1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382
|
/******************************************************************************
Copyright (c) 2007-2024, Intel Corp.
All rights reserved.
Redistribution and use in source and binary forms, with or without
modification, are permitted provided that the following conditions are met:
* Redistributions of source code must retain the above copyright notice,
this list of conditions and the following disclaimer.
* Redistributions in binary form must reproduce the above copyright
notice, this list of conditions and the following disclaimer in the
documentation and/or other materials provided with the distribution.
* Neither the name of Intel Corporation nor the names of its contributors
may be used to endorse or promote products derived from this software
without specific prior written permission.
THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS"
AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER OR CONTRIBUTORS BE
LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR
CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF
SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS
INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN
CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)
ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF
THE POSSIBILITY OF SUCH DAMAGE.
******************************************************************************/
#ifndef SQRT_MACROS_H
#define SQRT_MACROS_H
/* None of these macros screen for abnormal inputs.
They all assume positive finite values. */
#define NUM_FRAC_BITS 7
#define INDEX_MASK MAKE_MASK((NUM_FRAC_BITS + 1), 0)
#if (IEEE_FLOATING)
# define IF_IEEE_FLOATING(x) x
# define LOC_OF_EXPON ((BITS_PER_LS_INT_TYPE - 1) - B_EXP_WIDTH)
# define EXP_BITS_OF_ONE_HALF ((U_LS_INT_TYPE)(B_EXP_BIAS-B_NORM-1) << LOC_OF_EXPON)
# define EXPON_MASK MAKE_MASK(B_EXP_WIDTH, 0)
# define HI_EXP_BIT_MASK ((EXPON_MASK - 1) << LOC_OF_EXPON)
# define GET_SQRT_TABLE_INDEX(exp,index) \
index = (exp >> (LOC_OF_EXPON - NUM_FRAC_BITS)); \
index &= INDEX_MASK
# if ((ARCHITECTURE == alpha) && defined(HAS_LOAD_WRONG_STORE_SIZE_PENALTY))
# define V_UNION_64_BIT_STORE \
v.B_UNSIGNED_HI_64 = ((U_WORD)exp) >> 1
# define V_UNION_128_BIT_STORE \
v.B_UNSIGNED_HI_64 = ((U_WORD)exp) >> 1; \
v.B_UNSIGNED_LO_64 = 0
# else
# define V_UNION_64_BIT_STORE \
v.B_UNSIGNED_HI_64 = ((U_INT_64)(U_INT_32)exp) << 31
# define V_UNION_128_BIT_STORE \
v.B_UNSIGNED_HI_64 = ((U_INT_64)(U_INT_32)exp) << 31; \
v.B_UNSIGNED_LO_64 = 0
# endif
#else
# define IF_IEEE_FLOATING(x)
# define EXP_BITS_OF_ONE_HALF 0x4000
# define HI_EXP_BIT_MASK 0x7fe0
# define GET_SQRT_TABLE_INDEX(exp,index) \
index = ((exp << 3) | ((U_INT_32)exp >> 29)); \
index &= INDEX_MASK
# define V_UNION_64_BIT_STORE \
v.B_UNSIGNED_HI_64 = ((U_INT_64)(U_INT_32)exp) >> 1
# define V_UNION_128_BIT_STORE \
v.B_UNSIGNED_HI_64 = ((U_INT_64)(U_INT_32)exp) >> 1 ;\
v.B_UNSIGNED_LO_64 = 0
#endif
#if (ARCHITECTURE == alpha) || (BITS_PER_WORD == 64)
# if QUAD_PRECISION
# define STORE_EXP_TO_V_UNION \
V_UNION_128_BIT_STORE
# else
# define STORE_EXP_TO_V_UNION \
V_UNION_64_BIT_STORE
# endif
#else
# if QUAD_PRECISION
# define STORE_EXP_TO_V_UNION \
v.B_SIGNED_HI_32 = ((U_INT_32)exp) >> 1; \
v.B_SIGNED_LO1_32 = 0;\
v.B_SIGNED_LO2_32 = 0;\
v.B_SIGNED_LO3_32 = 0
# else
# define STORE_EXP_TO_V_UNION \
v.B_SIGNED_HI_32 = ((U_INT_32)exp) >> 1; \
v.B_SIGNED_LO_32 = 0
# endif
#endif
#if QUAD_PRECISION
# define NEWTONS_ITERATION \
a = y * scaled_x; \
b = a * y; \
b = one - b; \
b *= y; \
c = y + y; \
c += b; \
y = c * half
#else
# define NEWTONS_ITERATION
#endif
#if QUAD_PRECISION
# define NEWTONS_ITERATION_NO_SCALE(input) \
a = y * (B_TYPE)(input); \
b = a * y; \
b = one - b; \
b *= y; \
c = y + y; \
c += b; \
y = c * half
#else
# define NEWTONS_ITERATION_NO_SCALE(input)
#endif
typedef struct { float a, b; double c; } SQRT_COEF_STRUCT;
extern const SQRT_COEF_STRUCT D_SQRT_TABLE_NAME[];
#define SQRT_FIRST_PART(input) \
B_UNION u, v; \
B_TYPE y, a, b, c; \
B_TYPE scaled_x, half_scale; \
B_TYPE half = (B_TYPE)0.5; \
B_TYPE one = (B_TYPE)1.0; \
B_TYPE three = (B_TYPE)3.0; \
F_TYPE ulp, y_less_1_ulp, y_plus_1_ulp; \
F_TYPE f_type_y, truncated_y, truncated_product; \
LS_INT_TYPE exp; \
U_LS_INT_TYPE orig_rounding_mode; \
U_LS_INT_TYPE index; \
U_LS_INT_TYPE lo_exp_bit_and_hi_frac; \
U_LS_INT_TYPE hi_exp_mask = HI_EXP_BIT_MASK; \
U_LS_INT_TYPE exp_of_one_half = EXP_BITS_OF_ONE_HALF; \
u.f = (B_TYPE)(input); \
exp = u.B_HI_LS_INT_TYPE; \
B_COPY_SIGN_AND_EXP((B_TYPE)(input), half, y); \
GET_SQRT_TABLE_INDEX(exp,index); \
b = (B_TYPE)D_SQRT_TABLE_NAME[index].b; \
b *= y; \
c = (B_TYPE)D_SQRT_TABLE_NAME[index].c; \
lo_exp_bit_and_hi_frac = exp & ~hi_exp_mask; \
u.B_HI_LS_INT_TYPE = exp_of_one_half | lo_exp_bit_and_hi_frac; \
c += b; \
scaled_x = u.f; \
y *= y; \
a = (B_TYPE)D_SQRT_TABLE_NAME[index].a; \
exp ^= lo_exp_bit_and_hi_frac; \
exp += exp_of_one_half; \
y *= a; \
STORE_EXP_TO_V_UNION; \
y += c; \
half_scale = v.f
#define B_HALF_PREC_SQRT(input, result) { \
SQRT_FIRST_PART(input); \
a = scaled_x * y; \
b = half_scale + half_scale; \
y = a * b; \
(result) = (B_TYPE)y; \
}
#if (DYNAMIC_ROUNDING_MODES && !FAST_SQRT)
# define ESTABLISH_KNOWN_ROUNDING_MODE(old_mode) INIT_FPU_STATE_AND_ROUND_TO_ZERO(old_mode)
# define RESTORE_ORIGINAL_ROUNDING_MODE(old_mode) RESTORE_FPU_STATE(old_mode)
#else
# define ESTABLISH_KNOWN_ROUNDING_MODE(old_mode)
# define RESTORE_ORIGINAL_ROUNDING_MODE(old_mode)
#endif
#if !defined(F_MUL_CHOPPED)
# define F_MUL_CHOPPED(x,y,z) (z) = (x) * (y)
#endif
#if ( (F_PRECISION == 24) && PRECISION_BACKUP_AVAILABLE )
/* Make sure the last bit is correctly rounded by computing
a double-precision result, and then rounding it to single. */
# define ITERATE_AND_MAYBE_CHECK_LAST_BIT(input) \
a = y * scaled_x; \
b = a * y; \
c = a * half_scale; \
b = three - b; \
f_type_y = (F_TYPE)(c * b)
# define RESULT f_type_y
#else
# undef ULP_FACTOR
# if (F_PRECISION == 53)
# define ULP_FACTOR (F_TYPE)2.775557561562891351e-16 /* 1.25 * 2^(1 - F_PRECISION) */
# elif QUAD_PRECISION
# define ULP_FACTOR 1.9259299443872358530559779425849273185381e-34
# else
# error Unsupported F_PRECISION.
# endif
/* Newton's iteration for 1 / (nth root of x) is:
y' = y + [ (1 - x * y^n) * y / n ]
So, the iteration for 1 / sqrt(x) is:
y' = y + [ (1 - x * y^2) * y * 0.5 ]
If we want to do one iteration and multiply the result by x
and multiply the result by a scale factor we get:
y' = scale * x * ( y + [ (1 - x * y^2) * y * 0.5 ] )
y' = scale * x * y * ( 1 + [ (1 - x * y^2) * 0.5 ] )
y' = scale/2 * x * y * ( 2 + [ (1 - x * y^2) ] ) gives about 5/4 lsb error
y' = scale/2 * x * y * ( 3 - x * y^2 ) gives about 8/4 lsb error
So iterate to get better 1/sqrt(x) and multiply by x to get sqrt(x). */
# define ITERATE_AND_MAYBE_CHECK_LAST_BIT(input) \
a = y * scaled_x; \
ulp = ULP_FACTOR; \
b = a * y; \
c = a * half_scale; \
b = one - b; \
a = c + c; \
b = c * b; \
ulp *= c; \
y = a + b; \
y_less_1_ulp = y - ulp; \
ASSERT( y_less_1_ulp < y ); \
y_plus_1_ulp = y + ulp; \
ASSERT( y_plus_1_ulp > y ); \
ESTABLISH_KNOWN_ROUNDING_MODE(orig_rounding_mode); \
F_MUL_CHOPPED(y, y_less_1_ulp, a); \
F_MUL_CHOPPED(y, y_plus_1_ulp, b); \
RESTORE_ORIGINAL_ROUNDING_MODE(orig_rounding_mode); \
y = ((a >= input) ? y_less_1_ulp : y); \
y = ((b < input) ? y_plus_1_ulp : y); \
# define RESULT y
#endif
#if (SINGLE_PRECISION)
# define F_SQRT(input, result) { \
SQRT_FIRST_PART(input); \
a = scaled_x * y; \
b = half_scale + half_scale; \
y = a * b; \
(result) = (F_TYPE)y; \
}
# define F_PRECISE_SQRT(input, result) { \
SQRT_FIRST_PART(input); \
NEWTONS_ITERATION; \
NEWTONS_ITERATION; \
ITERATE_AND_MAYBE_CHECK_LAST_BIT(input); \
(result) = (F_TYPE) RESULT; \
}
# define F_SQRT_2_LSB(input,result) F_SQRT(input,result)
# define F_SQRT_2_LSB_NO_SCALE_FINISH_ITERATION(input) \
y *= (B_TYPE)(input) \
y += y;
#else
# define F_SQRT(input, result) { \
SQRT_FIRST_PART(input); \
NEWTONS_ITERATION;\
NEWTONS_ITERATION;\
a = scaled_x * y; \
b = a * y; \
c = a * half_scale; \
b = one - b; \
a = c + c; \
b = c * b; \
y = a + b; \
(result) = (F_TYPE)y; \
}
# define F_PRECISE_SQRT(input, result) { \
SQRT_FIRST_PART(input); \
NEWTONS_ITERATION;\
NEWTONS_ITERATION;\
ITERATE_AND_MAYBE_CHECK_LAST_BIT(input); \
(result) = (F_TYPE) RESULT; \
}
# define F_SQRT_2_LSB(input, result) { \
SQRT_FIRST_PART(input); \
NEWTONS_ITERATION; \
NEWTONS_ITERATION;\
a = scaled_x * y; \
b = a * y; \
c = a * half_scale; \
b = three - b; \
y = c * b; \
(result) = (F_TYPE)y; \
}
# define F_SQRT_2_LSB_NO_SCALE_FINISH_ITERATION(input) \
NEWTONS_ITERATION_NO_SCALE(input);\
NEWTONS_ITERATION_NO_SCALE(input);\
a = (B_TYPE)(input) * y; \
b = a * y; \
b = three - b; \
y = a * b
#endif
/* The F_SQRT_2_LSB_NO_SCALE macro avoids most scaling (i.e. 0.5 <= input < 2.0).
The input for the polynomial is still scaled, however, because the
coefficients have a scale factor built into them. */
#define F_SQRT_2_LSB_NO_SCALE_TIMES_2(input, result) { \
B_UNION u; \
B_TYPE y, a, b, c; \
B_TYPE half = (B_TYPE)0.5; \
B_TYPE one = (B_TYPE)1.0; \
B_TYPE three = (B_TYPE)3.0; \
LS_INT_TYPE exp; \
U_LS_INT_TYPE index; \
u.f = (B_TYPE)(input); \
exp = u.B_HI_LS_INT_TYPE; \
B_COPY_SIGN_AND_EXP((B_TYPE)(input), half, y); \
GET_SQRT_TABLE_INDEX(exp,index); \
b = (B_TYPE)D_SQRT_TABLE_NAME[index].b; \
b *= y; \
c = (B_TYPE)D_SQRT_TABLE_NAME[index].c; \
c += b; \
y *= y; \
a = (B_TYPE)D_SQRT_TABLE_NAME[index].a; \
y *= a; \
y += c; \
F_SQRT_2_LSB_NO_SCALE_FINISH_ITERATION(input); \
(result) = (F_TYPE)y; \
}
#endif /* SQRT_MACROS_H */
|