.file "asinf.s" // Copyright (c) 2000 - 2003, Intel Corporation // All rights reserved. // // Contributed 2000 by the Intel Numerics Group, Intel Corporation // // Redistribution and use in source and binary forms, with or without // modification, are permitted provided that the following conditions are // met: // // * Redistributions of source code must retain the above copyright // notice, this list of conditions and the following disclaimer. // // * Redistributions in binary form must reproduce the above copyright // notice, this list of conditions and the following disclaimer in the // documentation and/or other materials provided with the distribution. // // * The name of Intel Corporation may not be used to endorse or promote // products derived from this software without specific prior written // permission. // THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS // "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT // LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR // A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL INTEL OR ITS // CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, // EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, // PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR // PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY // OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY OR TORT (INCLUDING // NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS // SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. // // Intel Corporation is the author of this code, and requests that all // problem reports or change requests be submitted to it directly at // http://www.intel.com/software/products/opensource/libraries/num.htm. // History //============================================================== // 02/02/00 Initial version // 06/28/00 Improved speed // 06/31/00 Changed register allocation because of some duplicate macros // moved nan exit bundle up to gain a cycle. // 08/08/00 Improved speed by avoiding SIR flush. // 08/15/00 Bundle added after call to __libm_error_support to properly // set [the previously overwritten] GR_Parameter_RESULT. // 08/17/00 Changed predicate register macro-usage to direct predicate // names due to an assembler bug. // 10/17/00 Improved speed of x=0 and x=1 paths, set D flag if x denormal. // 03/13/01 Corrected sign of imm1 value in dep instruction. // 05/20/02 Cleaned up namespace and sf0 syntax // 02/06/03 Reordered header: .section, .global, .proc, .align // Description //========================================= // The asinf function computes the arc sine of x in the range [-pi,+pi]. // A doman error occurs for arguments not in the range [-1,+1]. // asinf(+-0) returns +-0 // asinf(x) returns a Nan and raises the invalid exception for |x| >1 // The acosf function returns the arc cosine in the range [0, +pi] radians. // A doman error occurs for arguments not in the range [-1,+1]. // acosf(1) returns +0 // acosf(x) returns a Nan and raises the invalid exception for |x| >1 // |x| <= sqrt(2)/2. get Ax and Bx // poly_p1 = x p1 // poly_p3 = x2 p4 + p3 // poly_p1 = x2 (poly_p1) + x = x2(x p1) + x // poly_p2 = x2( poly_p3) + p2 = x2(x2 p4 + p3) + p2 // poly_Ax = x5(x2( poly_p3) + p2) + x2(x p1) + x // = x5(x2(x2 p4 + p3) + p2) + x2(x p1) + x // poly_p7 = x2 p8 + p7 // poly_p5 = x2 p6 + p5 // poly_p7 = x4 p9 + (poly_p7) // poly_p7 = x4 p9 + (x2 p8 + p7) // poly_Bx = x4 (x4 p9 + (x2 p8 + p7)) + x2 p6 + p5 // answer1 = x11(x4 (x4 p9 + (x2 p8 + p7)) + x2 p6 + p5) + x5(x2(x2 p4 + p3) + p2) + x2(x p1) + x // = x19 p9 + x17 p8 + x15 p7 x13 p6 + x11 p5 + x9 p4 + x7 p3 + x5 p2 + x3 p1 + x // |x| > sqrt(2)/2 // Get z = sqrt(1-x2) // Get polynomial in t = 1-x2 // t2 = t t // t4 = t2 t2 // poly_p4 = t p5 + p4 // poly_p1 = t p1 + 1 // poly_p6 = t p7 + p6 // poly_p2 = t p3 + p2 // poly_p8 = t p9 + p8 // poly_p4 = t2 poly_p6 + poly_p4 // = t2 (t p7 + p6) + (t p5 + p4) // poly_p2 = t2 poly_p2 + poly_p1 // = t2 (t p3 + p2) + (t p1 + 1) // poly_p4 = t4 poly_p8 + poly_p4 // = t4 (t p9 + p8) + (t2 (t p7 + p6) + (t p5 + p4)) // P(t) = poly_p2 + t4 poly_p8 // = t2 (t p3 + p2) + (t p1 + 1) + t4 (t4 (t p9 + p8) + (t2 (t p7 + p6) + (t p5 + p4))) // = t3 p3 + t2 p2 + t p1 + 1 + t9 p9 + t8 p8 + t7 p7 + t6 p6 + t5 p5 + t4 p4 // answer2 = - sign(x) z P(t) + (sign(x) pi/2) // // Assembly macros //========================================= // predicate registers //asinf_pred_LEsqrt2by2 = p7 //asinf_pred_GTsqrt2by2 = p8 // integer registers ASINF_Addr1 = r33 ASINF_Addr2 = r34 ASINF_GR_1by2 = r35 ASINF_GR_3by2 = r36 ASINF_GR_5by2 = r37 GR_SAVE_B0 = r38 GR_SAVE_PFS = r39 GR_SAVE_GP = r40 GR_Parameter_X = r41 GR_Parameter_Y = r42 GR_Parameter_RESULT = r43 GR_Parameter_TAG = r44 // floating point registers asinf_y = f32 asinf_abs_x = f33 asinf_x2 = f34 asinf_sgn_x = f35 asinf_1by2 = f36 asinf_3by2 = f37 asinf_5by2 = f38 asinf_coeff_P3 = f39 asinf_coeff_P8 = f40 asinf_coeff_P1 = f41 asinf_coeff_P4 = f42 asinf_coeff_P5 = f43 asinf_coeff_P2 = f44 asinf_coeff_P7 = f45 asinf_coeff_P6 = f46 asinf_coeff_P9 = f47 asinf_x2 = f48 asinf_x3 = f49 asinf_x4 = f50 asinf_x8 = f51 asinf_x5 = f52 asinf_const_piby2 = f53 asinf_const_sqrt2by2 = f54 asinf_x11 = f55 asinf_poly_p1 = f56 asinf_poly_p3 = f57 asinf_sinf1 = f58 asinf_poly_p2 = f59 asinf_poly_Ax = f60 asinf_poly_p7 = f61 asinf_poly_p5 = f62 asinf_sgnx_t4 = f63 asinf_poly_Bx = f64 asinf_t = f65 asinf_yby2 = f66 asinf_B = f67 asinf_B2 = f68 asinf_Az = f69 asinf_dz = f70 asinf_Sz = f71 asinf_d2z = f72 asinf_Fz = f73 asinf_z = f74 asinf_sgnx_z = f75 asinf_t2 = f76 asinf_2poly_p4 = f77 asinf_2poly_p6 = f78 asinf_2poly_p1 = f79 asinf_2poly_p2 = f80 asinf_2poly_p8 = f81 asinf_t4 = f82 asinf_Pt = f83 asinf_sgnx_2poly_p2 = f84 asinf_sgn_x_piby2 = f85 asinf_poly_p7a = f86 asinf_2poly_p4a = f87 asinf_2poly_p4b = f88 asinf_2poly_p2a = f89 asinf_poly_p1a = f90 // Data tables //============================================================== RODATA .align 16 LOCAL_OBJECT_START(asinf_coeff_1_table) data8 0x3FC5555607DCF816 // P1 data8 0x3F9CF81AD9BAB2C6 // P4 data8 0x3FC59E0975074DF3 // P7 data8 0xBFA6F4CC2780AA1D // P6 data8 0x3FC2DD45292E93CB // P9 data8 0x3fe6a09e667f3bcd // sqrt(2)/2 LOCAL_OBJECT_END(asinf_coeff_1_table) LOCAL_OBJECT_START(asinf_coeff_2_table) data8 0x3FA6F108E31EFBA6 // P3 data8 0xBFCA31BF175D82A0 // P8 data8 0x3FA30C0337F6418B // P5 data8 0x3FB332C9266CB1F9 // P2 data8 0x3ff921fb54442d18 // pi_by_2 LOCAL_OBJECT_END(asinf_coeff_2_table) .section .text GLOBAL_LIBM_ENTRY(asinf) // Load the addresses of the two tables. // Then, load the coefficients and other constants. { .mfi alloc r32 = ar.pfs,1,8,4,0 fnma.s1 asinf_t = f8,f8,f1 dep.z ASINF_GR_1by2 = 0x3f,24,8 // 0x3f000000 } { .mfi addl ASINF_Addr1 = @ltoff(asinf_coeff_1_table),gp fma.s1 asinf_x2 = f8,f8,f0 addl ASINF_Addr2 = @ltoff(asinf_coeff_2_table),gp ;; } { .mfi ld8 ASINF_Addr1 = [ASINF_Addr1] fmerge.s asinf_abs_x = f1,f8 dep ASINF_GR_3by2 = -1,r0,22,8 // 0x3fc00000 } { .mlx nop.m 999 movl ASINF_GR_5by2 = 0x40200000;; } { .mfi setf.s asinf_1by2 = ASINF_GR_1by2 fmerge.s asinf_sgn_x = f8,f1 nop.i 999 } { .mfi ld8 ASINF_Addr2 = [ASINF_Addr2] nop.f 0 nop.i 999;; } { .mfi setf.s asinf_5by2 = ASINF_GR_5by2 fcmp.lt.s1 p11,p12 = f8,f0 nop.i 999;; } { .mmf ldfpd asinf_coeff_P1,asinf_coeff_P4 = [ASINF_Addr1],16 setf.s asinf_3by2 = ASINF_GR_3by2 fclass.m.unc p8,p0 = f8, 0xc3 ;; //@qnan | @snan } { .mfi ldfpd asinf_coeff_P7,asinf_coeff_P6 = [ASINF_Addr1],16 fma.s1 asinf_t2 = asinf_t,asinf_t,f0 nop.i 999 } { .mfi ldfpd asinf_coeff_P3,asinf_coeff_P8 = [ASINF_Addr2],16 fma.s1 asinf_x4 = asinf_x2,asinf_x2,f0 nop.i 999;; } { .mfi ldfpd asinf_coeff_P9,asinf_const_sqrt2by2 = [ASINF_Addr1] fclass.m.unc p10,p0 = f8, 0x07 //@zero nop.i 999 } { .mfi ldfpd asinf_coeff_P5,asinf_coeff_P2 = [ASINF_Addr2],16 fma.s1 asinf_x3 = f8,asinf_x2,f0 nop.i 999;; } { .mfi ldfd asinf_const_piby2 = [ASINF_Addr2] frsqrta.s1 asinf_B,p0 = asinf_t nop.i 999 } { .mfb nop.m 999 (p8) fma.s.s0 f8 = f8,f1,f0 (p8) br.ret.spnt b0 ;; // Exit if x=nan } { .mfb nop.m 999 fcmp.eq.s1 p6,p0 = asinf_abs_x,f1 (p10) br.ret.spnt b0 ;; // Exit if x=0 } { .mfi nop.m 999 fcmp.gt.s1 p9,p0 = asinf_abs_x,f1 nop.i 999;; } { .mfi nop.m 999 fma.s1 asinf_x8 = asinf_x4,asinf_x4,f0 nop.i 999 } { .mfb nop.m 999 fma.s1 asinf_t4 = asinf_t2,asinf_t2,f0 (p6) br.cond.spnt ASINF_ABS_ONE ;; // Branch if |x|=1 } { .mfi nop.m 999 fma.s1 asinf_x5 = asinf_x2,asinf_x3,f0 nop.i 999 } { .mfb (p9) mov GR_Parameter_TAG = 62 fma.s1 asinf_yby2 = asinf_t,asinf_1by2,f0 (p9) br.cond.spnt __libm_error_region ;; // Branch if |x|>1 } { .mfi nop.m 999 fma.s1 asinf_Az = asinf_t,asinf_B,f0 nop.i 999 } { .mfi nop.m 999 fma.s1 asinf_B2 = asinf_B,asinf_B,f0 nop.i 999;; } { .mfi nop.m 999 fma.s1 asinf_poly_p1 = f8,asinf_coeff_P1,f0 nop.i 999 } { .mfi nop.m 999 fma.s1 asinf_2poly_p1 = asinf_coeff_P1,asinf_t,f1 nop.i 999;; } { .mfi nop.m 999 fma.s1 asinf_poly_p3 = asinf_coeff_P4,asinf_x2,asinf_coeff_P3 nop.i 999 } { .mfi nop.m 999 fma.s1 asinf_2poly_p6 = asinf_coeff_P7,asinf_t,asinf_coeff_P6 nop.i 999;; } { .mfi nop.m 999 fma.s1 asinf_poly_p7 = asinf_x2,asinf_coeff_P8,asinf_coeff_P7 nop.i 999 } { .mfi nop.m 999 fma.s1 asinf_2poly_p2 = asinf_coeff_P3,asinf_t,asinf_coeff_P2 nop.i 999;; } { .mfi nop.m 999 fma.s1 asinf_poly_p5 = asinf_x2,asinf_coeff_P6,asinf_coeff_P5 nop.i 999 } { .mfi nop.m 999 fma.s1 asinf_2poly_p4 = asinf_coeff_P5,asinf_t,asinf_coeff_P4 nop.i 999;; } { .mfi nop.m 999 fma.d.s1 asinf_x11 = asinf_x8,asinf_x3,f0 nop.i 999 } { .mfi nop.m 999 fnma.s1 asinf_dz = asinf_B2,asinf_yby2,asinf_1by2 nop.i 999;; } { .mfi nop.m 999 fma.s1 asinf_poly_p1a = asinf_x2,asinf_poly_p1,f8 nop.i 999 } { .mfi nop.m 999 fma.s1 asinf_2poly_p8 = asinf_coeff_P9,asinf_t,asinf_coeff_P8 nop.i 999;; } // Get the absolute value of x and determine the region in which x lies { .mfi nop.m 999 fcmp.le.s1 p7,p8 = asinf_abs_x,asinf_const_sqrt2by2 nop.i 999 } { .mfi nop.m 999 fma.s1 asinf_poly_p2 = asinf_x2,asinf_poly_p3,asinf_coeff_P2 nop.i 999;; } { .mfi nop.m 999 fma.s1 asinf_poly_p7a = asinf_x4,asinf_coeff_P9,asinf_poly_p7 nop.i 999 } { .mfi nop.m 999 fma.s1 asinf_2poly_p2a = asinf_2poly_p2,asinf_t2,asinf_2poly_p1 nop.i 999;; } { .mfi nop.m 999 (p8) fma.s1 asinf_sgnx_t4 = asinf_sgn_x,asinf_t4,f0 nop.i 999 } { .mfi nop.m 999 (p8) fma.s1 asinf_2poly_p4a = asinf_2poly_p6,asinf_t2,asinf_2poly_p4 nop.i 999;; } { .mfi nop.m 999 (p8) fma.s1 asinf_Sz = asinf_5by2,asinf_dz,asinf_3by2 nop.i 999 } { .mfi nop.m 999 (p8) fma.s1 asinf_d2z = asinf_dz,asinf_dz,f0 nop.i 999;; } { .mfi nop.m 999 (p8) fma.s1 asinf_sgn_x_piby2 = asinf_sgn_x,asinf_const_piby2,f0 nop.i 999 } { .mfi nop.m 999 (p7) fma.d.s1 asinf_poly_Ax = asinf_x5,asinf_poly_p2,asinf_poly_p1a nop.i 999;; } { .mfi nop.m 999 (p7) fma.d.s1 asinf_poly_Bx = asinf_x4,asinf_poly_p7a,asinf_poly_p5 nop.i 999 } { .mfi nop.m 999 (p8) fma.s1 asinf_sgnx_2poly_p2 = asinf_sgn_x,asinf_2poly_p2a,f0 nop.i 999;; } { .mfi nop.m 999 fcmp.eq.s0 p6,p0 = f8,f0 // Only purpose is to set D if x denormal nop.i 999 } { .mfi nop.m 999 (p8) fma.s1 asinf_2poly_p4b = asinf_2poly_p8,asinf_t4,asinf_2poly_p4a nop.i 999;; } { .mfi nop.m 999 (p8) fma.s1 asinf_Fz = asinf_d2z,asinf_Sz,asinf_dz nop.i 999;; } { .mfi nop.m 999 (p8) fma.d.s1 asinf_Pt = asinf_2poly_p4b,asinf_sgnx_t4,asinf_sgnx_2poly_p2 nop.i 999;; } { .mfi nop.m 999 (p8) fma.d.s1 asinf_z = asinf_Az,asinf_Fz,asinf_Az nop.i 999;; } .pred.rel "mutex",p8,p7 //asinf_pred_GTsqrt2by2,asinf_pred_LEsqrt2by2 { .mfi nop.m 999 (p8) fnma.s.s0 f8 = asinf_z,asinf_Pt,asinf_sgn_x_piby2 nop.i 999 } { .mfb nop.m 999 (p7) fma.s.s0 f8 = asinf_x11,asinf_poly_Bx,asinf_poly_Ax br.ret.sptk b0 ;; } ASINF_ABS_ONE: // Here for short exit if |x|=1 { .mfb nop.m 999 fma.s.s0 f8 = asinf_sgn_x,asinf_const_piby2,f0 br.ret.sptk b0 } ;; GLOBAL_LIBM_END(asinf) // Stack operations when calling error support. // (1) (2) // sp -> + psp -> + // | | // | | <- GR_Y // | | // | <-GR_Y Y2->| // | | // | | <- GR_X // | | // sp-64 -> + sp -> + // save ar.pfs save b0 // save gp // Stack operations when calling error support. // (3) (call) (4) // psp -> + sp -> + // | | // R3 ->| <- GR_RESULT | -> f8 // | | // Y2 ->| <- GR_Y | // | | // X1 ->| | // | | // sp -> + + // restore gp // restore ar.pfs LOCAL_LIBM_ENTRY(__libm_error_region) .prologue { .mfi add GR_Parameter_Y=-32,sp // Parameter 2 value nop.f 999 .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 stfs [GR_Parameter_Y] = f1,16 // 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 { .mfi nop.m 0 frcpa.s0 f9,p0 = f0,f0 nop.i 0 };; { .mib stfs [GR_Parameter_X] = f8 // Store Parameter 1 on stack add GR_Parameter_RESULT = 0,GR_Parameter_Y nop.b 0 // Parameter 3 address } { .mib stfs [GR_Parameter_Y] = f9 // 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 nop.m 0 nop.m 0 add GR_Parameter_RESULT = 48,sp };; { .mmi ldfs 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 };; { .mib mov gp = GR_SAVE_GP // Restore gp mov ar.pfs = GR_SAVE_PFS // Restore ar.pfs br.ret.sptk b0 // Return };; LOCAL_LIBM_END(__libm_error_region) .type __libm_error_support#,@function .global __libm_error_support#