1.file "sincos.s" 2 3 4// Copyright (c) 2000 - 2005, Intel Corporation 5// All rights reserved. 6// 7// 8// Redistribution and use in source and binary forms, with or without 9// modification, are permitted provided that the following conditions are 10// met: 11// 12// * Redistributions of source code must retain the above copyright 13// notice, this list of conditions and the following disclaimer. 14// 15// * Redistributions in binary form must reproduce the above copyright 16// notice, this list of conditions and the following disclaimer in the 17// documentation and/or other materials provided with the distribution. 18// 19// * The name of Intel Corporation may not be used to endorse or promote 20// products derived from this software without specific prior written 21// permission. 22 23// THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS 24// "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT 25// LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR 26// A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL INTEL OR ITS 27// CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, 28// EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, 29// PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR 30// PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY 31// OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY OR TORT (INCLUDING 32// NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS 33// SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. 34// 35// Intel Corporation is the author of this code, and requests that all 36// problem reports or change requests be submitted to it directly at 37// http://www.intel.com/software/products/opensource/libraries/num.htm. 38// 39// History 40//============================================================== 41// 02/02/00 Initial version 42// 04/02/00 Unwind support added. 43// 06/16/00 Updated tables to enforce symmetry 44// 08/31/00 Saved 2 cycles in main path, and 9 in other paths. 45// 09/20/00 The updated tables regressed to an old version, so reinstated them 46// 10/18/00 Changed one table entry to ensure symmetry 47// 01/03/01 Improved speed, fixed flag settings for small arguments. 48// 02/18/02 Large arguments processing routine excluded 49// 05/20/02 Cleaned up namespace and sf0 syntax 50// 06/03/02 Insure inexact flag set for large arg result 51// 09/05/02 Work range is widened by reduction strengthen (3 parts of Pi/16) 52// 02/10/03 Reordered header: .section, .global, .proc, .align 53// 08/08/03 Improved performance 54// 10/28/04 Saved sincos_r_sincos to avoid clobber by dynamic loader 55// 03/31/05 Reformatted delimiters between data tables 56 57// API 58//============================================================== 59// double sin( double x); 60// double cos( double x); 61// 62// Overview of operation 63//============================================================== 64// 65// Step 1 66// ====== 67// Reduce x to region -1/2*pi/2^k ===== 0 ===== +1/2*pi/2^k where k=4 68// divide x by pi/2^k. 69// Multiply by 2^k/pi. 70// nfloat = Round result to integer (round-to-nearest) 71// 72// r = x - nfloat * pi/2^k 73// Do this as ((((x - nfloat * HIGH(pi/2^k))) - 74// nfloat * LOW(pi/2^k)) - 75// nfloat * LOWEST(pi/2^k) for increased accuracy. 76// pi/2^k is stored as two numbers that when added make pi/2^k. 77// pi/2^k = HIGH(pi/2^k) + LOW(pi/2^k) 78// HIGH and LOW parts are rounded to zero values, 79// and LOWEST is rounded to nearest one. 80// 81// x = (nfloat * pi/2^k) + r 82// r is small enough that we can use a polynomial approximation 83// and is referred to as the reduced argument. 84// 85// Step 3 86// ====== 87// Take the unreduced part and remove the multiples of 2pi. 88// So nfloat = nfloat (with lower k+1 bits cleared) + lower k+1 bits 89// 90// nfloat (with lower k+1 bits cleared) is a multiple of 2^(k+1) 91// N * 2^(k+1) 92// nfloat * pi/2^k = N * 2^(k+1) * pi/2^k + (lower k+1 bits) * pi/2^k 93// nfloat * pi/2^k = N * 2 * pi + (lower k+1 bits) * pi/2^k 94// nfloat * pi/2^k = N2pi + M * pi/2^k 95// 96// 97// Sin(x) = Sin((nfloat * pi/2^k) + r) 98// = Sin(nfloat * pi/2^k) * Cos(r) + Cos(nfloat * pi/2^k) * Sin(r) 99// 100// Sin(nfloat * pi/2^k) = Sin(N2pi + Mpi/2^k) 101// = Sin(N2pi)Cos(Mpi/2^k) + Cos(N2pi)Sin(Mpi/2^k) 102// = Sin(Mpi/2^k) 103// 104// Cos(nfloat * pi/2^k) = Cos(N2pi + Mpi/2^k) 105// = Cos(N2pi)Cos(Mpi/2^k) + Sin(N2pi)Sin(Mpi/2^k) 106// = Cos(Mpi/2^k) 107// 108// Sin(x) = Sin(Mpi/2^k) Cos(r) + Cos(Mpi/2^k) Sin(r) 109// 110// 111// Step 4 112// ====== 113// 0 <= M < 2^(k+1) 114// There are 2^(k+1) Sin entries in a table. 115// There are 2^(k+1) Cos entries in a table. 116// 117// Get Sin(Mpi/2^k) and Cos(Mpi/2^k) by table lookup. 118// 119// 120// Step 5 121// ====== 122// Calculate Cos(r) and Sin(r) by polynomial approximation. 123// 124// Cos(r) = 1 + r^2 q1 + r^4 q2 + r^6 q3 + ... = Series for Cos 125// Sin(r) = r + r^3 p1 + r^5 p2 + r^7 p3 + ... = Series for Sin 126// 127// and the coefficients q1, q2, ... and p1, p2, ... are stored in a table 128// 129// 130// Calculate 131// Sin(x) = Sin(Mpi/2^k) Cos(r) + Cos(Mpi/2^k) Sin(r) 132// 133// as follows 134// 135// S[m] = Sin(Mpi/2^k) and C[m] = Cos(Mpi/2^k) 136// rsq = r*r 137// 138// 139// P = p1 + r^2p2 + r^4p3 + r^6p4 140// Q = q1 + r^2q2 + r^4q3 + r^6q4 141// 142// rcub = r * rsq 143// Sin(r) = r + rcub * P 144// = r + r^3p1 + r^5p2 + r^7p3 + r^9p4 + ... = Sin(r) 145// 146// The coefficients are not exactly these values, but almost. 147// 148// p1 = -1/6 = -1/3! 149// p2 = 1/120 = 1/5! 150// p3 = -1/5040 = -1/7! 151// p4 = 1/362889 = 1/9! 152// 153// P = r + rcub * P 154// 155// Answer = S[m] Cos(r) + [Cm] P 156// 157// Cos(r) = 1 + rsq Q 158// Cos(r) = 1 + r^2 Q 159// Cos(r) = 1 + r^2 (q1 + r^2q2 + r^4q3 + r^6q4) 160// Cos(r) = 1 + r^2q1 + r^4q2 + r^6q3 + r^8q4 + ... 161// 162// S[m] Cos(r) = S[m](1 + rsq Q) 163// S[m] Cos(r) = S[m] + Sm rsq Q 164// S[m] Cos(r) = S[m] + s_rsq Q 165// Q = S[m] + s_rsq Q 166// 167// Then, 168// 169// Answer = Q + C[m] P 170 171 172// Registers used 173//============================================================== 174// general input registers: 175// r14 -> r26 176// r32 -> r35 177 178// predicate registers used: 179// p6 -> p11 180 181// floating-point registers used 182// f9 -> f15 183// f32 -> f61 184 185// Assembly macros 186//============================================================== 187sincos_NORM_f8 = f9 188sincos_W = f10 189sincos_int_Nfloat = f11 190sincos_Nfloat = f12 191 192sincos_r = f13 193sincos_rsq = f14 194sincos_rcub = f15 195sincos_save_tmp = f15 196 197sincos_Inv_Pi_by_16 = f32 198sincos_Pi_by_16_1 = f33 199sincos_Pi_by_16_2 = f34 200 201sincos_Inv_Pi_by_64 = f35 202 203sincos_Pi_by_16_3 = f36 204 205sincos_r_exact = f37 206 207sincos_Sm = f38 208sincos_Cm = f39 209 210sincos_P1 = f40 211sincos_Q1 = f41 212sincos_P2 = f42 213sincos_Q2 = f43 214sincos_P3 = f44 215sincos_Q3 = f45 216sincos_P4 = f46 217sincos_Q4 = f47 218 219sincos_P_temp1 = f48 220sincos_P_temp2 = f49 221 222sincos_Q_temp1 = f50 223sincos_Q_temp2 = f51 224 225sincos_P = f52 226sincos_Q = f53 227 228sincos_srsq = f54 229 230sincos_SIG_INV_PI_BY_16_2TO61 = f55 231sincos_RSHF_2TO61 = f56 232sincos_RSHF = f57 233sincos_2TOM61 = f58 234sincos_NFLOAT = f59 235sincos_W_2TO61_RSH = f60 236 237fp_tmp = f61 238 239///////////////////////////////////////////////////////////// 240 241sincos_GR_sig_inv_pi_by_16 = r14 242sincos_GR_rshf_2to61 = r15 243sincos_GR_rshf = r16 244sincos_GR_exp_2tom61 = r17 245sincos_GR_n = r18 246sincos_GR_m = r19 247sincos_GR_32m = r19 248sincos_GR_all_ones = r19 249sincos_AD_1 = r20 250sincos_AD_2 = r21 251sincos_exp_limit = r22 252sincos_r_signexp = r23 253sincos_r_17_ones = r24 254sincos_r_sincos = r25 255sincos_r_exp = r26 256 257GR_SAVE_PFS = r33 258GR_SAVE_B0 = r34 259GR_SAVE_GP = r35 260GR_SAVE_r_sincos = r36 261 262 263RODATA 264 265// Pi/16 parts 266.align 16 267LOCAL_OBJECT_START(double_sincos_pi) 268 data8 0xC90FDAA22168C234, 0x00003FFC // pi/16 1st part 269 data8 0xC4C6628B80DC1CD1, 0x00003FBC // pi/16 2nd part 270 data8 0xA4093822299F31D0, 0x00003F7A // pi/16 3rd part 271LOCAL_OBJECT_END(double_sincos_pi) 272 273// Coefficients for polynomials 274LOCAL_OBJECT_START(double_sincos_pq_k4) 275 data8 0x3EC71C963717C63A // P4 276 data8 0x3EF9FFBA8F191AE6 // Q4 277 data8 0xBF2A01A00F4E11A8 // P3 278 data8 0xBF56C16C05AC77BF // Q3 279 data8 0x3F8111111110F167 // P2 280 data8 0x3FA555555554DD45 // Q2 281 data8 0xBFC5555555555555 // P1 282 data8 0xBFDFFFFFFFFFFFFC // Q1 283LOCAL_OBJECT_END(double_sincos_pq_k4) 284 285// Sincos table (S[m], C[m]) 286LOCAL_OBJECT_START(double_sin_cos_beta_k4) 287 288data8 0x0000000000000000 , 0x00000000 // sin( 0 pi/16) S0 289data8 0x8000000000000000 , 0x00003fff // cos( 0 pi/16) C0 290// 291data8 0xc7c5c1e34d3055b3 , 0x00003ffc // sin( 1 pi/16) S1 292data8 0xfb14be7fbae58157 , 0x00003ffe // cos( 1 pi/16) C1 293// 294data8 0xc3ef1535754b168e , 0x00003ffd // sin( 2 pi/16) S2 295data8 0xec835e79946a3146 , 0x00003ffe // cos( 2 pi/16) C2 296// 297data8 0x8e39d9cd73464364 , 0x00003ffe // sin( 3 pi/16) S3 298data8 0xd4db3148750d181a , 0x00003ffe // cos( 3 pi/16) C3 299// 300data8 0xb504f333f9de6484 , 0x00003ffe // sin( 4 pi/16) S4 301data8 0xb504f333f9de6484 , 0x00003ffe // cos( 4 pi/16) C4 302// 303data8 0xd4db3148750d181a , 0x00003ffe // sin( 5 pi/16) C3 304data8 0x8e39d9cd73464364 , 0x00003ffe // cos( 5 pi/16) S3 305// 306data8 0xec835e79946a3146 , 0x00003ffe // sin( 6 pi/16) C2 307data8 0xc3ef1535754b168e , 0x00003ffd // cos( 6 pi/16) S2 308// 309data8 0xfb14be7fbae58157 , 0x00003ffe // sin( 7 pi/16) C1 310data8 0xc7c5c1e34d3055b3 , 0x00003ffc // cos( 7 pi/16) S1 311// 312data8 0x8000000000000000 , 0x00003fff // sin( 8 pi/16) C0 313data8 0x0000000000000000 , 0x00000000 // cos( 8 pi/16) S0 314// 315data8 0xfb14be7fbae58157 , 0x00003ffe // sin( 9 pi/16) C1 316data8 0xc7c5c1e34d3055b3 , 0x0000bffc // cos( 9 pi/16) -S1 317// 318data8 0xec835e79946a3146 , 0x00003ffe // sin(10 pi/16) C2 319data8 0xc3ef1535754b168e , 0x0000bffd // cos(10 pi/16) -S2 320// 321data8 0xd4db3148750d181a , 0x00003ffe // sin(11 pi/16) C3 322data8 0x8e39d9cd73464364 , 0x0000bffe // cos(11 pi/16) -S3 323// 324data8 0xb504f333f9de6484 , 0x00003ffe // sin(12 pi/16) S4 325data8 0xb504f333f9de6484 , 0x0000bffe // cos(12 pi/16) -S4 326// 327data8 0x8e39d9cd73464364 , 0x00003ffe // sin(13 pi/16) S3 328data8 0xd4db3148750d181a , 0x0000bffe // cos(13 pi/16) -C3 329// 330data8 0xc3ef1535754b168e , 0x00003ffd // sin(14 pi/16) S2 331data8 0xec835e79946a3146 , 0x0000bffe // cos(14 pi/16) -C2 332// 333data8 0xc7c5c1e34d3055b3 , 0x00003ffc // sin(15 pi/16) S1 334data8 0xfb14be7fbae58157 , 0x0000bffe // cos(15 pi/16) -C1 335// 336data8 0x0000000000000000 , 0x00000000 // sin(16 pi/16) S0 337data8 0x8000000000000000 , 0x0000bfff // cos(16 pi/16) -C0 338// 339data8 0xc7c5c1e34d3055b3 , 0x0000bffc // sin(17 pi/16) -S1 340data8 0xfb14be7fbae58157 , 0x0000bffe // cos(17 pi/16) -C1 341// 342data8 0xc3ef1535754b168e , 0x0000bffd // sin(18 pi/16) -S2 343data8 0xec835e79946a3146 , 0x0000bffe // cos(18 pi/16) -C2 344// 345data8 0x8e39d9cd73464364 , 0x0000bffe // sin(19 pi/16) -S3 346data8 0xd4db3148750d181a , 0x0000bffe // cos(19 pi/16) -C3 347// 348data8 0xb504f333f9de6484 , 0x0000bffe // sin(20 pi/16) -S4 349data8 0xb504f333f9de6484 , 0x0000bffe // cos(20 pi/16) -S4 350// 351data8 0xd4db3148750d181a , 0x0000bffe // sin(21 pi/16) -C3 352data8 0x8e39d9cd73464364 , 0x0000bffe // cos(21 pi/16) -S3 353// 354data8 0xec835e79946a3146 , 0x0000bffe // sin(22 pi/16) -C2 355data8 0xc3ef1535754b168e , 0x0000bffd // cos(22 pi/16) -S2 356// 357data8 0xfb14be7fbae58157 , 0x0000bffe // sin(23 pi/16) -C1 358data8 0xc7c5c1e34d3055b3 , 0x0000bffc // cos(23 pi/16) -S1 359// 360data8 0x8000000000000000 , 0x0000bfff // sin(24 pi/16) -C0 361data8 0x0000000000000000 , 0x00000000 // cos(24 pi/16) S0 362// 363data8 0xfb14be7fbae58157 , 0x0000bffe // sin(25 pi/16) -C1 364data8 0xc7c5c1e34d3055b3 , 0x00003ffc // cos(25 pi/16) S1 365// 366data8 0xec835e79946a3146 , 0x0000bffe // sin(26 pi/16) -C2 367data8 0xc3ef1535754b168e , 0x00003ffd // cos(26 pi/16) S2 368// 369data8 0xd4db3148750d181a , 0x0000bffe // sin(27 pi/16) -C3 370data8 0x8e39d9cd73464364 , 0x00003ffe // cos(27 pi/16) S3 371// 372data8 0xb504f333f9de6484 , 0x0000bffe // sin(28 pi/16) -S4 373data8 0xb504f333f9de6484 , 0x00003ffe // cos(28 pi/16) S4 374// 375data8 0x8e39d9cd73464364 , 0x0000bffe // sin(29 pi/16) -S3 376data8 0xd4db3148750d181a , 0x00003ffe // cos(29 pi/16) C3 377// 378data8 0xc3ef1535754b168e , 0x0000bffd // sin(30 pi/16) -S2 379data8 0xec835e79946a3146 , 0x00003ffe // cos(30 pi/16) C2 380// 381data8 0xc7c5c1e34d3055b3 , 0x0000bffc // sin(31 pi/16) -S1 382data8 0xfb14be7fbae58157 , 0x00003ffe // cos(31 pi/16) C1 383// 384data8 0x0000000000000000 , 0x00000000 // sin(32 pi/16) S0 385data8 0x8000000000000000 , 0x00003fff // cos(32 pi/16) C0 386LOCAL_OBJECT_END(double_sin_cos_beta_k4) 387 388.section .text 389 390//////////////////////////////////////////////////////// 391// There are two entry points: sin and cos 392 393 394// If from sin, p8 is true 395// If from cos, p9 is true 396 397GLOBAL_IEEE754_ENTRY(sin) 398 399{ .mlx 400 getf.exp sincos_r_signexp = f8 401 movl sincos_GR_sig_inv_pi_by_16 = 0xA2F9836E4E44152A // signd of 16/pi 402} 403{ .mlx 404 addl sincos_AD_1 = @ltoff(double_sincos_pi), gp 405 movl sincos_GR_rshf_2to61 = 0x47b8000000000000 // 1.1 2^(63+63-2) 406} 407;; 408 409{ .mfi 410 ld8 sincos_AD_1 = [sincos_AD_1] 411 fnorm.s0 sincos_NORM_f8 = f8 // Normalize argument 412 cmp.eq p8,p9 = r0, r0 // set p8 (clear p9) for sin 413} 414{ .mib 415 mov sincos_GR_exp_2tom61 = 0xffff-61 // exponent of scale 2^-61 416 mov sincos_r_sincos = 0x0 // sincos_r_sincos = 0 for sin 417 br.cond.sptk _SINCOS_COMMON // go to common part 418} 419;; 420 421GLOBAL_IEEE754_END(sin) 422libm_alias_double_other (__sin, sin) 423 424GLOBAL_IEEE754_ENTRY(cos) 425 426{ .mlx 427 getf.exp sincos_r_signexp = f8 428 movl sincos_GR_sig_inv_pi_by_16 = 0xA2F9836E4E44152A // signd of 16/pi 429} 430{ .mlx 431 addl sincos_AD_1 = @ltoff(double_sincos_pi), gp 432 movl sincos_GR_rshf_2to61 = 0x47b8000000000000 // 1.1 2^(63+63-2) 433} 434;; 435 436{ .mfi 437 ld8 sincos_AD_1 = [sincos_AD_1] 438 fnorm.s1 sincos_NORM_f8 = f8 // Normalize argument 439 cmp.eq p9,p8 = r0, r0 // set p9 (clear p8) for cos 440} 441{ .mib 442 mov sincos_GR_exp_2tom61 = 0xffff-61 // exp of scale 2^-61 443 mov sincos_r_sincos = 0x8 // sincos_r_sincos = 8 for cos 444 nop.b 999 445} 446;; 447 448//////////////////////////////////////////////////////// 449// All entry points end up here. 450// If from sin, sincos_r_sincos is 0 and p8 is true 451// If from cos, sincos_r_sincos is 8 = 2^(k-1) and p9 is true 452// We add sincos_r_sincos to N 453 454///////////// Common sin and cos part ////////////////// 455_SINCOS_COMMON: 456 457 458// Form two constants we need 459// 16/pi * 2^-2 * 2^63, scaled by 2^61 since we just loaded the significand 460// 1.1000...000 * 2^(63+63-2) to right shift int(W) into the low significand 461{ .mfi 462 setf.sig sincos_SIG_INV_PI_BY_16_2TO61 = sincos_GR_sig_inv_pi_by_16 463 fclass.m p6,p0 = f8, 0xe7 // if x = 0,inf,nan 464 mov sincos_exp_limit = 0x1001a 465} 466{ .mlx 467 setf.d sincos_RSHF_2TO61 = sincos_GR_rshf_2to61 468 movl sincos_GR_rshf = 0x43e8000000000000 // 1.1 2^63 469} // Right shift 470;; 471 472// Form another constant 473// 2^-61 for scaling Nfloat 474// 0x1001a is register_bias + 27. 475// So if f8 >= 2^27, go to large argument routines 476{ .mfi 477 alloc r32 = ar.pfs, 1, 4, 0, 0 478 fclass.m p11,p0 = f8, 0x0b // Test for x=unorm 479 mov sincos_GR_all_ones = -1 // For "inexect" constant create 480} 481{ .mib 482 setf.exp sincos_2TOM61 = sincos_GR_exp_2tom61 483 nop.i 999 484(p6) br.cond.spnt _SINCOS_SPECIAL_ARGS 485} 486;; 487 488// Load the two pieces of pi/16 489// Form another constant 490// 1.1000...000 * 2^63, the right shift constant 491{ .mmb 492 ldfe sincos_Pi_by_16_1 = [sincos_AD_1],16 493 setf.d sincos_RSHF = sincos_GR_rshf 494(p11) br.cond.spnt _SINCOS_UNORM // Branch if x=unorm 495} 496;; 497 498_SINCOS_COMMON2: 499// Return here if x=unorm 500// Create constant used to set inexact 501{ .mmi 502 ldfe sincos_Pi_by_16_2 = [sincos_AD_1],16 503 setf.sig fp_tmp = sincos_GR_all_ones 504 nop.i 999 505};; 506 507// Select exponent (17 lsb) 508{ .mfi 509 ldfe sincos_Pi_by_16_3 = [sincos_AD_1],16 510 nop.f 999 511 dep.z sincos_r_exp = sincos_r_signexp, 0, 17 512};; 513 514// Polynomial coefficients (Q4, P4, Q3, P3, Q2, Q1, P2, P1) loading 515// p10 is true if we must call routines to handle larger arguments 516// p10 is true if f8 exp is >= 0x1001a (2^27) 517{ .mmb 518 ldfpd sincos_P4,sincos_Q4 = [sincos_AD_1],16 519 cmp.ge p10,p0 = sincos_r_exp,sincos_exp_limit 520(p10) br.cond.spnt _SINCOS_LARGE_ARGS // Go to "large args" routine 521};; 522 523// sincos_W = x * sincos_Inv_Pi_by_16 524// Multiply x by scaled 16/pi and add large const to shift integer part of W to 525// rightmost bits of significand 526{ .mfi 527 ldfpd sincos_P3,sincos_Q3 = [sincos_AD_1],16 528 fma.s1 sincos_W_2TO61_RSH = sincos_NORM_f8,sincos_SIG_INV_PI_BY_16_2TO61,sincos_RSHF_2TO61 529 nop.i 999 530};; 531 532// get N = (int)sincos_int_Nfloat 533// sincos_NFLOAT = Round_Int_Nearest(sincos_W) 534// This is done by scaling back by 2^-61 and subtracting the shift constant 535{ .mmf 536 getf.sig sincos_GR_n = sincos_W_2TO61_RSH 537 ldfpd sincos_P2,sincos_Q2 = [sincos_AD_1],16 538 fms.s1 sincos_NFLOAT = sincos_W_2TO61_RSH,sincos_2TOM61,sincos_RSHF 539};; 540 541// sincos_r = -sincos_Nfloat * sincos_Pi_by_16_1 + x 542{ .mfi 543 ldfpd sincos_P1,sincos_Q1 = [sincos_AD_1],16 544 fnma.s1 sincos_r = sincos_NFLOAT, sincos_Pi_by_16_1, sincos_NORM_f8 545 nop.i 999 546};; 547 548// Add 2^(k-1) (which is in sincos_r_sincos) to N 549{ .mmi 550 add sincos_GR_n = sincos_GR_n, sincos_r_sincos 551;; 552// Get M (least k+1 bits of N) 553 and sincos_GR_m = 0x1f,sincos_GR_n 554 nop.i 999 555};; 556 557// sincos_r = sincos_r -sincos_Nfloat * sincos_Pi_by_16_2 558{ .mfi 559 nop.m 999 560 fnma.s1 sincos_r = sincos_NFLOAT, sincos_Pi_by_16_2, sincos_r 561 shl sincos_GR_32m = sincos_GR_m,5 562};; 563 564// Add 32*M to address of sin_cos_beta table 565// For sin denorm. - set uflow 566{ .mfi 567 add sincos_AD_2 = sincos_GR_32m, sincos_AD_1 568(p8) fclass.m.unc p10,p0 = f8,0x0b 569 nop.i 999 570};; 571 572// Load Sin and Cos table value using obtained index m (sincosf_AD_2) 573{ .mfi 574 ldfe sincos_Sm = [sincos_AD_2],16 575 nop.f 999 576 nop.i 999 577};; 578 579// get rsq = r*r 580{ .mfi 581 ldfe sincos_Cm = [sincos_AD_2] 582 fma.s1 sincos_rsq = sincos_r, sincos_r, f0 // r^2 = r*r 583 nop.i 999 584} 585{ .mfi 586 nop.m 999 587 fmpy.s0 fp_tmp = fp_tmp,fp_tmp // forces inexact flag 588 nop.i 999 589};; 590 591// sincos_r_exact = sincos_r -sincos_Nfloat * sincos_Pi_by_16_3 592{ .mfi 593 nop.m 999 594 fnma.s1 sincos_r_exact = sincos_NFLOAT, sincos_Pi_by_16_3, sincos_r 595 nop.i 999 596};; 597 598// Polynomials calculation 599// P_1 = P4*r^2 + P3 600// Q_2 = Q4*r^2 + Q3 601{ .mfi 602 nop.m 999 603 fma.s1 sincos_P_temp1 = sincos_rsq, sincos_P4, sincos_P3 604 nop.i 999 605} 606{ .mfi 607 nop.m 999 608 fma.s1 sincos_Q_temp1 = sincos_rsq, sincos_Q4, sincos_Q3 609 nop.i 999 610};; 611 612// get rcube = r^3 and S[m]*r^2 613{ .mfi 614 nop.m 999 615 fmpy.s1 sincos_srsq = sincos_Sm,sincos_rsq 616 nop.i 999 617} 618{ .mfi 619 nop.m 999 620 fmpy.s1 sincos_rcub = sincos_r_exact, sincos_rsq 621 nop.i 999 622};; 623 624// Polynomials calculation 625// Q_2 = Q_1*r^2 + Q2 626// P_1 = P_1*r^2 + P2 627{ .mfi 628 nop.m 999 629 fma.s1 sincos_Q_temp2 = sincos_rsq, sincos_Q_temp1, sincos_Q2 630 nop.i 999 631} 632{ .mfi 633 nop.m 999 634 fma.s1 sincos_P_temp2 = sincos_rsq, sincos_P_temp1, sincos_P2 635 nop.i 999 636};; 637 638// Polynomials calculation 639// Q = Q_2*r^2 + Q1 640// P = P_2*r^2 + P1 641{ .mfi 642 nop.m 999 643 fma.s1 sincos_Q = sincos_rsq, sincos_Q_temp2, sincos_Q1 644 nop.i 999 645} 646{ .mfi 647 nop.m 999 648 fma.s1 sincos_P = sincos_rsq, sincos_P_temp2, sincos_P1 649 nop.i 999 650};; 651 652// Get final P and Q 653// Q = Q*S[m]*r^2 + S[m] 654// P = P*r^3 + r 655{ .mfi 656 nop.m 999 657 fma.s1 sincos_Q = sincos_srsq,sincos_Q, sincos_Sm 658 nop.i 999 659} 660{ .mfi 661 nop.m 999 662 fma.s1 sincos_P = sincos_rcub,sincos_P, sincos_r_exact 663 nop.i 999 664};; 665 666// If sin(denormal), force underflow to be set 667{ .mfi 668 nop.m 999 669(p10) fmpy.d.s0 fp_tmp = sincos_NORM_f8,sincos_NORM_f8 670 nop.i 999 671};; 672 673// Final calculation 674// result = C[m]*P + Q 675{ .mfb 676 nop.m 999 677 fma.d.s0 f8 = sincos_Cm, sincos_P, sincos_Q 678 br.ret.sptk b0 // Exit for common path 679};; 680 681////////// x = 0/Inf/NaN path ////////////////// 682_SINCOS_SPECIAL_ARGS: 683.pred.rel "mutex",p8,p9 684// sin(+/-0) = +/-0 685// sin(Inf) = NaN 686// sin(NaN) = NaN 687{ .mfi 688 nop.m 999 689(p8) fma.d.s0 f8 = f8, f0, f0 // sin(+/-0,NaN,Inf) 690 nop.i 999 691} 692// cos(+/-0) = 1.0 693// cos(Inf) = NaN 694// cos(NaN) = NaN 695{ .mfb 696 nop.m 999 697(p9) fma.d.s0 f8 = f8, f0, f1 // cos(+/-0,NaN,Inf) 698 br.ret.sptk b0 // Exit for x = 0/Inf/NaN path 699};; 700 701_SINCOS_UNORM: 702// Here if x=unorm 703{ .mfb 704 getf.exp sincos_r_signexp = sincos_NORM_f8 // Get signexp of x 705 fcmp.eq.s0 p11,p0 = f8, f0 // Dummy op to set denorm flag 706 br.cond.sptk _SINCOS_COMMON2 // Return to main path 707};; 708 709GLOBAL_IEEE754_END(cos) 710libm_alias_double_other (__cos, cos) 711 712//////////// x >= 2^27 - large arguments routine call //////////// 713LOCAL_LIBM_ENTRY(__libm_callout_sincos) 714_SINCOS_LARGE_ARGS: 715.prologue 716{ .mfi 717 mov GR_SAVE_r_sincos = sincos_r_sincos // Save sin or cos 718 nop.f 999 719.save ar.pfs,GR_SAVE_PFS 720 mov GR_SAVE_PFS = ar.pfs 721} 722;; 723 724{ .mfi 725 mov GR_SAVE_GP = gp 726 nop.f 999 727.save b0, GR_SAVE_B0 728 mov GR_SAVE_B0 = b0 729} 730 731.body 732{ .mbb 733 setf.sig sincos_save_tmp = sincos_GR_all_ones// inexact set 734 nop.b 999 735(p8) br.call.sptk.many b0 = __libm_sin_large# // sin(large_X) 736 737};; 738 739{ .mbb 740 cmp.ne p9,p0 = GR_SAVE_r_sincos, r0 // set p9 if cos 741 nop.b 999 742(p9) br.call.sptk.many b0 = __libm_cos_large# // cos(large_X) 743};; 744 745{ .mfi 746 mov gp = GR_SAVE_GP 747 fma.d.s0 f8 = f8, f1, f0 // Round result to double 748 mov b0 = GR_SAVE_B0 749} 750// Force inexact set 751{ .mfi 752 nop.m 999 753 fmpy.s0 sincos_save_tmp = sincos_save_tmp, sincos_save_tmp 754 nop.i 999 755};; 756 757{ .mib 758 nop.m 999 759 mov ar.pfs = GR_SAVE_PFS 760 br.ret.sptk b0 // Exit for large arguments routine call 761};; 762 763LOCAL_LIBM_END(__libm_callout_sincos) 764 765.type __libm_sin_large#,@function 766.global __libm_sin_large# 767.type __libm_cos_large#,@function 768.global __libm_cos_large# 769