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glibc/sysdeps/ia64/fpu/s_cosl.S
Dmitry V. Levin a1b02ae763 Fix a few typos in comments 
Apply the following spelling fixes:
$ git grep -F -l 'relevent' |
  xargs sed -i 's/relevent/relevant/g'
$ git grep -F -l 'checked fot' |
  xargs sed -i 's/checked fot/checked for/g'
$ git grep -F -l "could't" |
  xargs sed -i "s/could't/couldn't/g"
$ git grep -F -l 'wheter' | grep -Fv ChangeLog.old |
  xargs sed -i 's/wheter/whether/g'
$ git grep -F -l 'neccessary' | grep -Fv ChangeLog.old |
  xargs sed -i 's/neccessary/necessary/g'
$ git grep -F -l 'ouput' |
  xargs sed -i 's/ouput/output/g'
$ git grep -F -w -l 'iput' |
  xargs sed -i 's/iput/input/g'

This is inspired by a gnulib bug report at
https://lists.gnu.org/archive/html/bug-gnulib/2019-01/msg00081.html

* argp/argp-help.c: Fix typo in comment.
* misc/sys/cdefs.h: Likewise.
* posix/regexec.c (sift_states_iter_mb): Likewise.
* socket/sockatmark.c: Likewise.
* socket/sys/socket.h: Likewise.
* sysdeps/ia64/fpu/libm_sincos_large.S: Likewise.
* sysdeps/ia64/fpu/libm_sincosl.S: Likewise.
* sysdeps/ia64/fpu/s_cosl.S: Likewise.
* sysdeps/ieee754/dbl-64/k_rem_pio2.c: Likewise.
* sysdeps/unix/sockatmark.c: Likewise.
* time/strptime_l.c: Likewise.
2019-01-12 13:44:51 +00:00

2368 lines
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.file "sincosl.s"
// Copyright (c) 2000 - 2004, 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 (hand-optimized)
// 04/04/00 Unwind support added
// 07/30/01 Improved speed on all paths
// 08/20/01 Fixed bundling typo
// 05/13/02 Changed interface to __libm_pi_by_2_reduce
// 02/10/03 Reordered header: .section, .global, .proc, .align;
// used data8 for long double table values
// 10/13/03 Corrected final .endp name to match .proc
// 10/26/04 Avoided using r14-31 as scratch so not clobbered by dynamic loader
//
//*********************************************************************
//
// Function: Combined sinl(x) and cosl(x), where
//
// sinl(x) = sine(x), for double-extended precision x values
// cosl(x) = cosine(x), for double-extended precision x values
//
//*********************************************************************
//
// Resources Used:
//
// Floating-Point Registers: f8 (Input and Return Value)
// f32-f99
//
// General Purpose Registers:
// r32-r58
//
// 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
//
// sinl(SNaN) = QNaN
// sinl(QNaN) = QNaN
// sinl(inf) = QNaN
// sinl(+/-0) = +/-0
// cosl(inf) = QNaN
// cosl(SNaN) = QNaN
// cosl(QNaN) = QNaN
// cosl(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
//
// cosl( Arg ) = sinl( (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
//
// sinl( Arg ) = (-1)^i_0 sinl( alpha ) if i_1 = 0,
// = (-1)^i_0 cosl( 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, sinl(r+c) and cosl(r+c) can be
// computed very easily by 2 or 3 terms of the Taylor series
// expansion as follows:
//
// Case 2:
// -------
//
// sinl(r + c) = r + c - r^3/6 accurately
// cosl(r + c) = 1 - 2^(-67) accurately
//
// Case 4:
// -------
//
// sinl(r + c) = r + c - r^3/6 + r^5/120 accurately
// cosl(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
//
// sinl(r + c) = sinl(r) + correction, or
// cosl(r + c) = cosl(r) + correction.
//
// Specifically,
//
// sinl(r + c) = sinl(r) + c sin'(r) + O(c^2)
// = sinl(r) + c cos (r) + O(c^2)
// = sinl(r) + c(1 - r^2/2) accurately.
// Similarly,
//
// cosl(r + c) = cosl(r) - c sinl(r) + O(c^2)
// = cosl(r) - c(r - r^3/6) accurately.
//
// We therefore concentrate on accurately calculating sinl(r) and
// cosl(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 sinl(r) and cosl(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
//
// sinl(Arg) = (-1)^i_0 * sinl(r + c) if i_1 = 0
// = (-1)^i_0 * cosl(r + c) if i_1 = 1
//
// can be accurately approximated by
//
// sinl(Arg) = (-1)^i_0 * [sinl(r) + c] if i_1 = 0
// = (-1)^i_0 * [cosl(r) - c*r] if i_1 = 1
//
// because |r| is small and thus the second terms in the correction
// are unnecessary.
//
// Finally, sinl(r) and cosl(r) are approximated by polynomials of
// moderate lengths.
//
// sinl(r) = r + S_1 r^3 + S_2 r^5 + ... + S_5 r^11
// cosl(r) = 1 + C_1 r^2 + C_2 r^4 + ... + C_5 r^10
//
// We can make use of predicates to selectively calculate
// sinl(r) or cosl(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,
//
// sinl(Arg) = (-1)^i_0 * sinl(r + c) if i_1 = 0
// = (-1)^i_0 * cosl(r + c) if i_1 = 1.
//
// Because |r| is now larger, we need one extra term in the
// correction. sinl(Arg) can be accurately approximated by
//
// sinl(Arg) = (-1)^i_0 * [sinl(r) + c(1-r^2/2)] if i_1 = 0
// = (-1)^i_0 * [cosl(r) - c*r*(1 - r^2/6)] i_1 = 1.
//
// Finally, sinl(r) and cosl(r) are approximated by polynomials of
// moderate lengths.
//
// sinl(r) = r + PP_1_hi r^3 + PP_1_lo r^3 +
// PP_2 r^5 + ... + PP_8 r^17
//
// cosl(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.
//
RODATA
.align 16
LOCAL_OBJECT_START(FSINCOSL_CONSTANTS)
sincosl_table_p:
data8 0xA2F9836E4E44152A, 0x00003FFE // Inv_pi_by_2
data8 0xC84D32B0CE81B9F1, 0x00004016 // P_0
data8 0xC90FDAA22168C235, 0x00003FFF // P_1
data8 0xECE675D1FC8F8CBB, 0x0000BFBD // P_2
data8 0xB7ED8FBBACC19C60, 0x0000BF7C // P_3
data8 0x8D848E89DBD171A1, 0x0000BFBF // d_1
data8 0xD5394C3618A66F8E, 0x0000BF7C // d_2
LOCAL_OBJECT_END(FSINCOSL_CONSTANTS)
LOCAL_OBJECT_START(sincosl_table_d)
data8 0xC90FDAA22168C234, 0x00003FFE // pi_by_4
data8 0xA397E5046EC6B45A, 0x00003FE7 // Inv_P_0
data4 0x3E000000, 0xBE000000 // 2^-3 and -2^-3
data4 0x2F000000, 0xAF000000 // 2^-33 and -2^-33
data4 0x9E000000, 0x00000000 // -2^-67
data4 0x00000000, 0x00000000 // pad
LOCAL_OBJECT_END(sincosl_table_d)
LOCAL_OBJECT_START(sincosl_table_pp)
data8 0xCC8ABEBCA21C0BC9, 0x00003FCE // PP_8
data8 0xD7468A05720221DA, 0x0000BFD6 // PP_7
data8 0xB092382F640AD517, 0x00003FDE // PP_6
data8 0xD7322B47D1EB75A4, 0x0000BFE5 // PP_5
data8 0xFFFFFFFFFFFFFFFE, 0x0000BFFD // C_1
data8 0xAAAA000000000000, 0x0000BFFC // PP_1_hi
data8 0xB8EF1D2ABAF69EEA, 0x00003FEC // PP_4
data8 0xD00D00D00D03BB69, 0x0000BFF2 // PP_3
data8 0x8888888888888962, 0x00003FF8 // PP_2
data8 0xAAAAAAAAAAAB0000, 0x0000BFEC // PP_1_lo
LOCAL_OBJECT_END(sincosl_table_pp)
LOCAL_OBJECT_START(sincosl_table_qq)
data8 0xD56232EFC2B0FE52, 0x00003FD2 // QQ_8
data8 0xC9C99ABA2B48DCA6, 0x0000BFDA // QQ_7
data8 0x8F76C6509C716658, 0x00003FE2 // QQ_6
data8 0x93F27DBAFDA8D0FC, 0x0000BFE9 // QQ_5
data8 0xAAAAAAAAAAAAAAAA, 0x0000BFFC // S_1
data8 0x8000000000000000, 0x0000BFFE // QQ_1
data8 0xD00D00D00C6E5041, 0x00003FEF // QQ_4
data8 0xB60B60B60B607F60, 0x0000BFF5 // QQ_3
data8 0xAAAAAAAAAAAAAA9B, 0x00003FFA // QQ_2
LOCAL_OBJECT_END(sincosl_table_qq)
LOCAL_OBJECT_START(sincosl_table_c)
data8 0xFFFFFFFFFFFFFFFE, 0x0000BFFD // C_1
data8 0xAAAAAAAAAAAA719F, 0x00003FFA // C_2
data8 0xB60B60B60356F994, 0x0000BFF5 // C_3
data8 0xD00CFFD5B2385EA9, 0x00003FEF // C_4
data8 0x93E4BD18292A14CD, 0x0000BFE9 // C_5
LOCAL_OBJECT_END(sincosl_table_c)
LOCAL_OBJECT_START(sincosl_table_s)
data8 0xAAAAAAAAAAAAAAAA, 0x0000BFFC // S_1
data8 0x88888888888868DB, 0x00003FF8 // S_2
data8 0xD00D00D0055EFD4B, 0x0000BFF2 // S_3
data8 0xB8EF1C5D839730B9, 0x00003FEC // S_4
data8 0xD71EA3A4E5B3F492, 0x0000BFE5 // S_5
data4 0x38800000, 0xB8800000 // two**-14 and -two**-14
LOCAL_OBJECT_END(sincosl_table_s)
FR_Input_X = f8
FR_Result = f8
FR_r = f8
FR_c = f9
FR_norm_x = f9
FR_inv_pi_2to63 = f10
FR_rshf_2to64 = f11
FR_2tom64 = f12
FR_rshf = f13
FR_N_float_signif = f14
FR_abs_x = f15
FR_Pi_by_4 = f34
FR_Two_to_M14 = f35
FR_Neg_Two_to_M14 = f36
FR_Two_to_M33 = 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_d_2 = f48
FR_tmp_result = 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 = f101
FR_PP_7 = f82
FR_QQ_7 = f102
FR_PP_6 = f83
FR_QQ_6 = f103
FR_PP_5 = f84
FR_QQ_5 = f104
FR_PP_4 = f85
FR_QQ_4 = f105
FR_PP_3 = f86
FR_QQ_3 = f106
FR_PP_2 = f87
FR_QQ_2 = f107
FR_QQ_1 = f108
FR_r_hi_sq = f88
FR_N_0_fix = f89
FR_Inv_P_0 = f90
FR_corr = f91
FR_poly = f92
FR_Neg_Two_to_M3 = f93
FR_Two_to_M3 = f94
FR_P_0 = f95
FR_C_lo = f96
FR_PP_1 = f97
FR_PP_1_lo = f98
FR_ArgPrime = f99
FR_inexact = f100
GR_exp_m2_to_m3= r36
GR_N_Inc = r37
GR_Sin_or_Cos = r38
GR_signexp_x = r40
GR_exp_x = r40
GR_exp_mask = r41
GR_exp_2_to_63 = r42
GR_exp_2_to_m3 = r43
GR_exp_2_to_24 = r44
GR_sig_inv_pi = r45
GR_rshf_2to64 = r46
GR_exp_2tom64 = r47
GR_rshf = r48
GR_ad_p = r49
GR_ad_d = r50
GR_ad_pp = r51
GR_ad_qq = r52
GR_ad_c = r53
GR_ad_s = r54
GR_ad_ce = r55
GR_ad_se = r56
GR_ad_m14 = r57
GR_ad_s1 = r58
// Added for unwind support
GR_SAVE_B0 = r39
GR_SAVE_GP = r40
GR_SAVE_PFS = r41
.section .text
GLOBAL_IEEE754_ENTRY(sinl)
{ .mlx
alloc r32 = ar.pfs,0,27,2,0
movl GR_sig_inv_pi = 0xa2f9836e4e44152a // significand of 1/pi
}
{ .mlx
mov GR_Sin_or_Cos = 0x0
movl GR_rshf_2to64 = 0x47e8000000000000 // 1.1000 2^(63+64)
}
;;
{ .mfi
addl GR_ad_p = @ltoff(FSINCOSL_CONSTANTS#), gp
fclass.m p6, p0 = FR_Input_X, 0x1E3 // Test x natval, nan, inf
mov GR_exp_2_to_m3 = 0xffff - 3 // Exponent of 2^-3
}
{ .mfb
nop.m 999
fnorm.s1 FR_norm_x = FR_Input_X // Normalize x
br.cond.sptk SINCOSL_CONTINUE
}
;;
GLOBAL_IEEE754_END(sinl)
libm_alias_ldouble_other (__sin, sin)
GLOBAL_IEEE754_ENTRY(cosl)
{ .mlx
alloc r32 = ar.pfs,0,27,2,0
movl GR_sig_inv_pi = 0xa2f9836e4e44152a // significand of 1/pi
}
{ .mlx
mov GR_Sin_or_Cos = 0x1
movl GR_rshf_2to64 = 0x47e8000000000000 // 1.1000 2^(63+64)
}
;;
{ .mfi
addl GR_ad_p = @ltoff(FSINCOSL_CONSTANTS#), gp
fclass.m p6, p0 = FR_Input_X, 0x1E3 // Test x natval, nan, inf
mov GR_exp_2_to_m3 = 0xffff - 3 // Exponent of 2^-3
}
{ .mfi
nop.m 999
fnorm.s1 FR_norm_x = FR_Input_X // Normalize x
nop.i 999
}
;;
SINCOSL_CONTINUE:
{ .mfi
setf.sig FR_inv_pi_2to63 = GR_sig_inv_pi // Form 1/pi * 2^63
nop.f 999
mov GR_exp_2tom64 = 0xffff - 64 // Scaling constant to compute N
}
{ .mlx
setf.d FR_rshf_2to64 = GR_rshf_2to64 // Form const 1.1000 * 2^(63+64)
movl GR_rshf = 0x43e8000000000000 // Form const 1.1000 * 2^63
}
;;
{ .mfi
ld8 GR_ad_p = [GR_ad_p] // Point to Inv_pi_by_2
fclass.m p7, p0 = FR_Input_X, 0x0b // Test x denormal
nop.i 999
}
;;
{ .mfi
getf.exp GR_signexp_x = FR_Input_X // Get sign and exponent of x
fclass.m p10, p0 = FR_Input_X, 0x007 // Test x zero
nop.i 999
}
{ .mib
mov GR_exp_mask = 0x1ffff // Exponent mask
nop.i 999
(p6) br.cond.spnt SINCOSL_SPECIAL // Branch if x natval, nan, inf
}
;;
{ .mfi
setf.exp FR_2tom64 = GR_exp_2tom64 // Form 2^-64 for scaling N_float
nop.f 0
add GR_ad_d = 0x70, GR_ad_p // Point to constant table d
}
{ .mib
setf.d FR_rshf = GR_rshf // Form right shift const 1.1000 * 2^63
mov GR_exp_m2_to_m3 = 0x2fffc // Form -(2^-3)
(p7) br.cond.spnt SINCOSL_DENORMAL // Branch if x denormal
}
;;
SINCOSL_COMMON:
{ .mfi
and GR_exp_x = GR_exp_mask, GR_signexp_x // Get exponent of x
fclass.nm p8, p0 = FR_Input_X, 0x1FF // Test x unsupported type
mov GR_exp_2_to_63 = 0xffff + 63 // Exponent of 2^63
}
{ .mib
add GR_ad_pp = 0x40, GR_ad_d // Point to constant table pp
mov GR_exp_2_to_24 = 0xffff + 24 // Exponent of 2^24
(p10) br.cond.spnt SINCOSL_ZERO // Branch if x zero
}
;;
{ .mfi
ldfe FR_Inv_pi_by_2 = [GR_ad_p], 16 // Load 2/pi
fcmp.eq.s0 p15, p0 = FR_Input_X, f0 // Dummy to set denormal
add GR_ad_qq = 0xa0, GR_ad_pp // Point to constant table qq
}
{ .mfi
ldfe FR_Pi_by_4 = [GR_ad_d], 16 // Load pi/4 for range test
nop.f 999
cmp.ge p10,p0 = GR_exp_x, GR_exp_2_to_63 // Is |x| >= 2^63
}
;;
{ .mfi
ldfe FR_P_0 = [GR_ad_p], 16 // Load P_0 for pi/4 <= |x| < 2^63
fmerge.s FR_abs_x = f1, FR_norm_x // |x|
add GR_ad_c = 0x90, GR_ad_qq // Point to constant table c
}
{ .mfi
ldfe FR_Inv_P_0 = [GR_ad_d], 16 // Load 1/P_0 for pi/4 <= |x| < 2^63
nop.f 999
cmp.ge p7,p0 = GR_exp_x, GR_exp_2_to_24 // Is |x| >= 2^24
}
;;
{ .mfi
ldfe FR_P_1 = [GR_ad_p], 16 // Load P_1 for pi/4 <= |x| < 2^63
nop.f 999
add GR_ad_s = 0x50, GR_ad_c // Point to constant table s
}
{ .mfi
ldfe FR_PP_8 = [GR_ad_pp], 16 // Load PP_8 for 2^-3 < |r| < pi/4
nop.f 999
nop.i 999
}
;;
{ .mfi
ldfe FR_P_2 = [GR_ad_p], 16 // Load P_2 for pi/4 <= |x| < 2^63
nop.f 999
add GR_ad_ce = 0x40, GR_ad_c // Point to end of constant table c
}
{ .mfi
ldfe FR_QQ_8 = [GR_ad_qq], 16 // Load QQ_8 for 2^-3 < |r| < pi/4
nop.f 999
nop.i 999
}
;;
{ .mfi
ldfe FR_QQ_7 = [GR_ad_qq], 16 // Load QQ_7 for 2^-3 < |r| < pi/4
fma.s1 FR_N_float_signif = FR_Input_X, FR_inv_pi_2to63, FR_rshf_2to64
add GR_ad_se = 0x40, GR_ad_s // Point to end of constant table s
}
{ .mib
ldfe FR_PP_7 = [GR_ad_pp], 16 // Load PP_7 for 2^-3 < |r| < pi/4
mov GR_ad_s1 = GR_ad_s // Save pointer to S_1
(p10) br.cond.spnt SINCOSL_ARG_TOO_LARGE // Branch if |x| >= 2^63
// Use Payne-Hanek Reduction
}
;;
{ .mfi
ldfe FR_P_3 = [GR_ad_p], 16 // Load P_3 for pi/4 <= |x| < 2^63
fmerge.se FR_r = FR_norm_x, FR_norm_x // r = x, in case |x| < pi/4
add GR_ad_m14 = 0x50, GR_ad_s // Point to constant table m14
}
{ .mfb
ldfps FR_Two_to_M3, FR_Neg_Two_to_M3 = [GR_ad_d], 8
fma.s1 FR_rsq = FR_norm_x, FR_norm_x, f0 // rsq = x*x, in case |x| < pi/4
(p7) br.cond.spnt SINCOSL_LARGER_ARG // Branch if 2^24 <= |x| < 2^63
// Use pre-reduction
}
;;
{ .mmf
ldfe FR_PP_6 = [GR_ad_pp], 16 // Load PP_6 for normal path
ldfe FR_QQ_6 = [GR_ad_qq], 16 // Load QQ_6 for normal path
fmerge.se FR_c = f0, f0 // c = 0 in case |x| < pi/4
}
;;
{ .mmf
ldfe FR_PP_5 = [GR_ad_pp], 16 // Load PP_5 for normal path
ldfe FR_QQ_5 = [GR_ad_qq], 16 // Load QQ_5 for normal path
nop.f 999
}
;;
// Here if 0 < |x| < 2^24
{ .mfi
ldfe FR_S_5 = [GR_ad_se], -16 // Load S_5 if i_1=0
fcmp.lt.s1 p6, p7 = FR_abs_x, FR_Pi_by_4 // Test |x| < pi/4
nop.i 999
}
{ .mfi
ldfe FR_C_5 = [GR_ad_ce], -16 // Load C_5 if i_1=1
fms.s1 FR_N_float = FR_N_float_signif, FR_2tom64, FR_rshf
nop.i 999
}
;;
{ .mmi
ldfe FR_S_4 = [GR_ad_se], -16 // Load S_4 if i_1=0
ldfe FR_C_4 = [GR_ad_ce], -16 // Load C_4 if i_1=1
nop.i 999
}
;;
//
// N = Arg * 2/pi
// Check if Arg < pi/4
//
//
// Case 2: Convert integer N_fix back to normalized floating-point value.
// Case 1: p8 is only affected when p6 is set
//
//
// Grab the integer part of N and call it N_fix
//
{ .mfi
(p7) ldfps FR_Two_to_M33, FR_Neg_Two_to_M33 = [GR_ad_d], 8
(p6) fma.s1 FR_r_cubed = FR_r, FR_rsq, f0 // r^3 if |x| < pi/4
(p6) mov GR_N_Inc = GR_Sin_or_Cos // N_Inc if |x| < pi/4
}
;;
// If |x| < pi/4, r = x and c = 0
// lf |x| < pi/4, is x < 2**(-3).
// r = Arg
// c = 0
{ .mmi
(p7) getf.sig GR_N_Inc = FR_N_float_signif
(p6) cmp.lt.unc p8,p0 = GR_exp_x, GR_exp_2_to_m3 // Is |x| < 2^-3
(p6) tbit.z p9,p10 = GR_N_Inc, 0 // p9 if i_1=0, N mod 4 = 0,1
// p10 if i_1=1, N mod 4 = 2,3
}
;;
//
// 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
//
{ .mbb
nop.m 999
(p8) br.cond.spnt SINCOSL_SMALL_R_0 // Branch if 0 < |x| < 2^-3
(p6) br.cond.spnt SINCOSL_NORMAL_R_0 // Branch if 2^-3 <= |x| < pi/4
}
;;
// Here if pi/4 <= |x| < 2^24
{ .mfi
ldfs FR_Neg_Two_to_M67 = [GR_ad_d], 8 // Load -2^-67
fnma.s1 FR_s = FR_N_float, FR_P_1, FR_Input_X // s = -N * P_1 + Arg
add GR_N_Inc = GR_N_Inc, GR_Sin_or_Cos // Adjust N_Inc for sin/cos
}
{ .mfi
nop.m 999
fma.s1 FR_w = FR_N_float, FR_P_2, f0 // w = N * P_2
nop.i 999
}
;;
{ .mfi
nop.m 999
fms.s1 FR_r = FR_s, f1, FR_w // r = s - w, assume |s| >= 2^-33
tbit.z p9,p10 = GR_N_Inc, 0 // p9 if i_1=0, N mod 4 = 0,1
// p10 if i_1=1, N mod 4 = 2,3
}
;;
{ .mfi
nop.m 999
fcmp.lt.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 // p6 if |s| >= 2^-33, else p7
nop.i 999
}
;;
{ .mfi
nop.m 999
fms.s1 FR_c = FR_s, f1, FR_r // c = s - r, for |s| >= 2^-33
nop.i 999
}
{ .mfi
nop.m 999
fma.s1 FR_rsq = FR_r, FR_r, f0 // rsq = r * r, for |s| >= 2^-33
nop.i 999
}
;;
{ .mfi
nop.m 999
(p7) fma.s1 FR_w = FR_N_float, FR_P_3, f0
nop.i 999
}
;;
{ .mmf
(p9) ldfe FR_C_1 = [GR_ad_pp], 16 // Load C_1 if i_1=0
(p10) ldfe FR_S_1 = [GR_ad_qq], 16 // Load S_1 if i_1=1
frcpa.s1 FR_r_hi, p15 = f1, FR_r // r_hi = frcpa(r)
}
;;
{ .mfi
nop.m 999
(p6) fcmp.lt.unc.s1 p8, p13 = FR_r, FR_Two_to_M3 // If big s, test r with 2^-3
nop.i 999
}
;;
{ .mfi
nop.m 999
(p7) fma.s1 FR_U_1 = FR_N_float, FR_P_2, FR_w
nop.i 999
}
;;
//
// For big s: r = s - w: No futher reduction is necessary
// For small s: w = N * P_3 (change sign) More reduction
//
{ .mfi
nop.m 999
(p8) fcmp.gt.s1 p8, p13 = FR_r, FR_Neg_Two_to_M3 // If big s, p8 if |r| < 2^-3
nop.i 999 ;;
}
{ .mfi
nop.m 999
(p9) fma.s1 FR_poly = FR_rsq, FR_PP_8, FR_PP_7 // poly = rsq*PP_8+PP_7 if i_1=0
nop.i 999
}
{ .mfi
nop.m 999
(p10) fma.s1 FR_poly = FR_rsq, FR_QQ_8, FR_QQ_7 // poly = rsq*QQ_8+QQ_7 if i_1=1
nop.i 999
}
;;
{ .mfi
nop.m 999
(p7) fms.s1 FR_r = FR_s, f1, FR_U_1
nop.i 999
}
;;
{ .mfi
nop.m 999
(p6) fma.s1 FR_r_cubed = FR_r, FR_rsq, f0 // rcubed = r * rsq
nop.i 999
}
;;
{ .mfi
//
// 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.
//
//
// For big s: c = c + w (w has not been negated.)
// For small s: r = S - U_1
//
nop.m 999
(p6) fms.s1 FR_c = FR_c, f1, FR_w
nop.i 999
}
{ .mbb
nop.m 999
(p8) br.cond.spnt SINCOSL_SMALL_R_1 // Branch if |s|>=2^-33, |r| < 2^-3,
// and pi/4 <= |x| < 2^24
(p13) br.cond.sptk SINCOSL_NORMAL_R_1 // Branch if |s|>=2^-33, |r| >= 2^-3,
// and pi/4 <= |x| < 2^24
}
;;
SINCOSL_S_TINY:
//
// Here if |s| < 2^-33, and pi/4 <= |x| < 2^24
//
{ .mfi
fms.s1 FR_U_2 = FR_N_float, FR_P_2, FR_U_1
//
// c = S - U_1
// r = S_1 * r
//
//
}
;;
{ .mmi
nop.m 999
//
// Get [i_0,i_1] - two lsb of N_fix_gr.
// Do dummy fmpy so inexact is always set.
//
tbit.z p9,p10 = GR_N_Inc, 0 // p9 if i_1=0, N mod 4 = 0,1
// p10 if i_1=1, N mod 4 = 2,3
}
;;
//
// For small s: U_2 = N * P_2 - U_1
// S_1 stored constant - grab the one stored with the
// coefficients.
//
{ .mfi
ldfe FR_S_1 = [GR_ad_s1], 16
//
// Check if i_1 and i_0 != 0
//
(p10) fma.s1 FR_poly = f0, f1, FR_Neg_Two_to_M67
tbit.z p11,p12 = GR_N_Inc, 1 // p11 if i_0=0, N mod 4 = 0,2
// p12 if i_0=1, N mod 4 = 1,3
}
;;
{ .mfi
nop.m 999
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
//
fma.s1 FR_rsq = FR_r, FR_r, f0
nop.i 999 ;;
}
{ .mfi
nop.m 999
fma.s1 FR_U_2 = FR_U_2, f1, FR_w
nop.i 999
}
{ .mfi
nop.m 999
fmerge.se FR_tmp_result = FR_r, FR_r
nop.i 999 ;;
}
{ .mfi
nop.m 999
(p10) fma.s1 FR_tmp_result = f0, f1, f1
nop.i 999 ;;
}
{ .mfi
nop.m 999
//
// FR_rsq = r * r
// Save r as the result.
//
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_tmp_result = FR_tmp_result, f1, f0
nop.i 999
}
{ .mfi
nop.m 999
fma.s1 FR_r = FR_S_1, FR_r, f0
nop.i 999 ;;
}
{ .mfi
nop.m 999
fma.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)
//
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.s0 FR_Result = FR_tmp_result, f1, FR_poly
nop.i 999 ;;
}
{ .mfb
nop.m 999
(p12) fms.s0 FR_Result = FR_tmp_result, f1, FR_poly
//
// if (i_0 == 0), Result = Result + poly
// else Result = Result - poly
//
br.ret.sptk b0 // Exit if |s| < 2^-33, and pi/4 <= |x| < 2^24
}
;;
SINCOSL_LARGER_ARG:
//
// Here if 2^24 <= |x| < 2^63
//
{ .mfi
ldfe FR_d_1 = [GR_ad_p], 16 // Load d_1 for |x| >= 2^24 path
fma.s1 FR_N_0 = FR_Input_X, FR_Inv_P_0, f0
nop.i 999
}
;;
//
// N_0 = Arg * Inv_P_0
//
// Load values 2**(-14) and -2**(-14)
{ .mmi
ldfps FR_Two_to_M14, FR_Neg_Two_to_M14 = [GR_ad_m14]
nop.i 999 ;;
}
{ .mfi
ldfe FR_d_2 = [GR_ad_p], 16 // Load d_2 for |x| >= 2^24 path
nop.f 999
nop.i 999 ;;
}
{ .mfi
nop.m 999
//
//
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
//
fcvt.xf FR_N_0 = FR_N_0_fix
nop.i 999 ;;
}
{ .mfi
nop.m 999
//
// Make N_0 the integer part
//
fnma.s1 FR_ArgPrime = FR_N_0, FR_P_0, FR_Input_X
nop.i 999
}
{ .mfi
nop.m 999
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
//
fma.s1 FR_N_float = FR_ArgPrime, FR_Inv_pi_by_2, f0
nop.i 999 ;;
}
{ .mfi
nop.m 999
//
// N = A' * 2/pi
//
fcvt.fx.s1 FR_N_fix = FR_N_float
nop.i 999 ;;
}
{ .mfi
nop.m 999
//
// N_fix is the integer part
//
fcvt.xf FR_N_float = FR_N_fix
nop.i 999 ;;
}
{ .mfi
getf.sig GR_N_Inc = FR_N_fix
nop.f 999
nop.i 999 ;;
}
{ .mii
nop.m 999
nop.i 999 ;;
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
//
fnma.s1 FR_s = FR_N_float, FR_P_1, FR_ArgPrime
nop.i 999
}
{ .mfi
nop.m 999
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
//
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 // p9 if |s| < 2^-14
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 ;;
}
{ .mmf
nop.m 999
nop.m 999
(p9) fms.s1 FR_U_lo = FR_N_0, FR_d_1, FR_U_hi
}
{ .mfi
nop.m 999
(p9) fmerge.s FR_V_hiabs = f0, FR_V_hi
nop.i 999 ;;
}
//{ .mfb
//(p9) fmerge.s f8= FR_V_lo,FR_V_lo
//(p9) br.ret.sptk b0
//}
//;;
{ .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 SINCOSL_SMALL_R // Branch if |r| < 2^-3
// and 2^24 <= |x| < 2^63
}
;;
{ .mib
nop.m 999
nop.i 999
(p13) br.cond.sptk SINCOSL_NORMAL_R // Branch if |r| >= 2^-3
// and 2^24 <= |x| < 2^63
}
;;
SINCOSL_LARGER_S_TINY:
//
// Here if |s| < 2^-14, and 2^24 <= |x| < 2^63
//
{ .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.
//
fcmp.ge.unc.s1 p7, p8 = FR_U_hiabs, FR_V_hiabs
}
{ .mfi
nop.m 999
//
// C_hi = S + A
//
fma.s1 FR_t = FR_U_lo, f1, FR_V_lo
}
;;
{ .mfi
nop.m 999
//
// t = U_lo + V_lo
//
//
(p7) fms.s1 FR_a = FR_U_hi, f1, FR_A
nop.i 999 ;;
}
{ .mfi
nop.m 999
(p8) fma.s1 FR_a = FR_V_hi, f1, FR_A
nop.i 999
}
;;
{ .mfi
//
// Is U_hiabs >= V_hiabs?
//
nop.m 999
fma.s1 FR_C_hi = FR_s, f1, FR_A
nop.i 999 ;;
}
{ .mmi
ldfe FR_C_1 = [GR_ad_c], 16 ;;
ldfe FR_C_2 = [GR_ad_c], 64
nop.i 999 ;;
}
//
// c = c + C_lo finished.
// Load C_2
//
{ .mfi
ldfe FR_S_1 = [GR_ad_s], 16
//
// C_lo = S - C_hi
//
fma.s1 FR_t = FR_t, f1, FR_w
nop.i 999 ;;
}
//
// 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
ldfe FR_S_2 = [GR_ad_s], 64
//
// t = t + w
//
(p7) fms.s1 FR_a = FR_a, f1, FR_V_hi
tbit.z p9,p10 = GR_N_Inc, 0 // p9 if i_1=0, N mod 4 = 0,1
// p10 if i_1=1, N mod 4 = 2,3
}
;;
{ .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
//
fms.s1 FR_C_lo = FR_s, f1, FR_C_hi
nop.i 999 ;;
}
{ .mfi
nop.m 999
(p8) fms.s1 FR_a = FR_U_hi, f1, FR_a
tbit.z p11,p12 = GR_N_Inc, 1 // p11 if i_0=0, N mod 4 = 0,2
// p12 if i_0=1, N mod 4 = 1,3
}
;;
{ .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.
//
fma.s1 FR_C_lo = FR_C_lo, f1, FR_A
nop.i 999 ;;
}
{ .mfi
nop.m 999
//
// C_lo = (S - C_hi) + A
//
fma.s1 FR_t = FR_t, f1, FR_a
nop.i 999 ;;
}
{ .mfi
nop.m 999
//
// t = t + a
//
fma.s1 FR_C_lo = FR_C_lo, f1, FR_t
nop.i 999 ;;
}
{ .mfi
nop.m 999
//
// C_lo = C_lo + t
//
fma.s1 FR_r = FR_C_hi, f1, FR_C_lo
nop.i 999 ;;
}
{ .mfi
nop.m 999
//
// Load S_2
//
fma.s1 FR_rsq = FR_r, FR_r, f0
nop.i 999
}
{ .mfi
nop.m 999
//
// r = C_hi + C_lo
//
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
//
(p9) fma.s1 FR_tmp_result = f0, f1, FR_r
nop.i 999 ;;
}
{ .mfi
nop.m 999
(p10) fma.s1 FR_tmp_result = f0, f1, f1
nop.i 999 ;;
}
{ .mfi
nop.m 999
//
// Compute r_cube = FR_rsq * r
//
(p9) fma.s1 FR_poly = FR_rsq, FR_S_2, FR_S_1
nop.i 999 ;;
}
{ .mfi
nop.m 999
(p10) 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 ?
//
fma.s1 FR_r_cubed = FR_rsq, FR_r, f0
nop.i 999 ;;
}
{ .mfi
nop.m 999
//
// c = C_hi - r
// Load C_1
//
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
//
(p12) fms.s1 FR_tmp_result = f0, f1, FR_tmp_result
nop.i 999 ;;
}
{ .mfi
nop.m 999
//
// if i_1 ==0: Result = r
// else Result = 1.0
//
(p9) fma.s1 FR_poly = FR_r_cubed, FR_poly, FR_c
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_0 !=0: Result = -Result
//
(p11) fma.s0 FR_Result = FR_tmp_result, f1, FR_poly
nop.i 999 ;;
}
{ .mfb
nop.m 999
(p12) fms.s0 FR_Result = FR_tmp_result, f1, FR_poly
//
// if i_0 == 0: Result = Result + poly
// else Result = Result - poly
//
br.ret.sptk b0 // Exit for |s| < 2^-14, and 2^24 <= |x| < 2^63
}
;;
SINCOSL_SMALL_R:
//
// Here if |r| < 2^-3
//
// Enter with r, c, and N_Inc computed
//
// Compare both i_1 and i_0 with 0.
// if i_1 == 0, set p9.
// if i_0 == 0, set p11.
//
{ .mfi
nop.m 999
fma.s1 FR_rsq = FR_r, FR_r, f0 // rsq = r * r
tbit.z p9,p10 = GR_N_Inc, 0 // p9 if i_1=0, N mod 4 = 0,1
// p10 if i_1=1, N mod 4 = 2,3
}
;;
{ .mmi
(p9) ldfe FR_S_5 = [GR_ad_se], -16 // Load S_5 if i_1=0
(p10) ldfe FR_C_5 = [GR_ad_ce], -16 // Load C_5 if i_1=1
nop.i 999
}
;;
{ .mmi
(p9) ldfe FR_S_4 = [GR_ad_se], -16 // Load S_4 if i_1=0
(p10) ldfe FR_C_4 = [GR_ad_ce], -16 // Load C_4 if i_1=1
nop.i 999
}
;;
SINCOSL_SMALL_R_0:
// Entry point for 2^-3 < |x| < pi/4
.pred.rel "mutex",p9,p10
SINCOSL_SMALL_R_1:
// Entry point for pi/4 < |x| < 2^24 and |r| < 2^-3
.pred.rel "mutex",p9,p10
{ .mfi
(p9) ldfe FR_S_3 = [GR_ad_se], -16 // Load S_3 if i_1=0
fma.s1 FR_Z = FR_rsq, FR_rsq, f0 // Z = rsq * rsq
nop.i 999
}
{ .mfi
(p10) ldfe FR_C_3 = [GR_ad_ce], -16 // Load C_3 if i_1=1
(p10) fnma.s1 FR_c = FR_c, FR_r, f0 // c = -c * r if i_1=0
nop.i 999
}
;;
{ .mmf
(p9) ldfe FR_S_2 = [GR_ad_se], -16 // Load S_2 if i_1=0
(p10) ldfe FR_C_2 = [GR_ad_ce], -16 // Load C_2 if i_1=1
(p10) fmerge.s FR_r = f1, f1
}
;;
{ .mmi
(p9) ldfe FR_S_1 = [GR_ad_se], -16 // Load S_1 if i_1=0
(p10) ldfe FR_C_1 = [GR_ad_ce], -16 // Load C_1 if i_1=1
nop.i 999
}
;;
{ .mfi
nop.m 999
(p9) fma.s1 FR_Z = FR_Z, FR_r, f0 // Z = Z * r if i_1=0
nop.i 999
}
;;
{ .mfi
nop.m 999
(p9) fma.s1 FR_poly_lo = FR_rsq, FR_S_5, FR_S_4 // poly_lo=rsq*S_5+S_4 if i_1=0
nop.i 999
}
{ .mfi
nop.m 999
(p10) fma.s1 FR_poly_lo = FR_rsq, FR_C_5, FR_C_4 // poly_lo=rsq*C_5+C_4 if i_1=1
nop.i 999
}
;;
{ .mfi
nop.m 999
(p9) fma.s1 FR_poly_hi = FR_rsq, FR_S_2, FR_S_1 // poly_hi=rsq*S_2+S_1 if i_1=0
nop.i 999
}
{ .mfi
nop.m 999
(p10) fma.s1 FR_poly_hi = FR_rsq, FR_C_2, FR_C_1 // poly_hi=rsq*C_2+C_1 if i_1=1
nop.i 999
}
;;
{ .mfi
nop.m 999
fma.s1 FR_Z = FR_Z, FR_rsq, f0 // Z = Z * rsq
nop.i 999
}
;;
{ .mfi
nop.m 999
(p9) fma.s1 FR_poly_lo = FR_rsq, FR_poly_lo, FR_S_3 // p_lo=p_lo*rsq+S_3, i_1=0
nop.i 999
}
{ .mfi
nop.m 999
(p10) fma.s1 FR_poly_lo = FR_rsq, FR_poly_lo, FR_C_3 // p_lo=p_lo*rsq+C_3, i_1=1
nop.i 999
}
;;
{ .mfi
nop.m 999
(p9) fma.s0 FR_inexact = FR_S_4, FR_S_4, f0 // Dummy op to set inexact
tbit.z p11,p12 = GR_N_Inc, 1 // p11 if i_0=0, N mod 4 = 0,2
// p12 if i_0=1, N mod 4 = 1,3
}
{ .mfi
nop.m 999
(p10) fma.s0 FR_inexact = FR_C_1, FR_C_1, f0 // Dummy op to set inexact
nop.i 999
}
;;
{ .mfi
nop.m 999
(p9) fma.s1 FR_poly_hi = FR_poly_hi, FR_rsq, f0 // p_hi=p_hi*rsq if i_1=0
nop.i 999
}
{ .mfi
nop.m 999
(p10) fma.s1 FR_poly_hi = FR_poly_hi, FR_rsq, f0 // p_hi=p_hi*rsq if i_1=1
nop.i 999
}
;;
{ .mfi
nop.m 999
fma.s1 FR_poly = FR_Z, FR_poly_lo, FR_c // poly=Z*poly_lo+c
nop.i 999
}
;;
{ .mfi
nop.m 999
(p9) fma.s1 FR_poly_hi = FR_r, FR_poly_hi, f0 // p_hi=r*p_hi if i_1=0
nop.i 999
}
;;
{ .mfi
nop.m 999
(p12) fms.s1 FR_r = f0, f1, FR_r // r = -r if i_0=1
nop.i 999
}
;;
{ .mfi
nop.m 999
fma.s1 FR_poly = FR_poly, f1, FR_poly_hi // poly=poly+poly_hi
nop.i 999
}
;;
//
// if (i_0 == 0) Result = r + poly
// if (i_0 != 0) Result = r - poly
//
{ .mfi
nop.m 999
(p11) fma.s0 FR_Result = FR_r, f1, FR_poly
nop.i 999
}
{ .mfb
nop.m 999
(p12) fms.s0 FR_Result = FR_r, f1, FR_poly
br.ret.sptk b0 // Exit for |r| < 2^-3
}
;;
SINCOSL_NORMAL_R:
//
// Here if 2^-3 <= |r| < pi/4
// THIS IS THE MAIN PATH
//
// Enter with r, c, and N_Inc having been computed
//
{ .mfi
ldfe FR_PP_6 = [GR_ad_pp], 16 // Load PP_6
fma.s1 FR_rsq = FR_r, FR_r, f0 // rsq = r * r
tbit.z p9,p10 = GR_N_Inc, 0 // p9 if i_1=0, N mod 4 = 0,1
// p10 if i_1=1, N mod 4 = 2,3
}
{ .mfi
ldfe FR_QQ_6 = [GR_ad_qq], 16 // Load QQ_6
nop.f 999
nop.i 999
}
;;
{ .mmi
(p9) ldfe FR_PP_5 = [GR_ad_pp], 16 // Load PP_5 if i_1=0
(p10) ldfe FR_QQ_5 = [GR_ad_qq], 16 // Load QQ_5 if i_1=1
nop.i 999
}
;;
SINCOSL_NORMAL_R_0:
// Entry for 2^-3 < |x| < pi/4
.pred.rel "mutex",p9,p10
{ .mmf
(p9) ldfe FR_C_1 = [GR_ad_pp], 16 // Load C_1 if i_1=0
(p10) ldfe FR_S_1 = [GR_ad_qq], 16 // Load S_1 if i_1=1
frcpa.s1 FR_r_hi, p6 = f1, FR_r // r_hi = frcpa(r)
}
;;
{ .mfi
nop.m 999
(p9) fma.s1 FR_poly = FR_rsq, FR_PP_8, FR_PP_7 // poly = rsq*PP_8+PP_7 if i_1=0
nop.i 999
}
{ .mfi
nop.m 999
(p10) fma.s1 FR_poly = FR_rsq, FR_QQ_8, FR_QQ_7 // poly = rsq*QQ_8+QQ_7 if i_1=1
nop.i 999
}
;;
{ .mfi
nop.m 999
fma.s1 FR_r_cubed = FR_r, FR_rsq, f0 // rcubed = r * rsq
nop.i 999
}
;;
SINCOSL_NORMAL_R_1:
// Entry for pi/4 <= |x| < 2^24
.pred.rel "mutex",p9,p10
{ .mmf
(p9) ldfe FR_PP_1 = [GR_ad_pp], 16 // Load PP_1_hi if i_1=0
(p10) ldfe FR_QQ_1 = [GR_ad_qq], 16 // Load QQ_1 if i_1=1
frcpa.s1 FR_r_hi, p6 = f1, FR_r_hi // r_hi = frpca(frcpa(r))
}
;;
{ .mfi
(p9) ldfe FR_PP_4 = [GR_ad_pp], 16 // Load PP_4 if i_1=0
(p9) fma.s1 FR_poly = FR_rsq, FR_poly, FR_PP_6 // poly = rsq*poly+PP_6 if i_1=0
nop.i 999
}
{ .mfi
(p10) ldfe FR_QQ_4 = [GR_ad_qq], 16 // Load QQ_4 if i_1=1
(p10) fma.s1 FR_poly = FR_rsq, FR_poly, FR_QQ_6 // poly = rsq*poly+QQ_6 if i_1=1
nop.i 999
}
;;
{ .mfi
nop.m 999
(p9) fma.s1 FR_corr = FR_C_1, FR_rsq, f0 // corr = C_1 * rsq if i_1=0
nop.i 999
}
{ .mfi
nop.m 999
(p10) fma.s1 FR_corr = FR_S_1, FR_r_cubed, FR_r // corr = S_1 * r^3 + r if i_1=1
nop.i 999
}
;;
{ .mfi
(p9) ldfe FR_PP_3 = [GR_ad_pp], 16 // Load PP_3 if i_1=0
fma.s1 FR_r_hi_sq = FR_r_hi, FR_r_hi, f0 // r_hi_sq = r_hi * r_hi
nop.i 999
}
{ .mfi
(p10) ldfe FR_QQ_3 = [GR_ad_qq], 16 // Load QQ_3 if i_1=1
fms.s1 FR_r_lo = FR_r, f1, FR_r_hi // r_lo = r - r_hi
nop.i 999
}
;;
{ .mfi
(p9) ldfe FR_PP_2 = [GR_ad_pp], 16 // Load PP_2 if i_1=0
(p9) fma.s1 FR_poly = FR_rsq, FR_poly, FR_PP_5 // poly = rsq*poly+PP_5 if i_1=0
nop.i 999
}
{ .mfi
(p10) ldfe FR_QQ_2 = [GR_ad_qq], 16 // Load QQ_2 if i_1=1
(p10) fma.s1 FR_poly = FR_rsq, FR_poly, FR_QQ_5 // poly = rsq*poly+QQ_5 if i_1=1
nop.i 999
}
;;
{ .mfi
(p9) ldfe FR_PP_1_lo = [GR_ad_pp], 16 // Load PP_1_lo if i_1=0
(p9) fma.s1 FR_corr = FR_corr, FR_c, FR_c // corr = corr * c + c if i_1=0
nop.i 999
}
{ .mfi
nop.m 999
(p10) fnma.s1 FR_corr = FR_corr, FR_c, f0 // corr = -corr * c if i_1=1
nop.i 999
}
;;
{ .mfi
nop.m 999
(p9) fma.s1 FR_U_lo = FR_r, FR_r_hi, FR_r_hi_sq // U_lo = r*r_hi+r_hi_sq, i_1=0
nop.i 999
}
{ .mfi
nop.m 999
(p10) fma.s1 FR_U_lo = FR_r_hi, f1, FR_r // U_lo = r_hi + r if i_1=1
nop.i 999
}
;;
{ .mfi
nop.m 999
(p9) fma.s1 FR_U_hi = FR_r_hi, FR_r_hi_sq, f0 // U_hi = r_hi*r_hi_sq if i_1=0
nop.i 999
}
{ .mfi
nop.m 999
(p10) fma.s1 FR_U_hi = FR_QQ_1, FR_r_hi_sq, f1 // U_hi = QQ_1*r_hi_sq+1, i_1=1
nop.i 999
}
;;
{ .mfi
nop.m 999
(p9) fma.s1 FR_poly = FR_rsq, FR_poly, FR_PP_4 // poly = poly*rsq+PP_4 if i_1=0
nop.i 999
}
{ .mfi
nop.m 999
(p10) fma.s1 FR_poly = FR_rsq, FR_poly, FR_QQ_4 // poly = poly*rsq+QQ_4 if i_1=1
nop.i 999
}
;;
{ .mfi
nop.m 999
(p9) fma.s1 FR_U_lo = FR_r, FR_r, FR_U_lo // U_lo = r * r + U_lo if i_1=0
nop.i 999
}
{ .mfi
nop.m 999
(p10) fma.s1 FR_U_lo = FR_r_lo, FR_U_lo, f0 // U_lo = r_lo * U_lo if i_1=1
nop.i 999
}
;;
{ .mfi
nop.m 999
(p9) fma.s1 FR_U_hi = FR_PP_1, FR_U_hi, f0 // U_hi = PP_1 * U_hi if i_1=0
nop.i 999
}
;;
{ .mfi
nop.m 999
(p9) fma.s1 FR_poly = FR_rsq, FR_poly, FR_PP_3 // poly = poly*rsq+PP_3 if i_1=0
nop.i 999
}
{ .mfi
nop.m 999
(p10) fma.s1 FR_poly = FR_rsq, FR_poly, FR_QQ_3 // poly = poly*rsq+QQ_3 if i_1=1
nop.i 999
}
;;
{ .mfi
nop.m 999
(p9) fma.s1 FR_U_lo = FR_r_lo, FR_U_lo, f0 // U_lo = r_lo * U_lo if i_1=0
nop.i 999
}
{ .mfi
nop.m 999
(p10) fma.s1 FR_U_lo = FR_QQ_1,FR_U_lo, f0 // U_lo = QQ_1 * U_lo if i_1=1
nop.i 999
}
;;
{ .mfi
nop.m 999
(p9) fma.s1 FR_U_hi = FR_r, f1, FR_U_hi // U_hi = r + U_hi if i_1=0
nop.i 999
}
;;
{ .mfi
nop.m 999
(p9) fma.s1 FR_poly = FR_rsq, FR_poly, FR_PP_2 // poly = poly*rsq+PP_2 if i_1=0
nop.i 999
}
{ .mfi
nop.m 999
(p10) fma.s1 FR_poly = FR_rsq, FR_poly, FR_QQ_2 // poly = poly*rsq+QQ_2 if i_1=1
nop.i 999
}
;;
{ .mfi
nop.m 999
(p9) fma.s1 FR_U_lo = FR_PP_1, FR_U_lo, f0 // U_lo = PP_1 * U_lo if i_1=0
nop.i 999
}
;;
{ .mfi
nop.m 999
(p9) fma.s1 FR_poly = FR_rsq, FR_poly, FR_PP_1_lo // poly =poly*rsq+PP1lo i_1=0
nop.i 999
}
{ .mfi
nop.m 999
(p10) fma.s1 FR_poly = FR_rsq, FR_poly, f0 // poly = poly*rsq if i_1=1
nop.i 999
}
;;
{ .mfi
nop.m 999
fma.s1 FR_V = FR_U_lo, f1, FR_corr // V = U_lo + corr
tbit.z p11,p12 = GR_N_Inc, 1 // p11 if i_0=0, N mod 4 = 0,2
// p12 if i_0=1, N mod 4 = 1,3
}
;;
{ .mfi
nop.m 999
(p9) fma.s0 FR_inexact = FR_PP_5, FR_PP_4, f0 // Dummy op to set inexact
nop.i 999
}
{ .mfi
nop.m 999
(p10) fma.s0 FR_inexact = FR_QQ_5, FR_QQ_5, f0 // Dummy op to set inexact
nop.i 999
}
;;
{ .mfi
nop.m 999
(p9) fma.s1 FR_poly = FR_r_cubed, FR_poly, f0 // poly = poly*r^3 if i_1=0
nop.i 999
}
{ .mfi
nop.m 999
(p10) fma.s1 FR_poly = FR_rsq, FR_poly, f0 // poly = poly*rsq if i_1=1
nop.i 999
}
;;
{ .mfi
nop.m 999
(p11) fma.s1 FR_tmp_result = f0, f1, f1// tmp_result=+1.0 if i_0=0
nop.i 999
}
{ .mfi
nop.m 999
(p12) fms.s1 FR_tmp_result = f0, f1, f1// tmp_result=-1.0 if i_0=1
nop.i 999
}
;;
{ .mfi
nop.m 999
fma.s1 FR_V = FR_poly, f1, FR_V // V = poly + V
nop.i 999
}
;;
// If i_0 = 0 Result = U_hi + V
// If i_0 = 1 Result = -U_hi - V
{ .mfi
nop.m 999
(p11) fma.s0 FR_Result = FR_tmp_result, FR_U_hi, FR_V
nop.i 999
}
{ .mfb
nop.m 999
(p12) fms.s0 FR_Result = FR_tmp_result, FR_U_hi, FR_V
br.ret.sptk b0 // Exit for 2^-3 <= |r| < pi/4
}
;;
SINCOSL_ZERO:
// Here if x = 0
{ .mfi
cmp.eq.unc p6, p7 = 0x1, GR_Sin_or_Cos
nop.f 999
nop.i 999
}
;;
{ .mfi
nop.m 999
(p7) fmerge.s FR_Result = FR_Input_X, FR_Input_X // If sin, result = input
nop.i 999
}
{ .mfb
nop.m 999
(p6) fma.s0 FR_Result = f1, f1, f0 // If cos, result=1.0
br.ret.sptk b0 // Exit for x=0
}
;;
SINCOSL_DENORMAL:
{ .mmb
getf.exp GR_signexp_x = FR_norm_x // Get sign and exponent of x
nop.m 999
br.cond.sptk SINCOSL_COMMON // Return to common code
}
;;
SINCOSL_SPECIAL:
{ .mfb
nop.m 999
//
// Path for Arg = +/- QNaN, SNaN, Inf
// Invalid can be raised. SNaNs
// become QNaNs
//
fmpy.s0 FR_Result = FR_Input_X, f0
br.ret.sptk b0 ;;
}
GLOBAL_IEEE754_END(cosl)
libm_alias_ldouble_other (__cos, cos)
// *******************************************************************
// *******************************************************************
// *******************************************************************
//
// Special Code to handle very large argument case.
// Call int __libm_pi_by_2_reduce(x,r,c) for |arguments| >= 2**63
// The interface is custom:
// On input:
// (Arg or x) is in f8
// On output:
// r is in f8
// c is in f9
// N is in r8
// Be sure to allocate at least 2 GP registers as output registers for
// __libm_pi_by_2_reduce. This routine uses r59-60. These are used as
// scratch registers within the __libm_pi_by_2_reduce routine (for speed).
//
// We know also that __libm_pi_by_2_reduce preserves f10-15, f71-127. We
// use this to eliminate save/restore of key fp registers in this calling
// function.
//
// *******************************************************************
// *******************************************************************
// *******************************************************************
LOCAL_LIBM_ENTRY(__libm_callout)
SINCOSL_ARG_TOO_LARGE:
.prologue
{ .mfi
nop.f 0
.save ar.pfs,GR_SAVE_PFS
mov GR_SAVE_PFS=ar.pfs // Save ar.pfs
};;
{ .mmi
setf.exp FR_Two_to_M3 = GR_exp_2_to_m3 // Form 2^-3
mov GR_SAVE_GP=gp // Save gp
.save b0, GR_SAVE_B0
mov GR_SAVE_B0=b0 // Save b0
};;
.body
//
// Call argument reduction with x in f8
// Returns with N in r8, r in f8, c in f9
// Assumes f71-127 are preserved across the call
//
{ .mib
setf.exp FR_Neg_Two_to_M3 = GR_exp_m2_to_m3 // Form -(2^-3)
nop.i 0
br.call.sptk b0=__libm_pi_by_2_reduce#
};;
{ .mfi
add GR_N_Inc = GR_Sin_or_Cos,r8
fcmp.lt.unc.s1 p6, p0 = FR_r, FR_Two_to_M3
mov b0 = GR_SAVE_B0 // Restore return address
};;
{ .mfi
mov gp = GR_SAVE_GP // Restore gp
(p6) fcmp.gt.unc.s1 p6, p0 = FR_r, FR_Neg_Two_to_M3
mov ar.pfs = GR_SAVE_PFS // Restore ar.pfs
};;
{ .mbb
nop.m 999
(p6) br.cond.spnt SINCOSL_SMALL_R // Branch if |r|< 2^-3 for |x| >= 2^63
br.cond.sptk SINCOSL_NORMAL_R // Branch if |r|>=2^-3 for |x| >= 2^63
};;
LOCAL_LIBM_END(__libm_callout)
.type __libm_pi_by_2_reduce#,@function
.global __libm_pi_by_2_reduce#
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