e_acos.s

glibc 2.9,最新版的C语言库函数

.file "acos.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.
// 08/30/00 Avoided bank conflicts on loads, shortened |x|=1 and x=0 paths,
//          fixed mfb split issue stalls.
// 05/20/02 Cleaned up namespace and sf0 syntax
// 08/02/02 New and much faster algorithm II
// 02/06/03 Reordered header: .section, .global, .proc, .align

// Description
//=========================================
// The acos function computes the principal value of the arc cosine of x.
// acos(0) returns Pi/2, acos(1) returns 0, acos(-1) returns Pi.
// A doman error occurs for arguments not in the range [-1,+1].
//
// The acos function returns the arc cosine in the range [0, Pi] radians.
//
// There are 8 paths:
// 1. x = +/-0.0
//    Return acos(x) = Pi/2 + x
//
// 2. 0.0 < |x| < 0.625
//    Return acos(x) = Pi/2 - 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 acos(x) = Pi/2 - asin(x) =
//                   = Pi/2 - 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:
//    acos(x) = Pi/2 - sign(x) * (Pi/2 - S3*PolB(R))
//
//    But for optimization purposes the reconstruction step is slightly
//    changed:
//    acos(x) = Cpi + sign(x)*PolB(R)*S2 - sign(x)*d2*S2*PolB(R)
//        where Cpi = 0 if x > 0 and Cpi = Pi if x < 0
//
// 4. |x| = 1.0
//    Return acos(1.0) = 0.0, acos(-1.0) = Pi
//
// 5. 1.0 < |x| <= +INF
//    A doman error occurs for arguments not in the range [-1,+1]
//
// 6. x = [S,Q]NaN
//    Return acos(x) = QNaN
//
// 7. x is denormal
//    Return acos(x) = Pi/2 - x,
//
// 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 -> f64

// General registers used:
// r3, r21 -> r31, r32 -> r38

// Predicate registers used:
// p0, p6 -> p14

//
// Assembly macros
//=========================================
// integer registers used
// scratch
rTblAddr                      = r3

rPiBy2Ptr                     = r21
rTmpPtr3                      = r22
rDenoBound                    = r23
rOne                          = r24
rAbsXBits                     = r25
rHalf                         = r26
r0625                         = r27
rSign                         = r28
rXBits                        = r29
rTmpPtr2                      = r30
rTmpPtr1                      = r31

// stacked
GR_SAVE_PFS                   = r32
GR_SAVE_B0                    = r33
GR_SAVE_GP                    = r34
GR_Parameter_X                = r35
GR_Parameter_Y                = r36
GR_Parameter_RESULT           = r37
GR_Parameter_TAG              = r38

// floating point registers used
FR_X                          = f10
FR_Y                          = f1
FR_RESULT                     = f8


// scratch
fXSqr                         = f6
fXCube                        = f7
fXQuadr                       = f9
f1pX                          = f10
f1mX                          = f11
f1pXRcp                       = f12
f1mXRcp                       = f13
fH                            = f14
fS                            = f15
// stacked
fA3                           = f32
fB1                           = f32
fA5                           = f33
fB2                           = f33
fA7                           = f34
fPiBy2                        = f34
fA9                           = f35
fA11                          = f36
fB10                          = f35
fB11                          = f36
fA13                          = f37
fA15                          = f38
fB4                           = f37
fB5                           = f38
fA17                          = f39
fA19                          = f40
fB6                           = f39
fB7                           = f40
fA21                          = f41
fA23                          = f42
fB3                           = f41
fB8                           = f42
fA25                          = f43
fA27                          = f44
fB9                           = f43
fB12                          = f44
fA29                          = f45
fA31                          = f46
fA33                          = f47
fA35                          = f48
fBaseP                        = f49
fB0                           = f50
fSignedS                      = f51
fD                            = f52
fHalf                         = f53
fR                            = f54
fCloseTo1Pol                  = f55
fSignX                        = f56
fDenoBound                    = f57
fNormX                        = f58
fX8                           = f59
fRSqr                         = f60
fRQuadr                       = f61
fR8                           = f62
fX16                          = f63
fCpi                          = f64

// Data tables
//==============================================================
RODATA
.align 16
LOCAL_OBJECT_START(acos_base_range_table)
// Ai: Polynomial coefficients for the acos(x), |x| < .625000
// Bi: Polynomial coefficients for the acos(x), |x| > .625000
data8 0xBFDAAB56C01AE468 //A29
data8 0x3FE1C470B76A5B2B //A31
data8 0xBFDC5FF82A0C4205 //A33
data8 0x3FC71FD88BFE93F0 //A35
data8 0xB504F333F9DE6487, 0x00003FFF //B0
data8 0xAAAAAAAAAAAAFC18, 0x00003FFC //A3
data8 0x3F9F1C71BC4A7823 //A9
data8 0x3F96E8BBAAB216B2 //A11
data8 0x3F91C4CA1F9F8A98 //A13
data8 0x3F8C9DDCEDEBE7A6 //A15
data8 0x3F877784442B1516 //A17
data8 0x3F859C0491802BA2 //A19
data8 0x9999999998C88B8F, 0x00003FFB //A5
data8 0x3F6BD7A9A660BF5E //A21
data8 0x3F9FC1659340419D //A23
data8 0xB6DB6DB798149BDF, 0x00003FFA //A7
data8 0xBFB3EF18964D3ED3 //A25
data8 0x3FCD285315542CF2 //A27
data8 0xF15BEEEFF7D2966A, 0x00003FFB //B1
data8 0x3EF0DDA376D10FB3 //B10
data8 0xBEB83CAFE05EBAC9 //B11
data8 0x3F65FFB67B513644 //B4
data8 0x3F5032FBB86A4501 //B5
data8 0x3F392162276C7CBA //B6
data8 0x3F2435949FD98BDF //B7
data8 0xD93923D7FA08341C, 0x00003FF9 //B2
data8 0x3F802995B6D90BDB //B3
data8 0x3F10DF86B341A63F //B8
data8 0xC90FDAA22168C235, 0x00003FFF // Pi/2
data8 0x3EFA3EBD6B0ECB9D //B9
data8 0x3EDE18BA080E9098 //B12
LOCAL_OBJECT_END(acos_base_range_table)

.section .text
GLOBAL_LIBM_ENTRY(acos)
acos_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(acos_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       acos_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       acos_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       acos_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       acos_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
}
{ .mfi
      nop.m              0
      fma.s1             fB7 = fB7, fR, fB6
      nop.i              0
}
;;
{ .mfi
      nop.m              0