.file "sincosf.s"// Copyright (c) 2000 - 2005, Intel Corporation// All rights reserved.//// Contributed 2000 by the Intel Numerics Group, Intel Corporation//// 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.//// * The name of Intel Corporation may not 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 INTEL OR ITS// 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.//// Intel Corporation is the author of this code, and requests that all// problem reports or change requests be submitted to it directly at// http://www.intel.com/software/products/opensource/libraries/num.htm.//// History//==============================================================// 02/02/00 Initial version// 04/02/00 Unwind support added.// 06/16/00 Updated tables to enforce symmetry// 08/31/00 Saved 2 cycles in main path, and 9 in other paths.// 09/20/00 The updated tables regressed to an old version, so reinstated them// 10/18/00 Changed one table entry to ensure symmetry// 01/03/01 Improved speed, fixed flag settings for small arguments.// 02/18/02 Large arguments processing routine excluded// 05/20/02 Cleaned up namespace and sf0 syntax// 06/03/02 Insure inexact flag set for large arg result// 09/05/02 Single precision version is made using double precision one as base// 02/10/03 Reordered header: .section, .global, .proc, .align// 03/31/05 Reformatted delimiters between data tables//// API//==============================================================// float sinf( float x);// float cosf( float x);//// Overview of operation//==============================================================//// Step 1// ======// Reduce x to region -1/2*pi/2^k ===== 0 ===== +1/2*pi/2^k  where k=4//    divide x by pi/2^k.//    Multiply by 2^k/pi.//    nfloat = Round result to integer (round-to-nearest)//// r = x -  nfloat * pi/2^k//    Do this as (x -  nfloat * HIGH(pi/2^k)) - nfloat * LOW(pi/2^k) //    for increased accuracy.//    pi/2^k is stored as two numbers that when added make pi/2^k.//       pi/2^k = HIGH(pi/2^k) + LOW(pi/2^k)//    HIGH part is rounded to zero, LOW - to nearest//// x = (nfloat * pi/2^k) + r//    r is small enough that we can use a polynomial approximation//    and is referred to as the reduced argument.//// Step 3// ======// Take the unreduced part and remove the multiples of 2pi.// So nfloat = nfloat (with lower k+1 bits cleared) + lower k+1 bits////    nfloat (with lower k+1 bits cleared) is a multiple of 2^(k+1)//    N * 2^(k+1)//    nfloat * pi/2^k = N * 2^(k+1) * pi/2^k + (lower k+1 bits) * pi/2^k//    nfloat * pi/2^k = N * 2 * pi + (lower k+1 bits) * pi/2^k//    nfloat * pi/2^k = N2pi + M * pi/2^k////// Sin(x) = Sin((nfloat * pi/2^k) + r)//        = Sin(nfloat * pi/2^k) * Cos(r) + Cos(nfloat * pi/2^k) * Sin(r)////          Sin(nfloat * pi/2^k) = Sin(N2pi + Mpi/2^k)//                               = Sin(N2pi)Cos(Mpi/2^k) + Cos(N2pi)Sin(Mpi/2^k)//                               = Sin(Mpi/2^k)////          Cos(nfloat * pi/2^k) = Cos(N2pi + Mpi/2^k)//                               = Cos(N2pi)Cos(Mpi/2^k) + Sin(N2pi)Sin(Mpi/2^k)//                               = Cos(Mpi/2^k)//// Sin(x) = Sin(Mpi/2^k) Cos(r) + Cos(Mpi/2^k) Sin(r)////// Step 4// ======// 0 <= M < 2^(k+1)// There are 2^(k+1) Sin entries in a table.// There are 2^(k+1) Cos entries in a table.//// Get Sin(Mpi/2^k) and Cos(Mpi/2^k) by table lookup.////// Step 5// ======// Calculate Cos(r) and Sin(r) by polynomial approximation.//// Cos(r) = 1 + r^2 q1  + r^4 q2  = Series for Cos// Sin(r) = r + r^3 p1  + r^5 p2  = Series for Sin//// and the coefficients q1, q2 and p1, p2 are stored in a table////// Calculate// Sin(x) = Sin(Mpi/2^k) Cos(r) + Cos(Mpi/2^k) Sin(r)//// as follows////    S[m] = Sin(Mpi/2^k) and C[m] = Cos(Mpi/2^k)//    rsq = r*r//////    P = P1 + r^2*P2//    Q = Q1 + r^2*Q2////       rcub = r * rsq//       Sin(r) = r + rcub * P//              = r + r^3p1  + r^5p2 = Sin(r)////            The coefficients are not exactly these values, but almost.////            p1 = -1/6  = -1/3!//            p2 = 1/120 =  1/5!//            p3 = -1/5040 = -1/7!//            p4 = 1/362889 = 1/9!////       P =  r + r^3 * P////    Answer = S[m] Cos(r) + C[m] P////       Cos(r) = 1 + rsq Q//       Cos(r) = 1 + r^2 Q//       Cos(r) = 1 + r^2 (q1 + r^2q2)//       Cos(r) = 1 + r^2q1 + r^4q2////       S[m] Cos(r) = S[m](1 + rsq Q)//       S[m] Cos(r) = S[m] + S[m] rsq Q//       S[m] Cos(r) = S[m] + s_rsq Q//       Q         = S[m] + s_rsq Q//// Then,////    Answer = Q + C[m] P// Registers used//==============================================================// general input registers:// r14 -> r19// r32 -> r45// predicate registers used:// p6 -> p14// floating-point registers used// f9 -> f15// f32 -> f61// Assembly macros//==============================================================sincosf_NORM_f8                 = f9sincosf_W                       = f10sincosf_int_Nfloat              = f11sincosf_Nfloat                  = f12sincosf_r                       = f13sincosf_rsq                     = f14sincosf_rcub                    = f15sincosf_save_tmp                = f15sincosf_Inv_Pi_by_16            = f32sincosf_Pi_by_16_1              = f33sincosf_Pi_by_16_2              = f34sincosf_Inv_Pi_by_64            = f35sincosf_Pi_by_16_3              = f36sincosf_r_exact                 = f37sincosf_Sm                      = f38sincosf_Cm                      = f39sincosf_P1                      = f40sincosf_Q1                      = f41sincosf_P2                      = f42sincosf_Q2                      = f43sincosf_P3                      = f44sincosf_Q3                      = f45sincosf_P4                      = f46sincosf_Q4                      = f47sincosf_P_temp1                 = f48sincosf_P_temp2                 = f49sincosf_Q_temp1                 = f50sincosf_Q_temp2                 = f51sincosf_P                       = f52sincosf_Q                       = f53sincosf_srsq                    = f54sincosf_SIG_INV_PI_BY_16_2TO61  = f55sincosf_RSHF_2TO61              = f56sincosf_RSHF                    = f57sincosf_2TOM61                  = f58sincosf_NFLOAT                  = f59sincosf_W_2TO61_RSH             = f60fp_tmp                          = f61/////////////////////////////////////////////////////////////sincosf_AD_1                    = r33sincosf_AD_2                    = r34sincosf_exp_limit               = r35sincosf_r_signexp               = r36sincosf_AD_beta_table           = r37sincosf_r_sincos                = r38sincosf_r_exp                   = r39sincosf_r_17_ones               = r40sincosf_GR_sig_inv_pi_by_16     = r14sincosf_GR_rshf_2to61           = r15sincosf_GR_rshf                 = r16sincosf_GR_exp_2tom61           = r17sincosf_GR_n                    = r18sincosf_GR_m                    = r19sincosf_GR_32m                  = r19sincosf_GR_all_ones             = r19gr_tmp                          = r41GR_SAVE_PFS                     = r41GR_SAVE_B0                      = r42GR_SAVE_GP                      = r43RODATA.align 16// Pi/16 partsLOCAL_OBJECT_START(double_sincosf_pi)   data8 0xC90FDAA22168C234, 0x00003FFC // pi/16 1st part   data8 0xC4C6628B80DC1CD1, 0x00003FBC // pi/16 2nd partLOCAL_OBJECT_END(double_sincosf_pi)// Coefficients for polynomialsLOCAL_OBJECT_START(double_sincosf_pq_k4)   data8 0x3F810FABB668E9A2 // P2   data8 0x3FA552E3D6DE75C9 // Q2   data8 0xBFC555554447BC7F // P1   data8 0xBFDFFFFFC447610A // Q1LOCAL_OBJECT_END(double_sincosf_pq_k4)// Sincos table (S[m], C[m])LOCAL_OBJECT_START(double_sin_cos_beta_k4)    data8 0x0000000000000000 // sin ( 0 Pi / 16 )    data8 0x3FF0000000000000 // cos ( 0 Pi / 16 )//    data8 0x3FC8F8B83C69A60B // sin ( 1 Pi / 16 )    data8 0x3FEF6297CFF75CB0 // cos ( 1 Pi / 16 )//    data8 0x3FD87DE2A6AEA963 // sin ( 2 Pi / 16 )    data8 0x3FED906BCF328D46 // cos ( 2 Pi / 16 )//    data8 0x3FE1C73B39AE68C8 // sin ( 3 Pi / 16 )    data8 0x3FEA9B66290EA1A3 // cos ( 3 Pi / 16 )//    data8 0x3FE6A09E667F3BCD // sin ( 4 Pi / 16 )    data8 0x3FE6A09E667F3BCD // cos ( 4 Pi / 16 )//    data8 0x3FEA9B66290EA1A3 // sin ( 5 Pi / 16 )    data8 0x3FE1C73B39AE68C8 // cos ( 5 Pi / 16 )//    data8 0x3FED906BCF328D46 // sin ( 6 Pi / 16 )    data8 0x3FD87DE2A6AEA963 // cos ( 6 Pi / 16 )//    data8 0x3FEF6297CFF75CB0 // sin ( 7 Pi / 16 )    data8 0x3FC8F8B83C69A60B // cos ( 7 Pi / 16 )//    data8 0x3FF0000000000000 // sin ( 8 Pi / 16 )    data8 0x0000000000000000 // cos ( 8 Pi / 16 )//    data8 0x3FEF6297CFF75CB0 // sin ( 9 Pi / 16 )    data8 0xBFC8F8B83C69A60B // cos ( 9 Pi / 16 )//    data8 0x3FED906BCF328D46 // sin ( 10 Pi / 16 )    data8 0xBFD87DE2A6AEA963 // cos ( 10 Pi / 16 )//    data8 0x3FEA9B66290EA1A3 // sin ( 11 Pi / 16 )    data8 0xBFE1C73B39AE68C8 // cos ( 11 Pi / 16 )//    data8 0x3FE6A09E667F3BCD // sin ( 12 Pi / 16 )    data8 0xBFE6A09E667F3BCD // cos ( 12 Pi / 16 )//    data8 0x3FE1C73B39AE68C8 // sin ( 13 Pi / 16 )    data8 0xBFEA9B66290EA1A3 // cos ( 13 Pi / 16 )//    data8 0x3FD87DE2A6AEA963 // sin ( 14 Pi / 16 )    data8 0xBFED906BCF328D46 // cos ( 14 Pi / 16 )//    data8 0x3FC8F8B83C69A60B // sin ( 15 Pi / 16 )    data8 0xBFEF6297CFF75CB0 // cos ( 15 Pi / 16 )//    data8 0x0000000000000000 // sin ( 16 Pi / 16 )    data8 0xBFF0000000000000 // cos ( 16 Pi / 16 )//    data8 0xBFC8F8B83C69A60B // sin ( 17 Pi / 16 )    data8 0xBFEF6297CFF75CB0 // cos ( 17 Pi / 16 )//    data8 0xBFD87DE2A6AEA963 // sin ( 18 Pi / 16 )    data8 0xBFED906BCF328D46 // cos ( 18 Pi / 16 )//    data8 0xBFE1C73B39AE68C8 // sin ( 19 Pi / 16 )    data8 0xBFEA9B66290EA1A3 // cos ( 19 Pi / 16 )//    data8 0xBFE6A09E667F3BCD // sin ( 20 Pi / 16 )    data8 0xBFE6A09E667F3BCD // cos ( 20 Pi / 16 )//    data8 0xBFEA9B66290EA1A3 // sin ( 21 Pi / 16 )    data8 0xBFE1C73B39AE68C8 // cos ( 21 Pi / 16 )//    data8 0xBFED906BCF328D46 // sin ( 22 Pi / 16 )    data8 0xBFD87DE2A6AEA963 // cos ( 22 Pi / 16 )//    data8 0xBFEF6297CFF75CB0 // sin ( 23 Pi / 16 )    data8 0xBFC8F8B83C69A60B // cos ( 23 Pi / 16 )//    data8 0xBFF0000000000000 // sin ( 24 Pi / 16 )    data8 0x0000000000000000 // cos ( 24 Pi / 16 )//    data8 0xBFEF6297CFF75CB0 // sin ( 25 Pi / 16 )    data8 0x3FC8F8B83C69A60B // cos ( 25 Pi / 16 )//