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Diffstat (limited to 'sysdeps/ia64/fpu/libm_sincosl.S')
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diff --git a/sysdeps/ia64/fpu/libm_sincosl.S b/sysdeps/ia64/fpu/libm_sincosl.S new file mode 100644 index 0000000..7fe92c5 --- /dev/null +++ b/sysdeps/ia64/fpu/libm_sincosl.S @@ -0,0 +1,2525 @@ +.file "libm_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: +// 05/13/02 Initial version of sincosl (based on libm's sinl and cosl) +// 02/10/03 Reordered header: .section, .global, .proc, .align; +// used data8 for long double table values +// 10/13/03 Corrected .file name +// 02/11/04 cisl is moved to the separate file. +// 10/26/04 Avoided using r14-31 as scratch so not clobbered by dynamic loader +// +//********************************************************************* +// +// Function: Combined sincosl routine with 3 different API's +// +// API's +//============================================================== +// 1) void sincosl(long double, long double*s, long double*c) +// 2) __libm_sincosl - internal LIBM function, that accepts +// argument in f8 and returns cosine through f8, sine through f9 +// +// +//********************************************************************* +// +// Resources Used: +// +// Floating-Point Registers: f8 (Input x and cosl return value), +// f9 (sinl returned) +// f32-f121 +// +// General Purpose Registers: +// r32-r61 +// +// Predicate Registers: p6-p15 +// +//********************************************************************* +// +// IEEE Special Conditions: +// +// Denormal fault raised on denormal inputs +// Overflow exceptions do not occur +// Underflow exceptions raised when appropriate for sincosl +// (No specialized error handling for this routine) +// Inexact raised when appropriate by algorithm +// +// sincosl(SNaN) = QNaN, QNaN +// sincosl(QNaN) = QNaN, QNaN +// sincosl(inf) = QNaN, QNaN +// sincosl(+/-0) = +/-0, 1 +// +//********************************************************************* +// +// Mathematical Description +// ======================== +// +// The computation of FSIN and FCOS performed in parallel. +// +// Arg = N pi/2 + alpha, |alpha| <= pi/4. +// +// 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 unneccessary. +// +// 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 shares the same code between FSIN and FCOS. +// The argument reduction description given +// previously is repeated below. +// +// +// Step 0. Initialization. +// +// 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 64 + +LOCAL_OBJECT_START(FSINCOSL_CONSTANTS) + +sincosl_table_p: +//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 0xDBD171A1, 0x8D848E89, 0x0000BFBF,0x00000000 // d_1 +//data4 0x18A66F8E, 0xD5394C36, 0x0000BF7C,0x00000000 // d_2 +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) +//data4 0x2168C234, 0xC90FDAA2, 0x00003FFE,0x00000000 // pi_by_4 +//data4 0x6EC6B45A, 0xA397E504, 0x00003FE7,0x00000000 // Inv_P_0 +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) +//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 +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) +//data4 0xC2B0FE52, 0xD56232EF, 0x00003FD2 // QQ_8 +//data4 0x2B48DCA6, 0xC9C99ABA, 0x0000BFDA // QQ_7 +//data4 0x9C716658, 0x8F76C650, 0x00003FE2 // QQ_6 +//data4 0xFDA8D0FC, 0x93F27DBA, 0x0000BFE9 // QQ_5 +//data4 0xAAAAAAAA, 0xAAAAAAAA, 0x0000BFFC // 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 +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) +//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 +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) +//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 +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_ResultS = f9 +FR_ResultC = 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_r6 = f32 +FR_r7 = f33 +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_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_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_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_d_2 = f93 +FR_P_0 = f95 +FR_C_lo = f96 +FR_PP_1 = f97 +FR_PP_1_lo = f98 +FR_ArgPrime = f99 +FR_inexact = f100 + +FR_Neg_Two_to_M3 = f109 +FR_Two_to_M3 = f110 + +FR_poly_hiS = f66 +FR_poly_hiC = f112 + +FR_poly_loS = f67 +FR_poly_loC = f113 + +FR_polyS = f92 +FR_polyC = f114 + +FR_cS = FR_c +FR_cC = f115 + +FR_corrS = f91 +FR_corrC = f116 + +FR_U_hiC = f117 +FR_U_loC = f118 + +FR_VS = f75 +FR_VC = f119 + +FR_FirstS = f120 +FR_FirstC = f121 + +FR_U_hiS = FR_U_hi +FR_U_loS = FR_U_lo + +FR_Tmp = f94 + + + + +sincos_pResSin = r34 +sincos_pResCos = r35 + +GR_exp_m2_to_m3= r36 +GR_N_Inc = r37 +GR_Cis = 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_N_SignS = r45 +GR_N_SignC = r46 +GR_N_SinCos = r47 + +GR_sig_inv_pi = r48 +GR_rshf_2to64 = r49 +GR_exp_2tom64 = r50 +GR_rshf = r51 +GR_ad_p = r52 +GR_ad_d = r53 +GR_ad_pp = r54 +GR_ad_qq = r55 +GR_ad_c = r56 +GR_ad_s = r57 +GR_ad_ce = r58 +GR_ad_se = r59 +GR_ad_m14 = r60 +GR_ad_s1 = r61 + +// For unwind support +GR_SAVE_B0 = r39 +GR_SAVE_GP = r40 +GR_SAVE_PFS = r41 + + +.section .text + +GLOBAL_IEEE754_ENTRY(sincosl) +{ .mlx ///////////////////////////// 1 ///////////////// + alloc r32 = ar.pfs,3,27,2,0 + movl GR_sig_inv_pi = 0xa2f9836e4e44152a // significand of 1/pi +} +{ .mlx + mov GR_N_Inc = 0x0 + movl GR_rshf_2to64 = 0x47e8000000000000 // 1.1000 2^(63+64) +};; + +{ .mfi ///////////////////////////// 2 ///////////////// + 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 + mov GR_Cis = 0x0 + fnorm.s1 FR_norm_x = FR_Input_X // Normalize x + br.cond.sptk _COMMON_SINCOSL +};; +GLOBAL_IEEE754_END(sincosl) + +GLOBAL_LIBM_ENTRY(__libm_sincosl) +{ .mlx ///////////////////////////// 1 ///////////////// + alloc r32 = ar.pfs,3,27,2,0 + movl GR_sig_inv_pi = 0xa2f9836e4e44152a // significand of 1/pi +} +{ .mlx + mov GR_N_Inc = 0x0 + movl GR_rshf_2to64 = 0x47e8000000000000 // 1.1000 2^(63+64) +};; + +{ .mfi ///////////////////////////// 2 ///////////////// + 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 + mov GR_Cis = 0x1 + fnorm.s1 FR_norm_x = FR_Input_X // Normalize x + nop.b 0 +};; + +_COMMON_SINCOSL: +{ .mfi ///////////////////////////// 3 ///////////////// + setf.sig FR_inv_pi_2to63 = GR_sig_inv_pi // Form 1/pi * 2^63 + nop.f 0 + 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 ///////////////////////////// 4 ///////////////// + 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 0 +};; + +{ .mfi ///////////////////////////// 5 ///////////////// + 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 0 +} +{ .mib + mov GR_exp_mask = 0x1ffff // Exponent mask + nop.i 0 +(p6) br.cond.spnt SINCOSL_SPECIAL // Branch if x natval, nan, inf +};; + +{ .mfi ///////////////////////////// 6 ///////////////// + 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_COMMON2: +{ .mfi ///////////////////////////// 7 ///////////////// + 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 ///////////////////////////// 8 ///////////////// + 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 0 + cmp.ge p10,p0 = GR_exp_x, GR_exp_2_to_63 // Is |x| >= 2^63 +};; + +{ .mfi ///////////////////////////// 9 ///////////////// + 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 0 + cmp.ge p7,p0 = GR_exp_x, GR_exp_2_to_24 // Is |x| >= 2^24 +};; + +{ .mfi ///////////////////////////// 10 ///////////////// + ldfe FR_P_1 = [GR_ad_p], 16 // Load P_1 for pi/4 <= |x| < 2^63 + nop.f 0 + 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 0 + nop.i 0 +};; + +{ .mfi ///////////////////////////// 11 ///////////////// + ldfe FR_P_2 = [GR_ad_p], 16 // Load P_2 for pi/4 <= |x| < 2^63 + nop.f 0 + 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 0 + nop.i 0 +};; + +{ .mfi ///////////////////////////// 12 ///////////////// + 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 ///////////////////////////// 13 ///////////////// + 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 ///////////////////////////// 14 ///////////////// + 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 ///////////////////////////// 15 ///////////////// + 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 0 +};; + +// Here if 0 < |x| < 2^24 +{ .mfi ///////////////////////////// 17 ///////////////// + 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 0 +} +{ .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 0 +};; + +{ .mmi ///////////////////////////// 18 ///////////////// + 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 0 +};; + +// +// 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 ///////////////////////////// 19 ///////////////// +(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 = 0x0 // N_IncS 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 ///////////////////////////// 20 ///////////////// +(p7) getf.sig GR_N_Inc = FR_N_float_signif + nop.m 0 +(p6) cmp.lt.unc p8,p0 = GR_exp_x, GR_exp_2_to_m3 // Is |x| < 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 ///////////////////////////// 21 ///////////////// + nop.m 0 +(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 + nop.i 0 +} +{ .mfi + nop.m 0 + fma.s1 FR_w = FR_N_float, FR_P_2, f0 // w = N * P_2 + nop.i 0 +};; + +{ .mfi + nop.m 0 + fms.s1 FR_r = FR_s, f1, FR_w // r = s - w, assume |s| >= 2^-33 + nop.i 0 +};; + +{ .mfi + nop.m 0 + fcmp.lt.s1 p7, p6 = FR_s, FR_Two_to_M33 + nop.i 0 +};; + +{ .mfi + nop.m 0 +(p7) fcmp.gt.s1 p7, p6 = FR_s, FR_Neg_Two_to_M33 // p6 if |s| >= 2^-33, else p7 + nop.i 0 +};; + +{ .mfi + nop.m 0 + fms.s1 FR_c = FR_s, f1, FR_r // c = s - r, for |s| >= 2^-33 + nop.i 0 +} +{ .mfi + nop.m 0 + fma.s1 FR_rsq = FR_r, FR_r, f0 // rsq = r * r, for |s| >= 2^-33 + nop.i 0 +};; + +{ .mfi + nop.m 0 +(p7) fma.s1 FR_w = FR_N_float, FR_P_3, f0 + nop.i 0 +};; + +{ .mmf + ldfe FR_C_1 = [GR_ad_pp], 16 // Load C_1 if i_1=0 + 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 0 +(p6) fcmp.lt.unc.s1 p8, p13 = FR_r, FR_Two_to_M3 // If big s, test r with 2^-3 + nop.i 0 +};; + +{ .mfi + nop.m 0 +(p7) fma.s1 FR_U_1 = FR_N_float, FR_P_2, FR_w + nop.i 0 +};; + +// +// 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 0 +(p8) fcmp.gt.s1 p8, p13 = FR_r, FR_Neg_Two_to_M3 // If big s, p8 if |r| < 2^-3 + nop.i 0 +};; + +{ .mfi + nop.m 0 + fma.s1 FR_polyS = FR_rsq, FR_PP_8, FR_PP_7 // poly = rsq*PP_8+PP_7 + nop.i 0 +} +{ .mfi + nop.m 0 + fma.s1 FR_polyC = FR_rsq, FR_QQ_8, FR_QQ_7 // poly = rsq*QQ_8+QQ_7 + nop.i 0 +};; + +{ .mfi + nop.m 0 +(p7) fms.s1 FR_r = FR_s, f1, FR_U_1 + nop.i 0 +};; + +{ .mfi + nop.m 0 +(p6) fma.s1 FR_r_cubed = FR_r, FR_rsq, f0 // rcubed = r * rsq + nop.i 0 +};; + +{ .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 0 +(p6) fms.s1 FR_c = FR_c, f1, FR_w + nop.i 0 +} +{ .mbb + nop.m 0 +(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 + and GR_N_SinCos = 0x1, GR_N_Inc + fms.s1 FR_U_2 = FR_N_float, FR_P_2, FR_U_1 + tbit.z p8,p12 = GR_N_Inc, 0 +};; + + +// +// 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 + fma.s1 FR_polyC = f0, f1, FR_Neg_Two_to_M67 + sub GR_N_SignS = GR_N_Inc, GR_N_SinCos +} +{ .mfi + add GR_N_SignC = GR_N_Inc, GR_N_SinCos + nop.f 0 + nop.i 0 +};; + +{ .mfi + nop.m 0 + fms.s1 FR_s = FR_s, f1, FR_r +(p8) tbit.z.unc p10,p11 = GR_N_SignC, 1 +} +{ .mfi + nop.m 0 + fma.s1 FR_rsq = FR_r, FR_r, f0 + nop.i 0 +};; + +{ .mfi + nop.m 0 + fma.s1 FR_U_2 = FR_U_2, f1, FR_w +(p8) tbit.z.unc p8,p9 = GR_N_SignS, 1 +};; + +{ .mfi + nop.m 0 + fmerge.se FR_FirstS = FR_r, FR_r +(p12) tbit.z.unc p14,p15 = GR_N_SignC, 1 +} +{ .mfi + nop.m 0 + fma.s1 FR_FirstC = f0, f1, f1 + nop.i 0 +};; + +{ .mfi + nop.m 0 + fms.s1 FR_c = FR_s, f1, FR_U_1 +(p12) tbit.z.unc p12,p13 = GR_N_SignS, 1 +};; + +{ .mfi + nop.m 0 + fma.s1 FR_r = FR_S_1, FR_r, f0 + nop.i 0 +};; + +{ .mfi + nop.m 0 + fma.s0 FR_S_1 = FR_S_1, FR_S_1, f0 + nop.i 0 +};; + +{ .mfi + nop.m 0 + fms.s1 FR_c = FR_c, f1, FR_U_2 + nop.i 0 +};; + +.pred.rel "mutex",p9,p15 +{ .mfi + nop.m 0 +(p9) fms.s0 FR_FirstS = f1, f0, FR_FirstS + nop.i 0 +} +{ .mfi + nop.m 0 +(p15) fms.s0 FR_FirstS = f1, f0, FR_FirstS + nop.i 0 +};; + +.pred.rel "mutex",p11,p13 +{ .mfi + nop.m 0 +(p11) fms.s0 FR_FirstC = f1, f0, FR_FirstC + nop.i 0 +} +{ .mfi + nop.m 0 +(p13) fms.s0 FR_FirstC = f1, f0, FR_FirstC + nop.i 0 +};; + +{ .mfi + nop.m 0 + fma.s1 FR_polyS = FR_r, FR_rsq, FR_c + nop.i 0 +};; + + +.pred.rel "mutex",p8,p9 +{ .mfi + nop.m 0 +(p8) fma.s0 FR_ResultS = FR_FirstS, f1, FR_polyS + nop.i 0 +} +{ .mfi + nop.m 0 +(p9) fms.s0 FR_ResultS = FR_FirstS, f1, FR_polyS + nop.i 0 +};; + +.pred.rel "mutex",p10,p11 +{ .mfi + nop.m 0 +(p10) fma.s0 FR_ResultC = FR_FirstC, f1, FR_polyC + nop.i 0 +} +{ .mfi + nop.m 0 +(p11) fms.s0 FR_ResultC = FR_FirstC, f1, FR_polyC + nop.i 0 +};; + + + +.pred.rel "mutex",p12,p13 +{ .mfi + nop.m 0 +(p12) fma.s0 FR_ResultS = FR_FirstC, f1, FR_polyC + nop.i 0 +} +{ .mfi + nop.m 0 +(p13) fms.s0 FR_ResultS = FR_FirstC, f1, FR_polyC + nop.i 0 +};; + +.pred.rel "mutex",p14,p15 +{ .mfi + nop.m 0 +(p14) fma.s0 FR_ResultC = FR_FirstS, f1, FR_polyS + nop.i 0 +} +{ .mfb + cmp.eq p10, p0 = 0x1, GR_Cis +(p15) fms.s0 FR_ResultC = FR_FirstS, f1, FR_polyS +(p10) br.ret.sptk b0 +};; + +{ .mmb // exit for sincosl + stfe [sincos_pResSin] = FR_ResultS + stfe [sincos_pResCos] = FR_ResultC + br.ret.sptk b0 +};; + + + + + + +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 // N_0 = Arg * Inv_P_0 + nop.i 0 +};; + +{ .mmi + ldfps FR_Two_to_M14, FR_Neg_Two_to_M14 = [GR_ad_m14] + nop.m 0 + nop.i 0 +};; + +{ .mfi + ldfe FR_d_2 = [GR_ad_p], 16 // Load d_2 for |x| >= 2^24 path + nop.f 0 + nop.i 0 +};; + +{ .mfi + nop.m 0 + fcvt.fx.s1 FR_N_0_fix = FR_N_0 // N_0_fix = integer part of N_0 + nop.i 0 +};; + +{ .mfi + nop.m 0 + fcvt.xf FR_N_0 = FR_N_0_fix // Make N_0 the integer part + nop.i 0 +};; + +{ .mfi + nop.m 0 + fnma.s1 FR_ArgPrime = FR_N_0, FR_P_0, FR_Input_X // Arg'=-N_0*P_0+Arg + nop.i 0 +} +{ .mfi + nop.m 0 + fma.s1 FR_w = FR_N_0, FR_d_1, f0 // w = N_0 * d_1 + nop.i 0 +};; + + +{ .mfi + nop.m 0 + fma.s1 FR_N_float = FR_ArgPrime, FR_Inv_pi_by_2, f0 // N = A' * 2/pi + nop.i 0 +};; + +{ .mfi + nop.m 0 + fcvt.fx.s1 FR_N_fix = FR_N_float // N_fix is the integer part + nop.i 0 +};; + +{ .mfi + nop.m 0 + fcvt.xf FR_N_float = FR_N_fix + nop.i 0 +};; + +{ .mfi + getf.sig GR_N_Inc = FR_N_fix // N is the integer part of + // the reduced-reduced argument + nop.f 0 + nop.i 0 +};; + +{ .mfi + nop.m 0 + fnma.s1 FR_s = FR_N_float, FR_P_1, FR_ArgPrime // s = -N*P_1 + Arg' + nop.i 0 +} +{ .mfi + nop.m 0 + fnma.s1 FR_w = FR_N_float, FR_P_2, FR_w // w = -N*P_2 + w + nop.i 0 +};; + +// +// For |s| > 2**(-14) r = S + w (r complete) +// Else U_hi = N_0 * d_1 +// +{ .mfi + nop.m 0 + fcmp.lt.unc.s1 p9, p8 = FR_s, FR_Two_to_M14 + nop.i 0 +};; + +{ .mfi + nop.m 0 +(p9) fcmp.gt.s1 p9, p8 = FR_s, FR_Neg_Two_to_M14 // p9 if |s| < 2^-14 + nop.i 0 +};; + +// +// Either S <= -2**(-14) or S >= 2**(-14) +// or -2**(-14) < s < 2**(-14) +// +{ .mfi + nop.m 0 +(p9) fma.s1 FR_V_hi = FR_N_float, FR_P_2, f0 + nop.i 0 +} +{ .mfi + nop.m 0 +(p9) fma.s1 FR_U_hi = FR_N_0, FR_d_1, f0 + nop.i 0 +};; + +{ .mfi + nop.m 0 +(p8) fma.s1 FR_r = FR_s, f1, FR_w + nop.i 0 +} +{ .mfi + nop.m 0 +(p9) fma.s1 FR_w = FR_N_float, FR_P_3, f0 + nop.i 0 +};; + +// +// We need abs of both U_hi and V_hi - don't +// worry about switched sign of V_hi. +// +// 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. +// +{ .mfi + nop.m 0 +(p9) fms.s1 FR_A = FR_U_hi, f1, FR_V_hi + nop.i 0 +} +{ .mfi + nop.m 0 +(p9) fnma.s1 FR_V_lo = FR_N_float, FR_P_2, FR_V_hi + nop.i 0 +};; + +{ .mfi + nop.m 0 +(p9) fmerge.s FR_V_hiabs = f0, FR_V_hi + nop.i 0 +} +{ .mfi + nop.m 0 +(p9) fms.s1 FR_U_lo = FR_N_0, FR_d_1, FR_U_hi // For small s: U_lo=N_0*d_1-U_hi + nop.i 0 +};; + +// +// 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. +// +{ .mfi + nop.m 0 +(p9) fmerge.s FR_U_hiabs = f0, FR_U_hi + nop.i 0 +} +{ .mfi + nop.m 0 +(p8) fms.s1 FR_c = FR_s, f1, FR_r // For big s: c = S - r + nop.i 0 +};; + +// +// 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. +// +{ .mfi + nop.m 0 +(p8) fcmp.lt.unc.s1 p12, p13 = FR_r, FR_Two_to_M3 + nop.i 0 +};; + +{ .mfi + nop.m 0 +(p12) fcmp.gt.s1 p12, p13 = FR_r, FR_Neg_Two_to_M3 + nop.i 0 +};; + +{ .mfi + nop.m 0 +(p8) fma.s1 FR_c = FR_c, f1, FR_w + nop.i 0 +} +{ .mfb + nop.m 0 +(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 0 + nop.i 0 +(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 +// +// 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. +// +{ .mfi + and GR_N_SinCos = 0x1, GR_N_Inc + fcmp.ge.unc.s1 p6, p7 = FR_U_hiabs, FR_V_hiabs + tbit.z p8,p12 = GR_N_Inc, 0 +} +{ .mfi + nop.m 0 + fma.s1 FR_t = FR_U_lo, f1, FR_V_lo // C_hi = S + A + nop.i 0 +};; + +{ .mfi + sub GR_N_SignS = GR_N_Inc, GR_N_SinCos +(p6) fms.s1 FR_a = FR_U_hi, f1, FR_A + add GR_N_SignC = GR_N_Inc, GR_N_SinCos +} +{ .mfi + nop.m 0 +(p7) fma.s1 FR_a = FR_V_hi, f1, FR_A + nop.i 0 +};; + +{ .mmf + ldfe FR_C_1 = [GR_ad_c], 16 + ldfe FR_S_1 = [GR_ad_s], 16 + fma.s1 FR_C_hi = FR_s, f1, FR_A +};; + +{ .mmi + ldfe FR_C_2 = [GR_ad_c], 64 + ldfe FR_S_2 = [GR_ad_s], 64 +(p8) tbit.z.unc p10,p11 = GR_N_SignC, 1 +};; + +// +// 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. +// +// For larger u than v: a = U_hi - A +// Else a = V_hi - A (do an add to account for missing (-) on V_hi +// +{ .mfi + nop.m 0 + fma.s1 FR_t = FR_t, f1, FR_w // t = t + w +(p8) tbit.z.unc p8,p9 = GR_N_SignS, 1 +} +{ .mfi + nop.m 0 +(p6) fms.s1 FR_a = FR_a, f1, FR_V_hi + nop.i 0 +};; + +// +// 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. +// +{ .mfi + nop.m 0 + fms.s1 FR_C_lo = FR_s, f1, FR_C_hi +(p12) tbit.z.unc p14,p15 = GR_N_SignC, 1 +} +{ .mfi + nop.m 0 +(p7) fms.s1 FR_a = FR_U_hi, f1, FR_a + nop.i 0 +};; + +{ .mfi + nop.m 0 + fma.s1 FR_C_lo = FR_C_lo, f1, FR_A // C_lo = (S - C_hi) + A +(p12) tbit.z.unc p12,p13 = GR_N_SignS, 1 +} +{ .mfi + nop.m 0 + fma.s1 FR_t = FR_t, f1, FR_a // t = t + a + nop.i 0 +};; + +{ .mfi + nop.m 0 + fma.s1 FR_r = FR_C_hi, f1, FR_C_lo + nop.i 0 +};; + +{ .mfi + nop.m 0 + fma.s1 FR_C_lo = FR_C_lo, f1, FR_t // C_lo = C_lo + t + nop.i 0 +};; + + +{ .mfi + nop.m 0 + fma.s1 FR_rsq = FR_r, FR_r, f0 + nop.i 0 +} +{ .mfi + nop.m 0 + fms.s1 FR_c = FR_C_hi, f1, FR_r + nop.i 0 +};; + +{ .mfi + nop.m 0 + fma.s1 FR_FirstS = f0, f1, FR_r + nop.i 0 +} +{ .mfi + nop.m 0 + fma.s1 FR_FirstC = f0, f1, f1 + nop.i 0 +};; + +{ .mfi + nop.m 0 + fma.s1 FR_polyS = FR_rsq, FR_S_2, FR_S_1 + nop.i 0 +} +{ .mfi + nop.m 0 + fma.s1 FR_polyC = FR_rsq, FR_C_2, FR_C_1 + nop.i 0 +};; + +{ .mfi + nop.m 0 + fma.s1 FR_r_cubed = FR_rsq, FR_r, f0 + nop.i 0 +} +{ .mfi + nop.m 0 + fma.s1 FR_c = FR_c, f1, FR_C_lo + nop.i 0 +};; + +.pred.rel "mutex",p9,p15 +{ .mfi + nop.m 0 +(p9) fms.s0 FR_FirstS = f1, f0, FR_FirstS + nop.i 0 +} +{ .mfi + nop.m 0 +(p15) fms.s0 FR_FirstS = f1, f0, FR_FirstS + nop.i 0 +};; + +.pred.rel "mutex",p11,p13 +{ .mfi + nop.m 0 +(p11) fms.s0 FR_FirstC = f1, f0, FR_FirstC + nop.i 0 +} +{ .mfi + nop.m 0 +(p13) fms.s0 FR_FirstC = f1, f0, FR_FirstC + nop.i 0 +};; + + +{ .mfi + nop.m 0 + fma.s1 FR_polyS = FR_r_cubed, FR_polyS, FR_c + nop.i 0 +} +{ .mfi + nop.m 0 + fma.s1 FR_polyC = FR_rsq, FR_polyC, f0 + nop.i 0 +};; + + + +.pred.rel "mutex",p8,p9 +{ .mfi + nop.m 0 +(p8) fma.s0 FR_ResultS = FR_FirstS, f1, FR_polyS + nop.i 0 +} +{ .mfi + nop.m 0 +(p9) fms.s0 FR_ResultS = FR_FirstS, f1, FR_polyS + nop.i 0 +};; + +.pred.rel "mutex",p10,p11 +{ .mfi + nop.m 0 +(p10) fma.s0 FR_ResultC = FR_FirstC, f1, FR_polyC + nop.i 0 +} +{ .mfi + nop.m 0 +(p11) fms.s0 FR_ResultC = FR_FirstC, f1, FR_polyC + nop.i 0 +};; + + + +.pred.rel "mutex",p12,p13 +{ .mfi + nop.m 0 +(p12) fma.s0 FR_ResultS = FR_FirstC, f1, FR_polyC + nop.i 0 +} +{ .mfi + nop.m 0 +(p13) fms.s0 FR_ResultS = FR_FirstC, f1, FR_polyC + nop.i 0 +};; + +.pred.rel "mutex",p14,p15 +{ .mfi + nop.m 0 +(p14) fma.s0 FR_ResultC = FR_FirstS, f1, FR_polyS + nop.i 0 +} +{ .mfb + cmp.eq p10, p0 = 0x1, GR_Cis +(p15) fms.s0 FR_ResultC = FR_FirstS, f1, FR_polyS +(p10) br.ret.sptk b0 +};; + + +{ .mmb // exit for sincosl + stfe [sincos_pResSin] = FR_ResultS + stfe [sincos_pResCos] = FR_ResultC + br.ret.sptk b0 +};; + + + +SINCOSL_SMALL_R: +// +// Here if |r| < 2^-3 +// +// Enter with r, c, and N_Inc computed +// +{ .mfi + nop.m 0 + fma.s1 FR_rsq = FR_r, FR_r, f0 // rsq = r * r + nop.i 0 +};; + +{ .mmi + ldfe FR_S_5 = [GR_ad_se], -16 // Load S_5 + ldfe FR_C_5 = [GR_ad_ce], -16 // Load C_5 + nop.i 0 +};; + +{ .mmi + ldfe FR_S_4 = [GR_ad_se], -16 // Load S_4 + ldfe FR_C_4 = [GR_ad_ce], -16 // Load C_4 + nop.i 0 +};; + +SINCOSL_SMALL_R_0: +// Entry point for 2^-3 < |x| < pi/4 +SINCOSL_SMALL_R_1: +// Entry point for pi/4 < |x| < 2^24 and |r| < 2^-3 +{ .mfi + ldfe FR_S_3 = [GR_ad_se], -16 // Load S_3 + fma.s1 FR_r6 = FR_rsq, FR_rsq, f0 // Z = rsq * rsq + tbit.z p7,p11 = GR_N_Inc, 0 +} +{ .mfi + ldfe FR_C_3 = [GR_ad_ce], -16 // Load C_3 + nop.f 0 + and GR_N_SinCos = 0x1, GR_N_Inc +};; + +{ .mfi + ldfe FR_S_2 = [GR_ad_se], -16 // Load S_2 + fnma.s1 FR_cC = FR_c, FR_r, f0 // c = -c * r + sub GR_N_SignS = GR_N_Inc, GR_N_SinCos +} +{ .mfi + ldfe FR_C_2 = [GR_ad_ce], -16 // Load C_2 + nop.f 0 + add GR_N_SignC = GR_N_Inc, GR_N_SinCos +};; + +{ .mmi + ldfe FR_S_1 = [GR_ad_se], -16 // Load S_1 + ldfe FR_C_1 = [GR_ad_ce], -16 // Load C_1 +(p7) tbit.z.unc p9,p10 = GR_N_SignC, 1 +};; + +{ .mfi + nop.m 0 + fma.s1 FR_r7 = FR_r6, FR_r, f0 // Z = Z * r +(p7) tbit.z.unc p7,p8 = GR_N_SignS, 1 +};; + +{ .mfi + nop.m 0 + fma.s1 FR_poly_loS = FR_rsq, FR_S_5, FR_S_4 // poly_lo=rsq*S_5+S_4 +(p11) tbit.z.unc p13,p14 = GR_N_SignC, 1 +} +{ .mfi + nop.m 0 + fma.s1 FR_poly_loC = FR_rsq, FR_C_5, FR_C_4 // poly_lo=rsq*C_5+C_4 + nop.i 0 +};; + +{ .mfi + nop.m 0 + fma.s1 FR_poly_hiS = FR_rsq, FR_S_2, FR_S_1 // poly_hi=rsq*S_2+S_1 +(p11) tbit.z.unc p11,p12 = GR_N_SignS, 1 +} +{ .mfi + nop.m 0 + fma.s1 FR_poly_hiC = FR_rsq, FR_C_2, FR_C_1 // poly_hi=rsq*C_2+C_1 + nop.i 0 +};; + +{ .mfi + nop.m 0 + fma.s0 FR_FirstS = FR_r, f1, f0 + nop.i 0 +} +{ .mfi + nop.m 0 + fma.s0 FR_FirstC = f1, f1, f0 + nop.i 0 +};; + + +{ .mfi + nop.m 0 + fma.s1 FR_r6 = FR_r6, FR_rsq, f0 + nop.i 0 +} +{ .mfi + nop.m 0 + fma.s1 FR_r7 = FR_r7, FR_rsq, f0 + nop.i 0 +};; + +{ .mfi + nop.m 0 + fma.s1 FR_poly_loS = FR_rsq, FR_poly_loS, FR_S_3 // p_lo=p_lo*rsq+S_3 + nop.i 0 +} +{ .mfi + nop.m 0 + fma.s1 FR_poly_loC = FR_rsq, FR_poly_loC, FR_C_3 // p_lo=p_lo*rsq+C_3 + nop.i 0 +};; + +{ .mfi + nop.m 0 + fma.s0 FR_inexact = FR_S_4, FR_S_4, f0 // Dummy op to set inexact + nop.i 0 +};; + +{ .mfi + nop.m 0 + fma.s1 FR_poly_hiS = FR_poly_hiS, FR_rsq, f0 // p_hi=p_hi*rsq + nop.i 0 +} +{ .mfi + nop.m 0 + fma.s1 FR_poly_hiC = FR_poly_hiC, FR_rsq, f0 // p_hi=p_hi*rsq + nop.i 0 +};; + +.pred.rel "mutex",p8,p14 +{ .mfi + nop.m 0 +(p8) fms.s0 FR_FirstS = f1, f0, FR_FirstS + nop.i 0 +} +{ .mfi + nop.m 0 +(p14) fms.s0 FR_FirstS = f1, f0, FR_FirstS + nop.i 0 +};; + +.pred.rel "mutex",p10,p12 +{ .mfi + nop.m 0 +(p10) fms.s0 FR_FirstC = f1, f0, FR_FirstC + nop.i 0 +} +{ .mfi + nop.m 0 +(p12) fms.s0 FR_FirstC = f1, f0, FR_FirstC + nop.i 0 +};; + +{ .mfi + nop.m 0 + fma.s1 FR_polyS = FR_r7, FR_poly_loS, FR_cS // poly=Z*poly_lo+c + nop.i 0 +} +{ .mfi + nop.m 0 + fma.s1 FR_polyC = FR_r6, FR_poly_loC, FR_cC // poly=Z*poly_lo+c + nop.i 0 +};; + +{ .mfi + nop.m 0 + fma.s1 FR_poly_hiS = FR_r, FR_poly_hiS, f0 // p_hi=r*p_hi + nop.i 0 +};; + + +{ .mfi + nop.m 0 + fma.s1 FR_polyS = FR_polyS, f1, FR_poly_hiS + nop.i 0 +} +{ .mfi + nop.m 0 + fma.s1 FR_polyC = FR_polyC, f1, FR_poly_hiC + nop.i 0 +};; + +.pred.rel "mutex",p7,p8 +{ .mfi + nop.m 0 +(p7) fma.s0 FR_ResultS = FR_FirstS, f1, FR_polyS + nop.i 0 +} +{ .mfi + nop.m 0 +(p8) fms.s0 FR_ResultS = FR_FirstS, f1, FR_polyS + nop.i 0 +};; + +.pred.rel "mutex",p9,p10 +{ .mfi + nop.m 0 +(p9) fma.s0 FR_ResultC = FR_FirstC, f1, FR_polyC + nop.i 0 +} +{ .mfi + nop.m 0 +(p10) fms.s0 FR_ResultC = FR_FirstC, f1, FR_polyC + nop.i 0 +};; + +.pred.rel "mutex",p11,p12 +{ .mfi + nop.m 0 +(p11) fma.s0 FR_ResultS = FR_FirstC, f1, FR_polyC + nop.i 0 +} +{ .mfi + nop.m 0 +(p12) fms.s0 FR_ResultS = FR_FirstC, f1, FR_polyC + nop.i 0 +};; + +.pred.rel "mutex",p13,p14 +{ .mfi + nop.m 0 +(p13) fma.s0 FR_ResultC = FR_FirstS, f1, FR_polyS + nop.i 0 +} +{ .mfb + cmp.eq p15, p0 = 0x1, GR_Cis +(p14) fms.s0 FR_ResultC = FR_FirstS, f1, FR_polyS +(p15) br.ret.sptk b0 +};; + + +{ .mmb // exit for sincosl + stfe [sincos_pResSin] = FR_ResultS + stfe [sincos_pResCos] = FR_ResultC + br.ret.sptk b0 +};; + + + + + + +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 + nop.i 0 +} +{ .mfi + ldfe FR_QQ_6 = [GR_ad_qq], 16 // Load QQ_6 + nop.f 0 + nop.i 0 +};; + +{ .mmi + ldfe FR_PP_5 = [GR_ad_pp], 16 // Load PP_5 + ldfe FR_QQ_5 = [GR_ad_qq], 16 // Load QQ_5 + nop.i 0 +};; + + + +SINCOSL_NORMAL_R_0: +// Entry for 2^-3 < |x| < pi/4 +.pred.rel "mutex",p9,p10 +{ .mmf + ldfe FR_C_1 = [GR_ad_pp], 16 // Load C_1 + ldfe FR_S_1 = [GR_ad_qq], 16 // Load S_1 + frcpa.s1 FR_r_hi, p6 = f1, FR_r // r_hi = frcpa(r) +};; + +{ .mfi + nop.m 0 + fma.s1 FR_polyS = FR_rsq, FR_PP_8, FR_PP_7 // poly = rsq*PP_8+PP_7 + nop.i 0 +} +{ .mfi + nop.m 0 + fma.s1 FR_polyC = FR_rsq, FR_QQ_8, FR_QQ_7 // poly = rsq*QQ_8+QQ_7 + nop.i 0 +};; + +{ .mfi + nop.m 0 + fma.s1 FR_r_cubed = FR_r, FR_rsq, f0 // rcubed = r * rsq + nop.i 0 +};; + + +SINCOSL_NORMAL_R_1: +// Entry for pi/4 <= |x| < 2^24 +.pred.rel "mutex",p9,p10 +{ .mmf + ldfe FR_PP_1 = [GR_ad_pp], 16 // Load PP_1_hi + ldfe FR_QQ_1 = [GR_ad_qq], 16 // Load QQ_1 + frcpa.s1 FR_r_hi, p6 = f1, FR_r_hi // r_hi = frpca(frcpa(r)) +};; + +{ .mfi + ldfe FR_PP_4 = [GR_ad_pp], 16 // Load PP_4 + fma.s1 FR_polyS = FR_rsq, FR_polyS, FR_PP_6 // poly = rsq*poly+PP_6 + and GR_N_SinCos = 0x1, GR_N_Inc +} +{ .mfi + ldfe FR_QQ_4 = [GR_ad_qq], 16 // Load QQ_4 + fma.s1 FR_polyC = FR_rsq, FR_polyC, FR_QQ_6 // poly = rsq*poly+QQ_6 + nop.i 0 +};; + +{ .mfi + nop.m 0 + fma.s1 FR_corrS = FR_C_1, FR_rsq, f0 // corr = C_1 * rsq + sub GR_N_SignS = GR_N_Inc, GR_N_SinCos +} +{ .mfi + nop.m 0 + fma.s1 FR_corrC = FR_S_1, FR_r_cubed, FR_r // corr = S_1 * r^3 + r + add GR_N_SignC = GR_N_Inc, GR_N_SinCos +};; + +{ .mfi + ldfe FR_PP_3 = [GR_ad_pp], 16 // Load PP_3 + fma.s1 FR_r_hi_sq = FR_r_hi, FR_r_hi, f0 // r_hi_sq = r_hi * r_hi + tbit.z p7,p11 = GR_N_Inc, 0 +} +{ .mfi + ldfe FR_QQ_3 = [GR_ad_qq], 16 // Load QQ_3 + fms.s1 FR_r_lo = FR_r, f1, FR_r_hi // r_lo = r - r_hi + nop.i 0 +};; + +{ .mfi + ldfe FR_PP_2 = [GR_ad_pp], 16 // Load PP_2 + fma.s1 FR_polyS = FR_rsq, FR_polyS, FR_PP_5 // poly = rsq*poly+PP_5 +(p7) tbit.z.unc p9,p10 = GR_N_SignC, 1 +} +{ .mfi + ldfe FR_QQ_2 = [GR_ad_qq], 16 // Load QQ_2 + fma.s1 FR_polyC = FR_rsq, FR_polyC, FR_QQ_5 // poly = rsq*poly+QQ_5 + nop.i 0 +};; + +{ .mfi + ldfe FR_PP_1_lo = [GR_ad_pp], 16 // Load PP_1_lo + fma.s1 FR_corrS = FR_corrS, FR_c, FR_c // corr = corr * c + c +(p7) tbit.z.unc p7,p8 = GR_N_SignS, 1 +} +{ .mfi + nop.m 0 + fnma.s1 FR_corrC = FR_corrC, FR_c, f0 // corr = -corr * c + nop.i 0 +};; + +{ .mfi + nop.m 0 + fma.s1 FR_U_loS = FR_r, FR_r_hi, FR_r_hi_sq // U_lo = r*r_hi+r_hi_sq +(p11) tbit.z.unc p13,p14 = GR_N_SignC, 1 +} +{ .mfi + nop.m 0 + fma.s1 FR_U_loC = FR_r_hi, f1, FR_r // U_lo = r_hi + r + nop.i 0 +};; + +{ .mfi + nop.m 0 + fma.s1 FR_U_hiS = FR_r_hi, FR_r_hi_sq, f0 // U_hi = r_hi*r_hi_sq +(p11) tbit.z.unc p11,p12 = GR_N_SignS, 1 +} +{ .mfi + nop.m 0 + fma.s1 FR_U_hiC = FR_QQ_1, FR_r_hi_sq, f1 // U_hi = QQ_1*r_hi_sq+1 + nop.i 0 +};; + +{ .mfi + nop.m 0 + fma.s1 FR_polyS = FR_rsq, FR_polyS, FR_PP_4 // poly = poly*rsq+PP_4 + nop.i 0 +} +{ .mfi + nop.m 0 + fma.s1 FR_polyC = FR_rsq, FR_polyC, FR_QQ_4 // poly = poly*rsq+QQ_4 + nop.i 0 +};; + +{ .mfi + nop.m 0 + fma.s1 FR_U_loS = FR_r, FR_r, FR_U_loS // U_lo = r * r + U_lo + nop.i 0 +} +{ .mfi + nop.m 0 + fma.s1 FR_U_loC = FR_r_lo, FR_U_loC, f0 // U_lo = r_lo * U_lo + nop.i 0 +};; + +{ .mfi + nop.m 0 + fma.s1 FR_U_hiS = FR_PP_1, FR_U_hiS, f0 // U_hi = PP_1 * U_hi + nop.i 0 +};; + +{ .mfi + nop.m 0 + fma.s1 FR_polyS = FR_rsq, FR_polyS, FR_PP_3 // poly = poly*rsq+PP_3 + nop.i 0 +} +{ .mfi + nop.m 0 + fma.s1 FR_polyC = FR_rsq, FR_polyC, FR_QQ_3 // poly = poly*rsq+QQ_3 + nop.i 0 +};; + +{ .mfi + nop.m 0 + fma.s1 FR_U_loS = FR_r_lo, FR_U_loS, f0 // U_lo = r_lo * U_lo + nop.i 0 +} +{ .mfi + nop.m 0 + fma.s1 FR_U_loC = FR_QQ_1,FR_U_loC, f0 // U_lo = QQ_1 * U_lo + nop.i 0 +};; + +{ .mfi + nop.m 0 + fma.s1 FR_U_hiS = FR_r, f1, FR_U_hiS // U_hi = r + U_hi + nop.i 0 +};; + +{ .mfi + nop.m 0 + fma.s1 FR_polyS = FR_rsq, FR_polyS, FR_PP_2 // poly = poly*rsq+PP_2 + nop.i 0 +} +{ .mfi + nop.m 0 + fma.s1 FR_polyC = FR_rsq, FR_polyC, FR_QQ_2 // poly = poly*rsq+QQ_2 + nop.i 0 +};; + +{ .mfi + nop.m 0 + fma.s1 FR_U_loS = FR_PP_1, FR_U_loS, f0 // U_lo = PP_1 * U_lo + nop.i 0 +};; + +{ .mfi + nop.m 0 + fma.s1 FR_polyS = FR_rsq, FR_polyS, FR_PP_1_lo // poly =poly*rsq+PP1lo + nop.i 0 +} +{ .mfi + nop.m 0 + fma.s1 FR_polyC = FR_rsq, FR_polyC, f0 // poly = poly*rsq + nop.i 0 +};; + + +.pred.rel "mutex",p8,p14 +{ .mfi + nop.m 0 +(p8) fms.s0 FR_U_hiS = f1, f0, FR_U_hiS + nop.i 0 +} +{ .mfi + nop.m 0 +(p14) fms.s0 FR_U_hiS = f1, f0, FR_U_hiS + nop.i 0 +};; + +.pred.rel "mutex",p10,p12 +{ .mfi + nop.m 0 +(p10) fms.s0 FR_U_hiC = f1, f0, FR_U_hiC + nop.i 0 +} +{ .mfi + nop.m 0 +(p12) fms.s0 FR_U_hiC = f1, f0, FR_U_hiC + nop.i 0 +};; + + +{ .mfi + nop.m 0 + fma.s1 FR_VS = FR_U_loS, f1, FR_corrS // V = U_lo + corr + nop.i 0 +} +{ .mfi + nop.m 0 + fma.s1 FR_VC = FR_U_loC, f1, FR_corrC // V = U_lo + corr + nop.i 0 +};; + +{ .mfi + nop.m 0 + fma.s0 FR_inexact = FR_PP_5, FR_PP_4, f0 // Dummy op to set inexact + nop.i 0 +};; + + +{ .mfi + nop.m 0 + fma.s1 FR_polyS = FR_r_cubed, FR_polyS, f0 // poly = poly*r^3 + nop.i 0 +} +{ .mfi + nop.m 0 + fma.s1 FR_polyC = FR_rsq, FR_polyC, f0 // poly = poly*rsq + nop.i 0 +};; + + +{ .mfi + nop.m 0 + fma.s1 FR_VS = FR_polyS, f1, FR_VS // V = poly + V + nop.i 0 +} +{ .mfi + nop.m 0 + fma.s1 FR_VC = FR_polyC, f1, FR_VC // V = poly + V + nop.i 0 +};; + + + +.pred.rel "mutex",p7,p8 +{ .mfi + nop.m 0 +(p7) fma.s0 FR_ResultS = FR_U_hiS, f1, FR_VS + nop.i 0 +} +{ .mfi + nop.m 0 +(p8) fms.s0 FR_ResultS = FR_U_hiS, f1, FR_VS + nop.i 0 +};; + +.pred.rel "mutex",p9,p10 +{ .mfi + nop.m 0 +(p9) fma.s0 FR_ResultC = FR_U_hiC, f1, FR_VC + nop.i 0 +} +{ .mfi + nop.m 0 +(p10) fms.s0 FR_ResultC = FR_U_hiC, f1, FR_VC + nop.i 0 +};; + + + +.pred.rel "mutex",p11,p12 +{ .mfi + nop.m 0 +(p11) fma.s0 FR_ResultS = FR_U_hiC, f1, FR_VC + nop.i 0 +} +{ .mfi + nop.m 0 +(p12) fms.s0 FR_ResultS = FR_U_hiC, f1, FR_VC + nop.i 0 +};; + +.pred.rel "mutex",p13,p14 +{ .mfi + nop.m 0 +(p13) fma.s0 FR_ResultC = FR_U_hiS, f1, FR_VS + nop.i 0 +} +{ .mfb + cmp.eq p15, p0 = 0x1, GR_Cis +(p14) fms.s0 FR_ResultC = FR_U_hiS, f1, FR_VS +(p15) br.ret.sptk b0 +};; + +{ .mmb // exit for sincosl + stfe [sincos_pResSin] = FR_ResultS + stfe [sincos_pResCos] = FR_ResultC + br.ret.sptk b0 +};; + + + + + +SINCOSL_ZERO: + +{ .mfi + nop.m 0 + fmerge.s FR_ResultS = FR_Input_X, FR_Input_X // If sin, result = input + nop.i 0 +} +{ .mfb + cmp.eq p15, p0 = 0x1, GR_Cis + fma.s0 FR_ResultC = f1, f1, f0 // If cos, result=1.0 +(p15) br.ret.sptk b0 +};; + +{ .mmb // exit for sincosl + stfe [sincos_pResSin] = FR_ResultS + stfe [sincos_pResCos] = FR_ResultC + br.ret.sptk b0 +};; + + +SINCOSL_DENORMAL: +{ .mmb + getf.exp GR_signexp_x = FR_norm_x // Get sign and exponent of x + nop.m 999 + br.cond.sptk SINCOSL_COMMON2 // Return to common code +} +;; + + +SINCOSL_SPECIAL: +// +// Path for Arg = +/- QNaN, SNaN, Inf +// Invalid can be raised. SNaNs +// become QNaNs +// +{ .mfi + cmp.eq p15, p0 = 0x1, GR_Cis + fmpy.s0 FR_ResultS = FR_Input_X, f0 + nop.i 0 +} +{ .mfb + nop.m 0 + fmpy.s0 FR_ResultC = FR_Input_X, f0 +(p15) br.ret.sptk b0 +};; + +{ .mmb // exit for sincosl + stfe [sincos_pResSin] = FR_ResultS + stfe [sincos_pResCos] = FR_ResultC + br.ret.sptk b0 +};; + +GLOBAL_LIBM_END(__libm_sincosl) + + +// ******************************************************************* +// ******************************************************************* +// ******************************************************************* +// +// 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 r62-63. 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 + mov GR_N_Inc = 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 0 +(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# |