.file "tancot.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 // 04/04/00 Unwind support added // 12/27/00 Improved speed // 02/21/01 Updated to call tanl // 05/30/02 Added cot // 02/10/03 Reordered header: .section, .global, .proc, .align // // API //============================================================== // double tan(double x); // double cot(double x); // // Overview of operation //============================================================== // If the input value in radians is |x| >= 1.xxxxx 2^10 call the // older slower version. // // The new algorithm is used when |x| <= 1.xxxxx 2^9. // // Represent the input X as Nfloat * pi/2 + r // where r can be negative and |r| <= pi/4 // // tan_W = x * 2/pi // Nfloat = round_int(tan_W) // // tan_r = x - Nfloat * (pi/2)_hi // a) tan_r = tan_r - Nfloat * (pi/2)_lo (for tan) // b) tan_r = Nfloat * (pi/2)_lo - tan_r (for cot) // // We have two paths: p8, when Nfloat is even and p9. when Nfloat is odd. // a) for tan: p8: tan(X) = tan(r) // p9: tan(X) = -cot(r) // b) for cot: p9: cot(X) = cot(r) // p8: cot(X) = -tan(r) // // Each is evaluated as a series. The p9 path requires 1/r. // // The coefficients used in the series are stored in a table as // are the pi constants. // // Registers used //============================================================== // // predicate registers used: // p6-12 // // floating-point registers used: // f10-15, f32-106 // f8, input // // general registers used // r14-26, r32-39 // // Assembly macros //============================================================== TAN_INV_PI_BY_2_2TO64 = f10 TAN_RSHF_2TO64 = f11 TAN_2TOM64 = f12 TAN_RSHF = f13 TAN_W_2TO64_RSH = f14 TAN_NFLOAT = f15 tan_Inv_Pi_by_2 = f32 tan_Pi_by_2_hi = f33 tan_Pi_by_2_lo = f34 tan_P0 = f35 tan_P1 = f36 tan_P2 = f37 tan_P3 = f38 tan_P4 = f39 tan_P5 = f40 tan_P6 = f41 tan_P7 = f42 tan_P8 = f43 tan_P9 = f44 tan_P10 = f45 tan_P11 = f46 tan_P12 = f47 tan_P13 = f48 tan_P14 = f49 tan_P15 = f50 tan_Q0 = f51 tan_Q1 = f52 tan_Q2 = f53 tan_Q3 = f54 tan_Q4 = f55 tan_Q5 = f56 tan_Q6 = f57 tan_Q7 = f58 tan_Q8 = f59 tan_Q9 = f60 tan_Q10 = f61 tan_r = f62 tan_rsq = f63 tan_rcube = f64 tan_v18 = f65 tan_v16 = f66 tan_v17 = f67 tan_v12 = f68 tan_v13 = f69 tan_v7 = f70 tan_v8 = f71 tan_v4 = f72 tan_v5 = f73 tan_v15 = f74 tan_v11 = f75 tan_v14 = f76 tan_v3 = f77 tan_v6 = f78 tan_v10 = f79 tan_v2 = f80 tan_v9 = f81 tan_v1 = f82 tan_int_Nfloat = f83 tan_Nfloat = f84 tan_NORM_f8 = f85 tan_W = f86 tan_y0 = f87 tan_d = f88 tan_y1 = f89 tan_dsq = f90 tan_y2 = f91 tan_d4 = f92 tan_inv_r = f93 tan_z1 = f94 tan_z2 = f95 tan_z3 = f96 tan_z4 = f97 tan_z5 = f98 tan_z6 = f99 tan_z7 = f100 tan_z8 = f101 tan_z9 = f102 tan_z10 = f103 tan_z11 = f104 tan_z12 = f105 arg_copy = f106 ///////////////////////////////////////////////////////////// tan_GR_sig_inv_pi_by_2 = r14 tan_GR_rshf_2to64 = r15 tan_GR_exp_2tom64 = r16 tan_GR_n = r17 tan_GR_rshf = r18 tan_AD = r19 tan_GR_10009 = r20 tan_GR_17_ones = r21 tan_GR_N_odd_even = r22 tan_GR_N = r23 tan_signexp = r24 tan_exp = r25 tan_ADQ = r26 GR_SAVE_B0 = r33 GR_SAVE_PFS = r34 GR_SAVE_GP = r35 GR_Parameter_X = r36 GR_Parameter_Y = r37 GR_Parameter_RESULT = r38 GR_Parameter_Tag = r39 RODATA .align 16 LOCAL_OBJECT_START(double_tan_constants) data8 0xC90FDAA22168C234, 0x00003FFF // pi/2 hi data8 0xBEEA54580DDEA0E1 // P14 data8 0x3ED3021ACE749A59 // P15 data8 0xBEF312BD91DC8DA1 // P12 data8 0x3EFAE9AFC14C5119 // P13 data8 0x3F2F342BF411E769 // P8 data8 0x3F1A60FC9F3B0227 // P9 data8 0x3EFF246E78E5E45B // P10 data8 0x3F01D9D2E782875C // P11 data8 0x3F8226E34C4499B6 // P4 data8 0x3F6D6D3F12C236AC // P5 data8 0x3F57DA1146DCFD8B // P6 data8 0x3F43576410FE3D75 // P7 data8 0x3FD5555555555555 // P0 data8 0x3FC11111111111C2 // P1 data8 0x3FABA1BA1BA0E850 // P2 data8 0x3F9664F4886725A7 // P3 LOCAL_OBJECT_END(double_tan_constants) LOCAL_OBJECT_START(double_Q_tan_constants) data8 0xC4C6628B80DC1CD1, 0x00003FBF // pi/2 lo data8 0x3E223A73BA576E48 // Q8 data8 0x3DF54AD8D1F2CA43 // Q9 data8 0x3EF66A8EE529A6AA // Q4 data8 0x3EC2281050410EE6 // Q5 data8 0x3E8D6BB992CC3CF5 // Q6 data8 0x3E57F88DE34832E4 // Q7 data8 0x3FD5555555555555 // Q0 data8 0x3F96C16C16C16DB8 // Q1 data8 0x3F61566ABBFFB489 // Q2 data8 0x3F2BBD77945C1733 // Q3 data8 0x3D927FB33E2B0E04 // Q10 LOCAL_OBJECT_END(double_Q_tan_constants) .section .text //////////////////////////////////////////////////////// LOCAL_LIBM_ENTRY(cot) // The initial fnorm will take any unmasked faults and // normalize any single/double unorms { .mlx cmp.eq p12, p11 = r0, r0 // set p12=1, p11=0 for cot movl tan_GR_sig_inv_pi_by_2 = 0xA2F9836E4E44152A // significand of 2/pi } { .mlx addl tan_AD = @ltoff(double_tan_constants), gp movl tan_GR_rshf_2to64 = 0x47e8000000000000 // 1.1000 2^(63+63+1) } ;; { .mlx mov tan_GR_exp_2tom64 = 0xffff-64 // exponent of scaling factor 2^-64 movl tan_GR_rshf = 0x43e8000000000000 // 1.1000 2^63 for right shift } { .mfb ld8 tan_AD = [tan_AD] fnorm.s0 tan_NORM_f8 = f8 br.cond.sptk COMMON_PATH } ;; LOCAL_LIBM_END(cot) GLOBAL_IEEE754_ENTRY(tan) // The initial fnorm will take any unmasked faults and // normalize any single/double unorms { .mlx cmp.eq p11, p12 = r0, r0 // set p11=1, p12=0 for tan movl tan_GR_sig_inv_pi_by_2 = 0xA2F9836E4E44152A // significand of 2/pi } { .mlx addl tan_AD = @ltoff(double_tan_constants), gp movl tan_GR_rshf_2to64 = 0x47e8000000000000 // 1.1000 2^(63+63+1) } ;; { .mlx mov tan_GR_exp_2tom64 = 0xffff-64 // exponent of scaling factor 2^-64 movl tan_GR_rshf = 0x43e8000000000000 // 1.1000 2^63 for right shift } { .mfi ld8 tan_AD = [tan_AD] fnorm.s0 tan_NORM_f8 = f8 nop.i 0 } ;; // Common path for both tan and cot COMMON_PATH: // Form two constants we need // 2/pi * 2^1 * 2^63, scaled by 2^64 since we just loaded the significand // 1.1000...000 * 2^(63+63+1) to right shift int(W) into the significand { .mmi setf.sig TAN_INV_PI_BY_2_2TO64 = tan_GR_sig_inv_pi_by_2 setf.d TAN_RSHF_2TO64 = tan_GR_rshf_2to64 mov tan_GR_17_ones = 0x1ffff ;; } // Form another constant // 2^-64 for scaling Nfloat // 1.1000...000 * 2^63, the right shift constant { .mmf setf.exp TAN_2TOM64 = tan_GR_exp_2tom64 adds tan_ADQ = double_Q_tan_constants - double_tan_constants, tan_AD (p11) fclass.m.unc p6,p0 = f8, 0x07 // Test for x=0 (tan) } ;; // Form another constant // 2^-64 for scaling Nfloat // 1.1000...000 * 2^63, the right shift constant { .mmf setf.d TAN_RSHF = tan_GR_rshf ldfe tan_Pi_by_2_hi = [tan_AD],16 fclass.m.unc p7,p0 = f8, 0x23 // Test for x=inf } ;; { .mfb ldfe tan_Pi_by_2_lo = [tan_ADQ],16 fclass.m.unc p8,p0 = f8, 0xc3 // Test for x=nan (p6) br.ret.spnt b0 ;; // Exit for x=0 (tan only) } { .mfi ldfpd tan_P14,tan_P15 = [tan_AD],16 (p7) frcpa.s0 f8,p9=f0,f0 // Set qnan indef if x=inf mov tan_GR_10009 = 0x10009 } { .mib ldfpd tan_Q8,tan_Q9 = [tan_ADQ],16 nop.i 999 (p7) br.ret.spnt b0 ;; // Exit for x=inf } { .mfi ldfpd tan_P12,tan_P13 = [tan_AD],16 (p12) fclass.m.unc p6,p0 = f8, 0x07 // Test for x=0 (cot) nop.i 999 } { .mfb ldfpd tan_Q4,tan_Q5 = [tan_ADQ],16 (p8) fma.d.s0 f8=f8,f1,f8 // Set qnan if x=nan (p8) br.ret.spnt b0 ;; // Exit for x=nan } { .mmf getf.exp tan_signexp = tan_NORM_f8 ldfpd tan_P8,tan_P9 = [tan_AD],16 fmerge.s arg_copy = f8, f8 ;; // Save input for error call } // Multiply x by scaled 2/pi and add large const to shift integer part of W to // rightmost bits of significand { .mmf alloc r32=ar.pfs,0,4,4,0 ldfpd tan_Q6,tan_Q7 = [tan_ADQ],16 fma.s1 TAN_W_2TO64_RSH = tan_NORM_f8,TAN_INV_PI_BY_2_2TO64,TAN_RSHF_2TO64 };; { .mmf ldfpd tan_P10,tan_P11 = [tan_AD],16 and tan_exp = tan_GR_17_ones, tan_signexp (p6) frcpa.s0 f8, p0 = f1, f8 ;; // cot(+-0) = +-Inf } // p7 is true if we must call DBX TAN // p7 is true if f8 exp is > 0x10009 (which includes all ones // NAN or inf) { .mmb ldfpd tan_Q0,tan_Q1 = [tan_ADQ],16 cmp.ge.unc p7,p0 = tan_exp,tan_GR_10009 (p7) br.cond.spnt TAN_DBX ;; } { .mmb ldfpd tan_P4,tan_P5 = [tan_AD],16 (p6) mov GR_Parameter_Tag = 226 // (cot) (p6) br.cond.spnt __libm_error_region ;; // call error support if cot(+-0) } { .mmi ldfpd tan_Q2,tan_Q3 = [tan_ADQ],16 nop.m 999 nop.i 999 ;; } // TAN_NFLOAT = Round_Int_Nearest(tan_W) { .mfi ldfpd tan_P6,tan_P7 = [tan_AD],16 fms.s1 TAN_NFLOAT = TAN_W_2TO64_RSH,TAN_2TOM64,TAN_RSHF nop.i 999 ;; } { .mfi ldfd tan_Q10 = [tan_ADQ] nop.f 999 nop.i 999 ;; } { .mfi ldfpd tan_P0,tan_P1 = [tan_AD],16 nop.f 999 nop.i 999 ;; } { .mmi getf.sig tan_GR_n = TAN_W_2TO64_RSH ldfpd tan_P2,tan_P3 = [tan_AD] nop.i 999 ;; } // tan_r = -tan_Nfloat * tan_Pi_by_2_hi + x { .mfi (p12) add tan_GR_n = 0x1, tan_GR_n // N = N + 1 (for cot) fnma.s1 tan_r = TAN_NFLOAT, tan_Pi_by_2_hi, tan_NORM_f8 nop.i 999 ;; } // p8 ==> even // p9 ==> odd { .mmi and tan_GR_N_odd_even = 0x1, tan_GR_n ;; nop.m 999 cmp.eq.unc p8,p9 = tan_GR_N_odd_even, r0 ;; } .pred.rel "mutex", p11, p12 // tan_r = tan_r -tan_Nfloat * tan_Pi_by_2_lo (tan) { .mfi nop.m 999 (p11) fnma.s1 tan_r = TAN_NFLOAT, tan_Pi_by_2_lo, tan_r nop.i 999 } // tan_r = -(tan_r -tan_Nfloat * tan_Pi_by_2_lo) (cot) { .mfi nop.m 999 (p12) fms.s1 tan_r = TAN_NFLOAT, tan_Pi_by_2_lo, tan_r nop.i 999 ;; } { .mfi nop.m 999 fma.s1 tan_rsq = tan_r, tan_r, f0 nop.i 999 ;; } { .mfi nop.m 999 (p9) frcpa.s1 tan_y0, p0 = f1,tan_r nop.i 999 ;; } { .mfi nop.m 999 (p8) fma.s1 tan_v18 = tan_rsq, tan_P15, tan_P14 nop.i 999 } { .mfi nop.m 999 (p8) fma.s1 tan_v4 = tan_rsq, tan_P1, tan_P0 nop.i 999 ;; } { .mfi nop.m 999 (p8) fma.s1 tan_v16 = tan_rsq, tan_P13, tan_P12 nop.i 999 } { .mfi nop.m 999 (p8) fma.s1 tan_v17 = tan_rsq, tan_rsq, f0 nop.i 999 ;; } { .mfi nop.m 999 (p8) fma.s1 tan_v12 = tan_rsq, tan_P9, tan_P8 nop.i 999 } { .mfi nop.m 999 (p8) fma.s1 tan_v13 = tan_rsq, tan_P11, tan_P10 nop.i 999 ;; } { .mfi nop.m 999 (p8) fma.s1 tan_v7 = tan_rsq, tan_P5, tan_P4 nop.i 999 } { .mfi nop.m 999 (p8) fma.s1 tan_v8 = tan_rsq, tan_P7, tan_P6 nop.i 999 ;; } { .mfi nop.m 999 (p9) fnma.s1 tan_d = tan_r, tan_y0, f1 nop.i 999 } { .mfi nop.m 999 (p8) fma.s1 tan_v5 = tan_rsq, tan_P3, tan_P2 nop.i 999 ;; } { .mfi nop.m 999 (p9) fma.s1 tan_z11 = tan_rsq, tan_Q9, tan_Q8 nop.i 999 } { .mfi nop.m 999 (p9) fma.s1 tan_z12 = tan_rsq, tan_rsq, f0 nop.i 999 ;; } { .mfi nop.m 999 (p8) fma.s1 tan_v15 = tan_v17, tan_v18, tan_v16 nop.i 999 } { .mfi nop.m 999 (p9) fma.s1 tan_z7 = tan_rsq, tan_Q5, tan_Q4 nop.i 999 ;; } { .mfi nop.m 999 (p8) fma.s1 tan_v11 = tan_v17, tan_v13, tan_v12 nop.i 999 } { .mfi nop.m 999 (p9) fma.s1 tan_z8 = tan_rsq, tan_Q7, tan_Q6 nop.i 999 ;; } { .mfi nop.m 999 (p8) fma.s1 tan_v14 = tan_v17, tan_v17, f0 nop.i 999 } { .mfi nop.m 999 (p9) fma.s1 tan_z3 = tan_rsq, tan_Q1, tan_Q0 nop.i 999 ;; } { .mfi nop.m 999 (p8) fma.s1 tan_v3 = tan_v17, tan_v5, tan_v4 nop.i 999 } { .mfi nop.m 999 (p8) fma.s1 tan_v6 = tan_v17, tan_v8, tan_v7 nop.i 999 ;; } { .mfi nop.m 999 (p9) fma.s1 tan_y1 = tan_y0, tan_d, tan_y0 nop.i 999 } { .mfi nop.m 999 (p9) fma.s1 tan_dsq = tan_d, tan_d, f0 nop.i 999 ;; } { .mfi nop.m 999 (p9) fma.s1 tan_z10 = tan_z12, tan_Q10, tan_z11 nop.i 999 } { .mfi nop.m 999 (p9) fma.s1 tan_z9 = tan_z12, tan_z12,f0 nop.i 999 ;; } { .mfi nop.m 999 (p9) fma.s1 tan_z4 = tan_rsq, tan_Q3, tan_Q2 nop.i 999 } { .mfi nop.m 999 (p9) fma.s1 tan_z6 = tan_z12, tan_z8, tan_z7 nop.i 999 ;; } { .mfi nop.m 999 (p8) fma.s1 tan_v10 = tan_v14, tan_v15, tan_v11 nop.i 999 ;; } { .mfi nop.m 999 (p9) fma.s1 tan_y2 = tan_y1, tan_d, tan_y0 nop.i 999 } { .mfi nop.m 999 (p9) fma.s1 tan_d4 = tan_dsq, tan_dsq, tan_d nop.i 999 ;; } { .mfi nop.m 999 (p8) fma.s1 tan_v2 = tan_v14, tan_v6, tan_v3 nop.i 999 } { .mfi nop.m 999 (p8) fma.s1 tan_v9 = tan_v14, tan_v14, f0 nop.i 999 ;; } { .mfi nop.m 999 (p9) fma.s1 tan_z2 = tan_z12, tan_z4, tan_z3 nop.i 999 } { .mfi nop.m 999 (p9) fma.s1 tan_z5 = tan_z9, tan_z10, tan_z6 nop.i 999 ;; } { .mfi nop.m 999 (p9) fma.s1 tan_inv_r = tan_d4, tan_y2, tan_y0 nop.i 999 } { .mfi nop.m 999 (p8) fma.s1 tan_rcube = tan_rsq, tan_r, f0 nop.i 999 ;; } { .mfi nop.m 999 (p8) fma.s1 tan_v1 = tan_v9, tan_v10, tan_v2 nop.i 999 } { .mfi nop.m 999 (p9) fma.s1 tan_z1 = tan_z9, tan_z5, tan_z2 nop.i 999 ;; } { .mfi nop.m 999 (p8) fma.d.s0 f8 = tan_v1, tan_rcube, tan_r nop.i 999 } { .mfb nop.m 999 (p9) fms.d.s0 f8 = tan_r, tan_z1, tan_inv_r br.ret.sptk b0 ;; } GLOBAL_IEEE754_END(tan) LOCAL_LIBM_ENTRY(__libm_callout) TAN_DBX: .prologue { .mfi nop.m 0 fmerge.s f9 = f0,f0 .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 { .mmb nop.m 999 nop.m 999 (p11) br.cond.sptk.many call_tanl ;; } // Here if we should call cotl { .mmb nop.m 999 nop.m 999 br.call.sptk.many b0=__libm_cotl# ;; } { .mfi mov gp = GR_SAVE_GP fnorm.d.s0 f8 = f8 mov b0 = GR_SAVE_B0 } ;; { .mib nop.m 999 mov ar.pfs = GR_SAVE_PFS br.ret.sptk b0 ;; } // Here if we should call tanl call_tanl: { .mmb nop.m 999 nop.m 999 br.call.sptk.many b0=__libm_tanl# ;; } { .mfi mov gp = GR_SAVE_GP fnorm.d.s0 f8 = f8 mov b0 = GR_SAVE_B0 } ;; { .mib nop.m 999 mov ar.pfs = GR_SAVE_PFS br.ret.sptk b0 ;; } LOCAL_LIBM_END(__libm_callout) .type __libm_tanl#,@function .global __libm_tanl# .type __libm_cotl#,@function .global __libm_cotl# LOCAL_LIBM_ENTRY(__libm_error_region) .prologue // (1) { .mfi add GR_Parameter_Y=-32,sp // Parameter 2 value 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 };; // (2) { .mmi stfd [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 // (3) { .mib stfd [GR_Parameter_X] = arg_copy // STORE Parameter 1 on stack add GR_Parameter_RESULT = 0,GR_Parameter_Y // Parameter 3 address nop.b 0 } { .mib stfd [GR_Parameter_Y] = f8 // 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 };; // (4) { .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 };; { .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#