.file "asin.s"// Copyright (c) 2000 - 2003 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// 08/17/00 New and much faster algorithm.// 08/31/00 Avoided bank conflicts on loads, shortened |x|=1 path,//          fixed mfb split issue stalls.// 12/19/00 Fixed small arg cases to force inexact, or inexact and underflow.// 08/02/02 New and much faster algorithm II// 02/06/03 Reordered header: .section, .global, .proc, .align// Description//=========================================// The asin function computes the principal value of the arc sine of x.// asin(0) returns 0, asin(1) returns pi/2, asin(-1) returns -pi/2.// A doman error occurs for arguments not in the range [-1,+1].//// The asin function returns the arc sine in the range [-pi/2, +pi/2] radians.//// There are 8 paths:// 1. x = +/-0.0//    Return asin(x) = +/-0.0//// 2. 0.0 < |x| < 0.625//    Return asin(x) = x + x^3 *PolA(x^2)//    where PolA(x^2) = A3 + A5*x^2 + A7*x^4 +...+ A35*x^32//// 3. 0.625 <=|x| < 1.0//    Return asin(x) = sign(x) * ( Pi/2 - sqrt(R) * PolB(R))//    Where R = 1 - |x|,//          PolB(R) = B0 + B1*R + B2*R^2 +...+B12*R^12////    sqrt(R) is approximated using the following sequence://        y0 = (1 + eps)/sqrt(R) - initial approximation by frsqrta,//             |eps| < 2^(-8)//        Then 3 iterations are used to refine the result://        H0 = 0.5*y0//        S0 = R*y0////        d0 = 0.5 - H0*S0//        H1 = H0 + d0*H0//        S1 = S0 + d0*S0////        d1 = 0.5 - H1*S1//        H2 = H1 + d0*H1//        S2 = S1 + d0*S1////        d2 = 0.5 - H2*S2//        S3 = S3 + d2*S3////        S3 approximates sqrt(R) with enough accuracy for this algorithm////    So, the result should be reconstracted as follows://    asin(x) = sign(x) * (Pi/2 - S3*PolB(R))////    But for optimization perposes the reconstruction step is slightly//    changed://    asin(x) = sign(x)*(Pi/2 - PolB(R)*S2) + sign(x)*d2*S2*PolB(R)//// 4. |x| = 1.0//    Return asin(x) = sign(x)*Pi/2//// 5. 1.0 < |x| <= +INF//    A doman error occurs for arguments not in the range [-1,+1]//// 6. x = [S,Q]NaN//    Return asin(x) = QNaN//// 7. x is denormal//    Return asin(x) = x + x^3,//// 8. x is unnormal//    Normalize input in f8 and return to the very beginning of the function//// Registers used//==============================================================// Floating Point registers used:// f8, input, output// f6, f7, f9 -> f15, f32 -> f63// General registers used:// r3, r21 -> r31, r32 -> r38// Predicate registers used:// p0, p6 -> p14//// Assembly macros//=========================================// integer registers used// scratchrTblAddr                      = r3rPiBy2Ptr                     = r21rTmpPtr3                      = r22rDenoBound                    = r23rOne                          = r24rAbsXBits                     = r25rHalf                         = r26r0625                         = r27rSign                         = r28rXBits                        = r29rTmpPtr2                      = r30rTmpPtr1                      = r31// stackedGR_SAVE_PFS                   = r32GR_SAVE_B0                    = r33GR_SAVE_GP                    = r34GR_Parameter_X                = r35GR_Parameter_Y                = r36GR_Parameter_RESULT           = r37GR_Parameter_TAG              = r38// floating point registers usedFR_X                          = f10FR_Y                          = f1FR_RESULT                     = f8// scratchfXSqr                         = f6fXCube                        = f7fXQuadr                       = f9f1pX                          = f10f1mX                          = f11f1pXRcp                       = f12f1mXRcp                       = f13fH                            = f14fS                            = f15// stackedfA3                           = f32fB1                           = f32fA5                           = f33fB2                           = f33fA7                           = f34fPiBy2                        = f34fA9                           = f35fA11                          = f36fB10                          = f35fB11                          = f36fA13                          = f37fA15                          = f38fB4                           = f37fB5                           = f38fA17                          = f39fA19                          = f40fB6                           = f39fB7                           = f40fA21                          = f41fA23                          = f42fB3                           = f41fB8                           = f42fA25                          = f43fA27                          = f44fB9                           = f43fB12                          = f44fA29                          = f45fA31                          = f46fA33                          = f47fA35                          = f48fBaseP                        = f49fB0                           = f50fSignedS                      = f51fD                            = f52fHalf                         = f53fR                            = f54fCloseTo1Pol                  = f55fSignX                        = f56fDenoBound                    = f57fNormX                        = f58fX8                           = f59fRSqr                         = f60fRQuadr                       = f61fR8                           = f62fX16                          = f63// Data tables//==============================================================RODATA.align 16LOCAL_OBJECT_START(asin_base_range_table)// Ai: Polynomial coefficients for the asin(x), |x| < .625000// Bi: Polynomial coefficients for the asin(x), |x| > .625000data8 0xBFDAAB56C01AE468 //A29data8 0x3FE1C470B76A5B2B //A31data8 0xBFDC5FF82A0C4205 //A33data8 0x3FC71FD88BFE93F0 //A35data8 0xB504F333F9DE6487, 0x00003FFF //B0data8 0xAAAAAAAAAAAAFC18, 0x00003FFC //A3data8 0x3F9F1C71BC4A7823 //A9data8 0x3F96E8BBAAB216B2 //A11data8 0x3F91C4CA1F9F8A98 //A13data8 0x3F8C9DDCEDEBE7A6 //A15data8 0x3F877784442B1516 //A17data8 0x3F859C0491802BA2 //A19data8 0x9999999998C88B8F, 0x00003FFB //A5data8 0x3F6BD7A9A660BF5E //A21data8 0x3F9FC1659340419D //A23data8 0xB6DB6DB798149BDF, 0x00003FFA //A7data8 0xBFB3EF18964D3ED3 //A25data8 0x3FCD285315542CF2 //A27data8 0xF15BEEEFF7D2966A, 0x00003FFB //B1data8 0x3EF0DDA376D10FB3 //B10data8 0xBEB83CAFE05EBAC9 //B11data8 0x3F65FFB67B513644 //B4data8 0x3F5032FBB86A4501 //B5data8 0x3F392162276C7CBA //B6data8 0x3F2435949FD98BDF //B7data8 0xD93923D7FA08341C, 0x00003FF9 //B2data8 0x3F802995B6D90BDB //B3data8 0x3F10DF86B341A63F //B8data8 0xC90FDAA22168C235, 0x00003FFF // Pi/2data8 0x3EFA3EBD6B0ECB9D //B9data8 0x3EDE18BA080E9098 //B12LOCAL_OBJECT_END(asin_base_range_table).section .textGLOBAL_LIBM_ENTRY(asin)asin_unnormal_back:{ .mfi      getf.d             rXBits = f8 // grab bits of input value      // set p12 = 1 if x is a NaN, denormal, or zero      fclass.m           p12, p0 = f8, 0xcf      adds               rSign = 1, r0}{ .mfi      addl               rTblAddr = @ltoff(asin_base_range_table),gp      // 1 - x = 1 - |x| for positive x      fms.s1             f1mX = f1, f1, f8      addl               rHalf = 0xFFFE, r0 // exponent of 1/2};;{ .mfi      addl               r0625 = 0x3FE4, r0 // high 16 bits of 0.625      // set p8 = 1 if x < 0      fcmp.lt.s1         p8, p9 = f8, f0      shl                rSign = rSign, 63 // sign bit}{ .mfi      // point to the beginning of the table      ld8                rTblAddr = [rTblAddr]      // 1 + x = 1 - |x| for negative x      fma.s1             f1pX = f1, f1, f8      adds               rOne = 0x3FF, r0};;{ .mfi      andcm              rAbsXBits = rXBits, rSign // bits of |x|      fmerge.s           fSignX = f8, f1 // signum(x)      shl                r0625 = r0625, 48 // bits of DP representation of 0.625}{ .mfb      setf.exp           fHalf = rHalf // load A2 to FP reg      fma.s1             fXSqr = f8, f8, f0 // x^2      // branch on special path if x is a NaN, denormal, or zero(p12) br.cond.spnt       asin_special};;{ .mfi      adds               rPiBy2Ptr = 272, rTblAddr      nop.f              0      shl                rOne = rOne, 52 // bits of 1.0}{ .mfi      adds               rTmpPtr1 = 16, rTblAddr      nop.f              0      // set p6 = 1 if |x| < 0.625      cmp.lt             p6, p7 = rAbsXBits, r0625};;{ .mfi      ldfpd              fA29, fA31 = [rTblAddr] // A29, fA31      // 1 - x = 1 - |x| for positive x(p9)  fms.s1             fR = f1, f1, f8      // point to coefficient of "near 1" polynomial(p7)  adds               rTmpPtr2 = 176, rTblAddr}{ .mfi      ldfpd              fA33, fA35 = [rTmpPtr1], 16 // A33, fA35      // 1 + x = 1 - |x| for negative x(p8)  fma.s1             fR = f1, f1, f8(p6)  adds               rTmpPtr2 = 48, rTblAddr};;{ .mfi      ldfe               fB0 = [rTmpPtr1], 16 // B0      nop.f              0      nop.i              0}{ .mib      adds               rTmpPtr3 = 16, rTmpPtr2      // set p10 = 1 if |x| = 1.0      cmp.eq             p10, p0 = rAbsXBits, rOne      // branch on special path for |x| = 1.0(p10) br.cond.spnt       asin_abs_1};;{ .mfi      ldfe               fA3 = [rTmpPtr2], 48 // A3 or B1      nop.f              0      adds               rTmpPtr1 = 64, rTmpPtr3}{ .mib      ldfpd              fA9, fA11 = [rTmpPtr3], 16 // A9, A11 or B10, B11      // set p11 = 1 if |x| > 1.0      cmp.gt             p11, p0 = rAbsXBits, rOne      // branch on special path for |x| > 1.0(p11) br.cond.spnt       asin_abs_gt_1};;{ .mfi      ldfpd              fA17, fA19 = [rTmpPtr2], 16 // A17, A19 or B6, B7      // initial approximation of 1 / sqrt(1 - x)      frsqrta.s1         f1mXRcp, p0 = f1mX      nop.i              0}{ .mfi      ldfpd              fA13, fA15 = [rTmpPtr3] // A13, A15 or B4, B5      fma.s1             fXCube = fXSqr, f8, f0 // x^3      nop.i              0};;{ .mfi      ldfe               fA5 = [rTmpPtr2], 48 // A5 or B2      // initial approximation of 1 / sqrt(1 + x)      frsqrta.s1         f1pXRcp, p0 = f1pX      nop.i              0}{ .mfi      ldfpd              fA21, fA23 = [rTmpPtr1], 16 // A21, A23 or B3, B8      fma.s1             fXQuadr = fXSqr, fXSqr, f0 // x^4      nop.i              0};;{ .mfi      ldfe               fA7 = [rTmpPtr1] // A7 or Pi/2      fma.s1             fRSqr = fR, fR, f0 // R^2      nop.i              0}{ .mfb      ldfpd              fA25, fA27 = [rTmpPtr2] // A25, A27 or B9, B12      nop.f              0(p6)  br.cond.spnt       asin_base_range;};;{ .mfi      nop.m              0(p9)  fma.s1             fH = fHalf, f1mXRcp, f0 // H0 for x > 0      nop.i              0}{ .mfi      nop.m              0(p9)  fma.s1             fS = f1mX, f1mXRcp, f0  // S0 for x > 0      nop.i              0};;{ .mfi      nop.m              0(p8)  fma.s1             fH = fHalf, f1pXRcp, f0 // H0 for x < 0      nop.i              0}{ .mfi      nop.m              0(p8)  fma.s1             fS = f1pX, f1pXRcp, f0  // S0 for x > 0      nop.i              0};;{ .mfi      nop.m              0      fma.s1             fRQuadr = fRSqr, fRSqr, f0 // R^4      nop.i              0};;{ .mfi      nop.m              0      fma.s1             fB11 = fB11, fR, fB10      nop.i              0}{ .mfi      nop.m              0      fma.s1             fB1 = fB1, fR, fB0      nop.i              0};;{ .mfi      nop.m              0      fma.s1             fB5 = fB5, fR, fB4      nop.i              0