n_argred.S revision 1.4.8.1
1 /* $NetBSD: n_argred.S,v 1.4.8.1 2000/08/14 21:24:15 ragge Exp $ */ 2 /* 3 * Copyright (c) 1985, 1993 4 * The Regents of the University of California. All rights reserved. 5 * 6 * Redistribution and use in source and binary forms, with or without 7 * modification, are permitted provided that the following conditions 8 * are met: 9 * 1. Redistributions of source code must retain the above copyright 10 * notice, this list of conditions and the following disclaimer. 11 * 2. Redistributions in binary form must reproduce the above copyright 12 * notice, this list of conditions and the following disclaimer in the 13 * documentation and/or other materials provided with the distribution. 14 * 3. All advertising materials mentioning features or use of this software 15 * must display the following acknowledgement: 16 * This product includes software developed by the University of 17 * California, Berkeley and its contributors. 18 * 4. Neither the name of the University nor the names of its contributors 19 * may be used to endorse or promote products derived from this software 20 * without specific prior written permission. 21 * 22 * THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND 23 * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE 24 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE 25 * ARE DISCLAIMED. IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE 26 * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL 27 * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS 28 * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) 29 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT 30 * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY 31 * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF 32 * SUCH DAMAGE. 33 * 34 * @(#)argred.s 8.1 (Berkeley) 6/4/93 35 */ 36 37 /* 38 * libm_argred implements Bob Corbett's argument reduction and 39 * libm_sincos implements Peter Tang's double precision sin/cos. 40 * 41 * Note: The two entry points libm_argred and libm_sincos are meant 42 * to be used only by _sin, _cos and _tan. 43 * 44 * method: true range reduction to [-pi/4,pi/4], P. Tang & B. Corbett 45 * S. McDonald, April 4, 1985 46 */ 47 #include <machine/asm.h> 48 .text 49 50 ENTRY(_libm_argred, 0) 51 /* 52 * Compare the argument with the largest possible that can 53 * be reduced by table lookup. r3 := |x| will be used in table_lookup . 54 */ 55 movd r0,r3 56 bgeq abs1 57 mnegd r3,r3 58 abs1: 59 cmpd r3,$0d+4.55530934770520019583e+01 60 blss small_arg 61 jsb trigred 62 ret 63 small_arg: 64 jsb table_lookup 65 ret 66 /* 67 * At this point, 68 * r0 contains the quadrant number, 0, 1, 2, or 3; 69 * r2/r1 contains the reduced argument as a D-format number; 70 * r3 contains a F-format extension to the reduced argument; 71 * r4 contains a 0 or 1 corresponding to a sin or cos entry. 72 */ 73 ENTRY(_libm_sincos, 0) 74 /* 75 * Compensate for a cosine entry by adding one to the quadrant number. 76 */ 77 addl2 r4,r0 78 /* 79 * Polyd clobbers r5-r0 ; save X in r7/r6 . 80 * This can be avoided by rewriting trigred . 81 */ 82 movd r1,r6 83 /* 84 * Likewise, save alpha in r8 . 85 * This can be avoided by rewriting trigred . 86 */ 87 movf r3,r8 88 /* 89 * Odd or even quadrant? cosine if odd, sine otherwise. 90 * Save floor(quadrant/2) in r9 ; it determines the final sign. 91 */ 92 rotl $-1,r0,r9 93 blss cosine 94 sine: 95 muld2 r1,r1 # Xsq = X * X 96 cmpw $0x2480,r1 # [zl] Xsq > 2^-56? 97 blss 1f # [zl] yes, go ahead and do polyd 98 clrq r1 # [zl] work around 11/780 FPA polyd bug 99 1: 100 polyd r1,$7,sin_coef # Q = P(Xsq) , of deg 7 101 mulf3 $0f3.0,r8,r4 # beta = 3 * alpha 102 mulf2 r0,r4 # beta = Q * beta 103 addf2 r8,r4 # beta = alpha + beta 104 muld2 r6,r0 # S(X) = X * Q 105 /* cvtfd r4,r4 ... r5 = 0 after a polyd. */ 106 addd2 r4,r0 # S(X) = beta + S(X) 107 addd2 r6,r0 # S(X) = X + S(X) 108 brb done 109 cosine: 110 muld2 r6,r6 # Xsq = X * X 111 beql zero_arg 112 mulf2 r1,r8 # beta = X * alpha 113 polyd r6,$7,cos_coef /* Q = P'(Xsq) , of deg 7 */ 114 subd3 r0,r8,r0 # beta = beta - Q 115 subw2 $0x80,r6 # Xsq = Xsq / 2 116 addd2 r0,r6 # Xsq = Xsq + beta 117 zero_arg: 118 subd3 r6,$0d1.0,r0 # C(X) = 1 - Xsq 119 done: 120 blbc r9,even 121 mnegd r0,r0 122 even: 123 ret 124 125 .data 126 .align 2 127 128 sin_coef: 129 .double 0d-7.53080332264191085773e-13 # s7 = 2^-29 -1.a7f2504ffc49f8.. 130 .double 0d+1.60573519267703489121e-10 # s6 = 2^-21 1.611adaede473c8.. 131 .double 0d-2.50520965150706067211e-08 # s5 = 2^-1a -1.ae644921ed8382.. 132 .double 0d+2.75573191800593885716e-06 # s4 = 2^-13 1.71de3a4b884278.. 133 .double 0d-1.98412698411850507950e-04 # s3 = 2^-0d -1.a01a01a0125e7d.. 134 .double 0d+8.33333333333325688985e-03 # s2 = 2^-07 1.11111111110e50 135 .double 0d-1.66666666666666664354e-01 # s1 = 2^-03 -1.55555555555554 136 .double 0d+0.00000000000000000000e+00 # s0 = 0 137 138 cos_coef: 139 .double 0d-1.13006966202629430300e-11 # s7 = 2^-25 -1.8D9BA04D1374BE.. 140 .double 0d+2.08746646574796004700e-09 # s6 = 2^-1D 1.1EE632650350BA.. 141 .double 0d-2.75573073031284417300e-07 # s5 = 2^-16 -1.27E4F31411719E.. 142 .double 0d+2.48015872682668025200e-05 # s4 = 2^-10 1.A01A0196B902E8.. 143 .double 0d-1.38888888888464709200e-03 # s3 = 2^-0A -1.6C16C16C11FACE.. 144 .double 0d+4.16666666666664761400e-02 # s2 = 2^-05 1.5555555555539E 145 .double 0d+0.00000000000000000000e+00 # s1 = 0 146 .double 0d+0.00000000000000000000e+00 # s0 = 0 147 148 /* 149 * Multiples of pi/2 expressed as the sum of three doubles, 150 * 151 * trailing: n * pi/2 , n = 0, 1, 2, ..., 29 152 * trailing[n] , 153 * 154 * middle: n * pi/2 , n = 0, 1, 2, ..., 29 155 * middle[n] , 156 * 157 * leading: n * pi/2 , n = 0, 1, 2, ..., 29 158 * leading[n] , 159 * 160 * where 161 * leading[n] := (n * pi/2) rounded, 162 * middle[n] := (n * pi/2 - leading[n]) rounded, 163 * trailing[n] := (( n * pi/2 - leading[n]) - middle[n]) rounded . 164 */ 165 trailing: 166 .double 0d+0.00000000000000000000e+00 # 0 * pi/2 trailing 167 .double 0d+4.33590506506189049611e-35 # 1 * pi/2 trailing 168 .double 0d+8.67181013012378099223e-35 # 2 * pi/2 trailing 169 .double 0d+1.30077151951856714215e-34 # 3 * pi/2 trailing 170 .double 0d+1.73436202602475619845e-34 # 4 * pi/2 trailing 171 .double 0d-1.68390735624352669192e-34 # 5 * pi/2 trailing 172 .double 0d+2.60154303903713428430e-34 # 6 * pi/2 trailing 173 .double 0d-8.16726343231148352150e-35 # 7 * pi/2 trailing 174 .double 0d+3.46872405204951239689e-34 # 8 * pi/2 trailing 175 .double 0d+3.90231455855570147991e-34 # 9 * pi/2 trailing 176 .double 0d-3.36781471248705338384e-34 # 10 * pi/2 trailing 177 .double 0d-1.06379439835298071785e-33 # 11 * pi/2 trailing 178 .double 0d+5.20308607807426856861e-34 # 12 * pi/2 trailing 179 .double 0d+5.63667658458045770509e-34 # 13 * pi/2 trailing 180 .double 0d-1.63345268646229670430e-34 # 14 * pi/2 trailing 181 .double 0d-1.19986217995610764801e-34 # 15 * pi/2 trailing 182 .double 0d+6.93744810409902479378e-34 # 16 * pi/2 trailing 183 .double 0d-8.03640094449267300110e-34 # 17 * pi/2 trailing 184 .double 0d+7.80462911711140295982e-34 # 18 * pi/2 trailing 185 .double 0d-7.16921993148029483506e-34 # 19 * pi/2 trailing 186 .double 0d-6.73562942497410676769e-34 # 20 * pi/2 trailing 187 .double 0d-6.30203891846791677593e-34 # 21 * pi/2 trailing 188 .double 0d-2.12758879670596143570e-33 # 22 * pi/2 trailing 189 .double 0d+2.53800212047402350390e-33 # 23 * pi/2 trailing 190 .double 0d+1.04061721561485371372e-33 # 24 * pi/2 trailing 191 .double 0d+6.11729905311472319056e-32 # 25 * pi/2 trailing 192 .double 0d+1.12733531691609154102e-33 # 26 * pi/2 trailing 193 .double 0d-3.70049587943078297272e-34 # 27 * pi/2 trailing 194 .double 0d-3.26690537292459340860e-34 # 28 * pi/2 trailing 195 .double 0d-1.14812616507957271361e-34 # 29 * pi/2 trailing 196 197 middle: 198 .double 0d+0.00000000000000000000e+00 # 0 * pi/2 middle 199 .double 0d+5.72118872610983179676e-18 # 1 * pi/2 middle 200 .double 0d+1.14423774522196635935e-17 # 2 * pi/2 middle 201 .double 0d-3.83475850529283316309e-17 # 3 * pi/2 middle 202 .double 0d+2.28847549044393271871e-17 # 4 * pi/2 middle 203 .double 0d-2.69052076007086676522e-17 # 5 * pi/2 middle 204 .double 0d-7.66951701058566632618e-17 # 6 * pi/2 middle 205 .double 0d-1.54628301484890040587e-17 # 7 * pi/2 middle 206 .double 0d+4.57695098088786543741e-17 # 8 * pi/2 middle 207 .double 0d+1.07001849766246313192e-16 # 9 * pi/2 middle 208 .double 0d-5.38104152014173353044e-17 # 10 * pi/2 middle 209 .double 0d-2.14622680169080983801e-16 # 11 * pi/2 middle 210 .double 0d-1.53390340211713326524e-16 # 12 * pi/2 middle 211 .double 0d-9.21580002543456677056e-17 # 13 * pi/2 middle 212 .double 0d-3.09256602969780081173e-17 # 14 * pi/2 middle 213 .double 0d+3.03066796603896507006e-17 # 15 * pi/2 middle 214 .double 0d+9.15390196177573087482e-17 # 16 * pi/2 middle 215 .double 0d+1.52771359575124969107e-16 # 17 * pi/2 middle 216 .double 0d+2.14003699532492626384e-16 # 18 * pi/2 middle 217 .double 0d-1.68853170360202329427e-16 # 19 * pi/2 middle 218 .double 0d-1.07620830402834670609e-16 # 20 * pi/2 middle 219 .double 0d+3.97700719404595604379e-16 # 21 * pi/2 middle 220 .double 0d-4.29245360338161967602e-16 # 22 * pi/2 middle 221 .double 0d-3.68013020380794313406e-16 # 23 * pi/2 middle 222 .double 0d-3.06780680423426653047e-16 # 24 * pi/2 middle 223 .double 0d-2.45548340466059054318e-16 # 25 * pi/2 middle 224 .double 0d-1.84316000508691335411e-16 # 26 * pi/2 middle 225 .double 0d-1.23083660551323675053e-16 # 27 * pi/2 middle 226 .double 0d-6.18513205939560162346e-17 # 28 * pi/2 middle 227 .double 0d-6.18980636588357585202e-19 # 29 * pi/2 middle 228 229 leading: 230 .double 0d+0.00000000000000000000e+00 # 0 * pi/2 leading 231 .double 0d+1.57079632679489661351e+00 # 1 * pi/2 leading 232 .double 0d+3.14159265358979322702e+00 # 2 * pi/2 leading 233 .double 0d+4.71238898038468989604e+00 # 3 * pi/2 leading 234 .double 0d+6.28318530717958645404e+00 # 4 * pi/2 leading 235 .double 0d+7.85398163397448312306e+00 # 5 * pi/2 leading 236 .double 0d+9.42477796076937979208e+00 # 6 * pi/2 leading 237 .double 0d+1.09955742875642763501e+01 # 7 * pi/2 leading 238 .double 0d+1.25663706143591729081e+01 # 8 * pi/2 leading 239 .double 0d+1.41371669411540694661e+01 # 9 * pi/2 leading 240 .double 0d+1.57079632679489662461e+01 # 10 * pi/2 leading 241 .double 0d+1.72787595947438630262e+01 # 11 * pi/2 leading 242 .double 0d+1.88495559215387595842e+01 # 12 * pi/2 leading 243 .double 0d+2.04203522483336561422e+01 # 13 * pi/2 leading 244 .double 0d+2.19911485751285527002e+01 # 14 * pi/2 leading 245 .double 0d+2.35619449019234492582e+01 # 15 * pi/2 leading 246 .double 0d+2.51327412287183458162e+01 # 16 * pi/2 leading 247 .double 0d+2.67035375555132423742e+01 # 17 * pi/2 leading 248 .double 0d+2.82743338823081389322e+01 # 18 * pi/2 leading 249 .double 0d+2.98451302091030359342e+01 # 19 * pi/2 leading 250 .double 0d+3.14159265358979324922e+01 # 20 * pi/2 leading 251 .double 0d+3.29867228626928286062e+01 # 21 * pi/2 leading 252 .double 0d+3.45575191894877260523e+01 # 22 * pi/2 leading 253 .double 0d+3.61283155162826226103e+01 # 23 * pi/2 leading 254 .double 0d+3.76991118430775191683e+01 # 24 * pi/2 leading 255 .double 0d+3.92699081698724157263e+01 # 25 * pi/2 leading 256 .double 0d+4.08407044966673122843e+01 # 26 * pi/2 leading 257 .double 0d+4.24115008234622088423e+01 # 27 * pi/2 leading 258 .double 0d+4.39822971502571054003e+01 # 28 * pi/2 leading 259 .double 0d+4.55530934770520019583e+01 # 29 * pi/2 leading 260 261 twoOverPi: 262 .double 0d+6.36619772367581343076e-01 263 .text 264 .align 1 265 266 table_lookup: 267 muld3 r3,twoOverPi,r0 268 cvtrdl r0,r0 # n = nearest int to ((2/pi)*|x|) rnded 269 mull3 $8,r0,r5 270 subd2 leading(r5),r3 # p = (|x| - leading n*pi/2) exactly 271 subd3 middle(r5),r3,r1 # q = (p - middle n*pi/2) rounded 272 subd2 r1,r3 # r = (p - q) 273 subd2 middle(r5),r3 # r = r - middle n*pi/2 274 subd2 trailing(r5),r3 # r = r - trailing n*pi/2 rounded 275 /* 276 * If the original argument was negative, 277 * negate the reduce argument and 278 * adjust the octant/quadrant number. 279 */ 280 tstw 4(ap) 281 bgeq abs2 282 mnegf r1,r1 283 mnegf r3,r3 284 /* subb3 r0,$8,r0 ...used for pi/4 reduction -S.McD */ 285 subb3 r0,$4,r0 286 abs2: 287 /* 288 * Clear all unneeded octant/quadrant bits. 289 */ 290 /* bicb2 $0xf8,r0 ...used for pi/4 reduction -S.McD */ 291 bicb2 $0xfc,r0 292 rsb 293 /* 294 * p.0 295 */ 296 .text 297 .align 2 298 /* 299 * Only 256 (actually 225) bits of 2/pi are needed for VAX double 300 * precision; this was determined by enumerating all the nearest 301 * machine integer multiples of pi/2 using continued fractions. 302 * (8a8d3673775b7ff7 required the most bits.) -S.McD 303 */ 304 .long 0 305 .long 0 306 .long 0xaef1586d 307 .long 0x9458eaf7 308 .long 0x10e4107f 309 .long 0xd8a5664f 310 .long 0x4d377036 311 .long 0x09d5f47d 312 .long 0x91054a7f 313 .long 0xbe60db93 314 bits2opi: 315 .long 0x00000028 316 .long 0 317 /* 318 * Note: wherever you see the word `octant', read `quadrant'. 319 * Currently this code is set up for pi/2 argument reduction. 320 * By uncommenting/commenting the appropriate lines, it will 321 * also serve as a pi/4 argument reduction code. 322 */ 323 324 /* p.1 325 * Trigred preforms argument reduction 326 * for the trigonometric functions. It 327 * takes one input argument, a D-format 328 * number in r1/r0 . The magnitude of 329 * the input argument must be greater 330 * than or equal to 1/2 . Trigred produces 331 * three results: the number of the octant 332 * occupied by the argument, the reduced 333 * argument, and an extension of the 334 * reduced argument. The octant number is 335 * returned in r0 . The reduced argument 336 * is returned as a D-format number in 337 * r2/r1 . An 8 bit extension of the 338 * reduced argument is returned as an 339 * F-format number in r3. 340 * p.2 341 */ 342 trigred: 343 /* 344 * Save the sign of the input argument. 345 */ 346 movw r0,-(sp) 347 /* 348 * Extract the exponent field. 349 */ 350 extzv $7,$7,r0,r2 351 /* 352 * Convert the fraction part of the input 353 * argument into a quadword integer. 354 */ 355 bicw2 $0xff80,r0 356 bisb2 $0x80,r0 # -S.McD 357 rotl $16,r0,r0 358 rotl $16,r1,r1 359 /* 360 * If r1 is negative, add 1 to r0 . This 361 * adjustment is made so that the two's 362 * complement multiplications done later 363 * will produce unsigned results. 364 */ 365 bgeq posmid 366 incl r0 367 posmid: 368 /* p.3 369 * 370 * Set r3 to the address of the first quadword 371 * used to obtain the needed portion of 2/pi . 372 * The address is longword aligned to ensure 373 * efficient access. 374 */ 375 ashl $-3,r2,r3 376 bicb2 $3,r3 377 mnegl r3,r3 378 movab bits2opi[r3],r3 379 /* 380 * Set r2 to the size of the shift needed to 381 * obtain the correct portion of 2/pi . 382 */ 383 bicb2 $0xe0,r2 384 /* p.4 385 * 386 * Move the needed 128 bits of 2/pi into 387 * r11 - r8 . Adjust the numbers to allow 388 * for unsigned multiplication. 389 */ 390 ashq r2,(r3),r10 391 392 subl2 $4,r3 393 ashq r2,(r3),r9 394 bgeq signoff1 395 incl r11 396 signoff1: 397 subl2 $4,r3 398 ashq r2,(r3),r8 399 bgeq signoff2 400 incl r10 401 signoff2: 402 subl2 $4,r3 403 ashq r2,(r3),r7 404 bgeq signoff3 405 incl r9 406 signoff3: 407 /* p.5 408 * 409 * Multiply the contents of r0/r1 by the 410 * slice of 2/pi in r11 - r8 . 411 */ 412 emul r0,r8,$0,r4 413 emul r0,r9,r5,r5 414 emul r0,r10,r6,r6 415 416 emul r1,r8,$0,r7 417 emul r1,r9,r8,r8 418 emul r1,r10,r9,r9 419 emul r1,r11,r10,r10 420 421 addl2 r4,r8 422 adwc r5,r9 423 adwc r6,r10 424 /* p.6 425 * 426 * If there are more than five leading zeros 427 * after the first two quotient bits or if there 428 * are more than five leading ones after the first 429 * two quotient bits, generate more fraction bits. 430 * Otherwise, branch to code to produce the result. 431 */ 432 bicl3 $0xc1ffffff,r10,r4 433 beql more1 434 cmpl $0x3e000000,r4 435 bneq result 436 more1: 437 /* p.7 438 * 439 * generate another 32 result bits. 440 */ 441 subl2 $4,r3 442 ashq r2,(r3),r5 443 bgeq signoff4 444 445 emul r1,r6,$0,r4 446 addl2 r1,r5 447 emul r0,r6,r5,r5 448 addl2 r0,r6 449 brb addbits1 450 451 signoff4: 452 emul r1,r6,$0,r4 453 emul r0,r6,r5,r5 454 455 addbits1: 456 addl2 r5,r7 457 adwc r6,r8 458 adwc $0,r9 459 adwc $0,r10 460 /* p.8 461 * 462 * Check for massive cancellation. 463 */ 464 bicl3 $0xc0000000,r10,r6 465 /* bneq more2 -S.McD Test was backwards */ 466 beql more2 467 cmpl $0x3fffffff,r6 468 bneq result 469 more2: 470 /* p.9 471 * 472 * If massive cancellation has occurred, 473 * generate another 24 result bits. 474 * Testing has shown there will always be 475 * enough bits after this point. 476 */ 477 subl2 $4,r3 478 ashq r2,(r3),r5 479 bgeq signoff5 480 481 emul r0,r6,r4,r5 482 addl2 r0,r6 483 brb addbits2 484 485 signoff5: 486 emul r0,r6,r4,r5 487 488 addbits2: 489 addl2 r6,r7 490 adwc $0,r8 491 adwc $0,r9 492 adwc $0,r10 493 /* p.10 494 * 495 * The following code produces the reduced 496 * argument from the product bits contained 497 * in r10 - r7 . 498 */ 499 result: 500 /* 501 * Extract the octant number from r10 . 502 */ 503 /* extzv $29,$3,r10,r0 ...used for pi/4 reduction -S.McD */ 504 extzv $30,$2,r10,r0 505 /* 506 * Clear the octant bits in r10 . 507 */ 508 /* bicl2 $0xe0000000,r10 ...used for pi/4 reduction -S.McD */ 509 bicl2 $0xc0000000,r10 510 /* 511 * Zero the sign flag. 512 */ 513 clrl r5 514 /* p.11 515 * 516 * Check to see if the fraction is greater than 517 * or equal to one-half. If it is, add one 518 * to the octant number, set the sign flag 519 * on, and replace the fraction with 1 minus 520 * the fraction. 521 */ 522 /* bitl $0x10000000,r10 ...used for pi/4 reduction -S.McD */ 523 bitl $0x20000000,r10 524 beql small 525 incl r0 526 incl r5 527 /* subl3 r10,$0x1fffffff,r10 ...used for pi/4 reduction -S.McD */ 528 subl3 r10,$0x3fffffff,r10 529 mcoml r9,r9 530 mcoml r8,r8 531 mcoml r7,r7 532 small: 533 /* p.12 534 * 535 * Test whether the first 29 bits of the ...used for pi/4 reduction -S.McD 536 * Test whether the first 30 bits of the 537 * fraction are zero. 538 */ 539 tstl r10 540 beql tiny 541 /* 542 * Find the position of the first one bit in r10 . 543 */ 544 cvtld r10,r1 545 extzv $7,$7,r1,r1 546 /* 547 * Compute the size of the shift needed. 548 */ 549 subl3 r1,$32,r6 550 /* 551 * Shift up the high order 64 bits of the 552 * product. 553 */ 554 ashq r6,r9,r10 555 ashq r6,r8,r9 556 brb mult 557 /* p.13 558 * 559 * Test to see if the sign bit of r9 is on. 560 */ 561 tiny: 562 tstl r9 563 bgeq tinier 564 /* 565 * If it is, shift the product bits up 32 bits. 566 */ 567 movl $32,r6 568 movq r8,r10 569 tstl r10 570 brb mult 571 /* p.14 572 * 573 * Test whether r9 is zero. It is probably 574 * impossible for both r10 and r9 to be 575 * zero, but until proven to be so, the test 576 * must be made. 577 */ 578 tinier: 579 beql zero 580 /* 581 * Find the position of the first one bit in r9 . 582 */ 583 cvtld r9,r1 584 extzv $7,$7,r1,r1 585 /* 586 * Compute the size of the shift needed. 587 */ 588 subl3 r1,$32,r1 589 addl3 $32,r1,r6 590 /* 591 * Shift up the high order 64 bits of the 592 * product. 593 */ 594 ashq r1,r8,r10 595 ashq r1,r7,r9 596 brb mult 597 /* p.15 598 * 599 * The following code sets the reduced 600 * argument to zero. 601 */ 602 zero: 603 clrl r1 604 clrl r2 605 clrl r3 606 brw return 607 /* p.16 608 * 609 * At this point, r0 contains the octant number, 610 * r6 indicates the number of bits the fraction 611 * has been shifted, r5 indicates the sign of 612 * the fraction, r11/r10 contain the high order 613 * 64 bits of the fraction, and the condition 614 * codes indicate where the sign bit of r10 615 * is on. The following code multiplies the 616 * fraction by pi/2 . 617 */ 618 mult: 619 /* 620 * Save r11/r10 in r4/r1 . -S.McD 621 */ 622 movl r11,r4 623 movl r10,r1 624 /* 625 * If the sign bit of r10 is on, add 1 to r11 . 626 */ 627 bgeq signoff6 628 incl r11 629 signoff6: 630 /* p.17 631 * 632 * Move pi/2 into r3/r2 . 633 */ 634 movq $0xc90fdaa22168c235,r2 635 /* 636 * Multiply the fraction by the portion of pi/2 637 * in r2 . 638 */ 639 emul r2,r10,$0,r7 640 emul r2,r11,r8,r7 641 /* 642 * Multiply the fraction by the portion of pi/2 643 * in r3 . 644 */ 645 emul r3,r10,$0,r9 646 emul r3,r11,r10,r10 647 /* 648 * Add the product bits together. 649 */ 650 addl2 r7,r9 651 adwc r8,r10 652 adwc $0,r11 653 /* 654 * Compensate for not sign extending r8 above.-S.McD 655 */ 656 tstl r8 657 bgeq signoff6a 658 decl r11 659 signoff6a: 660 /* 661 * Compensate for r11/r10 being unsigned. -S.McD 662 */ 663 addl2 r2,r10 664 adwc r3,r11 665 /* 666 * Compensate for r3/r2 being unsigned. -S.McD 667 */ 668 addl2 r1,r10 669 adwc r4,r11 670 /* p.18 671 * 672 * If the sign bit of r11 is zero, shift the 673 * product bits up one bit and increment r6 . 674 */ 675 blss signon 676 incl r6 677 ashq $1,r10,r10 678 tstl r9 679 bgeq signoff7 680 incl r10 681 signoff7: 682 signon: 683 /* p.19 684 * 685 * Shift the 56 most significant product 686 * bits into r9/r8 . The sign extension 687 * will be handled later. 688 */ 689 ashq $-8,r10,r8 690 /* 691 * Convert the low order 8 bits of r10 692 * into an F-format number. 693 */ 694 cvtbf r10,r3 695 /* 696 * If the result of the conversion was 697 * negative, add 1 to r9/r8 . 698 */ 699 bgeq chop 700 incl r8 701 adwc $0,r9 702 /* 703 * If r9 is now zero, branch to special 704 * code to handle that possibility. 705 */ 706 beql carryout 707 chop: 708 /* p.20 709 * 710 * Convert the number in r9/r8 into 711 * D-format number in r2/r1 . 712 */ 713 rotl $16,r8,r2 714 rotl $16,r9,r1 715 /* 716 * Set the exponent field to the appropriate 717 * value. Note that the extra bits created by 718 * sign extension are now eliminated. 719 */ 720 subw3 r6,$131,r6 721 insv r6,$7,$9,r1 722 /* 723 * Set the exponent field of the F-format 724 * number in r3 to the appropriate value. 725 */ 726 tstf r3 727 beql return 728 /* extzv $7,$8,r3,r4 -S.McD */ 729 extzv $7,$7,r3,r4 730 addw2 r4,r6 731 /* subw2 $217,r6 -S.McD */ 732 subw2 $64,r6 733 insv r6,$7,$8,r3 734 brb return 735 /* p.21 736 * 737 * The following code generates the appropriate 738 * result for the unlikely possibility that 739 * rounding the number in r9/r8 resulted in 740 * a carry out. 741 */ 742 carryout: 743 clrl r1 744 clrl r2 745 subw3 r6,$132,r6 746 insv r6,$7,$9,r1 747 tstf r3 748 beql return 749 extzv $7,$8,r3,r4 750 addw2 r4,r6 751 subw2 $218,r6 752 insv r6,$7,$8,r3 753 /* p.22 754 * 755 * The following code makes an needed 756 * adjustments to the signs of the 757 * results or to the octant number, and 758 * then returns. 759 */ 760 return: 761 /* 762 * Test if the fraction was greater than or 763 * equal to 1/2 . If so, negate the reduced 764 * argument. 765 */ 766 blbc r5,signoff8 767 mnegf r1,r1 768 mnegf r3,r3 769 signoff8: 770 /* p.23 771 * 772 * If the original argument was negative, 773 * negate the reduce argument and 774 * adjust the octant number. 775 */ 776 tstw (sp)+ 777 bgeq signoff9 778 mnegf r1,r1 779 mnegf r3,r3 780 /* subb3 r0,$8,r0 ...used for pi/4 reduction -S.McD */ 781 subb3 r0,$4,r0 782 signoff9: 783 /* 784 * Clear all unneeded octant bits. 785 * 786 * bicb2 $0xf8,r0 ...used for pi/4 reduction -S.McD */ 787 bicb2 $0xfc,r0 788 /* 789 * Return. 790 */ 791 rsb 792
Indexes created Thu Oct 02 01:09:59 GMT 2025