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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 | /*---------------------------------------------------------------------------+ | poly_sin.c | | | | Computation of an approximation of the sin function and the cosine | | function by a polynomial. | | | | Copyright (C) 1992,1993,1994,1997,1999 | | W. Metzenthen, 22 Parker St, Ormond, Vic 3163, Australia | | E-mail billm@melbpc.org.au | | | | | +---------------------------------------------------------------------------*/ #include "exception.h" #include "reg_constant.h" #include "fpu_emu.h" #include "fpu_system.h" #include "control_w.h" #include "poly.h" #define N_COEFF_P 4 #define N_COEFF_N 4 static const unsigned long long pos_terms_l[N_COEFF_P] = { 0xaaaaaaaaaaaaaaabLL, 0x00d00d00d00cf906LL, 0x000006b99159a8bbLL, 0x000000000d7392e6LL }; static const unsigned long long neg_terms_l[N_COEFF_N] = { 0x2222222222222167LL, 0x0002e3bc74aab624LL, 0x0000000b09229062LL, 0x00000000000c7973LL }; #define N_COEFF_PH 4 #define N_COEFF_NH 4 static const unsigned long long pos_terms_h[N_COEFF_PH] = { 0x0000000000000000LL, 0x05b05b05b05b0406LL, 0x000049f93edd91a9LL, 0x00000000c9c9ed62LL }; static const unsigned long long neg_terms_h[N_COEFF_NH] = { 0xaaaaaaaaaaaaaa98LL, 0x001a01a01a019064LL, 0x0000008f76c68a77LL, 0x0000000000d58f5eLL }; /*--- poly_sine() -----------------------------------------------------------+ | | +---------------------------------------------------------------------------*/ void poly_sine(FPU_REG *st0_ptr) { int exponent, echange; Xsig accumulator, argSqrd, argTo4; unsigned long fix_up, adj; unsigned long long fixed_arg; FPU_REG result; exponent = exponent(st0_ptr); accumulator.lsw = accumulator.midw = accumulator.msw = 0; /* Split into two ranges, for arguments below and above 1.0 */ /* The boundary between upper and lower is approx 0.88309101259 */ if ( (exponent < -1) || ((exponent == -1) && (st0_ptr->sigh <= 0xe21240aa)) ) { /* The argument is <= 0.88309101259 */ argSqrd.msw = st0_ptr->sigh; argSqrd.midw = st0_ptr->sigl; argSqrd.lsw = 0; mul64_Xsig(&argSqrd, &significand(st0_ptr)); shr_Xsig(&argSqrd, 2*(-1-exponent)); argTo4.msw = argSqrd.msw; argTo4.midw = argSqrd.midw; argTo4.lsw = argSqrd.lsw; mul_Xsig_Xsig(&argTo4, &argTo4); polynomial_Xsig(&accumulator, &XSIG_LL(argTo4), neg_terms_l, N_COEFF_N-1); mul_Xsig_Xsig(&accumulator, &argSqrd); negate_Xsig(&accumulator); polynomial_Xsig(&accumulator, &XSIG_LL(argTo4), pos_terms_l, N_COEFF_P-1); shr_Xsig(&accumulator, 2); /* Divide by four */ accumulator.msw |= 0x80000000; /* Add 1.0 */ mul64_Xsig(&accumulator, &significand(st0_ptr)); mul64_Xsig(&accumulator, &significand(st0_ptr)); mul64_Xsig(&accumulator, &significand(st0_ptr)); /* Divide by four, FPU_REG compatible, etc */ exponent = 3*exponent; /* The minimum exponent difference is 3 */ shr_Xsig(&accumulator, exponent(st0_ptr) - exponent); negate_Xsig(&accumulator); XSIG_LL(accumulator) += significand(st0_ptr); echange = round_Xsig(&accumulator); setexponentpos(&result, exponent(st0_ptr) + echange); } else { /* The argument is > 0.88309101259 */ /* We use sin(st(0)) = cos(pi/2-st(0)) */ fixed_arg = significand(st0_ptr); if ( exponent == 0 ) { /* The argument is >= 1.0 */ /* Put the binary point at the left. */ fixed_arg <<= 1; } /* pi/2 in hex is: 1.921fb54442d18469 898CC51701B839A2 52049C1 */ fixed_arg = 0x921fb54442d18469LL - fixed_arg; /* There is a special case which arises due to rounding, to fix here. */ if ( fixed_arg == 0xffffffffffffffffLL ) fixed_arg = 0; XSIG_LL(argSqrd) = fixed_arg; argSqrd.lsw = 0; mul64_Xsig(&argSqrd, &fixed_arg); XSIG_LL(argTo4) = XSIG_LL(argSqrd); argTo4.lsw = argSqrd.lsw; mul_Xsig_Xsig(&argTo4, &argTo4); polynomial_Xsig(&accumulator, &XSIG_LL(argTo4), neg_terms_h, N_COEFF_NH-1); mul_Xsig_Xsig(&accumulator, &argSqrd); negate_Xsig(&accumulator); polynomial_Xsig(&accumulator, &XSIG_LL(argTo4), pos_terms_h, N_COEFF_PH-1); negate_Xsig(&accumulator); mul64_Xsig(&accumulator, &fixed_arg); mul64_Xsig(&accumulator, &fixed_arg); shr_Xsig(&accumulator, 3); negate_Xsig(&accumulator); add_Xsig_Xsig(&accumulator, &argSqrd); shr_Xsig(&accumulator, 1); accumulator.lsw |= 1; /* A zero accumulator here would cause problems */ negate_Xsig(&accumulator); /* The basic computation is complete. Now fix the answer to compensate for the error due to the approximation used for pi/2 */ /* This has an exponent of -65 */ fix_up = 0x898cc517; /* The fix-up needs to be improved for larger args */ if ( argSqrd.msw & 0xffc00000 ) { /* Get about 32 bit precision in these: */ fix_up -= mul_32_32(0x898cc517, argSqrd.msw) / 6; } fix_up = mul_32_32(fix_up, LL_MSW(fixed_arg)); adj = accumulator.lsw; /* temp save */ accumulator.lsw -= fix_up; if ( accumulator.lsw > adj ) XSIG_LL(accumulator) --; echange = round_Xsig(&accumulator); setexponentpos(&result, echange - 1); } significand(&result) = XSIG_LL(accumulator); setsign(&result, getsign(st0_ptr)); FPU_copy_to_reg0(&result, TAG_Valid); #ifdef PARANOID if ( (exponent(&result) >= 0) && (significand(&result) > 0x8000000000000000LL) ) { EXCEPTION(EX_INTERNAL|0x150); } #endif /* PARANOID */ } /*--- poly_cos() ------------------------------------------------------------+ | | +---------------------------------------------------------------------------*/ void poly_cos(FPU_REG *st0_ptr) { FPU_REG result; long int exponent, exp2, echange; Xsig accumulator, argSqrd, fix_up, argTo4; unsigned long long fixed_arg; #ifdef PARANOID if ( (exponent(st0_ptr) > 0) || ((exponent(st0_ptr) == 0) && (significand(st0_ptr) > 0xc90fdaa22168c234LL)) ) { EXCEPTION(EX_Invalid); FPU_copy_to_reg0(&CONST_QNaN, TAG_Special); return; } #endif /* PARANOID */ exponent = exponent(st0_ptr); accumulator.lsw = accumulator.midw = accumulator.msw = 0; if ( (exponent < -1) || ((exponent == -1) && (st0_ptr->sigh <= 0xb00d6f54)) ) { /* arg is < 0.687705 */ argSqrd.msw = st0_ptr->sigh; argSqrd.midw = st0_ptr->sigl; argSqrd.lsw = 0; mul64_Xsig(&argSqrd, &significand(st0_ptr)); if ( exponent < -1 ) { /* shift the argument right by the required places */ shr_Xsig(&argSqrd, 2*(-1-exponent)); } argTo4.msw = argSqrd.msw; argTo4.midw = argSqrd.midw; argTo4.lsw = argSqrd.lsw; mul_Xsig_Xsig(&argTo4, &argTo4); polynomial_Xsig(&accumulator, &XSIG_LL(argTo4), neg_terms_h, N_COEFF_NH-1); mul_Xsig_Xsig(&accumulator, &argSqrd); negate_Xsig(&accumulator); polynomial_Xsig(&accumulator, &XSIG_LL(argTo4), pos_terms_h, N_COEFF_PH-1); negate_Xsig(&accumulator); mul64_Xsig(&accumulator, &significand(st0_ptr)); mul64_Xsig(&accumulator, &significand(st0_ptr)); shr_Xsig(&accumulator, -2*(1+exponent)); shr_Xsig(&accumulator, 3); negate_Xsig(&accumulator); add_Xsig_Xsig(&accumulator, &argSqrd); shr_Xsig(&accumulator, 1); /* It doesn't matter if accumulator is all zero here, the following code will work ok */ negate_Xsig(&accumulator); if ( accumulator.lsw & 0x80000000 ) XSIG_LL(accumulator) ++; if ( accumulator.msw == 0 ) { /* The result is 1.0 */ FPU_copy_to_reg0(&CONST_1, TAG_Valid); return; } else { significand(&result) = XSIG_LL(accumulator); /* will be a valid positive nr with expon = -1 */ setexponentpos(&result, -1); } } else { fixed_arg = significand(st0_ptr); if ( exponent == 0 ) { /* The argument is >= 1.0 */ /* Put the binary point at the left. */ fixed_arg <<= 1; } /* pi/2 in hex is: 1.921fb54442d18469 898CC51701B839A2 52049C1 */ fixed_arg = 0x921fb54442d18469LL - fixed_arg; /* There is a special case which arises due to rounding, to fix here. */ if ( fixed_arg == 0xffffffffffffffffLL ) fixed_arg = 0; exponent = -1; exp2 = -1; /* A shift is needed here only for a narrow range of arguments, i.e. for fixed_arg approx 2^-32, but we pick up more... */ if ( !(LL_MSW(fixed_arg) & 0xffff0000) ) { fixed_arg <<= 16; exponent -= 16; exp2 -= 16; } XSIG_LL(argSqrd) = fixed_arg; argSqrd.lsw = 0; mul64_Xsig(&argSqrd, &fixed_arg); if ( exponent < -1 ) { /* shift the argument right by the required places */ shr_Xsig(&argSqrd, 2*(-1-exponent)); } argTo4.msw = argSqrd.msw; argTo4.midw = argSqrd.midw; argTo4.lsw = argSqrd.lsw; mul_Xsig_Xsig(&argTo4, &argTo4); polynomial_Xsig(&accumulator, &XSIG_LL(argTo4), neg_terms_l, N_COEFF_N-1); mul_Xsig_Xsig(&accumulator, &argSqrd); negate_Xsig(&accumulator); polynomial_Xsig(&accumulator, &XSIG_LL(argTo4), pos_terms_l, N_COEFF_P-1); shr_Xsig(&accumulator, 2); /* Divide by four */ accumulator.msw |= 0x80000000; /* Add 1.0 */ mul64_Xsig(&accumulator, &fixed_arg); mul64_Xsig(&accumulator, &fixed_arg); mul64_Xsig(&accumulator, &fixed_arg); /* Divide by four, FPU_REG compatible, etc */ exponent = 3*exponent; /* The minimum exponent difference is 3 */ shr_Xsig(&accumulator, exp2 - exponent); negate_Xsig(&accumulator); XSIG_LL(accumulator) += fixed_arg; /* The basic computation is complete. Now fix the answer to compensate for the error due to the approximation used for pi/2 */ /* This has an exponent of -65 */ XSIG_LL(fix_up) = 0x898cc51701b839a2ll; fix_up.lsw = 0; /* The fix-up needs to be improved for larger args */ if ( argSqrd.msw & 0xffc00000 ) { /* Get about 32 bit precision in these: */ fix_up.msw -= mul_32_32(0x898cc517, argSqrd.msw) / 2; fix_up.msw += mul_32_32(0x898cc517, argTo4.msw) / 24; } exp2 += norm_Xsig(&accumulator); shr_Xsig(&accumulator, 1); /* Prevent overflow */ exp2++; shr_Xsig(&fix_up, 65 + exp2); add_Xsig_Xsig(&accumulator, &fix_up); echange = round_Xsig(&accumulator); setexponentpos(&result, exp2 + echange); significand(&result) = XSIG_LL(accumulator); } FPU_copy_to_reg0(&result, TAG_Valid); #ifdef PARANOID if ( (exponent(&result) >= 0) && (significand(&result) > 0x8000000000000000LL) ) { EXCEPTION(EX_INTERNAL|0x151); } #endif /* PARANOID */ } |