s_cos.s

glibc 2.9,最新版的C语言库函数

.file "sincos.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 Work range is widened by reduction strengthen (3 parts of Pi/16)
// 02/10/03 Reordered header: .section, .global, .proc, .align
// 08/08/03 Improved performance
// 10/28/04 Saved sincos_r_sincos to avoid clobber by dynamic loader 
// 03/31/05 Reformatted delimiters between data tables

// API
//==============================================================
// double sin( double x);
// double cos( double 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)) - 
//                        nfloat * LOWEST(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 and LOW parts are rounded to zero values, 
//    and LOWEST is rounded to nearest one.
//
// 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 + r^6 q3 + ... = Series for Cos
// Sin(r) = r + r^3 p1  + r^5 p2 + r^7 p3 + ... = 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 + r^4p3 + r^6p4
//    Q = q1 + r^2q2 + r^4q3 + r^6q4
//
//       rcub = r * rsq
//       Sin(r) = r + rcub * P
//              = r + r^3p1  + r^5p2 + r^7p3 + r^9p4 + ... = 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 + rcub * P
//
//    Answer = S[m] Cos(r) + [Cm] P
//
//       Cos(r) = 1 + rsq Q
//       Cos(r) = 1 + r^2 Q
//       Cos(r) = 1 + r^2 (q1 + r^2q2 + r^4q3 + r^6q4)
//       Cos(r) = 1 + r^2q1 + r^4q2 + r^6q3 + r^8q4 + ...
//
//       S[m] Cos(r) = S[m](1 + rsq Q)
//       S[m] Cos(r) = S[m] + Sm 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 -> r26
// r32 -> r35

// predicate registers used:
// p6 -> p11

// floating-point registers used
// f9 -> f15
// f32 -> f61

// Assembly macros
//==============================================================
sincos_NORM_f8                 = f9
sincos_W                       = f10
sincos_int_Nfloat              = f11
sincos_Nfloat                  = f12

sincos_r                       = f13
sincos_rsq                     = f14
sincos_rcub                    = f15
sincos_save_tmp                = f15

sincos_Inv_Pi_by_16            = f32
sincos_Pi_by_16_1              = f33
sincos_Pi_by_16_2              = f34

sincos_Inv_Pi_by_64            = f35

sincos_Pi_by_16_3              = f36

sincos_r_exact                 = f37

sincos_Sm                      = f38
sincos_Cm                      = f39

sincos_P1                      = f40
sincos_Q1                      = f41
sincos_P2                      = f42
sincos_Q2                      = f43
sincos_P3                      = f44
sincos_Q3                      = f45
sincos_P4                      = f46
sincos_Q4                      = f47

sincos_P_temp1                 = f48
sincos_P_temp2                 = f49

sincos_Q_temp1                 = f50
sincos_Q_temp2                 = f51

sincos_P                       = f52
sincos_Q                       = f53

sincos_srsq                    = f54

sincos_SIG_INV_PI_BY_16_2TO61  = f55
sincos_RSHF_2TO61              = f56
sincos_RSHF                    = f57
sincos_2TOM61                  = f58
sincos_NFLOAT                  = f59
sincos_W_2TO61_RSH             = f60

fp_tmp                         = f61

/////////////////////////////////////////////////////////////

sincos_GR_sig_inv_pi_by_16     = r14
sincos_GR_rshf_2to61           = r15
sincos_GR_rshf                 = r16
sincos_GR_exp_2tom61           = r17
sincos_GR_n                    = r18
sincos_GR_m                    = r19
sincos_GR_32m                  = r19
sincos_GR_all_ones             = r19
sincos_AD_1                    = r20
sincos_AD_2                    = r21
sincos_exp_limit               = r22
sincos_r_signexp               = r23
sincos_r_17_ones               = r24
sincos_r_sincos                = r25
sincos_r_exp                   = r26

GR_SAVE_PFS                    = r33
GR_SAVE_B0                     = r34
GR_SAVE_GP                     = r35
GR_SAVE_r_sincos               = r36


RODATA

// Pi/16 parts
.align 16
LOCAL_OBJECT_START(double_sincos_pi)
   data8 0xC90FDAA22168C234, 0x00003FFC // pi/16 1st part
   data8 0xC4C6628B80DC1CD1, 0x00003FBC // pi/16 2nd part
   data8 0xA4093822299F31D0, 0x00003F7A // pi/16 3rd part
LOCAL_OBJECT_END(double_sincos_pi)

// Coefficients for polynomials
LOCAL_OBJECT_START(double_sincos_pq_k4)
   data8 0x3EC71C963717C63A // P4
   data8 0x3EF9FFBA8F191AE6 // Q4
   data8 0xBF2A01A00F4E11A8 // P3
   data8 0xBF56C16C05AC77BF // Q3
   data8 0x3F8111111110F167 // P2
   data8 0x3FA555555554DD45 // Q2
   data8 0xBFC5555555555555 // P1
   data8 0xBFDFFFFFFFFFFFFC // Q1
LOCAL_OBJECT_END(double_sincos_pq_k4)

// Sincos table (S[m], C[m])
LOCAL_OBJECT_START(double_sin_cos_beta_k4)

data8 0x0000000000000000 , 0x00000000 // sin( 0 pi/16)  S0
data8 0x8000000000000000 , 0x00003fff // cos( 0 pi/16)  C0
//
data8 0xc7c5c1e34d3055b3 , 0x00003ffc // sin( 1 pi/16)  S1
data8 0xfb14be7fbae58157 , 0x00003ffe // cos( 1 pi/16)  C1
//
data8 0xc3ef1535754b168e , 0x00003ffd // sin( 2 pi/16)  S2
data8 0xec835e79946a3146 , 0x00003ffe // cos( 2 pi/16)  C2
//
data8 0x8e39d9cd73464364 , 0x00003ffe // sin( 3 pi/16)  S3
data8 0xd4db3148750d181a , 0x00003ffe // cos( 3 pi/16)  C3
//
data8 0xb504f333f9de6484 , 0x00003ffe // sin( 4 pi/16)  S4
data8 0xb504f333f9de6484 , 0x00003ffe // cos( 4 pi/16)  C4
//
data8 0xd4db3148750d181a , 0x00003ffe // sin( 5 pi/16)  C3
data8 0x8e39d9cd73464364 , 0x00003ffe // cos( 5 pi/16)  S3
//
data8 0xec835e79946a3146 , 0x00003ffe // sin( 6 pi/16)  C2
data8 0xc3ef1535754b168e , 0x00003ffd // cos( 6 pi/16)  S2
//
data8 0xfb14be7fbae58157 , 0x00003ffe // sin( 7 pi/16)  C1
data8 0xc7c5c1e34d3055b3 , 0x00003ffc // cos( 7 pi/16)  S1
//
data8 0x8000000000000000 , 0x00003fff // sin( 8 pi/16)  C0
data8 0x0000000000000000 , 0x00000000 // cos( 8 pi/16)  S0
//
data8 0xfb14be7fbae58157 , 0x00003ffe // sin( 9 pi/16)  C1
data8 0xc7c5c1e34d3055b3 , 0x0000bffc // cos( 9 pi/16)  -S1
//
data8 0xec835e79946a3146 , 0x00003ffe // sin(10 pi/16)  C2
data8 0xc3ef1535754b168e , 0x0000bffd // cos(10 pi/16)  -S2
//
data8 0xd4db3148750d181a , 0x00003ffe // sin(11 pi/16)  C3
data8 0x8e39d9cd73464364 , 0x0000bffe // cos(11 pi/16)  -S3
//
data8 0xb504f333f9de6484 , 0x00003ffe // sin(12 pi/16)  S4
data8 0xb504f333f9de6484 , 0x0000bffe // cos(12 pi/16)  -S4
//
data8 0x8e39d9cd73464364 , 0x00003ffe // sin(13 pi/16) S3
data8 0xd4db3148750d181a , 0x0000bffe // cos(13 pi/16) -C3
//
data8 0xc3ef1535754b168e , 0x00003ffd // sin(14 pi/16) S2
data8 0xec835e79946a3146 , 0x0000bffe // cos(14 pi/16) -C2
//
data8 0xc7c5c1e34d3055b3 , 0x00003ffc // sin(15 pi/16) S1
data8 0xfb14be7fbae58157 , 0x0000bffe // cos(15 pi/16) -C1
//
data8 0x0000000000000000 , 0x00000000 // sin(16 pi/16) S0
data8 0x8000000000000000 , 0x0000bfff // cos(16 pi/16) -C0
//
data8 0xc7c5c1e34d3055b3 , 0x0000bffc // sin(17 pi/16) -S1
data8 0xfb14be7fbae58157 , 0x0000bffe // cos(17 pi/16) -C1
//
data8 0xc3ef1535754b168e , 0x0000bffd // sin(18 pi/16) -S2
data8 0xec835e79946a3146 , 0x0000bffe // cos(18 pi/16) -C2
//
data8 0x8e39d9cd73464364 , 0x0000bffe // sin(19 pi/16) -S3
data8 0xd4db3148750d181a , 0x0000bffe // cos(19 pi/16) -C3
//
data8 0xb504f333f9de6484 , 0x0000bffe // sin(20 pi/16) -S4
data8 0xb504f333f9de6484 , 0x0000bffe // cos(20 pi/16) -S4
//
data8 0xd4db3148750d181a , 0x0000bffe // sin(21 pi/16) -C3
data8 0x8e39d9cd73464364 , 0x0000bffe // cos(21 pi/16) -S3
//
data8 0xec835e79946a3146 , 0x0000bffe // sin(22 pi/16) -C2
data8 0xc3ef1535754b168e , 0x0000bffd // cos(22 pi/16) -S2
//
data8 0xfb14be7fbae58157 , 0x0000bffe // sin(23 pi/16) -C1
data8 0xc7c5c1e34d3055b3 , 0x0000bffc // cos(23 pi/16) -S1
//
data8 0x8000000000000000 , 0x0000bfff // sin(24 pi/16) -C0
data8 0x0000000000000000 , 0x00000000 // cos(24 pi/16) S0
//
data8 0xfb14be7fbae58157 , 0x0000bffe // sin(25 pi/16) -C1
data8 0xc7c5c1e34d3055b3 , 0x00003ffc // cos(25 pi/16) S1
//
data8 0xec835e79946a3146 , 0x0000bffe // sin(26 pi/16) -C2
data8 0xc3ef1535754b168e , 0x00003ffd // cos(26 pi/16) S2
//
data8 0xd4db3148750d181a , 0x0000bffe // sin(27 pi/16) -C3
data8 0x8e39d9cd73464364 , 0x00003ffe // cos(27 pi/16) S3
//
data8 0xb504f333f9de6484 , 0x0000bffe // sin(28 pi/16) -S4
data8 0xb504f333f9de6484 , 0x00003ffe // cos(28 pi/16) S4
//
data8 0x8e39d9cd73464364 , 0x0000bffe // sin(29 pi/16) -S3
data8 0xd4db3148750d181a , 0x00003ffe // cos(29 pi/16) C3
//
data8 0xc3ef1535754b168e , 0x0000bffd // sin(30 pi/16) -S2
data8 0xec835e79946a3146 , 0x00003ffe // cos(30 pi/16) C2
//
data8 0xc7c5c1e34d3055b3 , 0x0000bffc // sin(31 pi/16) -S1
data8 0xfb14be7fbae58157 , 0x00003ffe // cos(31 pi/16) C1
//
data8 0x0000000000000000 , 0x00000000 // sin(32 pi/16) S0