.file "erfcf.s"// Copyright (c) 2002 - 2005, Intel Corporation// All rights reserved.//// Contributed 2002 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//==============================================================// 01/17/02  Initial version// 05/20/02  Cleaned up namespace and sf0 syntax// 02/06/03  Reordered header: .section, .global, .proc, .align// 03/31/05  Reformatted delimiters between data tables//// API//==============================================================// float erfcf(float)//// Overview of operation//==============================================================// 1. 0 <= x <= 10.06//    //    erfcf(x)  = P15(x) * exp( -x^2 )////    Comment:////    Let x(0)=0, x(i) = 2^(i), i=1,...3, x(4)= 10.06// //    Let x(i)<= x < x(i+1).//    We can find i as exponent of argument x (let i = 0 for 0<= x < 2  )// //    Let P15(x) - polynomial approximation of degree 15 for function//    erfcf(x) * exp( x^2) and x(i) <= x <= x(i+1), i = 0,1,2,3//    Polynomial coeffitients we have in the table erfc_p_table.////    So we can find result for erfcf(x) as above.//    Algorithm description for exp function see below.//                     // 2. -4.4 <= x < 0////    erfcf(x)  = 2.0 - erfcf(-x)//// 3. x > 10.06////    erfcf(x)  ~=~ 0.0//// 4. x < -4.4//            //    erfcf(x)  ~=~ 2.0// Special values //==============================================================// erfcf(+0)    = 1.0// erfcf(-0)    = 1.0// erfcf(+qnan) = +qnan // erfcf(-qnan) = -qnan // erfcf(+snan) = +qnan // erfcf(-snan) = -qnan // erfcf(-inf)  = 2.0 // erfcf(+inf)  = +0//==============================================================// Take double exp(double) from libm_64.//// Overview of operation//==============================================================// Take the input x. w is "how many log2/128 in x?"//  w = x * 128/log2//  n = int(w)//  x = n log2/128 + r + delta//  n = 128M + index_1 + 2^4 index_2//  x = M log2 + (log2/128) index_1 + (log2/8) index_2 + r + delta//  exp(x) = 2^M  2^(index_1/128)  2^(index_2/8) exp(r) exp(delta)//       Construct 2^M//       Get 2^(index_1/128) from table_1;//       Get 2^(index_2/8)   from table_2;//       Calculate exp(r) by series//          r = x - n (log2/128)_high//          delta = - n (log2/128)_low//       Calculate exp(delta) as 1 + delta//// Comment for erfcf://// Let exp(r) = 1 + x + 0.5*x^2 + (1/6)*x^3// Let delta  = 0.//==============================================================//// Registers used//==============================================================// Floating Point registers used: // f8, input// f6,f7,f9 -> f11,  f32 -> f92// General registers used: // r14 -> r22,r32 -> r50 // Predicate registers used:// p6 -> p15// Assembly macros//==============================================================EXP_AD_TB1             = r14exp_GR_sig_inv_ln2     = r15exp_TB1_size           = r16exp_GR_rshf_2to56      = r17exp_GR_exp_2tom56      = r18exp_GR_rshf            = r33EXP_AD_TB2             = r34EXP_AD_P               = r35exp_GR_N               = r36exp_GR_index_1         = r37exp_GR_index_2_16      = r38exp_GR_biased_M        = r39EXP_AD_T1              = r40EXP_AD_T2              = r41exp_TB2_size           = r42// GR for erfcf(x)//==============================================================GR_IndxPlusBias        = r19GR_ExpMask             = r20GR_BIAS                = r21GR_ShftPi_bias         = r22GR_P_POINT_1           = r43GR_P_POINT_2           = r44GR_P_POINT_3           = r45GR_P_POINT_4           = r46GR_ShftPi              = r47GR_EpsNorm             = r48GR_05                  = r49GR_1_by_6              = r50// GR for __libm_support call//==============================================================GR_SAVE_B0             = r43GR_SAVE_PFS            = r44GR_SAVE_GP             = r45GR_SAVE_SP             = r46GR_Parameter_X         = r47GR_Parameter_Y         = r48GR_Parameter_RESULT    = r49GR_Parameter_TAG       = r50// FR for exp(-x^2)//==============================================================FR_X                   = f10FR_Y                   = f1FR_RESULT              = f8EXP_2TOM56             = f6EXP_INV_LN2_2TO63      = f7EXP_W_2TO56_RSH        = f9exp_ln2_by_128_hi      = f11EXP_RSHF_2TO56         = f32 exp_ln2_by_128_lo      = f33 EXP_RSHF               = f34EXP_Nfloat             = f35 exp_r                  = f36exp_rsq                = f37EXP_2M                 = f38exp_S1                 = f39exp_T1                 = f40exp_P                  = f41exp_S                  = f42EXP_NORM_f8            = f43   exp_S2                 = f44exp_T2                 = f45// FR for erfcf(x)//==============================================================FR_AbsArg              = f46FR_Tmp                 = f47FR_Tmp1                = f48FR_Tmpf                = f49FR_NormX               = f50FR_A15                 = f51FR_A14                 = f52FR_A13                 = f53FR_A12                 = f54FR_A11                 = f55FR_A10                 = f56FR_A9                  = f57FR_A8                  = f58FR_A7                  = f59FR_A6                  = f60FR_A5                  = f61FR_A4                  = f62FR_A3                  = f63FR_A2                  = f64FR_A1                  = f65FR_A0                  = f66FR_P15_0_1             = f67FR_P15_1_1             = f68FR_P15_1_2             = f69FR_P15_2_1             = f70FR_P15_2_2             = f71FR_P15_3_1             = f72FR_P15_3_2             = f73FR_P15_4_1             = f74FR_P15_4_2             = f75FR_P15_7_1             = f76FR_P15_7_2             = f77FR_P15_8_1             = f78FR_P15_9_1             = f79FR_P15_9_2             = f80FR_P15_13_1            = f81FR_P15_14_1            = f82FR_P15_14_2            = f83FR_2                   = f84FR_05                  = f85FR_1_by_6              = f86FR_Pol                 = f87FR_Exp                 = f88FR_POS_ARG_ASYMP       = f89FR_NEG_ARG_ASYMP       = f90FR_UnfBound            = f91FR_EpsNorm             = f92// Data tables//==============================================================RODATA.align 16// ************* DO NOT CHANGE ORDER OF THESE TABLES ********************// double-extended 1/ln(2)// 3fff b8aa 3b29 5c17 f0bb be87fed0691d3e88// 3fff b8aa 3b29 5c17 f0bc // For speed the significand will be loaded directly with a movl and setf.sig//   and the exponent will be bias+63 instead of bias+0.  Thus subsequent//   computations need to scale appropriately.// The constant 128/ln(2) is needed for the computation of w.  This is also //   obtained by scaling the computations.//// Two shifting constants are loaded directly with movl and setf.d. //   1. EXP_RSHF_2TO56 = 1.1000..00 * 2^(63-7) //        This constant is added to x*1/ln2 to shift the integer part of//        x*128/ln2 into the rightmost bits of the significand.//        The result of this fma is EXP_W_2TO56_RSH.//   2. EXP_RSHF       = 1.1000..00 * 2^(63) //        This constant is subtracted from EXP_W_2TO56_RSH * 2^(-56) to give//        the integer part of w, n, as a floating-point number.//        The result of this fms is EXP_Nfloat.LOCAL_OBJECT_START(exp_table_1)data4 0x4120f5c3, 0x408ccccd      //POS_ARG_ASYMP = 10.06, NEG_ARG_ASYMP = 4.4data4 0x41131Cdf, 0x00800000     //UnfBound ~=~ 9.1, EpsNorm ~=~ 1.1754944e-38//data8 0xb17217f7d1cf79ab , 0x00003ff7                            // ln2/128 hidata8 0xc9e3b39803f2f6af , 0x00003fb7                            // ln2/128 lo//// Table 1 is 2^(index_1/128) where// index_1 goes from 0 to 15//data8 0x8000000000000000 , 0x00003FFFdata8 0x80B1ED4FD999AB6C , 0x00003FFFdata8 0x8164D1F3BC030773 , 0x00003FFFdata8 0x8218AF4373FC25EC , 0x00003FFFdata8 0x82CD8698AC2BA1D7 , 0x00003FFFdata8 0x8383594EEFB6EE37 , 0x00003FFFdata8 0x843A28C3ACDE4046 , 0x00003FFFdata8 0x84F1F656379C1A29 , 0x00003FFFdata8 0x85AAC367CC487B15 , 0x00003FFFdata8 0x8664915B923FBA04 , 0x00003FFFdata8 0x871F61969E8D1010 , 0x00003FFFdata8 0x87DB357FF698D792 , 0x00003FFFdata8 0x88980E8092DA8527 , 0x00003FFFdata8 0x8955EE03618E5FDD , 0x00003FFFdata8 0x8A14D575496EFD9A , 0x00003FFFdata8 0x8AD4C6452C728924 , 0x00003FFFLOCAL_OBJECT_END(exp_table_1)// Table 2 is 2^(index_1/8) where// index_2 goes from 0 to 7LOCAL_OBJECT_START(exp_table_2)data8 0x8000000000000000 , 0x00003FFFdata8 0x8B95C1E3EA8BD6E7 , 0x00003FFFdata8 0x9837F0518DB8A96F , 0x00003FFFdata8 0xA5FED6A9B15138EA , 0x00003FFFdata8 0xB504F333F9DE6484 , 0x00003FFFdata8 0xC5672A115506DADD , 0x00003FFFdata8 0xD744FCCAD69D6AF4 , 0x00003FFFdata8 0xEAC0C6E7DD24392F , 0x00003FFFLOCAL_OBJECT_END(exp_table_2)LOCAL_OBJECT_START(erfc_p_table)// Pol_0 data8 0xBEA3260C63CB0446             //A15 = -5.70673541831883454676e-07data8 0x3EE63D6178077654             //A14 = +1.06047480138940182343e-05data8 0xBF18646BC5FC70A7             //A13 = -9.30491237309283694347e-05data8 0x3F40F92F909117FE             //A12 = +5.17986512144075019133e-04data8 0xBF611344289DE1E6             //A11 = -2.08438217390159994419e-03data8 0x3F7AF9FE6AD16DC0             //A10 = +6.58606893292862351928e-03data8 0xBF91D219E196CBA7             //A9 = -1.74030345858217321001e-02data8 0x3FA4AFDDA355854C             //A8 = +4.04042493708041968315e-02data8 0xBFB5D465BB7025AE             //A7 = -8.52721769916999425445e-02data8 0x3FC54C15A95B717D             //A6 = +1.66384418195672549029e-01data8 0xBFD340A75B4B1AB5             //A5 = -3.00821150926292166899e-01data8 0x3FDFFFC0BFCD247F             //A4 = +4.99984919839853542841e-01data8 0xBFE81270C361852B             //A3 = -7.52251035312075583309e-01data8 0x3FEFFFFFC67295FC             //A2 = +9.99999892800303301771e-01data8 0xBFF20DD74F8CD2BF             //A1 = -1.12837916445020868099e+00data8 0x3FEFFFFFFFFE7C1D             //A0 = +9.99999999988975570714e-01// Pol_1 data8 0xBDE8EC4BDD953B56             //A15 = -1.81338928934942767144e-10data8 0x3E43607F269E2A1C             //A14 = +9.02309090272196442358e-09data8 0xBE8C4D9E69C10E02             //A13 = -2.10875261143659275328e-07data8 0x3EC9CF2F84566725             //A12 = +3.07671055805877356583e-06data8 0xBF007980B1B46A4D             //A11 = -3.14228438702169818945e-05data8 0x3F2F4C3AD6DEF24A             //A10 = +2.38783056770846320260e-04data8 0xBF56F5129F8D30FA             //A9 = -1.40120333363130546426e-03data8 0x3F7AA6C7ABFC38EE             //A8 = +6.50671002200751820429e-03data8 0xBF98E7522CB84BEF             //A7 = -2.43199195666185511109e-02data8 0x3FB2F68EB1C3D073             //A6 = +7.40746673580490638637e-02data8 0xBFC7C16055AC6385             //A5 = -1.85588876564704611769e-01data8 0x3FD8A707AEF5A440             //A4 = +3.85194702967570635211e-01data8 0xBFE547BFE39AE2EA             //A3 = -6.65008492032112467310e-01data8 0x3FEE7C91BDF13578             //A2 = +9.52706213932898128515e-01data8 0xBFF1CB5B61F8C589             //A1 = -1.11214769621105541214e+00data8 0x3FEFEA56BC81FD37             //A0 = +9.97355812243688815239e-01// Pol_2 data8 0xBD302724A12F46E0             //A15 = -5.73866382814058809406e-14data8 0x3D98889B75D3102E             //A14 = +5.57829983681360947356e-12data8 0xBDF16EA15074A1E9             //A13 = -2.53671153922423457844e-10data8 0x3E3EC6E688CFEE5F             //A12 = +7.16581828336436419561e-09data8 0xBE82E5ED44C52609             //A11 = -1.40802202239825487803e-07data8 0x3EC120BE5CE42353             //A10 = +2.04180535157522081699e-06data8 0xBEF7B8B0311A1911             //A9 = -2.26225266204633600888e-05data8 0x3F29A281F43FC238             //A8 = +1.95577968156184077632e-04data8 0xBF55E19858B3B7A4             //A7 = -1.33552434527526534043e-03data8 0x3F7DAC8C3D12E5FD             //A6 = +7.24463253680473816303e-03data8 0xBF9FF9C04613FB47             //A5 = -3.12261622211693854028e-02data8 0x3FBB3D5DBF9D9366             //A4 = +1.06405123978743883370e-01data8 0xBFD224DE9F62C258             //A3 = -2.83500342989133623476e-01data8 0x3FE28A95CB8C6D3E             //A2 = +5.79417131000276437708e-01data8 0xBFEC21205D358672             //A1 = -8.79043752717008257224e-01data8 0x3FEDAE44D5EDFE5B             //A0 = +9.27523057776805771830e-01// Pol_3 data8 0xBCA3BCA734AC82F1             //A15 = -1.36952437983096410260e-16data8 0x3D16740DC3990612             //A14 = +1.99425676175410093285e-14data8 0xBD77F4353812C46A             //A13 = -1.36162367755616790260e-12data8 0x3DCFD0BE13C73DB4             //A12 = +5.78718761040355136007e-11data8 0xBE1D728DF71189B4             //A11 = -1.71406885583934105120e-09data8 0x3E64252C8CB710B5             //A10 = +3.75233795940731111303e-08data8 0xBEA514B93180F33D             //A9 = -6.28261292774310809962e-07data8 0x3EE1381118CC7151             //A8 = +8.21066421390821904504e-06data8 0xBF1634404FB0FA72             //A7 = -8.47019436358372148764e-05data8 0x3F46B2CBBCF0EB32             //A6 = +6.92700845213200923490e-04data8 0xBF725C2B445E6D81             //A5 = -4.48243046949004063741e-03data8 0x3F974E7CFA4D89D9             //A4 = +2.27603462002522228717e-02data8 0xBFB6D7BAC2E342D1             //A3 = -8.92292714882032736443e-02data8 0x3FD0D156AD9CE2A6             //A2 = +2.62777013343603696631e-01data8 0xBFE1C228572AADB0             //A1 = -5.54950876471982857725e-01data8 0x3FE8A739F48B9A3B             //A0 = +7.70413377406675619766e-01LOCAL_OBJECT_END(erfc_p_table).section .textGLOBAL_LIBM_ENTRY(erfcf)// Form index i for table erfc_p_table as exponent of x // We use i + bias in real calculations { .mlx      getf.exp       GR_IndxPlusBias = f8          // (sign + exp + bias) of x      movl           exp_GR_sig_inv_ln2 = 0xb8aa3b295c17f0bc //signif.of 1/ln2}{ .mlx      addl           EXP_AD_TB1    = @ltoff(exp_table_1), gp      movl           exp_GR_rshf_2to56 = 0x4768000000000000 // 1.100 2^(63+56)};;// Form argument EXP_NORM_f8 for exp(-x^2){ .mfi      ld8            EXP_AD_TB1    = [EXP_AD_TB1]      fcmp.ge.s1     p6,p7 = f8, f0                     // p6: x >= 0 ,p7: x<0      mov            GR_BIAS = 0x0FFFF}{ .mfi      mov            exp_GR_exp_2tom56 = 0xffff-56      fnma.s1        EXP_NORM_f8   = f8, f8, f0                       //  -x^2      mov            GR_ExpMask  = 0x1ffff};;// Form two constants we need//  1/ln2 * 2^63  to compute  w = x * 1/ln2 * 128 //  1.1000..000 * 2^(63+63-7) to right shift int(w) into the significand// p9:  x = 0,+inf,-inf,nan,unnorm.// p10: x!= 0,+inf,-inf,nan,unnorm.{ .mfi      setf.sig       EXP_INV_LN2_2TO63 = exp_GR_sig_inv_ln2 // Form 1/ln2*2^63      fclass.m       p9,p10 = f8,0xef             shl            GR_ShftPi_bias = GR_BIAS, 7}{ .mfi      setf.d         EXP_RSHF_2TO56 = exp_GR_rshf_2to56 //Const 1.10*2^(63+56)      nop.f          0      and            GR_IndxPlusBias = GR_IndxPlusBias, GR_ExpMask // i + bias};;{ .mfi      alloc          r32 = ar.pfs, 0, 15, 4, 0(p6)  fma.s1         FR_AbsArg = f1, f0, f8                  // |x| if x >= 0      cmp.lt         p15,p0 = GR_IndxPlusBias, GR_BIAS//p15: i < 0 (for |x|<1)}{ .mlx      setf.exp       EXP_2TOM56 = exp_GR_exp_2tom56 //2^-56 for scaling Nfloat      movl           exp_GR_rshf = 0x43e8000000000000 //1.10 2^63,right shift.};;{ .mfi      ldfps          FR_POS_ARG_ASYMP, FR_NEG_ARG_ASYMP = [EXP_AD_TB1],8      nop.f          0(p15) mov            GR_IndxPlusBias = GR_BIAS            //Let i = 0 if i < 0}{ .mlx      mov            GR_P_POINT_3 = 0x1A0      movl           GR_05 = 0x3fe0000000000000};;// Form shift GR_ShftPi from the beginning of erfc_p_table // to the polynomial with number i{ .mfi      ldfps          FR_UnfBound, FR_EpsNorm = [EXP_AD_TB1],8      nop.f          0      shl            GR_ShftPi = GR_IndxPlusBias, 7