|
|
956 if (x_exp == EXP_MAX_P1) { // overflow
|
|
|
957 C1_hi = 0x7800000000000000ull; // +inf
|
|
|
958 C1_lo = 0x0ull;
|
|
|
959 x_exp = 0; // x_sign is preserved
|
|
|
960 // set overflow flag (the inexact flag was set too)
|
|
|
961 *pfpsf |= OVERFLOW_EXCEPTION;
|
|
|
962 }
|
|
|
963 }
|
|
|
964 } else if ((rnd_mode == ROUNDING_DOWN && !x_sign && y_sign) ||
|
|
|
965 (rnd_mode == ROUNDING_UP && x_sign && !y_sign) ||
|
|
|
966 (rnd_mode == ROUNDING_TO_ZERO
|
|
|
967 && x_sign != y_sign)) {
|
|
|
968 // subtract 1 ulp from C1
|
|
|
969 // Note: because delta >= P34 + 1 the result cannot be zero
|
|
|
970 C1_lo = C1_lo - 1;
|
|
|
971 if (C1_lo == 0xffffffffffffffffull)
|
|
|
972 C1_hi = C1_hi - 1;
|
|
|
973 // if the coefficient is 10^33 - 1 then make it 10^34 - 1 and
|
|
|
974 // decrease the exponent by 1 (because delta >= P34 + 1 the
|
|
|
975 // exponent will not become less than e_min)
|
|
|
976 // 10^33 - 1 = 0x0000314dc6448d9338c15b09ffffffff
|
|
|
977 // 10^34 - 1 = 0x0001ed09bead87c0378d8e63ffffffff
|
|
|
978 if (C1_hi == 0x0000314dc6448d93ull
|
|
|
979 && C1_lo == 0x38c15b09ffffffffull) {
|
|
|
980 // make C1 = 10^34 - 1
|
|
|
981 C1_hi = 0x0001ed09bead87c0ull;
|
|
|
982 C1_lo = 0x378d8e63ffffffffull;
|
|
|
983 x_exp = x_exp - EXP_P1;
|
|
|
984 }
|
|
|
985 } else {
|
|
|
986 ; // the result is already correct
|
|
|
987 }
|
|
|
988 }
|
|
|
989 // set the inexact flag
|
|
|
990 *pfpsf |= INEXACT_EXCEPTION;
|
|
|
991 // assemble the result
|
|
|
992 res.w[1] = x_sign | x_exp | C1_hi;
|
|
|
993 res.w[0] = C1_lo;
|
|
|
994 } else { // delta = P34
|
|
|
995 // in most cases, the smaller operand may be < or = or > 1/2 ulp of the
|
|
|
996 // larger operand
|
|
|
997 // however, the case C1 = 10^(q1-1) and x_sign != y_sign is special due
|
|
|
998 // to accuracy loss after subtraction, and will be treated separately
|
|
|
999 if (x_sign == y_sign || (q1 <= 20
|
|
|
1000 && (C1_hi != 0
|
|
|
1001 || C1_lo != ten2k64[q1 - 1]))
|
|
|
1002 || (q1 >= 21 && (C1_hi != ten2k128[q1 - 21].w[1]
|
|
|
1003 || C1_lo != ten2k128[q1 - 21].w[0]))) {
|
|
|
1004 // if x_sign == y_sign or C1 != 10^(q1-1)
|
|
|
1005 // compare C2 with 1/2 ulp = 5 * 10^(q2-1), the latter read from table
|
|
|
1006 // Note: cases q1<=19 and q1>=20 can be coalesced at some latency cost
|
|
|
1007 if (q2 <= 19) { // C2 and 5*10^(q2-1) both fit in 64 bits
|
|
|
1008 halfulp64 = midpoint64[q2 - 1]; // 5 * 10^(q2-1)
|
|
|
1009 if (C2_lo < halfulp64) { // n2 < 1/2 ulp (n1)
|
|
|
1010 // for RN the result is the operand with the larger magnitude,
|
|
|
1011 // possibly scaled up by 10^(P34-q1)
|
|
|
1012 // an overflow cannot occur in this case (rounding to nearest)
|
|
|
1013 if (q1 < P34) { // scale C1 up by 10^(P34-q1)
|
|
|
1014 // Note: because delta = P34 it is certain that
|
|
|
1015 // x_exp - ((UINT64)scale << 49) will stay above e_min
|
|
|
1016 scale = P34 - q1;
|
|
|
1017 if (q1 <= 19) { // C1 fits in 64 bits
|
|
|
1018 // 1 <= q1 <= 19 => 15 <= scale <= 33
|
|
|
1019 if (scale <= 19) { // 10^scale fits in 64 bits
|
|
|
1020 __mul_64x64_to_128MACH (C1, ten2k64[scale], C1_lo);
|
|
|
1021 } else { // if 20 <= scale <= 33
|
|
|
1022 // C1 * 10^scale = (C1 * 10^(scale-19)) * 10^19 where
|
|
|
1023 // (C1 * 10^(scale-19)) fits in 64 bits
|
|
|
1024 C1_lo = C1_lo * ten2k64[scale - 19];
|
|
|
1025 __mul_64x64_to_128MACH (C1, ten2k64[19], C1_lo);
|
|
|
1026 }
|
|
|
1027 } else { //if 20 <= q1 <= 33=P34-1 then C1 fits only in 128 bits
|
|
|
1028 // => 1 <= P34 - q1 <= 14 so 10^(P34-q1) fits in 64 bits
|
|
|
1029 C1.w[1] = C1_hi;
|
|
|
1030 C1.w[0] = C1_lo;
|
|
|
1031 // C1 = ten2k64[P34 - q1] * C1
|
|
|
1032 __mul_128x64_to_128 (C1, ten2k64[P34 - q1], C1);
|
|
|
1033 }
|
|
|
1034 x_exp = x_exp - ((UINT64) scale << 49);
|
|
|
1035 C1_hi = C1.w[1];
|
|
|
1036 C1_lo = C1.w[0];
|
|
|
1037 }
|
|
|
1038 if (rnd_mode != ROUNDING_TO_NEAREST) {
|
|
|
1039 if ((rnd_mode == ROUNDING_DOWN && x_sign && y_sign) ||
|
|
|
1040 (rnd_mode == ROUNDING_UP && !x_sign && !y_sign)) {
|
|
|
1041 // add 1 ulp and then check for overflow
|
|
|
1042 C1_lo = C1_lo + 1;
|
|
|
1043 if (C1_lo == 0) { // rounding overflow in the low 64 bits
|
|
|
1044 C1_hi = C1_hi + 1;
|
|
|
1045 }
|
|
|
1046 if (C1_hi == 0x0001ed09bead87c0ull
|
|
|
1047 && C1_lo == 0x378d8e6400000000ull) {
|
|
|
1048 // C1 = 10^34 => rounding overflow
|
|
|
1049 C1_hi = 0x0000314dc6448d93ull;
|
|
|
1050 C1_lo = 0x38c15b0a00000000ull; // 10^33
|
|
|
1051 x_exp = x_exp + EXP_P1;
|
|
|
1052 if (x_exp == EXP_MAX_P1) { // overflow
|
|
|
1053 C1_hi = 0x7800000000000000ull; // +inf
|
|
|
1054 C1_lo = 0x0ull;
|
|
|
1055 x_exp = 0; // x_sign is preserved
|
|
|
1056 // set overflow flag (the inexact flag was set too)
|
|
|
1057 *pfpsf |= OVERFLOW_EXCEPTION;
|
|
|
1058 }
|
|
|
1059 }
|
|
|
1060 } else
|
|
|
1061 if ((rnd_mode == ROUNDING_DOWN && !x_sign && y_sign)
|
|
|
1062 || (rnd_mode == ROUNDING_UP && x_sign && !y_sign)
|
|
|
1063 || (rnd_mode == ROUNDING_TO_ZERO
|
|
|
1064 && x_sign != y_sign)) {
|
|
|
1065 // subtract 1 ulp from C1
|
|
|
1066 // Note: because delta >= P34 + 1 the result cannot be zero
|
|
|
1067 C1_lo = C1_lo - 1;
|
|
|
1068 if (C1_lo == 0xffffffffffffffffull)
|
|
|
1069 C1_hi = C1_hi - 1;
|
|
|
1070 // if the coefficient is 10^33-1 then make it 10^34-1 and
|
|
|
1071 // decrease the exponent by 1 (because delta >= P34 + 1 the
|
|
|
1072 // exponent will not become less than e_min)
|
|
|
1073 // 10^33 - 1 = 0x0000314dc6448d9338c15b09ffffffff
|
|
|
1074 // 10^34 - 1 = 0x0001ed09bead87c0378d8e63ffffffff
|
|
|
1075 if (C1_hi == 0x0000314dc6448d93ull
|
|
|
1076 && C1_lo == 0x38c15b09ffffffffull) {
|
|
|
1077 // make C1 = 10^34 - 1
|
|
|
1078 C1_hi = 0x0001ed09bead87c0ull;
|
|
|
1079 C1_lo = 0x378d8e63ffffffffull;
|
|
|
1080 x_exp = x_exp - EXP_P1;
|
|
|
1081 }
|
|
|
1082 } else {
|
|
|
1083 ; // the result is already correct
|
|
|
1084 }
|
|
|
1085 }
|
|
|
1086 // set the inexact flag
|
|
|
1087 *pfpsf |= INEXACT_EXCEPTION;
|
|
|
1088 // assemble the result
|
|
|
1089 res.w[1] = x_sign | x_exp | C1_hi;
|
|
|
1090 res.w[0] = C1_lo;
|
|
|
1091 } else if ((C2_lo == halfulp64)
|
|
|
1092 && (q1 < P34 || ((C1_lo & 0x1) == 0))) {
|
|
|
1093 // n2 = 1/2 ulp (n1) and C1 is even
|
|
|
1094 // the result is the operand with the larger magnitude,
|
|
|
1095 // possibly scaled up by 10^(P34-q1)
|
|
|
1096 // an overflow cannot occur in this case (rounding to nearest)
|
|
|
1097 if (q1 < P34) { // scale C1 up by 10^(P34-q1)
|
|
|
1098 // Note: because delta = P34 it is certain that
|
|
|
1099 // x_exp - ((UINT64)scale << 49) will stay above e_min
|
|
|
1100 scale = P34 - q1;
|
|
|
1101 if (q1 <= 19) { // C1 fits in 64 bits
|
|
|
1102 // 1 <= q1 <= 19 => 15 <= scale <= 33
|
|
|
1103 if (scale <= 19) { // 10^scale fits in 64 bits
|
|
|
1104 __mul_64x64_to_128MACH (C1, ten2k64[scale], C1_lo);
|
|
|
1105 } else { // if 20 <= scale <= 33
|
|
|
1106 // C1 * 10^scale = (C1 * 10^(scale-19)) * 10^19 where
|
|
|
1107 // (C1 * 10^(scale-19)) fits in 64 bits
|
|
|
1108 C1_lo = C1_lo * ten2k64[scale - 19];
|
|
|
1109 __mul_64x64_to_128MACH (C1, ten2k64[19], C1_lo);
|
|
|
1110 }
|
|
|
1111 } else { //if 20 <= q1 <= 33=P34-1 then C1 fits only in 128 bits
|
|
|
1112 // => 1 <= P34 - q1 <= 14 so 10^(P34-q1) fits in 64 bits
|
|
|
1113 C1.w[1] = C1_hi;
|
|
|
1114 C1.w[0] = C1_lo;
|
|
|
1115 // C1 = ten2k64[P34 - q1] * C1
|
|
|
1116 __mul_128x64_to_128 (C1, ten2k64[P34 - q1], C1);
|
|
|
1117 }
|
|
|
1118 x_exp = x_exp - ((UINT64) scale << 49);
|
|
|
1119 C1_hi = C1.w[1];
|
|
|
1120 C1_lo = C1.w[0];
|
|
|
1121 }
|
|
|
1122 if ((rnd_mode == ROUNDING_TO_NEAREST && x_sign == y_sign
|
|
|
1123 && (C1_lo & 0x01)) || (rnd_mode == ROUNDING_TIES_AWAY
|
|
|
1124 && x_sign == y_sign)
|
|
|
1125 || (rnd_mode == ROUNDING_UP && !x_sign && !y_sign)
|
|
|
1126 || (rnd_mode == ROUNDING_DOWN && x_sign && y_sign)) {
|
|
|
1127 // add 1 ulp and then check for overflow
|
|
|
1128 C1_lo = C1_lo + 1;
|
|
|
1129 if (C1_lo == 0) { // rounding overflow in the low 64 bits
|
|
|
1130 C1_hi = C1_hi + 1;
|
|
|
1131 }
|
|
|
1132 if (C1_hi == 0x0001ed09bead87c0ull
|
|
|
1133 && C1_lo == 0x378d8e6400000000ull) {
|
|
|
1134 // C1 = 10^34 => rounding overflow
|
|
|
1135 C1_hi = 0x0000314dc6448d93ull;
|
|
|
1136 C1_lo = 0x38c15b0a00000000ull; // 10^33
|
|
|
1137 x_exp = x_exp + EXP_P1;
|
|
|
1138 if (x_exp == EXP_MAX_P1) { // overflow
|
|
|
1139 C1_hi = 0x7800000000000000ull; // +inf
|
|
|
1140 C1_lo = 0x0ull;
|
|
|
1141 x_exp = 0; // x_sign is preserved
|
|
|
1142 // set overflow flag (the inexact flag was set too)
|
|
|
1143 *pfpsf |= OVERFLOW_EXCEPTION;
|
|
|
1144 }
|
|
|
1145 }
|
|
|
1146 } else
|
|
|
1147 if ((rnd_mode == ROUNDING_TO_NEAREST && x_sign != y_sign
|
|
|
1148 && (C1_lo & 0x01)) || (rnd_mode == ROUNDING_DOWN
|
|
|
1149 && !x_sign && y_sign)
|
|
|
1150 || (rnd_mode == ROUNDING_UP && x_sign && !y_sign)
|
|
|
1151 || (rnd_mode == ROUNDING_TO_ZERO
|
|
|
1152 && x_sign != y_sign)) {
|
|
|
1153 // subtract 1 ulp from C1
|
|
|
1154 // Note: because delta >= P34 + 1 the result cannot be zero
|
|
|
1155 C1_lo = C1_lo - 1;
|
|
|
1156 if (C1_lo == 0xffffffffffffffffull)
|
|
|
1157 C1_hi = C1_hi - 1;
|
|
|
1158 // if the coefficient is 10^33 - 1 then make it 10^34 - 1
|
|
|
1159 // and decrease the exponent by 1 (because delta >= P34 + 1
|
|
|
1160 // the exponent will not become less than e_min)
|
|
|
1161 // 10^33 - 1 = 0x0000314dc6448d9338c15b09ffffffff
|
|
|
1162 // 10^34 - 1 = 0x0001ed09bead87c0378d8e63ffffffff
|
|
|
1163 if (C1_hi == 0x0000314dc6448d93ull
|
|
|
1164 && C1_lo == 0x38c15b09ffffffffull) {
|
|
|
1165 // make C1 = 10^34 - 1
|
|
|
1166 C1_hi = 0x0001ed09bead87c0ull;
|
|
|
1167 C1_lo = 0x378d8e63ffffffffull;
|
|
|
1168 x_exp = x_exp - EXP_P1;
|
|
|
1169 }
|
|
|
1170 } else {
|
|
|
1171 ; // the result is already correct
|
|
|
1172 }
|
|
|
1173 // set the inexact flag
|
|
|
1174 *pfpsf |= INEXACT_EXCEPTION;
|
|
|
1175 // assemble the result
|
|
|
1176 res.w[1] = x_sign | x_exp | C1_hi;
|
|
|
1177 res.w[0] = C1_lo;
|
|
|
1178 } else { // if C2_lo > halfulp64 ||
|
|
|
1179 // (C2_lo == halfulp64 && q1 == P34 && ((C1_lo & 0x1) == 1)), i.e.
|
|
|
1180 // 1/2 ulp(n1) < n2 < 1 ulp(n1) or n2 = 1/2 ulp(n1) and C1 odd
|
|
|
1181 // res = x+1 ulp if n1*n2 > 0 and res = x-1 ulp if n1*n2 < 0
|
|
|
1182 if (q1 < P34) { // then 1 ulp = 10^(e1+q1-P34) < 10^e1
|
|
|
1183 // Note: if (q1 == P34) then 1 ulp = 10^(e1+q1-P34) = 10^e1
|
|
|
1184 // because q1 < P34 we must first replace C1 by
|
|
|
1185 // C1 * 10^(P34-q1), and must decrease the exponent by
|
|
|
1186 // (P34-q1) (it will still be at least e_min)
|
|
|
1187 scale = P34 - q1;
|
|
|
1188 if (q1 <= 19) { // C1 fits in 64 bits
|
|
|
1189 // 1 <= q1 <= 19 => 15 <= scale <= 33
|
|
|
1190 if (scale <= 19) { // 10^scale fits in 64 bits
|
|
|
1191 __mul_64x64_to_128MACH (C1, ten2k64[scale], C1_lo);
|
|
|
1192 } else { // if 20 <= scale <= 33
|
|
|
1193 // C1 * 10^scale = (C1 * 10^(scale-19)) * 10^19 where
|
|
|
1194 // (C1 * 10^(scale-19)) fits in 64 bits
|
|
|
1195 C1_lo = C1_lo * ten2k64[scale - 19];
|
|
|
1196 __mul_64x64_to_128MACH (C1, ten2k64[19], C1_lo);
|
|
|
1197 }
|
|
|
1198 } else { //if 20 <= q1 <= 33=P34-1 then C1 fits only in 128 bits
|
|
|
1199 // => 1 <= P34 - q1 <= 14 so 10^(P34-q1) fits in 64 bits
|
|
|
1200 C1.w[1] = C1_hi;
|
|
|
1201 C1.w[0] = C1_lo;
|
|
|
1202 // C1 = ten2k64[P34 - q1] * C1
|
|
|
1203 __mul_128x64_to_128 (C1, ten2k64[P34 - q1], C1);
|
|
|
1204 }
|
|
|
1205 x_exp = x_exp - ((UINT64) scale << 49);
|
|
|
1206 C1_hi = C1.w[1];
|
|
|
1207 C1_lo = C1.w[0];
|
|
|
1208 // check for rounding overflow
|
|
|
1209 if (C1_hi == 0x0001ed09bead87c0ull
|
|
|
1210 && C1_lo == 0x378d8e6400000000ull) {
|
|
|
1211 // C1 = 10^34 => rounding overflow
|
|
|
1212 C1_hi = 0x0000314dc6448d93ull;
|
|
|
1213 C1_lo = 0x38c15b0a00000000ull; // 10^33
|
|
|
1214 x_exp = x_exp + EXP_P1;
|
|
|
1215 }
|
|
|
1216 }
|
|
|
1217 if ((rnd_mode == ROUNDING_TO_NEAREST && x_sign != y_sign)
|
|
|
1218 || (rnd_mode == ROUNDING_TIES_AWAY && x_sign != y_sign
|
|
|
1219 && C2_lo != halfulp64)
|
|
|
1220 || (rnd_mode == ROUNDING_DOWN && !x_sign && y_sign)
|
|
|
1221 || (rnd_mode == ROUNDING_UP && x_sign && !y_sign)
|
|
|
1222 || (rnd_mode == ROUNDING_TO_ZERO
|
|
|
1223 && x_sign != y_sign)) {
|
|
|
1224 // the result is x - 1
|
|
|
1225 // for RN n1 * n2 < 0; underflow not possible
|
|
|
1226 C1_lo = C1_lo - 1;
|
|
|
1227 if (C1_lo == 0xffffffffffffffffull)
|
|
|
1228 C1_hi--;
|
|
|
1229 // check if we crossed into the lower decade
|
|
|
1230 if (C1_hi == 0x0000314dc6448d93ull && C1_lo == 0x38c15b09ffffffffull) { // 10^33 - 1
|
|
|
1231 C1_hi = 0x0001ed09bead87c0ull; // 10^34 - 1
|
|
|
1232 C1_lo = 0x378d8e63ffffffffull;
|
|
|
1233 x_exp = x_exp - EXP_P1; // no underflow, because n1 >> n2
|
|
|
1234 }
|
|
|
1235 } else
|
|
|
1236 if ((rnd_mode == ROUNDING_TO_NEAREST
|
|
|
1237 && x_sign == y_sign)
|
|
|
1238 || (rnd_mode == ROUNDING_TIES_AWAY
|
|
|
1239 && x_sign == y_sign)
|
|
|
1240 || (rnd_mode == ROUNDING_DOWN && x_sign && y_sign)
|
|
|
1241 || (rnd_mode == ROUNDING_UP && !x_sign
|
|
|
1242 && !y_sign)) {
|
|
|
1243 // the result is x + 1
|
|
|
1244 // for RN x_sign = y_sign, i.e. n1*n2 > 0
|
|
|
1245 C1_lo = C1_lo + 1;
|
|
|
1246 if (C1_lo == 0) { // rounding overflow in the low 64 bits
|
|
|
1247 C1_hi = C1_hi + 1;
|
|
|
1248 }
|
|
|
1249 if (C1_hi == 0x0001ed09bead87c0ull
|
|
|
1250 && C1_lo == 0x378d8e6400000000ull) {
|
|
|
1251 // C1 = 10^34 => rounding overflow
|
|
|
1252 C1_hi = 0x0000314dc6448d93ull;
|
|
|
1253 C1_lo = 0x38c15b0a00000000ull; // 10^33
|
|
|
1254 x_exp = x_exp + EXP_P1;
|
|
|
1255 if (x_exp == EXP_MAX_P1) { // overflow
|
|
|
1256 C1_hi = 0x7800000000000000ull; // +inf
|
|
|
1257 C1_lo = 0x0ull;
|
|
|
1258 x_exp = 0; // x_sign is preserved
|
|
|
1259 // set the overflow flag
|
|
|
1260 *pfpsf |= OVERFLOW_EXCEPTION;
|
|
|
1261 }
|
|
|
1262 }
|
|
|
1263 } else {
|
|
|
1264 ; // the result is x
|
|
|
1265 }
|
|
|
1266 // set the inexact flag
|
|
|
1267 *pfpsf |= INEXACT_EXCEPTION;
|
|
|
1268 // assemble the result
|
|
|
1269 res.w[1] = x_sign | x_exp | C1_hi;
|
|
|
1270 res.w[0] = C1_lo;
|
|
|
1271 }
|
|
|
1272 } else { // if q2 >= 20 then 5*10^(q2-1) and C2 (the latter in
|
|
|
1273 // most cases) fit only in more than 64 bits
|
|
|
1274 halfulp128 = midpoint128[q2 - 20]; // 5 * 10^(q2-1)
|
|
|
1275 if ((C2_hi < halfulp128.w[1])
|
|
|
1276 || (C2_hi == halfulp128.w[1]
|
|
|
1277 && C2_lo < halfulp128.w[0])) {
|
|
|
1278 // n2 < 1/2 ulp (n1)
|
|
|
1279 // the result is the operand with the larger magnitude,
|
|
|
1280 // possibly scaled up by 10^(P34-q1)
|
|
|
1281 // an overflow cannot occur in this case (rounding to nearest)
|
|
|
1282 if (q1 < P34) { // scale C1 up by 10^(P34-q1)
|
|
|
1283 // Note: because delta = P34 it is certain that
|
|
|
1284 // x_exp - ((UINT64)scale << 49) will stay above e_min
|
|
|
1285 scale = P34 - q1;
|
|
|
1286 if (q1 <= 19) { // C1 fits in 64 bits
|
|
|
1287 // 1 <= q1 <= 19 => 15 <= scale <= 33
|
|
|
1288 if (scale <= 19) { // 10^scale fits in 64 bits
|
|
|
1289 __mul_64x64_to_128MACH (C1, ten2k64[scale], C1_lo);
|
|
|
1290 } else { // if 20 <= scale <= 33
|
|
|
1291 // C1 * 10^scale = (C1 * 10^(scale-19)) * 10^19 where
|
|
|
1292 // (C1 * 10^(scale-19)) fits in 64 bits
|
|
|
1293 C1_lo = C1_lo * ten2k64[scale - 19];
|
|
|
1294 __mul_64x64_to_128MACH (C1, ten2k64[19], C1_lo);
|
|
|
1295 }
|
|
|
1296 } else { //if 20 <= q1 <= 33=P34-1 then C1 fits only in 128 bits
|
|
|
1297 // => 1 <= P34 - q1 <= 14 so 10^(P34-q1) fits in 64 bits
|
|
|
1298 C1.w[1] = C1_hi;
|
|
|
1299 C1.w[0] = C1_lo;
|
|
|
1300 // C1 = ten2k64[P34 - q1] * C1
|
|
|
1301 __mul_128x64_to_128 (C1, ten2k64[P34 - q1], C1);
|
|
|
1302 }
|
|
|
1303 C1_hi = C1.w[1];
|
|
|
1304 C1_lo = C1.w[0];
|
|
|
1305 x_exp = x_exp - ((UINT64) scale << 49);
|
|
|
1306 }
|
|
|
1307 if (rnd_mode != ROUNDING_TO_NEAREST) {
|
|
|
1308 if ((rnd_mode == ROUNDING_DOWN && x_sign && y_sign) ||
|
|
|
1309 (rnd_mode == ROUNDING_UP && !x_sign && !y_sign)) {
|
|
|
1310 // add 1 ulp and then check for overflow
|
|
|
1311 C1_lo = C1_lo + 1;
|
|
|
1312 if (C1_lo == 0) { // rounding overflow in the low 64 bits
|
|
|
1313 C1_hi = C1_hi + 1;
|
|
|
1314 }
|
|
|
1315 if (C1_hi == 0x0001ed09bead87c0ull
|
|
|
1316 && C1_lo == 0x378d8e6400000000ull) {
|
|
|
1317 // C1 = 10^34 => rounding overflow
|
|
|
1318 C1_hi = 0x0000314dc6448d93ull;
|
|
|
1319 C1_lo = 0x38c15b0a00000000ull; // 10^33
|
|
|
1320 x_exp = x_exp + EXP_P1;
|
|
|
1321 if (x_exp == EXP_MAX_P1) { // overflow
|
|
|
1322 C1_hi = 0x7800000000000000ull; // +inf
|
|
|
1323 C1_lo = 0x0ull;
|
|
|
1324 x_exp = 0; // x_sign is preserved
|
|
|
1325 // set overflow flag (the inexact flag was set too)
|
|
|
1326 *pfpsf |= OVERFLOW_EXCEPTION;
|
|
|
1327 }
|
|
|
1328 }
|
|
|
1329 } else
|
|
|
1330 if ((rnd_mode == ROUNDING_DOWN && !x_sign && y_sign)
|
|
|
1331 || (rnd_mode == ROUNDING_UP && x_sign && !y_sign)
|
|
|
1332 || (rnd_mode == ROUNDING_TO_ZERO
|
|
|
1333 && x_sign != y_sign)) {
|
|
|
1334 // subtract 1 ulp from C1
|
|
|
1335 // Note: because delta >= P34 + 1 the result cannot be zero
|
|
|
1336 C1_lo = C1_lo - 1;
|
|
|
1337 if (C1_lo == 0xffffffffffffffffull)
|
|
|
1338 C1_hi = C1_hi - 1;
|
|
|
1339 // if the coefficient is 10^33-1 then make it 10^34-1 and
|
|
|
1340 // decrease the exponent by 1 (because delta >= P34 + 1 the
|
|
|
1341 // exponent will not become less than e_min)
|
|
|
1342 // 10^33 - 1 = 0x0000314dc6448d9338c15b09ffffffff
|
|
|
1343 // 10^34 - 1 = 0x0001ed09bead87c0378d8e63ffffffff
|
|
|
1344 if (C1_hi == 0x0000314dc6448d93ull
|
|
|
1345 && C1_lo == 0x38c15b09ffffffffull) {
|
|
|
1346 // make C1 = 10^34 - 1
|
|
|
1347 C1_hi = 0x0001ed09bead87c0ull;
|
|
|
1348 C1_lo = 0x378d8e63ffffffffull;
|
|
|
1349 x_exp = x_exp - EXP_P1;
|
|
|
1350 }
|
|
|
1351 } else {
|
|
|
1352 ; // the result is already correct
|
|
|
1353 }
|
|
|
1354 }
|
|
|
1355 // set the inexact flag
|
|
|
1356 *pfpsf |= INEXACT_EXCEPTION;
|
|
|
1357 // assemble the result
|
|
|
1358 res.w[1] = x_sign | x_exp | C1_hi;
|
|
|
1359 res.w[0] = C1_lo;
|
|
|
1360 } else if ((C2_hi == halfulp128.w[1]
|
|
|
1361 && C2_lo == halfulp128.w[0])
|
|
|
1362 && (q1 < P34 || ((C1_lo & 0x1) == 0))) {
|
|
|
1363 // midpoint & lsb in C1 is 0
|
|
|
1364 // n2 = 1/2 ulp (n1) and C1 is even
|
|
|
1365 // the result is the operand with the larger magnitude,
|
|
|
1366 // possibly scaled up by 10^(P34-q1)
|
|
|
1367 // an overflow cannot occur in this case (rounding to nearest)
|
|
|
1368 if (q1 < P34) { // scale C1 up by 10^(P34-q1)
|
|
|
1369 // Note: because delta = P34 it is certain that
|
|
|
1370 // x_exp - ((UINT64)scale << 49) will stay above e_min
|
|
|
1371 scale = P34 - q1;
|
|
|
1372 if (q1 <= 19) { // C1 fits in 64 bits
|
|
|
1373 // 1 <= q1 <= 19 => 15 <= scale <= 33
|
|
|
1374 if (scale <= 19) { // 10^scale fits in 64 bits
|
|
|
1375 __mul_64x64_to_128MACH (C1, ten2k64[scale], C1_lo);
|
|
|
1376 } else { // if 20 <= scale <= 33
|
|
|
1377 // C1 * 10^scale = (C1 * 10^(scale-19)) * 10^19 where
|
|
|
1378 // (C1 * 10^(scale-19)) fits in 64 bits
|
|
|
1379 C1_lo = C1_lo * ten2k64[scale - 19];
|
|
|
1380 __mul_64x64_to_128MACH (C1, ten2k64[19], C1_lo);
|
|
|
1381 }
|
|
|
1382 } else { //if 20 <= q1 <= 33=P34-1 then C1 fits only in 128 bits
|
|
|
1383 // => 1 <= P34 - q1 <= 14 so 10^(P34-q1) fits in 64 bits
|
|
|
1384 C1.w[1] = C1_hi;
|
|
|
1385 C1.w[0] = C1_lo;
|
|
|
1386 // C1 = ten2k64[P34 - q1] * C1
|
|
|
1387 __mul_128x64_to_128 (C1, ten2k64[P34 - q1], C1);
|
|
|
1388 }
|
|
|
1389 x_exp = x_exp - ((UINT64) scale << 49);
|
|
|
1390 C1_hi = C1.w[1];
|
|
|
1391 C1_lo = C1.w[0];
|
|
|
1392 }
|
|
|
1393 if (rnd_mode != ROUNDING_TO_NEAREST) {
|
|
|
1394 if ((rnd_mode == ROUNDING_TIES_AWAY && x_sign == y_sign)
|
|
|
1395 || (rnd_mode == ROUNDING_UP && !y_sign)) {
|
|
|
1396 // add 1 ulp and then check for overflow
|
|
|
1397 C1_lo = C1_lo + 1;
|
|
|
1398 if (C1_lo == 0) { // rounding overflow in the low 64 bits
|
|
|
1399 C1_hi = C1_hi + 1;
|
|
|
1400 }
|
|
|
1401 if (C1_hi == 0x0001ed09bead87c0ull
|
|
|
1402 && C1_lo == 0x378d8e6400000000ull) {
|
|
|
1403 // C1 = 10^34 => rounding overflow
|
|
|
1404 C1_hi = 0x0000314dc6448d93ull;
|
|
|
1405 C1_lo = 0x38c15b0a00000000ull; // 10^33
|
|
|
1406 x_exp = x_exp + EXP_P1;
|
|
|
1407 if (x_exp == EXP_MAX_P1) { // overflow
|
|
|
1408 C1_hi = 0x7800000000000000ull; // +inf
|
|
|
1409 C1_lo = 0x0ull;
|
|
|
1410 x_exp = 0; // x_sign is preserved
|
|
|
1411 // set overflow flag (the inexact flag was set too)
|
|
|
1412 *pfpsf |= OVERFLOW_EXCEPTION;
|
|
|
1413 }
|
|
|
1414 }
|
|
|
1415 } else if ((rnd_mode == ROUNDING_DOWN && y_sign)
|
|
|
1416 || (rnd_mode == ROUNDING_TO_ZERO
|
|
|
1417 && x_sign != y_sign)) {
|
|
|
1418 // subtract 1 ulp from C1
|
|
|
1419 // Note: because delta >= P34 + 1 the result cannot be zero
|
|
|
1420 C1_lo = C1_lo - 1;
|
|
|
1421 if (C1_lo == 0xffffffffffffffffull)
|
|
|
1422 C1_hi = C1_hi - 1;
|
|
|
1423 // if the coefficient is 10^33 - 1 then make it 10^34 - 1
|
|
|
1424 // and decrease the exponent by 1 (because delta >= P34 + 1
|
|
|
1425 // the exponent will not become less than e_min)
|
|
|
1426 // 10^33 - 1 = 0x0000314dc6448d9338c15b09ffffffff
|
|
|
1427 // 10^34 - 1 = 0x0001ed09bead87c0378d8e63ffffffff
|
|
|
1428 if (C1_hi == 0x0000314dc6448d93ull
|
|
|
1429 && C1_lo == 0x38c15b09ffffffffull) {
|
|
|
1430 // make C1 = 10^34 - 1
|
|
|
1431 C1_hi = 0x0001ed09bead87c0ull;
|
|
|
1432 C1_lo = 0x378d8e63ffffffffull;
|
|
|
1433 x_exp = x_exp - EXP_P1;
|
|
|
1434 }
|
|
|
1435 } else {
|
|
|
1436 ; // the result is already correct
|
|
|
1437 }
|
|
|
1438 }
|
|
|
1439 // set the inexact flag
|
|
|
1440 *pfpsf |= INEXACT_EXCEPTION;
|
|
|
1441 // assemble the result
|
|
|
1442 res.w[1] = x_sign | x_exp | C1_hi;
|
|
|
1443 res.w[0] = C1_lo;
|
|
|
1444 } else { // if C2 > halfulp128 ||
|
|
|
1445 // (C2 == halfulp128 && q1 == P34 && ((C1 & 0x1) == 1)), i.e.
|
|
|
1446 // 1/2 ulp(n1) < n2 < 1 ulp(n1) or n2 = 1/2 ulp(n1) and C1 odd
|
|
|
1447 // res = x+1 ulp if n1*n2 > 0 and res = x-1 ulp if n1*n2 < 0
|
|
|
1448 if (q1 < P34) { // then 1 ulp = 10^(e1+q1-P34) < 10^e1
|
|
|
1449 // Note: if (q1 == P34) then 1 ulp = 10^(e1+q1-P34) = 10^e1
|
|
|
1450 // because q1 < P34 we must first replace C1 by C1*10^(P34-q1),
|
|
|
1451 // and must decrease the exponent by (P34-q1) (it will still be
|
|
|
1452 // at least e_min)
|
|
|
1453 scale = P34 - q1;
|
|
|
1454 if (q1 <= 19) { // C1 fits in 64 bits
|
|
|
1455 // 1 <= q1 <= 19 => 15 <= scale <= 33
|
|
|
1456 if (scale <= 19) { // 10^scale fits in 64 bits
|
|
|
1457 __mul_64x64_to_128MACH (C1, ten2k64[scale], C1_lo);
|
|
|
1458 } else { // if 20 <= scale <= 33
|
|
|
1459 // C1 * 10^scale = (C1 * 10^(scale-19)) * 10^19 where
|
|
|
1460 // (C1 * 10^(scale-19)) fits in 64 bits
|
|
|
1461 C1_lo = C1_lo * ten2k64[scale - 19];
|
|
|
1462 __mul_64x64_to_128MACH (C1, ten2k64[19], C1_lo);
|
|
|
1463 }
|
|
|
1464 } else { //if 20 <= q1 <= 33=P34-1 then C1 fits only in 128 bits
|
|
|
1465 // => 1 <= P34 - q1 <= 14 so 10^(P34-q1) fits in 64 bits
|
|
|
1466 C1.w[1] = C1_hi;
|
|
|
1467 C1.w[0] = C1_lo;
|
|
|
1468 // C1 = ten2k64[P34 - q1] * C1
|
|
|
1469 __mul_128x64_to_128 (C1, ten2k64[P34 - q1], C1);
|
|
|
1470 }
|
|
|
1471 C1_hi = C1.w[1];
|
|
|
1472 C1_lo = C1.w[0];
|
|
|
1473 x_exp = x_exp - ((UINT64) scale << 49);
|
|
|
1474 }
|
|
|
1475 if ((rnd_mode == ROUNDING_TO_NEAREST && x_sign != y_sign)
|
|
|
1476 || (rnd_mode == ROUNDING_TIES_AWAY && x_sign != y_sign
|
|
|
1477 && (C2_hi != halfulp128.w[1]
|
|
|
1478 || C2_lo != halfulp128.w[0]))
|
|
|
1479 || (rnd_mode == ROUNDING_DOWN && !x_sign && y_sign)
|
|
|
1480 || (rnd_mode == ROUNDING_UP && x_sign && !y_sign)
|
|
|
1481 || (rnd_mode == ROUNDING_TO_ZERO
|
|
|
1482 && x_sign != y_sign)) {
|
|
|
1483 // the result is x - 1
|
|
|
1484 // for RN n1 * n2 < 0; underflow not possible
|
|
|
1485 C1_lo = C1_lo - 1;
|
|
|
1486 if (C1_lo == 0xffffffffffffffffull)
|
|
|
1487 C1_hi--;
|
|
|
1488 // check if we crossed into the lower decade
|
|
|
1489 if (C1_hi == 0x0000314dc6448d93ull && C1_lo == 0x38c15b09ffffffffull) { // 10^33 - 1
|
|
|
1490 C1_hi = 0x0001ed09bead87c0ull; // 10^34 - 1
|
|
|
1491 C1_lo = 0x378d8e63ffffffffull;
|
|
|
1492 x_exp = x_exp - EXP_P1; // no underflow, because n1 >> n2
|
|
|
1493 }
|
|
|
1494 } else
|
|
|
1495 if ((rnd_mode == ROUNDING_TO_NEAREST
|
|
|
1496 && x_sign == y_sign)
|
|
|
1497 || (rnd_mode == ROUNDING_TIES_AWAY
|
|
|
1498 && x_sign == y_sign)
|
|
|
1499 || (rnd_mode == ROUNDING_DOWN && x_sign && y_sign)
|
|
|
1500 || (rnd_mode == ROUNDING_UP && !x_sign
|
|
|
1501 && !y_sign)) {
|
|
|
1502 // the result is x + 1
|
|
|
1503 // for RN x_sign = y_sign, i.e. n1*n2 > 0
|
|
|
1504 C1_lo = C1_lo + 1;
|
|
|
1505 if (C1_lo == 0) { // rounding overflow in the low 64 bits
|
|
|
1506 C1_hi = C1_hi + 1;
|
|
|
1507 }
|
|
|
1508 if (C1_hi == 0x0001ed09bead87c0ull
|
|
|
1509 && C1_lo == 0x378d8e6400000000ull) {
|
|
|
1510 // C1 = 10^34 => rounding overflow
|
|
|
1511 C1_hi = 0x0000314dc6448d93ull;
|
|
|
1512 C1_lo = 0x38c15b0a00000000ull; // 10^33
|
|
|
1513 x_exp = x_exp + EXP_P1;
|
|
|
1514 if (x_exp == EXP_MAX_P1) { // overflow
|
|
|
1515 C1_hi = 0x7800000000000000ull; // +inf
|
|
|
1516 C1_lo = 0x0ull;
|
|
|
1517 x_exp = 0; // x_sign is preserved
|
|
|
1518 // set the overflow flag
|
|
|
1519 *pfpsf |= OVERFLOW_EXCEPTION;
|
|
|
1520 }
|
|
|
1521 }
|
|
|
1522 } else {
|
|
|
1523 ; // the result is x
|
|
|
1524 }
|
|
|
1525 // set the inexact flag
|
|
|
1526 *pfpsf |= INEXACT_EXCEPTION;
|
|
|
1527 // assemble the result
|
|
|
1528 res.w[1] = x_sign | x_exp | C1_hi;
|
|
|
1529 res.w[0] = C1_lo;
|
|
|
1530 }
|
|
|
1531 } // end q1 >= 20
|
|
|
1532 // end case where C1 != 10^(q1-1)
|
|
|
1533 } else { // C1 = 10^(q1-1) and x_sign != y_sign
|
|
|
1534 // instead of C' = (C1 * 10^(e1-e2) + C2)rnd,P34
|
|
|
1535 // calculate C' = C1 * 10^(e1-e2-x1) + (C2 * 10^(-x1))rnd,P34
|
|
|
1536 // where x1 = q2 - 1, 0 <= x1 <= P34 - 1
|
|
|
1537 // Because C1 = 10^(q1-1) and x_sign != y_sign, C' will have P34
|
|
|
1538 // digits and n = C' * 10^(e2+x1)
|
|
|
1539 // If the result has P34+1 digits, redo the steps above with x1+1
|
|
|
1540 // If the result has P34-1 digits or less, redo the steps above with
|
|
|
1541 // x1-1 but only if initially x1 >= 1
|
|
|
1542 // NOTE: these two steps can be improved, e.g we could guess if
|
|
|
1543 // P34+1 or P34-1 digits will be obtained by adding/subtracting
|
|
|
1544 // just the top 64 bits of the two operands
|
|
|
1545 // The result cannot be zero, and it cannot overflow
|
|
|
1546 x1 = q2 - 1; // 0 <= x1 <= P34-1
|
|
|
1547 // Calculate C1 * 10^(e1-e2-x1) where 1 <= e1-e2-x1 <= P34
|
|
|
1548 // scale = (int)(e1 >> 49) - (int)(e2 >> 49) - x1; 0 <= scale <= P34-1
|
|
|
1549 scale = P34 - q1 + 1; // scale=e1-e2-x1 = P34+1-q1; 1<=scale<=P34
|
|
|
1550 // either C1 or 10^(e1-e2-x1) may not fit is 64 bits,
|
|
|
1551 // but their product fits with certainty in 128 bits
|
|
|
1552 if (scale >= 20) { //10^(e1-e2-x1) doesn't fit in 64 bits, but C1 does
|
|
|
1553 __mul_128x64_to_128 (C1, C1_lo, ten2k128[scale - 20]);
|
|
|
1554 } else { // if (scale >= 1
|
|
|
1555 // if 1 <= scale <= 19 then 10^(e1-e2-x1) fits in 64 bits
|
|
|
1556 if (q1 <= 19) { // C1 fits in 64 bits
|
|
|
1557 __mul_64x64_to_128MACH (C1, C1_lo, ten2k64[scale]);
|
|
|
1558 } else { // q1 >= 20
|
|
|
1559 C1.w[1] = C1_hi;
|
|
|
1560 C1.w[0] = C1_lo;
|
|
|
1561 __mul_128x64_to_128 (C1, ten2k64[scale], C1);
|
|
|
1562 }
|
|
|
1563 }
|
|
|
1564 tmp64 = C1.w[0]; // C1.w[1], C1.w[0] contains C1 * 10^(e1-e2-x1)
|
|
|
1565
|
|
|
1566 // now round C2 to q2-x1 = 1 decimal digit
|
|
|
1567 // C2' = C2 + 1/2 * 10^x1 = C2 + 5 * 10^(x1-1)
|
|
|
1568 ind = x1 - 1; // -1 <= ind <= P34 - 2
|
|
|
1569 if (ind >= 0) { // if (x1 >= 1)
|
|
|
1570 C2.w[0] = C2_lo;
|
|
|
1571 C2.w[1] = C2_hi;
|
|
|
1572 if (ind <= 18) {
|
|
|
1573 C2.w[0] = C2.w[0] + midpoint64[ind];
|
|
|
1574 if (C2.w[0] < C2_lo)
|
|
|
1575 C2.w[1]++;
|
|
|
1576 } else { // 19 <= ind <= 32
|
|
|
1577 C2.w[0] = C2.w[0] + midpoint128[ind - 19].w[0];
|
|
|
1578 C2.w[1] = C2.w[1] + midpoint128[ind - 19].w[1];
|
|
|
1579 if (C2.w[0] < C2_lo)
|
|
|
1580 C2.w[1]++;
|
|
|
1581 }
|
|
|
1582 // the approximation of 10^(-x1) was rounded up to 118 bits
|
|
|
1583 __mul_128x128_to_256 (R256, C2, ten2mk128[ind]); // R256 = C2*, f2*
|
|
|
1584 // calculate C2* and f2*
|
|
|
1585 // C2* is actually floor(C2*) in this case
|
|
|
1586 // C2* and f2* need shifting and masking, as shown by
|
|
|
1587 // shiftright128[] and maskhigh128[]
|
|
|
1588 // the top Ex bits of 10^(-x1) are T* = ten2mk128trunc[ind], e.g.
|
|
|
1589 // if x1=1, T*=ten2mk128trunc[0]=0x19999999999999999999999999999999
|
|
|
1590 // if (0 < f2* < 10^(-x1)) then
|
|
|
1591 // if floor(C1+C2*) is even then C2* = floor(C2*) - logical right
|
|
|
1592 // shift; C2* has p decimal digits, correct by Prop. 1)
|
|
|
1593 // else if floor(C1+C2*) is odd C2* = floor(C2*)-1 (logical right
|
|
|
1594 // shift; C2* has p decimal digits, correct by Pr. 1)
|
|
|
1595 // else
|
|
|
1596 // C2* = floor(C2*) (logical right shift; C has p decimal digits,
|
|
|
1597 // correct by Property 1)
|
|
|
1598 // n = C2* * 10^(e2+x1)
|
|
|
1599
|
|
|
1600 if (ind <= 2) {
|
|
|
1601 highf2star.w[1] = 0x0;
|
|
|
1602 highf2star.w[0] = 0x0; // low f2* ok
|
|
|
1603 } else if (ind <= 21) {
|
|
|
1604 highf2star.w[1] = 0x0;
|
|
|
1605 highf2star.w[0] = R256.w[2] & maskhigh128[ind]; // low f2* ok
|
|
|
1606 } else {
|
|
|
1607 highf2star.w[1] = R256.w[3] & maskhigh128[ind];
|
|
|
1608 highf2star.w[0] = R256.w[2]; // low f2* is ok
|
|
|
1609 }
|
|
|
1610 // shift right C2* by Ex-128 = shiftright128[ind]
|
|
|
1611 if (ind >= 3) {
|
|
|
1612 shift = shiftright128[ind];
|
|
|
1613 if (shift < 64) { // 3 <= shift <= 63
|
|
|
1614 R256.w[2] =
|
|
|
1615 (R256.w[2] >> shift) | (R256.w[3] << (64 - shift));
|
|
|
1616 R256.w[3] = (R256.w[3] >> shift);
|
|
|
1617 } else { // 66 <= shift <= 102
|
|
|
1618 R256.w[2] = (R256.w[3] >> (shift - 64));
|
|
|
1619 R256.w[3] = 0x0ULL;
|
|
|
1620 }
|
|
|
1621 }
|
|
|
1622 // redundant
|
|
|
1623 is_inexact_lt_midpoint = 0;
|
|
|
1624 is_inexact_gt_midpoint = 0;
|
|
|
1625 is_midpoint_lt_even = 0;
|
|
|
1626 is_midpoint_gt_even = 0;
|
|
|
1627 // determine inexactness of the rounding of C2*
|
|
|
1628 // (cannot be followed by a second rounding)
|
|
|
1629 // if (0 < f2* - 1/2 < 10^(-x1)) then
|
|
|
1630 // the result is exact
|
|
|
1631 // else (if f2* - 1/2 > T* then)
|
|
|
1632 // the result of is inexact
|
|
|
1633 if (ind <= 2) {
|
|
|
1634 if (R256.w[1] > 0x8000000000000000ull ||
|
|
|
1635 (R256.w[1] == 0x8000000000000000ull
|
|
|
1636 && R256.w[0] > 0x0ull)) {
|
|
|
1637 // f2* > 1/2 and the result may be exact
|
|
|
1638 tmp64A = R256.w[1] - 0x8000000000000000ull; // f* - 1/2
|
|
|
1639 if ((tmp64A > ten2mk128trunc[ind].w[1]
|
|
|
1640 || (tmp64A == ten2mk128trunc[ind].w[1]
|
|
|
1641 && R256.w[0] >= ten2mk128trunc[ind].w[0]))) {
|
|
|
1642 // set the inexact flag
|
|
|
1643 *pfpsf |= INEXACT_EXCEPTION;
|
|
|
1644 // this rounding is applied to C2 only!
|
|
|
1645 // x_sign != y_sign
|
|
|
1646 is_inexact_gt_midpoint = 1;
|
|
|
1647 } // else the result is exact
|
|
|
1648 // rounding down, unless a midpoint in [ODD, EVEN]
|
|
|
1649 } else { // the result is inexact; f2* <= 1/2
|
|
|
1650 // set the inexact flag
|
|
|
1651 *pfpsf |= INEXACT_EXCEPTION;
|
|
|
1652 // this rounding is applied to C2 only!
|
|
|
1653 // x_sign != y_sign
|
|
|
1654 is_inexact_lt_midpoint = 1;
|
|
|
1655 }
|
|
|
1656 } else if (ind <= 21) { // if 3 <= ind <= 21
|
|
|
1657 if (highf2star.w[1] > 0x0 || (highf2star.w[1] == 0x0
|
|
|
1658 && highf2star.w[0] >
|
|
|
1659 onehalf128[ind])
|
|
|
1660 || (highf2star.w[1] == 0x0
|
|
|
1661 && highf2star.w[0] == onehalf128[ind]
|
|
|
1662 && (R256.w[1] || R256.w[0]))) {
|
|
|
1663 // f2* > 1/2 and the result may be exact
|
|
|
1664 // Calculate f2* - 1/2
|
|
|
1665 tmp64A = highf2star.w[0] - onehalf128[ind];
|
|
|
1666 tmp64B = highf2star.w[1];
|
|
|
1667 if (tmp64A > highf2star.w[0])
|
|
|
1668 tmp64B--;
|
|
|
1669 if (tmp64B || tmp64A
|
|
|
1670 || R256.w[1] > ten2mk128trunc[ind].w[1]
|
|
|
1671 || (R256.w[1] == ten2mk128trunc[ind].w[1]
|
|
|
1672 && R256.w[0] > ten2mk128trunc[ind].w[0])) {
|
|
|
1673 // set the inexact flag
|
|
|
1674 *pfpsf |= INEXACT_EXCEPTION;
|
|
|
1675 // this rounding is applied to C2 only!
|
|
|
1676 // x_sign != y_sign
|
|
|
1677 is_inexact_gt_midpoint = 1;
|
|
|
1678 } // else the result is exact
|
|
|
1679 } else { // the result is inexact; f2* <= 1/2
|
|
|
1680 // set the inexact flag
|
|
|
1681 *pfpsf |= INEXACT_EXCEPTION;
|
|
|
1682 // this rounding is applied to C2 only!
|
|
|
1683 // x_sign != y_sign
|
|
|
1684 is_inexact_lt_midpoint = 1;
|
|
|
1685 }
|
|
|
1686 } else { // if 22 <= ind <= 33
|
|
|
1687 if (highf2star.w[1] > onehalf128[ind]
|
|
|
1688 || (highf2star.w[1] == onehalf128[ind]
|
|
|
1689 && (highf2star.w[0] || R256.w[1]
|
|
|
1690 || R256.w[0]))) {
|
|
|
1691 // f2* > 1/2 and the result may be exact
|
|
|
1692 // Calculate f2* - 1/2
|
|
|
1693 // tmp64A = highf2star.w[0];
|
|
|
1694 tmp64B = highf2star.w[1] - onehalf128[ind];
|
|
|
1695 if (tmp64B || highf2star.w[0]
|
|
|
1696 || R256.w[1] > ten2mk128trunc[ind].w[1]
|
|
|
1697 || (R256.w[1] == ten2mk128trunc[ind].w[1]
|
|
|
1698 && R256.w[0] > ten2mk128trunc[ind].w[0])) {
|
|
|
1699 // set the inexact flag
|
|
|
1700 *pfpsf |= INEXACT_EXCEPTION;
|
|
|
1701 // this rounding is applied to C2 only!
|
|
|
1702 // x_sign != y_sign
|
|
|
1703 is_inexact_gt_midpoint = 1;
|
|
|
1704 } // else the result is exact
|
|
|
1705 } else { // the result is inexact; f2* <= 1/2
|
|
|
1706 // set the inexact flag
|
|
|
1707 *pfpsf |= INEXACT_EXCEPTION;
|
|
|
1708 // this rounding is applied to C2 only!
|
|
|
1709 // x_sign != y_sign
|
|
|
1710 is_inexact_lt_midpoint = 1;
|
|
|
1711 }
|
|
|
1712 }
|
|
|
1713 // check for midpoints after determining inexactness
|
|
|
1714 if ((R256.w[1] || R256.w[0]) && (highf2star.w[1] == 0)
|
|
|
1715 && (highf2star.w[0] == 0)
|
|
|
1716 && (R256.w[1] < ten2mk128trunc[ind].w[1]
|
|
|
1717 || (R256.w[1] == ten2mk128trunc[ind].w[1]
|
|
|
1718 && R256.w[0] <= ten2mk128trunc[ind].w[0]))) {
|
|
|
1719 // the result is a midpoint
|
|
|
1720 if ((tmp64 + R256.w[2]) & 0x01) { // MP in [EVEN, ODD]
|
|
|
1721 // if floor(C2*) is odd C = floor(C2*) - 1; the result may be 0
|
|
|
1722 R256.w[2]--;
|
|
|
1723 if (R256.w[2] == 0xffffffffffffffffull)
|
|
|
1724 R256.w[3]--;
|
|
|
1725 // this rounding is applied to C2 only!
|
|
|
1726 // x_sign != y_sign
|
|
|
1727 is_midpoint_lt_even = 1;
|
|
|
1728 is_inexact_lt_midpoint = 0;
|
|
|
1729 is_inexact_gt_midpoint = 0;
|
|
|
1730 } else {
|
|
|
1731 // else MP in [ODD, EVEN]
|
|
|
1732 // this rounding is applied to C2 only!
|
|
|
1733 // x_sign != y_sign
|
|
|
1734 is_midpoint_gt_even = 1;
|
|
|
1735 is_inexact_lt_midpoint = 0;
|
|
|
1736 is_inexact_gt_midpoint = 0;
|
|
|
1737 }
|
|
|
1738 }
|
|
|
1739 } else { // if (ind == -1) only when x1 = 0
|
|
|
1740 R256.w[2] = C2_lo;
|
|
|
1741 R256.w[3] = C2_hi;
|
|
|
1742 is_midpoint_lt_even = 0;
|
|
|
1743 is_midpoint_gt_even = 0;
|
|
|
1744 is_inexact_lt_midpoint = 0;
|
|
|
1745 is_inexact_gt_midpoint = 0;
|
|
|
1746 }
|
|
|
1747 // and now subtract C1 * 10^(e1-e2-x1) - (C2 * 10^(-x1))rnd,P34
|
|
|
1748 // because x_sign != y_sign this last operation is exact
|
|
|
1749 C1.w[0] = C1.w[0] - R256.w[2];
|
|
|
1750 C1.w[1] = C1.w[1] - R256.w[3];
|
|
|
1751 if (C1.w[0] > tmp64)
|
|
|
1752 C1.w[1]--; // borrow
|
|
|
1753 if (C1.w[1] >= 0x8000000000000000ull) { // negative coefficient!
|
|
|
1754 C1.w[0] = ~C1.w[0];
|
|
|
1755 C1.w[0]++;
|
|
|
1756 C1.w[1] = ~C1.w[1];
|
|
|
1757 if (C1.w[0] == 0x0)
|
|
|
1758 C1.w[1]++;
|
|
|
1759 tmp_sign = y_sign; // the result will have the sign of y
|
|
|
1760 } else {
|
|
|
1761 tmp_sign = x_sign;
|
|
|
1762 }
|
|
|
1763 // the difference has exactly P34 digits
|
|
|
1764 x_sign = tmp_sign;
|
|
|
1765 if (x1 >= 1)
|
|
|
1766 y_exp = y_exp + ((UINT64) x1 << 49);
|
|
|
1767 C1_hi = C1.w[1];
|
|
|
1768 C1_lo = C1.w[0];
|
|
|
1769 // general correction from RN to RA, RM, RP, RZ; result uses y_exp
|
|
|
1770 if (rnd_mode != ROUNDING_TO_NEAREST) {
|
|
|
1771 if ((!x_sign
|
|
|
1772 && ((rnd_mode == ROUNDING_UP && is_inexact_lt_midpoint)
|
|
|
1773 ||
|
|
|
1774 ((rnd_mode == ROUNDING_TIES_AWAY
|
|
|
1775 || rnd_mode == ROUNDING_UP)
|
|
|
1776 && is_midpoint_gt_even))) || (x_sign
|
|
|
1777 &&
|
|
|
1778 ((rnd_mode ==
|
|
|
1779 ROUNDING_DOWN
|
|
|
1780 &&
|
|
|
1781 is_inexact_lt_midpoint)
|
|
|
1782 ||
|
|
|
1783 ((rnd_mode ==
|
|
|
1784 ROUNDING_TIES_AWAY
|
|
|
1785 || rnd_mode ==
|
|
|
1786 ROUNDING_DOWN)
|
|
|
1787 &&
|
|
|
1788 is_midpoint_gt_even))))
|
|
|
1789 {
|
|
|
1790 // C1 = C1 + 1
|
|
|
1791 C1_lo = C1_lo + 1;
|
|
|
1792 if (C1_lo == 0) { // rounding overflow in the low 64 bits
|
|
|
1793 C1_hi = C1_hi + 1;
|
|
|
1794 }
|
|
|
1795 if (C1_hi == 0x0001ed09bead87c0ull
|
|
|
1796 && C1_lo == 0x378d8e6400000000ull) {
|
|
|
1797 // C1 = 10^34 => rounding overflow
|
|
|
1798 C1_hi = 0x0000314dc6448d93ull;
|
|
|
1799 C1_lo = 0x38c15b0a00000000ull; // 10^33
|
|
|
1800 y_exp = y_exp + EXP_P1;
|
|
|
1801 }
|
|
|
1802 } else if ((is_midpoint_lt_even || is_inexact_gt_midpoint)
|
|
|
1803 &&
|
|
|
1804 ((x_sign
|
|
|
1805 && (rnd_mode == ROUNDING_UP
|
|
|
1806 || rnd_mode == ROUNDING_TO_ZERO))
|
|
|
1807 || (!x_sign
|
|
|
1808 && (rnd_mode == ROUNDING_DOWN
|
|
|
1809 || rnd_mode == ROUNDING_TO_ZERO)))) {
|
|
|
1810 // C1 = C1 - 1
|
|
|
1811 C1_lo = C1_lo - 1;
|
|
|
1812 if (C1_lo == 0xffffffffffffffffull)
|
|
|
1813 C1_hi--;
|
|
|
1814 // check if we crossed into the lower decade
|
|
|
1815 if (C1_hi == 0x0000314dc6448d93ull && C1_lo == 0x38c15b09ffffffffull) { // 10^33 - 1
|
|
|
1816 C1_hi = 0x0001ed09bead87c0ull; // 10^34 - 1
|
|
|
1817 C1_lo = 0x378d8e63ffffffffull;
|
|
|
1818 y_exp = y_exp - EXP_P1;
|
|
|
1819 // no underflow, because delta + q2 >= P34 + 1
|
|
|
1820 }
|
|
|
1821 } else {
|
|
|
1822 ; // exact, the result is already correct
|
|
|
1823 }
|
|
|
1824 }
|
|
|
1825 // assemble the result
|
|
|
1826 res.w[1] = x_sign | y_exp | C1_hi;
|
|
|
1827 res.w[0] = C1_lo;
|
|
|
1828 }
|
|
|
1829 } // end delta = P34
|
|
|
1830 } else { // if (|delta| <= P34 - 1)
|
|
|
1831 if (delta >= 0) { // if (0 <= delta <= P34 - 1)
|
|
|
1832 if (delta <= P34 - 1 - q2) {
|
|
|
1833 // calculate C' directly; the result is exact
|
|
|
1834 // in this case 1<=q1<=P34-1, 1<=q2<=P34-1 and 0 <= e1-e2 <= P34-2
|
|
|
1835 // The coefficient of the result is C1 * 10^(e1-e2) + C2 and the
|
|
|
1836 // exponent is e2; either C1 or 10^(e1-e2) may not fit is 64 bits,
|
|
|
1837 // but their product fits with certainty in 128 bits (actually in 113)
|
|
|
1838 scale = delta - q1 + q2; // scale = (int)(e1 >> 49) - (int)(e2 >> 49)
|
|
|
1839
|
|
|
1840 if (scale >= 20) { // 10^(e1-e2) does not fit in 64 bits, but C1 does
|
|
|
1841 __mul_128x64_to_128 (C1, C1_lo, ten2k128[scale - 20]);
|
|
|
1842 C1_hi = C1.w[1];
|
|
|
1843 C1_lo = C1.w[0];
|
|
|
1844 } else if (scale >= 1) {
|
|
|
1845 // if 1 <= scale <= 19 then 10^(e1-e2) fits in 64 bits
|
|
|
1846 if (q1 <= 19) { // C1 fits in 64 bits
|
|
|
1847 __mul_64x64_to_128MACH (C1, C1_lo, ten2k64[scale]);
|
|
|
1848 } else { // q1 >= 20
|
|
|
1849 C1.w[1] = C1_hi;
|
|
|
1850 C1.w[0] = C1_lo;
|
|
|
1851 __mul_128x64_to_128 (C1, ten2k64[scale], C1);
|
|
|
1852 }
|
|
|
1853 C1_hi = C1.w[1];
|
|
|
1854 C1_lo = C1.w[0];
|
|
|
1855 } else { // if (scale == 0) C1 is unchanged
|
|
|
1856 C1.w[0] = C1_lo; // C1.w[1] = C1_hi;
|
|
|
1857 }
|
|
|
1858 // now add C2
|
|
|
1859 if (x_sign == y_sign) {
|
|
|
1860 // the result cannot overflow
|
|
|
1861 C1_lo = C1_lo + C2_lo;
|
|
|
1862 C1_hi = C1_hi + C2_hi;
|
|
|
1863 if (C1_lo < C1.w[0])
|
|
|
1864 C1_hi++;
|
|
|
1865 } else { // if x_sign != y_sign
|
|
|
1866 C1_lo = C1_lo - C2_lo;
|
|
|
1867 C1_hi = C1_hi - C2_hi;
|
|
|
1868 if (C1_lo > C1.w[0])
|
|
|
1869 C1_hi--;
|
|
|
1870 // the result can be zero, but it cannot overflow
|
|
|
1871 if (C1_lo == 0 && C1_hi == 0) {
|
|
|
1872 // assemble the result
|
|
|
1873 if (x_exp < y_exp)
|
|
|
1874 res.w[1] = x_exp;
|
|
|
1875 else
|
|
|
1876 res.w[1] = y_exp;
|
|
|
1877 res.w[0] = 0;
|
|
|
1878 if (rnd_mode == ROUNDING_DOWN) {
|
|
|
1879 res.w[1] |= 0x8000000000000000ull;
|
|
|
1880 }
|
|
|
1881 BID_SWAP128 (res);
|
|
|
1882 BID_RETURN (res);
|
|
|
1883 }
|
|
|
1884 if (C1_hi >= 0x8000000000000000ull) { // negative coefficient!
|
|
|
1885 C1_lo = ~C1_lo;
|
|
|
1886 C1_lo++;
|
|
|
1887 C1_hi = ~C1_hi;
|
|
|
1888 if (C1_lo == 0x0)
|
|
|
1889 C1_hi++;
|
|
|
1890 x_sign = y_sign; // the result will have the sign of y
|
|
|
1891 }
|
|
|
1892 }
|
|
|
1893 // assemble the result
|
|
|
1894 res.w[1] = x_sign | y_exp | C1_hi;
|
|
|
1895 res.w[0] = C1_lo;
|
|
|
1896 } else if (delta == P34 - q2) {
|
|
|
1897 // calculate C' directly; the result may be inexact if it requires
|
|
|
1898 // P34+1 decimal digits; in this case the 'cutoff' point for addition
|
|
|
1899 // is at the position of the lsb of C2, so 0 <= e1-e2 <= P34-1
|
|
|
1900 // The coefficient of the result is C1 * 10^(e1-e2) + C2 and the
|
|
|
1901 // exponent is e2; either C1 or 10^(e1-e2) may not fit is 64 bits,
|
|
|
1902 // but their product fits with certainty in 128 bits (actually in 113)
|
|
|
1903 scale = delta - q1 + q2; // scale = (int)(e1 >> 49) - (int)(e2 >> 49)
|
|
|
1904 if (scale >= 20) { // 10^(e1-e2) does not fit in 64 bits, but C1 does
|
|
|
1905 __mul_128x64_to_128 (C1, C1_lo, ten2k128[scale - 20]);
|
|
|
1906 } else if (scale >= 1) {
|
|
|
1907 // if 1 <= scale <= 19 then 10^(e1-e2) fits in 64 bits
|
|
|
1908 if (q1 <= 19) { // C1 fits in 64 bits
|
|
|
1909 __mul_64x64_to_128MACH (C1, C1_lo, ten2k64[scale]);
|
|
|
1910 } else { // q1 >= 20
|
|
|
1911 C1.w[1] = C1_hi;
|
|
|
1912 C1.w[0] = C1_lo;
|
|
|
1913 __mul_128x64_to_128 (C1, ten2k64[scale], C1);
|
|
|
1914 }
|
|
|
1915 } else { // if (scale == 0) C1 is unchanged
|
|
|
1916 C1.w[1] = C1_hi;
|
|
|
1917 C1.w[0] = C1_lo; // only the low part is necessary
|
|
|
1918 }
|
|
|
1919 C1_hi = C1.w[1];
|
|
|
1920 C1_lo = C1.w[0];
|
|
|
1921 // now add C2
|
|
|
1922 if (x_sign == y_sign) {
|
|
|
1923 // the result can overflow!
|
|
|
1924 C1_lo = C1_lo + C2_lo;
|
|
|
1925 C1_hi = C1_hi + C2_hi;
|
|
|
1926 if (C1_lo < C1.w[0])
|
|
|
1927 C1_hi++;
|
|
|
1928 // test for overflow, possible only when C1 >= 10^34
|
|
|
1929 if (C1_hi > 0x0001ed09bead87c0ull || (C1_hi == 0x0001ed09bead87c0ull && C1_lo >= 0x378d8e6400000000ull)) { // C1 >= 10^34
|
|
|
1930 // in this case q = P34 + 1 and x = q - P34 = 1, so multiply
|
|
|
1931 // C'' = C'+ 5 = C1 + 5 by k1 ~ 10^(-1) calculated for P34 + 1
|
|
|
1932 // decimal digits
|
|
|
1933 // Calculate C'' = C' + 1/2 * 10^x
|
|
|
1934 if (C1_lo >= 0xfffffffffffffffbull) { // low half add has carry
|
|
|
1935 C1_lo = C1_lo + 5;
|
|
|
1936 C1_hi = C1_hi + 1;
|
|
|
1937 } else {
|
|
|
1938 C1_lo = C1_lo + 5;
|
|
|
1939 }
|
|
|
1940 // the approximation of 10^(-1) was rounded up to 118 bits
|
|
|
1941 // 10^(-1) =~ 33333333333333333333333333333400 * 2^-129
|
|
|
1942 // 10^(-1) =~ 19999999999999999999999999999a00 * 2^-128
|
|
|
1943 C1.w[1] = C1_hi;
|
|
|
1944 C1.w[0] = C1_lo; // C''
|
|
|
1945 ten2m1.w[1] = 0x1999999999999999ull;
|
|
|
1946 ten2m1.w[0] = 0x9999999999999a00ull;
|
|
|
1947 __mul_128x128_to_256 (P256, C1, ten2m1); // P256 = C*, f*
|
|
|
1948 // C* is actually floor(C*) in this case
|
|
|
1949 // the top Ex = 128 bits of 10^(-1) are
|
|
|
1950 // T* = 0x00199999999999999999999999999999
|
|
|
1951 // if (0 < f* < 10^(-x)) then
|
|
|
1952 // if floor(C*) is even then C = floor(C*) - logical right
|
|
|
1953 // shift; C has p decimal digits, correct by Prop. 1)
|
|
|
1954 // else if floor(C*) is odd C = floor(C*) - 1 (logical right
|
|
|
1955 // shift; C has p decimal digits, correct by Pr. 1)
|
|
|
1956 // else
|
|
|
1957 // C = floor(C*) (logical right shift; C has p decimal digits,
|
|
|
1958 // correct by Property 1)
|
|
|
1959 // n = C * 10^(e2+x)
|
|
|
1960 if ((P256.w[1] || P256.w[0])
|
|
|
1961 && (P256.w[1] < 0x1999999999999999ull
|
|
|
1962 || (P256.w[1] == 0x1999999999999999ull
|
|
|
1963 && P256.w[0] <= 0x9999999999999999ull))) {
|
|
|
1964 // the result is a midpoint
|
|
|
1965 if (P256.w[2] & 0x01) {
|
|
|
1966 is_midpoint_gt_even = 1;
|
|
|
1967 // if floor(C*) is odd C = floor(C*) - 1; the result is not 0
|
|
|
1968 P256.w[2]--;
|
|
|
1969 if (P256.w[2] == 0xffffffffffffffffull)
|
|
|
1970 P256.w[3]--;
|
|
|
1971 } else {
|
|
|
1972 is_midpoint_lt_even = 1;
|
|
|
1973 }
|
|
|
1974 }
|
|
|
1975 // n = Cstar * 10^(e2+1)
|
|
|
1976 y_exp = y_exp + EXP_P1;
|
|
|
1977 // C* != 10^P because C* has P34 digits
|
|
|
1978 // check for overflow
|
|
|
1979 if (y_exp == EXP_MAX_P1
|
|
|
1980 && (rnd_mode == ROUNDING_TO_NEAREST
|
|
|
1981 || rnd_mode == ROUNDING_TIES_AWAY)) {
|
|
|
1982 // overflow for RN
|
|
|
1983 res.w[1] = x_sign | 0x7800000000000000ull; // +/-inf
|
|
|
1984 res.w[0] = 0x0ull;
|
|
|
1985 // set the inexact flag
|
|
|
1986 *pfpsf |= INEXACT_EXCEPTION;
|
|
|
1987 // set the overflow flag
|
|
|
1988 *pfpsf |= OVERFLOW_EXCEPTION;
|
|
|
1989 BID_SWAP128 (res);
|
|
|
1990 BID_RETURN (res);
|
|
|
1991 }
|
|
|
1992 // if (0 < f* - 1/2 < 10^(-x)) then
|
|
|
1993 // the result of the addition is exact
|
|
|
1994 // else
|
|
|
1995 // the result of the addition is inexact
|
|
|
1996 if (P256.w[1] > 0x8000000000000000ull || (P256.w[1] == 0x8000000000000000ull && P256.w[0] > 0x0ull)) { // the result may be exact
|
|
|
1997 tmp64 = P256.w[1] - 0x8000000000000000ull; // f* - 1/2
|
|
|
1998 if ((tmp64 > 0x1999999999999999ull
|
|
|
1999 || (tmp64 == 0x1999999999999999ull
|
|
|
2000 && P256.w[0] >= 0x9999999999999999ull))) {
|
|
|
2001 // set the inexact flag
|
|
|
2002 *pfpsf |= INEXACT_EXCEPTION;
|
|
|
2003 is_inexact = 1;
|
|
|
2004 } // else the result is exact
|
|
|
2005 } else { // the result is inexact
|
|
|
2006 // set the inexact flag
|
|
|
2007 *pfpsf |= INEXACT_EXCEPTION;
|
|
|
2008 is_inexact = 1;
|
|
|
2009 }
|
|
|
2010 C1_hi = P256.w[3];
|
|
|
2011 C1_lo = P256.w[2];
|
|
|
2012 if (!is_midpoint_gt_even && !is_midpoint_lt_even) {
|
|
|
2013 is_inexact_lt_midpoint = is_inexact
|
|
|
2014 && (P256.w[1] & 0x8000000000000000ull);
|
|
|
2015 is_inexact_gt_midpoint = is_inexact
|
|
|
2016 && !(P256.w[1] & 0x8000000000000000ull);
|
|
|
2017 }
|
|
|
2018 // general correction from RN to RA, RM, RP, RZ;
|
|
|
2019 // result uses y_exp
|
|
|
2020 if (rnd_mode != ROUNDING_TO_NEAREST) {
|
|
|
2021 if ((!x_sign
|
|
|
2022 &&
|
|
|
2023 ((rnd_mode == ROUNDING_UP
|
|
|
2024 && is_inexact_lt_midpoint)
|
|
|
2025 ||
|
|
|
2026 ((rnd_mode == ROUNDING_TIES_AWAY
|
|
|
2027 || rnd_mode == ROUNDING_UP)
|
|
|
2028 && is_midpoint_gt_even))) || (x_sign
|
|
|
2029 &&
|
|
|
2030 ((rnd_mode ==
|
|
|
2031 ROUNDING_DOWN
|
|
|
2032 &&
|
|
|
2033 is_inexact_lt_midpoint)
|
|
|
2034 ||
|
|
|
2035 ((rnd_mode ==
|
|
|
2036 ROUNDING_TIES_AWAY
|
|
|
2037 || rnd_mode ==
|
|
|
2038 ROUNDING_DOWN)
|
|
|
2039 &&
|
|
|
2040 is_midpoint_gt_even))))
|
|
|
2041 {
|
|
|
2042 // C1 = C1 + 1
|
|
|
2043 C1_lo = C1_lo + 1;
|
|
|
2044 if (C1_lo == 0) { // rounding overflow in the low 64 bits
|
|
|
2045 C1_hi = C1_hi + 1;
|
|
|
2046 }
|
|
|
2047 if (C1_hi == 0x0001ed09bead87c0ull
|
|
|
2048 && C1_lo == 0x378d8e6400000000ull) {
|
|
|
2049 // C1 = 10^34 => rounding overflow
|
|
|
2050 C1_hi = 0x0000314dc6448d93ull;
|
|
|
2051 C1_lo = 0x38c15b0a00000000ull; // 10^33
|
|
|
2052 y_exp = y_exp + EXP_P1;
|
|
|
2053 }
|
|
|
2054 } else
|
|
|
2055 if ((is_midpoint_lt_even || is_inexact_gt_midpoint)
|
|
|
2056 &&
|
|
|
2057 ((x_sign
|
|
|
2058 && (rnd_mode == ROUNDING_UP
|
|
|
2059 || rnd_mode == ROUNDING_TO_ZERO))
|
|
|
2060 || (!x_sign
|
|
|
2061 && (rnd_mode == ROUNDING_DOWN
|
|
|
2062 || rnd_mode == ROUNDING_TO_ZERO)))) {
|
|
|
2063 // C1 = C1 - 1
|
|
|
2064 C1_lo = C1_lo - 1;
|
|
|
2065 if (C1_lo == 0xffffffffffffffffull)
|
|
|
2066 C1_hi--;
|
|
|
2067 // check if we crossed into the lower decade
|
|
|
2068 if (C1_hi == 0x0000314dc6448d93ull && C1_lo == 0x38c15b09ffffffffull) { // 10^33 - 1
|
|
|
2069 C1_hi = 0x0001ed09bead87c0ull; // 10^34 - 1
|
|
|
2070 C1_lo = 0x378d8e63ffffffffull;
|
|
|
2071 y_exp = y_exp - EXP_P1;
|
|
|
2072 // no underflow, because delta + q2 >= P34 + 1
|
|
|
2073 }
|
|
|
2074 } else {
|
|
|
2075 ; // exact, the result is already correct
|
|
|
2076 }
|
|
|
2077 // in all cases check for overflow (RN and RA solved already)
|
|
|
2078 if (y_exp == EXP_MAX_P1) { // overflow
|
|
|
2079 if ((rnd_mode == ROUNDING_DOWN && x_sign) || // RM and res < 0
|
|
|
2080 (rnd_mode == ROUNDING_UP && !x_sign)) { // RP and res > 0
|
|
|
2081 C1_hi = 0x7800000000000000ull; // +inf
|
|
|
2082 C1_lo = 0x0ull;
|
|
|
2083 } else { // RM and res > 0, RP and res < 0, or RZ
|
|
|
2084 C1_hi = 0x5fffed09bead87c0ull;
|
|
|
2085 C1_lo = 0x378d8e63ffffffffull;
|
|
|
2086 }
|
|
|
2087 y_exp = 0; // x_sign is preserved
|
|
|
2088 // set the inexact flag (in case the exact addition was exact)
|
|
|
2089 *pfpsf |= INEXACT_EXCEPTION;
|
|
|
2090 // set the overflow flag
|
|
|
2091 *pfpsf |= OVERFLOW_EXCEPTION;
|
|
|
2092 }
|
|
|
2093 }
|
|
|
2094 } // else if (C1 < 10^34) then C1 is the coeff.; the result is exact
|
|
|
2095 } else { // if x_sign != y_sign the result is exact
|
|
|
2096 C1_lo = C1_lo - C2_lo;
|
|
|
2097 C1_hi = C1_hi - C2_hi;
|
|
|
2098 if (C1_lo > C1.w[0])
|
|
|
2099 C1_hi--;
|
|
|
2100 // the result can be zero, but it cannot overflow
|
|
|
2101 if (C1_lo == 0 && C1_hi == 0) {
|
|
|
2102 // assemble the result
|
|
|
2103 if (x_exp < y_exp)
|
|
|
2104 res.w[1] = x_exp;
|
|
|
2105 else
|
|
|
2106 res.w[1] = y_exp;
|
|
|
2107 res.w[0] = 0;
|
|
|
2108 if (rnd_mode == ROUNDING_DOWN) {
|
|
|
2109 res.w[1] |= 0x8000000000000000ull;
|
|
|
2110 }
|
|
|
2111 BID_SWAP128 (res);
|
|
|
2112 BID_RETURN (res);
|
|
|
2113 }
|
|
|
2114 if (C1_hi >= 0x8000000000000000ull) { // negative coefficient!
|
|
|
2115 C1_lo = ~C1_lo;
|
|
|
2116 C1_lo++;
|
|
|
2117 C1_hi = ~C1_hi;
|
|
|
2118 if (C1_lo == 0x0)
|
|
|
2119 C1_hi++;
|
|
|
2120 x_sign = y_sign; // the result will have the sign of y
|
|
|
2121 }
|
|
|
2122 }
|
|
|
2123 // assemble the result
|
|
|
2124 res.w[1] = x_sign | y_exp | C1_hi;
|
|
|
2125 res.w[0] = C1_lo;
|
|
|
2126 } else { // if (delta >= P34 + 1 - q2)
|
|
|
2127 // instead of C' = (C1 * 10^(e1-e2) + C2)rnd,P34
|
|
|
2128 // calculate C' = C1 * 10^(e1-e2-x1) + (C2 * 10^(-x1))rnd,P34
|
|
|
2129 // where x1 = q1 + e1 - e2 - P34, 1 <= x1 <= P34 - 1
|
|
|
2130 // In most cases C' will have P34 digits, and n = C' * 10^(e2+x1)
|
|
|
2131 // If the result has P34+1 digits, redo the steps above with x1+1
|
|
|
2132 // If the result has P34-1 digits or less, redo the steps above with
|
|
|
2133 // x1-1 but only if initially x1 >= 1
|
|
|
2134 // NOTE: these two steps can be improved, e.g we could guess if
|
|
|
2135 // P34+1 or P34-1 digits will be obtained by adding/subtracting just
|
|
|
2136 // the top 64 bits of the two operands
|
|
|
2137 // The result cannot be zero, but it can overflow
|
|
|
2138 x1 = delta + q2 - P34; // 1 <= x1 <= P34-1
|
|
|
2139 roundC2:
|
|
|
2140 // Calculate C1 * 10^(e1-e2-x1) where 0 <= e1-e2-x1 <= P34 - 1
|
|
|
2141 // scale = (int)(e1 >> 49) - (int)(e2 >> 49) - x1; 0 <= scale <= P34-1
|
|
|
2142 scale = delta - q1 + q2 - x1; // scale = e1 - e2 - x1 = P34 - q1
|
|
|
2143 // either C1 or 10^(e1-e2-x1) may not fit is 64 bits,
|
|
|
2144 // but their product fits with certainty in 128 bits (actually in 113)
|
|
|
2145 if (scale >= 20) { //10^(e1-e2-x1) doesn't fit in 64 bits, but C1 does
|
|
|
2146 __mul_128x64_to_128 (C1, C1_lo, ten2k128[scale - 20]);
|
|
|
2147 } else if (scale >= 1) {
|
|
|
2148 // if 1 <= scale <= 19 then 10^(e1-e2-x1) fits in 64 bits
|
|
|
2149 if (q1 <= 19) { // C1 fits in 64 bits
|
|
|
2150 __mul_64x64_to_128MACH (C1, C1_lo, ten2k64[scale]);
|
|
|
2151 } else { // q1 >= 20
|
|
|
2152 C1.w[1] = C1_hi;
|
|
|
2153 C1.w[0] = C1_lo;
|
|
|
2154 __mul_128x64_to_128 (C1, ten2k64[scale], C1);
|
|
|
2155 }
|
|
|
2156 } else { // if (scale == 0) C1 is unchanged
|
|
|
2157 C1.w[1] = C1_hi;
|
|
|
2158 C1.w[0] = C1_lo;
|
|
|
2159 }
|
|
|
2160 tmp64 = C1.w[0]; // C1.w[1], C1.w[0] contains C1 * 10^(e1-e2-x1)
|
|
|
2161
|
|
|
2162 // now round C2 to q2-x1 decimal digits, where 1<=x1<=q2-1<=P34-1
|
|
|
2163 // (but if we got here a second time after x1 = x1 - 1, then
|
|
|
2164 // x1 >= 0; note that for x1 = 0 C2 is unchanged)
|
|
|
2165 // C2' = C2 + 1/2 * 10^x1 = C2 + 5 * 10^(x1-1)
|
|
|
2166 ind = x1 - 1; // 0 <= ind <= q2-2<=P34-2=32; but note that if x1 = 0
|
|
|
2167 // during a second pass, then ind = -1
|
|
|
2168 if (ind >= 0) { // if (x1 >= 1)
|
|
|
2169 C2.w[0] = C2_lo;
|
|
|
2170 C2.w[1] = C2_hi;
|
|
|
2171 if (ind <= 18) {
|
|
|
2172 C2.w[0] = C2.w[0] + midpoint64[ind];
|
|
|
2173 if (C2.w[0] < C2_lo)
|
|
|
2174 C2.w[1]++;
|
|
|
2175 } else { // 19 <= ind <= 32
|
|
|
2176 C2.w[0] = C2.w[0] + midpoint128[ind - 19].w[0];
|
|
|
2177 C2.w[1] = C2.w[1] + midpoint128[ind - 19].w[1];
|
|
|
2178 if (C2.w[0] < C2_lo)
|
|
|
2179 C2.w[1]++;
|
|
|
2180 }
|
|
|
2181 // the approximation of 10^(-x1) was rounded up to 118 bits
|
|
|
2182 __mul_128x128_to_256 (R256, C2, ten2mk128[ind]); // R256 = C2*, f2*
|
|
|
2183 // calculate C2* and f2*
|
|
|
2184 // C2* is actually floor(C2*) in this case
|
|
|
2185 // C2* and f2* need shifting and masking, as shown by
|
|
|
2186 // shiftright128[] and maskhigh128[]
|
|
|
2187 // the top Ex bits of 10^(-x1) are T* = ten2mk128trunc[ind], e.g.
|
|
|
2188 // if x1=1, T*=ten2mk128trunc[0]=0x19999999999999999999999999999999
|
|
|
2189 // if (0 < f2* < 10^(-x1)) then
|
|
|
2190 // if floor(C1+C2*) is even then C2* = floor(C2*) - logical right
|
|
|
2191 // shift; C2* has p decimal digits, correct by Prop. 1)
|
|
|
2192 // else if floor(C1+C2*) is odd C2* = floor(C2*)-1 (logical right
|
|
|
2193 // shift; C2* has p decimal digits, correct by Pr. 1)
|
|
|
2194 // else
|
|
|
2195 // C2* = floor(C2*) (logical right shift; C has p decimal digits,
|
|
|
2196 // correct by Property 1)
|
|
|
2197 // n = C2* * 10^(e2+x1)
|
|
|
2198
|
|
|
2199 if (ind <= 2) {
|
|
|
2200 highf2star.w[1] = 0x0;
|
|
|
2201 highf2star.w[0] = 0x0; // low f2* ok
|
|
|
2202 } else if (ind <= 21) {
|
|
|
2203 highf2star.w[1] = 0x0;
|
|
|
2204 highf2star.w[0] = R256.w[2] & maskhigh128[ind]; // low f2* ok
|
|
|
2205 } else {
|
|
|
2206 highf2star.w[1] = R256.w[3] & maskhigh128[ind];
|
|
|
2207 highf2star.w[0] = R256.w[2]; // low f2* is ok
|
|
|
2208 }
|
|
|
2209 // shift right C2* by Ex-128 = shiftright128[ind]
|
|
|
2210 if (ind >= 3) {
|
|
|
2211 shift = shiftright128[ind];
|
|
|
2212 if (shift < 64) { // 3 <= shift <= 63
|
|
|
2213 R256.w[2] =
|
|
|
2214 (R256.w[2] >> shift) | (R256.w[3] << (64 - shift));
|
|
|
2215 R256.w[3] = (R256.w[3] >> shift);
|
|
|
2216 } else { // 66 <= shift <= 102
|
|
|
2217 R256.w[2] = (R256.w[3] >> (shift - 64));
|
|
|
2218 R256.w[3] = 0x0ULL;
|
|
|
2219 }
|
|
|
2220 }
|
|
|
2221 if (second_pass) {
|
|
|
2222 is_inexact_lt_midpoint = 0;
|
|
|
2223 is_inexact_gt_midpoint = 0;
|
|
|
2224 is_midpoint_lt_even = 0;
|
|
|
2225 is_midpoint_gt_even = 0;
|
|
|
2226 }
|
|
|
2227 // determine inexactness of the rounding of C2* (this may be
|
|
|
2228 // followed by a second rounding only if we get P34+1
|
|
|
2229 // decimal digits)
|
|
|
2230 // if (0 < f2* - 1/2 < 10^(-x1)) then
|
|
|
2231 // the result is exact
|
|
|
2232 // else (if f2* - 1/2 > T* then)
|
|
|
2233 // the result of is inexact
|
|
|
2234 if (ind <= 2) {
|
|
|
2235 if (R256.w[1] > 0x8000000000000000ull ||
|
|
|
2236 (R256.w[1] == 0x8000000000000000ull
|
|
|
2237 && R256.w[0] > 0x0ull)) {
|
|
|
2238 // f2* > 1/2 and the result may be exact
|
|
|
2239 tmp64A = R256.w[1] - 0x8000000000000000ull; // f* - 1/2
|
|
|
2240 if ((tmp64A > ten2mk128trunc[ind].w[1]
|
|
|
2241 || (tmp64A == ten2mk128trunc[ind].w[1]
|
|
|
2242 && R256.w[0] >= ten2mk128trunc[ind].w[0]))) {
|
|
|
2243 // set the inexact flag
|
|
|
2244 // *pfpsf |= INEXACT_EXCEPTION;
|
|
|
2245 tmp_inexact = 1; // may be set again during a second pass
|
|
|
2246 // this rounding is applied to C2 only!
|
|
|
2247 if (x_sign == y_sign)
|
|
|
2248 is_inexact_lt_midpoint = 1;
|
|
|
2249 else // if (x_sign != y_sign)
|
|
|
2250 is_inexact_gt_midpoint = 1;
|
|
|
2251 } // else the result is exact
|
|
|
2252 // rounding down, unless a midpoint in [ODD, EVEN]
|
|
|
2253 } else { // the result is inexact; f2* <= 1/2
|
|
|
2254 // set the inexact flag
|
|
|
2255 // *pfpsf |= INEXACT_EXCEPTION;
|
|
|
2256 tmp_inexact = 1; // just in case we will round a second time
|
|
|
2257 // rounding up, unless a midpoint in [EVEN, ODD]
|
|
|
2258 // this rounding is applied to C2 only!
|
|
|
2259 if (x_sign == y_sign)
|
|
|
2260 is_inexact_gt_midpoint = 1;
|
|
|
2261 else // if (x_sign != y_sign)
|
|
|
2262 is_inexact_lt_midpoint = 1;
|
|
|
2263 }
|
|
|
2264 } else if (ind <= 21) { // if 3 <= ind <= 21
|
|
|
2265 if (highf2star.w[1] > 0x0 || (highf2star.w[1] == 0x0
|
|
|
2266 && highf2star.w[0] >
|
|
|
2267 onehalf128[ind])
|
|
|
2268 || (highf2star.w[1] == 0x0
|
|
|
2269 && highf2star.w[0] == onehalf128[ind]
|
|
|
2270 && (R256.w[1] || R256.w[0]))) {
|
|
|
2271 // f2* > 1/2 and the result may be exact
|
|
|
2272 // Calculate f2* - 1/2
|
|
|
2273 tmp64A = highf2star.w[0] - onehalf128[ind];
|
|
|
2274 tmp64B = highf2star.w[1];
|
|
|
2275 if (tmp64A > highf2star.w[0])
|
|
|
2276 tmp64B--;
|
|
|
2277 if (tmp64B || tmp64A
|
|
|
2278 || R256.w[1] > ten2mk128trunc[ind].w[1]
|
|
|
2279 || (R256.w[1] == ten2mk128trunc[ind].w[1]
|
|
|
2280 && R256.w[0] > ten2mk128trunc[ind].w[0])) {
|
|
|
2281 // set the inexact flag
|
|
|
2282 // *pfpsf |= INEXACT_EXCEPTION;
|
|
|
2283 tmp_inexact = 1; // may be set again during a second pass
|
|
|
2284 // this rounding is applied to C2 only!
|
|
|
2285 if (x_sign == y_sign)
|
|
|
2286 is_inexact_lt_midpoint = 1;
|
|
|
2287 else // if (x_sign != y_sign)
|
|
|
2288 is_inexact_gt_midpoint = 1;
|
|
|
2289 } // else the result is exact
|
|
|
2290 } else { // the result is inexact; f2* <= 1/2
|
|
|
2291 // set the inexact flag
|
|
|
2292 // *pfpsf |= INEXACT_EXCEPTION;
|
|
|
2293 tmp_inexact = 1; // may be set again during a second pass
|
|
|
2294 // rounding up, unless a midpoint in [EVEN, ODD]
|
|
|
2295 // this rounding is applied to C2 only!
|
|
|
2296 if (x_sign == y_sign)
|
|
|
2297 is_inexact_gt_midpoint = 1;
|
|
|
2298 else // if (x_sign != y_sign)
|
|
|
2299 is_inexact_lt_midpoint = 1;
|
|
|
2300 }
|
|
|
2301 } else { // if 22 <= ind <= 33
|
|
|
2302 if (highf2star.w[1] > onehalf128[ind]
|
|
|
2303 || (highf2star.w[1] == onehalf128[ind]
|
|
|
2304 && (highf2star.w[0] || R256.w[1]
|
|
|
2305 || R256.w[0]))) {
|
|
|
2306 // f2* > 1/2 and the result may be exact
|
|
|
2307 // Calculate f2* - 1/2
|
|
|
2308 // tmp64A = highf2star.w[0];
|
|
|
2309 tmp64B = highf2star.w[1] - onehalf128[ind];
|
|
|
2310 if (tmp64B || highf2star.w[0]
|
|
|
2311 || R256.w[1] > ten2mk128trunc[ind].w[1]
|
|
|
2312 || (R256.w[1] == ten2mk128trunc[ind].w[1]
|
|
|
2313 && R256.w[0] > ten2mk128trunc[ind].w[0])) {
|
|
|
2314 // set the inexact flag
|
|
|
2315 // *pfpsf |= INEXACT_EXCEPTION;
|
|
|
2316 tmp_inexact = 1; // may be set again during a second pass
|
|
|
2317 // this rounding is applied to C2 only!
|
|
|
2318 if (x_sign == y_sign)
|
|
|
2319 is_inexact_lt_midpoint = 1;
|
|
|
2320 else // if (x_sign != y_sign)
|
|
|
2321 is_inexact_gt_midpoint = 1;
|
|
|
2322 } // else the result is exact
|
|
|
2323 } else { // the result is inexact; f2* <= 1/2
|
|
|
2324 // set the inexact flag
|
|
|
2325 // *pfpsf |= INEXACT_EXCEPTION;
|
|
|
2326 tmp_inexact = 1; // may be set again during a second pass
|
|
|
2327 // rounding up, unless a midpoint in [EVEN, ODD]
|
|
|
2328 // this rounding is applied to C2 only!
|
|
|
2329 if (x_sign == y_sign)
|
|
|
2330 is_inexact_gt_midpoint = 1;
|
|
|
2331 else // if (x_sign != y_sign)
|
|
|
2332 is_inexact_lt_midpoint = 1;
|
|
|
2333 }
|
|
|
2334 }
|
|
|
2335 // check for midpoints
|
|
|
2336 if ((R256.w[1] || R256.w[0]) && (highf2star.w[1] == 0)
|
|
|
2337 && (highf2star.w[0] == 0)
|
|
|
2338 && (R256.w[1] < ten2mk128trunc[ind].w[1]
|
|
|
2339 || (R256.w[1] == ten2mk128trunc[ind].w[1]
|
|
|
2340 && R256.w[0] <= ten2mk128trunc[ind].w[0]))) {
|
|
|
2341 // the result is a midpoint
|
|
|
2342 if ((tmp64 + R256.w[2]) & 0x01) { // MP in [EVEN, ODD]
|
|
|
2343 // if floor(C2*) is odd C = floor(C2*) - 1; the result may be 0
|
|
|
2344 R256.w[2]--;
|
|
|
2345 if (R256.w[2] == 0xffffffffffffffffull)
|
|
|
2346 R256.w[3]--;
|
|
|
2347 // this rounding is applied to C2 only!
|
|
|
2348 if (x_sign == y_sign)
|
|
|
2349 is_midpoint_gt_even = 1;
|
|
|
2350 else // if (x_sign != y_sign)
|
|
|
2351 is_midpoint_lt_even = 1;
|
|
|
2352 is_inexact_lt_midpoint = 0;
|
|
|
2353 is_inexact_gt_midpoint = 0;
|
|
|
2354 } else {
|
|
|
2355 // else MP in [ODD, EVEN]
|
|
|
2356 // this rounding is applied to C2 only!
|
|
|
2357 if (x_sign == y_sign)
|
|
|
2358 is_midpoint_lt_even = 1;
|
|
|
2359 else // if (x_sign != y_sign)
|
|
|
2360 is_midpoint_gt_even = 1;
|
|
|
2361 is_inexact_lt_midpoint = 0;
|
|
|
2362 is_inexact_gt_midpoint = 0;
|
|
|
2363 }
|
|
|
2364 }
|
|
|
2365 // end if (ind >= 0)
|
|
|
2366 } else { // if (ind == -1); only during a 2nd pass, and when x1 = 0
|
|
|
2367 R256.w[2] = C2_lo;
|
|
|
2368 R256.w[3] = C2_hi;
|
|
|
2369 tmp_inexact = 0;
|
|
|
2370 // to correct a possible setting to 1 from 1st pass
|
|
|
2371 if (second_pass) {
|
|
|
2372 is_midpoint_lt_even = 0;
|
|
|
2373 is_midpoint_gt_even = 0;
|
|
|
2374 is_inexact_lt_midpoint = 0;
|
|
|
2375 is_inexact_gt_midpoint = 0;
|
|
|
2376 }
|
|
|
2377 }
|
|
|
2378 // and now add/subtract C1 * 10^(e1-e2-x1) +/- (C2 * 10^(-x1))rnd,P34
|
|
|
2379 if (x_sign == y_sign) { // addition; could overflow
|
|
|
2380 // no second pass is possible this way (only for x_sign != y_sign)
|
|
|
2381 C1.w[0] = C1.w[0] + R256.w[2];
|
|
|
2382 C1.w[1] = C1.w[1] + R256.w[3];
|
|
|
2383 if (C1.w[0] < tmp64)
|
|
|
2384 C1.w[1]++; // carry
|
|
|
2385 // if the sum has P34+1 digits, i.e. C1>=10^34 redo the calculation
|
|
|
2386 // with x1=x1+1
|
|
|
2387 if (C1.w[1] > 0x0001ed09bead87c0ull || (C1.w[1] == 0x0001ed09bead87c0ull && C1.w[0] >= 0x378d8e6400000000ull)) { // C1 >= 10^34
|
|
|
2388 // chop off one more digit from the sum, but make sure there is
|
|
|
2389 // no double-rounding error (see table - double rounding logic)
|
|
|
2390 // now round C1 from P34+1 to P34 decimal digits
|
|
|
2391 // C1' = C1 + 1/2 * 10 = C1 + 5
|
|
|
2392 if (C1.w[0] >= 0xfffffffffffffffbull) { // low half add has carry
|
|
|
2393 C1.w[0] = C1.w[0] + 5;
|
|
|
2394 C1.w[1] = C1.w[1] + 1;
|
|
|
2395 } else {
|
|
|
2396 C1.w[0] = C1.w[0] + 5;
|
|
|
2397 }
|
|
|
2398 // the approximation of 10^(-1) was rounded up to 118 bits
|
|
|
2399 __mul_128x128_to_256 (Q256, C1, ten2mk128[0]); // Q256 = C1*, f1*
|
|
|
2400 // C1* is actually floor(C1*) in this case
|
|
|
2401 // the top 128 bits of 10^(-1) are
|
|
|
2402 // T* = ten2mk128trunc[0]=0x19999999999999999999999999999999
|
|
|
2403 // if (0 < f1* < 10^(-1)) then
|
|
|
2404 // if floor(C1*) is even then C1* = floor(C1*) - logical right
|
|
|
2405 // shift; C1* has p decimal digits, correct by Prop. 1)
|
|
|
2406 // else if floor(C1*) is odd C1* = floor(C1*) - 1 (logical right
|
|
|
2407 // shift; C1* has p decimal digits, correct by Pr. 1)
|
|
|
2408 // else
|
|
|
2409 // C1* = floor(C1*) (logical right shift; C has p decimal digits
|
|
|
2410 // correct by Property 1)
|
|
|
2411 // n = C1* * 10^(e2+x1+1)
|
|
|
2412 if ((Q256.w[1] || Q256.w[0])
|
|
|
2413 && (Q256.w[1] < ten2mk128trunc[0].w[1]
|
|
|
2414 || (Q256.w[1] == ten2mk128trunc[0].w[1]
|
|
|
2415 && Q256.w[0] <= ten2mk128trunc[0].w[0]))) {
|
|
|
2416 // the result is a midpoint
|
|
|
2417 if (is_inexact_lt_midpoint) { // for the 1st rounding
|
|
|
2418 is_inexact_gt_midpoint = 1;
|
|
|
2419 is_inexact_lt_midpoint = 0;
|
|
|
2420 is_midpoint_gt_even = 0;
|
|
|
2421 is_midpoint_lt_even = 0;
|
|
|
2422 } else if (is_inexact_gt_midpoint) { // for the 1st rounding
|
|
|
2423 Q256.w[2]--;
|
|
|
2424 if (Q256.w[2] == 0xffffffffffffffffull)
|
|
|
2425 Q256.w[3]--;
|
|
|
2426 is_inexact_gt_midpoint = 0;
|
|
|
2427 is_inexact_lt_midpoint = 1;
|
|
|
2428 is_midpoint_gt_even = 0;
|
|
|
2429 is_midpoint_lt_even = 0;
|
|
|
2430 } else if (is_midpoint_gt_even) { // for the 1st rounding
|
|
|
2431 // Note: cannot have is_midpoint_lt_even
|
|
|
2432 is_inexact_gt_midpoint = 0;
|
|
|
2433 is_inexact_lt_midpoint = 1;
|
|
|
2434 is_midpoint_gt_even = 0;
|
|
|
2435 is_midpoint_lt_even = 0;
|
|
|
2436 } else { // the first rounding must have been exact
|
|
|
2437 if (Q256.w[2] & 0x01) { // MP in [EVEN, ODD]
|
|
|
2438 // the truncated result is correct
|
|
|
2439 Q256.w[2]--;
|
|
|
2440 if (Q256.w[2] == 0xffffffffffffffffull)
|
|
|
2441 Q256.w[3]--;
|
|
|
2442 is_inexact_gt_midpoint = 0;
|
|
|
2443 is_inexact_lt_midpoint = 0;
|
|
|
2444 is_midpoint_gt_even = 1;
|
|
|
2445 is_midpoint_lt_even = 0;
|
|
|
2446 } else { // MP in [ODD, EVEN]
|
|
|
2447 is_inexact_gt_midpoint = 0;
|
|
|
2448 is_inexact_lt_midpoint = 0;
|
|
|
2449 is_midpoint_gt_even = 0;
|
|
|
2450 is_midpoint_lt_even = 1;
|
|
|
2451 }
|
|
|
2452 }
|
|
|
2453 tmp_inexact = 1; // in all cases
|
|
|
2454 } else { // the result is not a midpoint
|
|
|
2455 // determine inexactness of the rounding of C1 (the sum C1+C2*)
|
|
|
2456 // if (0 < f1* - 1/2 < 10^(-1)) then
|
|
|
2457 // the result is exact
|
|
|
2458 // else (if f1* - 1/2 > T* then)
|
|
|
2459 // the result of is inexact
|
|
|
2460 // ind = 0
|
|
|
2461 if (Q256.w[1] > 0x8000000000000000ull
|
|
|
2462 || (Q256.w[1] == 0x8000000000000000ull
|
|
|
2463 && Q256.w[0] > 0x0ull)) {
|
|
|
2464 // f1* > 1/2 and the result may be exact
|
|
|
2465 Q256.w[1] = Q256.w[1] - 0x8000000000000000ull; // f1* - 1/2
|
|
|
2466 if ((Q256.w[1] > ten2mk128trunc[0].w[1]
|
|
|
2467 || (Q256.w[1] == ten2mk128trunc[0].w[1]
|
|
|
2468 && Q256.w[0] > ten2mk128trunc[0].w[0]))) {
|
|
|
2469 is_inexact_gt_midpoint = 0;
|
|
|
2470 is_inexact_lt_midpoint = 1;
|
|
|
2471 is_midpoint_gt_even = 0;
|
|
|
2472 is_midpoint_lt_even = 0;
|
|
|
2473 // set the inexact flag
|
|
|
2474 tmp_inexact = 1;
|
|
|
2475 // *pfpsf |= INEXACT_EXCEPTION;
|
|
|
2476 } else { // else the result is exact for the 2nd rounding
|
|
|
2477 if (tmp_inexact) { // if the previous rounding was inexact
|
|
|
2478 if (is_midpoint_lt_even) {
|
|
|
2479 is_inexact_gt_midpoint = 1;
|
|
|
2480 is_midpoint_lt_even = 0;
|
|
|
2481 } else if (is_midpoint_gt_even) {
|
|
|
2482 is_inexact_lt_midpoint = 1;
|
|
|
2483 is_midpoint_gt_even = 0;
|
|
|
2484 } else {
|
|
|
2485 ; // no change
|
|
|
2486 }
|
|
|
2487 }
|
|
|
2488 }
|
|
|
2489 // rounding down, unless a midpoint in [ODD, EVEN]
|
|
|
2490 } else { // the result is inexact; f1* <= 1/2
|
|
|
2491 is_inexact_gt_midpoint = 1;
|
|
|
2492 is_inexact_lt_midpoint = 0;
|
|
|
2493 is_midpoint_gt_even = 0;
|
|
|
2494 is_midpoint_lt_even = 0;
|
|
|
2495 // set the inexact flag
|
|
|
2496 tmp_inexact = 1;
|
|
|
2497 // *pfpsf |= INEXACT_EXCEPTION;
|
|
|
2498 }
|
|
|
2499 } // end 'the result is not a midpoint'
|
|
|
2500 // n = C1 * 10^(e2+x1)
|
|
|
2501 C1.w[1] = Q256.w[3];
|
|
|
2502 C1.w[0] = Q256.w[2];
|
|
|
2503 y_exp = y_exp + ((UINT64) (x1 + 1) << 49);
|
|
|
2504 } else { // C1 < 10^34
|
|
|
2505 // C1.w[1] and C1.w[0] already set
|
|
|
2506 // n = C1 * 10^(e2+x1)
|
|
|
2507 y_exp = y_exp + ((UINT64) x1 << 49);
|
|
|
2508 }
|
|
|
2509 // check for overflow
|
|
|
2510 if (y_exp == EXP_MAX_P1
|
|
|
2511 && (rnd_mode == ROUNDING_TO_NEAREST
|
|
|
2512 || rnd_mode == ROUNDING_TIES_AWAY)) {
|
|
|
2513 res.w[1] = 0x7800000000000000ull | x_sign; // +/-inf
|
|
|
2514 res.w[0] = 0x0ull;
|
|
|
2515 // set the inexact flag
|
|
|
2516 *pfpsf |= INEXACT_EXCEPTION;
|
|
|
2517 // set the overflow flag
|
|
|
2518 *pfpsf |= OVERFLOW_EXCEPTION;
|
|
|
2519 BID_SWAP128 (res);
|
|
|
2520 BID_RETURN (res);
|
|
|
2521 } // else no overflow
|
|
|
2522 } else { // if x_sign != y_sign the result of this subtract. is exact
|
|
|
2523 C1.w[0] = C1.w[0] - R256.w[2];
|
|
|
2524 C1.w[1] = C1.w[1] - R256.w[3];
|
|
|
2525 if (C1.w[0] > tmp64)
|
|
|
2526 C1.w[1]--; // borrow
|
|
|
2527 if (C1.w[1] >= 0x8000000000000000ull) { // negative coefficient!
|
|
|
2528 C1.w[0] = ~C1.w[0];
|
|
|
2529 C1.w[0]++;
|
|
|
2530 C1.w[1] = ~C1.w[1];
|
|
|
2531 if (C1.w[0] == 0x0)
|
|
|
2532 C1.w[1]++;
|
|
|
2533 tmp_sign = y_sign;
|
|
|
2534 // the result will have the sign of y if last rnd
|
|
|
2535 } else {
|
|
|
2536 tmp_sign = x_sign;
|
|
|
2537 }
|
|
|
2538 // if the difference has P34-1 digits or less, i.e. C1 < 10^33 then
|
|
|
2539 // redo the calculation with x1=x1-1;
|
|
|
2540 // redo the calculation also if C1 = 10^33 and
|
|
|
2541 // (is_inexact_gt_midpoint or is_midpoint_lt_even);
|
|
|
2542 // (the last part should have really been
|
|
|
2543 // (is_inexact_lt_midpoint or is_midpoint_gt_even) from
|
|
|
2544 // the rounding of C2, but the position flags have been reversed)
|
|
|
2545 // 10^33 = 0x0000314dc6448d93 0x38c15b0a00000000
|
|
|
2546 if ((C1.w[1] < 0x0000314dc6448d93ull || (C1.w[1] == 0x0000314dc6448d93ull && C1.w[0] < 0x38c15b0a00000000ull)) || (C1.w[1] == 0x0000314dc6448d93ull && C1.w[0] == 0x38c15b0a00000000ull && (is_inexact_gt_midpoint || is_midpoint_lt_even))) { // C1=10^33
|
|
|
2547 x1 = x1 - 1; // x1 >= 0
|
|
|
2548 if (x1 >= 0) {
|
|
|
2549 // clear position flags and tmp_inexact
|
|
|
2550 is_midpoint_lt_even = 0;
|
|
|
2551 is_midpoint_gt_even = 0;
|
|
|
2552 is_inexact_lt_midpoint = 0;
|
|
|
2553 is_inexact_gt_midpoint = 0;
|
|
|
2554 tmp_inexact = 0;
|
|
|
2555 second_pass = 1;
|
|
|
2556 goto roundC2; // else result has less than P34 digits
|
|
|
2557 }
|
|
|
2558 }
|
|
|
2559 // if the coefficient of the result is 10^34 it means that this
|
|
|
2560 // must be the second pass, and we are done
|
|
|
2561 if (C1.w[1] == 0x0001ed09bead87c0ull && C1.w[0] == 0x378d8e6400000000ull) { // if C1 = 10^34
|
|
|
2562 C1.w[1] = 0x0000314dc6448d93ull; // C1 = 10^33
|
|
|
2563 C1.w[0] = 0x38c15b0a00000000ull;
|
|
|
2564 y_exp = y_exp + ((UINT64) 1 << 49);
|
|
|
2565 }
|
|
|
2566 x_sign = tmp_sign;
|
|
|
2567 if (x1 >= 1)
|
|
|
2568 y_exp = y_exp + ((UINT64) x1 << 49);
|
|
|
2569 // x1 = -1 is possible at the end of a second pass when the
|
|
|
2570 // first pass started with x1 = 1
|
|
|
2571 }
|
|
|
2572 C1_hi = C1.w[1];
|
|
|
2573 C1_lo = C1.w[0];
|
|
|
2574 // general correction from RN to RA, RM, RP, RZ; result uses y_exp
|
|
|
2575 if (rnd_mode != ROUNDING_TO_NEAREST) {
|
|
|
2576 if ((!x_sign
|
|
|
2577 && ((rnd_mode == ROUNDING_UP && is_inexact_lt_midpoint)
|
|
|
2578 ||
|
|
|
2579 ((rnd_mode == ROUNDING_TIES_AWAY
|
|
|
2580 || rnd_mode == ROUNDING_UP)
|
|
|
2581 && is_midpoint_gt_even))) || (x_sign
|
|
|
2582 &&
|
|
|
2583 ((rnd_mode ==
|
|
|
2584 ROUNDING_DOWN
|
|
|
2585 &&
|
|
|
2586 is_inexact_lt_midpoint)
|
|
|
2587 ||
|
|
|
2588 ((rnd_mode ==
|
|
|
2589 ROUNDING_TIES_AWAY
|
|
|
2590 || rnd_mode ==
|
|
|
2591 ROUNDING_DOWN)
|
|
|
2592 &&
|
|
|
2593 is_midpoint_gt_even))))
|
|
|
2594 {
|
|
|
2595 // C1 = C1 + 1
|
|
|
2596 C1_lo = C1_lo + 1;
|
|
|
2597 if (C1_lo == 0) { // rounding overflow in the low 64 bits
|
|
|
2598 C1_hi = C1_hi + 1;
|
|
|
2599 }
|
|
|
2600 if (C1_hi == 0x0001ed09bead87c0ull
|
|
|
2601 && C1_lo == 0x378d8e6400000000ull) {
|
|
|
2602 // C1 = 10^34 => rounding overflow
|
|
|
2603 C1_hi = 0x0000314dc6448d93ull;
|
|
|
2604 C1_lo = 0x38c15b0a00000000ull; // 10^33
|
|
|
2605 y_exp = y_exp + EXP_P1;
|
|
|
2606 }
|
|
|
2607 } else if ((is_midpoint_lt_even || is_inexact_gt_midpoint)
|
|
|
2608 &&
|
|
|
2609 ((x_sign
|
|
|
2610 && (rnd_mode == ROUNDING_UP
|
|
|
2611 || rnd_mode == ROUNDING_TO_ZERO))
|
|
|
2612 || (!x_sign
|
|
|
2613 && (rnd_mode == ROUNDING_DOWN
|
|
|
2614 || rnd_mode == ROUNDING_TO_ZERO)))) {
|
|
|
2615 // C1 = C1 - 1
|
|
|
2616 C1_lo = C1_lo - 1;
|
|
|
2617 if (C1_lo == 0xffffffffffffffffull)
|
|
|
2618 C1_hi--;
|
|
|
2619 // check if we crossed into the lower decade
|
|
|
2620 if (C1_hi == 0x0000314dc6448d93ull && C1_lo == 0x38c15b09ffffffffull) { // 10^33 - 1
|
|
|
2621 C1_hi = 0x0001ed09bead87c0ull; // 10^34 - 1
|
|
|
2622 C1_lo = 0x378d8e63ffffffffull;
|
|
|
2623 y_exp = y_exp - EXP_P1;
|
|
|
2624 // no underflow, because delta + q2 >= P34 + 1
|
|
|
2625 }
|
|
|
2626 } else {
|
|
|
2627 ; // exact, the result is already correct
|
|
|
2628 }
|
|
|
2629 // in all cases check for overflow (RN and RA solved already)
|
|
|
2630 if (y_exp == EXP_MAX_P1) { // overflow
|
|
|
2631 if ((rnd_mode == ROUNDING_DOWN && x_sign) || // RM and res < 0
|
|
|
2632 (rnd_mode == ROUNDING_UP && !x_sign)) { // RP and res > 0
|
|
|
2633 C1_hi = 0x7800000000000000ull; // +inf
|
|
|
2634 C1_lo = 0x0ull;
|
|
|
2635 } else { // RM and res > 0, RP and res < 0, or RZ
|
|
|
2636 C1_hi = 0x5fffed09bead87c0ull;
|
|
|
2637 C1_lo = 0x378d8e63ffffffffull;
|
|
|
2638 }
|
|
|
2639 y_exp = 0; // x_sign is preserved
|
|
|
2640 // set the inexact flag (in case the exact addition was exact)
|
|
|
2641 *pfpsf |= INEXACT_EXCEPTION;
|
|
|
2642 // set the overflow flag
|
|
|
2643 *pfpsf |= OVERFLOW_EXCEPTION;
|
|
|
2644 }
|
|
|
2645 }
|
|
|
2646 // assemble the result
|
|
|
2647 res.w[1] = x_sign | y_exp | C1_hi;
|
|
|
2648 res.w[0] = C1_lo;
|
|
|
2649 if (tmp_inexact)
|
|
|
2650 *pfpsf |= INEXACT_EXCEPTION;
|
|
|
2651 }
|
|
|
2652 } else { // if (-P34 + 1 <= delta <= -1) <=> 1 <= -delta <= P34 - 1
|
|
|
2653 // NOTE: the following, up to "} else { // if x_sign != y_sign
|
|
|
2654 // the result is exact" is identical to "else if (delta == P34 - q2) {"
|
|
|
2655 // from above; also, the code is not symmetric: a+b and b+a may take
|
|
|
2656 // different paths (need to unify eventually!)
|
|
|
2657 // calculate C' = C2 + C1 * 10^(e1-e2) directly; the result may be
|
|
|
2658 // inexact if it requires P34 + 1 decimal digits; in either case the
|
|
|
2659 // 'cutoff' point for addition is at the position of the lsb of C2
|
|
|
2660 // The coefficient of the result is C1 * 10^(e1-e2) + C2 and the
|
|
|
2661 // exponent is e2; either C1 or 10^(e1-e2) may not fit is 64 bits,
|
|
|
2662 // but their product fits with certainty in 128 bits (actually in 113)
|
|
|
2663 // Note that 0 <= e1 - e2 <= P34 - 2
|
|
|
2664 // -P34 + 1 <= delta <= -1 <=> -P34 + 1 <= delta <= -1 <=>
|
|
|
2665 // -P34 + 1 <= q1 + e1 - q2 - e2 <= -1 <=>
|
|
|
2666 // q2 - q1 - P34 + 1 <= e1 - e2 <= q2 - q1 - 1 <=>
|
|
|
2667 // 1 - P34 - P34 + 1 <= e1-e2 <= P34 - 1 - 1 => 0 <= e1-e2 <= P34 - 2
|
|
|
2668 scale = delta - q1 + q2; // scale = (int)(e1 >> 49) - (int)(e2 >> 49)
|
|
|
2669 if (scale >= 20) { // 10^(e1-e2) does not fit in 64 bits, but C1 does
|
|
|
2670 __mul_128x64_to_128 (C1, C1_lo, ten2k128[scale - 20]);
|
|
|
2671 } else if (scale >= 1) {
|
|
|
2672 // if 1 <= scale <= 19 then 10^(e1-e2) fits in 64 bits
|
|
|
2673 if (q1 <= 19) { // C1 fits in 64 bits
|
|
|
2674 __mul_64x64_to_128MACH (C1, C1_lo, ten2k64[scale]);
|
|
|
2675 } else { // q1 >= 20
|
|
|
2676 C1.w[1] = C1_hi;
|
|
|
2677 C1.w[0] = C1_lo;
|
|
|
2678 __mul_128x64_to_128 (C1, ten2k64[scale], C1);
|
|
|
2679 }
|
|
|
2680 } else { // if (scale == 0) C1 is unchanged
|
|
|
2681 C1.w[1] = C1_hi;
|
|
|
2682 C1.w[0] = C1_lo; // only the low part is necessary
|
|
|
2683 }
|
|
|
2684 C1_hi = C1.w[1];
|
|
|
2685 C1_lo = C1.w[0];
|
|
|
2686 // now add C2
|
|
|
2687 if (x_sign == y_sign) {
|
|
|
2688 // the result can overflow!
|
|
|
2689 C1_lo = C1_lo + C2_lo;
|
|
|
2690 C1_hi = C1_hi + C2_hi;
|
|
|
2691 if (C1_lo < C1.w[0])
|
|
|
2692 C1_hi++;
|
|
|
2693 // test for overflow, possible only when C1 >= 10^34
|
|
|
2694 if (C1_hi > 0x0001ed09bead87c0ull || (C1_hi == 0x0001ed09bead87c0ull && C1_lo >= 0x378d8e6400000000ull)) { // C1 >= 10^34
|
|
|
2695 // in this case q = P34 + 1 and x = q - P34 = 1, so multiply
|
|
|
2696 // C'' = C'+ 5 = C1 + 5 by k1 ~ 10^(-1) calculated for P34 + 1
|
|
|
2697 // decimal digits
|
|
|
2698 // Calculate C'' = C' + 1/2 * 10^x
|
|
|
2699 if (C1_lo >= 0xfffffffffffffffbull) { // low half add has carry
|
|
|
2700 C1_lo = C1_lo + 5;
|
|
|
2701 C1_hi = C1_hi + 1;
|
|
|
2702 } else {
|
|
|
2703 C1_lo = C1_lo + 5;
|
|
|
2704 }
|
|
|
2705 // the approximation of 10^(-1) was rounded up to 118 bits
|
|
|
2706 // 10^(-1) =~ 33333333333333333333333333333400 * 2^-129
|
|
|
2707 // 10^(-1) =~ 19999999999999999999999999999a00 * 2^-128
|
|
|
2708 C1.w[1] = C1_hi;
|
|
|
2709 C1.w[0] = C1_lo; // C''
|
|
|
2710 ten2m1.w[1] = 0x1999999999999999ull;
|
|
|
2711 ten2m1.w[0] = 0x9999999999999a00ull;
|
|
|
2712 __mul_128x128_to_256 (P256, C1, ten2m1); // P256 = C*, f*
|
|
|
2713 // C* is actually floor(C*) in this case
|
|
|
2714 // the top Ex = 128 bits of 10^(-1) are
|
|
|
2715 // T* = 0x00199999999999999999999999999999
|
|
|
2716 // if (0 < f* < 10^(-x)) then
|
|
|
2717 // if floor(C*) is even then C = floor(C*) - logical right
|
|
|
2718 // shift; C has p decimal digits, correct by Prop. 1)
|
|
|
2719 // else if floor(C*) is odd C = floor(C*) - 1 (logical right
|
|
|
2720 // shift; C has p decimal digits, correct by Pr. 1)
|
|
|
2721 // else
|
|
|
2722 // C = floor(C*) (logical right shift; C has p decimal digits,
|
|
|
2723 // correct by Property 1)
|
|
|
2724 // n = C * 10^(e2+x)
|
|
|
2725 if ((P256.w[1] || P256.w[0])
|
|
|
2726 && (P256.w[1] < 0x1999999999999999ull
|
|
|
2727 || (P256.w[1] == 0x1999999999999999ull
|
|
|
2728 && P256.w[0] <= 0x9999999999999999ull))) {
|
|
|
2729 // the result is a midpoint
|
|
|
2730 if (P256.w[2] & 0x01) {
|
|
|
2731 is_midpoint_gt_even = 1;
|
|
|
2732 // if floor(C*) is odd C = floor(C*) - 1; the result is not 0
|
|
|
2733 P256.w[2]--;
|
|
|
2734 if (P256.w[2] == 0xffffffffffffffffull)
|
|
|
2735 P256.w[3]--;
|
|
|
2736 } else {
|
|
|
2737 is_midpoint_lt_even = 1;
|
|
|
2738 }
|
|
|
2739 }
|
|
|
2740 // n = Cstar * 10^(e2+1)
|
|
|
2741 y_exp = y_exp + EXP_P1;
|
|
|
2742 // C* != 10^P34 because C* has P34 digits
|
|
|
2743 // check for overflow
|
|
|
2744 if (y_exp == EXP_MAX_P1
|
|
|
2745 && (rnd_mode == ROUNDING_TO_NEAREST
|
|
|
2746 || rnd_mode == ROUNDING_TIES_AWAY)) {
|
|
|
2747 // overflow for RN
|
|
|
2748 res.w[1] = x_sign | 0x7800000000000000ull; // +/-inf
|
|
|
2749 res.w[0] = 0x0ull;
|
|
|
2750 // set the inexact flag
|
|
|
2751 *pfpsf |= INEXACT_EXCEPTION;
|
|
|
2752 // set the overflow flag
|
|
|
2753 *pfpsf |= OVERFLOW_EXCEPTION;
|
|
|
2754 BID_SWAP128 (res);
|
|
|
2755 BID_RETURN (res);
|
|
|
2756 }
|
|
|
2757 // if (0 < f* - 1/2 < 10^(-x)) then
|
|
|
2758 // the result of the addition is exact
|
|
|
2759 // else
|
|
|
2760 // the result of the addition is inexact
|
|
|
2761 if (P256.w[1] > 0x8000000000000000ull || (P256.w[1] == 0x8000000000000000ull && P256.w[0] > 0x0ull)) { // the result may be exact
|
|
|
2762 tmp64 = P256.w[1] - 0x8000000000000000ull; // f* - 1/2
|
|
|
2763 if ((tmp64 > 0x1999999999999999ull
|
|
|
2764 || (tmp64 == 0x1999999999999999ull
|
|
|
2765 && P256.w[0] >= 0x9999999999999999ull))) {
|
|
|
2766 // set the inexact flag
|
|
|
2767 *pfpsf |= INEXACT_EXCEPTION;
|
|
|
2768 is_inexact = 1;
|
|
|
2769 } // else the result is exact
|
|
|
2770 } else { // the result is inexact
|
|
|
2771 // set the inexact flag
|
|
|
2772 *pfpsf |= INEXACT_EXCEPTION;
|
|
|
2773 is_inexact = 1;
|
|
|
2774 }
|
|
|
2775 C1_hi = P256.w[3];
|
|
|
2776 C1_lo = P256.w[2];
|
|
|
2777 if (!is_midpoint_gt_even && !is_midpoint_lt_even) {
|
|
|
2778 is_inexact_lt_midpoint = is_inexact
|
|
|
2779 && (P256.w[1] & 0x8000000000000000ull);
|
|
|
2780 is_inexact_gt_midpoint = is_inexact
|
|
|
2781 && !(P256.w[1] & 0x8000000000000000ull);
|
|
|
2782 }
|
|
|
2783 // general correction from RN to RA, RM, RP, RZ; result uses y_exp
|
|
|
2784 if (rnd_mode != ROUNDING_TO_NEAREST) {
|
|
|
2785 if ((!x_sign
|
|
|
2786 && ((rnd_mode == ROUNDING_UP
|
|
|
2787 && is_inexact_lt_midpoint)
|
|
|
2788 || ((rnd_mode == ROUNDING_TIES_AWAY
|
|
|
2789 || rnd_mode == ROUNDING_UP)
|
|
|
2790 && is_midpoint_gt_even))) || (x_sign
|
|
|
2791 &&
|
|
|
2792 ((rnd_mode ==
|
|
|
2793 ROUNDING_DOWN
|
|
|
2794 &&
|
|
|
2795 is_inexact_lt_midpoint)
|
|
|
2796 ||
|
|
|
2797 ((rnd_mode ==
|
|
|
2798 ROUNDING_TIES_AWAY
|
|
|
2799 || rnd_mode
|
|
|
2800 ==
|
|
|
2801 ROUNDING_DOWN)
|
|
|
2802 &&
|
|
|
2803 is_midpoint_gt_even))))
|
|
|
2804 {
|
|
|
2805 // C1 = C1 + 1
|
|
|
2806 C1_lo = C1_lo + 1;
|
|
|
2807 if (C1_lo == 0) { // rounding overflow in the low 64 bits
|
|
|
2808 C1_hi = C1_hi + 1;
|
|
|
2809 }
|
|
|
2810 if (C1_hi == 0x0001ed09bead87c0ull
|
|
|
2811 && C1_lo == 0x378d8e6400000000ull) {
|
|
|
2812 // C1 = 10^34 => rounding overflow
|
|
|
2813 C1_hi = 0x0000314dc6448d93ull;
|
|
|
2814 C1_lo = 0x38c15b0a00000000ull; // 10^33
|
|
|
2815 y_exp = y_exp + EXP_P1;
|
|
|
2816 }
|
|
|
2817 } else
|
|
|
2818 if ((is_midpoint_lt_even || is_inexact_gt_midpoint) &&
|
|
|
2819 ((x_sign && (rnd_mode == ROUNDING_UP ||
|
|
|
2820 rnd_mode == ROUNDING_TO_ZERO)) ||
|
|
|
2821 (!x_sign && (rnd_mode == ROUNDING_DOWN ||
|
|
|
2822 rnd_mode == ROUNDING_TO_ZERO)))) {
|
|
|
2823 // C1 = C1 - 1
|
|
|
2824 C1_lo = C1_lo - 1;
|
|
|
2825 if (C1_lo == 0xffffffffffffffffull)
|
|
|
2826 C1_hi--;
|
|
|
2827 // check if we crossed into the lower decade
|
|
|
2828 if (C1_hi == 0x0000314dc6448d93ull && C1_lo == 0x38c15b09ffffffffull) { // 10^33 - 1
|
|
|
2829 C1_hi = 0x0001ed09bead87c0ull; // 10^34 - 1
|
|
|
2830 C1_lo = 0x378d8e63ffffffffull;
|
|
|
2831 y_exp = y_exp - EXP_P1;
|
|
|
2832 // no underflow, because delta + q2 >= P34 + 1
|
|
|
2833 }
|
|
|
2834 } else {
|
|
|
2835 ; // exact, the result is already correct
|
|
|
2836 }
|
|
|
2837 // in all cases check for overflow (RN and RA solved already)
|
|
|
2838 if (y_exp == EXP_MAX_P1) { // overflow
|
|
|
2839 if ((rnd_mode == ROUNDING_DOWN && x_sign) || // RM and res < 0
|
|
|
2840 (rnd_mode == ROUNDING_UP && !x_sign)) { // RP and res > 0
|
|
|
2841 C1_hi = 0x7800000000000000ull; // +inf
|
|
|
2842 C1_lo = 0x0ull;
|
|
|
2843 } else { // RM and res > 0, RP and res < 0, or RZ
|
|
|
2844 C1_hi = 0x5fffed09bead87c0ull;
|
|
|
2845 C1_lo = 0x378d8e63ffffffffull;
|
|
|
2846 }
|
|
|
2847 y_exp = 0; // x_sign is preserved
|
|
|
2848 // set the inexact flag (in case the exact addition was exact)
|
|
|
2849 *pfpsf |= INEXACT_EXCEPTION;
|
|
|
2850 // set the overflow flag
|
|
|
2851 *pfpsf |= OVERFLOW_EXCEPTION;
|
|
|
2852 }
|
|
|
2853 }
|
|
|
2854 } // else if (C1 < 10^34) then C1 is the coeff.; the result is exact
|
|
|
2855 // assemble the result
|
|
|
2856 res.w[1] = x_sign | y_exp | C1_hi;
|
|
|
2857 res.w[0] = C1_lo;
|
|
|
2858 } else { // if x_sign != y_sign the result is exact
|
|
|
2859 C1_lo = C2_lo - C1_lo;
|
|
|
2860 C1_hi = C2_hi - C1_hi;
|
|
|
2861 if (C1_lo > C2_lo)
|
|
|
2862 C1_hi--;
|
|
|
2863 if (C1_hi >= 0x8000000000000000ull) { // negative coefficient!
|
|
|
2864 C1_lo = ~C1_lo;
|
|
|
2865 C1_lo++;
|
|
|
2866 C1_hi = ~C1_hi;
|
|
|
2867 if (C1_lo == 0x0)
|
|
|
2868 C1_hi++;
|
|
|
2869 x_sign = y_sign; // the result will have the sign of y
|
|
|
2870 }
|
|
|
2871 // the result can be zero, but it cannot overflow
|
|
|
2872 if (C1_lo == 0 && C1_hi == 0) {
|
|
|
2873 // assemble the result
|
|
|
2874 if (x_exp < y_exp)
|
|
|
2875 res.w[1] = x_exp;
|
|
|
2876 else
|
|
|
2877 res.w[1] = y_exp;
|
|
|
2878 res.w[0] = 0;
|
|
|
2879 if (rnd_mode == ROUNDING_DOWN) {
|
|
|
2880 res.w[1] |= 0x8000000000000000ull;
|
|
|
2881 }
|
|
|
2882 BID_SWAP128 (res);
|
|
|
2883 BID_RETURN (res);
|
|
|
2884 }
|
|
|
2885 // assemble the result
|
|
|
2886 res.w[1] = y_sign | y_exp | C1_hi;
|
|
|
2887 res.w[0] = C1_lo;
|
|
|
2888 }
|
|
|
2889 }
|
|
|
2890 }
|
|
|
2891 BID_SWAP128 (res);
|
|
|
2892 BID_RETURN (res)
|
|
|
2893 }
|
|
|
2894 }
|
|
|
2895
|
|
|
2896
|
|
|
2897
|
|
|
2898 // bid128_sub stands for bid128qq_sub
|
|
|
2899
|
|
|
2900 /*****************************************************************************
|
|
|
2901 * BID128 sub
|
|
|
2902 ****************************************************************************/
|
|
|
2903
|
|
|
2904 #if DECIMAL_CALL_BY_REFERENCE
|
|
|
2905 void
|
|
|
2906 bid128_sub (UINT128 * pres, UINT128 * px, UINT128 * py
|
|
|
2907 _RND_MODE_PARAM _EXC_FLAGS_PARAM _EXC_MASKS_PARAM
|
|
|
2908 _EXC_INFO_PARAM) {
|
|
|
2909 UINT128 x = *px, y = *py;
|
|
|
2910 #if !DECIMAL_GLOBAL_ROUNDING
|
|
|
2911 unsigned int rnd_mode = *prnd_mode;
|
|
|
2912 #endif
|
|
|
2913 #else
|
|
|
2914 UINT128
|
|
|
2915 bid128_sub (UINT128 x, UINT128 y
|
|
|
2916 _RND_MODE_PARAM _EXC_FLAGS_PARAM _EXC_MASKS_PARAM
|
|
|
2917 _EXC_INFO_PARAM) {
|
|
|
2918 #endif
|
|
|
2919
|
|
|
2920 UINT128 res;
|
|
|
2921 UINT64 y_sign;
|
|
|
2922
|
|
|
2923 if ((y.w[HIGH_128W] & MASK_NAN) != MASK_NAN) { // y is not NAN
|
|
|
2924 // change its sign
|
|
|
2925 y_sign = y.w[HIGH_128W] & MASK_SIGN; // 0 for positive, MASK_SIGN for negative
|
|
|
2926 if (y_sign)
|
|
|
2927 y.w[HIGH_128W] = y.w[HIGH_128W] & 0x7fffffffffffffffull;
|
|
|
2928 else
|
|
|
2929 y.w[HIGH_128W] = y.w[HIGH_128W] | 0x8000000000000000ull;
|
|
|
2930 }
|
|
|
2931 #if DECIMAL_CALL_BY_REFERENCE
|
|
|
2932 bid128_add (&res, &x, &y
|
|
|
2933 _RND_MODE_ARG _EXC_FLAGS_ARG _EXC_MASKS_ARG
|
|
|
2934 _EXC_INFO_ARG);
|
|
|
2935 #else
|
|
|
2936 res = bid128_add (x, y
|
|
|
2937 _RND_MODE_ARG _EXC_FLAGS_ARG _EXC_MASKS_ARG
|
|
|
2938 _EXC_INFO_ARG);
|
|
|
2939 #endif
|
|
|
2940 BID_RETURN (res);
|
|
|
2941 }
|
