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windows-server-2003/base/crts/fpw32/tran/ia64/sincos.s

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.file "sincos.s"
// Copyright (c) 2000, Intel Corporation
// All rights reserved.
//
// Contributed 2/2/2000 by John Harrison, Ted Kubaska, Bob Norin, Shane Story,
// and Ping Tak Peter Tang of the Computational Software Lab, Intel Corporation.
//
// WARRANTY DISCLAIMER
//
// 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://developer.intel.com/opensource.
//
// History
//==============================================================
// 2/02/00 Initial revision
// 4/02/00 Unwind support added.
// 6/16/00 Updated tables to enforce symmetry
// 8/31/00 Saved 2 cycles in main path, and 9 in other paths.
// 9/20/00 The updated tables regressed to an old version, so reinstated them
// 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
// 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)
//
// 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
//
// Sm = Sin(Mpi/2^k) and Cm = 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 = Sm 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 + ...
//
// Sm Cos(r) = Sm(1 + rsq Q)
// Sm Cos(r) = Sm + Sm rsq Q
// Sm Cos(r) = Sm + s_rsq Q
// Q = Sm + s_rsq Q
//
// Then,
//
// Answer = Q + Cm P
// Registers used
//==============================================================
// general input registers:
// r32 -> r45
// predicate registers used:
// p6 -> p13
// floating-point registers used: 31
// f9 -> f15
// f32 -> f54
// Assembly macros
//==============================================================
sind_W = f10
sind_int_Nfloat = f11
sind_Nfloat = f12
sind_r = f13
sind_rsq = f14
sind_rcub = f15
sind_Inv_Pi_by_16 = f32
sind_Pi_by_16_hi = f33
sind_Pi_by_16_lo = f34
sind_Inv_Pi_by_64 = f35
sind_Pi_by_64_hi = f36
sind_Pi_by_64_lo = f37
sind_Sm = f38
sind_Cm = f39
sind_P1 = f40
sind_Q1 = f41
sind_P2 = f42
sind_Q2 = f43
sind_P3 = f44
sind_Q3 = f45
sind_P4 = f46
sind_Q4 = f47
sind_P_temp1 = f48
sind_P_temp2 = f49
sind_Q_temp1 = f50
sind_Q_temp2 = f51
sind_P = f52
sind_Q = f53
sind_srsq = f54
/////////////////////////////////////////////////////////////
sind_r_signexp = r36
sind_AD_beta_table = r37
sind_r_sincos = r38
sind_r_exp = r39
sind_r_17_ones = r40
GR_SAVE_PFS = r41
GR_SAVE_B0 = r42
GR_SAVE_GP = r43
.data
.align 16
double_sind_pi:
data8 0xA2F9836E4E44152A, 0x00004001 // 16/pi
// c90fdaa22168c234
data8 0xC90FDAA22168C234, 0x00003FFC // pi/16 hi
// c4c6628b80dc1cd1 29024e088a
data8 0xC4C6628B80DC1CD1, 0x00003FBC
double_sind_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
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 0xc3ef1535754b168c , 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
data8 0x8000000000000000 , 0x00003fff // cos(32 pi/16) C0
.align 32
.global sin#
.global cos#
////////////////////////////////////////////////////////
// There are two entry points: sin and cos
// If from sin, p8 is true
// If from cos, p9 is true
.section .text
.proc sin#
.align 32
sin:
// The initial fnorm will take any unmasked faults and
// normalize any single/double unorms
{ .mfi
alloc r32=ar.pfs,1,13,0,0
(p0) fnorm f8 = f8
(p0) cmp.eq.unc p8,p9 = r0, r0
}
{ .mib
(p0) addl r33 = @ltoff(double_sind_pi), gp
(p0) mov sind_r_sincos = 0x0
(p0) br.sptk SIND_SINCOS ;;
}
.endp sin
.section .text
.proc cos#
.align 32
cos:
// The initial fnorm will take any unmasked faults and
// normalize any single/double unorms
{ .mfi
alloc r32=ar.pfs,1,13,0,0
(p0) fnorm f8 = f8
(p0) cmp.eq.unc p9,p8 = r0, r0
}
{ .mib
(p0) addl r33 = @ltoff(double_sind_pi), gp
(p0) mov sind_r_sincos = 0x8
(p0) br.sptk SIND_SINCOS ;;
}
////////////////////////////////////////////////////////
// All entry points end up here.
// If from sin, sind_r_sincos is 0 and p8 is true
// If from cos, sind_r_sincos is 8 = 2^(k-1) and p9 is true
// We add sind_r_sincos to N
SIND_SINCOS:
{ .mmi
ld8 r33 = [r33]
(p0) addl r34 = @ltoff(double_sind_pq_k4), gp
(p0) mov sind_r_17_ones = 0x1ffff
}
;;
{ .mfi
ld8 r34 = [r34]
nop.f 999
nop.i 999 ;;
}
// 0x10009 is register_bias + 10.
// So if f8 > 2^10 = Gamma, go to DBX
{ .mii
(p0) ldfe sind_Inv_Pi_by_16 = [r33],16
(p0) mov r35 = 0x10009
nop.i 999 ;;
}
// Start loading P, Q coefficients
{ .mmi
(p0) ldfpd sind_P4,sind_Q4 = [r34],16
(p0) addl sind_AD_beta_table = @ltoff(double_sin_cos_beta_k4), gp
nop.i 999 ;;
}
// SIN(0)
{ .mfi
ld8 sind_AD_beta_table = [sind_AD_beta_table]
(p8) fclass.m.unc p6,p0 = f8, 0x07
nop.i 999 ;;
}
// COS(0)
{ .mfi
(p0) getf.exp sind_r_signexp = f8
(p9) fclass.m.unc p7,p0 = f8, 0x07
nop.i 999
}
{ .mfi
(p0) ldfe sind_Pi_by_16_hi = [r33],16
nop.f 999
nop.i 999 ;;
}
{ .mfb
(p0) ldfe sind_Pi_by_16_lo = [r33],16
nop.f 999
(p6) br.ret.spnt b0 ;;
}
{ .mfb
(p0) and sind_r_exp = sind_r_17_ones, sind_r_signexp
(p7) fmerge.s f8 = f1,f1
(p7) br.ret.spnt b0 ;;
}
// p10 is true if we must call DBX SIN
// p10 is true if f8 exp is > 0x10009 (which includes all ones
// NAN or inf)
{ .mib
(p0) ldfpd sind_P3,sind_Q3 = [r34],16
(p0) cmp.ge.unc p10,p0 = sind_r_exp,r35
(p10) br.cond.spnt SIND_DBX ;;
}
{ .mfi
(p0) ldfpd sind_P2,sind_Q2 = [r34],16
nop.f 999
nop.i 999 ;;
}
// sind_W = x * sind_Inv_Pi_by_16
{ .mfi
(p0) ldfpd sind_P1,sind_Q1 = [r34]
(p0) fma.s1 sind_W = f8, sind_Inv_Pi_by_16, f0
nop.i 999 ;;
}
// sind_int_Nfloat = Round_Int_Nearest(sind_W)
// sind_r = -sind_Nfloat * sind_Pi_by_16_hi + x
// sind_r = sind_r -sind_Nfloat * sind_Pi_by_16_lo
{ .mfi
nop.m 999
(p0) fcvt.fx.s1 sind_int_Nfloat = sind_W
nop.i 999 ;;
}
{ .mfi
nop.m 999
(p0) fcvt.xf sind_Nfloat = sind_int_Nfloat
nop.i 999 ;;
}
// get N = (int)sind_int_Nfloat
// Add 2^(k-1) (which is in sind_r_sincos) to N
{ .mfi
(p0) getf.sig r43 = sind_int_Nfloat
nop.f 999
nop.i 999 ;;
}
{ .mmi
(p0) add r43 = r43, sind_r_sincos ;;
(p0) and r44 = 0x1f,r43
nop.i 999 ;;
}
// Get M (least k+1 bits of N)
// Add 32*M to address of sin_cos_beta table
{ .mfi
nop.m 999
(p0) fnma.s1 sind_r = sind_Nfloat, sind_Pi_by_16_hi, f8
(p0) shl r44 = r44,5 ;;
}
{ .mmi
(p0) add r45 = r44, sind_AD_beta_table
nop.m 999
nop.i 999 ;;
}
{ .mmi
(p0) ldfe sind_Sm = [r45],16 ;;
(p0) ldfe sind_Cm = [r45]
nop.i 999 ;;
}
{ .mfi
nop.m 999
(p0) fnma.s1 sind_r = sind_Nfloat, sind_Pi_by_16_lo, sind_r
nop.i 999 ;;
}
// get rsq
{ .mfi
nop.m 999
(p0) fma.s1 sind_rsq = sind_r, sind_r, f0
nop.i 999 ;;
}
// form P and Q series
{ .mfi
nop.m 999
(p0) fma.s1 sind_P_temp1 = sind_rsq, sind_P4, sind_P3
nop.i 999
}
{ .mfi
nop.m 999
(p0) fma.s1 sind_Q_temp1 = sind_rsq, sind_Q4, sind_Q3
nop.i 999 ;;
}
// get rcube and sm*rsq
{ .mfi
nop.m 999
(p0) fmpy.s1 sind_srsq = sind_Sm,sind_rsq
nop.i 999
}
{ .mfi
nop.m 999
(p0) fmpy.s1 sind_rcub = sind_r, sind_rsq
nop.i 999 ;;
}
{ .mfi
nop.m 999
(p0) fma.s1 sind_Q_temp2 = sind_rsq, sind_Q_temp1, sind_Q2
nop.i 999
}
{ .mfi
nop.m 999
(p0) fma.s1 sind_P_temp2 = sind_rsq, sind_P_temp1, sind_P2
nop.i 999 ;;
}
{ .mfi
nop.m 999
(p0) fma.s1 sind_Q = sind_rsq, sind_Q_temp2, sind_Q1
nop.i 999
}
{ .mfi
nop.m 999
(p0) fma.s1 sind_P = sind_rsq, sind_P_temp2, sind_P1
nop.i 999 ;;
}
// Get final P and Q
{ .mfi
nop.m 999
(p0) fma.s1 sind_Q = sind_srsq,sind_Q, sind_Sm
nop.i 999
}
{ .mfi
nop.m 999
(p0) fma.s1 sind_P = sind_rcub,sind_P, sind_r
nop.i 999 ;;
}
// Final calculation
{ .mfb
nop.m 999
(p0) fma.d f8 = sind_Cm, sind_P, sind_Q
(p0) br.ret.sptk b0 ;;
}
.endp cos#
.proc __libm_callout_1
__libm_callout_1:
SIND_DBX:
.prologue
{ .mfi
nop.m 0
nop.f 0
.save ar.pfs,GR_SAVE_PFS
mov GR_SAVE_PFS=ar.pfs
}
;;
{ .mfi
mov GR_SAVE_GP=gp
nop.f 0
.save b0, GR_SAVE_B0
mov GR_SAVE_B0=b0
}
.body
{ .mbb
nop.m 999
(p9) br.cond.spnt COSD_DBX
(p8) br.call.spnt.many b0=__libm_sin_double_dbx# ;;
}
;;
// if we come out of __libm_sin_double_dbx#
// we want to ensure that p9 is false.
{ .mii
nop.m 999
nop.i 999
(p0) cmp.eq.unc p8,p9 = r0,r0
;;
}
COSD_DBX:
{ .mib
nop.m 999
nop.i 999
(p9) br.call.spnt.many b0=__libm_cos_double_dbx# ;;
}
{ .mfi
(p0) mov gp = GR_SAVE_GP
nop.f 999
(p0) mov b0 = GR_SAVE_B0
}
;;
{ .mib
nop.m 999
(p0) mov ar.pfs = GR_SAVE_PFS
(p0) br.ret.sptk b0 ;;
}
.endp __libm_callout_1
// ====================================================================
// ====================================================================
// These functions calculate the sin and cos for inputs
// greater than 2^10
// __libm_sin_double_dbx# and __libm_cos_double_dbx#
//*********************************************************************
//*********************************************************************
//
// Function: Combined sin(x) and cos(x), where
//
// sin(x) = sine(x), for double precision x values
// cos(x) = cosine(x), for double precision x values
//
//*********************************************************************
//
// Accuracy: Within .7 ulps for 80-bit floating point values
// Very accurate for double precision values
//
//*********************************************************************
//
// Resources Used:
//
// Floating-Point Registers: f8 (Input and Return Value)
// f32-f99
//
// General Purpose Registers:
// r32-r43
// r44-r45 (Used to pass arguments to pi_by_2 reduce routine)
//
// Predicate Registers: p6-p13
//
//*********************************************************************
//
// IEEE Special Conditions:
//
// Denormal fault raised on denormal inputs
// Overflow exceptions do not occur
// Underflow exceptions raised when appropriate for sin
// (No specialized error handling for this routine)
// Inexact raised when appropriate by algorithm
//
// sin(SNaN) = QNaN
// sin(QNaN) = QNaN
// sin(inf) = QNaN
// sin(+/-0) = +/-0
// cos(inf) = QNaN
// cos(SNaN) = QNaN
// cos(QNaN) = QNaN
// cos(0) = 1
//
//*********************************************************************
//
// Mathematical Description
// ========================
//
// The computation of FSIN and FCOS is best handled in one piece of
// code. The main reason is that given any argument Arg, computation
// of trigonometric functions first calculate N and an approximation
// to alpha where
//
// Arg = N pi/2 + alpha, |alpha| <= pi/4.
//
// Since
//
// cos( Arg ) = sin( (N+1) pi/2 + alpha ),
//
// therefore, the code for computing sine will produce cosine as long
// as 1 is added to N immediately after the argument reduction
// process.
//
// Let M = N if sine
// N+1 if cosine.
//
// Now, given
//
// Arg = M pi/2 + alpha, |alpha| <= pi/4,
//
// let I = M mod 4, or I be the two lsb of M when M is represented
// as 2's complement. I = [i_0 i_1]. Then
//
// sin( Arg ) = (-1)^i_0 sin( alpha ) if i_1 = 0,
// = (-1)^i_0 cos( alpha ) if i_1 = 1.
//
// For example:
// if M = -1, I = 11
// sin ((-pi/2 + alpha) = (-1) cos (alpha)
// if M = 0, I = 00
// sin (alpha) = sin (alpha)
// if M = 1, I = 01
// sin (pi/2 + alpha) = cos (alpha)
// if M = 2, I = 10
// sin (pi + alpha) = (-1) sin (alpha)
// if M = 3, I = 11
// sin ((3/2)pi + alpha) = (-1) cos (alpha)
//
// The value of alpha is obtained by argument reduction and
// represented by two working precision numbers r and c where
//
// alpha = r + c accurately.
//
// The reduction method is described in a previous write up.
// The argument reduction scheme identifies 4 cases. For Cases 2
// and 4, because |alpha| is small, sin(r+c) and cos(r+c) can be
// computed very easily by 2 or 3 terms of the Taylor series
// expansion as follows:
//
// Case 2:
// -------
//
// sin(r + c) = r + c - r^3/6 accurately
// cos(r + c) = 1 - 2^(-67) accurately
//
// Case 4:
// -------
//
// sin(r + c) = r + c - r^3/6 + r^5/120 accurately
// cos(r + c) = 1 - r^2/2 + r^4/24 accurately
//
// The only cases left are Cases 1 and 3 of the argument reduction
// procedure. These two cases will be merged since after the
// argument is reduced in either cases, we have the reduced argument
// represented as r + c and that the magnitude |r + c| is not small
// enough to allow the usage of a very short approximation.
//
// The required calculation is either
//
// sin(r + c) = sin(r) + correction, or
// cos(r + c) = cos(r) + correction.
//
// Specifically,
//
// sin(r + c) = sin(r) + c sin'(r) + O(c^2)
// = sin(r) + c cos (r) + O(c^2)
// = sin(r) + c(1 - r^2/2) accurately.
// Similarly,
//
// cos(r + c) = cos(r) - c sin(r) + O(c^2)
// = cos(r) - c(r - r^3/6) accurately.
//
// We therefore concentrate on accurately calculating sin(r) and
// cos(r) for a working-precision number r, |r| <= pi/4 to within
// 0.1% or so.
//
// The greatest challenge of this task is that the second terms of
// the Taylor series
//
// r - r^3/3! + r^r/5! - ...
//
// and
//
// 1 - r^2/2! + r^4/4! - ...
//
// are not very small when |r| is close to pi/4 and the rounding
// errors will be a concern if simple polynomial accumulation is
// used. When |r| < 2^-3, however, the second terms will be small
// enough (6 bits or so of right shift) that a normal Horner
// recurrence suffices. Hence there are two cases that we consider
// in the accurate computation of sin(r) and cos(r), |r| <= pi/4.
//
// Case small_r: |r| < 2^(-3)
// --------------------------
//
// Since Arg = M pi/4 + r + c accurately, and M mod 4 is [i_0 i_1],
// we have
//
// sin(Arg) = (-1)^i_0 * sin(r + c) if i_1 = 0
// = (-1)^i_0 * cos(r + c) if i_1 = 1
//
// can be accurately approximated by
//
// sin(Arg) = (-1)^i_0 * [sin(r) + c] if i_1 = 0
// = (-1)^i_0 * [cos(r) - c*r] if i_1 = 1
//
// because |r| is small and thus the second terms in the correction
// are unneccessary.
//
// Finally, sin(r) and cos(r) are approximated by polynomials of
// moderate lengths.
//
// sin(r) = r + S_1 r^3 + S_2 r^5 + ... + S_5 r^11
// cos(r) = 1 + C_1 r^2 + C_2 r^4 + ... + C_5 r^10
//
// We can make use of predicates to selectively calculate
// sin(r) or cos(r) based on i_1.
//
// Case normal_r: 2^(-3) <= |r| <= pi/4
// ------------------------------------
//
// This case is more likely than the previous one if one considers
// r to be uniformly distributed in [-pi/4 pi/4]. Again,
//
// sin(Arg) = (-1)^i_0 * sin(r + c) if i_1 = 0
// = (-1)^i_0 * cos(r + c) if i_1 = 1.
//
// Because |r| is now larger, we need one extra term in the
// correction. sin(Arg) can be accurately approximated by
//
// sin(Arg) = (-1)^i_0 * [sin(r) + c(1-r^2/2)] if i_1 = 0
// = (-1)^i_0 * [cos(r) - c*r*(1 - r^2/6)] i_1 = 1.
//
// Finally, sin(r) and cos(r) are approximated by polynomials of
// moderate lengths.
//
// sin(r) = r + PP_1_hi r^3 + PP_1_lo r^3 +
// PP_2 r^5 + ... + PP_8 r^17
//
// cos(r) = 1 + QQ_1 r^2 + QQ_2 r^4 + ... + QQ_8 r^16
//
// where PP_1_hi is only about 16 bits long and QQ_1 is -1/2.
// The crux in accurate computation is to calculate
//
// r + PP_1_hi r^3 or 1 + QQ_1 r^2
//
// accurately as two pieces: U_hi and U_lo. The way to achieve this
// is to obtain r_hi as a 10 sig. bit number that approximates r to
// roughly 8 bits or so of accuracy. (One convenient way is
//
// r_hi := frcpa( frcpa( r ) ).)
//
// This way,
//
// r + PP_1_hi r^3 = r + PP_1_hi r_hi^3 +
// PP_1_hi (r^3 - r_hi^3)
// = [r + PP_1_hi r_hi^3] +
// [PP_1_hi (r - r_hi)
// (r^2 + r_hi r + r_hi^2) ]
// = U_hi + U_lo
//
// Since r_hi is only 10 bit long and PP_1_hi is only 16 bit long,
// PP_1_hi * r_hi^3 is only at most 46 bit long and thus computed
// exactly. Furthermore, r and PP_1_hi r_hi^3 are of opposite sign
// and that there is no more than 8 bit shift off between r and
// PP_1_hi * r_hi^3. Hence the sum, U_hi, is representable and thus
// calculated without any error. Finally, the fact that
//
// |U_lo| <= 2^(-8) |U_hi|
//
// says that U_hi + U_lo is approximating r + PP_1_hi r^3 to roughly
// 8 extra bits of accuracy.
//
// Similarly,
//
// 1 + QQ_1 r^2 = [1 + QQ_1 r_hi^2] +
// [QQ_1 (r - r_hi)(r + r_hi)]
// = U_hi + U_lo.
//
// Summarizing, we calculate r_hi = frcpa( frcpa( r ) ).
//
// If i_1 = 0, then
//
// U_hi := r + PP_1_hi * r_hi^3
// U_lo := PP_1_hi * (r - r_hi) * (r^2 + r*r_hi + r_hi^2)
// poly := PP_1_lo r^3 + PP_2 r^5 + ... + PP_8 r^17
// correction := c * ( 1 + C_1 r^2 )
//
// Else ...i_1 = 1
//
// U_hi := 1 + QQ_1 * r_hi * r_hi
// U_lo := QQ_1 * (r - r_hi) * (r + r_hi)
// poly := QQ_2 * r^4 + QQ_3 * r^6 + ... + QQ_8 r^16
// correction := -c * r * (1 + S_1 * r^2)
//
// End
//
// Finally,
//
// V := poly + ( U_lo + correction )
//
// / U_hi + V if i_0 = 0
// result := |
// \ (-U_hi) - V if i_0 = 1
//
// It is important that in the last step, negation of U_hi is
// performed prior to the subtraction which is to be performed in
// the user-set rounding mode.
//
//
// Algorithmic Description
// =======================
//
// The argument reduction algorithm is tightly integrated into FSIN
// and FCOS which share the same code. The following is complete and
// self-contained. The argument reduction description given
// previously is repeated below.
//
//
// Step 0. Initialization.
//
// If FSIN is invoked, set N_inc := 0; else if FCOS is invoked,
// set N_inc := 1.
//
// Step 1. Check for exceptional and special cases.
//
// * If Arg is +-0, +-inf, NaN, NaT, go to Step 10 for special
// handling.
// * If |Arg| < 2^24, go to Step 2 for reduction of moderate
// arguments. This is the most likely case.
// * If |Arg| < 2^63, go to Step 8 for pre-reduction of large
// arguments.
// * If |Arg| >= 2^63, go to Step 10 for special handling.
//
// Step 2. Reduction of moderate arguments.
//
// If |Arg| < pi/4 ...quick branch
// N_fix := N_inc (integer)
// r := Arg
// c := 0.0
// Branch to Step 4, Case_1_complete
// Else ...cf. argument reduction
// N := Arg * two_by_PI (fp)
// N_fix := fcvt.fx( N ) (int)
// N := fcvt.xf( N_fix )
// N_fix := N_fix + N_inc
// s := Arg - N * P_1 (first piece of pi/2)
// w := -N * P_2 (second piece of pi/2)
//
// If |s| >= 2^(-33)
// go to Step 3, Case_1_reduce
// Else
// go to Step 7, Case_2_reduce
// Endif
// Endif
//
// Step 3. Case_1_reduce.
//
// r := s + w
// c := (s - r) + w ...observe order
//
// Step 4. Case_1_complete
//
// ...At this point, the reduced argument alpha is
// ...accurately represented as r + c.
// If |r| < 2^(-3), go to Step 6, small_r.
//
// Step 5. Normal_r.
//
// Let [i_0 i_1] by the 2 lsb of N_fix.
// FR_rsq := r * r
// r_hi := frcpa( frcpa( r ) )
// r_lo := r - r_hi
//
// If i_1 = 0, then
// poly := r*FR_rsq*(PP_1_lo + FR_rsq*(PP_2 + ... FR_rsq*PP_8))
// U_hi := r + PP_1_hi*r_hi*r_hi*r_hi ...any order
// U_lo := PP_1_hi*r_lo*(r*r + r*r_hi + r_hi*r_hi)
// correction := c + c*C_1*FR_rsq ...any order
// Else
// poly := FR_rsq*FR_rsq*(QQ_2 + FR_rsq*(QQ_3 + ... + FR_rsq*QQ_8))
// U_hi := 1 + QQ_1 * r_hi * r_hi ...any order
// U_lo := QQ_1 * r_lo * (r + r_hi)
// correction := -c*(r + S_1*FR_rsq*r) ...any order
// Endif
//
// V := poly + (U_lo + correction) ...observe order
//
// result := (i_0 == 0? 1.0 : -1.0)
//
// Last instruction in user-set rounding mode
//
// result := (i_0 == 0? result*U_hi + V :
// result*U_hi - V)
//
// Return
//
// Step 6. Small_r.
//
// ...Use flush to zero mode without causing exception
// Let [i_0 i_1] be the two lsb of N_fix.
//
// FR_rsq := r * r
//
// If i_1 = 0 then
// z := FR_rsq*FR_rsq; z := FR_rsq*z *r
// poly_lo := S_3 + FR_rsq*(S_4 + FR_rsq*S_5)
// poly_hi := r*FR_rsq*(S_1 + FR_rsq*S_2)
// correction := c
// result := r
// Else
// z := FR_rsq*FR_rsq; z := FR_rsq*z
// poly_lo := C_3 + FR_rsq*(C_4 + FR_rsq*C_5)
// poly_hi := FR_rsq*(C_1 + FR_rsq*C_2)
// correction := -c*r
// result := 1
// Endif
//
// poly := poly_hi + (z * poly_lo + correction)
//
// If i_0 = 1, result := -result
//
// Last operation. Perform in user-set rounding mode
//
// result := (i_0 == 0? result + poly :
// result - poly )
// Return
//
// Step 7. Case_2_reduce.
//
// ...Refer to the write up for argument reduction for
// ...rationale. The reduction algorithm below is taken from
// ...argument reduction description and integrated this.
//
// w := N*P_3
// U_1 := N*P_2 + w ...FMA
// U_2 := (N*P_2 - U_1) + w ...2 FMA
// ...U_1 + U_2 is N*(P_2+P_3) accurately
//
// r := s - U_1
// c := ( (s - r) - U_1 ) - U_2
//
// ...The mathematical sum r + c approximates the reduced
// ...argument accurately. Note that although compared to
// ...Case 1, this case requires much more work to reduce
// ...the argument, the subsequent calculation needed for
// ...any of the trigonometric function is very little because
// ...|alpha| < 1.01*2^(-33) and thus two terms of the
// ...Taylor series expansion suffices.
//
// If i_1 = 0 then
// poly := c + S_1 * r * r * r ...any order
// result := r
// Else
// poly := -2^(-67)
// result := 1.0
// Endif
//
// If i_0 = 1, result := -result
//
// Last operation. Perform in user-set rounding mode
//
// result := (i_0 == 0? result + poly :
// result - poly )
//
// Return
//
//
// Step 8. Pre-reduction of large arguments.
//
// ...Again, the following reduction procedure was described
// ...in the separate write up for argument reduction, which
// ...is tightly integrated here.
// N_0 := Arg * Inv_P_0
// N_0_fix := fcvt.fx( N_0 )
// N_0 := fcvt.xf( N_0_fix)
// Arg' := Arg - N_0 * P_0
// w := N_0 * d_1
// N := Arg' * two_by_PI
// N_fix := fcvt.fx( N )
// N := fcvt.xf( N_fix )
// N_fix := N_fix + N_inc
//
// s := Arg' - N * P_1
// w := w - N * P_2
//
// If |s| >= 2^(-14)
// go to Step 3
// Else
// go to Step 9
// Endif
//
// Step 9. Case_4_reduce.
//
// ...first obtain N_0*d_1 and -N*P_2 accurately
// U_hi := N_0 * d_1 V_hi := -N*P_2
// U_lo := N_0 * d_1 - U_hi V_lo := -N*P_2 - U_hi ...FMAs
//
// ...compute the contribution from N_0*d_1 and -N*P_3
// w := -N*P_3
// w := w + N_0*d_2
// t := U_lo + V_lo + w ...any order
//
// ...at this point, the mathematical value
// ...s + U_hi + V_hi + t approximates the true reduced argument
// ...accurately. Just need to compute this accurately.
//
// ...Calculate U_hi + V_hi accurately:
// A := U_hi + V_hi
// if |U_hi| >= |V_hi| then
// a := (U_hi - A) + V_hi
// else
// a := (V_hi - A) + U_hi
// endif
// ...order in computing "a" must be observed. This branch is
// ...best implemented by predicates.
// ...A + a is U_hi + V_hi accurately. Moreover, "a" is
// ...much smaller than A: |a| <= (1/2)ulp(A).
//
// ...Just need to calculate s + A + a + t
// C_hi := s + A t := t + a
// C_lo := (s - C_hi) + A
// C_lo := C_lo + t
//
// ...Final steps for reduction
// r := C_hi + C_lo
// c := (C_hi - r) + C_lo
//
// ...At this point, we have r and c
// ...And all we need is a couple of terms of the corresponding
// ...Taylor series.
//
// If i_1 = 0
// poly := c + r*FR_rsq*(S_1 + FR_rsq*S_2)
// result := r
// Else
// poly := FR_rsq*(C_1 + FR_rsq*C_2)
// result := 1
// Endif
//
// If i_0 = 1, result := -result
//
// Last operation. Perform in user-set rounding mode
//
// result := (i_0 == 0? result + poly :
// result - poly )
// Return
//
// Large Arguments: For arguments above 2**63, a Payne-Hanek
// style argument reduction is used and pi_by_2 reduce is called.
//
.data
.align 64
FSINCOS_CONSTANTS:
data4 0x4B800000, 0xCB800000, 0x00000000,0x00000000 // two**24, -two**24
data4 0x4E44152A, 0xA2F9836E, 0x00003FFE,0x00000000 // Inv_pi_by_2
data4 0xCE81B9F1, 0xC84D32B0, 0x00004016,0x00000000 // P_0
data4 0x2168C235, 0xC90FDAA2, 0x00003FFF,0x00000000 // P_1
data4 0xFC8F8CBB, 0xECE675D1, 0x0000BFBD,0x00000000 // P_2
data4 0xACC19C60, 0xB7ED8FBB, 0x0000BF7C,0x00000000 // P_3
data4 0x5F000000, 0xDF000000, 0x00000000,0x00000000 // two_to_63, -two_to_63
data4 0x6EC6B45A, 0xA397E504, 0x00003FE7,0x00000000 // Inv_P_0
data4 0xDBD171A1, 0x8D848E89, 0x0000BFBF,0x00000000 // d_1
data4 0x18A66F8E, 0xD5394C36, 0x0000BF7C,0x00000000 // d_2
data4 0x2168C234, 0xC90FDAA2, 0x00003FFE,0x00000000 // pi_by_4
data4 0x2168C234, 0xC90FDAA2, 0x0000BFFE,0x00000000 // neg_pi_by_4
data4 0x3E000000, 0xBE000000, 0x00000000,0x00000000 // two**-3, -two**-3
data4 0x2F000000, 0xAF000000, 0x9E000000,0x00000000 // two**-33, -two**-33, -two**-67
data4 0xA21C0BC9, 0xCC8ABEBC, 0x00003FCE,0x00000000 // PP_8
data4 0x720221DA, 0xD7468A05, 0x0000BFD6,0x00000000 // PP_7
data4 0x640AD517, 0xB092382F, 0x00003FDE,0x00000000 // PP_6
data4 0xD1EB75A4, 0xD7322B47, 0x0000BFE5,0x00000000 // PP_5
data4 0xFFFFFFFE, 0xFFFFFFFF, 0x0000BFFD,0x00000000 // C_1
data4 0x00000000, 0xAAAA0000, 0x0000BFFC,0x00000000 // PP_1_hi
data4 0xBAF69EEA, 0xB8EF1D2A, 0x00003FEC,0x00000000 // PP_4
data4 0x0D03BB69, 0xD00D00D0, 0x0000BFF2,0x00000000 // PP_3
data4 0x88888962, 0x88888888, 0x00003FF8,0x00000000 // PP_2
data4 0xAAAB0000, 0xAAAAAAAA, 0x0000BFEC,0x00000000 // PP_1_lo
data4 0xC2B0FE52, 0xD56232EF, 0x00003FD2,0x00000000 // QQ_8
data4 0x2B48DCA6, 0xC9C99ABA, 0x0000BFDA,0x00000000 // QQ_7
data4 0x9C716658, 0x8F76C650, 0x00003FE2,0x00000000 // QQ_6
data4 0xFDA8D0FC, 0x93F27DBA, 0x0000BFE9,0x00000000 // QQ_5
data4 0xAAAAAAAA, 0xAAAAAAAA, 0x0000BFFC,0x00000000 // S_1
data4 0x00000000, 0x80000000, 0x0000BFFE,0x00000000 // QQ_1
data4 0x0C6E5041, 0xD00D00D0, 0x00003FEF,0x00000000 // QQ_4
data4 0x0B607F60, 0xB60B60B6, 0x0000BFF5,0x00000000 // QQ_3
data4 0xAAAAAA9B, 0xAAAAAAAA, 0x00003FFA,0x00000000 // QQ_2
data4 0xFFFFFFFE, 0xFFFFFFFF, 0x0000BFFD,0x00000000 // C_1
data4 0xAAAA719F, 0xAAAAAAAA, 0x00003FFA,0x00000000 // C_2
data4 0x0356F994, 0xB60B60B6, 0x0000BFF5,0x00000000 // C_3
data4 0xB2385EA9, 0xD00CFFD5, 0x00003FEF,0x00000000 // C_4
data4 0x292A14CD, 0x93E4BD18, 0x0000BFE9,0x00000000 // C_5
data4 0xAAAAAAAA, 0xAAAAAAAA, 0x0000BFFC,0x00000000 // S_1
data4 0x888868DB, 0x88888888, 0x00003FF8,0x00000000 // S_2
data4 0x055EFD4B, 0xD00D00D0, 0x0000BFF2,0x00000000 // S_3
data4 0x839730B9, 0xB8EF1C5D, 0x00003FEC,0x00000000 // S_4
data4 0xE5B3F492, 0xD71EA3A4, 0x0000BFE5,0x00000000 // S_5
data4 0x38800000, 0xB8800000, 0x00000000 // two**-14, -two**-14
FR_Input_X = f8
FR_Neg_Two_to_M3 = f32
FR_Two_to_63 = f32
FR_Two_to_24 = f33
FR_Pi_by_4 = f33
FR_Two_to_M14 = f34
FR_Two_to_M33 = f35
FR_Neg_Two_to_24 = f36
FR_Neg_Pi_by_4 = f36
FR_Neg_Two_to_M14 = f37
FR_Neg_Two_to_M33 = f38
FR_Neg_Two_to_M67 = f39
FR_Inv_pi_by_2 = f40
FR_N_float = f41
FR_N_fix = f42
FR_P_1 = f43
FR_P_2 = f44
FR_P_3 = f45
FR_s = f46
FR_w = f47
FR_c = f48
FR_r = f49
FR_Z = f50
FR_A = f51
FR_a = f52
FR_t = f53
FR_U_1 = f54
FR_U_2 = f55
FR_C_1 = f56
FR_C_2 = f57
FR_C_3 = f58
FR_C_4 = f59
FR_C_5 = f60
FR_S_1 = f61
FR_S_2 = f62
FR_S_3 = f63
FR_S_4 = f64
FR_S_5 = f65
FR_poly_hi = f66
FR_poly_lo = f67
FR_r_hi = f68
FR_r_lo = f69
FR_rsq = f70
FR_r_cubed = f71
FR_C_hi = f72
FR_N_0 = f73
FR_d_1 = f74
FR_V = f75
FR_V_hi = f75
FR_V_lo = f76
FR_U_hi = f77
FR_U_lo = f78
FR_U_hiabs = f79
FR_V_hiabs = f80
FR_PP_8 = f81
FR_QQ_8 = f81
FR_PP_7 = f82
FR_QQ_7 = f82
FR_PP_6 = f83
FR_QQ_6 = f83
FR_PP_5 = f84
FR_QQ_5 = f84
FR_PP_4 = f85
FR_QQ_4 = f85
FR_PP_3 = f86
FR_QQ_3 = f86
FR_PP_2 = f87
FR_QQ_2 = f87
FR_QQ_1 = f88
FR_N_0_fix = f89
FR_Inv_P_0 = f90
FR_corr = f91
FR_poly = f92
FR_d_2 = f93
FR_Two_to_M3 = f94
FR_Neg_Two_to_63 = f94
FR_P_0 = f95
FR_C_lo = f96
FR_PP_1 = f97
FR_PP_1_lo = f98
FR_ArgPrime = f99
GR_Table_Base = r32
GR_Table_Base1 = r33
GR_i_0 = r34
GR_i_1 = r35
GR_N_Inc = r36
GR_Sin_or_Cos = r37
GR_SAVE_B0 = r39
GR_SAVE_GP = r40
GR_SAVE_PFS = r41
.section .text
.proc __libm_sin_double_dbx#
.align 64
__libm_sin_double_dbx:
{ .mlx
alloc GR_Table_Base = ar.pfs,0,12,2,0
(p0) movl GR_Sin_or_Cos = 0x0 ;;
}
{ .mmi
nop.m 999
(p0) addl GR_Table_Base = @ltoff(FSINCOS_CONSTANTS#), gp
nop.i 999
}
;;
{ .mmi
ld8 GR_Table_Base = [GR_Table_Base]
nop.m 999
nop.i 999
}
;;
{ .mib
nop.m 999
nop.i 999
(p0) br.cond.sptk SINCOS_CONTINUE ;;
}
.endp __libm_sin_double_dbx#
.section .text
.proc __libm_cos_double_dbx#
__libm_cos_double_dbx:
{ .mlx
alloc GR_Table_Base= ar.pfs,0,12,2,0
(p0) movl GR_Sin_or_Cos = 0x1 ;;
}
{ .mmi
nop.m 999
(p0) addl GR_Table_Base = @ltoff(FSINCOS_CONSTANTS#), gp
nop.i 999
}
;;
{ .mmi
ld8 GR_Table_Base = [GR_Table_Base]
nop.m 999
nop.i 999
}
;;
//
// Load Table Address
//
SINCOS_CONTINUE:
{ .mmi
(p0) add GR_Table_Base1 = 96, GR_Table_Base
(p0) ldfs FR_Two_to_24 = [GR_Table_Base], 4
nop.i 999
}
;;
{ .mmi
nop.m 999
//
// Load 2**24, load 2**63.
//
(p0) ldfs FR_Neg_Two_to_24 = [GR_Table_Base], 12
(p0) mov r41 = ar.pfs ;;
}
{ .mfi
(p0) ldfs FR_Two_to_63 = [GR_Table_Base1], 4
//
// Check for unnormals - unsupported operands. We do not want
// to generate denormal exception
// Check for NatVals, QNaNs, SNaNs, +/-Infs
// Check for EM unsupporteds
// Check for Zero
//
(p0) fclass.m.unc p6, p8 = FR_Input_X, 0x1E3
(p0) mov r40 = gp ;;
}
{ .mfi
nop.m 999
(p0) fclass.nm.unc p8, p0 = FR_Input_X, 0x1FF
// GR_Sin_or_Cos denotes
(p0) mov r39 = b0
}
{ .mfb
(p0) ldfs FR_Neg_Two_to_63 = [GR_Table_Base1], 12
(p0) fclass.m.unc p10, p0 = FR_Input_X, 0x007
(p6) br.cond.spnt SINCOS_SPECIAL ;;
}
{ .mib
nop.m 999
nop.i 999
(p8) br.cond.spnt SINCOS_SPECIAL ;;
}
{ .mib
nop.m 999
nop.i 999
//
// Branch if +/- NaN, Inf.
// Load -2**24, load -2**63.
//
(p10) br.cond.spnt SINCOS_ZERO ;;
}
{ .mmb
(p0) ldfe FR_Inv_pi_by_2 = [GR_Table_Base], 16
(p0) ldfe FR_Inv_P_0 = [GR_Table_Base1], 16
nop.b 999 ;;
}
{ .mmb
nop.m 999
(p0) ldfe FR_d_1 = [GR_Table_Base1], 16
nop.b 999 ;;
}
//
// Raise possible denormal operand flag with useful fcmp
// Is x <= -2**63
// Load Inv_P_0 for pre-reduction
// Load Inv_pi_by_2
//
{ .mmb
(p0) ldfe FR_P_0 = [GR_Table_Base], 16
(p0) ldfe FR_d_2 = [GR_Table_Base1], 16
nop.b 999 ;;
}
//
// Load P_0
// Load d_1
// Is x >= 2**63
// Is x <= -2**24?
//
{ .mmi
(p0) ldfe FR_P_1 = [GR_Table_Base], 16 ;;
//
// Load P_1
// Load d_2
// Is x >= 2**24?
//
(p0) ldfe FR_P_2 = [GR_Table_Base], 16
nop.i 999 ;;
}
{ .mmf
nop.m 999
(p0) ldfe FR_P_3 = [GR_Table_Base], 16
(p0) fcmp.le.unc.s1 p7, p8 = FR_Input_X, FR_Neg_Two_to_24
}
{ .mfi
nop.m 999
//
// Branch if +/- zero.
// Decide about the paths to take:
// If -2**24 < FR_Input_X < 2**24 - CASE 1 OR 2
// OTHERWISE - CASE 3 OR 4
//
(p0) fcmp.le.unc.s0 p10, p11 = FR_Input_X, FR_Neg_Two_to_63
nop.i 999 ;;
}
{ .mfi
nop.m 999
(p8) fcmp.ge.s1 p7, p0 = FR_Input_X, FR_Two_to_24
nop.i 999
}
{ .mfi
(p0) ldfe FR_Pi_by_4 = [GR_Table_Base1], 16
(p11) fcmp.ge.s1 p10, p0 = FR_Input_X, FR_Two_to_63
nop.i 999 ;;
}
{ .mmi
(p0) ldfe FR_Neg_Pi_by_4 = [GR_Table_Base1], 16 ;;
(p0) ldfs FR_Two_to_M3 = [GR_Table_Base1], 4
nop.i 999 ;;
}
{ .mib
(p0) ldfs FR_Neg_Two_to_M3 = [GR_Table_Base1], 12
nop.i 999
//
// Load P_2
// Load P_3
// Load pi_by_4
// Load neg_pi_by_4
// Load 2**(-3)
// Load -2**(-3).
//
(p10) br.cond.spnt SINCOS_ARG_TOO_LARGE ;;
}
{ .mib
nop.m 999
nop.i 999
//
// Branch out if x >= 2**63. Use Payne-Hanek Reduction
//
(p7) br.cond.spnt SINCOS_LARGER_ARG ;;
}
{ .mfi
nop.m 999
//
// Branch if Arg <= -2**24 or Arg >= 2**24 and use pre-reduction.
//
(p0) fma.s1 FR_N_float = FR_Input_X, FR_Inv_pi_by_2, f0
nop.i 999 ;;
}
{ .mfi
nop.m 999
(p0) fcmp.lt.unc.s1 p6, p7 = FR_Input_X, FR_Pi_by_4
nop.i 999 ;;
}
{ .mfi
nop.m 999
//
// Select the case when |Arg| < pi/4
// Else Select the case when |Arg| >= pi/4
//
(p0) fcvt.fx.s1 FR_N_fix = FR_N_float
nop.i 999 ;;
}
{ .mfi
nop.m 999
//
// N = Arg * 2/pi
// Check if Arg < pi/4
//
(p6) fcmp.gt.s1 p6, p7 = FR_Input_X, FR_Neg_Pi_by_4
nop.i 999 ;;
}
//
// Case 2: Convert integer N_fix back to normalized floating-point value.
// Case 1: p8 is only affected when p6 is set
//
{ .mfi
(p7) ldfs FR_Two_to_M33 = [GR_Table_Base1], 4
//
// Grab the integer part of N and call it N_fix
//
(p6) fmerge.se FR_r = FR_Input_X, FR_Input_X
// If |x| < pi/4, r = x and c = 0
// lf |x| < pi/4, is x < 2**(-3).
// r = Arg
// c = 0
(p6) mov GR_N_Inc = GR_Sin_or_Cos ;;
}
{ .mmf
nop.m 999
(p7) ldfs FR_Neg_Two_to_M33 = [GR_Table_Base1], 4
(p6) fmerge.se FR_c = f0, f0
}
{ .mfi
nop.m 999
(p6) fcmp.lt.unc.s1 p8, p9 = FR_Input_X, FR_Two_to_M3
nop.i 999 ;;
}
{ .mfi
nop.m 999
//
// lf |x| < pi/4, is -2**(-3)< x < 2**(-3) - set p8.
// If |x| >= pi/4,
// Create the right N for |x| < pi/4 and otherwise
// Case 2: Place integer part of N in GP register
//
(p7) fcvt.xf FR_N_float = FR_N_fix
nop.i 999 ;;
}
{ .mmf
nop.m 999
(p7) getf.sig GR_N_Inc = FR_N_fix
(p8) fcmp.gt.s1 p8, p0 = FR_Input_X, FR_Neg_Two_to_M3 ;;
}
{ .mib
nop.m 999
nop.i 999
//
// Load 2**(-33), -2**(-33)
//
(p8) br.cond.spnt SINCOS_SMALL_R ;;
}
{ .mib
nop.m 999
nop.i 999
(p6) br.cond.sptk SINCOS_NORMAL_R ;;
}
//
// if |x| < pi/4, branch based on |x| < 2**(-3) or otherwise.
//
//
// In this branch, |x| >= pi/4.
//
{ .mfi
(p0) ldfs FR_Neg_Two_to_M67 = [GR_Table_Base1], 8
//
// Load -2**(-67)
//
(p0) fnma.s1 FR_s = FR_N_float, FR_P_1, FR_Input_X
//
// w = N * P_2
// s = -N * P_1 + Arg
//
(p0) add GR_N_Inc = GR_N_Inc, GR_Sin_or_Cos
}
{ .mfi
nop.m 999
(p0) fma.s1 FR_w = FR_N_float, FR_P_2, f0
nop.i 999 ;;
}
{ .mfi
nop.m 999
//
// Adjust N_fix by N_inc to determine whether sine or
// cosine is being calculated
//
(p0) fcmp.lt.unc.s1 p7, p6 = FR_s, FR_Two_to_M33
nop.i 999 ;;
}
{ .mfi
nop.m 999
(p7) fcmp.gt.s1 p7, p6 = FR_s, FR_Neg_Two_to_M33
nop.i 999 ;;
}
{ .mfi
nop.m 999
// Remember x >= pi/4.
// Is s <= -2**(-33) or s >= 2**(-33) (p6)
// or -2**(-33) < s < 2**(-33) (p7)
(p6) fms.s1 FR_r = FR_s, f1, FR_w
nop.i 999
}
{ .mfi
nop.m 999
(p7) fma.s1 FR_w = FR_N_float, FR_P_3, f0
nop.i 999 ;;
}
{ .mfi
nop.m 999
(p7) fma.s1 FR_U_1 = FR_N_float, FR_P_2, FR_w
nop.i 999
}
{ .mfi
nop.m 999
(p6) fms.s1 FR_c = FR_s, f1, FR_r
nop.i 999 ;;
}
{ .mfi
nop.m 999
//
// For big s: r = s - w: No futher reduction is necessary
// For small s: w = N * P_3 (change sign) More reduction
//
(p6) fcmp.lt.unc.s1 p8, p9 = FR_r, FR_Two_to_M3
nop.i 999 ;;
}
{ .mfi
nop.m 999
(p8) fcmp.gt.s1 p8, p9 = FR_r, FR_Neg_Two_to_M3
nop.i 999 ;;
}
{ .mfi
nop.m 999
(p7) fms.s1 FR_r = FR_s, f1, FR_U_1
nop.i 999
}
{ .mfb
nop.m 999
//
// For big s: Is |r| < 2**(-3)?
// For big s: c = S - r
// For small s: U_1 = N * P_2 + w
//
// If p8 is set, prepare to branch to Small_R.
// If p9 is set, prepare to branch to Normal_R.
// For big s, r is complete here.
//
(p6) fms.s1 FR_c = FR_c, f1, FR_w
//
// For big s: c = c + w (w has not been negated.)
// For small s: r = S - U_1
//
(p8) br.cond.spnt SINCOS_SMALL_R ;;
}
{ .mib
nop.m 999
nop.i 999
(p9) br.cond.sptk SINCOS_NORMAL_R ;;
}
{ .mfi
(p7) add GR_Table_Base1 = 224, GR_Table_Base1
//
// Branch to SINCOS_SMALL_R or SINCOS_NORMAL_R
//
(p7) fms.s1 FR_U_2 = FR_N_float, FR_P_2, FR_U_1
//
// c = S - U_1
// r = S_1 * r
//
//
(p7) extr.u GR_i_1 = GR_N_Inc, 0, 1
}
{ .mmi
nop.m 999 ;;
//
// Get [i_0,i_1] - two lsb of N_fix_gr.
// Do dummy fmpy so inexact is always set.
//
(p7) cmp.eq.unc p9, p10 = 0x0, GR_i_1
(p7) extr.u GR_i_0 = GR_N_Inc, 1, 1 ;;
}
//
// For small s: U_2 = N * P_2 - U_1
// S_1 stored constant - grab the one stored with the
// coefficients.
//
{ .mfi
(p7) ldfe FR_S_1 = [GR_Table_Base1], 16
//
// Check if i_1 and i_0 != 0
//
(p10) fma.s1 FR_poly = f0, f1, FR_Neg_Two_to_M67
(p7) cmp.eq.unc p11, p12 = 0x0, GR_i_0 ;;
}
{ .mfi
nop.m 999
(p7) fms.s1 FR_s = FR_s, f1, FR_r
nop.i 999
}
{ .mfi
nop.m 999
//
// S = S - r
// U_2 = U_2 + w
// load S_1
//
(p7) fma.s1 FR_rsq = FR_r, FR_r, f0
nop.i 999 ;;
}
{ .mfi
nop.m 999
(p7) fma.s1 FR_U_2 = FR_U_2, f1, FR_w
nop.i 999
}
{ .mfi
nop.m 999
(p7) fmerge.se FR_Input_X = FR_r, FR_r
nop.i 999 ;;
}
{ .mfi
nop.m 999
(p10) fma.s1 FR_Input_X = f0, f1, f1
nop.i 999 ;;
}
{ .mfi
nop.m 999
//
// FR_rsq = r * r
// Save r as the result.
//
(p7) fms.s1 FR_c = FR_s, f1, FR_U_1
nop.i 999 ;;
}
{ .mfi
nop.m 999
//
// if ( i_1 ==0) poly = c + S_1*r*r*r
// else Result = 1
//
(p12) fnma.s1 FR_Input_X = FR_Input_X, f1, f0
nop.i 999
}
{ .mfi
nop.m 999
(p7) fma.s1 FR_r = FR_S_1, FR_r, f0
nop.i 999 ;;
}
{ .mfi
nop.m 999
(p7) fma.d.s0 FR_S_1 = FR_S_1, FR_S_1, f0
nop.i 999 ;;
}
{ .mfi
nop.m 999
//
// If i_1 != 0, poly = 2**(-67)
//
(p7) fms.s1 FR_c = FR_c, f1, FR_U_2
nop.i 999 ;;
}
{ .mfi
nop.m 999
//
// c = c - U_2
//
(p9) fma.s1 FR_poly = FR_r, FR_rsq, FR_c
nop.i 999 ;;
}
{ .mfi
nop.m 999
//
// i_0 != 0, so Result = -Result
//
(p11) fma.d.s0 FR_Input_X = FR_Input_X, f1, FR_poly
nop.i 999 ;;
}
{ .mfb
nop.m 999
(p12) fms.d.s0 FR_Input_X = FR_Input_X, f1, FR_poly
//
// if (i_0 == 0), Result = Result + poly
// else Result = Result - poly
//
(p0) br.ret.sptk b0 ;;
}
SINCOS_LARGER_ARG:
{ .mfi
nop.m 999
(p0) fma.s1 FR_N_0 = FR_Input_X, FR_Inv_P_0, f0
nop.i 999
}
;;
// This path for argument > 2*24
// Adjust table_ptr1 to beginning of table.
//
{ .mmi
nop.m 999
(p0) addl GR_Table_Base = @ltoff(FSINCOS_CONSTANTS#), gp
nop.i 999
}
;;
{ .mmi
ld8 GR_Table_Base = [GR_Table_Base]
nop.m 999
nop.i 999
}
;;
//
// Point to 2*-14
// N_0 = Arg * Inv_P_0
//
{ .mmi
(p0) add GR_Table_Base = 688, GR_Table_Base ;;
(p0) ldfs FR_Two_to_M14 = [GR_Table_Base], 4
nop.i 999 ;;
}
{ .mfi
(p0) ldfs FR_Neg_Two_to_M14 = [GR_Table_Base], 0
nop.f 999
nop.i 999 ;;
}
{ .mfi
nop.m 999
//
// Load values 2**(-14) and -2**(-14)
//
(p0) fcvt.fx.s1 FR_N_0_fix = FR_N_0
nop.i 999 ;;
}
{ .mfi
nop.m 999
//
// N_0_fix = integer part of N_0
//
(p0) fcvt.xf FR_N_0 = FR_N_0_fix
nop.i 999 ;;
}
{ .mfi
nop.m 999
//
// Make N_0 the integer part
//
(p0) fnma.s1 FR_ArgPrime = FR_N_0, FR_P_0, FR_Input_X
nop.i 999
}
{ .mfi
nop.m 999
(p0) fma.s1 FR_w = FR_N_0, FR_d_1, f0
nop.i 999 ;;
}
{ .mfi
nop.m 999
//
// Arg' = -N_0 * P_0 + Arg
// w = N_0 * d_1
//
(p0) fma.s1 FR_N_float = FR_ArgPrime, FR_Inv_pi_by_2, f0
nop.i 999 ;;
}
{ .mfi
nop.m 999
//
// N = A' * 2/pi
//
(p0) fcvt.fx.s1 FR_N_fix = FR_N_float
nop.i 999 ;;
}
{ .mfi
nop.m 999
//
// N_fix is the integer part
//
(p0) fcvt.xf FR_N_float = FR_N_fix
nop.i 999 ;;
}
{ .mfi
(p0) getf.sig GR_N_Inc = FR_N_fix
nop.f 999
nop.i 999 ;;
}
{ .mii
nop.m 999
nop.i 999 ;;
(p0) add GR_N_Inc = GR_N_Inc, GR_Sin_or_Cos ;;
}
{ .mfi
nop.m 999
//
// N is the integer part of the reduced-reduced argument.
// Put the integer in a GP register
//
(p0) fnma.s1 FR_s = FR_N_float, FR_P_1, FR_ArgPrime
nop.i 999
}
{ .mfi
nop.m 999
(p0) fnma.s1 FR_w = FR_N_float, FR_P_2, FR_w
nop.i 999 ;;
}
{ .mfi
nop.m 999
//
// s = -N*P_1 + Arg'
// w = -N*P_2 + w
// N_fix_gr = N_fix_gr + N_inc
//
(p0) fcmp.lt.unc.s1 p9, p8 = FR_s, FR_Two_to_M14
nop.i 999 ;;
}
{ .mfi
nop.m 999
(p9) fcmp.gt.s1 p9, p8 = FR_s, FR_Neg_Two_to_M14
nop.i 999 ;;
}
{ .mfi
nop.m 999
//
// For |s| > 2**(-14) r = S + w (r complete)
// Else U_hi = N_0 * d_1
//
(p9) fma.s1 FR_V_hi = FR_N_float, FR_P_2, f0
nop.i 999
}
{ .mfi
nop.m 999
(p9) fma.s1 FR_U_hi = FR_N_0, FR_d_1, f0
nop.i 999 ;;
}
{ .mfi
nop.m 999
//
// Either S <= -2**(-14) or S >= 2**(-14)
// or -2**(-14) < s < 2**(-14)
//
(p8) fma.s1 FR_r = FR_s, f1, FR_w
nop.i 999
}
{ .mfi
nop.m 999
(p9) fma.s1 FR_w = FR_N_float, FR_P_3, f0
nop.i 999 ;;
}
{ .mfi
nop.m 999
//
// We need abs of both U_hi and V_hi - don't
// worry about switched sign of V_hi.
//
(p9) fms.s1 FR_A = FR_U_hi, f1, FR_V_hi
nop.i 999
}
{ .mfi
nop.m 999
//
// Big s: finish up c = (S - r) + w (c complete)
// Case 4: A = U_hi + V_hi
// Note: Worry about switched sign of V_hi, so subtract instead of add.
//
(p9) fnma.s1 FR_V_lo = FR_N_float, FR_P_2, FR_V_hi
nop.i 999 ;;
}
{ .mfi
nop.m 999
(p9) fms.s1 FR_U_lo = FR_N_0, FR_d_1, FR_U_hi
nop.i 999 ;;
}
{ .mfi
nop.m 999
(p9) fmerge.s FR_V_hiabs = f0, FR_V_hi
nop.i 999
}
{ .mfi
nop.m 999
// For big s: c = S - r
// For small s do more work: U_lo = N_0 * d_1 - U_hi
//
(p9) fmerge.s FR_U_hiabs = f0, FR_U_hi
nop.i 999 ;;
}
{ .mfi
nop.m 999
//
// For big s: Is |r| < 2**(-3)
// For big s: if p12 set, prepare to branch to Small_R.
// For big s: If p13 set, prepare to branch to Normal_R.
//
(p8) fms.s1 FR_c = FR_s, f1, FR_r
nop.i 999
}
{ .mfi
nop.m 999
//
// For small S: V_hi = N * P_2
// w = N * P_3
// Note the product does not include the (-) as in the writeup
// so (-) missing for V_hi and w.
//
(p8) fcmp.lt.unc.s1 p12, p13 = FR_r, FR_Two_to_M3
nop.i 999 ;;
}
{ .mfi
nop.m 999
(p12) fcmp.gt.s1 p12, p13 = FR_r, FR_Neg_Two_to_M3
nop.i 999 ;;
}
{ .mfi
nop.m 999
(p8) fma.s1 FR_c = FR_c, f1, FR_w
nop.i 999
}
{ .mfb
nop.m 999
(p9) fms.s1 FR_w = FR_N_0, FR_d_2, FR_w
(p12) br.cond.spnt SINCOS_SMALL_R ;;
}
{ .mib
nop.m 999
nop.i 999
(p13) br.cond.sptk SINCOS_NORMAL_R ;;
}
{ .mfi
nop.m 999
//
// Big s: Vector off when |r| < 2**(-3). Recall that p8 will be true.
// The remaining stuff is for Case 4.
// Small s: V_lo = N * P_2 + U_hi (U_hi is in place of V_hi in writeup)
// Note: the (-) is still missing for V_lo.
// Small s: w = w + N_0 * d_2
// Note: the (-) is now incorporated in w.
//
(p9) fcmp.ge.unc.s1 p10, p11 = FR_U_hiabs, FR_V_hiabs
(p0) extr.u GR_i_1 = GR_N_Inc, 0, 1 ;;
}
{ .mfi
nop.m 999
//
// C_hi = S + A
//
(p9) fma.s1 FR_t = FR_U_lo, f1, FR_V_lo
(p0) extr.u GR_i_0 = GR_N_Inc, 1, 1 ;;
}
{ .mfi
nop.m 999
//
// t = U_lo + V_lo
//
//
(p10) fms.s1 FR_a = FR_U_hi, f1, FR_A
nop.i 999 ;;
}
{ .mfi
nop.m 999
(p11) fma.s1 FR_a = FR_V_hi, f1, FR_A
nop.i 999
}
;;
{ .mmi
nop.m 999
(p0) addl GR_Table_Base = @ltoff(FSINCOS_CONSTANTS#), gp
nop.i 999
}
;;
{ .mmi
ld8 GR_Table_Base = [GR_Table_Base]
nop.m 999
nop.i 999
}
;;
{ .mfi
(p0) add GR_Table_Base = 528, GR_Table_Base
//
// Is U_hiabs >= V_hiabs?
//
(p9) fma.s1 FR_C_hi = FR_s, f1, FR_A
nop.i 999 ;;
}
{ .mmi
(p0) ldfe FR_C_1 = [GR_Table_Base], 16 ;;
(p0) ldfe FR_C_2 = [GR_Table_Base], 64
nop.i 999 ;;
}
{ .mmf
nop.m 999
//
// c = c + C_lo finished.
// Load C_2
//
(p0) ldfe FR_S_1 = [GR_Table_Base], 16
//
// C_lo = S - C_hi
//
(p0) fma.s1 FR_t = FR_t, f1, FR_w ;;
}
//
// r and c have been computed.
// Make sure ftz mode is set - should be automatic when using wre
// |r| < 2**(-3)
// Get [i_0,i_1] - two lsb of N_fix.
// Load S_1
//
{ .mfi
(p0) ldfe FR_S_2 = [GR_Table_Base], 64
//
// t = t + w
//
(p10) fms.s1 FR_a = FR_a, f1, FR_V_hi
(p0) cmp.eq.unc p9, p10 = 0x0, GR_i_0
}
{ .mfi
nop.m 999
//
// For larger u than v: a = U_hi - A
// Else a = V_hi - A (do an add to account for missing (-) on V_hi
//
(p0) fms.s1 FR_C_lo = FR_s, f1, FR_C_hi
nop.i 999 ;;
}
{ .mfi
nop.m 999
(p11) fms.s1 FR_a = FR_U_hi, f1, FR_a
(p0) cmp.eq.unc p11, p12 = 0x0, GR_i_1
}
{ .mfi
nop.m 999
//
// If u > v: a = (U_hi - A) + V_hi
// Else a = (V_hi - A) + U_hi
// In each case account for negative missing from V_hi.
//
(p0) fma.s1 FR_C_lo = FR_C_lo, f1, FR_A
nop.i 999 ;;
}
{ .mfi
nop.m 999
//
// C_lo = (S - C_hi) + A
//
(p0) fma.s1 FR_t = FR_t, f1, FR_a
nop.i 999 ;;
}
{ .mfi
nop.m 999
//
// t = t + a
//
(p0) fma.s1 FR_C_lo = FR_C_lo, f1, FR_t
nop.i 999 ;;
}
{ .mfi
nop.m 999
//
// C_lo = C_lo + t
// Adjust Table_Base to beginning of table
//
(p0) fma.s1 FR_r = FR_C_hi, f1, FR_C_lo
nop.i 999 ;;
}
{ .mfi
nop.m 999
//
// Load S_2
//
(p0) fma.s1 FR_rsq = FR_r, FR_r, f0
nop.i 999
}
{ .mfi
nop.m 999
//
// Table_Base points to C_1
// r = C_hi + C_lo
//
(p0) fms.s1 FR_c = FR_C_hi, f1, FR_r
nop.i 999 ;;
}
{ .mfi
nop.m 999
//
// if i_1 ==0: poly = S_2 * FR_rsq + S_1
// else poly = C_2 * FR_rsq + C_1
//
(p11) fma.s1 FR_Input_X = f0, f1, FR_r
nop.i 999 ;;
}
{ .mfi
nop.m 999
(p12) fma.s1 FR_Input_X = f0, f1, f1
nop.i 999 ;;
}
{ .mfi
nop.m 999
//
// Compute r_cube = FR_rsq * r
//
(p11) fma.s1 FR_poly = FR_rsq, FR_S_2, FR_S_1
nop.i 999 ;;
}
{ .mfi
nop.m 999
(p12) fma.s1 FR_poly = FR_rsq, FR_C_2, FR_C_1
nop.i 999
}
{ .mfi
nop.m 999
//
// Compute FR_rsq = r * r
// Is i_1 == 0 ?
//
(p0) fma.s1 FR_r_cubed = FR_rsq, FR_r, f0
nop.i 999 ;;
}
{ .mfi
nop.m 999
//
// c = C_hi - r
// Load C_1
//
(p0) fma.s1 FR_c = FR_c, f1, FR_C_lo
nop.i 999
}
{ .mfi
nop.m 999
//
// if i_1 ==0: poly = r_cube * poly + c
// else poly = FR_rsq * poly
//
(p10) fms.s1 FR_Input_X = f0, f1, FR_Input_X
nop.i 999 ;;
}
{ .mfi
nop.m 999
//
// if i_1 ==0: Result = r
// else Result = 1.0
//
(p11) fma.s1 FR_poly = FR_r_cubed, FR_poly, FR_c
nop.i 999 ;;
}
{ .mfi
nop.m 999
(p12) fma.s1 FR_poly = FR_rsq, FR_poly, f0
nop.i 999 ;;
}
{ .mfi
nop.m 999
//
// if i_0 !=0: Result = -Result
//
(p9) fma.d.s0 FR_Input_X = FR_Input_X, f1, FR_poly
nop.i 999 ;;
}
{ .mfb
nop.m 999
(p10) fms.d.s0 FR_Input_X = FR_Input_X, f1, FR_poly
//
// if i_0 == 0: Result = Result + poly
// else Result = Result - poly
//
(p0) br.ret.sptk b0 ;;
}
SINCOS_SMALL_R:
{ .mii
nop.m 999
(p0) extr.u GR_i_1 = GR_N_Inc, 0, 1 ;;
//
//
// Compare both i_1 and i_0 with 0.
// if i_1 == 0, set p9.
// if i_0 == 0, set p11.
//
(p0) cmp.eq.unc p9, p10 = 0x0, GR_i_1 ;;
}
{ .mfi
nop.m 999
(p0) fma.s1 FR_rsq = FR_r, FR_r, f0
(p0) extr.u GR_i_0 = GR_N_Inc, 1, 1 ;;
}
{ .mfi
nop.m 999
//
// Z = Z * FR_rsq
//
(p10) fnma.s1 FR_c = FR_c, FR_r, f0
(p0) cmp.eq.unc p11, p12 = 0x0, GR_i_0
}
;;
// ******************************************************************
// ******************************************************************
// ******************************************************************
// r and c have been computed.
// We know whether this is the sine or cosine routine.
// Make sure ftz mode is set - should be automatic when using wre
// |r| < 2**(-3)
//
// Set table_ptr1 to beginning of constant table.
// Get [i_0,i_1] - two lsb of N_fix_gr.
//
{ .mmi
nop.m 999
(p0) addl GR_Table_Base = @ltoff(FSINCOS_CONSTANTS#), gp
nop.i 999
}
;;
{ .mmi
ld8 GR_Table_Base = [GR_Table_Base]
nop.m 999
nop.i 999
}
;;
//
// Set table_ptr1 to point to S_5.
// Set table_ptr1 to point to C_5.
// Compute FR_rsq = r * r
//
{ .mfi
(p9) add GR_Table_Base = 672, GR_Table_Base
(p10) fmerge.s FR_r = f1, f1
(p10) add GR_Table_Base = 592, GR_Table_Base ;;
}
//
// Set table_ptr1 to point to S_5.
// Set table_ptr1 to point to C_5.
//
{ .mmi
(p9) ldfe FR_S_5 = [GR_Table_Base], -16 ;;
//
// if (i_1 == 0) load S_5
// if (i_1 != 0) load C_5
//
(p9) ldfe FR_S_4 = [GR_Table_Base], -16
nop.i 999 ;;
}
{ .mmf
(p10) ldfe FR_C_5 = [GR_Table_Base], -16
//
// Z = FR_rsq * FR_rsq
//
(p9) ldfe FR_S_3 = [GR_Table_Base], -16
//
// Compute FR_rsq = r * r
// if (i_1 == 0) load S_4
// if (i_1 != 0) load C_4
//
(p0) fma.s1 FR_Z = FR_rsq, FR_rsq, f0 ;;
}
//
// if (i_1 == 0) load S_3
// if (i_1 != 0) load C_3
//
{ .mmi
(p9) ldfe FR_S_2 = [GR_Table_Base], -16 ;;
//
// if (i_1 == 0) load S_2
// if (i_1 != 0) load C_2
//
(p9) ldfe FR_S_1 = [GR_Table_Base], -16
nop.i 999
}
{ .mmi
(p10) ldfe FR_C_4 = [GR_Table_Base], -16 ;;
(p10) ldfe FR_C_3 = [GR_Table_Base], -16
nop.i 999 ;;
}
{ .mmi
(p10) ldfe FR_C_2 = [GR_Table_Base], -16 ;;
(p10) ldfe FR_C_1 = [GR_Table_Base], -16
nop.i 999
}
{ .mfi
nop.m 999
//
// if (i_1 != 0):
// poly_lo = FR_rsq * C_5 + C_4
// poly_hi = FR_rsq * C_2 + C_1
//
(p9) fma.s1 FR_Z = FR_Z, FR_r, f0
nop.i 999 ;;
}
{ .mfi
nop.m 999
//
// if (i_1 == 0) load S_1
// if (i_1 != 0) load C_1
//
(p9) fma.s1 FR_poly_lo = FR_rsq, FR_S_5, FR_S_4
nop.i 999
}
{ .mfi
nop.m 999
//
// c = -c * r
// dummy fmpy's to flag inexact.
//
(p9) fma.d.s0 FR_S_4 = FR_S_4, FR_S_4, f0
nop.i 999 ;;
}
{ .mfi
nop.m 999
//
// poly_lo = FR_rsq * poly_lo + C_3
// poly_hi = FR_rsq * poly_hi
//
(p0) fma.s1 FR_Z = FR_Z, FR_rsq, f0
nop.i 999 ;;
}
{ .mfi
nop.m 999
(p9) fma.s1 FR_poly_hi = FR_rsq, FR_S_2, FR_S_1
nop.i 999
}
{ .mfi
nop.m 999
//
// if (i_1 == 0):
// poly_lo = FR_rsq * S_5 + S_4
// poly_hi = FR_rsq * S_2 + S_1
//
(p10) fma.s1 FR_poly_lo = FR_rsq, FR_C_5, FR_C_4
nop.i 999 ;;
}
{ .mfi
nop.m 999
//
// if (i_1 == 0):
// Z = Z * r for only one of the small r cases - not there
// in original implementation notes.
//
(p9) fma.s1 FR_poly_lo = FR_rsq, FR_poly_lo, FR_S_3
nop.i 999 ;;
}
{ .mfi
nop.m 999
(p10) fma.s1 FR_poly_hi = FR_rsq, FR_C_2, FR_C_1
nop.i 999
}
{ .mfi
nop.m 999
(p10) fma.d.s0 FR_C_1 = FR_C_1, FR_C_1, f0
nop.i 999 ;;
}
{ .mfi
nop.m 999
(p9) fma.s1 FR_poly_hi = FR_poly_hi, FR_rsq, f0
nop.i 999
}
{ .mfi
nop.m 999
//
// poly_lo = FR_rsq * poly_lo + S_3
// poly_hi = FR_rsq * poly_hi
//
(p10) fma.s1 FR_poly_lo = FR_rsq, FR_poly_lo, FR_C_3
nop.i 999 ;;
}
{ .mfi
nop.m 999
(p10) fma.s1 FR_poly_hi = FR_poly_hi, FR_rsq, f0
nop.i 999 ;;
}
{ .mfi
nop.m 999
//
// if (i_1 == 0): dummy fmpy's to flag inexact
// r = 1
//
(p9) fma.s1 FR_poly_hi = FR_r, FR_poly_hi, f0
nop.i 999
}
{ .mfi
nop.m 999
//
// poly_hi = r * poly_hi
//
(p0) fma.s1 FR_poly = FR_Z, FR_poly_lo, FR_c
nop.i 999 ;;
}
{ .mfi
nop.m 999
(p12) fms.s1 FR_r = f0, f1, FR_r
nop.i 999 ;;
}
{ .mfi
nop.m 999
//
// poly_hi = Z * poly_lo + c
// if i_0 == 1: r = -r
//
(p0) fma.s1 FR_poly = FR_poly, f1, FR_poly_hi
nop.i 999 ;;
}
{ .mfi
nop.m 999
(p12) fms.d.s0 FR_Input_X = FR_r, f1, FR_poly
nop.i 999
}
{ .mfb
nop.m 999
//
// poly = poly + poly_hi
//
(p11) fma.d.s0 FR_Input_X = FR_r, f1, FR_poly
//
// if (i_0 == 0) Result = r + poly
// if (i_0 != 0) Result = r - poly
//
(p0) br.ret.sptk b0 ;;
}
SINCOS_NORMAL_R:
{ .mii
nop.m 999
(p0) extr.u GR_i_1 = GR_N_Inc, 0, 1 ;;
//
// Set table_ptr1 and table_ptr2 to base address of
// constant table.
(p0) cmp.eq.unc p9, p10 = 0x0, GR_i_1 ;;
}
{ .mfi
nop.m 999
(p0) fma.s1 FR_rsq = FR_r, FR_r, f0
(p0) extr.u GR_i_0 = GR_N_Inc, 1, 1 ;;
}
{ .mfi
nop.m 999
(p0) frcpa.s1 FR_r_hi, p6 = f1, FR_r
(p0) cmp.eq.unc p11, p12 = 0x0, GR_i_0
}
;;
// ******************************************************************
// ******************************************************************
// ******************************************************************
//
// r and c have been computed.
// We known whether this is the sine or cosine routine.
// Make sure ftz mode is set - should be automatic when using wre
// Get [i_0,i_1] - two lsb of N_fix_gr alone.
//
{ .mmi
nop.m 999
(p0) addl GR_Table_Base = @ltoff(FSINCOS_CONSTANTS#), gp
nop.i 999
}
;;
{ .mmi
ld8 GR_Table_Base = [GR_Table_Base]
nop.m 999
nop.i 999
}
;;
{ .mfi
(p10) add GR_Table_Base = 384, GR_Table_Base
(p12) fms.s1 FR_Input_X = f0, f1, f1
(p9) add GR_Table_Base = 224, GR_Table_Base ;;
}
{ .mmf
nop.m 999
(p10) ldfe FR_QQ_8 = [GR_Table_Base], 16
//
// if (i_1==0) poly = poly * FR_rsq + PP_1_lo
// else poly = FR_rsq * poly
//
(p11) fma.s1 FR_Input_X = f0, f1, f1 ;;
}
{ .mmf
(p10) ldfe FR_QQ_7 = [GR_Table_Base], 16
//
// Adjust table pointers based on i_0
// Compute rsq = r * r
//
(p9) ldfe FR_PP_8 = [GR_Table_Base], 16
(p0) fma.s1 FR_r_cubed = FR_r, FR_rsq, f0 ;;
}
{ .mmf
(p9) ldfe FR_PP_7 = [GR_Table_Base], 16
(p10) ldfe FR_QQ_6 = [GR_Table_Base], 16
//
// Load PP_8 and QQ_8; PP_7 and QQ_7
//
(p0) frcpa.s1 FR_r_hi, p6 = f1, FR_r_hi ;;
}
//
// if (i_1==0) poly = PP_7 + FR_rsq * PP_8.
// else poly = QQ_7 + FR_rsq * QQ_8.
//
{ .mmb
(p9) ldfe FR_PP_6 = [GR_Table_Base], 16
(p10) ldfe FR_QQ_5 = [GR_Table_Base], 16
nop.b 999 ;;
}
{ .mmb
(p9) ldfe FR_PP_5 = [GR_Table_Base], 16
(p10) ldfe FR_S_1 = [GR_Table_Base], 16
nop.b 999 ;;
}
{ .mmb
(p10) ldfe FR_QQ_1 = [GR_Table_Base], 16
(p9) ldfe FR_C_1 = [GR_Table_Base], 16
nop.b 999 ;;
}
{ .mmi
(p10) ldfe FR_QQ_4 = [GR_Table_Base], 16 ;;
(p9) ldfe FR_PP_1 = [GR_Table_Base], 16
nop.i 999 ;;
}
{ .mmf
(p10) ldfe FR_QQ_3 = [GR_Table_Base], 16
//
// if (i_1=0) corr = corr + c*c
// else corr = corr * c
//
(p9) ldfe FR_PP_4 = [GR_Table_Base], 16
(p10) fma.s1 FR_poly = FR_rsq, FR_QQ_8, FR_QQ_7 ;;
}
//
// if (i_1=0) poly = rsq * poly + PP_5
// else poly = rsq * poly + QQ_5
// Load PP_4 or QQ_4
//
{ .mmf
(p9) ldfe FR_PP_3 = [GR_Table_Base], 16
(p10) ldfe FR_QQ_2 = [GR_Table_Base], 16
//
// r_hi = frcpa(frcpa(r)).
// r_cube = r * FR_rsq.
//
(p9) fma.s1 FR_poly = FR_rsq, FR_PP_8, FR_PP_7 ;;
}
//
// Do dummy multiplies so inexact is always set.
//
{ .mfi
(p9) ldfe FR_PP_2 = [GR_Table_Base], 16
//
// r_lo = r - r_hi
//
(p9) fma.s1 FR_U_lo = FR_r_hi, FR_r_hi, f0
nop.i 999 ;;
}
{ .mmf
nop.m 999
(p9) ldfe FR_PP_1_lo = [GR_Table_Base], 16
(p10) fma.s1 FR_corr = FR_S_1, FR_r_cubed, FR_r
}
{ .mfi
nop.m 999
(p10) fma.s1 FR_poly = FR_rsq, FR_poly, FR_QQ_6
nop.i 999 ;;
}
{ .mfi
nop.m 999
//
// if (i_1=0) U_lo = r_hi * r_hi
// else U_lo = r_hi + r
//
(p9) fma.s1 FR_corr = FR_C_1, FR_rsq, f0
nop.i 999 ;;
}
{ .mfi
nop.m 999
//
// if (i_1=0) corr = C_1 * rsq
// else corr = S_1 * r_cubed + r
//
(p9) fma.s1 FR_poly = FR_rsq, FR_poly, FR_PP_6
nop.i 999
}
{ .mfi
nop.m 999
(p10) fma.s1 FR_U_lo = FR_r_hi, f1, FR_r
nop.i 999 ;;
}
{ .mfi
nop.m 999
//
// if (i_1=0) U_hi = r_hi + U_hi
// else U_hi = QQ_1 * U_hi + 1
//
(p9) fma.s1 FR_U_lo = FR_r, FR_r_hi, FR_U_lo
nop.i 999
}
{ .mfi
nop.m 999
//
// U_hi = r_hi * r_hi
//
(p0) fms.s1 FR_r_lo = FR_r, f1, FR_r_hi
nop.i 999 ;;
}
{ .mfi
nop.m 999
//
// Load PP_1, PP_6, PP_5, and C_1
// Load QQ_1, QQ_6, QQ_5, and S_1
//
(p0) fma.s1 FR_U_hi = FR_r_hi, FR_r_hi, f0
nop.i 999 ;;
}
{ .mfi
nop.m 999
(p10) fma.s1 FR_poly = FR_rsq, FR_poly, FR_QQ_5
nop.i 999
}
{ .mfi
nop.m 999
(p10) fnma.s1 FR_corr = FR_corr, FR_c, f0
nop.i 999 ;;
}
{ .mfi
nop.m 999
//
// if (i_1=0) U_lo = r * r_hi + U_lo
// else U_lo = r_lo * U_lo
//
(p9) fma.s1 FR_corr = FR_corr, FR_c, FR_c
nop.i 999 ;;
}
{ .mfi
nop.m 999
(p9) fma.s1 FR_poly = FR_rsq, FR_poly, FR_PP_5
nop.i 999
}
{ .mfi
nop.m 999
//
// if (i_1 =0) U_hi = r + U_hi
// if (i_1 =0) U_lo = r_lo * U_lo
//
//
(p9) fma.d.s0 FR_PP_5 = FR_PP_5, FR_PP_4, f0
nop.i 999 ;;
}
{ .mfi
nop.m 999
(p9) fma.s1 FR_U_lo = FR_r, FR_r, FR_U_lo
nop.i 999
}
{ .mfi
nop.m 999
(p10) fma.s1 FR_U_lo = FR_r_lo, FR_U_lo, f0
nop.i 999 ;;
}
{ .mfi
nop.m 999
//
// if (i_1=0) poly = poly * rsq + PP_6
// else poly = poly * rsq + QQ_6
//
(p9) fma.s1 FR_U_hi = FR_r_hi, FR_U_hi, f0
nop.i 999 ;;
}
{ .mfi
nop.m 999
(p10) fma.s1 FR_poly = FR_rsq, FR_poly, FR_QQ_4
nop.i 999
}
{ .mfi
nop.m 999
(p10) fma.s1 FR_U_hi = FR_QQ_1, FR_U_hi, f1
nop.i 999 ;;
}
{ .mfi
nop.m 999
(p10) fma.d.s0 FR_QQ_5 = FR_QQ_5, FR_QQ_5, f0
nop.i 999 ;;
}
{ .mfi
nop.m 999
//
// if (i_1!=0) U_hi = PP_1 * U_hi
// if (i_1!=0) U_lo = r * r + U_lo
// Load PP_3 or QQ_3
//
(p9) fma.s1 FR_poly = FR_rsq, FR_poly, FR_PP_4
nop.i 999 ;;
}
{ .mfi
nop.m 999
(p9) fma.s1 FR_U_lo = FR_r_lo, FR_U_lo, f0
nop.i 999
}
{ .mfi
nop.m 999
(p10) fma.s1 FR_U_lo = FR_QQ_1,FR_U_lo, f0
nop.i 999 ;;
}
{ .mfi
nop.m 999
(p9) fma.s1 FR_U_hi = FR_PP_1, FR_U_hi, f0
nop.i 999 ;;
}
{ .mfi
nop.m 999
(p10) fma.s1 FR_poly = FR_rsq, FR_poly, FR_QQ_3
nop.i 999 ;;
}
{ .mfi
nop.m 999
//
// Load PP_2, QQ_2
//
(p9) fma.s1 FR_poly = FR_rsq, FR_poly, FR_PP_3
nop.i 999 ;;
}
{ .mfi
nop.m 999
//
// if (i_1==0) poly = FR_rsq * poly + PP_3
// else poly = FR_rsq * poly + QQ_3
// Load PP_1_lo
//
(p9) fma.s1 FR_U_lo = FR_PP_1, FR_U_lo, f0
nop.i 999 ;;
}
{ .mfi
nop.m 999
//
// if (i_1 =0) poly = poly * rsq + pp_r4
// else poly = poly * rsq + qq_r4
//
(p9) fma.s1 FR_U_hi = FR_r, f1, FR_U_hi
nop.i 999 ;;
}
{ .mfi
nop.m 999
(p10) fma.s1 FR_poly = FR_rsq, FR_poly, FR_QQ_2
nop.i 999 ;;
}
{ .mfi
nop.m 999
//
// if (i_1==0) U_lo = PP_1_hi * U_lo
// else U_lo = QQ_1 * U_lo
//
(p9) fma.s1 FR_poly = FR_rsq, FR_poly, FR_PP_2
nop.i 999 ;;
}
{ .mfi
nop.m 999
//
// if (i_0==0) Result = 1
// else Result = -1
//
(p0) fma.s1 FR_V = FR_U_lo, f1, FR_corr
nop.i 999 ;;
}
{ .mfi
nop.m 999
(p10) fma.s1 FR_poly = FR_rsq, FR_poly, f0
nop.i 999 ;;
}
{ .mfi
nop.m 999
//
// if (i_1==0) poly = FR_rsq * poly + PP_2
// else poly = FR_rsq * poly + QQ_2
//
(p9) fma.s1 FR_poly = FR_rsq, FR_poly, FR_PP_1_lo
nop.i 999 ;;
}
{ .mfi
nop.m 999
(p10) fma.s1 FR_poly = FR_rsq, FR_poly, f0
nop.i 999 ;;
}
{ .mfi
nop.m 999
//
// V = U_lo + corr
//
(p9) fma.s1 FR_poly = FR_r_cubed, FR_poly, f0
nop.i 999 ;;
}
{ .mfi
nop.m 999
//
// if (i_1==0) poly = r_cube * poly
// else poly = FR_rsq * poly
//
(p0) fma.s1 FR_V = FR_poly, f1, FR_V
nop.i 999 ;;
}
{ .mfi
nop.m 999
(p12) fms.d.s0 FR_Input_X = FR_Input_X, FR_U_hi, FR_V
nop.i 999
}
{ .mfb
nop.m 999
//
// V = V + poly
//
(p11) fma.d.s0 FR_Input_X = FR_Input_X, FR_U_hi, FR_V
//
// if (i_0==0) Result = Result * U_hi + V
// else Result = Result * U_hi - V
//
(p0) br.ret.sptk b0 ;;
}
//
// If cosine, FR_Input_X = 1
// If sine, FR_Input_X = +/-Zero (Input FR_Input_X)
// Results are exact, no exceptions
//
SINCOS_ZERO:
{ .mmb
(p0) cmp.eq.unc p6, p7 = 0x1, GR_Sin_or_Cos
nop.m 999
nop.b 999 ;;
}
{ .mfi
nop.m 999
(p7) fmerge.s FR_Input_X = FR_Input_X, FR_Input_X
nop.i 999
}
{ .mfb
nop.m 999
(p6) fmerge.s FR_Input_X = f1, f1
(p0) br.ret.sptk b0 ;;
}
SINCOS_SPECIAL:
//
// Path for Arg = +/- QNaN, SNaN, Inf
// Invalid can be raised. SNaNs
// become QNaNs
//
{ .mfb
nop.m 999
(p0) fmpy.d.s0 FR_Input_X = FR_Input_X, f0
(p0) br.ret.sptk b0 ;;
}
.endp __libm_cos_double_dbx#
//
// Call int pi_by_2_reduce(double* x, double *y)
// for |arguments| >= 2**63
// Address to save r and c as double
//
//
// psp sp+64
// sp+48 -> f0 c
// r45 sp+32 -> f0 r
// r44 -> sp+16 -> InputX
// sp sp -> scratch provided to callee
.proc __libm_callout_2
__libm_callout_2:
SINCOS_ARG_TOO_LARGE:
.prologue
{ .mfi
add r45=-32,sp // Parameter: r address
nop.f 0
.save ar.pfs,GR_SAVE_PFS
mov GR_SAVE_PFS=ar.pfs // Save ar.pfs
}
{ .mfi
.fframe 64
add sp=-64,sp // Create new stack
nop.f 0
mov GR_SAVE_GP=gp // Save gp
};;
{ .mmi
stfe [r45] = f0,16 // Clear Parameter r on stack
add r44 = 16,sp // Parameter x address
.save b0, GR_SAVE_B0
mov GR_SAVE_B0=b0 // Save b0
};;
.body
{ .mib
stfe [r45] = f0,-16 // Clear Parameter c on stack
nop.i 0
nop.b 0
}
{ .mib
stfe [r44] = FR_Input_X // Store Parameter x on stack
nop.i 0
(p0) br.call.sptk b0=__libm_pi_by_2_reduce# ;;
};;
{ .mii
(p0) ldfe FR_Input_X =[r44],16
//
// Get r and c off stack
//
(p0) adds GR_Table_Base1 = -16, GR_Table_Base1
//
// Get r and c off stack
//
(p0) add GR_N_Inc = GR_Sin_or_Cos,r8 ;;
}
{ .mmb
(p0) ldfe FR_r =[r45],16
//
// Get X off the stack
// Readjust Table ptr
//
(p0) ldfs FR_Two_to_M3 = [GR_Table_Base1],4
nop.b 999 ;;
}
{ .mmb
(p0) ldfs FR_Neg_Two_to_M3 = [GR_Table_Base1],0
(p0) ldfe FR_c =[r45]
nop.b 999 ;;
}
{ .mfi
.restore
add sp = 64,sp // Restore stack pointer
(p0) fcmp.lt.unc.s1 p6, p0 = FR_r, FR_Two_to_M3
mov b0 = GR_SAVE_B0 // Restore return address
};;
{ .mib
mov gp = GR_SAVE_GP // Restore gp
mov ar.pfs = GR_SAVE_PFS // Restore ar.pfs
nop.b 0
};;
{ .mfi
nop.m 999
(p6) fcmp.gt.unc.s1 p6, p0 = FR_r, FR_Neg_Two_to_M3
nop.i 999 ;;
}
{ .mib
nop.m 999
nop.i 999
(p6) br.cond.spnt SINCOS_SMALL_R ;;
}
{ .mib
nop.m 999
nop.i 999
(p0) br.cond.sptk SINCOS_NORMAL_R ;;
}
.endp __libm_callout_2
.type __libm_pi_by_2_reduce#,@function
.global __libm_pi_by_2_reduce#
.type __libm_sin_double_dbx#,@function
.global __libm_sin_double_dbx#
.type __libm_cos_double_dbx#,@function
.global __libm_cos_double_dbx#