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Diffstat (limited to 'sysdeps/ia64/fpu/e_asin.S')
| -rw-r--r-- | sysdeps/ia64/fpu/e_asin.S | 1466 |
1 files changed, 711 insertions, 755 deletions
diff --git a/sysdeps/ia64/fpu/e_asin.S b/sysdeps/ia64/fpu/e_asin.S index bb4c242fb2..398079eae4 100644 --- a/sysdeps/ia64/fpu/e_asin.S +++ b/sysdeps/ia64/fpu/e_asin.S @@ -1,10 +1,10 @@ .file "asin.s" -// Copyright (C) 2000, 2001, Intel Corporation + +// Copyright (c) 2000 - 2003 Intel Corporation // All rights reserved. // -// Contributed 2/2/2000 by John Harrison, Ted Kubaska, Bob Norin, Shane Story, -// and Ping Tak Peter Tang of the Computational Software Lab, Intel Corporation. +// 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 @@ -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 @@ -35,818 +35,776 @@ // // Intel Corporation is the author of this code, and requests that all // problem reports or change requests be submitted to it directly at -// http://developer.intel.com/opensource. +// http://www.intel.com/software/products/opensource/libraries/num.htm. // History //============================================================== -// 2/02/00 Initial version -// 8/17/00 New and much faster algorithm. -// 8/31/00 Avoided bank conflicts on loads, shortened |x|=1 path, +// 02/02/00 Initial version +// 08/17/00 New and much faster algorithm. +// 08/31/00 Avoided bank conflicts on loads, shortened |x|=1 path, // fixed mfb split issue stalls. // 12/19/00 Fixed small arg cases to force inexact, or inexact and underflow. +// 08/02/02 New and much faster algorithm II +// 02/06/03 Reordered header: .section, .global, .proc, .align // Description //========================================= -// The asin function computes the principle value of the arc sine of x. +// The asin function computes the principal value of the arc sine of x. // asin(0) returns 0, asin(1) returns pi/2, asin(-1) returns -pi/2. // A doman error occurs for arguments not in the range [-1,+1]. - +// // The asin function returns the arc sine in the range [-pi/2, +pi/2] radians. +// +// There are 8 paths: +// 1. x = +/-0.0 +// Return asin(x) = +/-0.0 +// +// 2. 0.0 < |x| < 0.625 +// Return asin(x) = x + x^3 *PolA(x^2) +// where PolA(x^2) = A3 + A5*x^2 + A7*x^4 +...+ A35*x^32 +// +// 3. 0.625 <=|x| < 1.0 +// Return asin(x) = sign(x) * ( Pi/2 - sqrt(R) * PolB(R)) +// Where R = 1 - |x|, +// PolB(R) = B0 + B1*R + B2*R^2 +...+B12*R^12 +// +// sqrt(R) is approximated using the following sequence: +// y0 = (1 + eps)/sqrt(R) - initial approximation by frsqrta, +// |eps| < 2^(-8) +// Then 3 iterations are used to refine the result: +// H0 = 0.5*y0 +// S0 = R*y0 +// +// d0 = 0.5 - H0*S0 +// H1 = H0 + d0*H0 +// S1 = S0 + d0*S0 +// +// d1 = 0.5 - H1*S1 +// H2 = H1 + d0*H1 +// S2 = S1 + d0*S1 +// +// d2 = 0.5 - H2*S2 +// S3 = S3 + d2*S3 +// +// S3 approximates sqrt(R) with enough accuracy for this algorithm +// +// So, the result should be reconstracted as follows: +// asin(x) = sign(x) * (Pi/2 - S3*PolB(R)) +// +// But for optimization perposes the reconstruction step is slightly +// changed: +// asin(x) = sign(x)*(Pi/2 - PolB(R)*S2) + sign(x)*d2*S2*PolB(R) +// +// 4. |x| = 1.0 +// Return asin(x) = sign(x)*Pi/2 +// +// 5. 1.0 < |x| <= +INF +// A doman error occurs for arguments not in the range [-1,+1] +// +// 6. x = [S,Q]NaN +// Return asin(x) = QNaN +// +// 7. x is denormal +// Return asin(x) = x + x^3, +// +// 8. x is unnormal +// Normalize input in f8 and return to the very beginning of the function +// +// Registers used +//============================================================== +// Floating Point registers used: +// f8, input, output +// f6, f7, f9 -> f15, f32 -> f63 -#include "libm_support.h" +// General registers used: +// r3, r21 -> r31, r32 -> r38 + +// Predicate registers used: +// p0, p6 -> p14 // // Assembly macros //========================================= - - -// predicate registers -//asin_pred_LEsqrt2by2 = p7 -//asin_pred_GTsqrt2by2 = p8 - -// integer registers -ASIN_Addr1 = r33 -ASIN_Addr2 = r34 -ASIN_FFFE = r35 -ASIN_lnorm_sig = r36 -ASIN_snorm_exp = r37 - -GR_SAVE_B0 = r36 -GR_SAVE_PFS = r37 -GR_SAVE_GP = r38 - -GR_Parameter_X = r39 -GR_Parameter_Y = r40 -GR_Parameter_RESULT = r41 -GR_Parameter_Tag = r42 - -// floating point registers -asin_coeff_P1 = f32 -asin_coeff_P2 = f33 -asin_coeff_P3 = f34 -asin_coeff_P4 = f35 - -asin_coeff_P5 = f36 -asin_coeff_P6 = f37 -asin_coeff_P7 = f38 -asin_coeff_P8 = f39 -asin_coeff_P9 = f40 - -asin_coeff_P10 = f41 -asin_coeff_P11 = f42 -asin_coeff_P12 = f43 -asin_coeff_P13 = f44 -asin_coeff_P14 = f45 - -asin_coeff_P15 = f46 -asin_coeff_P16 = f47 -asin_coeff_P17 = f48 -asin_coeff_P18 = f49 -asin_coeff_P19 = f50 - -asin_coeff_P20 = f51 -asin_coeff_P21 = f52 -asin_const_sqrt2by2 = f53 -asin_const_piby2 = f54 -asin_abs_x = f55 - -asin_tx = f56 -asin_tx2 = f57 -asin_tx3 = f58 -asin_tx4 = f59 -asin_tx8 = f60 - -asin_tx11 = f61 -asin_1poly_p8 = f62 -asin_1poly_p19 = f63 -asin_1poly_p4 = f64 -asin_1poly_p15 = f65 - -asin_1poly_p6 = f66 -asin_1poly_p17 = f67 -asin_1poly_p0 = f68 -asin_1poly_p11 = f69 -asin_1poly_p2 = f70 - -asin_1poly_p13 = f71 -asin_series_tx = f72 -asin_t = f73 -asin_t2 = f74 -asin_t3 = f75 - -asin_t4 = f76 -asin_t8 = f77 -asin_t11 = f78 -asin_poly_p8 = f79 -asin_poly_p19 = f80 - -asin_poly_p4 = f81 -asin_poly_p15 = f82 -asin_poly_p6 = f83 -asin_poly_p17 = f84 -asin_poly_p0 = f85 - -asin_poly_p11 = f86 -asin_poly_p2 = f87 -asin_poly_p13 = f88 -asin_series_t = f89 -asin_1by2 = f90 - -asin_3by2 = f91 -asin_5by2 = f92 -asin_11by4 = f93 -asin_35by8 = f94 -asin_63by8 = f95 - -asin_231by16 = f96 -asin_y0 = f97 -asin_H0 = f98 -asin_S0 = f99 -asin_d = f100 - -asin_l1 = f101 -asin_d2 = f102 -asin_T0 = f103 -asin_d1 = f104 -asin_e0 = f105 - -asin_l2 = f106 -asin_d3 = f107 -asin_T3 = f108 -asin_S1 = f109 -asin_e1 = f110 - -asin_z = f111 -answer2 = f112 -asin_sgn_x = f113 -asin_429by16 = f114 -asin_18by4 = f115 - -asin_3by4 = f116 -asin_l3 = f117 -asin_T6 = f118 -asin_eps_exp = f119 -asin_eps_sig = f120 -asin_eps = f120 - +// integer registers used +// scratch +rTblAddr = r3 + +rPiBy2Ptr = r21 +rTmpPtr3 = r22 +rDenoBound = r23 +rOne = r24 +rAbsXBits = r25 +rHalf = r26 +r0625 = r27 +rSign = r28 +rXBits = r29 +rTmpPtr2 = r30 +rTmpPtr1 = r31 + +// stacked +GR_SAVE_PFS = r32 +GR_SAVE_B0 = r33 +GR_SAVE_GP = r34 +GR_Parameter_X = r35 +GR_Parameter_Y = r36 +GR_Parameter_RESULT = r37 +GR_Parameter_TAG = r38 + +// floating point registers used +FR_X = f10 +FR_Y = f1 +FR_RESULT = f8 + + +// scratch +fXSqr = f6 +fXCube = f7 +fXQuadr = f9 +f1pX = f10 +f1mX = f11 +f1pXRcp = f12 +f1mXRcp = f13 +fH = f14 +fS = f15 +// stacked +fA3 = f32 +fB1 = f32 +fA5 = f33 +fB2 = f33 +fA7 = f34 +fPiBy2 = f34 +fA9 = f35 +fA11 = f36 +fB10 = f35 +fB11 = f36 +fA13 = f37 +fA15 = f38 +fB4 = f37 +fB5 = f38 +fA17 = f39 +fA19 = f40 +fB6 = f39 +fB7 = f40 +fA21 = f41 +fA23 = f42 +fB3 = f41 +fB8 = f42 +fA25 = f43 +fA27 = f44 +fB9 = f43 +fB12 = f44 +fA29 = f45 +fA31 = f46 +fA33 = f47 +fA35 = f48 +fBaseP = f49 +fB0 = f50 +fSignedS = f51 +fD = f52 +fHalf = f53 +fR = f54 +fCloseTo1Pol = f55 +fSignX = f56 +fDenoBound = f57 +fNormX = f58 +fX8 = f59 +fRSqr = f60 +fRQuadr = f61 +fR8 = f62 +fX16 = f63 // Data tables //============================================================== - -#ifdef _LIBC -.rodata -#else -.data -#endif - +RODATA .align 16 - -asin_coeff_1_table: -ASM_TYPE_DIRECTIVE(asin_coeff_1_table,@object) -data8 0xE4E7E0A423A21249 , 0x00003FF8 //P7 -data8 0xC2F7EE0200FCE2A5 , 0x0000C003 //P18 -data8 0xB745D7F6C65C20E0 , 0x00003FF9 //P5 -data8 0xF75E381A323D4D94 , 0x0000C002 //P16 -data8 0x8959C2629C1024C0 , 0x0000C002 //P20 -data8 0xAFF68E7D241292C5 , 0x00003FF8 //P9 -data8 0xB6DB6DB7260AC30D , 0x00003FFA //P3 -data8 0xD0417CE2B41CB7BF , 0x0000C000 //P14 -data8 0x81D570FEA724E3E4 , 0x0000BFFD //P12 -data8 0xAAAAAAAAAAAAC277 , 0x00003FFC //P1 -data8 0xF534912FF3E7B76F , 0x00003FFF //P21 -data8 0xc90fdaa22168c235 , 0x00003fff // pi/2 -data8 0x0000000000000000 , 0x00000000 // pad to avoid data bank conflict -ASM_SIZE_DIRECTIVE(asin_coeff_1_table) - - -asin_coeff_2_table: -ASM_TYPE_DIRECTIVE(asin_coeff_2_table,@object) -data8 0x8E26AF5F29B39A2A , 0x00003FF9 //P6 -data8 0xB4F118A4B1015470 , 0x00004003 //P17 -data8 0xF8E38E10C25990E0 , 0x00003FF9 //P4 -data8 0x80F50489AEF1CAC6 , 0x00004002 //P15 -data8 0x92728015172CFE1C , 0x00004003 //P19 -data8 0xBBC3D831D4595971 , 0x00003FF8 //P8 -data8 0x999999999952A5C3 , 0x00003FFB //P2 -data8 0x855576BE6F0975EC , 0x00003FFF //P13 -data8 0xF12420E778077D89 , 0x00003FFA //P11 -data8 0xB6590FF4D23DE003 , 0x00003FF3 //P10 -data8 0xb504f333f9de6484 , 0x00003ffe // sqrt(2)/2 -ASM_SIZE_DIRECTIVE(asin_coeff_2_table) +LOCAL_OBJECT_START(asin_base_range_table) +// Ai: Polynomial coefficients for the asin(x), |x| < .625000 +// Bi: Polynomial coefficients for the asin(x), |x| > .625000 +data8 0xBFDAAB56C01AE468 //A29 +data8 0x3FE1C470B76A5B2B //A31 +data8 0xBFDC5FF82A0C4205 //A33 +data8 0x3FC71FD88BFE93F0 //A35 +data8 0xB504F333F9DE6487, 0x00003FFF //B0 +data8 0xAAAAAAAAAAAAFC18, 0x00003FFC //A3 +data8 0x3F9F1C71BC4A7823 //A9 +data8 0x3F96E8BBAAB216B2 //A11 +data8 0x3F91C4CA1F9F8A98 //A13 +data8 0x3F8C9DDCEDEBE7A6 //A15 +data8 0x3F877784442B1516 //A17 +data8 0x3F859C0491802BA2 //A19 +data8 0x9999999998C88B8F, 0x00003FFB //A5 +data8 0x3F6BD7A9A660BF5E //A21 +data8 0x3F9FC1659340419D //A23 +data8 0xB6DB6DB798149BDF, 0x00003FFA //A7 +data8 0xBFB3EF18964D3ED3 //A25 +data8 0x3FCD285315542CF2 //A27 +data8 0xF15BEEEFF7D2966A, 0x00003FFB //B1 +data8 0x3EF0DDA376D10FB3 //B10 +data8 0xBEB83CAFE05EBAC9 //B11 +data8 0x3F65FFB67B513644 //B4 +data8 0x3F5032FBB86A4501 //B5 +data8 0x3F392162276C7CBA //B6 +data8 0x3F2435949FD98BDF //B7 +data8 0xD93923D7FA08341C, 0x00003FF9 //B2 +data8 0x3F802995B6D90BDB //B3 +data8 0x3F10DF86B341A63F //B8 +data8 0xC90FDAA22168C235, 0x00003FFF // Pi/2 +data8 0x3EFA3EBD6B0ECB9D //B9 +data8 0x3EDE18BA080E9098 //B12 +LOCAL_OBJECT_END(asin_base_range_table) - -.align 32 -.global asin - .section .text -.proc asin -.align 32 - - -asin: - -{ .mfi - alloc r32 = ar.pfs,1,6,4,0 - fma.s1 asin_tx = f8,f8,f0 - addl ASIN_Addr2 = @ltoff(asin_coeff_2_table),gp -} -{ .mfi - mov ASIN_FFFE = 0xFFFE - fnma.s1 asin_t = f8,f8,f1 - addl ASIN_Addr1 = @ltoff(asin_coeff_1_table),gp +GLOBAL_LIBM_ENTRY(asin) +asin_unnormal_back: +{ .mfi + getf.d rXBits = f8 // grab bits of input value + // set p12 = 1 if x is a NaN, denormal, or zero + fclass.m p12, p0 = f8, 0xcf + adds rSign = 1, r0 +} +{ .mfi + addl rTblAddr = @ltoff(asin_base_range_table),gp + // 1 - x = 1 - |x| for positive x + fms.s1 f1mX = f1, f1, f8 + addl rHalf = 0xFFFE, r0 // exponent of 1/2 } ;; - - -{ .mfi - setf.exp asin_1by2 = ASIN_FFFE - fmerge.s asin_abs_x = f1,f8 - nop.i 999 ;; -} - -{ .mmf - ld8 ASIN_Addr1 = [ASIN_Addr1] - ld8 ASIN_Addr2 = [ASIN_Addr2] - fmerge.s asin_sgn_x = f8,f1 ;; -} - - -{ .mfi - ldfe asin_coeff_P7 = [ASIN_Addr1],16 - fma.s1 asin_tx2 = asin_tx,asin_tx,f0 - nop.i 999 -} -{ .mfi - ldfe asin_coeff_P6 = [ASIN_Addr2],16 - fma.s1 asin_t2 = asin_t,asin_t,f0 - nop.i 999;; +{ .mfi + addl r0625 = 0x3FE4, r0 // high 16 bits of 0.625 + // set p8 = 1 if x < 0 + fcmp.lt.s1 p8, p9 = f8, f0 + shl rSign = rSign, 63 // sign bit } - - -{ .mmf - ldfe asin_coeff_P18 = [ASIN_Addr1],16 - ldfe asin_coeff_P17 = [ASIN_Addr2],16 - fclass.m.unc p8,p0 = f8, 0xc3 //@qnan |@snan -} -;; - -{ .mmf - ldfe asin_coeff_P5 = [ASIN_Addr1],16 - ldfe asin_coeff_P4 = [ASIN_Addr2],16 - frsqrta.s1 asin_y0,p0 = asin_t -} -;; - -{ .mfi - ldfe asin_coeff_P16 = [ASIN_Addr1],16 - fcmp.gt.s1 p9,p0 = asin_abs_x,f1 - nop.i 999 -} -{ .mfb - ldfe asin_coeff_P15 = [ASIN_Addr2],16 -(p8) fma.d f8 = f8,f1,f0 -(p8) br.ret.spnt b0 +{ .mfi + // point to the beginning of the table + ld8 rTblAddr = [rTblAddr] + // 1 + x = 1 - |x| for negative x + fma.s1 f1pX = f1, f1, f8 + adds rOne = 0x3FF, r0 } ;; - - -{ .mmf - ldfe asin_coeff_P20 = [ASIN_Addr1],16 - ldfe asin_coeff_P19 = [ASIN_Addr2],16 - fclass.m.unc p8,p0 = f8, 0x07 //@zero -} -;; - - -{ .mfi - ldfe asin_coeff_P9 = [ASIN_Addr1],16 - fma.s1 asin_t4 = asin_t2,asin_t2,f0 -(p9) mov GR_Parameter_Tag = 61 -} -{ .mfi - ldfe asin_coeff_P8 = [ASIN_Addr2],16 - fma.s1 asin_3by2 = asin_1by2,f1,f1 - nop.i 999;; +{ .mfi + andcm rAbsXBits = rXBits, rSign // bits of |x| + fmerge.s fSignX = f8, f1 // signum(x) + shl r0625 = r0625, 48 // bits of DP representation of 0.625 } - - -{ .mfi - ldfe asin_coeff_P2 = [ASIN_Addr2],16 - fma.s1 asin_tx4 = asin_tx2,asin_tx2,f0 - nop.i 999 -} -{ .mfb - ldfe asin_coeff_P3 = [ASIN_Addr1],16 - fma.s1 asin_t3 = asin_t,asin_t2,f0 -(p8) br.ret.spnt b0 +{ .mfb + setf.exp fHalf = rHalf // load A2 to FP reg + fma.s1 fXSqr = f8, f8, f0 // x^2 + // branch on special path if x is a NaN, denormal, or zero +(p12) br.cond.spnt asin_special } ;; - - -{ .mfi - ldfe asin_coeff_P13 = [ASIN_Addr2],16 - fma.s1 asin_H0 = asin_y0,asin_1by2,f0 - nop.i 999 -} -{ .mfb - ldfe asin_coeff_P14 = [ASIN_Addr1],16 - fma.s1 asin_S0 = asin_y0,asin_t,f0 -(p9) br.cond.spnt __libm_error_region +{ .mfi + adds rPiBy2Ptr = 272, rTblAddr + nop.f 0 + shl rOne = rOne, 52 // bits of 1.0 +} +{ .mfi + adds rTmpPtr1 = 16, rTblAddr + nop.f 0 + // set p6 = 1 if |x| < 0.625 + cmp.lt p6, p7 = rAbsXBits, r0625 } ;; - - -{ .mfi - ldfe asin_coeff_P11 = [ASIN_Addr2],16 - fcmp.eq.s1 p6,p0 = asin_abs_x,f1 - nop.i 999 -} -{ .mfi - ldfe asin_coeff_P12 = [ASIN_Addr1],16 - fma.s1 asin_tx3 = asin_tx,asin_tx2,f0 - nop.i 999;; +{ .mfi + ldfpd fA29, fA31 = [rTblAddr] // A29, fA31 + // 1 - x = 1 - |x| for positive x +(p9) fms.s1 fR = f1, f1, f8 + // point to coefficient of "near 1" polynomial +(p7) adds rTmpPtr2 = 176, rTblAddr } - - -{ .mfi - ldfe asin_coeff_P10 = [ASIN_Addr2],16 - fma.s1 asin_1poly_p6 = asin_tx,asin_coeff_P7,asin_coeff_P6 - nop.i 999 -} -{ .mfi - ldfe asin_coeff_P1 = [ASIN_Addr1],16 - fma.s1 asin_poly_p6 = asin_t,asin_coeff_P7,asin_coeff_P6 - nop.i 999;; +{ .mfi + ldfpd fA33, fA35 = [rTmpPtr1], 16 // A33, fA35 + // 1 + x = 1 - |x| for negative x +(p8) fma.s1 fR = f1, f1, f8 +(p6) adds rTmpPtr2 = 48, rTblAddr } - - -{ .mfi - ldfe asin_const_sqrt2by2 = [ASIN_Addr2],16 - fma.s1 asin_5by2 = asin_3by2,f1,f1 - nop.i 999 -} -{ .mfi - ldfe asin_coeff_P21 = [ASIN_Addr1],16 - fma.s1 asin_11by4 = asin_3by2,asin_3by2,asin_1by2 - nop.i 999;; +;; +{ .mfi + ldfe fB0 = [rTmpPtr1], 16 // B0 + nop.f 0 + nop.i 0 } - - -{ .mfi - ldfe asin_const_piby2 = [ASIN_Addr1],16 - fma.s1 asin_poly_p17 = asin_t,asin_coeff_P18,asin_coeff_P17 - nop.i 999 -} -{ .mfb - nop.m 999 - fma.s1 asin_3by4 = asin_3by2,asin_1by2,f0 -(p6) br.cond.spnt L(ASIN_ABS_1) // Branch to short exit if |x|=1 +{ .mib + adds rTmpPtr3 = 16, rTmpPtr2 + // set p10 = 1 if |x| = 1.0 + cmp.eq p10, p0 = rAbsXBits, rOne + // branch on special path for |x| = 1.0 +(p10) br.cond.spnt asin_abs_1 } ;; - - -{ .mfi - addl ASIN_lnorm_sig = -0x1,r0 // Form significand 0xffffffffffffffff - fma.s1 asin_poly_p15 = asin_t,asin_coeff_P16,asin_coeff_P15 - nop.i 999 -} -{ .mfi - addl ASIN_snorm_exp = 0x0c001,r0 // Form small exponent - fnma.s1 asin_d = asin_S0,asin_H0,asin_1by2 - nop.i 999;; +{ .mfi + ldfe fA3 = [rTmpPtr2], 48 // A3 or B1 + nop.f 0 + adds rTmpPtr1 = 64, rTmpPtr3 } - - -// Form the exponent and significand of a small number -{ .mfi - setf.sig asin_eps_sig = ASIN_lnorm_sig - fma.s1 asin_poly_p19 = asin_t,asin_coeff_P20,asin_coeff_P19 - nop.i 999 -} -{ .mfi - setf.exp asin_eps_exp = ASIN_snorm_exp - fma.s1 asin_poly_p4 = asin_t,asin_coeff_P5,asin_coeff_P4 - nop.i 999;; +{ .mib + ldfpd fA9, fA11 = [rTmpPtr3], 16 // A9, A11 or B10, B11 + // set p11 = 1 if |x| > 1.0 + cmp.gt p11, p0 = rAbsXBits, rOne + // branch on special path for |x| > 1.0 +(p11) br.cond.spnt asin_abs_gt_1 } - - -{ .mfi - nop.m 999 - fma.s1 asin_1poly_p17 = asin_tx,asin_coeff_P18,asin_coeff_P17 - nop.i 999 -} -{ .mfi - nop.m 999 - fma.s1 asin_poly_p8 = asin_t,asin_coeff_P9,asin_coeff_P8 - nop.i 999;; +;; +{ .mfi + ldfpd fA17, fA19 = [rTmpPtr2], 16 // A17, A19 or B6, B7 + // initial approximation of 1 / sqrt(1 - x) + frsqrta.s1 f1mXRcp, p0 = f1mX + nop.i 0 } - - -{ .mfi - nop.m 999 - fms.s1 asin_35by8 = asin_5by2,asin_11by4,asin_5by2 - nop.i 999 -} -{ .mfi - nop.m 999 - fma.s1 asin_63by8 = asin_5by2,asin_11by4,f1 - nop.i 999;; +{ .mfi + ldfpd fA13, fA15 = [rTmpPtr3] // A13, A15 or B4, B5 + fma.s1 fXCube = fXSqr, f8, f0 // x^3 + nop.i 0 } - - -{ .mfi - nop.m 999 - fma.s1 asin_poly_p13 = asin_t,asin_coeff_P14,asin_coeff_P13 - nop.i 999 -} -{ .mfi - nop.m 999 - fma.s1 asin_18by4 = asin_3by2,asin_5by2,asin_3by4 - nop.i 999;; +;; +{ .mfi + ldfe fA5 = [rTmpPtr2], 48 // A5 or B2 + // initial approximation of 1 / sqrt(1 + x) + frsqrta.s1 f1pXRcp, p0 = f1pX + nop.i 0 } - - -{ .mfi - nop.m 999 - fma.s1 asin_l1 = asin_5by2,asin_d,asin_3by2 - nop.i 999 -} -{ .mfi - nop.m 999 - fma.s1 asin_d2 = asin_d,asin_d,f0 - nop.i 999;; +{ .mfi + ldfpd fA21, fA23 = [rTmpPtr1], 16 // A21, A23 or B3, B8 + fma.s1 fXQuadr = fXSqr, fXSqr, f0 // x^4 + nop.i 0 } - - -{ .mfi - nop.m 999 - fma.s1 asin_poly_p15 = asin_t2,asin_poly_p17,asin_poly_p15 - nop.i 999 -} -{ .mfi - nop.m 999 - fma.s1 asin_T0 = asin_d,asin_S0,f0 - nop.i 999;; +;; +{ .mfi + ldfe fA7 = [rTmpPtr1] // A7 or Pi/2 + fma.s1 fRSqr = fR, fR, f0 // R^2 + nop.i 0 } - - -{ .mfi - nop.m 999 - fma.s1 asin_poly_p19 = asin_t2,asin_coeff_P21,asin_poly_p19 - nop.i 999 -} -{ .mfi - nop.m 999 - fma.s1 asin_poly_p4 = asin_t2,asin_poly_p6,asin_poly_p4 - nop.i 999;; +{ .mfb + ldfpd fA25, fA27 = [rTmpPtr2] // A25, A27 or B9, B12 + nop.f 0 +(p6) br.cond.spnt asin_base_range; } +;; - -{ .mfi - nop.m 999 - fma.s1 asin_d1 = asin_35by8,asin_d,f0 - nop.i 999 -} -{ .mfi - nop.m 999 - fma.s1 asin_231by16 = asin_3by2,asin_35by8,asin_63by8 - nop.i 999;; +{ .mfi + nop.m 0 +(p9) fma.s1 fH = fHalf, f1mXRcp, f0 // H0 for x > 0 + nop.i 0 } - - -{ .mfi - nop.m 999 - fma.s1 asin_poly_p2 = asin_t,asin_coeff_P3,asin_coeff_P2 - nop.i 999 -} -{ .mfi - nop.m 999 - fma.s1 asin_poly_p8 = asin_t2,asin_coeff_P10,asin_poly_p8 - nop.i 999;; +{ .mfi + nop.m 0 +(p9) fma.s1 fS = f1mX, f1mXRcp, f0 // S0 for x > 0 + nop.i 0 } - - -{ .mfi - nop.m 999 - fma.s1 asin_poly_p11 = asin_t,asin_coeff_P12,asin_coeff_P11 - nop.i 999 -} -{ .mfi - nop.m 999 - fma.s1 asin_e0 = asin_d2,asin_l1,asin_d - nop.i 999;; +;; +{ .mfi + nop.m 0 +(p8) fma.s1 fH = fHalf, f1pXRcp, f0 // H0 for x < 0 + nop.i 0 } - - -{ .mfi - nop.m 999 - fma.s1 asin_1poly_p15 = asin_tx,asin_coeff_P16,asin_coeff_P15 - nop.i 999 -} -{ .mfi - nop.m 999 - fma.s1 asin_poly_p0 = asin_t,asin_coeff_P1,f1 - nop.i 999;; +{ .mfi + nop.m 0 +(p8) fma.s1 fS = f1pX, f1pXRcp, f0 // S0 for x > 0 + nop.i 0 } - - -{ .mfi - nop.m 999 - fma.s1 asin_1poly_p19 = asin_tx,asin_coeff_P20,asin_coeff_P19 - nop.i 999 -} -{ .mfi - nop.m 999 - fma.s1 asin_1poly_p4 = asin_tx,asin_coeff_P5,asin_coeff_P4 - nop.i 999;; +;; +{ .mfi + nop.m 0 + fma.s1 fRQuadr = fRSqr, fRSqr, f0 // R^4 + nop.i 0 } - - -{ .mfi - nop.m 999 - fma.s1 asin_1poly_p8 = asin_tx,asin_coeff_P9,asin_coeff_P8 - nop.i 999 -} -{ .mfi - nop.m 999 - fma.s1 asin_l2 = asin_231by16,asin_d,asin_63by8 - nop.i 999;; +;; +{ .mfi + nop.m 0 + fma.s1 fB11 = fB11, fR, fB10 + nop.i 0 } - - -{ .mfi - nop.m 999 - fma.s1 asin_d3 = asin_d2,asin_d,f0 - nop.i 999 -} -{ .mfi - nop.m 999 - fma.s1 asin_T3 = asin_d2,asin_T0,f0 - nop.i 999;; +{ .mfi + nop.m 0 + fma.s1 fB1 = fB1, fR, fB0 + nop.i 0 } - - -{ .mfi - nop.m 999 - fma.s1 asin_429by16 = asin_18by4,asin_11by4,asin_231by16 - nop.i 999 -} -{ .mfi - nop.m 999 - fma.s1 asin_S1 = asin_e0,asin_S0,asin_S0 - nop.i 999;; +;; +{ .mfi + nop.m 0 + fma.s1 fB5 = fB5, fR, fB4 + nop.i 0 } - - -{ .mfi - nop.m 999 - fma.s1 asin_poly_p4 = asin_t4,asin_poly_p8,asin_poly_p4 - nop.i 999 -} -{ .mfi - nop.m 999 - fma.s1 asin_poly_p15 = asin_t4,asin_poly_p19,asin_poly_p15 - nop.i 999;; +{ .mfi + nop.m 0 + fma.s1 fB7 = fB7, fR, fB6 + nop.i 0 } - - -{ .mfi - nop.m 999 - fma.s1 asin_poly_p0 = asin_t2,asin_poly_p2,asin_poly_p0 - nop.i 999 -} -{ .mfi - nop.m 999 - fma.s1 asin_poly_p11 = asin_t2,asin_poly_p13,asin_poly_p11 - nop.i 999;; +;; +{ .mfi + nop.m 0 + fma.s1 fB3 = fB3, fR, fB2 + nop.i 0 } - - -{ .mfi - nop.m 999 - fma.s1 asin_t8 = asin_t4,asin_t4,f0 - nop.i 999 -} -{ .mfi - nop.m 999 - fma.s1 asin_e1 = asin_d2,asin_l2,asin_d1 - nop.i 999;; +;; +{ .mfi + nop.m 0 + fnma.s1 fD = fH, fS, fHalf // d0 = 1/2 - H0*S0 + nop.i 0 } - - -{ .mfi - nop.m 999 - fma.s1 asin_1poly_p4 = asin_tx2,asin_1poly_p6,asin_1poly_p4 - nop.i 999 -} -{ .mfi - nop.m 999 - fma.s1 asin_1poly_p15 = asin_tx2,asin_1poly_p17,asin_1poly_p15 - nop.i 999;; +;; +{ .mfi + nop.m 0 + fma.s1 fR8 = fRQuadr, fRQuadr, f0 // R^4 + nop.i 0 } - - -{ .mfi - nop.m 999 - fma.s1 asin_1poly_p8 = asin_tx2,asin_coeff_P10,asin_1poly_p8 - nop.i 999 -} -{ .mfi - nop.m 999 - fma.s1 asin_1poly_p19 = asin_tx2,asin_coeff_P21,asin_1poly_p19 - nop.i 999;; +{ .mfi + nop.m 0 + fma.s1 fB9 = fB9, fR, fB8 + nop.i 0 } - - -{ .mfi - nop.m 999 - fma.s1 asin_1poly_p2 = asin_tx,asin_coeff_P3,asin_coeff_P2 - nop.i 999 -} -{ .mfi - nop.m 999 - fma.s1 asin_1poly_p13 = asin_tx,asin_coeff_P14,asin_coeff_P13 - nop.i 999;; +;; +{.mfi + nop.m 0 + fma.s1 fB12 = fB12, fRSqr, fB11 + nop.i 0 } - - -{ .mfi - nop.m 999 - fma.s1 asin_1poly_p0 = asin_tx,asin_coeff_P1,f1 - nop.i 999 -} -{ .mfi - nop.m 999 - fma.s1 asin_1poly_p11 = asin_tx,asin_coeff_P12,asin_coeff_P11 - nop.i 999;; +{.mfi + nop.m 0 + fma.s1 fB7 = fB7, fRSqr, fB5 + nop.i 0 } - - -{ .mfi - nop.m 999 - fma.s1 asin_l3 = asin_429by16,asin_d,f0 - nop.i 999 -} -{ .mfi - nop.m 999 - fma.s1 asin_z = asin_e1,asin_T3,asin_S1 - nop.i 999;; +;; +{.mfi + nop.m 0 + fma.s1 fB3 = fB3, fRSqr, fB1 + nop.i 0 } - - -{ .mfi - nop.m 999 - fma.s1 asin_poly_p11 = asin_t4,asin_poly_p15,asin_poly_p11 - nop.i 999 -} -{ .mfi - nop.m 999 - fma.s1 asin_T6 = asin_T3,asin_d3,f0 - nop.i 999;; +;; +{ .mfi + nop.m 0 + fma.s1 fH = fH, fD, fH // H1 = H0 + H0*d0 + nop.i 0 } - - -{ .mfi - nop.m 999 - fma.s1 asin_t11 = asin_t8,asin_t3,f0 - nop.i 999 -} -{ .mfi - nop.m 999 - fma.s1 asin_poly_p0 = asin_t4,asin_poly_p4,asin_poly_p0 - nop.i 999;; +{ .mfi + nop.m 0 + fma.s1 fS = fS, fD, fS // S1 = S0 + S0*d0 + nop.i 0 } - - -{ .mfi - nop.m 999 - fma.s1 asin_1poly_p4 = asin_tx4,asin_1poly_p8,asin_1poly_p4 - nop.i 999 -} -{ .mfi - nop.m 999 - fma.s1 asin_1poly_p15 = asin_tx4,asin_1poly_p19,asin_1poly_p15 - nop.i 999;; +;; +{.mfi + nop.m 0 + fma.s1 fPiBy2 = fPiBy2, fSignX, f0 // signum(x)*Pi/2 + nop.i 0 } - - -{ .mfi - nop.m 999 - fma.s1 asin_1poly_p0 = asin_tx2,asin_1poly_p2,asin_1poly_p0 - nop.i 999 -} -{ .mfi - nop.m 999 - fma.s1 asin_1poly_p11 = asin_tx2,asin_1poly_p13,asin_1poly_p11 - nop.i 999;; +;; +{ .mfi + nop.m 0 + fma.s1 fB12 = fB12, fRSqr, fB9 + nop.i 0 } - - -{ .mfi - nop.m 999 -// fcmp.le.s1 asin_pred_LEsqrt2by2,asin_pred_GTsqrt2by2 = asin_abs_x,asin_const_sqrt2by2 - fcmp.le.s1 p7,p8 = asin_abs_x,asin_const_sqrt2by2 - nop.i 999 -} -{ .mfi - nop.m 999 - fma.s1 asin_tx8 = asin_tx4,asin_tx4,f0 - nop.i 999;; +{ .mfi + nop.m 0 + fma.s1 fB7 = fB7, fRQuadr, fB3 + nop.i 0 } - - -// Form a small number to force inexact flag for small args -{ .mfi - nop.m 999 - fmerge.se asin_eps = asin_eps_exp,asin_eps_sig - nop.i 999 -} -{ .mfi - nop.m 999 - fma.s1 asin_z = asin_l3,asin_T6,asin_z - nop.i 999;; -} - -{ .mfi - nop.m 999 - fma.s1 asin_series_t = asin_t11,asin_poly_p11,asin_poly_p0 - nop.i 999;; -} - -{ .mfi - nop.m 999 - fma.s1 asin_1poly_p0 = asin_tx4,asin_1poly_p4,asin_1poly_p0 - nop.i 999 -} -{ .mfi - nop.m 999 - fma.s1 asin_1poly_p11 = asin_tx4,asin_1poly_p15,asin_1poly_p11 - nop.i 999;; +;; +{.mfi + nop.m 0 + fnma.s1 fD = fH, fS, fHalf // d1 = 1/2 - H1*S1 + nop.i 0 +} +{ .mfi + nop.m 0 + fnma.s1 fSignedS = fSignX, fS, f0 // -signum(x)*S1 + nop.i 0 +} +;; +{ .mfi + nop.m 0 + fma.s1 fCloseTo1Pol = fB12, fR8, fB7 + nop.i 0 +} +;; +{ .mfi + nop.m 0 + fma.s1 fH = fH, fD, fH // H2 = H1 + H1*d1 + nop.i 0 +} +{ .mfi + nop.m 0 + fma.s1 fS = fS, fD, fS // S2 = S1 + S1*d1 + nop.i 0 +} +;; +{ .mfi + nop.m 0 + // -signum(x)* S2 = -signum(x)*(S1 + S1*d1) + fma.s1 fSignedS = fSignedS, fD, fSignedS + nop.i 0 +} +;; +{.mfi + nop.m 0 + fnma.s1 fD = fH, fS, fHalf // d2 = 1/2 - H2*S2 + nop.i 0 +} +;; +{ .mfi + nop.m 0 + // signum(x)*(Pi/2 - PolB*S2) + fma.s1 fPiBy2 = fSignedS, fCloseTo1Pol, fPiBy2 + nop.i 0 +} +{ .mfi + nop.m 0 + // -signum(x)*PolB * S2 + fma.s1 fCloseTo1Pol = fSignedS, fCloseTo1Pol, f0 + nop.i 0 +} +;; +{ .mfb + nop.m 0 + // final result for 0.625 <= |x| < 1 + fma.d.s0 f8 = fCloseTo1Pol, fD, fPiBy2 + // exit here for 0.625 <= |x| < 1 + br.ret.sptk b0 } +;; - -{ .mfi - nop.m 999 - fma.s1 asin_tx11 = asin_tx8,asin_tx3,f0 - nop.i 999;; -} - -{ .mfi - nop.m 999 -//(asin_pred_GTsqrt2by2) fnma.s1 answer2 = asin_z,asin_series_t,asin_const_piby2 -(p8) fnma.s1 answer2 = asin_z,asin_series_t,asin_const_piby2 - nop.i 999;; -} - -{ .mfi - nop.m 999 - fma.s1 asin_series_tx = asin_tx11,asin_1poly_p11,asin_1poly_p0 - nop.i 999;; -} - -{ .mfi - nop.m 999 -//(asin_pred_GTsqrt2by2) fma.d f8 = asin_sgn_x,answer2,f0 -(p8) fma.d f8 = asin_sgn_x,answer2,f0 - nop.i 999;; -} - -// asin_eps is added only to force inexact and possibly underflow flag -// in case asin_series_tx is zero -// -{ .mfi - nop.m 999 -(p7) fma.d asin_eps = f8,asin_series_tx,asin_eps - nop.i 999 -} -{ .mfb - nop.m 999 -//(asin_pred_LEsqrt2by2) fma.d f8 = f8,asin_series_tx,f0 -(p7) fma.d f8 = f8,asin_series_tx,f0 - br.ret.sptk b0 -} + +// here if |x| < 0.625 +.align 32 +asin_base_range: +{ .mfi + nop.m 0 + fma.s1 fA33 = fA33, fXSqr, fA31 + nop.i 0 +} +{ .mfi + nop.m 0 + fma.s1 fA15 = fA15, fXSqr, fA13 + nop.i 0 +} +;; +{ .mfi + nop.m 0 + fma.s1 fA29 = fA29, fXSqr, fA27 + nop.i 0 +} +{ .mfi + nop.m 0 + fma.s1 fA25 = fA25, fXSqr, fA23 + nop.i 0 +} +;; +{ .mfi + nop.m 0 + fma.s1 fA21 = fA21, fXSqr, fA19 + nop.i 0 +} +{ .mfi + nop.m 0 + fma.s1 fA9 = fA9, fXSqr, fA7 + nop.i 0 +} +;; +{ .mfi + nop.m 0 + fma.s1 fA5 = fA5, fXSqr, fA3 + nop.i 0 +} +;; +{ .mfi + nop.m 0 + fma.s1 fA35 = fA35, fXQuadr, fA33 + nop.i 0 +} +{ .mfi + nop.m 0 + fma.s1 fA17 = fA17, fXQuadr, fA15 + nop.i 0 +} +;; +{ .mfi + nop.m 0 + fma.s1 fX8 = fXQuadr, fXQuadr, f0 // x^8 + nop.i 0 +} +{ .mfi + nop.m 0 + fma.s1 fA25 = fA25, fXQuadr, fA21 + nop.i 0 +} +;; +{ .mfi + nop.m 0 + fma.s1 fA9 = fA9, fXQuadr, fA5 + nop.i 0 +} +;; +{ .mfi + nop.m 0 + fma.s1 fA35 = fA35, fXQuadr, fA29 + nop.i 0 +} +{ .mfi + nop.m 0 + fma.s1 fA17 = fA17, fXSqr, fA11 + nop.i 0 +} +;; +{ .mfi + nop.m 0 + fma.s1 fX16 = fX8, fX8, f0 // x^16 + nop.i 0 +} +;; +{ .mfi + nop.m 0 + fma.s1 fA35 = fA35, fX8, fA25 + nop.i 0 +} +{ .mfi + nop.m 0 + fma.s1 fA17 = fA17, fX8, fA9 + nop.i 0 +} +;; +{ .mfi + nop.m 0 + fma.s1 fBaseP = fA35, fX16, fA17 + nop.i 0 +} +;; +{ .mfb + nop.m 0 + // final result for |x| < 0.625 + fma.d.s0 f8 = fBaseP, fXCube, f8 + // exit here for |x| < 0.625 path + br.ret.sptk b0 +} ;; +// here if |x| = 1 +// asin(x) = sign(x) * Pi/2 +.align 32 +asin_abs_1: +{ .mfi + ldfe fPiBy2 = [rPiBy2Ptr] // Pi/2 + nop.f 0 + nop.i 0 +} +;; +{.mfb + nop.m 0 + // result for |x| = 1.0 + fma.d.s0 f8 = fPiBy2, fSignX, f0 + // exit here for |x| = 1.0 + br.ret.sptk b0 +} +;; -L(ASIN_ABS_1): -// Here for short exit if |x|=1 -{ .mfb - nop.m 999 - fma.d f8 = asin_sgn_x,asin_const_piby2,f0 - br.ret.sptk b0 -} +// here if x is a NaN, denormal, or zero +.align 32 +asin_special: +{ .mfi + nop.m 0 + // set p12 = 1 if x is a NaN + fclass.m p12, p0 = f8, 0xc3 + nop.i 0 +} +{ .mlx + nop.m 0 + // smallest positive DP normalized number + movl rDenoBound = 0x0010000000000000 +} +;; +{ .mfi + nop.m 0 + // set p13 = 1 if x = 0.0 + fclass.m p13, p0 = f8, 0x07 + nop.i 0 +} +{ .mfi + nop.m 0 + fnorm.s1 fNormX = f8 + nop.i 0 +} +;; +{ .mfb + // load smallest normal to FP reg + setf.d fDenoBound = rDenoBound + // answer if x is a NaN +(p12) fma.d.s0 f8 = f8,f1,f0 + // exit here if x is a NaN +(p12) br.ret.spnt b0 +} +;; +{ .mfb + nop.m 0 + nop.f 0 + // exit here if x = 0.0 +(p13) br.ret.spnt b0 +} +;; +// if we still here then x is denormal or unnormal +{ .mfi + nop.m 0 + // absolute value of normalized x + fmerge.s fNormX = f1, fNormX + nop.i 0 +} +;; +{ .mfi + nop.m 0 + // set p14 = 1 if normalized x is greater than or + // equal to the smallest denormalized value + // So, if p14 is set to 1 it means that we deal with + // unnormal rather than with "true" denormal + fcmp.ge.s1 p14, p0 = fNormX, fDenoBound + nop.i 0 +} +;; +{ .mfi + nop.m 0 +(p14) fcmp.eq.s0 p6, p0 = f8, f0 // Set D flag if x unnormal + nop.i 0 +} +{ .mfb + nop.m 0 + // normalize unnormal input +(p14) fnorm.s1 f8 = f8 + // return to the main path +(p14) br.cond.sptk asin_unnormal_back +} +;; +// if we still here it means that input is "true" denormal +{ .mfb + nop.m 0 + // final result if x is denormal + fma.d.s0 f8 = f8, fXSqr, f8 + // exit here if x is denormal + br.ret.sptk b0 +} ;; +// here if |x| > 1.0 +// error handler should be called +.align 32 +asin_abs_gt_1: +{ .mfi + alloc r32 = ar.pfs, 0, 3, 4, 0 // get some registers + fmerge.s FR_X = f8,f8 + nop.i 0 +} +{ .mfb + mov GR_Parameter_TAG = 61 // error code + frcpa.s0 FR_RESULT, p0 = f0,f0 + // call error handler routine + br.cond.sptk __libm_error_region +} +;; +GLOBAL_LIBM_END(asin) -.endp asin -ASM_SIZE_DIRECTIVE(asin) -.proc __libm_error_region -__libm_error_region: +LOCAL_LIBM_ENTRY(__libm_error_region) .prologue { .mfi add GR_Parameter_Y=-32,sp // Parameter 2 value - nop.f 999 + nop.f 0 .save ar.pfs,GR_SAVE_PFS mov GR_SAVE_PFS=ar.pfs // Save ar.pfs } @@ -857,28 +815,29 @@ __libm_error_region: mov GR_SAVE_GP=gp // Save gp };; { .mmi - stfs [GR_Parameter_Y] = f1,16 // Store Parameter 2 on stack + stfd [GR_Parameter_Y] = FR_Y,16 // STORE Parameter 2 on stack add GR_Parameter_X = 16,sp // Parameter 1 address .save b0, GR_SAVE_B0 mov GR_SAVE_B0=b0 // Save b0 };; - .body - frcpa.s0 f9,p0 = f0,f0 -;; - { .mib - stfd [GR_Parameter_X] = f8 // Store Parameter 1 on stack - add GR_Parameter_RESULT = 0,GR_Parameter_Y - nop.b 0 // Parameter 3 address + stfd [GR_Parameter_X] = FR_X // STORE Parameter 1 on stack + add GR_Parameter_RESULT = 0,GR_Parameter_Y // Parameter 3 address + nop.b 0 } { .mib - stfd [GR_Parameter_Y] = f9,-16 // Store Parameter 3 on stack - adds r32 = 48,sp - br.call.sptk b0=__libm_error_support# // Call error handling function + stfd [GR_Parameter_Y] = FR_RESULT // STORE Parameter 3 on stack + add GR_Parameter_Y = -16,GR_Parameter_Y + br.call.sptk b0=__libm_error_support# // Call error handling function };; { .mmi - ldfd f8 = [r32] // Get return result off stack + add GR_Parameter_RESULT = 48,sp + nop.m 0 + nop.i 0 +};; +{ .mmi + ldfd f8 = [GR_Parameter_RESULT] // Get return result off stack .restore sp add sp = 64,sp // Restore stack pointer mov b0 = GR_SAVE_B0 // Restore return address @@ -887,11 +846,8 @@ __libm_error_region: mov gp = GR_SAVE_GP // Restore gp mov ar.pfs = GR_SAVE_PFS // Restore ar.pfs br.ret.sptk b0 // Return - };; -.endp __libm_error_region -ASM_SIZE_DIRECTIVE(__libm_error_region) - -.type __libm_error_support,@function -.global __libm_error_support +LOCAL_LIBM_END(__libm_error_region) +.type __libm_error_support#,@function +.global __libm_error_support# |