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83ffe9cd | 1 | /* Copyright (C) 2007-2023 Free Software Foundation, Inc. |
200359e8 L |
2 | |
3 | This file is part of GCC. | |
4 | ||
5 | GCC is free software; you can redistribute it and/or modify it under | |
6 | the terms of the GNU General Public License as published by the Free | |
748086b7 | 7 | Software Foundation; either version 3, or (at your option) any later |
200359e8 L |
8 | version. |
9 | ||
200359e8 L |
10 | GCC is distributed in the hope that it will be useful, but WITHOUT ANY |
11 | WARRANTY; without even the implied warranty of MERCHANTABILITY or | |
12 | FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License | |
13 | for more details. | |
14 | ||
748086b7 JJ |
15 | Under Section 7 of GPL version 3, you are granted additional |
16 | permissions described in the GCC Runtime Library Exception, version | |
17 | 3.1, as published by the Free Software Foundation. | |
18 | ||
19 | You should have received a copy of the GNU General Public License and | |
20 | a copy of the GCC Runtime Library Exception along with this program; | |
21 | see the files COPYING3 and COPYING.RUNTIME respectively. If not, see | |
22 | <http://www.gnu.org/licenses/>. */ | |
200359e8 L |
23 | |
24 | #define BID_128RES | |
25 | ||
26 | #include "bid_internal.h" | |
27 | ||
28 | /***************************************************************************** | |
29 | * BID128_round_integral_exact | |
30 | ****************************************************************************/ | |
31 | ||
b2a00c89 | 32 | BID128_FUNCTION_ARG1 (bid128_round_integral_exact, x) |
200359e8 | 33 | |
b2a00c89 L |
34 | UINT128 res = { {0xbaddbaddbaddbaddull, 0xbaddbaddbaddbaddull} |
35 | }; | |
36 | UINT64 x_sign; | |
37 | UINT64 x_exp; | |
38 | int exp; // unbiased exponent | |
200359e8 | 39 | // Note: C1.w[1], C1.w[0] represent x_signif_hi, x_signif_lo (all are UINT64) |
b2a00c89 L |
40 | UINT64 tmp64; |
41 | BID_UI64DOUBLE tmp1; | |
42 | unsigned int x_nr_bits; | |
43 | int q, ind, shift; | |
44 | UINT128 C1; | |
45 | UINT256 fstar; | |
46 | UINT256 P256; | |
200359e8 L |
47 | |
48 | // check for NaN or Infinity | |
b2a00c89 L |
49 | if ((x.w[1] & MASK_SPECIAL) == MASK_SPECIAL) { |
50 | // x is special | |
51 | if ((x.w[1] & MASK_NAN) == MASK_NAN) { // x is NAN | |
52 | // if x = NaN, then res = Q (x) | |
53 | // check first for non-canonical NaN payload | |
54 | if (((x.w[1] & 0x00003fffffffffffull) > 0x0000314dc6448d93ull) || | |
55 | (((x.w[1] & 0x00003fffffffffffull) == 0x0000314dc6448d93ull) && | |
56 | (x.w[0] > 0x38c15b09ffffffffull))) { | |
57 | x.w[1] = x.w[1] & 0xffffc00000000000ull; | |
58 | x.w[0] = 0x0ull; | |
59 | } | |
60 | if ((x.w[1] & MASK_SNAN) == MASK_SNAN) { // x is SNAN | |
61 | // set invalid flag | |
62 | *pfpsf |= INVALID_EXCEPTION; | |
63 | // return quiet (x) | |
64 | res.w[1] = x.w[1] & 0xfc003fffffffffffull; // clear out also G[6]-G[16] | |
65 | res.w[0] = x.w[0]; | |
66 | } else { // x is QNaN | |
67 | // return x | |
68 | res.w[1] = x.w[1] & 0xfc003fffffffffffull; // clear out G[6]-G[16] | |
69 | res.w[0] = x.w[0]; | |
200359e8 | 70 | } |
b2a00c89 L |
71 | BID_RETURN (res) |
72 | } else { // x is not a NaN, so it must be infinity | |
73 | if ((x.w[1] & MASK_SIGN) == 0x0ull) { // x is +inf | |
74 | // return +inf | |
75 | res.w[1] = 0x7800000000000000ull; | |
76 | res.w[0] = 0x0000000000000000ull; | |
77 | } else { // x is -inf | |
78 | // return -inf | |
79 | res.w[1] = 0xf800000000000000ull; | |
80 | res.w[0] = 0x0000000000000000ull; | |
81 | } | |
82 | BID_RETURN (res); | |
200359e8 | 83 | } |
b2a00c89 | 84 | } |
200359e8 | 85 | // unpack x |
b2a00c89 L |
86 | x_sign = x.w[1] & MASK_SIGN; // 0 for positive, MASK_SIGN for negative |
87 | C1.w[1] = x.w[1] & MASK_COEFF; | |
88 | C1.w[0] = x.w[0]; | |
200359e8 | 89 | |
b2a00c89 L |
90 | // check for non-canonical values (treated as zero) |
91 | if ((x.w[1] & 0x6000000000000000ull) == 0x6000000000000000ull) { // G0_G1=11 | |
92 | // non-canonical | |
93 | x_exp = (x.w[1] << 2) & MASK_EXP; // biased and shifted left 49 bits | |
94 | C1.w[1] = 0; // significand high | |
95 | C1.w[0] = 0; // significand low | |
96 | } else { // G0_G1 != 11 | |
97 | x_exp = x.w[1] & MASK_EXP; // biased and shifted left 49 bits | |
98 | if (C1.w[1] > 0x0001ed09bead87c0ull || | |
99 | (C1.w[1] == 0x0001ed09bead87c0ull | |
100 | && C1.w[0] > 0x378d8e63ffffffffull)) { | |
101 | // x is non-canonical if coefficient is larger than 10^34 -1 | |
200359e8 L |
102 | C1.w[1] = 0; |
103 | C1.w[0] = 0; | |
b2a00c89 L |
104 | } else { // canonical |
105 | ; | |
200359e8 | 106 | } |
b2a00c89 L |
107 | } |
108 | ||
200359e8 | 109 | // test for input equal to zero |
b2a00c89 L |
110 | if ((C1.w[1] == 0x0ull) && (C1.w[0] == 0x0ull)) { |
111 | // x is 0 | |
112 | // return 0 preserving the sign bit and the preferred exponent | |
113 | // of MAX(Q(x), 0) | |
114 | if (x_exp <= (0x1820ull << 49)) { | |
115 | res.w[1] = (x.w[1] & 0x8000000000000000ull) | 0x3040000000000000ull; | |
116 | } else { | |
117 | res.w[1] = x_sign | x_exp; | |
200359e8 | 118 | } |
b2a00c89 L |
119 | res.w[0] = 0x0000000000000000ull; |
120 | BID_RETURN (res); | |
121 | } | |
200359e8 L |
122 | // x is not special and is not zero |
123 | ||
b2a00c89 L |
124 | switch (rnd_mode) { |
125 | case ROUNDING_TO_NEAREST: | |
126 | case ROUNDING_TIES_AWAY: | |
127 | // if (exp <= -(p+1)) return 0.0 | |
128 | if (x_exp <= 0x2ffa000000000000ull) { // 0x2ffa000000000000ull == -35 | |
129 | res.w[1] = x_sign | 0x3040000000000000ull; | |
130 | res.w[0] = 0x0000000000000000ull; | |
131 | *pfpsf |= INEXACT_EXCEPTION; | |
200359e8 L |
132 | BID_RETURN (res); |
133 | } | |
b2a00c89 L |
134 | break; |
135 | case ROUNDING_DOWN: | |
136 | // if (exp <= -p) return -1.0 or +0.0 | |
137 | if (x_exp <= 0x2ffc000000000000ull) { // 0x2ffa000000000000ull == -34 | |
138 | if (x_sign) { | |
139 | // if negative, return negative 1, because we know coefficient | |
140 | // is non-zero (would have been caught above) | |
141 | res.w[1] = 0xb040000000000000ull; | |
142 | res.w[0] = 0x0000000000000001ull; | |
143 | } else { | |
144 | // if positive, return positive 0, because we know coefficient is | |
145 | // non-zero (would have been caught above) | |
146 | res.w[1] = 0x3040000000000000ull; | |
200359e8 | 147 | res.w[0] = 0x0000000000000000ull; |
200359e8 | 148 | } |
b2a00c89 L |
149 | *pfpsf |= INEXACT_EXCEPTION; |
150 | BID_RETURN (res); | |
200359e8 | 151 | } |
b2a00c89 L |
152 | break; |
153 | case ROUNDING_UP: | |
154 | // if (exp <= -p) return -0.0 or +1.0 | |
155 | if (x_exp <= 0x2ffc000000000000ull) { // 0x2ffc000000000000ull == -34 | |
156 | if (x_sign) { | |
157 | // if negative, return negative 0, because we know the coefficient | |
158 | // is non-zero (would have been caught above) | |
159 | res.w[1] = 0xb040000000000000ull; | |
160 | res.w[0] = 0x0000000000000000ull; | |
200359e8 | 161 | } else { |
b2a00c89 L |
162 | // if positive, return positive 1, because we know coefficient is |
163 | // non-zero (would have been caught above) | |
164 | res.w[1] = 0x3040000000000000ull; | |
165 | res.w[0] = 0x0000000000000001ull; | |
200359e8 | 166 | } |
b2a00c89 | 167 | *pfpsf |= INEXACT_EXCEPTION; |
200359e8 L |
168 | BID_RETURN (res); |
169 | } | |
b2a00c89 L |
170 | break; |
171 | case ROUNDING_TO_ZERO: | |
172 | // if (exp <= -p) return -0.0 or +0.0 | |
173 | if (x_exp <= 0x2ffc000000000000ull) { // 0x2ffc000000000000ull == -34 | |
200359e8 L |
174 | res.w[1] = x_sign | 0x3040000000000000ull; |
175 | res.w[0] = 0x0000000000000000ull; | |
b2a00c89 | 176 | *pfpsf |= INEXACT_EXCEPTION; |
200359e8 L |
177 | BID_RETURN (res); |
178 | } | |
b2a00c89 L |
179 | break; |
180 | } | |
181 | ||
200359e8 L |
182 | // q = nr. of decimal digits in x |
183 | // determine first the nr. of bits in x | |
b2a00c89 L |
184 | if (C1.w[1] == 0) { |
185 | if (C1.w[0] >= 0x0020000000000000ull) { // x >= 2^53 | |
186 | // split the 64-bit value in two 32-bit halves to avoid rounding errors | |
187 | if (C1.w[0] >= 0x0000000100000000ull) { // x >= 2^32 | |
188 | tmp1.d = (double) (C1.w[0] >> 32); // exact conversion | |
189 | x_nr_bits = | |
190 | 33 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); | |
191 | } else { // x < 2^32 | |
192 | tmp1.d = (double) (C1.w[0]); // exact conversion | |
200359e8 L |
193 | x_nr_bits = |
194 | 1 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); | |
195 | } | |
b2a00c89 L |
196 | } else { // if x < 2^53 |
197 | tmp1.d = (double) C1.w[0]; // exact conversion | |
200359e8 | 198 | x_nr_bits = |
b2a00c89 | 199 | 1 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); |
200359e8 | 200 | } |
b2a00c89 L |
201 | } else { // C1.w[1] != 0 => nr. bits = 64 + nr_bits (C1.w[1]) |
202 | tmp1.d = (double) C1.w[1]; // exact conversion | |
203 | x_nr_bits = | |
204 | 65 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); | |
205 | } | |
206 | ||
207 | q = nr_digits[x_nr_bits - 1].digits; | |
208 | if (q == 0) { | |
209 | q = nr_digits[x_nr_bits - 1].digits1; | |
210 | if (C1.w[1] > nr_digits[x_nr_bits - 1].threshold_hi || | |
211 | (C1.w[1] == nr_digits[x_nr_bits - 1].threshold_hi && | |
212 | C1.w[0] >= nr_digits[x_nr_bits - 1].threshold_lo)) | |
213 | q++; | |
214 | } | |
215 | exp = (x_exp >> 49) - 6176; | |
216 | if (exp >= 0) { // -exp <= 0 | |
217 | // the argument is an integer already | |
218 | res.w[1] = x.w[1]; | |
219 | res.w[0] = x.w[0]; | |
220 | BID_RETURN (res); | |
221 | } | |
222 | // exp < 0 | |
223 | switch (rnd_mode) { | |
224 | case ROUNDING_TO_NEAREST: | |
225 | if ((q + exp) >= 0) { // exp < 0 and 1 <= -exp <= q | |
226 | // need to shift right -exp digits from the coefficient; exp will be 0 | |
200359e8 L |
227 | ind = -exp; // 1 <= ind <= 34; ind is a synonym for 'x' |
228 | // chop off ind digits from the lower part of C1 | |
229 | // C1 = C1 + 1/2 * 10^x where the result C1 fits in 127 bits | |
230 | tmp64 = C1.w[0]; | |
231 | if (ind <= 19) { | |
b2a00c89 | 232 | C1.w[0] = C1.w[0] + midpoint64[ind - 1]; |
200359e8 | 233 | } else { |
b2a00c89 L |
234 | C1.w[0] = C1.w[0] + midpoint128[ind - 20].w[0]; |
235 | C1.w[1] = C1.w[1] + midpoint128[ind - 20].w[1]; | |
200359e8 L |
236 | } |
237 | if (C1.w[0] < tmp64) | |
238 | C1.w[1]++; | |
239 | // calculate C* and f* | |
240 | // C* is actually floor(C*) in this case | |
241 | // C* and f* need shifting and masking, as shown by | |
b2a00c89 | 242 | // shiftright128[] and maskhigh128[] |
200359e8 | 243 | // 1 <= x <= 34 |
b2a00c89 | 244 | // kx = 10^(-x) = ten2mk128[ind - 1] |
200359e8 L |
245 | // C* = (C1 + 1/2 * 10^x) * 10^(-x) |
246 | // the approximation of 10^(-x) was rounded up to 118 bits | |
b2a00c89 | 247 | __mul_128x128_to_256 (P256, C1, ten2mk128[ind - 1]); |
200359e8 | 248 | // determine the value of res and fstar |
b2a00c89 L |
249 | |
250 | // determine inexactness of the rounding of C* | |
251 | // if (0 < f* - 1/2 < 10^(-x)) then | |
252 | // the result is exact | |
253 | // else // if (f* - 1/2 > T*) then | |
254 | // the result is inexact | |
255 | // Note: we are going to use ten2mk128[] instead of ten2mk128trunc[] | |
256 | ||
200359e8 | 257 | if (ind - 1 <= 2) { // 0 <= ind - 1 <= 2 => shift = 0 |
b2a00c89 | 258 | // redundant shift = shiftright128[ind - 1]; // shift = 0 |
200359e8 L |
259 | res.w[1] = P256.w[3]; |
260 | res.w[0] = P256.w[2]; | |
261 | // redundant fstar.w[3] = 0; | |
262 | // redundant fstar.w[2] = 0; | |
b2a00c89 L |
263 | fstar.w[1] = P256.w[1]; |
264 | fstar.w[0] = P256.w[0]; | |
200359e8 L |
265 | // fraction f* < 10^(-x) <=> midpoint |
266 | // f* is in the right position to be compared with | |
b2a00c89 | 267 | // 10^(-x) from ten2mk128[] |
200359e8 L |
268 | // if 0 < fstar < 10^(-x), subtract 1 if odd (for rounding to even) |
269 | if ((res.w[0] & 0x0000000000000001ull) && // is result odd? | |
b2a00c89 L |
270 | ((fstar.w[1] < (ten2mk128[ind - 1].w[1])) |
271 | || ((fstar.w[1] == ten2mk128[ind - 1].w[1]) | |
272 | && (fstar.w[0] < ten2mk128[ind - 1].w[0])))) { | |
200359e8 L |
273 | // subract 1 to make even |
274 | if (res.w[0]-- == 0) { | |
275 | res.w[1]--; | |
276 | } | |
277 | } | |
b2a00c89 L |
278 | if (fstar.w[1] > 0x8000000000000000ull || |
279 | (fstar.w[1] == 0x8000000000000000ull | |
280 | && fstar.w[0] > 0x0ull)) { | |
281 | // f* > 1/2 and the result may be exact | |
282 | tmp64 = fstar.w[1] - 0x8000000000000000ull; // f* - 1/2 | |
283 | if (tmp64 > ten2mk128[ind - 1].w[1] || | |
284 | (tmp64 == ten2mk128[ind - 1].w[1] && | |
285 | fstar.w[0] >= ten2mk128[ind - 1].w[0])) { | |
286 | // set the inexact flag | |
287 | *pfpsf |= INEXACT_EXCEPTION; | |
288 | } // else the result is exact | |
289 | } else { // the result is inexact; f2* <= 1/2 | |
290 | // set the inexact flag | |
291 | *pfpsf |= INEXACT_EXCEPTION; | |
292 | } | |
200359e8 | 293 | } else if (ind - 1 <= 21) { // 3 <= ind - 1 <= 21 => 3 <= shift <= 63 |
b2a00c89 | 294 | shift = shiftright128[ind - 1]; // 3 <= shift <= 63 |
200359e8 L |
295 | res.w[1] = (P256.w[3] >> shift); |
296 | res.w[0] = (P256.w[3] << (64 - shift)) | (P256.w[2] >> shift); | |
297 | // redundant fstar.w[3] = 0; | |
b2a00c89 | 298 | fstar.w[2] = P256.w[2] & maskhigh128[ind - 1]; |
200359e8 L |
299 | fstar.w[1] = P256.w[1]; |
300 | fstar.w[0] = P256.w[0]; | |
301 | // fraction f* < 10^(-x) <=> midpoint | |
302 | // f* is in the right position to be compared with | |
b2a00c89 | 303 | // 10^(-x) from ten2mk128[] |
200359e8 | 304 | if ((res.w[0] & 0x0000000000000001ull) && // is result odd? |
b2a00c89 L |
305 | fstar.w[2] == 0 && (fstar.w[1] < ten2mk128[ind - 1].w[1] || |
306 | (fstar.w[1] == ten2mk128[ind - 1].w[1] && | |
307 | fstar.w[0] < ten2mk128[ind - 1].w[0]))) { | |
200359e8 L |
308 | // subract 1 to make even |
309 | if (res.w[0]-- == 0) { | |
310 | res.w[1]--; | |
311 | } | |
312 | } | |
b2a00c89 L |
313 | if (fstar.w[2] > onehalf128[ind - 1] || |
314 | (fstar.w[2] == onehalf128[ind - 1] | |
315 | && (fstar.w[1] || fstar.w[0]))) { | |
316 | // f2* > 1/2 and the result may be exact | |
317 | // Calculate f2* - 1/2 | |
318 | tmp64 = fstar.w[2] - onehalf128[ind - 1]; | |
319 | if (tmp64 || fstar.w[1] > ten2mk128[ind - 1].w[1] || | |
320 | (fstar.w[1] == ten2mk128[ind - 1].w[1] && | |
321 | fstar.w[0] >= ten2mk128[ind - 1].w[0])) { | |
322 | // set the inexact flag | |
323 | *pfpsf |= INEXACT_EXCEPTION; | |
324 | } // else the result is exact | |
325 | } else { // the result is inexact; f2* <= 1/2 | |
326 | // set the inexact flag | |
327 | *pfpsf |= INEXACT_EXCEPTION; | |
328 | } | |
200359e8 | 329 | } else { // 22 <= ind - 1 <= 33 |
b2a00c89 | 330 | shift = shiftright128[ind - 1] - 64; // 2 <= shift <= 38 |
200359e8 L |
331 | res.w[1] = 0; |
332 | res.w[0] = P256.w[3] >> shift; | |
b2a00c89 | 333 | fstar.w[3] = P256.w[3] & maskhigh128[ind - 1]; |
200359e8 L |
334 | fstar.w[2] = P256.w[2]; |
335 | fstar.w[1] = P256.w[1]; | |
336 | fstar.w[0] = P256.w[0]; | |
337 | // fraction f* < 10^(-x) <=> midpoint | |
338 | // f* is in the right position to be compared with | |
b2a00c89 | 339 | // 10^(-x) from ten2mk128[] |
200359e8 | 340 | if ((res.w[0] & 0x0000000000000001ull) && // is result odd? |
b2a00c89 L |
341 | fstar.w[3] == 0 && fstar.w[2] == 0 && |
342 | (fstar.w[1] < ten2mk128[ind - 1].w[1] || | |
343 | (fstar.w[1] == ten2mk128[ind - 1].w[1] && | |
344 | fstar.w[0] < ten2mk128[ind - 1].w[0]))) { | |
200359e8 L |
345 | // subract 1 to make even |
346 | if (res.w[0]-- == 0) { | |
347 | res.w[1]--; | |
348 | } | |
349 | } | |
b2a00c89 L |
350 | if (fstar.w[3] > onehalf128[ind - 1] || |
351 | (fstar.w[3] == onehalf128[ind - 1] && | |
352 | (fstar.w[2] || fstar.w[1] || fstar.w[0]))) { | |
353 | // f2* > 1/2 and the result may be exact | |
354 | // Calculate f2* - 1/2 | |
355 | tmp64 = fstar.w[3] - onehalf128[ind - 1]; | |
356 | if (tmp64 || fstar.w[2] || fstar.w[1] > ten2mk128[ind - 1].w[1] | |
357 | || (fstar.w[1] == ten2mk128[ind - 1].w[1] | |
358 | && fstar.w[0] >= ten2mk128[ind - 1].w[0])) { | |
359 | // set the inexact flag | |
360 | *pfpsf |= INEXACT_EXCEPTION; | |
361 | } // else the result is exact | |
362 | } else { // the result is inexact; f2* <= 1/2 | |
363 | // set the inexact flag | |
364 | *pfpsf |= INEXACT_EXCEPTION; | |
365 | } | |
200359e8 L |
366 | } |
367 | res.w[1] = x_sign | 0x3040000000000000ull | res.w[1]; | |
368 | BID_RETURN (res); | |
369 | } else { // if ((q + exp) < 0) <=> q < -exp | |
370 | // the result is +0 or -0 | |
371 | res.w[1] = x_sign | 0x3040000000000000ull; | |
372 | res.w[0] = 0x0000000000000000ull; | |
b2a00c89 | 373 | *pfpsf |= INEXACT_EXCEPTION; |
200359e8 L |
374 | BID_RETURN (res); |
375 | } | |
b2a00c89 L |
376 | break; |
377 | case ROUNDING_TIES_AWAY: | |
378 | if ((q + exp) >= 0) { // exp < 0 and 1 <= -exp <= q | |
379 | // need to shift right -exp digits from the coefficient; exp will be 0 | |
380 | ind = -exp; // 1 <= ind <= 34; ind is a synonym for 'x' | |
381 | // chop off ind digits from the lower part of C1 | |
382 | // C1 = C1 + 1/2 * 10^x where the result C1 fits in 127 bits | |
383 | tmp64 = C1.w[0]; | |
384 | if (ind <= 19) { | |
385 | C1.w[0] = C1.w[0] + midpoint64[ind - 1]; | |
386 | } else { | |
387 | C1.w[0] = C1.w[0] + midpoint128[ind - 20].w[0]; | |
388 | C1.w[1] = C1.w[1] + midpoint128[ind - 20].w[1]; | |
389 | } | |
390 | if (C1.w[0] < tmp64) | |
391 | C1.w[1]++; | |
392 | // calculate C* and f* | |
393 | // C* is actually floor(C*) in this case | |
394 | // C* and f* need shifting and masking, as shown by | |
395 | // shiftright128[] and maskhigh128[] | |
396 | // 1 <= x <= 34 | |
397 | // kx = 10^(-x) = ten2mk128[ind - 1] | |
398 | // C* = (C1 + 1/2 * 10^x) * 10^(-x) | |
399 | // the approximation of 10^(-x) was rounded up to 118 bits | |
400 | __mul_128x128_to_256 (P256, C1, ten2mk128[ind - 1]); | |
401 | // the top Ex bits of 10^(-x) are T* = ten2mk128trunc[ind], e.g. | |
402 | // if x=1, T*=ten2mk128trunc[0]=0x19999999999999999999999999999999 | |
403 | // if (0 < f* < 10^(-x)) then the result is a midpoint | |
404 | // if floor(C*) is even then C* = floor(C*) - logical right | |
405 | // shift; C* has p decimal digits, correct by Prop. 1) | |
406 | // else if floor(C*) is odd C* = floor(C*)-1 (logical right | |
407 | // shift; C* has p decimal digits, correct by Pr. 1) | |
408 | // else | |
409 | // C* = floor(C*) (logical right shift; C has p decimal digits, | |
410 | // correct by Property 1) | |
411 | // n = C* * 10^(e+x) | |
200359e8 | 412 | |
b2a00c89 L |
413 | // determine also the inexactness of the rounding of C* |
414 | // if (0 < f* - 1/2 < 10^(-x)) then | |
415 | // the result is exact | |
416 | // else // if (f* - 1/2 > T*) then | |
417 | // the result is inexact | |
418 | // Note: we are going to use ten2mk128[] instead of ten2mk128trunc[] | |
419 | // shift right C* by Ex-128 = shiftright128[ind] | |
420 | if (ind - 1 <= 2) { // 0 <= ind - 1 <= 2 => shift = 0 | |
421 | // redundant shift = shiftright128[ind - 1]; // shift = 0 | |
422 | res.w[1] = P256.w[3]; | |
423 | res.w[0] = P256.w[2]; | |
424 | // redundant fstar.w[3] = 0; | |
425 | // redundant fstar.w[2] = 0; | |
426 | fstar.w[1] = P256.w[1]; | |
427 | fstar.w[0] = P256.w[0]; | |
428 | if (fstar.w[1] > 0x8000000000000000ull || | |
429 | (fstar.w[1] == 0x8000000000000000ull | |
430 | && fstar.w[0] > 0x0ull)) { | |
431 | // f* > 1/2 and the result may be exact | |
432 | tmp64 = fstar.w[1] - 0x8000000000000000ull; // f* - 1/2 | |
433 | if ((tmp64 > ten2mk128[ind - 1].w[1] || | |
434 | (tmp64 == ten2mk128[ind - 1].w[1] && | |
435 | fstar.w[0] >= ten2mk128[ind - 1].w[0]))) { | |
436 | // set the inexact flag | |
437 | *pfpsf |= INEXACT_EXCEPTION; | |
438 | } // else the result is exact | |
439 | } else { // the result is inexact; f2* <= 1/2 | |
440 | // set the inexact flag | |
441 | *pfpsf |= INEXACT_EXCEPTION; | |
442 | } | |
443 | } else if (ind - 1 <= 21) { // 3 <= ind - 1 <= 21 => 3 <= shift <= 63 | |
444 | shift = shiftright128[ind - 1]; // 3 <= shift <= 63 | |
445 | res.w[1] = (P256.w[3] >> shift); | |
446 | res.w[0] = (P256.w[3] << (64 - shift)) | (P256.w[2] >> shift); | |
447 | // redundant fstar.w[3] = 0; | |
448 | fstar.w[2] = P256.w[2] & maskhigh128[ind - 1]; | |
449 | fstar.w[1] = P256.w[1]; | |
450 | fstar.w[0] = P256.w[0]; | |
451 | if (fstar.w[2] > onehalf128[ind - 1] || | |
452 | (fstar.w[2] == onehalf128[ind - 1] | |
453 | && (fstar.w[1] || fstar.w[0]))) { | |
454 | // f2* > 1/2 and the result may be exact | |
455 | // Calculate f2* - 1/2 | |
456 | tmp64 = fstar.w[2] - onehalf128[ind - 1]; | |
457 | if (tmp64 || fstar.w[1] > ten2mk128[ind - 1].w[1] || | |
458 | (fstar.w[1] == ten2mk128[ind - 1].w[1] && | |
459 | fstar.w[0] >= ten2mk128[ind - 1].w[0])) { | |
460 | // set the inexact flag | |
461 | *pfpsf |= INEXACT_EXCEPTION; | |
462 | } // else the result is exact | |
463 | } else { // the result is inexact; f2* <= 1/2 | |
464 | // set the inexact flag | |
465 | *pfpsf |= INEXACT_EXCEPTION; | |
200359e8 | 466 | } |
b2a00c89 L |
467 | } else { // 22 <= ind - 1 <= 33 |
468 | shift = shiftright128[ind - 1] - 64; // 2 <= shift <= 38 | |
469 | res.w[1] = 0; | |
470 | res.w[0] = P256.w[3] >> shift; | |
471 | fstar.w[3] = P256.w[3] & maskhigh128[ind - 1]; | |
472 | fstar.w[2] = P256.w[2]; | |
473 | fstar.w[1] = P256.w[1]; | |
474 | fstar.w[0] = P256.w[0]; | |
475 | if (fstar.w[3] > onehalf128[ind - 1] || | |
476 | (fstar.w[3] == onehalf128[ind - 1] && | |
477 | (fstar.w[2] || fstar.w[1] || fstar.w[0]))) { | |
478 | // f2* > 1/2 and the result may be exact | |
479 | // Calculate f2* - 1/2 | |
480 | tmp64 = fstar.w[3] - onehalf128[ind - 1]; | |
481 | if (tmp64 || fstar.w[2] || fstar.w[1] > ten2mk128[ind - 1].w[1] | |
482 | || (fstar.w[1] == ten2mk128[ind - 1].w[1] | |
483 | && fstar.w[0] >= ten2mk128[ind - 1].w[0])) { | |
484 | // set the inexact flag | |
485 | *pfpsf |= INEXACT_EXCEPTION; | |
486 | } // else the result is exact | |
487 | } else { // the result is inexact; f2* <= 1/2 | |
488 | // set the inexact flag | |
489 | *pfpsf |= INEXACT_EXCEPTION; | |
200359e8 | 490 | } |
200359e8 | 491 | } |
b2a00c89 L |
492 | // if the result was a midpoint, it was already rounded away from zero |
493 | res.w[1] |= x_sign | 0x3040000000000000ull; | |
200359e8 | 494 | BID_RETURN (res); |
b2a00c89 L |
495 | } else { // if ((q + exp) < 0) <=> q < -exp |
496 | // the result is +0 or -0 | |
497 | res.w[1] = x_sign | 0x3040000000000000ull; | |
498 | res.w[0] = 0x0000000000000000ull; | |
499 | *pfpsf |= INEXACT_EXCEPTION; | |
200359e8 L |
500 | BID_RETURN (res); |
501 | } | |
b2a00c89 L |
502 | break; |
503 | case ROUNDING_DOWN: | |
504 | if ((q + exp) > 0) { // exp < 0 and 1 <= -exp < q | |
505 | // need to shift right -exp digits from the coefficient; exp will be 0 | |
200359e8 L |
506 | ind = -exp; // 1 <= ind <= 34; ind is a synonym for 'x' |
507 | // (number of digits to be chopped off) | |
508 | // chop off ind digits from the lower part of C1 | |
509 | // FOR ROUND_TO_NEAREST, WE ADD 1/2 ULP(y) then truncate | |
510 | // FOR ROUND_TO_ZERO, WE DON'T NEED TO ADD 1/2 ULP | |
511 | // FOR ROUND_TO_POSITIVE_INFINITY, WE TRUNCATE, THEN ADD 1 IF POSITIVE | |
512 | // FOR ROUND_TO_NEGATIVE_INFINITY, WE TRUNCATE, THEN ADD 1 IF NEGATIVE | |
b2a00c89 | 513 | // tmp64 = C1.w[0]; |
200359e8 | 514 | // if (ind <= 19) { |
b2a00c89 | 515 | // C1.w[0] = C1.w[0] + midpoint64[ind - 1]; |
200359e8 | 516 | // } else { |
b2a00c89 L |
517 | // C1.w[0] = C1.w[0] + midpoint128[ind - 20].w[0]; |
518 | // C1.w[1] = C1.w[1] + midpoint128[ind - 20].w[1]; | |
200359e8 L |
519 | // } |
520 | // if (C1.w[0] < tmp64) C1.w[1]++; | |
521 | // if carry-out from C1.w[0], increment C1.w[1] | |
522 | // calculate C* and f* | |
523 | // C* is actually floor(C*) in this case | |
524 | // C* and f* need shifting and masking, as shown by | |
b2a00c89 | 525 | // shiftright128[] and maskhigh128[] |
200359e8 | 526 | // 1 <= x <= 34 |
b2a00c89 | 527 | // kx = 10^(-x) = ten2mk128[ind - 1] |
200359e8 L |
528 | // C* = (C1 + 1/2 * 10^x) * 10^(-x) |
529 | // the approximation of 10^(-x) was rounded up to 118 bits | |
b2a00c89 | 530 | __mul_128x128_to_256 (P256, C1, ten2mk128[ind - 1]); |
200359e8 L |
531 | if (ind - 1 <= 2) { // 0 <= ind - 1 <= 2 => shift = 0 |
532 | res.w[1] = P256.w[3]; | |
533 | res.w[0] = P256.w[2]; | |
b2a00c89 L |
534 | // redundant fstar.w[3] = 0; |
535 | // redundant fstar.w[2] = 0; | |
536 | // redundant fstar.w[1] = P256.w[1]; | |
537 | // redundant fstar.w[0] = P256.w[0]; | |
538 | // fraction f* > 10^(-x) <=> inexact | |
539 | // f* is in the right position to be compared with | |
540 | // 10^(-x) from ten2mk128[] | |
541 | if ((P256.w[1] > ten2mk128[ind - 1].w[1]) | |
542 | || (P256.w[1] == ten2mk128[ind - 1].w[1] | |
543 | && (P256.w[0] >= ten2mk128[ind - 1].w[0]))) { | |
544 | *pfpsf |= INEXACT_EXCEPTION; | |
545 | // if positive, the truncated value is already the correct result | |
546 | if (x_sign) { // if negative | |
200359e8 L |
547 | if (++res.w[0] == 0) { |
548 | res.w[1]++; | |
549 | } | |
550 | } | |
551 | } | |
552 | } else if (ind - 1 <= 21) { // 3 <= ind - 1 <= 21 => 3 <= shift <= 63 | |
b2a00c89 | 553 | shift = shiftright128[ind - 1]; // 0 <= shift <= 102 |
200359e8 L |
554 | res.w[1] = (P256.w[3] >> shift); |
555 | res.w[0] = (P256.w[3] << (64 - shift)) | (P256.w[2] >> shift); | |
b2a00c89 L |
556 | // redundant fstar.w[3] = 0; |
557 | fstar.w[2] = P256.w[2] & maskhigh128[ind - 1]; | |
558 | fstar.w[1] = P256.w[1]; | |
559 | fstar.w[0] = P256.w[0]; | |
560 | // fraction f* > 10^(-x) <=> inexact | |
561 | // f* is in the right position to be compared with | |
562 | // 10^(-x) from ten2mk128[] | |
563 | if (fstar.w[2] || fstar.w[1] > ten2mk128[ind - 1].w[1] || | |
564 | (fstar.w[1] == ten2mk128[ind - 1].w[1] && | |
565 | fstar.w[0] >= ten2mk128[ind - 1].w[0])) { | |
566 | *pfpsf |= INEXACT_EXCEPTION; | |
567 | // if positive, the truncated value is already the correct result | |
568 | if (x_sign) { // if negative | |
200359e8 L |
569 | if (++res.w[0] == 0) { |
570 | res.w[1]++; | |
571 | } | |
572 | } | |
573 | } | |
574 | } else { // 22 <= ind - 1 <= 33 | |
b2a00c89 | 575 | shift = shiftright128[ind - 1] - 64; // 2 <= shift <= 38 |
200359e8 L |
576 | res.w[1] = 0; |
577 | res.w[0] = P256.w[3] >> shift; | |
b2a00c89 L |
578 | fstar.w[3] = P256.w[3] & maskhigh128[ind - 1]; |
579 | fstar.w[2] = P256.w[2]; | |
580 | fstar.w[1] = P256.w[1]; | |
581 | fstar.w[0] = P256.w[0]; | |
582 | // fraction f* > 10^(-x) <=> inexact | |
583 | // f* is in the right position to be compared with | |
584 | // 10^(-x) from ten2mk128[] | |
585 | if (fstar.w[3] || fstar.w[2] | |
586 | || fstar.w[1] > ten2mk128[ind - 1].w[1] | |
587 | || (fstar.w[1] == ten2mk128[ind - 1].w[1] | |
588 | && fstar.w[0] >= ten2mk128[ind - 1].w[0])) { | |
589 | *pfpsf |= INEXACT_EXCEPTION; | |
590 | // if positive, the truncated value is already the correct result | |
591 | if (x_sign) { // if negative | |
200359e8 L |
592 | if (++res.w[0] == 0) { |
593 | res.w[1]++; | |
594 | } | |
595 | } | |
596 | } | |
597 | } | |
598 | res.w[1] = x_sign | 0x3040000000000000ull | res.w[1]; | |
599 | BID_RETURN (res); | |
600 | } else { // if exp < 0 and q + exp <= 0 | |
601 | if (x_sign) { // negative rounds down to -1.0 | |
602 | res.w[1] = 0xb040000000000000ull; | |
603 | res.w[0] = 0x0000000000000001ull; | |
604 | } else { // positive rpunds down to +0.0 | |
605 | res.w[1] = 0x3040000000000000ull; | |
606 | res.w[0] = 0x0000000000000000ull; | |
607 | } | |
b2a00c89 | 608 | *pfpsf |= INEXACT_EXCEPTION; |
200359e8 L |
609 | BID_RETURN (res); |
610 | } | |
b2a00c89 L |
611 | break; |
612 | case ROUNDING_UP: | |
613 | if ((q + exp) > 0) { // exp < 0 and 1 <= -exp < q | |
200359e8 L |
614 | // need to shift right -exp digits from the coefficient; exp will be 0 |
615 | ind = -exp; // 1 <= ind <= 34; ind is a synonym for 'x' | |
616 | // (number of digits to be chopped off) | |
617 | // chop off ind digits from the lower part of C1 | |
618 | // FOR ROUND_TO_NEAREST, WE ADD 1/2 ULP(y) then truncate | |
619 | // FOR ROUND_TO_ZERO, WE DON'T NEED TO ADD 1/2 ULP | |
620 | // FOR ROUND_TO_POSITIVE_INFINITY, WE TRUNCATE, THEN ADD 1 IF POSITIVE | |
621 | // FOR ROUND_TO_NEGATIVE_INFINITY, WE TRUNCATE, THEN ADD 1 IF NEGATIVE | |
622 | // tmp64 = C1.w[0]; | |
623 | // if (ind <= 19) { | |
b2a00c89 | 624 | // C1.w[0] = C1.w[0] + midpoint64[ind - 1]; |
200359e8 | 625 | // } else { |
b2a00c89 L |
626 | // C1.w[0] = C1.w[0] + midpoint128[ind - 20].w[0]; |
627 | // C1.w[1] = C1.w[1] + midpoint128[ind - 20].w[1]; | |
200359e8 L |
628 | // } |
629 | // if (C1.w[0] < tmp64) C1.w[1]++; | |
630 | // if carry-out from C1.w[0], increment C1.w[1] | |
631 | // calculate C* and f* | |
632 | // C* is actually floor(C*) in this case | |
633 | // C* and f* need shifting and masking, as shown by | |
b2a00c89 | 634 | // shiftright128[] and maskhigh128[] |
200359e8 | 635 | // 1 <= x <= 34 |
b2a00c89 | 636 | // kx = 10^(-x) = ten2mk128[ind - 1] |
200359e8 L |
637 | // C* = C1 * 10^(-x) |
638 | // the approximation of 10^(-x) was rounded up to 118 bits | |
b2a00c89 | 639 | __mul_128x128_to_256 (P256, C1, ten2mk128[ind - 1]); |
200359e8 L |
640 | if (ind - 1 <= 2) { // 0 <= ind - 1 <= 2 => shift = 0 |
641 | res.w[1] = P256.w[3]; | |
642 | res.w[0] = P256.w[2]; | |
b2a00c89 L |
643 | // redundant fstar.w[3] = 0; |
644 | // redundant fstar.w[2] = 0; | |
645 | // redundant fstar.w[1] = P256.w[1]; | |
646 | // redundant fstar.w[0] = P256.w[0]; | |
647 | // fraction f* > 10^(-x) <=> inexact | |
648 | // f* is in the right position to be compared with | |
649 | // 10^(-x) from ten2mk128[] | |
650 | if ((P256.w[1] > ten2mk128[ind - 1].w[1]) | |
651 | || (P256.w[1] == ten2mk128[ind - 1].w[1] | |
652 | && (P256.w[0] >= ten2mk128[ind - 1].w[0]))) { | |
653 | *pfpsf |= INEXACT_EXCEPTION; | |
654 | // if negative, the truncated value is already the correct result | |
655 | if (!x_sign) { // if positive | |
200359e8 L |
656 | if (++res.w[0] == 0) { |
657 | res.w[1]++; | |
658 | } | |
659 | } | |
660 | } | |
661 | } else if (ind - 1 <= 21) { // 3 <= ind - 1 <= 21 => 3 <= shift <= 63 | |
b2a00c89 | 662 | shift = shiftright128[ind - 1]; // 3 <= shift <= 63 |
200359e8 L |
663 | res.w[1] = (P256.w[3] >> shift); |
664 | res.w[0] = (P256.w[3] << (64 - shift)) | (P256.w[2] >> shift); | |
b2a00c89 L |
665 | // redundant fstar.w[3] = 0; |
666 | fstar.w[2] = P256.w[2] & maskhigh128[ind - 1]; | |
667 | fstar.w[1] = P256.w[1]; | |
668 | fstar.w[0] = P256.w[0]; | |
669 | // fraction f* > 10^(-x) <=> inexact | |
670 | // f* is in the right position to be compared with | |
671 | // 10^(-x) from ten2mk128[] | |
672 | if (fstar.w[2] || fstar.w[1] > ten2mk128[ind - 1].w[1] || | |
673 | (fstar.w[1] == ten2mk128[ind - 1].w[1] && | |
674 | fstar.w[0] >= ten2mk128[ind - 1].w[0])) { | |
675 | *pfpsf |= INEXACT_EXCEPTION; | |
676 | // if negative, the truncated value is already the correct result | |
677 | if (!x_sign) { // if positive | |
200359e8 L |
678 | if (++res.w[0] == 0) { |
679 | res.w[1]++; | |
680 | } | |
681 | } | |
682 | } | |
683 | } else { // 22 <= ind - 1 <= 33 | |
b2a00c89 | 684 | shift = shiftright128[ind - 1] - 64; // 2 <= shift <= 38 |
200359e8 L |
685 | res.w[1] = 0; |
686 | res.w[0] = P256.w[3] >> shift; | |
b2a00c89 L |
687 | fstar.w[3] = P256.w[3] & maskhigh128[ind - 1]; |
688 | fstar.w[2] = P256.w[2]; | |
689 | fstar.w[1] = P256.w[1]; | |
690 | fstar.w[0] = P256.w[0]; | |
691 | // fraction f* > 10^(-x) <=> inexact | |
692 | // f* is in the right position to be compared with | |
693 | // 10^(-x) from ten2mk128[] | |
694 | if (fstar.w[3] || fstar.w[2] | |
695 | || fstar.w[1] > ten2mk128[ind - 1].w[1] | |
696 | || (fstar.w[1] == ten2mk128[ind - 1].w[1] | |
697 | && fstar.w[0] >= ten2mk128[ind - 1].w[0])) { | |
698 | *pfpsf |= INEXACT_EXCEPTION; | |
699 | // if negative, the truncated value is already the correct result | |
700 | if (!x_sign) { // if positive | |
200359e8 L |
701 | if (++res.w[0] == 0) { |
702 | res.w[1]++; | |
703 | } | |
704 | } | |
705 | } | |
706 | } | |
707 | res.w[1] = x_sign | 0x3040000000000000ull | res.w[1]; | |
708 | BID_RETURN (res); | |
709 | } else { // if exp < 0 and q + exp <= 0 | |
710 | if (x_sign) { // negative rounds up to -0.0 | |
711 | res.w[1] = 0xb040000000000000ull; | |
712 | res.w[0] = 0x0000000000000000ull; | |
713 | } else { // positive rpunds up to +1.0 | |
714 | res.w[1] = 0x3040000000000000ull; | |
715 | res.w[0] = 0x0000000000000001ull; | |
716 | } | |
b2a00c89 | 717 | *pfpsf |= INEXACT_EXCEPTION; |
200359e8 L |
718 | BID_RETURN (res); |
719 | } | |
b2a00c89 L |
720 | break; |
721 | case ROUNDING_TO_ZERO: | |
722 | if ((q + exp) > 0) { // exp < 0 and 1 <= -exp < q | |
723 | // need to shift right -exp digits from the coefficient; exp will be 0 | |
200359e8 L |
724 | ind = -exp; // 1 <= ind <= 34; ind is a synonym for 'x' |
725 | // (number of digits to be chopped off) | |
726 | // chop off ind digits from the lower part of C1 | |
727 | // FOR ROUND_TO_NEAREST, WE ADD 1/2 ULP(y) then truncate | |
728 | // FOR ROUND_TO_ZERO, WE DON'T NEED TO ADD 1/2 ULP | |
729 | // FOR ROUND_TO_POSITIVE_INFINITY, WE TRUNCATE, THEN ADD 1 IF POSITIVE | |
730 | // FOR ROUND_TO_NEGATIVE_INFINITY, WE TRUNCATE, THEN ADD 1 IF NEGATIVE | |
731 | //tmp64 = C1.w[0]; | |
732 | // if (ind <= 19) { | |
b2a00c89 | 733 | // C1.w[0] = C1.w[0] + midpoint64[ind - 1]; |
200359e8 | 734 | // } else { |
b2a00c89 L |
735 | // C1.w[0] = C1.w[0] + midpoint128[ind - 20].w[0]; |
736 | // C1.w[1] = C1.w[1] + midpoint128[ind - 20].w[1]; | |
200359e8 L |
737 | // } |
738 | // if (C1.w[0] < tmp64) C1.w[1]++; | |
739 | // if carry-out from C1.w[0], increment C1.w[1] | |
740 | // calculate C* and f* | |
741 | // C* is actually floor(C*) in this case | |
742 | // C* and f* need shifting and masking, as shown by | |
b2a00c89 | 743 | // shiftright128[] and maskhigh128[] |
200359e8 | 744 | // 1 <= x <= 34 |
b2a00c89 | 745 | // kx = 10^(-x) = ten2mk128[ind - 1] |
200359e8 L |
746 | // C* = (C1 + 1/2 * 10^x) * 10^(-x) |
747 | // the approximation of 10^(-x) was rounded up to 118 bits | |
b2a00c89 | 748 | __mul_128x128_to_256 (P256, C1, ten2mk128[ind - 1]); |
200359e8 L |
749 | if (ind - 1 <= 2) { // 0 <= ind - 1 <= 2 => shift = 0 |
750 | res.w[1] = P256.w[3]; | |
751 | res.w[0] = P256.w[2]; | |
b2a00c89 L |
752 | // redundant fstar.w[3] = 0; |
753 | // redundant fstar.w[2] = 0; | |
754 | // redundant fstar.w[1] = P256.w[1]; | |
755 | // redundant fstar.w[0] = P256.w[0]; | |
756 | // fraction f* > 10^(-x) <=> inexact | |
757 | // f* is in the right position to be compared with | |
758 | // 10^(-x) from ten2mk128[] | |
759 | if ((P256.w[1] > ten2mk128[ind - 1].w[1]) | |
760 | || (P256.w[1] == ten2mk128[ind - 1].w[1] | |
761 | && (P256.w[0] >= ten2mk128[ind - 1].w[0]))) { | |
762 | *pfpsf |= INEXACT_EXCEPTION; | |
763 | } | |
200359e8 | 764 | } else if (ind - 1 <= 21) { // 3 <= ind - 1 <= 21 => 3 <= shift <= 63 |
b2a00c89 | 765 | shift = shiftright128[ind - 1]; // 3 <= shift <= 63 |
200359e8 L |
766 | res.w[1] = (P256.w[3] >> shift); |
767 | res.w[0] = (P256.w[3] << (64 - shift)) | (P256.w[2] >> shift); | |
b2a00c89 L |
768 | // redundant fstar.w[3] = 0; |
769 | fstar.w[2] = P256.w[2] & maskhigh128[ind - 1]; | |
770 | fstar.w[1] = P256.w[1]; | |
771 | fstar.w[0] = P256.w[0]; | |
772 | // fraction f* > 10^(-x) <=> inexact | |
773 | // f* is in the right position to be compared with | |
774 | // 10^(-x) from ten2mk128[] | |
775 | if (fstar.w[2] || fstar.w[1] > ten2mk128[ind - 1].w[1] || | |
776 | (fstar.w[1] == ten2mk128[ind - 1].w[1] && | |
777 | fstar.w[0] >= ten2mk128[ind - 1].w[0])) { | |
778 | *pfpsf |= INEXACT_EXCEPTION; | |
779 | } | |
200359e8 | 780 | } else { // 22 <= ind - 1 <= 33 |
b2a00c89 | 781 | shift = shiftright128[ind - 1] - 64; // 2 <= shift <= 38 |
200359e8 L |
782 | res.w[1] = 0; |
783 | res.w[0] = P256.w[3] >> shift; | |
b2a00c89 L |
784 | fstar.w[3] = P256.w[3] & maskhigh128[ind - 1]; |
785 | fstar.w[2] = P256.w[2]; | |
786 | fstar.w[1] = P256.w[1]; | |
787 | fstar.w[0] = P256.w[0]; | |
788 | // fraction f* > 10^(-x) <=> inexact | |
789 | // f* is in the right position to be compared with | |
790 | // 10^(-x) from ten2mk128[] | |
791 | if (fstar.w[3] || fstar.w[2] | |
792 | || fstar.w[1] > ten2mk128[ind - 1].w[1] | |
793 | || (fstar.w[1] == ten2mk128[ind - 1].w[1] | |
794 | && fstar.w[0] >= ten2mk128[ind - 1].w[0])) { | |
795 | *pfpsf |= INEXACT_EXCEPTION; | |
796 | } | |
200359e8 L |
797 | } |
798 | res.w[1] = x_sign | 0x3040000000000000ull | res.w[1]; | |
799 | BID_RETURN (res); | |
800 | } else { // if exp < 0 and q + exp <= 0 the result is +0 or -0 | |
801 | res.w[1] = x_sign | 0x3040000000000000ull; | |
802 | res.w[0] = 0x0000000000000000ull; | |
b2a00c89 | 803 | *pfpsf |= INEXACT_EXCEPTION; |
200359e8 L |
804 | BID_RETURN (res); |
805 | } | |
b2a00c89 L |
806 | break; |
807 | } | |
808 | ||
809 | BID_RETURN (res); | |
200359e8 L |
810 | } |
811 | ||
812 | /***************************************************************************** | |
b2a00c89 | 813 | * BID128_round_integral_nearest_even |
200359e8 L |
814 | ****************************************************************************/ |
815 | ||
b2a00c89 | 816 | BID128_FUNCTION_ARG1_NORND (bid128_round_integral_nearest_even, x) |
200359e8 | 817 | |
b2a00c89 L |
818 | UINT128 res; |
819 | UINT64 x_sign; | |
820 | UINT64 x_exp; | |
821 | int exp; // unbiased exponent | |
822 | // Note: C1.w[1], C1.w[0] represent x_signif_hi, x_signif_lo (all are UINT64) | |
823 | UINT64 tmp64; | |
824 | BID_UI64DOUBLE tmp1; | |
825 | unsigned int x_nr_bits; | |
826 | int q, ind, shift; | |
827 | UINT128 C1; | |
828 | // UINT128 res is C* at first - represents up to 34 decimal digits ~ 113 bits | |
829 | UINT256 fstar; | |
830 | UINT256 P256; | |
200359e8 L |
831 | |
832 | // check for NaN or Infinity | |
b2a00c89 | 833 | if ((x.w[1] & MASK_SPECIAL) == MASK_SPECIAL) { |
200359e8 | 834 | // x is special |
b2a00c89 L |
835 | if ((x.w[1] & MASK_NAN) == MASK_NAN) { // x is NAN |
836 | // if x = NaN, then res = Q (x) | |
837 | // check first for non-canonical NaN payload | |
838 | if (((x.w[1] & 0x00003fffffffffffull) > 0x0000314dc6448d93ull) || | |
839 | (((x.w[1] & 0x00003fffffffffffull) == 0x0000314dc6448d93ull) && | |
840 | (x.w[0] > 0x38c15b09ffffffffull))) { | |
841 | x.w[1] = x.w[1] & 0xffffc00000000000ull; | |
842 | x.w[0] = 0x0ull; | |
843 | } | |
844 | if ((x.w[1] & MASK_SNAN) == MASK_SNAN) { // x is SNAN | |
845 | // set invalid flag | |
846 | *pfpsf |= INVALID_EXCEPTION; | |
847 | // return quiet (x) | |
848 | res.w[1] = x.w[1] & 0xfc003fffffffffffull; // clear out also G[6]-G[16] | |
849 | res.w[0] = x.w[0]; | |
850 | } else { // x is QNaN | |
851 | // return x | |
852 | res.w[1] = x.w[1] & 0xfc003fffffffffffull; // clear out G[6]-G[16] | |
853 | res.w[0] = x.w[0]; | |
854 | } | |
855 | BID_RETURN (res) | |
856 | } else { // x is not a NaN, so it must be infinity | |
857 | if ((x.w[1] & MASK_SIGN) == 0x0ull) { // x is +inf | |
858 | // return +inf | |
859 | res.w[1] = 0x7800000000000000ull; | |
860 | res.w[0] = 0x0000000000000000ull; | |
861 | } else { // x is -inf | |
862 | // return -inf | |
863 | res.w[1] = 0xf800000000000000ull; | |
864 | res.w[0] = 0x0000000000000000ull; | |
200359e8 | 865 | } |
b2a00c89 L |
866 | BID_RETURN (res); |
867 | } | |
868 | } | |
200359e8 | 869 | // unpack x |
b2a00c89 L |
870 | x_sign = x.w[1] & MASK_SIGN; // 0 for positive, MASK_SIGN for negative |
871 | C1.w[1] = x.w[1] & MASK_COEFF; | |
872 | C1.w[0] = x.w[0]; | |
200359e8 | 873 | |
b2a00c89 L |
874 | // check for non-canonical values (treated as zero) |
875 | if ((x.w[1] & 0x6000000000000000ull) == 0x6000000000000000ull) { // G0_G1=11 | |
876 | // non-canonical | |
877 | x_exp = (x.w[1] << 2) & MASK_EXP; // biased and shifted left 49 bits | |
878 | C1.w[1] = 0; // significand high | |
879 | C1.w[0] = 0; // significand low | |
880 | } else { // G0_G1 != 11 | |
881 | x_exp = x.w[1] & MASK_EXP; // biased and shifted left 49 bits | |
882 | if (C1.w[1] > 0x0001ed09bead87c0ull || | |
883 | (C1.w[1] == 0x0001ed09bead87c0ull | |
884 | && C1.w[0] > 0x378d8e63ffffffffull)) { | |
885 | // x is non-canonical if coefficient is larger than 10^34 -1 | |
200359e8 L |
886 | C1.w[1] = 0; |
887 | C1.w[0] = 0; | |
b2a00c89 L |
888 | } else { // canonical |
889 | ; | |
200359e8 | 890 | } |
b2a00c89 L |
891 | } |
892 | ||
200359e8 | 893 | // test for input equal to zero |
b2a00c89 L |
894 | if ((C1.w[1] == 0x0ull) && (C1.w[0] == 0x0ull)) { |
895 | // x is 0 | |
896 | // return 0 preserving the sign bit and the preferred exponent | |
897 | // of MAX(Q(x), 0) | |
898 | if (x_exp <= (0x1820ull << 49)) { | |
899 | res.w[1] = (x.w[1] & 0x8000000000000000ull) | 0x3040000000000000ull; | |
900 | } else { | |
901 | res.w[1] = x_sign | x_exp; | |
200359e8 | 902 | } |
b2a00c89 L |
903 | res.w[0] = 0x0000000000000000ull; |
904 | BID_RETURN (res); | |
905 | } | |
200359e8 L |
906 | // x is not special and is not zero |
907 | ||
b2a00c89 L |
908 | // if (exp <= -(p+1)) return 0 |
909 | if (x_exp <= 0x2ffa000000000000ull) { // 0x2ffa000000000000ull == -35 | |
910 | res.w[1] = x_sign | 0x3040000000000000ull; | |
911 | res.w[0] = 0x0000000000000000ull; | |
912 | BID_RETURN (res); | |
913 | } | |
200359e8 L |
914 | // q = nr. of decimal digits in x |
915 | // determine first the nr. of bits in x | |
b2a00c89 L |
916 | if (C1.w[1] == 0) { |
917 | if (C1.w[0] >= 0x0020000000000000ull) { // x >= 2^53 | |
918 | // split the 64-bit value in two 32-bit halves to avoid rounding errors | |
919 | if (C1.w[0] >= 0x0000000100000000ull) { // x >= 2^32 | |
920 | tmp1.d = (double) (C1.w[0] >> 32); // exact conversion | |
921 | x_nr_bits = | |
922 | 33 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); | |
923 | } else { // x < 2^32 | |
924 | tmp1.d = (double) (C1.w[0]); // exact conversion | |
200359e8 L |
925 | x_nr_bits = |
926 | 1 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); | |
927 | } | |
b2a00c89 L |
928 | } else { // if x < 2^53 |
929 | tmp1.d = (double) C1.w[0]; // exact conversion | |
200359e8 | 930 | x_nr_bits = |
b2a00c89 | 931 | 1 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); |
200359e8 | 932 | } |
b2a00c89 L |
933 | } else { // C1.w[1] != 0 => nr. bits = 64 + nr_bits (C1.w[1]) |
934 | tmp1.d = (double) C1.w[1]; // exact conversion | |
935 | x_nr_bits = | |
936 | 65 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); | |
937 | } | |
938 | ||
939 | q = nr_digits[x_nr_bits - 1].digits; | |
940 | if (q == 0) { | |
941 | q = nr_digits[x_nr_bits - 1].digits1; | |
942 | if (C1.w[1] > nr_digits[x_nr_bits - 1].threshold_hi | |
943 | || (C1.w[1] == nr_digits[x_nr_bits - 1].threshold_hi && | |
944 | C1.w[0] >= nr_digits[x_nr_bits - 1].threshold_lo)) | |
945 | q++; | |
946 | } | |
947 | exp = (x_exp >> 49) - 6176; | |
948 | if (exp >= 0) { // -exp <= 0 | |
949 | // the argument is an integer already | |
950 | res.w[1] = x.w[1]; | |
951 | res.w[0] = x.w[0]; | |
952 | BID_RETURN (res); | |
953 | } else if ((q + exp) >= 0) { // exp < 0 and 1 <= -exp <= q | |
954 | // need to shift right -exp digits from the coefficient; the exp will be 0 | |
955 | ind = -exp; // 1 <= ind <= 34; ind is a synonym for 'x' | |
956 | // chop off ind digits from the lower part of C1 | |
957 | // C1 = C1 + 1/2 * 10^x where the result C1 fits in 127 bits | |
958 | tmp64 = C1.w[0]; | |
959 | if (ind <= 19) { | |
960 | C1.w[0] = C1.w[0] + midpoint64[ind - 1]; | |
961 | } else { | |
962 | C1.w[0] = C1.w[0] + midpoint128[ind - 20].w[0]; | |
963 | C1.w[1] = C1.w[1] + midpoint128[ind - 20].w[1]; | |
200359e8 | 964 | } |
b2a00c89 L |
965 | if (C1.w[0] < tmp64) |
966 | C1.w[1]++; | |
967 | // calculate C* and f* | |
968 | // C* is actually floor(C*) in this case | |
969 | // C* and f* need shifting and masking, as shown by | |
970 | // shiftright128[] and maskhigh128[] | |
971 | // 1 <= x <= 34 | |
972 | // kx = 10^(-x) = ten2mk128[ind - 1] | |
973 | // C* = (C1 + 1/2 * 10^x) * 10^(-x) | |
974 | // the approximation of 10^(-x) was rounded up to 118 bits | |
975 | __mul_128x128_to_256 (P256, C1, ten2mk128[ind - 1]); | |
976 | // determine the value of res and fstar | |
977 | if (ind - 1 <= 2) { // 0 <= ind - 1 <= 2 => shift = 0 | |
978 | // redundant shift = shiftright128[ind - 1]; // shift = 0 | |
979 | res.w[1] = P256.w[3]; | |
980 | res.w[0] = P256.w[2]; | |
981 | // redundant fstar.w[3] = 0; | |
982 | // redundant fstar.w[2] = 0; | |
983 | // redundant fstar.w[1] = P256.w[1]; | |
984 | // redundant fstar.w[0] = P256.w[0]; | |
985 | // fraction f* < 10^(-x) <=> midpoint | |
986 | // f* is in the right position to be compared with | |
987 | // 10^(-x) from ten2mk128[] | |
988 | // if 0 < fstar < 10^(-x), subtract 1 if odd (for rounding to even) | |
989 | if ((res.w[0] & 0x0000000000000001ull) && // is result odd? | |
990 | ((P256.w[1] < (ten2mk128[ind - 1].w[1])) | |
991 | || ((P256.w[1] == ten2mk128[ind - 1].w[1]) | |
992 | && (P256.w[0] < ten2mk128[ind - 1].w[0])))) { | |
993 | // subract 1 to make even | |
994 | if (res.w[0]-- == 0) { | |
995 | res.w[1]--; | |
996 | } | |
200359e8 | 997 | } |
b2a00c89 L |
998 | } else if (ind - 1 <= 21) { // 3 <= ind - 1 <= 21 => 3 <= shift <= 63 |
999 | shift = shiftright128[ind - 1]; // 3 <= shift <= 63 | |
1000 | res.w[1] = (P256.w[3] >> shift); | |
1001 | res.w[0] = (P256.w[3] << (64 - shift)) | (P256.w[2] >> shift); | |
1002 | // redundant fstar.w[3] = 0; | |
1003 | fstar.w[2] = P256.w[2] & maskhigh128[ind - 1]; | |
1004 | fstar.w[1] = P256.w[1]; | |
1005 | fstar.w[0] = P256.w[0]; | |
1006 | // fraction f* < 10^(-x) <=> midpoint | |
1007 | // f* is in the right position to be compared with | |
1008 | // 10^(-x) from ten2mk128[] | |
1009 | if ((res.w[0] & 0x0000000000000001ull) && // is result odd? | |
1010 | fstar.w[2] == 0 && (fstar.w[1] < ten2mk128[ind - 1].w[1] || | |
1011 | (fstar.w[1] == ten2mk128[ind - 1].w[1] && | |
1012 | fstar.w[0] < ten2mk128[ind - 1].w[0]))) { | |
1013 | // subract 1 to make even | |
1014 | if (res.w[0]-- == 0) { | |
1015 | res.w[1]--; | |
1016 | } | |
200359e8 | 1017 | } |
b2a00c89 L |
1018 | } else { // 22 <= ind - 1 <= 33 |
1019 | shift = shiftright128[ind - 1] - 64; // 2 <= shift <= 38 | |
1020 | res.w[1] = 0; | |
1021 | res.w[0] = P256.w[3] >> shift; | |
1022 | fstar.w[3] = P256.w[3] & maskhigh128[ind - 1]; | |
1023 | fstar.w[2] = P256.w[2]; | |
1024 | fstar.w[1] = P256.w[1]; | |
1025 | fstar.w[0] = P256.w[0]; | |
1026 | // fraction f* < 10^(-x) <=> midpoint | |
1027 | // f* is in the right position to be compared with | |
1028 | // 10^(-x) from ten2mk128[] | |
1029 | if ((res.w[0] & 0x0000000000000001ull) && // is result odd? | |
1030 | fstar.w[3] == 0 && fstar.w[2] == 0 | |
1031 | && (fstar.w[1] < ten2mk128[ind - 1].w[1] | |
1032 | || (fstar.w[1] == ten2mk128[ind - 1].w[1] | |
1033 | && fstar.w[0] < ten2mk128[ind - 1].w[0]))) { | |
1034 | // subract 1 to make even | |
1035 | if (res.w[0]-- == 0) { | |
1036 | res.w[1]--; | |
1037 | } | |
1038 | } | |
1039 | } | |
1040 | res.w[1] = x_sign | 0x3040000000000000ull | res.w[1]; | |
1041 | BID_RETURN (res); | |
1042 | } else { // if ((q + exp) < 0) <=> q < -exp | |
1043 | // the result is +0 or -0 | |
1044 | res.w[1] = x_sign | 0x3040000000000000ull; | |
1045 | res.w[0] = 0x0000000000000000ull; | |
1046 | BID_RETURN (res); | |
1047 | } | |
1048 | } | |
1049 | ||
1050 | /***************************************************************************** | |
1051 | * BID128_round_integral_negative | |
1052 | ****************************************************************************/ | |
1053 | ||
1054 | BID128_FUNCTION_ARG1_NORND (bid128_round_integral_negative, x) | |
1055 | ||
1056 | UINT128 res; | |
1057 | UINT64 x_sign; | |
1058 | UINT64 x_exp; | |
1059 | int exp; // unbiased exponent | |
1060 | // Note: C1.w[1], C1.w[0] represent x_signif_hi, x_signif_lo | |
1061 | // (all are UINT64) | |
1062 | BID_UI64DOUBLE tmp1; | |
1063 | unsigned int x_nr_bits; | |
1064 | int q, ind, shift; | |
1065 | UINT128 C1; | |
1066 | // UINT128 res is C* at first - represents up to 34 decimal digits ~ | |
1067 | // 113 bits | |
1068 | UINT256 fstar; | |
1069 | UINT256 P256; | |
1070 | ||
1071 | // check for NaN or Infinity | |
1072 | if ((x.w[1] & MASK_SPECIAL) == MASK_SPECIAL) { | |
1073 | // x is special | |
1074 | if ((x.w[1] & MASK_NAN) == MASK_NAN) { // x is NAN | |
1075 | // if x = NaN, then res = Q (x) | |
1076 | // check first for non-canonical NaN payload | |
1077 | if (((x.w[1] & 0x00003fffffffffffull) > 0x0000314dc6448d93ull) || | |
1078 | (((x.w[1] & 0x00003fffffffffffull) == 0x0000314dc6448d93ull) && | |
1079 | (x.w[0] > 0x38c15b09ffffffffull))) { | |
1080 | x.w[1] = x.w[1] & 0xffffc00000000000ull; | |
1081 | x.w[0] = 0x0ull; | |
1082 | } | |
1083 | if ((x.w[1] & MASK_SNAN) == MASK_SNAN) { // x is SNAN | |
1084 | // set invalid flag | |
1085 | *pfpsf |= INVALID_EXCEPTION; | |
1086 | // return quiet (x) | |
1087 | res.w[1] = x.w[1] & 0xfc003fffffffffffull; // clear out also G[6]-G[16] | |
1088 | res.w[0] = x.w[0]; | |
1089 | } else { // x is QNaN | |
1090 | // return x | |
1091 | res.w[1] = x.w[1] & 0xfc003fffffffffffull; // clear out G[6]-G[16] | |
1092 | res.w[0] = x.w[0]; | |
1093 | } | |
1094 | BID_RETURN (res) | |
1095 | } else { // x is not a NaN, so it must be infinity | |
1096 | if ((x.w[1] & MASK_SIGN) == 0x0ull) { // x is +inf | |
1097 | // return +inf | |
1098 | res.w[1] = 0x7800000000000000ull; | |
200359e8 | 1099 | res.w[0] = 0x0000000000000000ull; |
b2a00c89 L |
1100 | } else { // x is -inf |
1101 | // return -inf | |
1102 | res.w[1] = 0xf800000000000000ull; | |
1103 | res.w[0] = 0x0000000000000000ull; | |
1104 | } | |
1105 | BID_RETURN (res); | |
1106 | } | |
1107 | } | |
1108 | // unpack x | |
1109 | x_sign = x.w[1] & MASK_SIGN; // 0 for positive, MASK_SIGN for negative | |
1110 | C1.w[1] = x.w[1] & MASK_COEFF; | |
1111 | C1.w[0] = x.w[0]; | |
1112 | ||
1113 | // check for non-canonical values (treated as zero) | |
1114 | if ((x.w[1] & 0x6000000000000000ull) == 0x6000000000000000ull) { // G0_G1=11 | |
1115 | // non-canonical | |
1116 | x_exp = (x.w[1] << 2) & MASK_EXP; // biased and shifted left 49 bits | |
1117 | C1.w[1] = 0; // significand high | |
1118 | C1.w[0] = 0; // significand low | |
1119 | } else { // G0_G1 != 11 | |
1120 | x_exp = x.w[1] & MASK_EXP; // biased and shifted left 49 bits | |
1121 | if (C1.w[1] > 0x0001ed09bead87c0ull || | |
1122 | (C1.w[1] == 0x0001ed09bead87c0ull | |
1123 | && C1.w[0] > 0x378d8e63ffffffffull)) { | |
1124 | // x is non-canonical if coefficient is larger than 10^34 -1 | |
1125 | C1.w[1] = 0; | |
1126 | C1.w[0] = 0; | |
1127 | } else { // canonical | |
1128 | ; | |
1129 | } | |
1130 | } | |
1131 | ||
1132 | // test for input equal to zero | |
1133 | if ((C1.w[1] == 0x0ull) && (C1.w[0] == 0x0ull)) { | |
1134 | // x is 0 | |
1135 | // return 0 preserving the sign bit and the preferred exponent | |
1136 | // of MAX(Q(x), 0) | |
1137 | if (x_exp <= (0x1820ull << 49)) { | |
1138 | res.w[1] = (x.w[1] & 0x8000000000000000ull) | 0x3040000000000000ull; | |
1139 | } else { | |
1140 | res.w[1] = x_sign | x_exp; | |
1141 | } | |
1142 | res.w[0] = 0x0000000000000000ull; | |
1143 | BID_RETURN (res); | |
1144 | } | |
1145 | // x is not special and is not zero | |
1146 | ||
1147 | // if (exp <= -p) return -1.0 or +0.0 | |
1148 | if (x_exp <= 0x2ffc000000000000ull) { // 0x2ffc000000000000ull == -34 | |
1149 | if (x_sign) { | |
1150 | // if negative, return negative 1, because we know the coefficient | |
1151 | // is non-zero (would have been caught above) | |
1152 | res.w[1] = 0xb040000000000000ull; | |
1153 | res.w[0] = 0x0000000000000001ull; | |
1154 | } else { | |
1155 | // if positive, return positive 0, because we know coefficient is | |
1156 | // non-zero (would have been caught above) | |
1157 | res.w[1] = 0x3040000000000000ull; | |
1158 | res.w[0] = 0x0000000000000000ull; | |
1159 | } | |
1160 | BID_RETURN (res); | |
1161 | } | |
1162 | // q = nr. of decimal digits in x | |
1163 | // determine first the nr. of bits in x | |
1164 | if (C1.w[1] == 0) { | |
1165 | if (C1.w[0] >= 0x0020000000000000ull) { // x >= 2^53 | |
1166 | // split the 64-bit value in two 32-bit halves to avoid rounding errors | |
1167 | if (C1.w[0] >= 0x0000000100000000ull) { // x >= 2^32 | |
1168 | tmp1.d = (double) (C1.w[0] >> 32); // exact conversion | |
1169 | x_nr_bits = | |
1170 | 33 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); | |
1171 | } else { // x < 2^32 | |
1172 | tmp1.d = (double) (C1.w[0]); // exact conversion | |
1173 | x_nr_bits = | |
1174 | 1 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); | |
1175 | } | |
1176 | } else { // if x < 2^53 | |
1177 | tmp1.d = (double) C1.w[0]; // exact conversion | |
1178 | x_nr_bits = | |
1179 | 1 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); | |
1180 | } | |
1181 | } else { // C1.w[1] != 0 => nr. bits = 64 + nr_bits (C1.w[1]) | |
1182 | tmp1.d = (double) C1.w[1]; // exact conversion | |
1183 | x_nr_bits = | |
1184 | 65 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); | |
1185 | } | |
1186 | ||
1187 | q = nr_digits[x_nr_bits - 1].digits; | |
1188 | if (q == 0) { | |
1189 | q = nr_digits[x_nr_bits - 1].digits1; | |
1190 | if (C1.w[1] > nr_digits[x_nr_bits - 1].threshold_hi || | |
1191 | (C1.w[1] == nr_digits[x_nr_bits - 1].threshold_hi && | |
1192 | C1.w[0] >= nr_digits[x_nr_bits - 1].threshold_lo)) | |
1193 | q++; | |
1194 | } | |
1195 | exp = (x_exp >> 49) - 6176; | |
1196 | if (exp >= 0) { // -exp <= 0 | |
1197 | // the argument is an integer already | |
1198 | res.w[1] = x.w[1]; | |
1199 | res.w[0] = x.w[0]; | |
1200 | BID_RETURN (res); | |
1201 | } else if ((q + exp) > 0) { // exp < 0 and 1 <= -exp < q | |
1202 | // need to shift right -exp digits from the coefficient; the exp will be 0 | |
1203 | ind = -exp; // 1 <= ind <= 34; ind is a synonym for 'x' | |
1204 | // (number of digits to be chopped off) | |
1205 | // chop off ind digits from the lower part of C1 | |
1206 | // FOR ROUND_TO_NEAREST, WE ADD 1/2 ULP(y) then truncate | |
1207 | // FOR ROUND_TO_ZERO, WE DON'T NEED TO ADD 1/2 ULP | |
1208 | // FOR ROUND_TO_POSITIVE_INFINITY, WE TRUNCATE, THEN ADD 1 IF POSITIVE | |
1209 | // FOR ROUND_TO_NEGATIVE_INFINITY, WE TRUNCATE, THEN ADD 1 IF NEGATIVE | |
1210 | //tmp64 = C1.w[0]; | |
1211 | // if (ind <= 19) { | |
1212 | // C1.w[0] = C1.w[0] + midpoint64[ind - 1]; | |
1213 | // } else { | |
1214 | // C1.w[0] = C1.w[0] + midpoint128[ind - 20].w[0]; | |
1215 | // C1.w[1] = C1.w[1] + midpoint128[ind - 20].w[1]; | |
1216 | // } | |
1217 | // if (C1.w[0] < tmp64) C1.w[1]++; | |
1218 | // if carry-out from C1.w[0], increment C1.w[1] | |
1219 | // calculate C* and f* | |
1220 | // C* is actually floor(C*) in this case | |
1221 | // C* and f* need shifting and masking, as shown by | |
1222 | // shiftright128[] and maskhigh128[] | |
1223 | // 1 <= x <= 34 | |
1224 | // kx = 10^(-x) = ten2mk128[ind - 1] | |
1225 | // C* = (C1 + 1/2 * 10^x) * 10^(-x) | |
1226 | // the approximation of 10^(-x) was rounded up to 118 bits | |
1227 | __mul_128x128_to_256 (P256, C1, ten2mk128[ind - 1]); | |
1228 | if (ind - 1 <= 2) { // 0 <= ind - 1 <= 2 => shift = 0 | |
1229 | res.w[1] = P256.w[3]; | |
1230 | res.w[0] = P256.w[2]; | |
1231 | // if positive, the truncated value is already the correct result | |
1232 | if (x_sign) { // if negative | |
1233 | // redundant fstar.w[3] = 0; | |
1234 | // redundant fstar.w[2] = 0; | |
1235 | // redundant fstar.w[1] = P256.w[1]; | |
1236 | // redundant fstar.w[0] = P256.w[0]; | |
1237 | // fraction f* > 10^(-x) <=> inexact | |
1238 | // f* is in the right position to be compared with | |
1239 | // 10^(-x) from ten2mk128[] | |
1240 | if ((P256.w[1] > ten2mk128[ind - 1].w[1]) | |
1241 | || (P256.w[1] == ten2mk128[ind - 1].w[1] | |
1242 | && (P256.w[0] >= ten2mk128[ind - 1].w[0]))) { | |
1243 | if (++res.w[0] == 0) { | |
1244 | res.w[1]++; | |
1245 | } | |
1246 | } | |
1247 | } | |
1248 | } else if (ind - 1 <= 21) { // 3 <= ind - 1 <= 21 => 3 <= shift <= 63 | |
1249 | shift = shiftright128[ind - 1]; // 0 <= shift <= 102 | |
1250 | res.w[1] = (P256.w[3] >> shift); | |
1251 | res.w[0] = (P256.w[3] << (64 - shift)) | (P256.w[2] >> shift); | |
1252 | // if positive, the truncated value is already the correct result | |
1253 | if (x_sign) { // if negative | |
1254 | // redundant fstar.w[3] = 0; | |
1255 | fstar.w[2] = P256.w[2] & maskhigh128[ind - 1]; | |
1256 | fstar.w[1] = P256.w[1]; | |
1257 | fstar.w[0] = P256.w[0]; | |
1258 | // fraction f* > 10^(-x) <=> inexact | |
1259 | // f* is in the right position to be compared with | |
1260 | // 10^(-x) from ten2mk128[] | |
1261 | if (fstar.w[2] || fstar.w[1] > ten2mk128[ind - 1].w[1] || | |
1262 | (fstar.w[1] == ten2mk128[ind - 1].w[1] && | |
1263 | fstar.w[0] >= ten2mk128[ind - 1].w[0])) { | |
1264 | if (++res.w[0] == 0) { | |
1265 | res.w[1]++; | |
1266 | } | |
1267 | } | |
1268 | } | |
1269 | } else { // 22 <= ind - 1 <= 33 | |
1270 | shift = shiftright128[ind - 1] - 64; // 2 <= shift <= 38 | |
1271 | res.w[1] = 0; | |
1272 | res.w[0] = P256.w[3] >> shift; | |
1273 | // if positive, the truncated value is already the correct result | |
1274 | if (x_sign) { // if negative | |
1275 | fstar.w[3] = P256.w[3] & maskhigh128[ind - 1]; | |
1276 | fstar.w[2] = P256.w[2]; | |
1277 | fstar.w[1] = P256.w[1]; | |
1278 | fstar.w[0] = P256.w[0]; | |
1279 | // fraction f* > 10^(-x) <=> inexact | |
1280 | // f* is in the right position to be compared with | |
1281 | // 10^(-x) from ten2mk128[] | |
1282 | if (fstar.w[3] || fstar.w[2] | |
1283 | || fstar.w[1] > ten2mk128[ind - 1].w[1] | |
1284 | || (fstar.w[1] == ten2mk128[ind - 1].w[1] | |
1285 | && fstar.w[0] >= ten2mk128[ind - 1].w[0])) { | |
1286 | if (++res.w[0] == 0) { | |
1287 | res.w[1]++; | |
1288 | } | |
1289 | } | |
1290 | } | |
1291 | } | |
1292 | res.w[1] = x_sign | 0x3040000000000000ull | res.w[1]; | |
1293 | BID_RETURN (res); | |
1294 | } else { // if exp < 0 and q + exp <= 0 | |
1295 | if (x_sign) { // negative rounds down to -1.0 | |
1296 | res.w[1] = 0xb040000000000000ull; | |
1297 | res.w[0] = 0x0000000000000001ull; | |
1298 | } else { // positive rpunds down to +0.0 | |
1299 | res.w[1] = 0x3040000000000000ull; | |
1300 | res.w[0] = 0x0000000000000000ull; | |
1301 | } | |
1302 | BID_RETURN (res); | |
1303 | } | |
1304 | } | |
1305 | ||
1306 | /***************************************************************************** | |
1307 | * BID128_round_integral_positive | |
1308 | ****************************************************************************/ | |
1309 | ||
1310 | BID128_FUNCTION_ARG1_NORND (bid128_round_integral_positive, x) | |
1311 | ||
1312 | UINT128 res; | |
1313 | UINT64 x_sign; | |
1314 | UINT64 x_exp; | |
1315 | int exp; // unbiased exponent | |
1316 | // Note: C1.w[1], C1.w[0] represent x_signif_hi, x_signif_lo | |
1317 | // (all are UINT64) | |
1318 | BID_UI64DOUBLE tmp1; | |
1319 | unsigned int x_nr_bits; | |
1320 | int q, ind, shift; | |
1321 | UINT128 C1; | |
1322 | // UINT128 res is C* at first - represents up to 34 decimal digits ~ | |
1323 | // 113 bits | |
1324 | UINT256 fstar; | |
1325 | UINT256 P256; | |
1326 | ||
1327 | // check for NaN or Infinity | |
1328 | if ((x.w[1] & MASK_SPECIAL) == MASK_SPECIAL) { | |
1329 | // x is special | |
1330 | if ((x.w[1] & MASK_NAN) == MASK_NAN) { // x is NAN | |
1331 | // if x = NaN, then res = Q (x) | |
1332 | // check first for non-canonical NaN payload | |
1333 | if (((x.w[1] & 0x00003fffffffffffull) > 0x0000314dc6448d93ull) || | |
1334 | (((x.w[1] & 0x00003fffffffffffull) == 0x0000314dc6448d93ull) && | |
1335 | (x.w[0] > 0x38c15b09ffffffffull))) { | |
1336 | x.w[1] = x.w[1] & 0xffffc00000000000ull; | |
1337 | x.w[0] = 0x0ull; | |
1338 | } | |
1339 | if ((x.w[1] & MASK_SNAN) == MASK_SNAN) { // x is SNAN | |
1340 | // set invalid flag | |
1341 | *pfpsf |= INVALID_EXCEPTION; | |
1342 | // return quiet (x) | |
1343 | res.w[1] = x.w[1] & 0xfc003fffffffffffull; // clear out also G[6]-G[16] | |
1344 | res.w[0] = x.w[0]; | |
1345 | } else { // x is QNaN | |
1346 | // return x | |
1347 | res.w[1] = x.w[1] & 0xfc003fffffffffffull; // clear out G[6]-G[16] | |
1348 | res.w[0] = x.w[0]; | |
1349 | } | |
1350 | BID_RETURN (res) | |
1351 | } else { // x is not a NaN, so it must be infinity | |
1352 | if ((x.w[1] & MASK_SIGN) == 0x0ull) { // x is +inf | |
1353 | // return +inf | |
1354 | res.w[1] = 0x7800000000000000ull; | |
1355 | res.w[0] = 0x0000000000000000ull; | |
1356 | } else { // x is -inf | |
1357 | // return -inf | |
1358 | res.w[1] = 0xf800000000000000ull; | |
1359 | res.w[0] = 0x0000000000000000ull; | |
1360 | } | |
1361 | BID_RETURN (res); | |
1362 | } | |
1363 | } | |
1364 | // unpack x | |
1365 | x_sign = x.w[1] & MASK_SIGN; // 0 for positive, MASK_SIGN for negative | |
1366 | C1.w[1] = x.w[1] & MASK_COEFF; | |
1367 | C1.w[0] = x.w[0]; | |
1368 | ||
1369 | // check for non-canonical values (treated as zero) | |
1370 | if ((x.w[1] & 0x6000000000000000ull) == 0x6000000000000000ull) { // G0_G1=11 | |
1371 | // non-canonical | |
1372 | x_exp = (x.w[1] << 2) & MASK_EXP; // biased and shifted left 49 bits | |
1373 | C1.w[1] = 0; // significand high | |
1374 | C1.w[0] = 0; // significand low | |
1375 | } else { // G0_G1 != 11 | |
1376 | x_exp = x.w[1] & MASK_EXP; // biased and shifted left 49 bits | |
1377 | if (C1.w[1] > 0x0001ed09bead87c0ull || | |
1378 | (C1.w[1] == 0x0001ed09bead87c0ull | |
1379 | && C1.w[0] > 0x378d8e63ffffffffull)) { | |
1380 | // x is non-canonical if coefficient is larger than 10^34 -1 | |
1381 | C1.w[1] = 0; | |
1382 | C1.w[0] = 0; | |
1383 | } else { // canonical | |
1384 | ; | |
1385 | } | |
1386 | } | |
1387 | ||
1388 | // test for input equal to zero | |
1389 | if ((C1.w[1] == 0x0ull) && (C1.w[0] == 0x0ull)) { | |
1390 | // x is 0 | |
1391 | // return 0 preserving the sign bit and the preferred exponent | |
1392 | // of MAX(Q(x), 0) | |
1393 | if (x_exp <= (0x1820ull << 49)) { | |
1394 | res.w[1] = (x.w[1] & 0x8000000000000000ull) | 0x3040000000000000ull; | |
1395 | } else { | |
1396 | res.w[1] = x_sign | x_exp; | |
1397 | } | |
1398 | res.w[0] = 0x0000000000000000ull; | |
1399 | BID_RETURN (res); | |
1400 | } | |
1401 | // x is not special and is not zero | |
1402 | ||
1403 | // if (exp <= -p) return -0.0 or +1.0 | |
1404 | if (x_exp <= 0x2ffc000000000000ull) { // 0x2ffc000000000000ull == -34 | |
1405 | if (x_sign) { | |
1406 | // if negative, return negative 0, because we know the coefficient | |
1407 | // is non-zero (would have been caught above) | |
1408 | res.w[1] = 0xb040000000000000ull; | |
1409 | res.w[0] = 0x0000000000000000ull; | |
1410 | } else { | |
1411 | // if positive, return positive 1, because we know coefficient is | |
1412 | // non-zero (would have been caught above) | |
1413 | res.w[1] = 0x3040000000000000ull; | |
1414 | res.w[0] = 0x0000000000000001ull; | |
1415 | } | |
1416 | BID_RETURN (res); | |
1417 | } | |
1418 | // q = nr. of decimal digits in x | |
1419 | // determine first the nr. of bits in x | |
1420 | if (C1.w[1] == 0) { | |
1421 | if (C1.w[0] >= 0x0020000000000000ull) { // x >= 2^53 | |
1422 | // split 64-bit value in two 32-bit halves to avoid rounding errors | |
1423 | if (C1.w[0] >= 0x0000000100000000ull) { // x >= 2^32 | |
1424 | tmp1.d = (double) (C1.w[0] >> 32); // exact conversion | |
1425 | x_nr_bits = | |
1426 | 33 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); | |
1427 | } else { // x < 2^32 | |
1428 | tmp1.d = (double) (C1.w[0]); // exact conversion | |
1429 | x_nr_bits = | |
1430 | 1 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); | |
1431 | } | |
1432 | } else { // if x < 2^53 | |
1433 | tmp1.d = (double) C1.w[0]; // exact conversion | |
1434 | x_nr_bits = | |
1435 | 1 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); | |
1436 | } | |
1437 | } else { // C1.w[1] != 0 => nr. bits = 64 + nr_bits (C1.w[1]) | |
1438 | tmp1.d = (double) C1.w[1]; // exact conversion | |
1439 | x_nr_bits = | |
1440 | 65 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); | |
1441 | } | |
1442 | ||
1443 | q = nr_digits[x_nr_bits - 1].digits; | |
1444 | if (q == 0) { | |
1445 | q = nr_digits[x_nr_bits - 1].digits1; | |
1446 | if (C1.w[1] > nr_digits[x_nr_bits - 1].threshold_hi || | |
1447 | (C1.w[1] == nr_digits[x_nr_bits - 1].threshold_hi && | |
1448 | C1.w[0] >= nr_digits[x_nr_bits - 1].threshold_lo)) | |
1449 | q++; | |
1450 | } | |
1451 | exp = (x_exp >> 49) - 6176; | |
1452 | if (exp >= 0) { // -exp <= 0 | |
1453 | // the argument is an integer already | |
1454 | res.w[1] = x.w[1]; | |
1455 | res.w[0] = x.w[0]; | |
1456 | BID_RETURN (res); | |
1457 | } else if ((q + exp) > 0) { // exp < 0 and 1 <= -exp < q | |
1458 | // need to shift right -exp digits from the coefficient; exp will be 0 | |
1459 | ind = -exp; // 1 <= ind <= 34; ind is a synonym for 'x' | |
1460 | // (number of digits to be chopped off) | |
1461 | // chop off ind digits from the lower part of C1 | |
1462 | // FOR ROUND_TO_NEAREST, WE ADD 1/2 ULP(y) then truncate | |
1463 | // FOR ROUND_TO_ZERO, WE DON'T NEED TO ADD 1/2 ULP | |
1464 | // FOR ROUND_TO_POSITIVE_INFINITY, WE TRUNCATE, THEN ADD 1 IF POSITIVE | |
1465 | // FOR ROUND_TO_NEGATIVE_INFINITY, WE TRUNCATE, THEN ADD 1 IF NEGATIVE | |
1466 | // tmp64 = C1.w[0]; | |
1467 | // if (ind <= 19) { | |
1468 | // C1.w[0] = C1.w[0] + midpoint64[ind - 1]; | |
1469 | // } else { | |
1470 | // C1.w[0] = C1.w[0] + midpoint128[ind - 20].w[0]; | |
1471 | // C1.w[1] = C1.w[1] + midpoint128[ind - 20].w[1]; | |
1472 | // } | |
1473 | // if (C1.w[0] < tmp64) C1.w[1]++; | |
1474 | // if carry-out from C1.w[0], increment C1.w[1] | |
1475 | // calculate C* and f* | |
1476 | // C* is actually floor(C*) in this case | |
1477 | // C* and f* need shifting and masking, as shown by | |
1478 | // shiftright128[] and maskhigh128[] | |
1479 | // 1 <= x <= 34 | |
1480 | // kx = 10^(-x) = ten2mk128[ind - 1] | |
1481 | // C* = C1 * 10^(-x) | |
1482 | // the approximation of 10^(-x) was rounded up to 118 bits | |
1483 | __mul_128x128_to_256 (P256, C1, ten2mk128[ind - 1]); | |
1484 | if (ind - 1 <= 2) { // 0 <= ind - 1 <= 2 => shift = 0 | |
1485 | res.w[1] = P256.w[3]; | |
1486 | res.w[0] = P256.w[2]; | |
1487 | // if negative, the truncated value is already the correct result | |
1488 | if (!x_sign) { // if positive | |
1489 | // redundant fstar.w[3] = 0; | |
1490 | // redundant fstar.w[2] = 0; | |
1491 | // redundant fstar.w[1] = P256.w[1]; | |
1492 | // redundant fstar.w[0] = P256.w[0]; | |
1493 | // fraction f* > 10^(-x) <=> inexact | |
1494 | // f* is in the right position to be compared with | |
1495 | // 10^(-x) from ten2mk128[] | |
1496 | if ((P256.w[1] > ten2mk128[ind - 1].w[1]) | |
1497 | || (P256.w[1] == ten2mk128[ind - 1].w[1] | |
1498 | && (P256.w[0] >= ten2mk128[ind - 1].w[0]))) { | |
1499 | if (++res.w[0] == 0) { | |
1500 | res.w[1]++; | |
1501 | } | |
1502 | } | |
1503 | } | |
1504 | } else if (ind - 1 <= 21) { // 3 <= ind - 1 <= 21 => 3 <= shift <= 63 | |
1505 | shift = shiftright128[ind - 1]; // 3 <= shift <= 63 | |
1506 | res.w[1] = (P256.w[3] >> shift); | |
1507 | res.w[0] = (P256.w[3] << (64 - shift)) | (P256.w[2] >> shift); | |
1508 | // if negative, the truncated value is already the correct result | |
1509 | if (!x_sign) { // if positive | |
1510 | // redundant fstar.w[3] = 0; | |
1511 | fstar.w[2] = P256.w[2] & maskhigh128[ind - 1]; | |
1512 | fstar.w[1] = P256.w[1]; | |
1513 | fstar.w[0] = P256.w[0]; | |
1514 | // fraction f* > 10^(-x) <=> inexact | |
1515 | // f* is in the right position to be compared with | |
1516 | // 10^(-x) from ten2mk128[] | |
1517 | if (fstar.w[2] || fstar.w[1] > ten2mk128[ind - 1].w[1] || | |
1518 | (fstar.w[1] == ten2mk128[ind - 1].w[1] && | |
1519 | fstar.w[0] >= ten2mk128[ind - 1].w[0])) { | |
1520 | if (++res.w[0] == 0) { | |
1521 | res.w[1]++; | |
1522 | } | |
1523 | } | |
1524 | } | |
1525 | } else { // 22 <= ind - 1 <= 33 | |
1526 | shift = shiftright128[ind - 1] - 64; // 2 <= shift <= 38 | |
1527 | res.w[1] = 0; | |
1528 | res.w[0] = P256.w[3] >> shift; | |
1529 | // if negative, the truncated value is already the correct result | |
1530 | if (!x_sign) { // if positive | |
1531 | fstar.w[3] = P256.w[3] & maskhigh128[ind - 1]; | |
1532 | fstar.w[2] = P256.w[2]; | |
1533 | fstar.w[1] = P256.w[1]; | |
1534 | fstar.w[0] = P256.w[0]; | |
1535 | // fraction f* > 10^(-x) <=> inexact | |
1536 | // f* is in the right position to be compared with | |
1537 | // 10^(-x) from ten2mk128[] | |
1538 | if (fstar.w[3] || fstar.w[2] | |
1539 | || fstar.w[1] > ten2mk128[ind - 1].w[1] | |
1540 | || (fstar.w[1] == ten2mk128[ind - 1].w[1] | |
1541 | && fstar.w[0] >= ten2mk128[ind - 1].w[0])) { | |
1542 | if (++res.w[0] == 0) { | |
1543 | res.w[1]++; | |
1544 | } | |
1545 | } | |
1546 | } | |
1547 | } | |
1548 | res.w[1] = x_sign | 0x3040000000000000ull | res.w[1]; | |
1549 | BID_RETURN (res); | |
1550 | } else { // if exp < 0 and q + exp <= 0 | |
1551 | if (x_sign) { // negative rounds up to -0.0 | |
1552 | res.w[1] = 0xb040000000000000ull; | |
1553 | res.w[0] = 0x0000000000000000ull; | |
1554 | } else { // positive rpunds up to +1.0 | |
1555 | res.w[1] = 0x3040000000000000ull; | |
1556 | res.w[0] = 0x0000000000000001ull; | |
1557 | } | |
1558 | BID_RETURN (res); | |
1559 | } | |
1560 | } | |
1561 | ||
1562 | /***************************************************************************** | |
1563 | * BID128_round_integral_zero | |
1564 | ****************************************************************************/ | |
1565 | ||
1566 | BID128_FUNCTION_ARG1_NORND (bid128_round_integral_zero, x) | |
1567 | ||
1568 | UINT128 res; | |
1569 | UINT64 x_sign; | |
1570 | UINT64 x_exp; | |
1571 | int exp; // unbiased exponent | |
1572 | // Note: C1.w[1], C1.w[0] represent x_signif_hi, x_signif_lo | |
1573 | // (all are UINT64) | |
1574 | BID_UI64DOUBLE tmp1; | |
1575 | unsigned int x_nr_bits; | |
1576 | int q, ind, shift; | |
1577 | UINT128 C1; | |
1578 | // UINT128 res is C* at first - represents up to 34 decimal digits ~ | |
1579 | // 113 bits | |
1580 | UINT256 P256; | |
1581 | ||
1582 | // check for NaN or Infinity | |
1583 | if ((x.w[1] & MASK_SPECIAL) == MASK_SPECIAL) { | |
1584 | // x is special | |
1585 | if ((x.w[1] & MASK_NAN) == MASK_NAN) { // x is NAN | |
1586 | // if x = NaN, then res = Q (x) | |
1587 | // check first for non-canonical NaN payload | |
1588 | if (((x.w[1] & 0x00003fffffffffffull) > 0x0000314dc6448d93ull) || | |
1589 | (((x.w[1] & 0x00003fffffffffffull) == 0x0000314dc6448d93ull) && | |
1590 | (x.w[0] > 0x38c15b09ffffffffull))) { | |
1591 | x.w[1] = x.w[1] & 0xffffc00000000000ull; | |
1592 | x.w[0] = 0x0ull; | |
200359e8 | 1593 | } |
b2a00c89 L |
1594 | if ((x.w[1] & MASK_SNAN) == MASK_SNAN) { // x is SNAN |
1595 | // set invalid flag | |
1596 | *pfpsf |= INVALID_EXCEPTION; | |
1597 | // return quiet (x) | |
1598 | res.w[1] = x.w[1] & 0xfc003fffffffffffull; // clear out also G[6]-G[16] | |
1599 | res.w[0] = x.w[0]; | |
1600 | } else { // x is QNaN | |
1601 | // return x | |
1602 | res.w[1] = x.w[1] & 0xfc003fffffffffffull; // clear out G[6]-G[16] | |
1603 | res.w[0] = x.w[0]; | |
1604 | } | |
1605 | BID_RETURN (res) | |
1606 | } else { // x is not a NaN, so it must be infinity | |
1607 | if ((x.w[1] & MASK_SIGN) == 0x0ull) { // x is +inf | |
1608 | // return +inf | |
1609 | res.w[1] = 0x7800000000000000ull; | |
1610 | res.w[0] = 0x0000000000000000ull; | |
1611 | } else { // x is -inf | |
1612 | // return -inf | |
1613 | res.w[1] = 0xf800000000000000ull; | |
1614 | res.w[0] = 0x0000000000000000ull; | |
1615 | } | |
1616 | BID_RETURN (res); | |
1617 | } | |
1618 | } | |
1619 | // unpack x | |
1620 | x_sign = x.w[1] & MASK_SIGN; // 0 for positive, MASK_SIGN for negative | |
1621 | C1.w[1] = x.w[1] & MASK_COEFF; | |
1622 | C1.w[0] = x.w[0]; | |
1623 | ||
1624 | // check for non-canonical values (treated as zero) | |
1625 | if ((x.w[1] & 0x6000000000000000ull) == 0x6000000000000000ull) { // G0_G1=11 | |
1626 | // non-canonical | |
1627 | x_exp = (x.w[1] << 2) & MASK_EXP; // biased and shifted left 49 bits | |
1628 | C1.w[1] = 0; // significand high | |
1629 | C1.w[0] = 0; // significand low | |
1630 | } else { // G0_G1 != 11 | |
1631 | x_exp = x.w[1] & MASK_EXP; // biased and shifted left 49 bits | |
1632 | if (C1.w[1] > 0x0001ed09bead87c0ull || | |
1633 | (C1.w[1] == 0x0001ed09bead87c0ull | |
1634 | && C1.w[0] > 0x378d8e63ffffffffull)) { | |
1635 | // x is non-canonical if coefficient is larger than 10^34 -1 | |
1636 | C1.w[1] = 0; | |
1637 | C1.w[0] = 0; | |
1638 | } else { // canonical | |
1639 | ; | |
1640 | } | |
1641 | } | |
1642 | ||
1643 | // test for input equal to zero | |
1644 | if ((C1.w[1] == 0x0ull) && (C1.w[0] == 0x0ull)) { | |
1645 | // x is 0 | |
1646 | // return 0 preserving the sign bit and the preferred exponent | |
1647 | // of MAX(Q(x), 0) | |
1648 | if (x_exp <= (0x1820ull << 49)) { | |
1649 | res.w[1] = (x.w[1] & 0x8000000000000000ull) | 0x3040000000000000ull; | |
1650 | } else { | |
1651 | res.w[1] = x_sign | x_exp; | |
1652 | } | |
1653 | res.w[0] = 0x0000000000000000ull; | |
1654 | BID_RETURN (res); | |
1655 | } | |
1656 | // x is not special and is not zero | |
1657 | ||
1658 | // if (exp <= -p) return -0.0 or +0.0 | |
1659 | if (x_exp <= 0x2ffc000000000000ull) { // 0x2ffc000000000000ull == -34 | |
1660 | res.w[1] = x_sign | 0x3040000000000000ull; | |
1661 | res.w[0] = 0x0000000000000000ull; | |
1662 | BID_RETURN (res); | |
1663 | } | |
1664 | // q = nr. of decimal digits in x | |
1665 | // determine first the nr. of bits in x | |
1666 | if (C1.w[1] == 0) { | |
1667 | if (C1.w[0] >= 0x0020000000000000ull) { // x >= 2^53 | |
1668 | // split the 64-bit value in two 32-bit halves to avoid rounding errors | |
1669 | if (C1.w[0] >= 0x0000000100000000ull) { // x >= 2^32 | |
1670 | tmp1.d = (double) (C1.w[0] >> 32); // exact conversion | |
1671 | x_nr_bits = | |
1672 | 33 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); | |
1673 | } else { // x < 2^32 | |
1674 | tmp1.d = (double) (C1.w[0]); // exact conversion | |
1675 | x_nr_bits = | |
1676 | 1 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); | |
1677 | } | |
1678 | } else { // if x < 2^53 | |
1679 | tmp1.d = (double) C1.w[0]; // exact conversion | |
1680 | x_nr_bits = | |
1681 | 1 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); | |
1682 | } | |
1683 | } else { // C1.w[1] != 0 => nr. bits = 64 + nr_bits (C1.w[1]) | |
1684 | tmp1.d = (double) C1.w[1]; // exact conversion | |
1685 | x_nr_bits = | |
1686 | 65 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); | |
1687 | } | |
1688 | ||
1689 | q = nr_digits[x_nr_bits - 1].digits; | |
1690 | if (q == 0) { | |
1691 | q = nr_digits[x_nr_bits - 1].digits1; | |
1692 | if (C1.w[1] > nr_digits[x_nr_bits - 1].threshold_hi || | |
1693 | (C1.w[1] == nr_digits[x_nr_bits - 1].threshold_hi && | |
1694 | C1.w[0] >= nr_digits[x_nr_bits - 1].threshold_lo)) | |
1695 | q++; | |
1696 | } | |
1697 | exp = (x_exp >> 49) - 6176; | |
1698 | if (exp >= 0) { // -exp <= 0 | |
1699 | // the argument is an integer already | |
1700 | res.w[1] = x.w[1]; | |
1701 | res.w[0] = x.w[0]; | |
1702 | BID_RETURN (res); | |
1703 | } else if ((q + exp) > 0) { // exp < 0 and 1 <= -exp < q | |
1704 | // need to shift right -exp digits from the coefficient; the exp will be 0 | |
1705 | ind = -exp; // 1 <= ind <= 34; ind is a synonym for 'x' | |
1706 | // (number of digits to be chopped off) | |
1707 | // chop off ind digits from the lower part of C1 | |
1708 | // FOR ROUND_TO_NEAREST, WE ADD 1/2 ULP(y) then truncate | |
1709 | // FOR ROUND_TO_ZERO, WE DON'T NEED TO ADD 1/2 ULP | |
1710 | // FOR ROUND_TO_POSITIVE_INFINITY, WE TRUNCATE, THEN ADD 1 IF POSITIVE | |
1711 | // FOR ROUND_TO_NEGATIVE_INFINITY, WE TRUNCATE, THEN ADD 1 IF NEGATIVE | |
1712 | //tmp64 = C1.w[0]; | |
1713 | // if (ind <= 19) { | |
1714 | // C1.w[0] = C1.w[0] + midpoint64[ind - 1]; | |
1715 | // } else { | |
1716 | // C1.w[0] = C1.w[0] + midpoint128[ind - 20].w[0]; | |
1717 | // C1.w[1] = C1.w[1] + midpoint128[ind - 20].w[1]; | |
1718 | // } | |
1719 | // if (C1.w[0] < tmp64) C1.w[1]++; | |
1720 | // if carry-out from C1.w[0], increment C1.w[1] | |
1721 | // calculate C* and f* | |
1722 | // C* is actually floor(C*) in this case | |
1723 | // C* and f* need shifting and masking, as shown by | |
1724 | // shiftright128[] and maskhigh128[] | |
1725 | // 1 <= x <= 34 | |
1726 | // kx = 10^(-x) = ten2mk128[ind - 1] | |
1727 | // C* = (C1 + 1/2 * 10^x) * 10^(-x) | |
1728 | // the approximation of 10^(-x) was rounded up to 118 bits | |
1729 | __mul_128x128_to_256 (P256, C1, ten2mk128[ind - 1]); | |
1730 | if (ind - 1 <= 2) { // 0 <= ind - 1 <= 2 => shift = 0 | |
1731 | res.w[1] = P256.w[3]; | |
1732 | res.w[0] = P256.w[2]; | |
1733 | } else if (ind - 1 <= 21) { // 3 <= ind - 1 <= 21 => 3 <= shift <= 63 | |
1734 | shift = shiftright128[ind - 1]; // 3 <= shift <= 63 | |
1735 | res.w[1] = (P256.w[3] >> shift); | |
1736 | res.w[0] = (P256.w[3] << (64 - shift)) | (P256.w[2] >> shift); | |
1737 | } else { // 22 <= ind - 1 <= 33 | |
1738 | shift = shiftright128[ind - 1] - 64; // 2 <= shift <= 38 | |
1739 | res.w[1] = 0; | |
1740 | res.w[0] = P256.w[3] >> shift; | |
1741 | } | |
1742 | res.w[1] = x_sign | 0x3040000000000000ull | res.w[1]; | |
1743 | BID_RETURN (res); | |
1744 | } else { // if exp < 0 and q + exp <= 0 the result is +0 or -0 | |
1745 | res.w[1] = x_sign | 0x3040000000000000ull; | |
1746 | res.w[0] = 0x0000000000000000ull; | |
1747 | BID_RETURN (res); | |
1748 | } | |
1749 | } | |
1750 | ||
1751 | /***************************************************************************** | |
1752 | * BID128_round_integral_nearest_away | |
1753 | ****************************************************************************/ | |
1754 | ||
1755 | BID128_FUNCTION_ARG1_NORND (bid128_round_integral_nearest_away, x) | |
1756 | ||
1757 | UINT128 res; | |
1758 | UINT64 x_sign; | |
1759 | UINT64 x_exp; | |
1760 | int exp; // unbiased exponent | |
1761 | // Note: C1.w[1], C1.w[0] represent x_signif_hi, x_signif_lo | |
1762 | // (all are UINT64) | |
1763 | UINT64 tmp64; | |
1764 | BID_UI64DOUBLE tmp1; | |
1765 | unsigned int x_nr_bits; | |
1766 | int q, ind, shift; | |
1767 | UINT128 C1; | |
1768 | // UINT128 res is C* at first - represents up to 34 decimal digits ~ | |
1769 | // 113 bits | |
1770 | // UINT256 fstar; | |
1771 | UINT256 P256; | |
1772 | ||
1773 | // check for NaN or Infinity | |
1774 | if ((x.w[1] & MASK_SPECIAL) == MASK_SPECIAL) { | |
1775 | // x is special | |
1776 | if ((x.w[1] & MASK_NAN) == MASK_NAN) { // x is NAN | |
1777 | // if x = NaN, then res = Q (x) | |
1778 | // check first for non-canonical NaN payload | |
1779 | if (((x.w[1] & 0x00003fffffffffffull) > 0x0000314dc6448d93ull) || | |
1780 | (((x.w[1] & 0x00003fffffffffffull) == 0x0000314dc6448d93ull) && | |
1781 | (x.w[0] > 0x38c15b09ffffffffull))) { | |
1782 | x.w[1] = x.w[1] & 0xffffc00000000000ull; | |
1783 | x.w[0] = 0x0ull; | |
1784 | } | |
1785 | if ((x.w[1] & MASK_SNAN) == MASK_SNAN) { // x is SNAN | |
1786 | // set invalid flag | |
1787 | *pfpsf |= INVALID_EXCEPTION; | |
1788 | // return quiet (x) | |
1789 | res.w[1] = x.w[1] & 0xfc003fffffffffffull; // clear out also G[6]-G[16] | |
1790 | res.w[0] = x.w[0]; | |
1791 | } else { // x is QNaN | |
1792 | // return x | |
1793 | res.w[1] = x.w[1] & 0xfc003fffffffffffull; // clear out G[6]-G[16] | |
1794 | res.w[0] = x.w[0]; | |
1795 | } | |
1796 | BID_RETURN (res) | |
1797 | } else { // x is not a NaN, so it must be infinity | |
1798 | if ((x.w[1] & MASK_SIGN) == 0x0ull) { // x is +inf | |
1799 | // return +inf | |
1800 | res.w[1] = 0x7800000000000000ull; | |
1801 | res.w[0] = 0x0000000000000000ull; | |
1802 | } else { // x is -inf | |
1803 | // return -inf | |
1804 | res.w[1] = 0xf800000000000000ull; | |
1805 | res.w[0] = 0x0000000000000000ull; | |
1806 | } | |
1807 | BID_RETURN (res); | |
1808 | } | |
1809 | } | |
1810 | // unpack x | |
1811 | x_sign = x.w[1] & MASK_SIGN; // 0 for positive, MASK_SIGN for negative | |
1812 | C1.w[1] = x.w[1] & MASK_COEFF; | |
1813 | C1.w[0] = x.w[0]; | |
1814 | ||
1815 | // check for non-canonical values (treated as zero) | |
1816 | if ((x.w[1] & 0x6000000000000000ull) == 0x6000000000000000ull) { // G0_G1=11 | |
1817 | // non-canonical | |
1818 | x_exp = (x.w[1] << 2) & MASK_EXP; // biased and shifted left 49 bits | |
1819 | C1.w[1] = 0; // significand high | |
1820 | C1.w[0] = 0; // significand low | |
1821 | } else { // G0_G1 != 11 | |
1822 | x_exp = x.w[1] & MASK_EXP; // biased and shifted left 49 bits | |
1823 | if (C1.w[1] > 0x0001ed09bead87c0ull || | |
1824 | (C1.w[1] == 0x0001ed09bead87c0ull | |
1825 | && C1.w[0] > 0x378d8e63ffffffffull)) { | |
1826 | // x is non-canonical if coefficient is larger than 10^34 -1 | |
1827 | C1.w[1] = 0; | |
1828 | C1.w[0] = 0; | |
1829 | } else { // canonical | |
1830 | ; | |
1831 | } | |
1832 | } | |
1833 | ||
1834 | // test for input equal to zero | |
1835 | if ((C1.w[1] == 0x0ull) && (C1.w[0] == 0x0ull)) { | |
1836 | // x is 0 | |
1837 | // return 0 preserving the sign bit and the preferred exponent | |
1838 | // of MAX(Q(x), 0) | |
1839 | if (x_exp <= (0x1820ull << 49)) { | |
1840 | res.w[1] = (x.w[1] & 0x8000000000000000ull) | 0x3040000000000000ull; | |
1841 | } else { | |
1842 | res.w[1] = x_sign | x_exp; | |
1843 | } | |
1844 | res.w[0] = 0x0000000000000000ull; | |
1845 | BID_RETURN (res); | |
1846 | } | |
1847 | // x is not special and is not zero | |
1848 | ||
1849 | // if (exp <= -(p+1)) return 0.0 | |
1850 | if (x_exp <= 0x2ffa000000000000ull) { // 0x2ffa000000000000ull == -35 | |
1851 | res.w[1] = x_sign | 0x3040000000000000ull; | |
1852 | res.w[0] = 0x0000000000000000ull; | |
1853 | BID_RETURN (res); | |
1854 | } | |
1855 | // q = nr. of decimal digits in x | |
1856 | // determine first the nr. of bits in x | |
1857 | if (C1.w[1] == 0) { | |
1858 | if (C1.w[0] >= 0x0020000000000000ull) { // x >= 2^53 | |
1859 | // split the 64-bit value in two 32-bit halves to avoid rounding errors | |
1860 | if (C1.w[0] >= 0x0000000100000000ull) { // x >= 2^32 | |
1861 | tmp1.d = (double) (C1.w[0] >> 32); // exact conversion | |
1862 | x_nr_bits = | |
1863 | 33 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); | |
1864 | } else { // x < 2^32 | |
1865 | tmp1.d = (double) (C1.w[0]); // exact conversion | |
1866 | x_nr_bits = | |
1867 | 1 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); | |
1868 | } | |
1869 | } else { // if x < 2^53 | |
1870 | tmp1.d = (double) C1.w[0]; // exact conversion | |
1871 | x_nr_bits = | |
1872 | 1 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); | |
1873 | } | |
1874 | } else { // C1.w[1] != 0 => nr. bits = 64 + nr_bits (C1.w[1]) | |
1875 | tmp1.d = (double) C1.w[1]; // exact conversion | |
1876 | x_nr_bits = | |
1877 | 65 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); | |
1878 | } | |
1879 | ||
1880 | q = nr_digits[x_nr_bits - 1].digits; | |
1881 | if (q == 0) { | |
1882 | q = nr_digits[x_nr_bits - 1].digits1; | |
1883 | if (C1.w[1] > nr_digits[x_nr_bits - 1].threshold_hi || | |
1884 | (C1.w[1] == nr_digits[x_nr_bits - 1].threshold_hi && | |
1885 | C1.w[0] >= nr_digits[x_nr_bits - 1].threshold_lo)) | |
1886 | q++; | |
1887 | } | |
1888 | exp = (x_exp >> 49) - 6176; | |
1889 | if (exp >= 0) { // -exp <= 0 | |
1890 | // the argument is an integer already | |
1891 | res.w[1] = x.w[1]; | |
1892 | res.w[0] = x.w[0]; | |
1893 | BID_RETURN (res); | |
1894 | } else if ((q + exp) >= 0) { // exp < 0 and 1 <= -exp <= q | |
1895 | // need to shift right -exp digits from the coefficient; the exp will be 0 | |
1896 | ind = -exp; // 1 <= ind <= 34; ind is a synonym for 'x' | |
1897 | // chop off ind digits from the lower part of C1 | |
1898 | // C1 = C1 + 1/2 * 10^x where the result C1 fits in 127 bits | |
1899 | tmp64 = C1.w[0]; | |
1900 | if (ind <= 19) { | |
1901 | C1.w[0] = C1.w[0] + midpoint64[ind - 1]; | |
1902 | } else { | |
1903 | C1.w[0] = C1.w[0] + midpoint128[ind - 20].w[0]; | |
1904 | C1.w[1] = C1.w[1] + midpoint128[ind - 20].w[1]; | |
1905 | } | |
1906 | if (C1.w[0] < tmp64) | |
1907 | C1.w[1]++; | |
1908 | // calculate C* and f* | |
1909 | // C* is actually floor(C*) in this case | |
1910 | // C* and f* need shifting and masking, as shown by | |
1911 | // shiftright128[] and maskhigh128[] | |
1912 | // 1 <= x <= 34 | |
1913 | // kx = 10^(-x) = ten2mk128[ind - 1] | |
1914 | // C* = (C1 + 1/2 * 10^x) * 10^(-x) | |
1915 | // the approximation of 10^(-x) was rounded up to 118 bits | |
1916 | __mul_128x128_to_256 (P256, C1, ten2mk128[ind - 1]); | |
1917 | // the top Ex bits of 10^(-x) are T* = ten2mk128trunc[ind], e.g. | |
1918 | // if x=1, T*=ten2mk128trunc[0]=0x19999999999999999999999999999999 | |
1919 | // if (0 < f* < 10^(-x)) then the result is a midpoint | |
1920 | // if floor(C*) is even then C* = floor(C*) - logical right | |
1921 | // shift; C* has p decimal digits, correct by Prop. 1) | |
1922 | // else if floor(C*) is odd C* = floor(C*)-1 (logical right | |
1923 | // shift; C* has p decimal digits, correct by Pr. 1) | |
1924 | // else | |
1925 | // C* = floor(C*) (logical right shift; C has p decimal digits, | |
1926 | // correct by Property 1) | |
1927 | // n = C* * 10^(e+x) | |
1928 | ||
1929 | // shift right C* by Ex-128 = shiftright128[ind] | |
1930 | if (ind - 1 <= 2) { // 0 <= ind - 1 <= 2 => shift = 0 | |
1931 | res.w[1] = P256.w[3]; | |
1932 | res.w[0] = P256.w[2]; | |
1933 | } else if (ind - 1 <= 21) { // 3 <= ind - 1 <= 21 => 3 <= shift <= 63 | |
1934 | shift = shiftright128[ind - 1]; // 3 <= shift <= 63 | |
1935 | res.w[0] = (P256.w[3] << (64 - shift)) | (P256.w[2] >> shift); | |
1936 | res.w[1] = (P256.w[3] >> shift); | |
1937 | } else { // 22 <= ind - 1 <= 33 | |
1938 | shift = shiftright128[ind - 1]; // 2 <= shift <= 38 | |
1939 | res.w[1] = 0; | |
1940 | res.w[0] = (P256.w[3] >> (shift - 64)); // 2 <= shift - 64 <= 38 | |
1941 | } | |
1942 | // if the result was a midpoint, it was already rounded away from zero | |
1943 | res.w[1] |= x_sign | 0x3040000000000000ull; | |
1944 | BID_RETURN (res); | |
1945 | } else { // if ((q + exp) < 0) <=> q < -exp | |
1946 | // the result is +0 or -0 | |
1947 | res.w[1] = x_sign | 0x3040000000000000ull; | |
1948 | res.w[0] = 0x0000000000000000ull; | |
1949 | BID_RETURN (res); | |
1950 | } | |
200359e8 | 1951 | } |