n_argred.S revision 1.2
1 /* $NetBSD: n_argred.S,v 1.2 1998/10/31 02:06:01 matt 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 .text 48 .globl libm$argred 49 .type libm$argred,@label 50 .globl libm$sincos 51 .type libm$sincos,@label 52 .align 1 53 54 libm$argred: 55 /* 56 * Compare the argument with the largest possible that can 57 * be reduced by table lookup. r3 := |x| will be used in table_lookup . 58 */ 59 movd r0,r3 60 bgeq abs1 61 mnegd r3,r3 62 abs1: 63 cmpd r3,$0d+4.55530934770520019583e+01 64 blss small_arg 65 jsb trigred 66 rsb 67 small_arg: 68 jsb table_lookup 69 rsb 70 /* 71 * At this point, 72 * r0 contains the quadrant number, 0, 1, 2, or 3; 73 * r2/r1 contains the reduced argument as a D-format number; 74 * r3 contains a F-format extension to the reduced argument; 75 * r4 contains a 0 or 1 corresponding to a sin or cos entry. 76 */ 77 libm$sincos: 78 /* 79 * Compensate for a cosine entry by adding one to the quadrant number. 80 */ 81 addl2 r4,r0 82 /* 83 * Polyd clobbers r5-r0 ; save X in r7/r6 . 84 * This can be avoided by rewriting trigred . 85 */ 86 movd r1,r6 87 /* 88 * Likewise, save alpha in r8 . 89 * This can be avoided by rewriting trigred . 90 */ 91 movf r3,r8 92 /* 93 * Odd or even quadrant? cosine if odd, sine otherwise. 94 * Save floor(quadrant/2) in r9 ; it determines the final sign. 95 */ 96 rotl $-1,r0,r9 97 blss cosine 98 sine: 99 muld2 r1,r1 # Xsq = X * X 100 cmpw $0x2480,r1 # [zl] Xsq > 2^-56? 101 blss 1f # [zl] yes, go ahead and do polyd 102 clrq r1 # [zl] work around 11/780 FPA polyd bug 103 1: 104 polyd r1,$7,sin_coef # Q = P(Xsq) , of deg 7 105 mulf3 $0f3.0,r8,r4 # beta = 3 * alpha 106 mulf2 r0,r4 # beta = Q * beta 107 addf2 r8,r4 # beta = alpha + beta 108 muld2 r6,r0 # S(X) = X * Q 109 /* cvtfd r4,r4 ... r5 = 0 after a polyd. */ 110 addd2 r4,r0 # S(X) = beta + S(X) 111 addd2 r6,r0 # S(X) = X + S(X) 112 brb done 113 cosine: 114 muld2 r6,r6 # Xsq = X * X 115 beql zero_arg 116 mulf2 r1,r8 # beta = X * alpha 117 polyd r6,$7,cos_coef /* Q = P'(Xsq) , of deg 7 */ 118 subd3 r0,r8,r0 # beta = beta - Q 119 subw2 $0x80,r6 # Xsq = Xsq / 2 120 addd2 r0,r6 # Xsq = Xsq + beta 121 zero_arg: 122 subd3 r6,$0d1.0,r0 # C(X) = 1 - Xsq 123 done: 124 blbc r9,even 125 mnegd r0,r0 126 even: 127 rsb 128 129 .data 130 .align 2 131 132 sin_coef: 133 .double 0d-7.53080332264191085773e-13 # s7 = 2^-29 -1.a7f2504ffc49f8.. 134 .double 0d+1.60573519267703489121e-10 # s6 = 2^-21 1.611adaede473c8.. 135 .double 0d-2.50520965150706067211e-08 # s5 = 2^-1a -1.ae644921ed8382.. 136 .double 0d+2.75573191800593885716e-06 # s4 = 2^-13 1.71de3a4b884278.. 137 .double 0d-1.98412698411850507950e-04 # s3 = 2^-0d -1.a01a01a0125e7d.. 138 .double 0d+8.33333333333325688985e-03 # s2 = 2^-07 1.11111111110e50 139 .double 0d-1.66666666666666664354e-01 # s1 = 2^-03 -1.55555555555554 140 .double 0d+0.00000000000000000000e+00 # s0 = 0 141 142 cos_coef: 143 .double 0d-1.13006966202629430300e-11 # s7 = 2^-25 -1.8D9BA04D1374BE.. 144 .double 0d+2.08746646574796004700e-09 # s6 = 2^-1D 1.1EE632650350BA.. 145 .double 0d-2.75573073031284417300e-07 # s5 = 2^-16 -1.27E4F31411719E.. 146 .double 0d+2.48015872682668025200e-05 # s4 = 2^-10 1.A01A0196B902E8.. 147 .double 0d-1.38888888888464709200e-03 # s3 = 2^-0A -1.6C16C16C11FACE.. 148 .double 0d+4.16666666666664761400e-02 # s2 = 2^-05 1.5555555555539E 149 .double 0d+0.00000000000000000000e+00 # s1 = 0 150 .double 0d+0.00000000000000000000e+00 # s0 = 0 151 152 /* 153 * Multiples of pi/2 expressed as the sum of three doubles, 154 * 155 * trailing: n * pi/2 , n = 0, 1, 2, ..., 29 156 * trailing[n] , 157 * 158 * middle: n * pi/2 , n = 0, 1, 2, ..., 29 159 * middle[n] , 160 * 161 * leading: n * pi/2 , n = 0, 1, 2, ..., 29 162 * leading[n] , 163 * 164 * where 165 * leading[n] := (n * pi/2) rounded, 166 * middle[n] := (n * pi/2 - leading[n]) rounded, 167 * trailing[n] := (( n * pi/2 - leading[n]) - middle[n]) rounded . 168 */ 169 trailing: 170 .double 0d+0.00000000000000000000e+00 # 0 * pi/2 trailing 171 .double 0d+4.33590506506189049611e-35 # 1 * pi/2 trailing 172 .double 0d+8.67181013012378099223e-35 # 2 * pi/2 trailing 173 .double 0d+1.30077151951856714215e-34 # 3 * pi/2 trailing 174 .double 0d+1.73436202602475619845e-34 # 4 * pi/2 trailing 175 .double 0d-1.68390735624352669192e-34 # 5 * pi/2 trailing 176 .double 0d+2.60154303903713428430e-34 # 6 * pi/2 trailing 177 .double 0d-8.16726343231148352150e-35 # 7 * pi/2 trailing 178 .double 0d+3.46872405204951239689e-34 # 8 * pi/2 trailing 179 .double 0d+3.90231455855570147991e-34 # 9 * pi/2 trailing 180 .double 0d-3.36781471248705338384e-34 # 10 * pi/2 trailing 181 .double 0d-1.06379439835298071785e-33 # 11 * pi/2 trailing 182 .double 0d+5.20308607807426856861e-34 # 12 * pi/2 trailing 183 .double 0d+5.63667658458045770509e-34 # 13 * pi/2 trailing 184 .double 0d-1.63345268646229670430e-34 # 14 * pi/2 trailing 185 .double 0d-1.19986217995610764801e-34 # 15 * pi/2 trailing 186 .double 0d+6.93744810409902479378e-34 # 16 * pi/2 trailing 187 .double 0d-8.03640094449267300110e-34 # 17 * pi/2 trailing 188 .double 0d+7.80462911711140295982e-34 # 18 * pi/2 trailing 189 .double 0d-7.16921993148029483506e-34 # 19 * pi/2 trailing 190 .double 0d-6.73562942497410676769e-34 # 20 * pi/2 trailing 191 .double 0d-6.30203891846791677593e-34 # 21 * pi/2 trailing 192 .double 0d-2.12758879670596143570e-33 # 22 * pi/2 trailing 193 .double 0d+2.53800212047402350390e-33 # 23 * pi/2 trailing 194 .double 0d+1.04061721561485371372e-33 # 24 * pi/2 trailing 195 .double 0d+6.11729905311472319056e-32 # 25 * pi/2 trailing 196 .double 0d+1.12733531691609154102e-33 # 26 * pi/2 trailing 197 .double 0d-3.70049587943078297272e-34 # 27 * pi/2 trailing 198 .double 0d-3.26690537292459340860e-34 # 28 * pi/2 trailing 199 .double 0d-1.14812616507957271361e-34 # 29 * pi/2 trailing 200 201 middle: 202 .double 0d+0.00000000000000000000e+00 # 0 * pi/2 middle 203 .double 0d+5.72118872610983179676e-18 # 1 * pi/2 middle 204 .double 0d+1.14423774522196635935e-17 # 2 * pi/2 middle 205 .double 0d-3.83475850529283316309e-17 # 3 * pi/2 middle 206 .double 0d+2.28847549044393271871e-17 # 4 * pi/2 middle 207 .double 0d-2.69052076007086676522e-17 # 5 * pi/2 middle 208 .double 0d-7.66951701058566632618e-17 # 6 * pi/2 middle 209 .double 0d-1.54628301484890040587e-17 # 7 * pi/2 middle 210 .double 0d+4.57695098088786543741e-17 # 8 * pi/2 middle 211 .double 0d+1.07001849766246313192e-16 # 9 * pi/2 middle 212 .double 0d-5.38104152014173353044e-17 # 10 * pi/2 middle 213 .double 0d-2.14622680169080983801e-16 # 11 * pi/2 middle 214 .double 0d-1.53390340211713326524e-16 # 12 * pi/2 middle 215 .double 0d-9.21580002543456677056e-17 # 13 * pi/2 middle 216 .double 0d-3.09256602969780081173e-17 # 14 * pi/2 middle 217 .double 0d+3.03066796603896507006e-17 # 15 * pi/2 middle 218 .double 0d+9.15390196177573087482e-17 # 16 * pi/2 middle 219 .double 0d+1.52771359575124969107e-16 # 17 * pi/2 middle 220 .double 0d+2.14003699532492626384e-16 # 18 * pi/2 middle 221 .double 0d-1.68853170360202329427e-16 # 19 * pi/2 middle 222 .double 0d-1.07620830402834670609e-16 # 20 * pi/2 middle 223 .double 0d+3.97700719404595604379e-16 # 21 * pi/2 middle 224 .double 0d-4.29245360338161967602e-16 # 22 * pi/2 middle 225 .double 0d-3.68013020380794313406e-16 # 23 * pi/2 middle 226 .double 0d-3.06780680423426653047e-16 # 24 * pi/2 middle 227 .double 0d-2.45548340466059054318e-16 # 25 * pi/2 middle 228 .double 0d-1.84316000508691335411e-16 # 26 * pi/2 middle 229 .double 0d-1.23083660551323675053e-16 # 27 * pi/2 middle 230 .double 0d-6.18513205939560162346e-17 # 28 * pi/2 middle 231 .double 0d-6.18980636588357585202e-19 # 29 * pi/2 middle 232 233 leading: 234 .double 0d+0.00000000000000000000e+00 # 0 * pi/2 leading 235 .double 0d+1.57079632679489661351e+00 # 1 * pi/2 leading 236 .double 0d+3.14159265358979322702e+00 # 2 * pi/2 leading 237 .double 0d+4.71238898038468989604e+00 # 3 * pi/2 leading 238 .double 0d+6.28318530717958645404e+00 # 4 * pi/2 leading 239 .double 0d+7.85398163397448312306e+00 # 5 * pi/2 leading 240 .double 0d+9.42477796076937979208e+00 # 6 * pi/2 leading 241 .double 0d+1.09955742875642763501e+01 # 7 * pi/2 leading 242 .double 0d+1.25663706143591729081e+01 # 8 * pi/2 leading 243 .double 0d+1.41371669411540694661e+01 # 9 * pi/2 leading 244 .double 0d+1.57079632679489662461e+01 # 10 * pi/2 leading 245 .double 0d+1.72787595947438630262e+01 # 11 * pi/2 leading 246 .double 0d+1.88495559215387595842e+01 # 12 * pi/2 leading 247 .double 0d+2.04203522483336561422e+01 # 13 * pi/2 leading 248 .double 0d+2.19911485751285527002e+01 # 14 * pi/2 leading 249 .double 0d+2.35619449019234492582e+01 # 15 * pi/2 leading 250 .double 0d+2.51327412287183458162e+01 # 16 * pi/2 leading 251 .double 0d+2.67035375555132423742e+01 # 17 * pi/2 leading 252 .double 0d+2.82743338823081389322e+01 # 18 * pi/2 leading 253 .double 0d+2.98451302091030359342e+01 # 19 * pi/2 leading 254 .double 0d+3.14159265358979324922e+01 # 20 * pi/2 leading 255 .double 0d+3.29867228626928286062e+01 # 21 * pi/2 leading 256 .double 0d+3.45575191894877260523e+01 # 22 * pi/2 leading 257 .double 0d+3.61283155162826226103e+01 # 23 * pi/2 leading 258 .double 0d+3.76991118430775191683e+01 # 24 * pi/2 leading 259 .double 0d+3.92699081698724157263e+01 # 25 * pi/2 leading 260 .double 0d+4.08407044966673122843e+01 # 26 * pi/2 leading 261 .double 0d+4.24115008234622088423e+01 # 27 * pi/2 leading 262 .double 0d+4.39822971502571054003e+01 # 28 * pi/2 leading 263 .double 0d+4.55530934770520019583e+01 # 29 * pi/2 leading 264 265 twoOverPi: 266 .double 0d+6.36619772367581343076e-01 267 .text 268 .align 1 269 270 table_lookup: 271 muld3 r3,twoOverPi,r0 272 cvtrdl r0,r0 # n = nearest int to ((2/pi)*|x|) rnded 273 mull3 $8,r0,r5 274 subd2 leading(r5),r3 # p = (|x| - leading n*pi/2) exactly 275 subd3 middle(r5),r3,r1 # q = (p - middle n*pi/2) rounded 276 subd2 r1,r3 # r = (p - q) 277 subd2 middle(r5),r3 # r = r - middle n*pi/2 278 subd2 trailing(r5),r3 # r = r - trailing n*pi/2 rounded 279 /* 280 * If the original argument was negative, 281 * negate the reduce argument and 282 * adjust the octant/quadrant number. 283 */ 284 tstw 4(ap) 285 bgeq abs2 286 mnegf r1,r1 287 mnegf r3,r3 288 /* subb3 r0,$8,r0 ...used for pi/4 reduction -S.McD */ 289 subb3 r0,$4,r0 290 abs2: 291 /* 292 * Clear all unneeded octant/quadrant bits. 293 */ 294 /* bicb2 $0xf8,r0 ...used for pi/4 reduction -S.McD */ 295 bicb2 $0xfc,r0 296 rsb 297 /* 298 * p.0 299 */ 300 .text 301 .align 2 302 /* 303 * Only 256 (actually 225) bits of 2/pi are needed for VAX double 304 * precision; this was determined by enumerating all the nearest 305 * machine integer multiples of pi/2 using continued fractions. 306 * (8a8d3673775b7ff7 required the most bits.) -S.McD 307 */ 308 .long 0 309 .long 0 310 .long 0xaef1586d 311 .long 0x9458eaf7 312 .long 0x10e4107f 313 .long 0xd8a5664f 314 .long 0x4d377036 315 .long 0x09d5f47d 316 .long 0x91054a7f 317 .long 0xbe60db93 318 bits2opi: 319 .long 0x00000028 320 .long 0 321 /* 322 * Note: wherever you see the word `octant', read `quadrant'. 323 * Currently this code is set up for pi/2 argument reduction. 324 * By uncommenting/commenting the appropriate lines, it will 325 * also serve as a pi/4 argument reduction code. 326 */ 327 328 /* p.1 329 * Trigred preforms argument reduction 330 * for the trigonometric functions. It 331 * takes one input argument, a D-format 332 * number in r1/r0 . The magnitude of 333 * the input argument must be greater 334 * than or equal to 1/2 . Trigred produces 335 * three results: the number of the octant 336 * occupied by the argument, the reduced 337 * argument, and an extension of the 338 * reduced argument. The octant number is 339 * returned in r0 . The reduced argument 340 * is returned as a D-format number in 341 * r2/r1 . An 8 bit extension of the 342 * reduced argument is returned as an 343 * F-format number in r3. 344 * p.2 345 */ 346 trigred: 347 /* 348 * Save the sign of the input argument. 349 */ 350 movw r0,-(sp) 351 /* 352 * Extract the exponent field. 353 */ 354 extzv $7,$7,r0,r2 355 /* 356 * Convert the fraction part of the input 357 * argument into a quadword integer. 358 */ 359 bicw2 $0xff80,r0 360 bisb2 $0x80,r0 # -S.McD 361 rotl $16,r0,r0 362 rotl $16,r1,r1 363 /* 364 * If r1 is negative, add 1 to r0 . This 365 * adjustment is made so that the two's 366 * complement multiplications done later 367 * will produce unsigned results. 368 */ 369 bgeq posmid 370 incl r0 371 posmid: 372 /* p.3 373 * 374 * Set r3 to the address of the first quadword 375 * used to obtain the needed portion of 2/pi . 376 * The address is longword aligned to ensure 377 * efficient access. 378 */ 379 ashl $-3,r2,r3 380 bicb2 $3,r3 381 subl3 r3,$bits2opi,r3 382 /* 383 * Set r2 to the size of the shift needed to 384 * obtain the correct portion of 2/pi . 385 */ 386 bicb2 $0xe0,r2 387 /* p.4 388 * 389 * Move the needed 128 bits of 2/pi into 390 * r11 - r8 . Adjust the numbers to allow 391 * for unsigned multiplication. 392 */ 393 ashq r2,(r3),r10 394 395 subl2 $4,r3 396 ashq r2,(r3),r9 397 bgeq signoff1 398 incl r11 399 signoff1: 400 subl2 $4,r3 401 ashq r2,(r3),r8 402 bgeq signoff2 403 incl r10 404 signoff2: 405 subl2 $4,r3 406 ashq r2,(r3),r7 407 bgeq signoff3 408 incl r9 409 signoff3: 410 /* p.5 411 * 412 * Multiply the contents of r0/r1 by the 413 * slice of 2/pi in r11 - r8 . 414 */ 415 emul r0,r8,$0,r4 416 emul r0,r9,r5,r5 417 emul r0,r10,r6,r6 418 419 emul r1,r8,$0,r7 420 emul r1,r9,r8,r8 421 emul r1,r10,r9,r9 422 emul r1,r11,r10,r10 423 424 addl2 r4,r8 425 adwc r5,r9 426 adwc r6,r10 427 /* p.6 428 * 429 * If there are more than five leading zeros 430 * after the first two quotient bits or if there 431 * are more than five leading ones after the first 432 * two quotient bits, generate more fraction bits. 433 * Otherwise, branch to code to produce the result. 434 */ 435 bicl3 $0xc1ffffff,r10,r4 436 beql more1 437 cmpl $0x3e000000,r4 438 bneq result 439 more1: 440 /* p.7 441 * 442 * generate another 32 result bits. 443 */ 444 subl2 $4,r3 445 ashq r2,(r3),r5 446 bgeq signoff4 447 448 emul r1,r6,$0,r4 449 addl2 r1,r5 450 emul r0,r6,r5,r5 451 addl2 r0,r6 452 brb addbits1 453 454 signoff4: 455 emul r1,r6,$0,r4 456 emul r0,r6,r5,r5 457 458 addbits1: 459 addl2 r5,r7 460 adwc r6,r8 461 adwc $0,r9 462 adwc $0,r10 463 /* p.8 464 * 465 * Check for massive cancellation. 466 */ 467 bicl3 $0xc0000000,r10,r6 468 /* bneq more2 -S.McD Test was backwards */ 469 beql more2 470 cmpl $0x3fffffff,r6 471 bneq result 472 more2: 473 /* p.9 474 * 475 * If massive cancellation has occurred, 476 * generate another 24 result bits. 477 * Testing has shown there will always be 478 * enough bits after this point. 479 */ 480 subl2 $4,r3 481 ashq r2,(r3),r5 482 bgeq signoff5 483 484 emul r0,r6,r4,r5 485 addl2 r0,r6 486 brb addbits2 487 488 signoff5: 489 emul r0,r6,r4,r5 490 491 addbits2: 492 addl2 r6,r7 493 adwc $0,r8 494 adwc $0,r9 495 adwc $0,r10 496 /* p.10 497 * 498 * The following code produces the reduced 499 * argument from the product bits contained 500 * in r10 - r7 . 501 */ 502 result: 503 /* 504 * Extract the octant number from r10 . 505 */ 506 /* extzv $29,$3,r10,r0 ...used for pi/4 reduction -S.McD */ 507 extzv $30,$2,r10,r0 508 /* 509 * Clear the octant bits in r10 . 510 */ 511 /* bicl2 $0xe0000000,r10 ...used for pi/4 reduction -S.McD */ 512 bicl2 $0xc0000000,r10 513 /* 514 * Zero the sign flag. 515 */ 516 clrl r5 517 /* p.11 518 * 519 * Check to see if the fraction is greater than 520 * or equal to one-half. If it is, add one 521 * to the octant number, set the sign flag 522 * on, and replace the fraction with 1 minus 523 * the fraction. 524 */ 525 /* bitl $0x10000000,r10 ...used for pi/4 reduction -S.McD */ 526 bitl $0x20000000,r10 527 beql small 528 incl r0 529 incl r5 530 /* subl3 r10,$0x1fffffff,r10 ...used for pi/4 reduction -S.McD */ 531 subl3 r10,$0x3fffffff,r10 532 mcoml r9,r9 533 mcoml r8,r8 534 mcoml r7,r7 535 small: 536 /* p.12 537 * 538 * Test whether the first 29 bits of the ...used for pi/4 reduction -S.McD 539 * Test whether the first 30 bits of the 540 * fraction are zero. 541 */ 542 tstl r10 543 beql tiny 544 /* 545 * Find the position of the first one bit in r10 . 546 */ 547 cvtld r10,r1 548 extzv $7,$7,r1,r1 549 /* 550 * Compute the size of the shift needed. 551 */ 552 subl3 r1,$32,r6 553 /* 554 * Shift up the high order 64 bits of the 555 * product. 556 */ 557 ashq r6,r9,r10 558 ashq r6,r8,r9 559 brb mult 560 /* p.13 561 * 562 * Test to see if the sign bit of r9 is on. 563 */ 564 tiny: 565 tstl r9 566 bgeq tinier 567 /* 568 * If it is, shift the product bits up 32 bits. 569 */ 570 movl $32,r6 571 movq r8,r10 572 tstl r10 573 brb mult 574 /* p.14 575 * 576 * Test whether r9 is zero. It is probably 577 * impossible for both r10 and r9 to be 578 * zero, but until proven to be so, the test 579 * must be made. 580 */ 581 tinier: 582 beql zero 583 /* 584 * Find the position of the first one bit in r9 . 585 */ 586 cvtld r9,r1 587 extzv $7,$7,r1,r1 588 /* 589 * Compute the size of the shift needed. 590 */ 591 subl3 r1,$32,r1 592 addl3 $32,r1,r6 593 /* 594 * Shift up the high order 64 bits of the 595 * product. 596 */ 597 ashq r1,r8,r10 598 ashq r1,r7,r9 599 brb mult 600 /* p.15 601 * 602 * The following code sets the reduced 603 * argument to zero. 604 */ 605 zero: 606 clrl r1 607 clrl r2 608 clrl r3 609 brw return 610 /* p.16 611 * 612 * At this point, r0 contains the octant number, 613 * r6 indicates the number of bits the fraction 614 * has been shifted, r5 indicates the sign of 615 * the fraction, r11/r10 contain the high order 616 * 64 bits of the fraction, and the condition 617 * codes indicate where the sign bit of r10 618 * is on. The following code multiplies the 619 * fraction by pi/2 . 620 */ 621 mult: 622 /* 623 * Save r11/r10 in r4/r1 . -S.McD 624 */ 625 movl r11,r4 626 movl r10,r1 627 /* 628 * If the sign bit of r10 is on, add 1 to r11 . 629 */ 630 bgeq signoff6 631 incl r11 632 signoff6: 633 /* p.17 634 * 635 * Move pi/2 into r3/r2 . 636 */ 637 movq $0xc90fdaa22168c235,r2 638 /* 639 * Multiply the fraction by the portion of pi/2 640 * in r2 . 641 */ 642 emul r2,r10,$0,r7 643 emul r2,r11,r8,r7 644 /* 645 * Multiply the fraction by the portion of pi/2 646 * in r3 . 647 */ 648 emul r3,r10,$0,r9 649 emul r3,r11,r10,r10 650 /* 651 * Add the product bits together. 652 */ 653 addl2 r7,r9 654 adwc r8,r10 655 adwc $0,r11 656 /* 657 * Compensate for not sign extending r8 above.-S.McD 658 */ 659 tstl r8 660 bgeq signoff6a 661 decl r11 662 signoff6a: 663 /* 664 * Compensate for r11/r10 being unsigned. -S.McD 665 */ 666 addl2 r2,r10 667 adwc r3,r11 668 /* 669 * Compensate for r3/r2 being unsigned. -S.McD 670 */ 671 addl2 r1,r10 672 adwc r4,r11 673 /* p.18 674 * 675 * If the sign bit of r11 is zero, shift the 676 * product bits up one bit and increment r6 . 677 */ 678 blss signon 679 incl r6 680 ashq $1,r10,r10 681 tstl r9 682 bgeq signoff7 683 incl r10 684 signoff7: 685 signon: 686 /* p.19 687 * 688 * Shift the 56 most significant product 689 * bits into r9/r8 . The sign extension 690 * will be handled later. 691 */ 692 ashq $-8,r10,r8 693 /* 694 * Convert the low order 8 bits of r10 695 * into an F-format number. 696 */ 697 cvtbf r10,r3 698 /* 699 * If the result of the conversion was 700 * negative, add 1 to r9/r8 . 701 */ 702 bgeq chop 703 incl r8 704 adwc $0,r9 705 /* 706 * If r9 is now zero, branch to special 707 * code to handle that possibility. 708 */ 709 beql carryout 710 chop: 711 /* p.20 712 * 713 * Convert the number in r9/r8 into 714 * D-format number in r2/r1 . 715 */ 716 rotl $16,r8,r2 717 rotl $16,r9,r1 718 /* 719 * Set the exponent field to the appropriate 720 * value. Note that the extra bits created by 721 * sign extension are now eliminated. 722 */ 723 subw3 r6,$131,r6 724 insv r6,$7,$9,r1 725 /* 726 * Set the exponent field of the F-format 727 * number in r3 to the appropriate value. 728 */ 729 tstf r3 730 beql return 731 /* extzv $7,$8,r3,r4 -S.McD */ 732 extzv $7,$7,r3,r4 733 addw2 r4,r6 734 /* subw2 $217,r6 -S.McD */ 735 subw2 $64,r6 736 insv r6,$7,$8,r3 737 brb return 738 /* p.21 739 * 740 * The following code generates the appropriate 741 * result for the unlikely possibility that 742 * rounding the number in r9/r8 resulted in 743 * a carry out. 744 */ 745 carryout: 746 clrl r1 747 clrl r2 748 subw3 r6,$132,r6 749 insv r6,$7,$9,r1 750 tstf r3 751 beql return 752 extzv $7,$8,r3,r4 753 addw2 r4,r6 754 subw2 $218,r6 755 insv r6,$7,$8,r3 756 /* p.22 757 * 758 * The following code makes an needed 759 * adjustments to the signs of the 760 * results or to the octant number, and 761 * then returns. 762 */ 763 return: 764 /* 765 * Test if the fraction was greater than or 766 * equal to 1/2 . If so, negate the reduced 767 * argument. 768 */ 769 blbc r5,signoff8 770 mnegf r1,r1 771 mnegf r3,r3 772 signoff8: 773 /* p.23 774 * 775 * If the original argument was negative, 776 * negate the reduce argument and 777 * adjust the octant number. 778 */ 779 tstw (sp)+ 780 bgeq signoff9 781 mnegf r1,r1 782 mnegf r3,r3 783 /* subb3 r0,$8,r0 ...used for pi/4 reduction -S.McD */ 784 subb3 r0,$4,r0 785 signoff9: 786 /* 787 * Clear all unneeded octant bits. 788 * 789 * bicb2 $0xf8,r0 ...used for pi/4 reduction -S.McD */ 790 bicb2 $0xfc,r0 791 /* 792 * Return. 793 */ 794 rsb 795
Indexes created Tue Sep 30 20:09:53 GMT 2025