diff options
Diffstat (limited to 'sysdeps/ia64/fpu/s_cos.S')
| -rw-r--r-- | sysdeps/ia64/fpu/s_cos.S | 3482 |
1 files changed, 3099 insertions, 383 deletions
diff --git a/sysdeps/ia64/fpu/s_cos.S b/sysdeps/ia64/fpu/s_cos.S index 84c177abab..6540aec724 100644 --- a/sysdeps/ia64/fpu/s_cos.S +++ b/sysdeps/ia64/fpu/s_cos.S @@ -1,10 +1,10 @@ .file "sincos.s" - -// Copyright (c) 2000 - 2003, Intel Corporation +// Copyright (C) 2000, 2001, Intel Corporation // All rights reserved. // -// Contributed 2000 by the Intel Numerics Group, Intel Corporation +// Contributed 2/2/2000 by John Harrison, Ted Kubaska, Bob Norin, Shane Story, +// and Ping Tak Peter Tang of the Computational Software Lab, Intel Corporation. // // Redistribution and use in source and binary forms, with or without // modification, are permitted provided that the following conditions are @@ -20,7 +20,7 @@ // * 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 @@ -35,22 +35,17 @@ // // 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. +// http://developer.intel.com/opensource. // // History //============================================================== -// 02/02/00 Initial version -// 04/02/00 Unwind support added. -// 06/16/00 Updated tables to enforce symmetry -// 08/31/00 Saved 2 cycles in main path, and 9 in other paths. -// 09/20/00 The updated tables regressed to an old version, so reinstated them +// 2/02/00 Initial revision +// 4/02/00 Unwind support added. +// 6/16/00 Updated tables to enforce symmetry +// 8/31/00 Saved 2 cycles in main path, and 9 in other paths. +// 9/20/00 The updated tables regressed to an old version, so reinstated them // 10/18/00 Changed one table entry to ensure symmetry -// 01/03/01 Improved speed, fixed flag settings for small arguments. -// 02/18/02 Large arguments processing routine excluded -// 05/20/02 Cleaned up namespace and sf0 syntax -// 06/03/02 Insure inexact flag set for large arg result -// 09/05/02 Work range is widened by reduction strengthen (3 parts of Pi/16) -// 02/10/03 Reordered header: .section, .global, .proc, .align +// 1/03/01 Improved speed, fixed flag settings for small arguments. // API //============================================================== @@ -68,13 +63,9 @@ // nfloat = Round result to integer (round-to-nearest) // // r = x - nfloat * pi/2^k -// Do this as ((((x - nfloat * HIGH(pi/2^k))) - -// nfloat * LOW(pi/2^k)) - -// nfloat * LOWEST(pi/2^k) for increased accuracy. +// Do this as (x - nfloat * HIGH(pi/2^k)) - nfloat * LOW(pi/2^k) for increased accuracy. // pi/2^k is stored as two numbers that when added make pi/2^k. // pi/2^k = HIGH(pi/2^k) + LOW(pi/2^k) -// HIGH and LOW parts are rounded to zero values, -// and LOWEST is rounded to nearest one. // // x = (nfloat * pi/2^k) + r // r is small enough that we can use a polynomial approximation @@ -130,7 +121,7 @@ // // as follows // -// S[m] = Sin(Mpi/2^k) and C[m] = Cos(Mpi/2^k) +// Sm = Sin(Mpi/2^k) and Cm = Cos(Mpi/2^k) // rsq = r*r // // @@ -150,22 +141,23 @@ // // P = r + rcub * P // -// Answer = S[m] Cos(r) + [Cm] P +// Answer = Sm Cos(r) + Cm P // // Cos(r) = 1 + rsq Q // Cos(r) = 1 + r^2 Q // Cos(r) = 1 + r^2 (q1 + r^2q2 + r^4q3 + r^6q4) // Cos(r) = 1 + r^2q1 + r^4q2 + r^6q3 + r^8q4 + ... // -// S[m] Cos(r) = S[m](1 + rsq Q) -// S[m] Cos(r) = S[m] + Sm rsq Q -// S[m] Cos(r) = S[m] + s_rsq Q -// Q = S[m] + s_rsq Q +// Sm Cos(r) = Sm(1 + rsq Q) +// Sm Cos(r) = Sm + Sm rsq Q +// Sm Cos(r) = Sm + s_rsq Q +// Q = Sm + s_rsq Q // // Then, // -// Answer = Q + C[m] P +// Answer = Q + Cm P +#include "libm_support.h" // Registers used //============================================================== @@ -182,97 +174,99 @@ // Assembly macros //============================================================== -sincos_NORM_f8 = f9 -sincos_W = f10 -sincos_int_Nfloat = f11 -sincos_Nfloat = f12 +sind_NORM_f8 = f9 +sind_W = f10 +sind_int_Nfloat = f11 +sind_Nfloat = f12 -sincos_r = f13 -sincos_rsq = f14 -sincos_rcub = f15 -sincos_save_tmp = f15 +sind_r = f13 +sind_rsq = f14 +sind_rcub = f15 -sincos_Inv_Pi_by_16 = f32 -sincos_Pi_by_16_1 = f33 -sincos_Pi_by_16_2 = f34 +sind_Inv_Pi_by_16 = f32 +sind_Pi_by_16_hi = f33 +sind_Pi_by_16_lo = f34 -sincos_Inv_Pi_by_64 = f35 +sind_Inv_Pi_by_64 = f35 +sind_Pi_by_64_hi = f36 +sind_Pi_by_64_lo = f37 -sincos_Pi_by_16_3 = f36 +sind_Sm = f38 +sind_Cm = f39 -sincos_r_exact = f37 +sind_P1 = f40 +sind_Q1 = f41 +sind_P2 = f42 +sind_Q2 = f43 +sind_P3 = f44 +sind_Q3 = f45 +sind_P4 = f46 +sind_Q4 = f47 -sincos_Sm = f38 -sincos_Cm = f39 +sind_P_temp1 = f48 +sind_P_temp2 = f49 -sincos_P1 = f40 -sincos_Q1 = f41 -sincos_P2 = f42 -sincos_Q2 = f43 -sincos_P3 = f44 -sincos_Q3 = f45 -sincos_P4 = f46 -sincos_Q4 = f47 +sind_Q_temp1 = f50 +sind_Q_temp2 = f51 -sincos_P_temp1 = f48 -sincos_P_temp2 = f49 +sind_P = f52 +sind_Q = f53 -sincos_Q_temp1 = f50 -sincos_Q_temp2 = f51 +sind_srsq = f54 -sincos_P = f52 -sincos_Q = f53 +sind_SIG_INV_PI_BY_16_2TO61 = f55 +sind_RSHF_2TO61 = f56 +sind_RSHF = f57 +sind_2TOM61 = f58 +sind_NFLOAT = f59 +sind_W_2TO61_RSH = f60 -sincos_srsq = f54 - -sincos_SIG_INV_PI_BY_16_2TO61 = f55 -sincos_RSHF_2TO61 = f56 -sincos_RSHF = f57 -sincos_2TOM61 = f58 -sincos_NFLOAT = f59 -sincos_W_2TO61_RSH = f60 - -fp_tmp = f61 +fp_tmp = f61 ///////////////////////////////////////////////////////////// -sincos_AD_1 = r33 -sincos_AD_2 = r34 -sincos_exp_limit = r35 -sincos_r_signexp = r36 -sincos_AD_beta_table = r37 -sincos_r_sincos = r38 +sind_AD_1 = r33 +sind_AD_2 = r34 +sind_exp_limit = r35 +sind_r_signexp = r36 +sind_AD_beta_table = r37 +sind_r_sincos = r38 -sincos_r_exp = r39 -sincos_r_17_ones = r40 +sind_r_exp = r39 +sind_r_17_ones = r40 -sincos_GR_sig_inv_pi_by_16 = r14 -sincos_GR_rshf_2to61 = r15 -sincos_GR_rshf = r16 -sincos_GR_exp_2tom61 = r17 -sincos_GR_n = r18 -sincos_GR_m = r19 -sincos_GR_32m = r19 -sincos_GR_all_ones = r19 +sind_GR_sig_inv_pi_by_16 = r14 +sind_GR_rshf_2to61 = r15 +sind_GR_rshf = r16 +sind_GR_exp_2tom61 = r17 +sind_GR_n = r18 +sind_GR_m = r19 +sind_GR_32m = r19 -gr_tmp = r41 -GR_SAVE_PFS = r41 -GR_SAVE_B0 = r42 -GR_SAVE_GP = r43 +gr_tmp = r41 +GR_SAVE_PFS = r41 +GR_SAVE_B0 = r42 +GR_SAVE_GP = r43 -RODATA +#ifdef _LIBC +.rodata +#else +.data +#endif -// Pi/16 parts .align 16 -LOCAL_OBJECT_START(double_sincos_pi) - data8 0xC90FDAA22168C234, 0x00003FFC // pi/16 1st part - data8 0xC4C6628B80DC1CD1, 0x00003FBC // pi/16 2nd part - data8 0xA4093822299F31D0, 0x00003F7A // pi/16 3rd part -LOCAL_OBJECT_END(double_sincos_pi) - -// Coefficients for polynomials -LOCAL_OBJECT_START(double_sincos_pq_k4) +double_sind_pi: +ASM_TYPE_DIRECTIVE(double_sind_pi,@object) +// data8 0xA2F9836E4E44152A, 0x00004001 // 16/pi (significand loaded w/ setf) +// c90fdaa22168c234 + data8 0xC90FDAA22168C234, 0x00003FFC // pi/16 hi +// c4c6628b80dc1cd1 29024e088a + data8 0xC4C6628B80DC1CD1, 0x00003FBC // pi/16 lo +ASM_SIZE_DIRECTIVE(double_sind_pi) + +double_sind_pq_k4: +ASM_TYPE_DIRECTIVE(double_sind_pq_k4,@object) data8 0x3EC71C963717C63A // P4 data8 0x3EF9FFBA8F191AE6 // Q4 data8 0xBF2A01A00F4E11A8 // P3 @@ -281,119 +275,125 @@ LOCAL_OBJECT_START(double_sincos_pq_k4) data8 0x3FA555555554DD45 // Q2 data8 0xBFC5555555555555 // P1 data8 0xBFDFFFFFFFFFFFFC // Q1 -LOCAL_OBJECT_END(double_sincos_pq_k4) +ASM_SIZE_DIRECTIVE(double_sind_pq_k4) -// Sincos table (S[m], C[m]) -LOCAL_OBJECT_START(double_sin_cos_beta_k4) +double_sin_cos_beta_k4: +ASM_TYPE_DIRECTIVE(double_sin_cos_beta_k4,@object) data8 0x0000000000000000 , 0x00000000 // sin( 0 pi/16) S0 data8 0x8000000000000000 , 0x00003fff // cos( 0 pi/16) C0 -// + data8 0xc7c5c1e34d3055b3 , 0x00003ffc // sin( 1 pi/16) S1 data8 0xfb14be7fbae58157 , 0x00003ffe // cos( 1 pi/16) C1 -// + data8 0xc3ef1535754b168e , 0x00003ffd // sin( 2 pi/16) S2 data8 0xec835e79946a3146 , 0x00003ffe // cos( 2 pi/16) C2 -// + data8 0x8e39d9cd73464364 , 0x00003ffe // sin( 3 pi/16) S3 data8 0xd4db3148750d181a , 0x00003ffe // cos( 3 pi/16) C3 -// + data8 0xb504f333f9de6484 , 0x00003ffe // sin( 4 pi/16) S4 data8 0xb504f333f9de6484 , 0x00003ffe // cos( 4 pi/16) C4 -// -// + + data8 0xd4db3148750d181a , 0x00003ffe // sin( 5 pi/16) C3 data8 0x8e39d9cd73464364 , 0x00003ffe // cos( 5 pi/16) S3 -// + data8 0xec835e79946a3146 , 0x00003ffe // sin( 6 pi/16) C2 data8 0xc3ef1535754b168e , 0x00003ffd // cos( 6 pi/16) S2 -// + data8 0xfb14be7fbae58157 , 0x00003ffe // sin( 7 pi/16) C1 data8 0xc7c5c1e34d3055b3 , 0x00003ffc // cos( 7 pi/16) S1 -// + data8 0x8000000000000000 , 0x00003fff // sin( 8 pi/16) C0 data8 0x0000000000000000 , 0x00000000 // cos( 8 pi/16) S0 -// -// + + data8 0xfb14be7fbae58157 , 0x00003ffe // sin( 9 pi/16) C1 data8 0xc7c5c1e34d3055b3 , 0x0000bffc // cos( 9 pi/16) -S1 -// + data8 0xec835e79946a3146 , 0x00003ffe // sin(10 pi/16) C2 data8 0xc3ef1535754b168e , 0x0000bffd // cos(10 pi/16) -S2 -// + data8 0xd4db3148750d181a , 0x00003ffe // sin(11 pi/16) C3 data8 0x8e39d9cd73464364 , 0x0000bffe // cos(11 pi/16) -S3 -// + data8 0xb504f333f9de6484 , 0x00003ffe // sin(12 pi/16) S4 data8 0xb504f333f9de6484 , 0x0000bffe // cos(12 pi/16) -S4 -// -// + + data8 0x8e39d9cd73464364 , 0x00003ffe // sin(13 pi/16) S3 data8 0xd4db3148750d181a , 0x0000bffe // cos(13 pi/16) -C3 -// + data8 0xc3ef1535754b168e , 0x00003ffd // sin(14 pi/16) S2 data8 0xec835e79946a3146 , 0x0000bffe // cos(14 pi/16) -C2 -// + data8 0xc7c5c1e34d3055b3 , 0x00003ffc // sin(15 pi/16) S1 data8 0xfb14be7fbae58157 , 0x0000bffe // cos(15 pi/16) -C1 -// + data8 0x0000000000000000 , 0x00000000 // sin(16 pi/16) S0 data8 0x8000000000000000 , 0x0000bfff // cos(16 pi/16) -C0 -// -// + + data8 0xc7c5c1e34d3055b3 , 0x0000bffc // sin(17 pi/16) -S1 data8 0xfb14be7fbae58157 , 0x0000bffe // cos(17 pi/16) -C1 -// + data8 0xc3ef1535754b168e , 0x0000bffd // sin(18 pi/16) -S2 data8 0xec835e79946a3146 , 0x0000bffe // cos(18 pi/16) -C2 -// + data8 0x8e39d9cd73464364 , 0x0000bffe // sin(19 pi/16) -S3 data8 0xd4db3148750d181a , 0x0000bffe // cos(19 pi/16) -C3 -// + data8 0xb504f333f9de6484 , 0x0000bffe // sin(20 pi/16) -S4 data8 0xb504f333f9de6484 , 0x0000bffe // cos(20 pi/16) -S4 -// -// + + data8 0xd4db3148750d181a , 0x0000bffe // sin(21 pi/16) -C3 data8 0x8e39d9cd73464364 , 0x0000bffe // cos(21 pi/16) -S3 -// + data8 0xec835e79946a3146 , 0x0000bffe // sin(22 pi/16) -C2 data8 0xc3ef1535754b168e , 0x0000bffd // cos(22 pi/16) -S2 -// + data8 0xfb14be7fbae58157 , 0x0000bffe // sin(23 pi/16) -C1 data8 0xc7c5c1e34d3055b3 , 0x0000bffc // cos(23 pi/16) -S1 -// + data8 0x8000000000000000 , 0x0000bfff // sin(24 pi/16) -C0 data8 0x0000000000000000 , 0x00000000 // cos(24 pi/16) S0 -// -// + + data8 0xfb14be7fbae58157 , 0x0000bffe // sin(25 pi/16) -C1 data8 0xc7c5c1e34d3055b3 , 0x00003ffc // cos(25 pi/16) S1 -// + data8 0xec835e79946a3146 , 0x0000bffe // sin(26 pi/16) -C2 data8 0xc3ef1535754b168e , 0x00003ffd // cos(26 pi/16) S2 -// + data8 0xd4db3148750d181a , 0x0000bffe // sin(27 pi/16) -C3 data8 0x8e39d9cd73464364 , 0x00003ffe // cos(27 pi/16) S3 -// + data8 0xb504f333f9de6484 , 0x0000bffe // sin(28 pi/16) -S4 data8 0xb504f333f9de6484 , 0x00003ffe // cos(28 pi/16) S4 -// -// + + data8 0x8e39d9cd73464364 , 0x0000bffe // sin(29 pi/16) -S3 data8 0xd4db3148750d181a , 0x00003ffe // cos(29 pi/16) C3 -// + data8 0xc3ef1535754b168e , 0x0000bffd // sin(30 pi/16) -S2 data8 0xec835e79946a3146 , 0x00003ffe // cos(30 pi/16) C2 -// + data8 0xc7c5c1e34d3055b3 , 0x0000bffc // sin(31 pi/16) -S1 data8 0xfb14be7fbae58157 , 0x00003ffe // cos(31 pi/16) C1 -// + data8 0x0000000000000000 , 0x00000000 // sin(32 pi/16) S0 data8 0x8000000000000000 , 0x00003fff // cos(32 pi/16) C0 -LOCAL_OBJECT_END(double_sin_cos_beta_k4) +ASM_SIZE_DIRECTIVE(double_sin_cos_beta_k4) -.section .text +.align 32 +.global sin# +.global cos# +#ifdef _LIBC +.global __sin# +.global __cos# +#endif //////////////////////////////////////////////////////// // There are two entry points: sin and cos @@ -402,63 +402,85 @@ LOCAL_OBJECT_END(double_sin_cos_beta_k4) // If from sin, p8 is true // If from cos, p9 is true -GLOBAL_IEEE754_ENTRY(sin) +.section .text +.proc sin# +#ifdef _LIBC +.proc __sin# +#endif +.align 32 + +sin: +#ifdef _LIBC +__sin: +#endif { .mlx - alloc r32 = ar.pfs, 1, 13, 0, 0 - movl sincos_GR_sig_inv_pi_by_16 = 0xA2F9836E4E44152A // signd of 16/pi + alloc r32=ar.pfs,1,13,0,0 + movl sind_GR_sig_inv_pi_by_16 = 0xA2F9836E4E44152A // significand of 16/pi } { .mlx - addl sincos_AD_1 = @ltoff(double_sincos_pi), gp - movl sincos_GR_rshf_2to61 = 0x47b8000000000000 // 1.1 2^(63+63-2) + addl sind_AD_1 = @ltoff(double_sind_pi), gp + movl sind_GR_rshf_2to61 = 0x47b8000000000000 // 1.1000 2^(63+63-2) } ;; { .mfi - ld8 sincos_AD_1 = [sincos_AD_1] - fnorm.s0 sincos_NORM_f8 = f8 // Normalize argument - cmp.eq p8,p9 = r0, r0 // set p8 (clear p9) for sin + ld8 sind_AD_1 = [sind_AD_1] + fnorm sind_NORM_f8 = f8 + cmp.eq p8,p9 = r0, r0 } { .mib - mov sincos_GR_exp_2tom61 = 0xffff-61 // exponent of scale 2^-61 - mov sincos_r_sincos = 0x0 // sincos_r_sincos = 0 for sin - br.cond.sptk _SINCOS_COMMON // go to common part + mov sind_GR_exp_2tom61 = 0xffff-61 // exponent of scaling factor 2^-61 + mov sind_r_sincos = 0x0 + br.cond.sptk L(SIND_SINCOS) } ;; -GLOBAL_IEEE754_END(sin) -GLOBAL_IEEE754_ENTRY(cos) +.endp sin +ASM_SIZE_DIRECTIVE(sin) + + +.section .text +.proc cos# +#ifdef _LIBC +.proc __cos# +#endif +.align 32 +cos: +#ifdef _LIBC +__cos: +#endif { .mlx - alloc r32 = ar.pfs, 1, 13, 0, 0 - movl sincos_GR_sig_inv_pi_by_16 = 0xA2F9836E4E44152A // signd of 16/pi + alloc r32=ar.pfs,1,13,0,0 + movl sind_GR_sig_inv_pi_by_16 = 0xA2F9836E4E44152A // significand of 16/pi } { .mlx - addl sincos_AD_1 = @ltoff(double_sincos_pi), gp - movl sincos_GR_rshf_2to61 = 0x47b8000000000000 // 1.1 2^(63+63-2) + addl sind_AD_1 = @ltoff(double_sind_pi), gp + movl sind_GR_rshf_2to61 = 0x47b8000000000000 // 1.1000 2^(63+63-2) } ;; { .mfi - ld8 sincos_AD_1 = [sincos_AD_1] - fnorm.s1 sincos_NORM_f8 = f8 // Normalize argument - cmp.eq p9,p8 = r0, r0 // set p9 (clear p8) for cos + ld8 sind_AD_1 = [sind_AD_1] + fnorm.s1 sind_NORM_f8 = f8 + cmp.eq p9,p8 = r0, r0 } { .mib - mov sincos_GR_exp_2tom61 = 0xffff-61 // exp of scale 2^-61 - mov sincos_r_sincos = 0x8 // sincos_r_sincos = 8 for cos - nop.b 999 + mov sind_GR_exp_2tom61 = 0xffff-61 // exponent of scaling factor 2^-61 + mov sind_r_sincos = 0x8 + br.cond.sptk L(SIND_SINCOS) } ;; + //////////////////////////////////////////////////////// // All entry points end up here. -// If from sin, sincos_r_sincos is 0 and p8 is true -// If from cos, sincos_r_sincos is 8 = 2^(k-1) and p9 is true -// We add sincos_r_sincos to N +// If from sin, sind_r_sincos is 0 and p8 is true +// If from cos, sind_r_sincos is 8 = 2^(k-1) and p9 is true +// We add sind_r_sincos to N -///////////// Common sin and cos part ////////////////// -_SINCOS_COMMON: +L(SIND_SINCOS): // Form two constants we need @@ -466,320 +488,3014 @@ _SINCOS_COMMON: // 1.1000...000 * 2^(63+63-2) to right shift int(W) into the low significand // fcmp used to set denormal, and invalid on snans { .mfi - setf.sig sincos_SIG_INV_PI_BY_16_2TO61 = sincos_GR_sig_inv_pi_by_16 - fclass.m p6,p0 = f8, 0xe7 // if x = 0,inf,nan - mov sincos_exp_limit = 0x1001a + setf.sig sind_SIG_INV_PI_BY_16_2TO61 = sind_GR_sig_inv_pi_by_16 + fcmp.eq.s0 p12,p0=f8,f0 + mov sind_r_17_ones = 0x1ffff } { .mlx - setf.d sincos_RSHF_2TO61 = sincos_GR_rshf_2to61 - movl sincos_GR_rshf = 0x43e8000000000000 // 1.1 2^63 -} // Right shift + setf.d sind_RSHF_2TO61 = sind_GR_rshf_2to61 + movl sind_GR_rshf = 0x43e8000000000000 // 1.1000 2^63 for right shift +} ;; // Form another constant // 2^-61 for scaling Nfloat -// 0x1001a is register_bias + 27. -// So if f8 >= 2^27, go to large argument routines -{ .mmi - getf.exp sincos_r_signexp = f8 - setf.exp sincos_2TOM61 = sincos_GR_exp_2tom61 - addl gr_tmp = -1,r0 // For "inexect" constant create +// 0x10009 is register_bias + 10. +// So if f8 > 2^10 = Gamma, go to DBX +{ .mfi + setf.exp sind_2TOM61 = sind_GR_exp_2tom61 + fclass.m p13,p0 = f8, 0x23 // Test for x inf + mov sind_exp_limit = 0x10009 } ;; // Load the two pieces of pi/16 // Form another constant // 1.1000...000 * 2^63, the right shift constant -{ .mmb - ldfe sincos_Pi_by_16_1 = [sincos_AD_1],16 - setf.d sincos_RSHF = sincos_GR_rshf -(p6) br.cond.spnt _SINCOS_SPECIAL_ARGS +{ .mmf + ldfe sind_Pi_by_16_hi = [sind_AD_1],16 + setf.d sind_RSHF = sind_GR_rshf + fclass.m p14,p0 = f8, 0xc3 // Test for x nan } ;; -{ .mmi - ldfe sincos_Pi_by_16_2 = [sincos_AD_1],16 - setf.sig fp_tmp = gr_tmp // constant for inexact set - nop.i 999 -};; +{ .mfi + ldfe sind_Pi_by_16_lo = [sind_AD_1],16 +(p13) frcpa.s0 f8,p12=f0,f0 // force qnan indef for x=inf + addl gr_tmp = -1,r0 +} +{ .mfb + addl sind_AD_beta_table = @ltoff(double_sin_cos_beta_k4), gp + nop.f 999 +(p13) br.ret.spnt b0 ;; // Exit for x=inf +} +// Start loading P, Q coefficients +// SIN(0) { .mfi - ldfe sincos_Pi_by_16_3 = [sincos_AD_1],16 - nop.f 999 - nop.i 999 -};; + ldfpd sind_P4,sind_Q4 = [sind_AD_1],16 +(p8) fclass.m.unc p6,p0 = f8, 0x07 // Test for sin(0) + nop.i 999 +} +{ .mfb + addl sind_AD_beta_table = @ltoff(double_sin_cos_beta_k4), gp +(p14) fma.d f8=f8,f1,f0 // qnan for x=nan +(p14) br.ret.spnt b0 ;; // Exit for x=nan +} -// Polynomial coefficients (Q4, P4, Q3, P3, Q2, Q1, P2, P1) loading -{ .mmi - ldfpd sincos_P4,sincos_Q4 = [sincos_AD_1],16 - nop.m 999 - nop.i 999 -};; -// Select exponent (17 lsb) -{ .mmi - ldfpd sincos_P3,sincos_Q3 = [sincos_AD_1],16 - nop.m 999 - dep.z sincos_r_exp = sincos_r_signexp, 0, 17 +// COS(0) +{ .mfi + getf.exp sind_r_signexp = f8 +(p9) fclass.m.unc p7,p0 = f8, 0x07 // Test for sin(0) + nop.i 999 +} +{ .mfi + ld8 sind_AD_beta_table = [sind_AD_beta_table] + nop.f 999 + nop.i 999 ;; } -;; -// p10 is true if we must call routines to handle larger arguments -// p10 is true if f8 exp is >= 0x1001a (2^27) { .mmb - ldfpd sincos_P2,sincos_Q2 = [sincos_AD_1],16 - cmp.ge p10,p0 = sincos_r_exp,sincos_exp_limit -(p10) br.cond.spnt _SINCOS_LARGE_ARGS // Go to "large args" routine -};; + ldfpd sind_P3,sind_Q3 = [sind_AD_1],16 + setf.sig fp_tmp = gr_tmp // Create constant such that fmpy sets inexact +(p6) br.ret.spnt b0 ;; +} + +{ .mfb + and sind_r_exp = sind_r_17_ones, sind_r_signexp +(p7) fmerge.s f8 = f1,f1 +(p7) br.ret.spnt b0 ;; +} -// sincos_W = x * sincos_Inv_Pi_by_16 +// p10 is true if we must call routines to handle larger arguments +// p10 is true if f8 exp is > 0x10009 + +{ .mfi + ldfpd sind_P2,sind_Q2 = [sind_AD_1],16 + nop.f 999 + cmp.ge p10,p0 = sind_r_exp,sind_exp_limit +} +;; + +// sind_W = x * sind_Inv_Pi_by_16 // Multiply x by scaled 16/pi and add large const to shift integer part of W to // rightmost bits of significand { .mfi - ldfpd sincos_P1,sincos_Q1 = [sincos_AD_1],16 - fma.s1 sincos_W_2TO61_RSH = sincos_NORM_f8,sincos_SIG_INV_PI_BY_16_2TO61,sincos_RSHF_2TO61 - nop.i 999 -};; + ldfpd sind_P1,sind_Q1 = [sind_AD_1] + fma.s1 sind_W_2TO61_RSH = sind_NORM_f8,sind_SIG_INV_PI_BY_16_2TO61,sind_RSHF_2TO61 + nop.i 999 +} +{ .mbb +(p10) cmp.ne.unc p11,p12=sind_r_sincos,r0 // p11 call __libm_cos_double_dbx + // p12 call __libm_sin_double_dbx +(p11) br.cond.spnt L(COSD_DBX) +(p12) br.cond.spnt L(SIND_DBX) +} +;; + -// sincos_NFLOAT = Round_Int_Nearest(sincos_W) +// sind_NFLOAT = Round_Int_Nearest(sind_W) // This is done by scaling back by 2^-61 and subtracting the shift constant { .mfi - nop.m 999 - fms.s1 sincos_NFLOAT = sincos_W_2TO61_RSH,sincos_2TOM61,sincos_RSHF - nop.i 999 -};; + nop.m 999 + fms.s1 sind_NFLOAT = sind_W_2TO61_RSH,sind_2TOM61,sind_RSHF + nop.i 999 ;; +} -// get N = (int)sincos_int_Nfloat +// get N = (int)sind_int_Nfloat { .mfi - getf.sig sincos_GR_n = sincos_W_2TO61_RSH - nop.f 999 - nop.i 999 -};; + getf.sig sind_GR_n = sind_W_2TO61_RSH + nop.f 999 + nop.i 999 ;; +} -// Add 2^(k-1) (which is in sincos_r_sincos) to N -// sincos_r = -sincos_Nfloat * sincos_Pi_by_16_1 + x +// Add 2^(k-1) (which is in sind_r_sincos) to N +// sind_r = -sind_Nfloat * sind_Pi_by_16_hi + x +// sind_r = sind_r -sind_Nfloat * sind_Pi_by_16_lo { .mfi - add sincos_GR_n = sincos_GR_n, sincos_r_sincos - fnma.s1 sincos_r = sincos_NFLOAT, sincos_Pi_by_16_1, sincos_NORM_f8 - nop.i 999 -};; + add sind_GR_n = sind_GR_n, sind_r_sincos + fnma.s1 sind_r = sind_NFLOAT, sind_Pi_by_16_hi, sind_NORM_f8 + nop.i 999 ;; +} + // Get M (least k+1 bits of N) { .mmi - and sincos_GR_m = 0x1f,sincos_GR_n;; - nop.m 999 - shl sincos_GR_32m = sincos_GR_m,5 -};; + and sind_GR_m = 0x1f,sind_GR_n ;; + nop.m 999 + shl sind_GR_32m = sind_GR_m,5 ;; +} // Add 32*M to address of sin_cos_beta table +{ .mmi + add sind_AD_2 = sind_GR_32m, sind_AD_beta_table + nop.m 999 + nop.i 999 ;; +} + { .mfi - add sincos_AD_2 = sincos_GR_32m, sincos_AD_1 -(p8) fclass.m.unc p10,p0 = f8,0x0b // For sin denorm. - set uflow - nop.i 999 -};; + ldfe sind_Sm = [sind_AD_2],16 +(p8) fclass.m.unc p10,p0=f8,0x0b // If sin, note denormal input to set uflow + nop.i 999 ;; +} -// Load Sin and Cos table value using obtained index m (sincosf_AD_2) { .mfi - ldfe sincos_Sm = [sincos_AD_2],16 -(p9) fclass.m.unc p11,p0 = f8,0x0b // For cos denorm - set denorm - nop.i 999 -};; + ldfe sind_Cm = [sind_AD_2] + fnma.s1 sind_r = sind_NFLOAT, sind_Pi_by_16_lo, sind_r + nop.i 999 ;; +} -// sincos_r = sincos_r -sincos_Nfloat * sincos_Pi_by_16_2 +// get rsq { .mfi - ldfe sincos_Cm = [sincos_AD_2] - fnma.s1 sincos_r = sincos_NFLOAT, sincos_Pi_by_16_2, sincos_r - nop.i 999 -};; + nop.m 999 + fma.s1 sind_rsq = sind_r, sind_r, f0 + nop.i 999 +} +{ .mfi + nop.m 999 + fmpy.s0 fp_tmp = fp_tmp,fp_tmp // fmpy forces inexact flag + nop.i 999 ;; +} -// get rsq = r*r +// form P and Q series { .mfi - nop.m 999 - fma.s1 sincos_rsq = sincos_r, sincos_r, f0 // r^2 = r*r - nop.i 999 + nop.m 999 + fma.s1 sind_P_temp1 = sind_rsq, sind_P4, sind_P3 + nop.i 999 } + { .mfi - nop.m 999 - fmpy.s0 fp_tmp = fp_tmp,fp_tmp // forces inexact flag - nop.i 999 -};; + nop.m 999 + fma.s1 sind_Q_temp1 = sind_rsq, sind_Q4, sind_Q3 + nop.i 999 ;; +} -// sincos_r_exact = sincos_r -sincos_Nfloat * sincos_Pi_by_16_3 +// get rcube and sm*rsq { .mfi - nop.m 999 - fnma.s1 sincos_r_exact = sincos_NFLOAT, sincos_Pi_by_16_3, sincos_r - nop.i 999 -};; + nop.m 999 + fmpy.s1 sind_srsq = sind_Sm,sind_rsq + nop.i 999 +} -// Polynomials calculation -// P_1 = P4*r^2 + P3 -// Q_2 = Q4*r^2 + Q3 { .mfi - nop.m 999 - fma.s1 sincos_P_temp1 = sincos_rsq, sincos_P4, sincos_P3 - nop.i 999 + nop.m 999 + fmpy.s1 sind_rcub = sind_r, sind_rsq + nop.i 999 ;; } + { .mfi - nop.m 999 - fma.s1 sincos_Q_temp1 = sincos_rsq, sincos_Q4, sincos_Q3 - nop.i 999 -};; + nop.m 999 + fma.s1 sind_Q_temp2 = sind_rsq, sind_Q_temp1, sind_Q2 + nop.i 999 +} -// get rcube = r^3 and S[m]*r^2 { .mfi - nop.m 999 - fmpy.s1 sincos_srsq = sincos_Sm,sincos_rsq - nop.i 999 + nop.m 999 + fma.s1 sind_P_temp2 = sind_rsq, sind_P_temp1, sind_P2 + nop.i 999 ;; } + { .mfi - nop.m 999 - fmpy.s1 sincos_rcub = sincos_r_exact, sincos_rsq - nop.i 999 -};; + nop.m 999 + fma.s1 sind_Q = sind_rsq, sind_Q_temp2, sind_Q1 + nop.i 999 +} -// Polynomials calculation -// Q_2 = Q_1*r^2 + Q2 -// P_1 = P_1*r^2 + P2 { .mfi - nop.m 999 - fma.s1 sincos_Q_temp2 = sincos_rsq, sincos_Q_temp1, sincos_Q2 - nop.i 999 + nop.m 999 + fma.s1 sind_P = sind_rsq, sind_P_temp2, sind_P1 + nop.i 999 ;; } + +// Get final P and Q { .mfi - nop.m 999 - fma.s1 sincos_P_temp2 = sincos_rsq, sincos_P_temp1, sincos_P2 - nop.i 999 -};; + nop.m 999 + fma.s1 sind_Q = sind_srsq,sind_Q, sind_Sm + nop.i 999 +} -// Polynomials calculation -// Q = Q_2*r^2 + Q1 -// P = P_2*r^2 + P1 { .mfi - nop.m 999 - fma.s1 sincos_Q = sincos_rsq, sincos_Q_temp2, sincos_Q1 - nop.i 999 + nop.m 999 + fma.s1 sind_P = sind_rcub,sind_P, sind_r + nop.i 999 ;; } +// If sin(denormal), force inexact to be set { .mfi - nop.m 999 - fma.s1 sincos_P = sincos_rsq, sincos_P_temp2, sincos_P1 - nop.i 999 -};; + nop.m 999 +(p10) fmpy.d.s0 fp_tmp = f8,f8 + nop.i 999 ;; +} -// Get final P and Q -// Q = Q*S[m]*r^2 + S[m] -// P = P*r^3 + r +// Final calculation +{ .mfb + nop.m 999 + fma.d f8 = sind_Cm, sind_P, sind_Q + br.ret.sptk b0 ;; +} +.endp cos# +ASM_SIZE_DIRECTIVE(cos#) + + + +.proc __libm_callout_1s +__libm_callout_1s: +L(SIND_DBX): +.prologue { .mfi - nop.m 999 - fma.s1 sincos_Q = sincos_srsq,sincos_Q, sincos_Sm - nop.i 999 + nop.m 0 + nop.f 0 +.save ar.pfs,GR_SAVE_PFS + mov GR_SAVE_PFS=ar.pfs } +;; { .mfi - nop.m 999 - fma.s1 sincos_P = sincos_rcub,sincos_P, sincos_r_exact - nop.i 999 -};; + mov GR_SAVE_GP=gp + nop.f 0 +.save b0, GR_SAVE_B0 + mov GR_SAVE_B0=b0 +} + +.body +{ .mib + nop.m 999 + nop.i 999 + br.call.sptk.many b0=__libm_sin_double_dbx# ;; +} +;; + -// If sin(denormal), force underflow to be set -.pred.rel "mutex",p10,p11 { .mfi - nop.m 999 -(p10) fmpy.d.s0 fp_tmp = f8,f8 // forces underflow flag - nop.i 999 // for denormal sine args + mov gp = GR_SAVE_GP + nop.f 999 + mov b0 = GR_SAVE_B0 } +;; + +{ .mib + nop.m 999 + mov ar.pfs = GR_SAVE_PFS + br.ret.sptk b0 ;; +} +.endp __libm_callout_1s +ASM_SIZE_DIRECTIVE(__libm_callout_1s) + + +.proc __libm_callout_1c +__libm_callout_1c: +L(COSD_DBX): +.prologue { .mfi - nop.m 999 -(p11) fma.d.s0 fp_tmp = f8,f1, f8 // forces denormal flag - nop.i 999 // for denormal cosine args -};; + nop.m 0 + nop.f 0 +.save ar.pfs,GR_SAVE_PFS + mov GR_SAVE_PFS=ar.pfs +} +;; +{ .mfi + mov GR_SAVE_GP=gp + nop.f 0 +.save b0, GR_SAVE_B0 + mov GR_SAVE_B0=b0 +} + +.body +{ .mib + nop.m 999 + nop.i 999 + br.call.sptk.many b0=__libm_cos_double_dbx# ;; +} +;; + + +{ .mfi + mov gp = GR_SAVE_GP + nop.f 999 + mov b0 = GR_SAVE_B0 +} +;; + +{ .mib + nop.m 999 + mov ar.pfs = GR_SAVE_PFS + br.ret.sptk b0 ;; +} +.endp __libm_callout_1c +ASM_SIZE_DIRECTIVE(__libm_callout_1c) + + +// ==================================================================== +// ==================================================================== + +// These functions calculate the sin and cos for inputs +// greater than 2^10 +// __libm_sin_double_dbx# and __libm_cos_double_dbx# + +// ********************************************************************* +// ********************************************************************* +// +// Function: Combined sin(x) and cos(x), where +// +// sin(x) = sine(x), for double precision x values +// cos(x) = cosine(x), for double precision x values +// +// ********************************************************************* +// +// Accuracy: Within .7 ulps for 80-bit floating point values +// Very accurate for double precision values +// +// ********************************************************************* +// +// Resources Used: +// +// Floating-Point Registers: f8 (Input and Return Value) +// f32-f99 +// +// General Purpose Registers: +// r32-r43 +// r44-r45 (Used to pass arguments to pi_by_2 reduce routine) +// +// Predicate Registers: p6-p13 +// +// ********************************************************************* +// +// IEEE Special Conditions: +// +// Denormal fault raised on denormal inputs +// Overflow exceptions do not occur +// Underflow exceptions raised when appropriate for sin +// (No specialized error handling for this routine) +// Inexact raised when appropriate by algorithm +// +// sin(SNaN) = QNaN +// sin(QNaN) = QNaN +// sin(inf) = QNaN +// sin(+/-0) = +/-0 +// cos(inf) = QNaN +// cos(SNaN) = QNaN +// cos(QNaN) = QNaN +// cos(0) = 1 +// +// ********************************************************************* +// +// Mathematical Description +// ======================== +// +// The computation of FSIN and FCOS is best handled in one piece of +// code. The main reason is that given any argument Arg, computation +// of trigonometric functions first calculate N and an approximation +// to alpha where +// +// Arg = N pi/2 + alpha, |alpha| <= pi/4. +// +// Since +// +// cos( Arg ) = sin( (N+1) pi/2 + alpha ), +// +// therefore, the code for computing sine will produce cosine as long +// as 1 is added to N immediately after the argument reduction +// process. +// +// Let M = N if sine +// N+1 if cosine. +// +// Now, given +// +// Arg = M pi/2 + alpha, |alpha| <= pi/4, +// +// let I = M mod 4, or I be the two lsb of M when M is represented +// as 2's complement. I = [i_0 i_1]. Then +// +// sin( Arg ) = (-1)^i_0 sin( alpha ) if i_1 = 0, +// = (-1)^i_0 cos( alpha ) if i_1 = 1. +// +// For example: +// if M = -1, I = 11 +// sin ((-pi/2 + alpha) = (-1) cos (alpha) +// if M = 0, I = 00 +// sin (alpha) = sin (alpha) +// if M = 1, I = 01 +// sin (pi/2 + alpha) = cos (alpha) +// if M = 2, I = 10 +// sin (pi + alpha) = (-1) sin (alpha) +// if M = 3, I = 11 +// sin ((3/2)pi + alpha) = (-1) cos (alpha) +// +// The value of alpha is obtained by argument reduction and +// represented by two working precision numbers r and c where +// +// alpha = r + c accurately. +// +// The reduction method is described in a previous write up. +// The argument reduction scheme identifies 4 cases. For Cases 2 +// and 4, because |alpha| is small, sin(r+c) and cos(r+c) can be +// computed very easily by 2 or 3 terms of the Taylor series +// expansion as follows: +// +// Case 2: +// ------- +// +// sin(r + c) = r + c - r^3/6 accurately +// cos(r + c) = 1 - 2^(-67) accurately +// +// Case 4: +// ------- +// +// sin(r + c) = r + c - r^3/6 + r^5/120 accurately +// cos(r + c) = 1 - r^2/2 + r^4/24 accurately +// +// The only cases left are Cases 1 and 3 of the argument reduction +// procedure. These two cases will be merged since after the +// argument is reduced in either cases, we have the reduced argument +// represented as r + c and that the magnitude |r + c| is not small +// enough to allow the usage of a very short approximation. +// +// The required calculation is either +// +// sin(r + c) = sin(r) + correction, or +// cos(r + c) = cos(r) + correction. +// +// Specifically, +// +// sin(r + c) = sin(r) + c sin'(r) + O(c^2) +// = sin(r) + c cos (r) + O(c^2) +// = sin(r) + c(1 - r^2/2) accurately. +// Similarly, +// +// cos(r + c) = cos(r) - c sin(r) + O(c^2) +// = cos(r) - c(r - r^3/6) accurately. +// +// We therefore concentrate on accurately calculating sin(r) and +// cos(r) for a working-precision number r, |r| <= pi/4 to within +// 0.1% or so. +// +// The greatest challenge of this task is that the second terms of +// the Taylor series +// +// r - r^3/3! + r^r/5! - ... +// +// and +// +// 1 - r^2/2! + r^4/4! - ... +// +// are not very small when |r| is close to pi/4 and the rounding +// errors will be a concern if simple polynomial accumulation is +// used. When |r| < 2^-3, however, the second terms will be small +// enough (6 bits or so of right shift) that a normal Horner +// recurrence suffices. Hence there are two cases that we consider +// in the accurate computation of sin(r) and cos(r), |r| <= pi/4. +// +// Case small_r: |r| < 2^(-3) +// -------------------------- +// +// Since Arg = M pi/4 + r + c accurately, and M mod 4 is [i_0 i_1], +// we have +// +// sin(Arg) = (-1)^i_0 * sin(r + c) if i_1 = 0 +// = (-1)^i_0 * cos(r + c) if i_1 = 1 +// +// can be accurately approximated by +// +// sin(Arg) = (-1)^i_0 * [sin(r) + c] if i_1 = 0 +// = (-1)^i_0 * [cos(r) - c*r] if i_1 = 1 +// +// because |r| is small and thus the second terms in the correction +// are unneccessary. +// +// Finally, sin(r) and cos(r) are approximated by polynomials of +// moderate lengths. +// +// sin(r) = r + S_1 r^3 + S_2 r^5 + ... + S_5 r^11 +// cos(r) = 1 + C_1 r^2 + C_2 r^4 + ... + C_5 r^10 +// +// We can make use of predicates to selectively calculate +// sin(r) or cos(r) based on i_1. +// +// Case normal_r: 2^(-3) <= |r| <= pi/4 +// ------------------------------------ +// +// This case is more likely than the previous one if one considers +// r to be uniformly distributed in [-pi/4 pi/4]. Again, +// +// sin(Arg) = (-1)^i_0 * sin(r + c) if i_1 = 0 +// = (-1)^i_0 * cos(r + c) if i_1 = 1. +// +// Because |r| is now larger, we need one extra term in the +// correction. sin(Arg) can be accurately approximated by +// +// sin(Arg) = (-1)^i_0 * [sin(r) + c(1-r^2/2)] if i_1 = 0 +// = (-1)^i_0 * [cos(r) - c*r*(1 - r^2/6)] i_1 = 1. +// +// Finally, sin(r) and cos(r) are approximated by polynomials of +// moderate lengths. +// +// sin(r) = r + PP_1_hi r^3 + PP_1_lo r^3 + +// PP_2 r^5 + ... + PP_8 r^17 +// +// cos(r) = 1 + QQ_1 r^2 + QQ_2 r^4 + ... + QQ_8 r^16 +// +// where PP_1_hi is only about 16 bits long and QQ_1 is -1/2. +// The crux in accurate computation is to calculate +// +// r + PP_1_hi r^3 or 1 + QQ_1 r^2 +// +// accurately as two pieces: U_hi and U_lo. The way to achieve this +// is to obtain r_hi as a 10 sig. bit number that approximates r to +// roughly 8 bits or so of accuracy. (One convenient way is +// +// r_hi := frcpa( frcpa( r ) ).) +// +// This way, +// +// r + PP_1_hi r^3 = r + PP_1_hi r_hi^3 + +// PP_1_hi (r^3 - r_hi^3) +// = [r + PP_1_hi r_hi^3] + +// [PP_1_hi (r - r_hi) +// (r^2 + r_hi r + r_hi^2) ] +// = U_hi + U_lo +// +// Since r_hi is only 10 bit long and PP_1_hi is only 16 bit long, +// PP_1_hi * r_hi^3 is only at most 46 bit long and thus computed +// exactly. Furthermore, r and PP_1_hi r_hi^3 are of opposite sign +// and that there is no more than 8 bit shift off between r and +// PP_1_hi * r_hi^3. Hence the sum, U_hi, is representable and thus +// calculated without any error. Finally, the fact that +// +// |U_lo| <= 2^(-8) |U_hi| +// +// says that U_hi + U_lo is approximating r + PP_1_hi r^3 to roughly +// 8 extra bits of accuracy. +// +// Similarly, +// +// 1 + QQ_1 r^2 = [1 + QQ_1 r_hi^2] + +// [QQ_1 (r - r_hi)(r + r_hi)] +// = U_hi + U_lo. +// +// Summarizing, we calculate r_hi = frcpa( frcpa( r ) ). +// +// If i_1 = 0, then +// +// U_hi := r + PP_1_hi * r_hi^3 +// U_lo := PP_1_hi * (r - r_hi) * (r^2 + r*r_hi + r_hi^2) +// poly := PP_1_lo r^3 + PP_2 r^5 + ... + PP_8 r^17 +// correction := c * ( 1 + C_1 r^2 ) +// +// Else ...i_1 = 1 +// +// U_hi := 1 + QQ_1 * r_hi * r_hi +// U_lo := QQ_1 * (r - r_hi) * (r + r_hi) +// poly := QQ_2 * r^4 + QQ_3 * r^6 + ... + QQ_8 r^16 +// correction := -c * r * (1 + S_1 * r^2) +// +// End +// +// Finally, +// +// V := poly + ( U_lo + correction ) +// +// / U_hi + V if i_0 = 0 +// result := | +// \ (-U_hi) - V if i_0 = 1 +// +// It is important that in the last step, negation of U_hi is +// performed prior to the subtraction which is to be performed in +// the user-set rounding mode. +// +// +// Algorithmic Description +// ======================= +// +// The argument reduction algorithm is tightly integrated into FSIN +// and FCOS which share the same code. The following is complete and +// self-contained. The argument reduction description given +// previously is repeated below. +// +// +// Step 0. Initialization. +// +// If FSIN is invoked, set N_inc := 0; else if FCOS is invoked, +// set N_inc := 1. +// +// Step 1. Check for exceptional and special cases. +// +// * If Arg is +-0, +-inf, NaN, NaT, go to Step 10 for special +// handling. +// * If |Arg| < 2^24, go to Step 2 for reduction of moderate +// arguments. This is the most likely case. +// * If |Arg| < 2^63, go to Step 8 for pre-reduction of large +// arguments. +// * If |Arg| >= 2^63, go to Step 10 for special handling. +// +// Step 2. Reduction of moderate arguments. +// +// If |Arg| < pi/4 ...quick branch +// N_fix := N_inc (integer) +// r := Arg +// c := 0.0 +// Branch to Step 4, Case_1_complete +// Else ...cf. argument reduction +// N := Arg * two_by_PI (fp) +// N_fix := fcvt.fx( N ) (int) +// N := fcvt.xf( N_fix ) +// N_fix := N_fix + N_inc +// s := Arg - N * P_1 (first piece of pi/2) +// w := -N * P_2 (second piece of pi/2) +// +// If |s| >= 2^(-33) +// go to Step 3, Case_1_reduce +// Else +// go to Step 7, Case_2_reduce +// Endif +// Endif +// +// Step 3. Case_1_reduce. +// +// r := s + w +// c := (s - r) + w ...observe order +// +// Step 4. Case_1_complete +// +// ...At this point, the reduced argument alpha is +// ...accurately represented as r + c. +// If |r| < 2^(-3), go to Step 6, small_r. +// +// Step 5. Normal_r. +// +// Let [i_0 i_1] by the 2 lsb of N_fix. +// FR_rsq := r * r +// r_hi := frcpa( frcpa( r ) ) +// r_lo := r - r_hi +// +// If i_1 = 0, then +// poly := r*FR_rsq*(PP_1_lo + FR_rsq*(PP_2 + ... FR_rsq*PP_8)) +// U_hi := r + PP_1_hi*r_hi*r_hi*r_hi ...any order +// U_lo := PP_1_hi*r_lo*(r*r + r*r_hi + r_hi*r_hi) +// correction := c + c*C_1*FR_rsq ...any order +// Else +// poly := FR_rsq*FR_rsq*(QQ_2 + FR_rsq*(QQ_3 + ... + FR_rsq*QQ_8)) +// U_hi := 1 + QQ_1 * r_hi * r_hi ...any order +// U_lo := QQ_1 * r_lo * (r + r_hi) +// correction := -c*(r + S_1*FR_rsq*r) ...any order +// Endif +// +// V := poly + (U_lo + correction) ...observe order +// +// result := (i_0 == 0? 1.0 : -1.0) +// +// Last instruction in user-set rounding mode +// +// result := (i_0 == 0? result*U_hi + V : +// result*U_hi - V) +// +// Return +// +// Step 6. Small_r. +// +// ...Use flush to zero mode without causing exception +// Let [i_0 i_1] be the two lsb of N_fix. +// +// FR_rsq := r * r +// +// If i_1 = 0 then +// z := FR_rsq*FR_rsq; z := FR_rsq*z *r +// poly_lo := S_3 + FR_rsq*(S_4 + FR_rsq*S_5) +// poly_hi := r*FR_rsq*(S_1 + FR_rsq*S_2) +// correction := c +// result := r +// Else +// z := FR_rsq*FR_rsq; z := FR_rsq*z +// poly_lo := C_3 + FR_rsq*(C_4 + FR_rsq*C_5) +// poly_hi := FR_rsq*(C_1 + FR_rsq*C_2) +// correction := -c*r +// result := 1 +// Endif +// +// poly := poly_hi + (z * poly_lo + correction) +// +// If i_0 = 1, result := -result +// +// Last operation. Perform in user-set rounding mode +// +// result := (i_0 == 0? result + poly : +// result - poly ) +// Return +// +// Step 7. Case_2_reduce. +// +// ...Refer to the write up for argument reduction for +// ...rationale. The reduction algorithm below is taken from +// ...argument reduction description and integrated this. +// +// w := N*P_3 +// U_1 := N*P_2 + w ...FMA +// U_2 := (N*P_2 - U_1) + w ...2 FMA +// ...U_1 + U_2 is N*(P_2+P_3) accurately +// +// r := s - U_1 +// c := ( (s - r) - U_1 ) - U_2 +// +// ...The mathematical sum r + c approximates the reduced +// ...argument accurately. Note that although compared to +// ...Case 1, this case requires much more work to reduce +// ...the argument, the subsequent calculation needed for +// ...any of the trigonometric function is very little because +// ...|alpha| < 1.01*2^(-33) and thus two terms of the +// ...Taylor series expansion suffices. +// +// If i_1 = 0 then +// poly := c + S_1 * r * r * r ...any order +// result := r +// Else +// poly := -2^(-67) +// result := 1.0 +// Endif +// +// If i_0 = 1, result := -result +// +// Last operation. Perform in user-set rounding mode +// +// result := (i_0 == 0? result + poly : +// result - poly ) +// +// Return +// +// +// Step 8. Pre-reduction of large arguments. +// +// ...Again, the following reduction procedure was described +// ...in the separate write up for argument reduction, which +// ...is tightly integrated here. + +// N_0 := Arg * Inv_P_0 +// N_0_fix := fcvt.fx( N_0 ) +// N_0 := fcvt.xf( N_0_fix) + +// Arg' := Arg - N_0 * P_0 +// w := N_0 * d_1 +// N := Arg' * two_by_PI +// N_fix := fcvt.fx( N ) +// N := fcvt.xf( N_fix ) +// N_fix := N_fix + N_inc +// +// s := Arg' - N * P_1 +// w := w - N * P_2 +// +// If |s| >= 2^(-14) +// go to Step 3 +// Else +// go to Step 9 +// Endif +// +// Step 9. Case_4_reduce. +// +// ...first obtain N_0*d_1 and -N*P_2 accurately +// U_hi := N_0 * d_1 V_hi := -N*P_2 +// U_lo := N_0 * d_1 - U_hi V_lo := -N*P_2 - U_hi ...FMAs +// +// ...compute the contribution from N_0*d_1 and -N*P_3 +// w := -N*P_3 +// w := w + N_0*d_2 +// t := U_lo + V_lo + w ...any order +// +// ...at this point, the mathematical value +// ...s + U_hi + V_hi + t approximates the true reduced argument +// ...accurately. Just need to compute this accurately. +// +// ...Calculate U_hi + V_hi accurately: +// A := U_hi + V_hi +// if |U_hi| >= |V_hi| then +// a := (U_hi - A) + V_hi +// else +// a := (V_hi - A) + U_hi +// endif +// ...order in computing "a" must be observed. This branch is +// ...best implemented by predicates. +// ...A + a is U_hi + V_hi accurately. Moreover, "a" is +// ...much smaller than A: |a| <= (1/2)ulp(A). +// +// ...Just need to calculate s + A + a + t +// C_hi := s + A t := t + a +// C_lo := (s - C_hi) + A +// C_lo := C_lo + t +// +// ...Final steps for reduction +// r := C_hi + C_lo +// c := (C_hi - r) + C_lo +// +// ...At this point, we have r and c +// ...And all we need is a couple of terms of the corresponding +// ...Taylor series. +// +// If i_1 = 0 +// poly := c + r*FR_rsq*(S_1 + FR_rsq*S_2) +// result := r +// Else +// poly := FR_rsq*(C_1 + FR_rsq*C_2) +// result := 1 +// Endif +// +// If i_0 = 1, result := -result +// +// Last operation. Perform in user-set rounding mode +// +// result := (i_0 == 0? result + poly : +// result - poly ) +// Return +// +// Large Arguments: For arguments above 2**63, a Payne-Hanek +// style argument reduction is used and pi_by_2 reduce is called. +// + + +#ifdef _LIBC +.rodata +#else +.data +#endif +.align 64 + +FSINCOS_CONSTANTS: +ASM_TYPE_DIRECTIVE(FSINCOS_CONSTANTS,@object) +data4 0x4B800000, 0xCB800000, 0x00000000,0x00000000 // two**24, -two**24 +data4 0x4E44152A, 0xA2F9836E, 0x00003FFE,0x00000000 // Inv_pi_by_2 +data4 0xCE81B9F1, 0xC84D32B0, 0x00004016,0x00000000 // P_0 +data4 0x2168C235, 0xC90FDAA2, 0x00003FFF,0x00000000 // P_1 +data4 0xFC8F8CBB, 0xECE675D1, 0x0000BFBD,0x00000000 // P_2 +data4 0xACC19C60, 0xB7ED8FBB, 0x0000BF7C,0x00000000 // P_3 +data4 0x5F000000, 0xDF000000, 0x00000000,0x00000000 // two_to_63, -two_to_63 +data4 0x6EC6B45A, 0xA397E504, 0x00003FE7,0x00000000 // Inv_P_0 +data4 0xDBD171A1, 0x8D848E89, 0x0000BFBF,0x00000000 // d_1 +data4 0x18A66F8E, 0xD5394C36, 0x0000BF7C,0x00000000 // d_2 +data4 0x2168C234, 0xC90FDAA2, 0x00003FFE,0x00000000 // pi_by_4 +data4 0x2168C234, 0xC90FDAA2, 0x0000BFFE,0x00000000 // neg_pi_by_4 +data4 0x3E000000, 0xBE000000, 0x00000000,0x00000000 // two**-3, -two**-3 +data4 0x2F000000, 0xAF000000, 0x9E000000,0x00000000 // two**-33, -two**-33, -two**-67 +data4 0xA21C0BC9, 0xCC8ABEBC, 0x00003FCE,0x00000000 // PP_8 +data4 0x720221DA, 0xD7468A05, 0x0000BFD6,0x00000000 // PP_7 +data4 0x640AD517, 0xB092382F, 0x00003FDE,0x00000000 // PP_6 +data4 0xD1EB75A4, 0xD7322B47, 0x0000BFE5,0x00000000 // PP_5 +data4 0xFFFFFFFE, 0xFFFFFFFF, 0x0000BFFD,0x00000000 // C_1 +data4 0x00000000, 0xAAAA0000, 0x0000BFFC,0x00000000 // PP_1_hi +data4 0xBAF69EEA, 0xB8EF1D2A, 0x00003FEC,0x00000000 // PP_4 +data4 0x0D03BB69, 0xD00D00D0, 0x0000BFF2,0x00000000 // PP_3 +data4 0x88888962, 0x88888888, 0x00003FF8,0x00000000 // PP_2 +data4 0xAAAB0000, 0xAAAAAAAA, 0x0000BFEC,0x00000000 // PP_1_lo +data4 0xC2B0FE52, 0xD56232EF, 0x00003FD2,0x00000000 // QQ_8 +data4 0x2B48DCA6, 0xC9C99ABA, 0x0000BFDA,0x00000000 // QQ_7 +data4 0x9C716658, 0x8F76C650, 0x00003FE2,0x00000000 // QQ_6 +data4 0xFDA8D0FC, 0x93F27DBA, 0x0000BFE9,0x00000000 // QQ_5 +data4 0xAAAAAAAA, 0xAAAAAAAA, 0x0000BFFC,0x00000000 // S_1 +data4 0x00000000, 0x80000000, 0x0000BFFE,0x00000000 // QQ_1 +data4 0x0C6E5041, 0xD00D00D0, 0x00003FEF,0x00000000 // QQ_4 +data4 0x0B607F60, 0xB60B60B6, 0x0000BFF5,0x00000000 // QQ_3 +data4 0xAAAAAA9B, 0xAAAAAAAA, 0x00003FFA,0x00000000 // QQ_2 +data4 0xFFFFFFFE, 0xFFFFFFFF, 0x0000BFFD,0x00000000 // C_1 +data4 0xAAAA719F, 0xAAAAAAAA, 0x00003FFA,0x00000000 // C_2 +data4 0x0356F994, 0xB60B60B6, 0x0000BFF5,0x00000000 // C_3 +data4 0xB2385EA9, 0xD00CFFD5, 0x00003FEF,0x00000000 // C_4 +data4 0x292A14CD, 0x93E4BD18, 0x0000BFE9,0x00000000 // C_5 +data4 0xAAAAAAAA, 0xAAAAAAAA, 0x0000BFFC,0x00000000 // S_1 +data4 0x888868DB, 0x88888888, 0x00003FF8,0x00000000 // S_2 +data4 0x055EFD4B, 0xD00D00D0, 0x0000BFF2,0x00000000 // S_3 +data4 0x839730B9, 0xB8EF1C5D, 0x00003FEC,0x00000000 // S_4 +data4 0xE5B3F492, 0xD71EA3A4, 0x0000BFE5,0x00000000 // S_5 +data4 0x38800000, 0xB8800000, 0x00000000 // two**-14, -two**-14 +ASM_SIZE_DIRECTIVE(FSINCOS_CONSTANTS) + +FR_Input_X = f8 +FR_Neg_Two_to_M3 = f32 +FR_Two_to_63 = f32 +FR_Two_to_24 = f33 +FR_Pi_by_4 = f33 +FR_Two_to_M14 = f34 +FR_Two_to_M33 = f35 +FR_Neg_Two_to_24 = f36 +FR_Neg_Pi_by_4 = f36 +FR_Neg_Two_to_M14 = f37 +FR_Neg_Two_to_M33 = f38 +FR_Neg_Two_to_M67 = f39 +FR_Inv_pi_by_2 = f40 +FR_N_float = f41 +FR_N_fix = f42 +FR_P_1 = f43 +FR_P_2 = f44 +FR_P_3 = f45 +FR_s = f46 +FR_w = f47 +FR_c = f48 +FR_r = f49 +FR_Z = f50 +FR_A = f51 +FR_a = f52 +FR_t = f53 +FR_U_1 = f54 +FR_U_2 = f55 +FR_C_1 = f56 +FR_C_2 = f57 +FR_C_3 = f58 +FR_C_4 = f59 +FR_C_5 = f60 +FR_S_1 = f61 +FR_S_2 = f62 +FR_S_3 = f63 +FR_S_4 = f64 +FR_S_5 = f65 +FR_poly_hi = f66 +FR_poly_lo = f67 +FR_r_hi = f68 +FR_r_lo = f69 +FR_rsq = f70 +FR_r_cubed = f71 +FR_C_hi = f72 +FR_N_0 = f73 +FR_d_1 = f74 +FR_V = f75 +FR_V_hi = f75 +FR_V_lo = f76 +FR_U_hi = f77 +FR_U_lo = f78 +FR_U_hiabs = f79 +FR_V_hiabs = f80 +FR_PP_8 = f81 +FR_QQ_8 = f81 +FR_PP_7 = f82 +FR_QQ_7 = f82 +FR_PP_6 = f83 +FR_QQ_6 = f83 +FR_PP_5 = f84 +FR_QQ_5 = f84 +FR_PP_4 = f85 +FR_QQ_4 = f85 +FR_PP_3 = f86 +FR_QQ_3 = f86 +FR_PP_2 = f87 +FR_QQ_2 = f87 +FR_QQ_1 = f88 +FR_N_0_fix = f89 +FR_Inv_P_0 = f90 +FR_corr = f91 +FR_poly = f92 +FR_d_2 = f93 +FR_Two_to_M3 = f94 +FR_Neg_Two_to_63 = f94 +FR_P_0 = f95 +FR_C_lo = f96 +FR_PP_1 = f97 +FR_PP_1_lo = f98 +FR_ArgPrime = f99 + +GR_Table_Base = r32 +GR_Table_Base1 = r33 +GR_i_0 = r34 +GR_i_1 = r35 +GR_N_Inc = r36 +GR_Sin_or_Cos = r37 + +GR_SAVE_B0 = r39 +GR_SAVE_GP = r40 +GR_SAVE_PFS = r41 + +.section .text +.proc __libm_sin_double_dbx# +.align 64 +__libm_sin_double_dbx: + +{ .mlx +alloc GR_Table_Base = ar.pfs,0,12,2,0 + movl GR_Sin_or_Cos = 0x0 ;; +} + +{ .mmi + nop.m 999 + addl GR_Table_Base = @ltoff(FSINCOS_CONSTANTS#), gp + nop.i 999 +} +;; + +{ .mmi + ld8 GR_Table_Base = [GR_Table_Base] + nop.m 999 + nop.i 999 +} +;; + + +{ .mib + nop.m 999 + nop.i 999 + br.cond.sptk L(SINCOS_CONTINUE) ;; +} + +.endp __libm_sin_double_dbx# +ASM_SIZE_DIRECTIVE(__libm_sin_double_dbx) + +.section .text +.proc __libm_cos_double_dbx# +__libm_cos_double_dbx: + +{ .mlx +alloc GR_Table_Base= ar.pfs,0,12,2,0 + movl GR_Sin_or_Cos = 0x1 ;; +} + +{ .mmi + nop.m 999 + addl GR_Table_Base = @ltoff(FSINCOS_CONSTANTS#), gp + nop.i 999 +} +;; + +{ .mmi + ld8 GR_Table_Base = [GR_Table_Base] + nop.m 999 + nop.i 999 +} +;; + +// +// Load Table Address +// +L(SINCOS_CONTINUE): + +{ .mmi + add GR_Table_Base1 = 96, GR_Table_Base + ldfs FR_Two_to_24 = [GR_Table_Base], 4 + nop.i 999 +} +;; + +{ .mmi + nop.m 999 +// +// Load 2**24, load 2**63. +// + ldfs FR_Neg_Two_to_24 = [GR_Table_Base], 12 + mov r41 = ar.pfs ;; +} + +{ .mfi + ldfs FR_Two_to_63 = [GR_Table_Base1], 4 +// +// Check for unnormals - unsupported operands. We do not want +// to generate denormal exception +// Check for NatVals, QNaNs, SNaNs, +/-Infs +// Check for EM unsupporteds +// Check for Zero +// + fclass.m.unc p6, p8 = FR_Input_X, 0x1E3 + mov r40 = gp ;; +} + +{ .mfi + nop.m 999 + fclass.nm.unc p8, p0 = FR_Input_X, 0x1FF +// GR_Sin_or_Cos denotes + mov r39 = b0 +} -// Final calculation -// result = C[m]*P + Q { .mfb - nop.m 999 - fma.d.s0 f8 = sincos_Cm, sincos_P, sincos_Q - br.ret.sptk b0 // Exit for common path -};; + ldfs FR_Neg_Two_to_63 = [GR_Table_Base1], 12 + fclass.m.unc p10, p0 = FR_Input_X, 0x007 +(p6) br.cond.spnt L(SINCOS_SPECIAL) ;; +} + +{ .mib + nop.m 999 + nop.i 999 +(p8) br.cond.spnt L(SINCOS_SPECIAL) ;; +} + +{ .mib + nop.m 999 + nop.i 999 +// +// Branch if +/- NaN, Inf. +// Load -2**24, load -2**63. +// +(p10) br.cond.spnt L(SINCOS_ZERO) ;; +} + +{ .mmb + ldfe FR_Inv_pi_by_2 = [GR_Table_Base], 16 + ldfe FR_Inv_P_0 = [GR_Table_Base1], 16 + nop.b 999 ;; +} + +{ .mmb + nop.m 999 + ldfe FR_d_1 = [GR_Table_Base1], 16 + nop.b 999 ;; +} +// +// Raise possible denormal operand flag with useful fcmp +// Is x <= -2**63 +// Load Inv_P_0 for pre-reduction +// Load Inv_pi_by_2 +// + +{ .mmb + ldfe FR_P_0 = [GR_Table_Base], 16 + ldfe FR_d_2 = [GR_Table_Base1], 16 + nop.b 999 ;; +} +// +// Load P_0 +// Load d_1 +// Is x >= 2**63 +// Is x <= -2**24? +// + +{ .mmi + ldfe FR_P_1 = [GR_Table_Base], 16 ;; +// +// Load P_1 +// Load d_2 +// Is x >= 2**24? +// + ldfe FR_P_2 = [GR_Table_Base], 16 + nop.i 999 ;; +} + +{ .mmf + nop.m 999 + ldfe FR_P_3 = [GR_Table_Base], 16 + fcmp.le.unc.s1 p7, p8 = FR_Input_X, FR_Neg_Two_to_24 +} + +{ .mfi + nop.m 999 +// +// Branch if +/- zero. +// Decide about the paths to take: +// If -2**24 < FR_Input_X < 2**24 - CASE 1 OR 2 +// OTHERWISE - CASE 3 OR 4 +// + fcmp.le.unc.s0 p10, p11 = FR_Input_X, FR_Neg_Two_to_63 + nop.i 999 ;; +} + +{ .mfi + nop.m 999 +(p8) fcmp.ge.s1 p7, p0 = FR_Input_X, FR_Two_to_24 + nop.i 999 +} + +{ .mfi + ldfe FR_Pi_by_4 = [GR_Table_Base1], 16 +(p11) fcmp.ge.s1 p10, p0 = FR_Input_X, FR_Two_to_63 + nop.i 999 ;; +} + +{ .mmi + ldfe FR_Neg_Pi_by_4 = [GR_Table_Base1], 16 ;; + ldfs FR_Two_to_M3 = [GR_Table_Base1], 4 + nop.i 999 ;; +} + +{ .mib + ldfs FR_Neg_Two_to_M3 = [GR_Table_Base1], 12 + nop.i 999 +// +// Load P_2 +// Load P_3 +// Load pi_by_4 +// Load neg_pi_by_4 +// Load 2**(-3) +// Load -2**(-3). +// +(p10) br.cond.spnt L(SINCOS_ARG_TOO_LARGE) ;; +} + +{ .mib + nop.m 999 + nop.i 999 +// +// Branch out if x >= 2**63. Use Payne-Hanek Reduction +// +(p7) br.cond.spnt L(SINCOS_LARGER_ARG) ;; +} + +{ .mfi + nop.m 999 +// +// Branch if Arg <= -2**24 or Arg >= 2**24 and use pre-reduction. +// + fma.s1 FR_N_float = FR_Input_X, FR_Inv_pi_by_2, f0 + nop.i 999 ;; +} + +{ .mfi + nop.m 999 + fcmp.lt.unc.s1 p6, p7 = FR_Input_X, FR_Pi_by_4 + nop.i 999 ;; +} + +{ .mfi + nop.m 999 +// +// Select the case when |Arg| < pi/4 +// Else Select the case when |Arg| >= pi/4 +// + fcvt.fx.s1 FR_N_fix = FR_N_float + nop.i 999 ;; +} + +{ .mfi + nop.m 999 +// +// N = Arg * 2/pi +// Check if Arg < pi/4 +// +(p6) fcmp.gt.s1 p6, p7 = FR_Input_X, FR_Neg_Pi_by_4 + nop.i 999 ;; +} +// +// Case 2: Convert integer N_fix back to normalized floating-point value. +// Case 1: p8 is only affected when p6 is set +// + +{ .mfi +(p7) ldfs FR_Two_to_M33 = [GR_Table_Base1], 4 +// +// Grab the integer part of N and call it N_fix +// +(p6) fmerge.se FR_r = FR_Input_X, FR_Input_X +// If |x| < pi/4, r = x and c = 0 +// lf |x| < pi/4, is x < 2**(-3). +// r = Arg +// c = 0 +(p6) mov GR_N_Inc = GR_Sin_or_Cos ;; +} + +{ .mmf + nop.m 999 +(p7) ldfs FR_Neg_Two_to_M33 = [GR_Table_Base1], 4 +(p6) fmerge.se FR_c = f0, f0 +} + +{ .mfi + nop.m 999 +(p6) fcmp.lt.unc.s1 p8, p9 = FR_Input_X, FR_Two_to_M3 + nop.i 999 ;; +} + +{ .mfi + nop.m 999 +// +// lf |x| < pi/4, is -2**(-3)< x < 2**(-3) - set p8. +// If |x| >= pi/4, +// Create the right N for |x| < pi/4 and otherwise +// Case 2: Place integer part of N in GP register +// +(p7) fcvt.xf FR_N_float = FR_N_fix + nop.i 999 ;; +} + +{ .mmf + nop.m 999 +(p7) getf.sig GR_N_Inc = FR_N_fix +(p8) fcmp.gt.s1 p8, p0 = FR_Input_X, FR_Neg_Two_to_M3 ;; +} + +{ .mib + nop.m 999 + nop.i 999 +// +// Load 2**(-33), -2**(-33) +// +(p8) br.cond.spnt L(SINCOS_SMALL_R) ;; +} + +{ .mib + nop.m 999 + nop.i 999 +(p6) br.cond.sptk L(SINCOS_NORMAL_R) ;; +} +// +// if |x| < pi/4, branch based on |x| < 2**(-3) or otherwise. +// +// +// In this branch, |x| >= pi/4. +// -////////// x = 0/Inf/NaN path ////////////////// -_SINCOS_SPECIAL_ARGS: -.pred.rel "mutex",p8,p9 -// sin(+/-0) = +/-0 -// sin(Inf) = NaN -// sin(NaN) = NaN { .mfi - nop.m 999 -(p8) fma.d.s0 f8 = f8, f0, f0 // sin(+/-0,NaN,Inf) - nop.i 999 + ldfs FR_Neg_Two_to_M67 = [GR_Table_Base1], 8 +// +// Load -2**(-67) +// + fnma.s1 FR_s = FR_N_float, FR_P_1, FR_Input_X +// +// w = N * P_2 +// s = -N * P_1 + Arg +// + add GR_N_Inc = GR_N_Inc, GR_Sin_or_Cos +} + +{ .mfi + nop.m 999 + fma.s1 FR_w = FR_N_float, FR_P_2, f0 + nop.i 999 ;; +} + +{ .mfi + nop.m 999 +// +// Adjust N_fix by N_inc to determine whether sine or +// cosine is being calculated +// + fcmp.lt.unc.s1 p7, p6 = FR_s, FR_Two_to_M33 + nop.i 999 ;; +} + +{ .mfi + nop.m 999 +(p7) fcmp.gt.s1 p7, p6 = FR_s, FR_Neg_Two_to_M33 + nop.i 999 ;; +} + +{ .mfi + nop.m 999 +// Remember x >= pi/4. +// Is s <= -2**(-33) or s >= 2**(-33) (p6) +// or -2**(-33) < s < 2**(-33) (p7) +(p6) fms.s1 FR_r = FR_s, f1, FR_w + nop.i 999 +} + +{ .mfi + nop.m 999 +(p7) fma.s1 FR_w = FR_N_float, FR_P_3, f0 + nop.i 999 ;; } -// cos(+/-0) = 1.0 -// cos(Inf) = NaN -// cos(NaN) = NaN + +{ .mfi + nop.m 999 +(p7) fma.s1 FR_U_1 = FR_N_float, FR_P_2, FR_w + nop.i 999 +} + +{ .mfi + nop.m 999 +(p6) fms.s1 FR_c = FR_s, f1, FR_r + nop.i 999 ;; +} + +{ .mfi + nop.m 999 +// +// For big s: r = s - w: No futher reduction is necessary +// For small s: w = N * P_3 (change sign) More reduction +// +(p6) fcmp.lt.unc.s1 p8, p9 = FR_r, FR_Two_to_M3 + nop.i 999 ;; +} + +{ .mfi + nop.m 999 +(p8) fcmp.gt.s1 p8, p9 = FR_r, FR_Neg_Two_to_M3 + nop.i 999 ;; +} + +{ .mfi + nop.m 999 +(p7) fms.s1 FR_r = FR_s, f1, FR_U_1 + nop.i 999 +} + { .mfb - nop.m 999 -(p9) fma.d.s0 f8 = f8, f0, f1 // cos(+/-0,NaN,Inf) - br.ret.sptk b0 // Exit for x = 0/Inf/NaN path -};; + nop.m 999 +// +// For big s: Is |r| < 2**(-3)? +// For big s: c = S - r +// For small s: U_1 = N * P_2 + w +// +// If p8 is set, prepare to branch to Small_R. +// If p9 is set, prepare to branch to Normal_R. +// For big s, r is complete here. +// +(p6) fms.s1 FR_c = FR_c, f1, FR_w +// +// For big s: c = c + w (w has not been negated.) +// For small s: r = S - U_1 +// +(p8) br.cond.spnt L(SINCOS_SMALL_R) ;; +} + +{ .mib + nop.m 999 + nop.i 999 +(p9) br.cond.sptk L(SINCOS_NORMAL_R) ;; +} + +{ .mfi +(p7) add GR_Table_Base1 = 224, GR_Table_Base1 +// +// Branch to SINCOS_SMALL_R or SINCOS_NORMAL_R +// +(p7) fms.s1 FR_U_2 = FR_N_float, FR_P_2, FR_U_1 +// +// c = S - U_1 +// r = S_1 * r +// +// +(p7) extr.u GR_i_1 = GR_N_Inc, 0, 1 +} + +{ .mmi + nop.m 999 ;; +// +// Get [i_0,i_1] - two lsb of N_fix_gr. +// Do dummy fmpy so inexact is always set. +// +(p7) cmp.eq.unc p9, p10 = 0x0, GR_i_1 +(p7) extr.u GR_i_0 = GR_N_Inc, 1, 1 ;; +} +// +// For small s: U_2 = N * P_2 - U_1 +// S_1 stored constant - grab the one stored with the +// coefficients. +// + +{ .mfi +(p7) ldfe FR_S_1 = [GR_Table_Base1], 16 +// +// Check if i_1 and i_0 != 0 +// +(p10) fma.s1 FR_poly = f0, f1, FR_Neg_Two_to_M67 +(p7) cmp.eq.unc p11, p12 = 0x0, GR_i_0 ;; +} + +{ .mfi + nop.m 999 +(p7) fms.s1 FR_s = FR_s, f1, FR_r + nop.i 999 +} + +{ .mfi + nop.m 999 +// +// S = S - r +// U_2 = U_2 + w +// load S_1 +// +(p7) fma.s1 FR_rsq = FR_r, FR_r, f0 + nop.i 999 ;; +} + +{ .mfi + nop.m 999 +(p7) fma.s1 FR_U_2 = FR_U_2, f1, FR_w + nop.i 999 +} -GLOBAL_IEEE754_END(cos) -//////////// x >= 2^27 - large arguments routine call //////////// -LOCAL_LIBM_ENTRY(__libm_callout_sincos) -_SINCOS_LARGE_ARGS: -.prologue { .mfi - mov sincos_GR_all_ones = -1 // 0xffffffff - nop.f 999 -.save ar.pfs,GR_SAVE_PFS - mov GR_SAVE_PFS = ar.pfs + nop.m 999 +(p7) fmerge.se FR_Input_X = FR_r, FR_r + nop.i 999 ;; +} + +{ .mfi + nop.m 999 +(p10) fma.s1 FR_Input_X = f0, f1, f1 + nop.i 999 ;; +} + +{ .mfi + nop.m 999 +// +// FR_rsq = r * r +// Save r as the result. +// +(p7) fms.s1 FR_c = FR_s, f1, FR_U_1 + nop.i 999 ;; +} + +{ .mfi + nop.m 999 +// +// if ( i_1 ==0) poly = c + S_1*r*r*r +// else Result = 1 +// +(p12) fnma.s1 FR_Input_X = FR_Input_X, f1, f0 + nop.i 999 +} + +{ .mfi + nop.m 999 +(p7) fma.s1 FR_r = FR_S_1, FR_r, f0 + nop.i 999 ;; +} + +{ .mfi + nop.m 999 +(p7) fma.d.s0 FR_S_1 = FR_S_1, FR_S_1, f0 + nop.i 999 ;; +} + +{ .mfi + nop.m 999 +// +// If i_1 != 0, poly = 2**(-67) +// +(p7) fms.s1 FR_c = FR_c, f1, FR_U_2 + nop.i 999 ;; +} + +{ .mfi + nop.m 999 +// +// c = c - U_2 +// +(p9) fma.s1 FR_poly = FR_r, FR_rsq, FR_c + nop.i 999 ;; +} + +{ .mfi + nop.m 999 +// +// i_0 != 0, so Result = -Result +// +(p11) fma.d.s0 FR_Input_X = FR_Input_X, f1, FR_poly + nop.i 999 ;; +} + +{ .mfb + nop.m 999 +(p12) fms.d.s0 FR_Input_X = FR_Input_X, f1, FR_poly +// +// if (i_0 == 0), Result = Result + poly +// else Result = Result - poly +// + br.ret.sptk b0 ;; +} +L(SINCOS_LARGER_ARG): + +{ .mfi + nop.m 999 + fma.s1 FR_N_0 = FR_Input_X, FR_Inv_P_0, f0 + nop.i 999 +} +;; + +// This path for argument > 2*24 +// Adjust table_ptr1 to beginning of table. +// + +{ .mmi + nop.m 999 + addl GR_Table_Base = @ltoff(FSINCOS_CONSTANTS#), gp + nop.i 999 +} +;; + +{ .mmi + ld8 GR_Table_Base = [GR_Table_Base] + nop.m 999 + nop.i 999 } ;; + +// +// Point to 2*-14 +// N_0 = Arg * Inv_P_0 +// + +{ .mmi + add GR_Table_Base = 688, GR_Table_Base ;; + ldfs FR_Two_to_M14 = [GR_Table_Base], 4 + nop.i 999 ;; +} + { .mfi - mov GR_SAVE_GP = gp - nop.f 999 -.save b0, GR_SAVE_B0 - mov GR_SAVE_B0 = b0 + ldfs FR_Neg_Two_to_M14 = [GR_Table_Base], 0 + nop.f 999 + nop.i 999 ;; } -.body -{ .mbb - setf.sig sincos_save_tmp = sincos_GR_all_ones// inexact set - nop.b 999 -(p8) br.call.sptk.many b0 = __libm_sin_large# // sin(large_X) +{ .mfi + nop.m 999 +// +// Load values 2**(-14) and -2**(-14) +// + 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 ;; +} -{ .mbb - cmp.ne p9,p0 = sincos_r_sincos, r0 // set p9 if cos - nop.b 999 -(p9) br.call.sptk.many b0 = __libm_cos_large# // cos(large_X) -};; +{ .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 + nop.i 999 ;; +} + +{ .mfi + nop.m 999 +// +// For |s| > 2**(-14) r = S + w (r complete) +// Else U_hi = N_0 * d_1 +// +(p9) fma.s1 FR_V_hi = FR_N_float, FR_P_2, f0 + nop.i 999 +} + +{ .mfi + nop.m 999 +(p9) fma.s1 FR_U_hi = FR_N_0, FR_d_1, f0 + nop.i 999 ;; +} + +{ .mfi + nop.m 999 +// +// Either S <= -2**(-14) or S >= 2**(-14) +// or -2**(-14) < s < 2**(-14) +// +(p8) fma.s1 FR_r = FR_s, f1, FR_w + nop.i 999 +} + +{ .mfi + nop.m 999 +(p9) fma.s1 FR_w = FR_N_float, FR_P_3, f0 + nop.i 999 ;; +} + +{ .mfi + nop.m 999 +// +// We need abs of both U_hi and V_hi - don't +// worry about switched sign of V_hi. +// +(p9) fms.s1 FR_A = FR_U_hi, f1, FR_V_hi + nop.i 999 +} + +{ .mfi + nop.m 999 +// +// Big s: finish up c = (S - r) + w (c complete) +// Case 4: A = U_hi + V_hi +// Note: Worry about switched sign of V_hi, so subtract instead of add. +// +(p9) fnma.s1 FR_V_lo = FR_N_float, FR_P_2, FR_V_hi + nop.i 999 ;; +} + +{ .mfi + nop.m 999 +(p9) fms.s1 FR_U_lo = FR_N_0, FR_d_1, FR_U_hi + nop.i 999 ;; +} + +{ .mfi + nop.m 999 +(p9) fmerge.s FR_V_hiabs = f0, FR_V_hi + nop.i 999 +} + +{ .mfi + nop.m 999 +// For big s: c = S - r +// For small s do more work: U_lo = N_0 * d_1 - U_hi +// +(p9) fmerge.s FR_U_hiabs = f0, FR_U_hi + nop.i 999 ;; +} + +{ .mfi + nop.m 999 +// +// For big s: Is |r| < 2**(-3) +// For big s: if p12 set, prepare to branch to Small_R. +// For big s: If p13 set, prepare to branch to Normal_R. +// +(p8) fms.s1 FR_c = FR_s, f1, FR_r + nop.i 999 +} + +{ .mfi + nop.m 999 +// +// For small S: V_hi = N * P_2 +// w = N * P_3 +// Note the product does not include the (-) as in the writeup +// so (-) missing for V_hi and w. +// +(p8) fcmp.lt.unc.s1 p12, p13 = FR_r, FR_Two_to_M3 + nop.i 999 ;; +} { .mfi - mov gp = GR_SAVE_GP - fma.d.s0 f8 = f8, f1, f0 // Round result to double - mov b0 = GR_SAVE_B0 + nop.m 999 +(p12) fcmp.gt.s1 p12, p13 = FR_r, FR_Neg_Two_to_M3 + nop.i 999 ;; } -// Force inexact set + { .mfi - nop.m 999 - fmpy.s0 sincos_save_tmp = sincos_save_tmp, sincos_save_tmp - nop.i 999 + 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 L(SINCOS_SMALL_R) ;; +} + +{ .mib + nop.m 999 + nop.i 999 +(p13) br.cond.sptk L(SINCOS_NORMAL_R) ;; +} + +{ .mfi + nop.m 999 +// +// Big s: Vector off when |r| < 2**(-3). Recall that p8 will be true. +// The remaining stuff is for Case 4. +// Small s: V_lo = N * P_2 + U_hi (U_hi is in place of V_hi in writeup) +// Note: the (-) is still missing for V_lo. +// Small s: w = w + N_0 * d_2 +// Note: the (-) is now incorporated in w. +// +(p9) fcmp.ge.unc.s1 p10, p11 = FR_U_hiabs, FR_V_hiabs + extr.u GR_i_1 = GR_N_Inc, 0, 1 ;; +} + +{ .mfi + nop.m 999 +// +// C_hi = S + A +// +(p9) fma.s1 FR_t = FR_U_lo, f1, FR_V_lo + extr.u GR_i_0 = GR_N_Inc, 1, 1 ;; +} + +{ .mfi + nop.m 999 +// +// t = U_lo + V_lo +// +// +(p10) fms.s1 FR_a = FR_U_hi, f1, FR_A + nop.i 999 ;; +} + +{ .mfi + nop.m 999 +(p11) fma.s1 FR_a = FR_V_hi, f1, FR_A + nop.i 999 +} +;; + +{ .mmi + nop.m 999 + addl GR_Table_Base = @ltoff(FSINCOS_CONSTANTS#), gp + nop.i 999 +} +;; + +{ .mmi + ld8 GR_Table_Base = [GR_Table_Base] + nop.m 999 + nop.i 999 +} +;; + + +{ .mfi + add GR_Table_Base = 528, GR_Table_Base +// +// Is U_hiabs >= V_hiabs? +// +(p9) fma.s1 FR_C_hi = FR_s, f1, FR_A + nop.i 999 ;; +} + +{ .mmi + ldfe FR_C_1 = [GR_Table_Base], 16 ;; + ldfe FR_C_2 = [GR_Table_Base], 64 + nop.i 999 ;; +} + +{ .mmf + nop.m 999 +// +// c = c + C_lo finished. +// Load C_2 +// + ldfe FR_S_1 = [GR_Table_Base], 16 +// +// C_lo = S - C_hi +// + fma.s1 FR_t = FR_t, f1, FR_w ;; +} +// +// r and c have been computed. +// Make sure ftz mode is set - should be automatic when using wre +// |r| < 2**(-3) +// Get [i_0,i_1] - two lsb of N_fix. +// Load S_1 +// + +{ .mfi + ldfe FR_S_2 = [GR_Table_Base], 64 +// +// t = t + w +// +(p10) fms.s1 FR_a = FR_a, f1, FR_V_hi + cmp.eq.unc p9, p10 = 0x0, GR_i_0 +} + +{ .mfi + nop.m 999 +// +// For larger u than v: a = U_hi - A +// Else a = V_hi - A (do an add to account for missing (-) on V_hi +// + fms.s1 FR_C_lo = FR_s, f1, FR_C_hi + nop.i 999 ;; +} + +{ .mfi + nop.m 999 +(p11) fms.s1 FR_a = FR_U_hi, f1, FR_a + cmp.eq.unc p11, p12 = 0x0, GR_i_1 +} + +{ .mfi + nop.m 999 +// +// If u > v: a = (U_hi - A) + V_hi +// Else a = (V_hi - A) + U_hi +// In each case account for negative missing from V_hi. +// + 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 +// Adjust Table_Base to beginning of table +// + 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 +// +// Table_Base points to C_1 +// 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 +// +(p11) fma.s1 FR_Input_X = f0, f1, FR_r + nop.i 999 ;; +} + +{ .mfi + nop.m 999 +(p12) fma.s1 FR_Input_X = f0, f1, f1 + nop.i 999 ;; +} + +{ .mfi + nop.m 999 +// +// Compute r_cube = FR_rsq * r +// +(p11) fma.s1 FR_poly = FR_rsq, FR_S_2, FR_S_1 + nop.i 999 ;; +} + +{ .mfi + nop.m 999 +(p12) fma.s1 FR_poly = FR_rsq, FR_C_2, FR_C_1 + nop.i 999 +} + +{ .mfi + nop.m 999 +// +// Compute FR_rsq = r * r +// Is i_1 == 0 ? +// + 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 +// +(p10) fms.s1 FR_Input_X = f0, f1, FR_Input_X + nop.i 999 ;; +} + +{ .mfi + nop.m 999 +// +// if i_1 ==0: Result = r +// else Result = 1.0 +// +(p11) fma.s1 FR_poly = FR_r_cubed, FR_poly, FR_c + nop.i 999 ;; +} + +{ .mfi + nop.m 999 +(p12) fma.s1 FR_poly = FR_rsq, FR_poly, f0 + nop.i 999 ;; +} + +{ .mfi + nop.m 999 +// +// if i_0 !=0: Result = -Result +// +(p9) fma.d.s0 FR_Input_X = FR_Input_X, f1, FR_poly + nop.i 999 ;; +} + +{ .mfb + nop.m 999 +(p10) fms.d.s0 FR_Input_X = FR_Input_X, f1, FR_poly +// +// if i_0 == 0: Result = Result + poly +// else Result = Result - poly +// + br.ret.sptk b0 ;; +} +L(SINCOS_SMALL_R): + +{ .mii + nop.m 999 + extr.u GR_i_1 = GR_N_Inc, 0, 1 ;; +// +// +// Compare both i_1 and i_0 with 0. +// if i_1 == 0, set p9. +// if i_0 == 0, set p11. +// + cmp.eq.unc p9, p10 = 0x0, GR_i_1 ;; +} + +{ .mfi + nop.m 999 + fma.s1 FR_rsq = FR_r, FR_r, f0 + extr.u GR_i_0 = GR_N_Inc, 1, 1 ;; +} + +{ .mfi + nop.m 999 +// +// Z = Z * FR_rsq +// +(p10) fnma.s1 FR_c = FR_c, FR_r, f0 + cmp.eq.unc p11, p12 = 0x0, GR_i_0 +} +;; + +// ****************************************************************** +// ****************************************************************** +// ****************************************************************** +// r and c have been computed. +// We know whether this is the sine or cosine routine. +// Make sure ftz mode is set - should be automatic when using wre +// |r| < 2**(-3) +// +// Set table_ptr1 to beginning of constant table. +// Get [i_0,i_1] - two lsb of N_fix_gr. +// + +{ .mmi + nop.m 999 + addl GR_Table_Base = @ltoff(FSINCOS_CONSTANTS#), gp + nop.i 999 +} +;; + +{ .mmi + ld8 GR_Table_Base = [GR_Table_Base] + nop.m 999 + nop.i 999 +} +;; + + +// +// Set table_ptr1 to point to S_5. +// Set table_ptr1 to point to C_5. +// Compute FR_rsq = r * r +// + +{ .mfi +(p9) add GR_Table_Base = 672, GR_Table_Base +(p10) fmerge.s FR_r = f1, f1 +(p10) add GR_Table_Base = 592, GR_Table_Base ;; +} +// +// Set table_ptr1 to point to S_5. +// Set table_ptr1 to point to C_5. +// + +{ .mmi +(p9) ldfe FR_S_5 = [GR_Table_Base], -16 ;; +// +// if (i_1 == 0) load S_5 +// if (i_1 != 0) load C_5 +// +(p9) ldfe FR_S_4 = [GR_Table_Base], -16 + nop.i 999 ;; +} + +{ .mmf +(p10) ldfe FR_C_5 = [GR_Table_Base], -16 +// +// Z = FR_rsq * FR_rsq +// +(p9) ldfe FR_S_3 = [GR_Table_Base], -16 +// +// Compute FR_rsq = r * r +// if (i_1 == 0) load S_4 +// if (i_1 != 0) load C_4 +// + fma.s1 FR_Z = FR_rsq, FR_rsq, f0 ;; +} +// +// if (i_1 == 0) load S_3 +// if (i_1 != 0) load C_3 +// + +{ .mmi +(p9) ldfe FR_S_2 = [GR_Table_Base], -16 ;; +// +// if (i_1 == 0) load S_2 +// if (i_1 != 0) load C_2 +// +(p9) ldfe FR_S_1 = [GR_Table_Base], -16 + nop.i 999 +} + +{ .mmi +(p10) ldfe FR_C_4 = [GR_Table_Base], -16 ;; +(p10) ldfe FR_C_3 = [GR_Table_Base], -16 + nop.i 999 ;; +} + +{ .mmi +(p10) ldfe FR_C_2 = [GR_Table_Base], -16 ;; +(p10) ldfe FR_C_1 = [GR_Table_Base], -16 + nop.i 999 +} + +{ .mfi + nop.m 999 +// +// if (i_1 != 0): +// poly_lo = FR_rsq * C_5 + C_4 +// poly_hi = FR_rsq * C_2 + C_1 +// +(p9) fma.s1 FR_Z = FR_Z, FR_r, f0 + nop.i 999 ;; +} + +{ .mfi + nop.m 999 +// +// if (i_1 == 0) load S_1 +// if (i_1 != 0) load C_1 +// +(p9) fma.s1 FR_poly_lo = FR_rsq, FR_S_5, FR_S_4 + nop.i 999 +} + +{ .mfi + nop.m 999 +// +// c = -c * r +// dummy fmpy's to flag inexact. +// +(p9) fma.d.s0 FR_S_4 = FR_S_4, FR_S_4, f0 + nop.i 999 ;; +} + +{ .mfi + nop.m 999 +// +// poly_lo = FR_rsq * poly_lo + C_3 +// poly_hi = FR_rsq * poly_hi +// + fma.s1 FR_Z = FR_Z, FR_rsq, f0 + nop.i 999 ;; +} + +{ .mfi + nop.m 999 +(p9) fma.s1 FR_poly_hi = FR_rsq, FR_S_2, FR_S_1 + nop.i 999 +} + +{ .mfi + nop.m 999 +// +// if (i_1 == 0): +// poly_lo = FR_rsq * S_5 + S_4 +// poly_hi = FR_rsq * S_2 + S_1 +// +(p10) fma.s1 FR_poly_lo = FR_rsq, FR_C_5, FR_C_4 + nop.i 999 ;; +} + +{ .mfi + nop.m 999 +// +// if (i_1 == 0): +// Z = Z * r for only one of the small r cases - not there +// in original implementation notes. +// +(p9) fma.s1 FR_poly_lo = FR_rsq, FR_poly_lo, FR_S_3 + nop.i 999 ;; +} + +{ .mfi + nop.m 999 +(p10) fma.s1 FR_poly_hi = FR_rsq, FR_C_2, FR_C_1 + nop.i 999 +} + +{ .mfi + nop.m 999 +(p10) fma.d.s0 FR_C_1 = FR_C_1, FR_C_1, f0 + nop.i 999 ;; +} + +{ .mfi + nop.m 999 +(p9) fma.s1 FR_poly_hi = FR_poly_hi, FR_rsq, f0 + nop.i 999 +} + +{ .mfi + nop.m 999 +// +// poly_lo = FR_rsq * poly_lo + S_3 +// poly_hi = FR_rsq * poly_hi +// +(p10) fma.s1 FR_poly_lo = FR_rsq, FR_poly_lo, FR_C_3 + nop.i 999 ;; +} + +{ .mfi + nop.m 999 +(p10) fma.s1 FR_poly_hi = FR_poly_hi, FR_rsq, f0 + nop.i 999 ;; +} + +{ .mfi + nop.m 999 +// +// if (i_1 == 0): dummy fmpy's to flag inexact +// r = 1 +// +(p9) fma.s1 FR_poly_hi = FR_r, FR_poly_hi, f0 + nop.i 999 +} + +{ .mfi + nop.m 999 +// +// poly_hi = r * poly_hi +// + fma.s1 FR_poly = FR_Z, FR_poly_lo, FR_c + nop.i 999 ;; +} + +{ .mfi + nop.m 999 +(p12) fms.s1 FR_r = f0, f1, FR_r + nop.i 999 ;; +} + +{ .mfi + nop.m 999 +// +// poly_hi = Z * poly_lo + c +// if i_0 == 1: r = -r +// + fma.s1 FR_poly = FR_poly, f1, FR_poly_hi + nop.i 999 ;; +} + +{ .mfi + nop.m 999 +(p12) fms.d.s0 FR_Input_X = FR_r, f1, FR_poly + nop.i 999 +} + +{ .mfb + nop.m 999 +// +// poly = poly + poly_hi +// +(p11) fma.d.s0 FR_Input_X = FR_r, f1, FR_poly +// +// if (i_0 == 0) Result = r + poly +// if (i_0 != 0) Result = r - poly +// + br.ret.sptk b0 ;; +} +L(SINCOS_NORMAL_R): + +{ .mii + nop.m 999 + extr.u GR_i_1 = GR_N_Inc, 0, 1 ;; +// +// Set table_ptr1 and table_ptr2 to base address of +// constant table. + cmp.eq.unc p9, p10 = 0x0, GR_i_1 ;; +} + +{ .mfi + nop.m 999 + fma.s1 FR_rsq = FR_r, FR_r, f0 + extr.u GR_i_0 = GR_N_Inc, 1, 1 ;; +} + +{ .mfi + nop.m 999 + frcpa.s1 FR_r_hi, p6 = f1, FR_r + cmp.eq.unc p11, p12 = 0x0, GR_i_0 +} +;; + +// ****************************************************************** +// ****************************************************************** +// ****************************************************************** +// +// r and c have been computed. +// We known whether this is the sine or cosine routine. +// Make sure ftz mode is set - should be automatic when using wre +// Get [i_0,i_1] - two lsb of N_fix_gr alone. +// + +{ .mmi + nop.m 999 + addl GR_Table_Base = @ltoff(FSINCOS_CONSTANTS#), gp + nop.i 999 +} +;; + +{ .mmi + ld8 GR_Table_Base = [GR_Table_Base] + nop.m 999 + nop.i 999 +} +;; + + +{ .mfi +(p10) add GR_Table_Base = 384, GR_Table_Base +(p12) fms.s1 FR_Input_X = f0, f1, f1 +(p9) add GR_Table_Base = 224, GR_Table_Base ;; +} + +{ .mmf + nop.m 999 +(p10) ldfe FR_QQ_8 = [GR_Table_Base], 16 +// +// if (i_1==0) poly = poly * FR_rsq + PP_1_lo +// else poly = FR_rsq * poly +// +(p11) fma.s1 FR_Input_X = f0, f1, f1 ;; +} + +{ .mmf +(p10) ldfe FR_QQ_7 = [GR_Table_Base], 16 +// +// Adjust table pointers based on i_0 +// Compute rsq = r * r +// +(p9) ldfe FR_PP_8 = [GR_Table_Base], 16 + fma.s1 FR_r_cubed = FR_r, FR_rsq, f0 ;; +} + +{ .mmf +(p9) ldfe FR_PP_7 = [GR_Table_Base], 16 +(p10) ldfe FR_QQ_6 = [GR_Table_Base], 16 +// +// Load PP_8 and QQ_8; PP_7 and QQ_7 +// + frcpa.s1 FR_r_hi, p6 = f1, FR_r_hi ;; +} +// +// if (i_1==0) poly = PP_7 + FR_rsq * PP_8. +// else poly = QQ_7 + FR_rsq * QQ_8. +// + +{ .mmb +(p9) ldfe FR_PP_6 = [GR_Table_Base], 16 +(p10) ldfe FR_QQ_5 = [GR_Table_Base], 16 + nop.b 999 ;; +} + +{ .mmb +(p9) ldfe FR_PP_5 = [GR_Table_Base], 16 +(p10) ldfe FR_S_1 = [GR_Table_Base], 16 + nop.b 999 ;; +} + +{ .mmb +(p10) ldfe FR_QQ_1 = [GR_Table_Base], 16 +(p9) ldfe FR_C_1 = [GR_Table_Base], 16 + nop.b 999 ;; +} + +{ .mmi +(p10) ldfe FR_QQ_4 = [GR_Table_Base], 16 ;; +(p9) ldfe FR_PP_1 = [GR_Table_Base], 16 + nop.i 999 ;; +} + +{ .mmf +(p10) ldfe FR_QQ_3 = [GR_Table_Base], 16 +// +// if (i_1=0) corr = corr + c*c +// else corr = corr * c +// +(p9) ldfe FR_PP_4 = [GR_Table_Base], 16 +(p10) fma.s1 FR_poly = FR_rsq, FR_QQ_8, FR_QQ_7 ;; +} +// +// if (i_1=0) poly = rsq * poly + PP_5 +// else poly = rsq * poly + QQ_5 +// Load PP_4 or QQ_4 +// + +{ .mmf +(p9) ldfe FR_PP_3 = [GR_Table_Base], 16 +(p10) ldfe FR_QQ_2 = [GR_Table_Base], 16 +// +// r_hi = frcpa(frcpa(r)). +// r_cube = r * FR_rsq. +// +(p9) fma.s1 FR_poly = FR_rsq, FR_PP_8, FR_PP_7 ;; +} +// +// Do dummy multiplies so inexact is always set. +// + +{ .mfi +(p9) ldfe FR_PP_2 = [GR_Table_Base], 16 +// +// r_lo = r - r_hi +// +(p9) fma.s1 FR_U_lo = FR_r_hi, FR_r_hi, f0 + nop.i 999 ;; +} + +{ .mmf + nop.m 999 +(p9) ldfe FR_PP_1_lo = [GR_Table_Base], 16 +(p10) fma.s1 FR_corr = FR_S_1, FR_r_cubed, FR_r +} + +{ .mfi + nop.m 999 +(p10) fma.s1 FR_poly = FR_rsq, FR_poly, FR_QQ_6 + nop.i 999 ;; +} + +{ .mfi + nop.m 999 +// +// if (i_1=0) U_lo = r_hi * r_hi +// else U_lo = r_hi + r +// +(p9) fma.s1 FR_corr = FR_C_1, FR_rsq, f0 + nop.i 999 ;; +} + +{ .mfi + nop.m 999 +// +// if (i_1=0) corr = C_1 * rsq +// else corr = S_1 * r_cubed + r +// +(p9) fma.s1 FR_poly = FR_rsq, FR_poly, FR_PP_6 + nop.i 999 +} + +{ .mfi + nop.m 999 +(p10) fma.s1 FR_U_lo = FR_r_hi, f1, FR_r + nop.i 999 ;; +} + +{ .mfi + nop.m 999 +// +// if (i_1=0) U_hi = r_hi + U_hi +// else U_hi = QQ_1 * U_hi + 1 +// +(p9) fma.s1 FR_U_lo = FR_r, FR_r_hi, FR_U_lo + nop.i 999 +} + +{ .mfi + nop.m 999 +// +// U_hi = r_hi * r_hi +// + fms.s1 FR_r_lo = FR_r, f1, FR_r_hi + nop.i 999 ;; +} + +{ .mfi + nop.m 999 +// +// Load PP_1, PP_6, PP_5, and C_1 +// Load QQ_1, QQ_6, QQ_5, and S_1 +// + fma.s1 FR_U_hi = FR_r_hi, FR_r_hi, f0 + nop.i 999 ;; +} + +{ .mfi + nop.m 999 +(p10) fma.s1 FR_poly = FR_rsq, FR_poly, FR_QQ_5 + nop.i 999 +} + +{ .mfi + nop.m 999 +(p10) fnma.s1 FR_corr = FR_corr, FR_c, f0 + nop.i 999 ;; +} + +{ .mfi + nop.m 999 +// +// if (i_1=0) U_lo = r * r_hi + U_lo +// else U_lo = r_lo * U_lo +// +(p9) fma.s1 FR_corr = FR_corr, FR_c, FR_c + nop.i 999 ;; +} + +{ .mfi + nop.m 999 +(p9) fma.s1 FR_poly = FR_rsq, FR_poly, FR_PP_5 + nop.i 999 +} + +{ .mfi + nop.m 999 +// +// if (i_1 =0) U_hi = r + U_hi +// if (i_1 =0) U_lo = r_lo * U_lo +// +// +(p9) fma.d.s0 FR_PP_5 = FR_PP_5, FR_PP_4, f0 + nop.i 999 ;; +} + +{ .mfi + nop.m 999 +(p9) fma.s1 FR_U_lo = FR_r, FR_r, FR_U_lo + nop.i 999 +} + +{ .mfi + nop.m 999 +(p10) fma.s1 FR_U_lo = FR_r_lo, FR_U_lo, f0 + nop.i 999 ;; +} + +{ .mfi + nop.m 999 +// +// if (i_1=0) poly = poly * rsq + PP_6 +// else poly = poly * rsq + QQ_6 +// +(p9) fma.s1 FR_U_hi = FR_r_hi, FR_U_hi, f0 + nop.i 999 ;; +} + +{ .mfi + nop.m 999 +(p10) fma.s1 FR_poly = FR_rsq, FR_poly, FR_QQ_4 + nop.i 999 +} + +{ .mfi + nop.m 999 +(p10) fma.s1 FR_U_hi = FR_QQ_1, FR_U_hi, f1 + nop.i 999 ;; +} + +{ .mfi + nop.m 999 +(p10) fma.d.s0 FR_QQ_5 = FR_QQ_5, FR_QQ_5, f0 + nop.i 999 ;; +} + +{ .mfi + nop.m 999 +// +// if (i_1!=0) U_hi = PP_1 * U_hi +// if (i_1!=0) U_lo = r * r + U_lo +// Load PP_3 or QQ_3 +// +(p9) fma.s1 FR_poly = FR_rsq, FR_poly, FR_PP_4 + nop.i 999 ;; +} + +{ .mfi + nop.m 999 +(p9) fma.s1 FR_U_lo = FR_r_lo, FR_U_lo, f0 + nop.i 999 +} + +{ .mfi + nop.m 999 +(p10) fma.s1 FR_U_lo = FR_QQ_1,FR_U_lo, f0 + nop.i 999 ;; +} + +{ .mfi + nop.m 999 +(p9) fma.s1 FR_U_hi = FR_PP_1, FR_U_hi, f0 + nop.i 999 ;; +} + +{ .mfi + nop.m 999 +(p10) fma.s1 FR_poly = FR_rsq, FR_poly, FR_QQ_3 + nop.i 999 ;; +} + +{ .mfi + nop.m 999 +// +// Load PP_2, QQ_2 +// +(p9) fma.s1 FR_poly = FR_rsq, FR_poly, FR_PP_3 + nop.i 999 ;; +} + +{ .mfi + nop.m 999 +// +// if (i_1==0) poly = FR_rsq * poly + PP_3 +// else poly = FR_rsq * poly + QQ_3 +// Load PP_1_lo +// +(p9) fma.s1 FR_U_lo = FR_PP_1, FR_U_lo, f0 + nop.i 999 ;; +} + +{ .mfi + nop.m 999 +// +// if (i_1 =0) poly = poly * rsq + pp_r4 +// else poly = poly * rsq + qq_r4 +// +(p9) fma.s1 FR_U_hi = FR_r, f1, FR_U_hi + nop.i 999 ;; +} + +{ .mfi + nop.m 999 +(p10) fma.s1 FR_poly = FR_rsq, FR_poly, FR_QQ_2 + nop.i 999 ;; +} + +{ .mfi + nop.m 999 +// +// if (i_1==0) U_lo = PP_1_hi * U_lo +// else U_lo = QQ_1 * U_lo +// +(p9) fma.s1 FR_poly = FR_rsq, FR_poly, FR_PP_2 + nop.i 999 ;; +} + +{ .mfi + nop.m 999 +// +// if (i_0==0) Result = 1 +// else Result = -1 +// + fma.s1 FR_V = FR_U_lo, f1, FR_corr + nop.i 999 ;; +} + +{ .mfi + nop.m 999 +(p10) fma.s1 FR_poly = FR_rsq, FR_poly, f0 + nop.i 999 ;; +} + +{ .mfi + nop.m 999 +// +// if (i_1==0) poly = FR_rsq * poly + PP_2 +// else poly = FR_rsq * poly + QQ_2 +// +(p9) fma.s1 FR_poly = FR_rsq, FR_poly, FR_PP_1_lo + nop.i 999 ;; +} + +{ .mfi + nop.m 999 +(p10) fma.s1 FR_poly = FR_rsq, FR_poly, f0 + nop.i 999 ;; +} + +{ .mfi + nop.m 999 +// +// V = U_lo + corr +// +(p9) fma.s1 FR_poly = FR_r_cubed, FR_poly, f0 + nop.i 999 ;; +} + +{ .mfi + nop.m 999 +// +// if (i_1==0) poly = r_cube * poly +// else poly = FR_rsq * poly +// + fma.s1 FR_V = FR_poly, f1, FR_V + nop.i 999 ;; +} + +{ .mfi + nop.m 999 +(p12) fms.d.s0 FR_Input_X = FR_Input_X, FR_U_hi, FR_V + nop.i 999 +} + +{ .mfb + nop.m 999 +// +// V = V + poly +// +(p11) fma.d.s0 FR_Input_X = FR_Input_X, FR_U_hi, FR_V +// +// if (i_0==0) Result = Result * U_hi + V +// else Result = Result * U_hi - V +// + br.ret.sptk b0 ;; +} + +// +// If cosine, FR_Input_X = 1 +// If sine, FR_Input_X = +/-Zero (Input FR_Input_X) +// Results are exact, no exceptions +// +L(SINCOS_ZERO): + +{ .mmb + cmp.eq.unc p6, p7 = 0x1, GR_Sin_or_Cos + nop.m 999 + nop.b 999 ;; +} + +{ .mfi + nop.m 999 +(p7) fmerge.s FR_Input_X = FR_Input_X, FR_Input_X + nop.i 999 +} + +{ .mfb + nop.m 999 +(p6) fmerge.s FR_Input_X = f1, f1 + br.ret.sptk b0 ;; +} + +L(SINCOS_SPECIAL): + +// +// Path for Arg = +/- QNaN, SNaN, Inf +// Invalid can be raised. SNaNs +// become QNaNs +// + +{ .mfb + nop.m 999 + fmpy.d.s0 FR_Input_X = FR_Input_X, f0 + br.ret.sptk b0 ;; +} +.endp __libm_cos_double_dbx# +ASM_SIZE_DIRECTIVE(__libm_cos_double_dbx#) + + + +// +// Call int pi_by_2_reduce(double* x, double *y) +// for |arguments| >= 2**63 +// Address to save r and c as double +// +// +// psp sp+64 +// sp+48 -> f0 c +// r45 sp+32 -> f0 r +// r44 -> sp+16 -> InputX +// sp sp -> scratch provided to callee + + + +.proc __libm_callout_2 +__libm_callout_2: +L(SINCOS_ARG_TOO_LARGE): + +.prologue +{ .mfi + add r45=-32,sp // Parameter: r address + nop.f 0 +.save ar.pfs,GR_SAVE_PFS + mov GR_SAVE_PFS=ar.pfs // Save ar.pfs +} +{ .mfi +.fframe 64 + add sp=-64,sp // Create new stack + nop.f 0 + mov GR_SAVE_GP=gp // Save gp +};; +{ .mmi + stfe [r45] = f0,16 // Clear Parameter r on stack + add r44 = 16,sp // Parameter x address +.save b0, GR_SAVE_B0 + mov GR_SAVE_B0=b0 // Save b0 +};; +.body +{ .mib + stfe [r45] = f0,-16 // Clear Parameter c on stack + nop.i 0 + nop.b 0 +} +{ .mib + stfe [r44] = FR_Input_X // Store Parameter x on stack + nop.i 0 + br.call.sptk b0=__libm_pi_by_2_reduce# ;; };; + +{ .mii + ldfe FR_Input_X =[r44],16 +// +// Get r and c off stack +// + adds GR_Table_Base1 = -16, GR_Table_Base1 +// +// Get r and c off stack +// + add GR_N_Inc = GR_Sin_or_Cos,r8 ;; +} +{ .mmb + ldfe FR_r =[r45],16 +// +// Get X off the stack +// Readjust Table ptr +// + ldfs FR_Two_to_M3 = [GR_Table_Base1],4 + nop.b 999 ;; +} +{ .mmb + ldfs FR_Neg_Two_to_M3 = [GR_Table_Base1],0 + ldfe FR_c =[r45] + nop.b 999 ;; +} + +{ .mfi +.restore sp + add sp = 64,sp // Restore stack pointer + fcmp.lt.unc.s1 p6, p0 = FR_r, FR_Two_to_M3 + mov b0 = GR_SAVE_B0 // Restore return address +};; { .mib - nop.m 999 - mov ar.pfs = GR_SAVE_PFS - br.ret.sptk b0 // Exit for large arguments routine call + mov gp = GR_SAVE_GP // Restore gp + mov ar.pfs = GR_SAVE_PFS // Restore ar.pfs + nop.b 0 };; -LOCAL_LIBM_END(__libm_callout_sincos) -.type __libm_sin_large#,@function -.global __libm_sin_large# -.type __libm_cos_large#,@function -.global __libm_cos_large# +{ .mfi + nop.m 999 +(p6) fcmp.gt.unc.s1 p6, p0 = FR_r, FR_Neg_Two_to_M3 + nop.i 999 ;; +} + +{ .mib + nop.m 999 + nop.i 999 +(p6) br.cond.spnt L(SINCOS_SMALL_R) ;; +} + +{ .mib + nop.m 999 + nop.i 999 + br.cond.sptk L(SINCOS_NORMAL_R) ;; +} + +.endp __libm_callout_2 +ASM_SIZE_DIRECTIVE(__libm_callout_2) + +.type __libm_pi_by_2_reduce#,@function +.global __libm_pi_by_2_reduce# + +.type __libm_sin_double_dbx#,@function +.global __libm_sin_double_dbx# +.type __libm_cos_double_dbx#,@function +.global __libm_cos_double_dbx# |