.file "libm_sincosf.s"// Copyright (c) 2002 - 2005, Intel Corporation// All rights reserved.//// Contributed 2002 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//==============================================================// 02/01/02 Initial version// 02/18/02 Large arguments processing routine is excluded.//          External interface entry points are added// 02/26/02 Added temporary return of results in r8, r9// 03/13/02 Corrected restore of predicate registers// 03/19/02 Added stack unwind around call to __libm_cisf_large// 09/05/02 Work range is widened by reduction strengthen (2 parts of Pi/16)// 02/10/03 Reordered header: .section, .global, .proc, .align// 02/11/04 cisf is moved to the separate file.// 03/31/05 Reformatted delimiters between data tables// API//==============================================================// 1) void sincosf(float, float*s, float*c)// 2) __libm_sincosf - internal LIBM function, that accepts//    argument in f8 and returns cosine through f8, sine through f9//// 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^2p2//    Q = q1 + r^2q2////       rcub = r * rsq//       Sin(r) = r + rcub * P//              = r + r^3p1  + r^5p2 = Sin(r)////       P =  r + rcub * 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 -> r49// predicate registers used:// p6 -> p14// floating-point registers used// f9 -> f15// f32 -> f100// Assembly macros//==============================================================cisf_Arg                     = f8cisf_Sin_res                 = f9cisf_Cos_res                 = f8cisf_NORM_f8                 = f10cisf_W                       = f11cisf_int_Nfloat              = f12cisf_Nfloat                  = f13cisf_r                       = f14cisf_r_exact                 = f68cisf_rsq                     = f15cisf_rcub                    = f32cisf_Inv_Pi_by_16            = f33cisf_Pi_by_16_hi             = f34cisf_Pi_by_16_lo             = f35cisf_Inv_Pi_by_64            = f36cisf_Pi_by_64_hi             = f37cisf_Pi_by_64_lo             = f38cisf_P1                      = f39cisf_Q1                      = f40cisf_P2                      = f41cisf_Q2                      = f42cisf_P3                      = f43cisf_Q3                      = f44cisf_P4                      = f45cisf_Q4                      = f46cisf_P_temp1                 = f47cisf_P_temp2                 = f48cisf_Q_temp1                 = f49cisf_Q_temp2                 = f50cisf_P                       = f51cisf_SIG_INV_PI_BY_16_2TO61  = f52cisf_RSHF_2TO61              = f53cisf_RSHF                    = f54cisf_2TOM61                  = f55cisf_NFLOAT                  = f56cisf_W_2TO61_RSH             = f57cisf_tmp                     = f58cisf_Sm_sin                  = f59cisf_Cm_sin                  = f60cisf_Sm_cos                  = f61cisf_Cm_cos                  = f62cisf_srsq_sin                = f63cisf_srsq_cos                = f64cisf_Q_sin                   = f65cisf_Q_cos                   = f66cisf_Q                       = f67/////////////////////////////////////////////////////////////cisf_pResSin                 = r33cisf_pResCos                 = r34cisf_exp_limit               = r35cisf_r_signexp               = r36cisf_AD_beta_table           = r37cisf_r_sincos                = r38cisf_r_exp                   = r39cisf_r_17_ones               = r40cisf_GR_sig_inv_pi_by_16     = r14cisf_GR_rshf_2to61           = r15cisf_GR_rshf                 = r16cisf_GR_exp_2tom61           = r17cisf_GR_n                    = r18cisf_GR_n_sin                = r19cisf_GR_m_sin                = r41cisf_GR_32m_sin              = r41cisf_GR_n_cos                = r42cisf_GR_m_cos                = r43cisf_GR_32m_cos              = r43cisf_AD_2_sin                = r44cisf_AD_2_cos                = r45cisf_gr_tmp                  = r46GR_SAVE_B0                   = r47GR_SAVE_GP                   = r48rB0_SAVED                    = r49GR_SAVE_PFS                  = r50GR_SAVE_PR                   = r51cisf_AD_1                    = r52RODATA.align 16// Pi/16 partsLOCAL_OBJECT_START(double_cisf_pi)   data8 0xC90FDAA22168C234, 0x00003FFC // pi/16 1st part   data8 0xC4C6628B80DC1CD1, 0x00003FBC // pi/16 2nd partLOCAL_OBJECT_END(double_cisf_pi)// Coefficients for polynomialsLOCAL_OBJECT_START(double_cisf_pq_k4)   data8 0x3F810FABB668E9A2 // P2   data8 0x3FA552E3D6DE75C9 // Q2   data8 0xBFC555554447BC7F // P1   data8 0xBFDFFFFFC447610A // Q1LOCAL_OBJECT_END(double_cisf_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 )