.file "log1p.s"// Copyright (c) 2000 - 2005, 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// 08/15/00 Bundle added after call to __libm_error_support to properly//          set [the previously overwritten] GR_Parameter_RESULT.// 06/29/01 Improved speed of all paths// 05/20/02 Cleaned up namespace and sf0 syntax// 10/02/02 Improved performance by basing on log algorithm// 02/10/03 Reordered header: .section, .global, .proc, .align// 04/18/03 Eliminate possible WAW dependency warning// 03/31/05 Reformatted delimiters between data tables//// API//==============================================================// double log1p(double)//// log1p(x) = log(x+1)//// Overview of operation//==============================================================// Background// ----------//// This algorithm is based on fact that// log1p(x) = log(1+x) and// log(a b) = log(a) + log(b).// In our case we have 1+x = 2^N f, where 1 <= f < 2.// So//   log(1+x) = log(2^N f) = log(2^N) + log(f) = n*log(2) + log(f)//// To calculate log(f) we do following//   log(f) = log(f * frcpa(f) / frcpa(f)) =//          = log(f * frcpa(f)) + log(1/frcpa(f))//// According to definition of IA-64's frcpa instruction it's a// floating point that approximates 1/f using a lookup on the// top of 8 bits of the input number's + 1 significand with relative// error < 2^(-8.886). So we have following//// |(1/f - frcpa(f)) / (1/f))| = |1 - f*frcpa(f)| < 1/256//// and//// log(f) = log(f * frcpa(f)) + log(1/frcpa(f)) =//        = log(1 + r) + T//// The first value can be computed by polynomial P(r) approximating// log(1 + r) on |r| < 1/256 and the second is precomputed tabular// value defined by top 8 bit of f.//// Finally we have that  log(1+x) ~ (N*log(2) + T) + P(r)//// Note that if input argument is close to 0.0 (in our case it means// that |x| < 1/256) we can use just polynomial approximation// because 1+x = 2^0 * f = f = 1 + r and// log(1+x) = log(1 + r) ~ P(r)////// Implementation// --------------//// 1. |x| >= 2^(-8), and x > -1//   InvX = frcpa(x+1)//   r = InvX*(x+1) - 1//   P(r) = r*((r*A3 - A2) + r^4*((A4 + r*A5) + r^2*(A6 + r*A7)),//   all coefficients are calcutated in quad and rounded to double//   precision. A7,A6,A5,A4 are stored in memory whereas A3 and A2//   created with setf.////   N = float(n) where n is true unbiased exponent of x////   T is tabular value of log(1/frcpa(x)) calculated in quad precision//   and represented by two floating-point numbers 64-bit Thi and 32-bit Tlo.//   To load Thi,Tlo we get bits from 55 to 62 of register format significand//   as index and calculate two addresses//     ad_Thi = Thi_table_base_addr + 8 * index//     ad_Tlo = Tlo_table_base_addr + 4 * index////   L1 (log(2)) is calculated in quad//   precision and represented by two floating-point 64-bit numbers L1hi,L1lo//   stored in memory.////   And final result = ((L1hi*N + Thi) + (N*L1lo + Tlo)) + P(r)////// 2. 2^(-80) <= |x| < 2^(-8)//   r = x//   P(r) = r*((r*A3 - A2) + r^4*((A4 + r*A5) + r^2*(A6 + r*A7)),//   A7,A6,A5,A4,A3,A2 are the same as in case |x| >= 1/256////   And final results//     log(1+x)   = P(r)//// 3. 0 < |x| < 2^(-80)//   Although log1p(x) is basically x, we would like to preserve the inexactness//   nature as well as consistent behavior under different rounding modes.//   We can do this by computing the result as////     log1p(x) = x - x*x//////    Note: NaT, any NaNs, +/-INF, +/-0, negatives and unnormalized numbers are//          filtered and processed on special branches.////// Special values//==============================================================//// log1p(-1)    = -inf            // Call error support//// log1p(+qnan) = +qnan// log1p(-qnan) = -qnan// log1p(+snan) = +qnan// log1p(-snan) = -qnan//// log1p(x),x<-1= QNAN Indefinite // Call error support// log1p(-inf)  = QNAN Indefinite// log1p(+inf)  = +inf// log1p(+/-0)  = +/-0////// Registers used//==============================================================// Floating Point registers used:// f8, input// f7 -> f15,  f32 -> f40//// General registers used:// r8  -> r11// r14 -> r20//// Predicate registers used:// p6 -> p12// Assembly macros//==============================================================GR_TAG                 = r8GR_ad_1                = r8GR_ad_2                = r9GR_Exp                 = r10GR_N                   = r11GR_signexp_x           = r14GR_exp_mask            = r15GR_exp_bias            = r16GR_05                  = r17GR_A3                  = r18GR_Sig                 = r19GR_Ind                 = r19GR_exp_x               = r20GR_SAVE_B0             = r33GR_SAVE_PFS            = r34GR_SAVE_GP             = r35GR_SAVE_SP             = r36GR_Parameter_X         = r37GR_Parameter_Y         = r38GR_Parameter_RESULT    = r39GR_Parameter_TAG       = r40FR_NormX               = f7FR_RcpX                = f9FR_r                   = f10FR_r2                  = f11FR_r4                  = f12FR_N                   = f13FR_Ln2hi               = f14FR_Ln2lo               = f15FR_A7                  = f32FR_A6                  = f33FR_A5                  = f34FR_A4                  = f35FR_A3                  = f36FR_A2                  = f37FR_Thi                 = f38FR_NxLn2hipThi         = f38FR_NxLn2pT             = f38FR_Tlo                 = f39FR_NxLn2lopTlo         = f39FR_Xp1                 = f40FR_Y                   = f1FR_X                   = f10FR_RESULT              = f8// Data//==============================================================RODATA.align 16LOCAL_OBJECT_START(log_data)// coefficients of polynomial approximationdata8 0x3FC2494104381A8E // A7data8 0xBFC5556D556BBB69 // A6data8 0x3FC999999988B5E9 // A5data8 0xBFCFFFFFFFF6FFF5 // A4//// hi parts of ln(1/frcpa(1+i/256)), i=0...255data8 0x3F60040155D5889D // 0data8 0x3F78121214586B54 // 1data8 0x3F841929F96832EF // 2data8 0x3F8C317384C75F06 // 3data8 0x3F91A6B91AC73386 // 4data8 0x3F95BA9A5D9AC039 // 5data8 0x3F99D2A8074325F3 // 6data8 0x3F9D6B2725979802 // 7data8 0x3FA0C58FA19DFAA9 // 8data8 0x3FA2954C78CBCE1A // 9data8 0x3FA4A94D2DA96C56 // 10data8 0x3FA67C94F2D4BB58 // 11data8 0x3FA85188B630F068 // 12data8 0x3FAA6B8ABE73AF4C // 13data8 0x3FAC441E06F72A9E // 14data8 0x3FAE1E6713606D06 // 15data8 0x3FAFFA6911AB9300 // 16data8 0x3FB0EC139C5DA600 // 17data8 0x3FB1DBD2643D190B // 18data8 0x3FB2CC7284FE5F1C // 19data8 0x3FB3BDF5A7D1EE64 // 20data8 0x3FB4B05D7AA012E0 // 21data8 0x3FB580DB7CEB5701 // 22data8 0x3FB674F089365A79 // 23data8 0x3FB769EF2C6B568D // 24data8 0x3FB85FD927506A47 // 25data8 0x3FB9335E5D594988 // 26data8 0x3FBA2B0220C8E5F4 // 27data8 0x3FBB0004AC1A86AB // 28data8 0x3FBBF968769FCA10 // 29data8 0x3FBCCFEDBFEE13A8 // 30data8 0x3FBDA727638446A2 // 31data8 0x3FBEA3257FE10F79 // 32data8 0x3FBF7BE9FEDBFDE5 // 33data8 0x3FC02AB352FF25F3 // 34data8 0x3FC097CE579D204C // 35data8 0x3FC1178E8227E47B // 36data8 0x3FC185747DBECF33 // 37data8 0x3FC1F3B925F25D41 // 38data8 0x3FC2625D1E6DDF56 // 39data8 0x3FC2D1610C868139 // 40data8 0x3FC340C59741142E // 41data8 0x3FC3B08B6757F2A9 // 42data8 0x3FC40DFB08378003 // 43data8 0x3FC47E74E8CA5F7C // 44data8 0x3FC4EF51F6466DE4 // 45data8 0x3FC56092E02BA516 // 46data8 0x3FC5D23857CD74D4 // 47data8 0x3FC6313A37335D76 // 48data8 0x3FC6A399DABBD383 // 49data8 0x3FC70337DD3CE41A // 50data8 0x3FC77654128F6127 // 51data8 0x3FC7E9D82A0B022D // 52data8 0x3FC84A6B759F512E // 53data8 0x3FC8AB47D5F5A30F // 54data8 0x3FC91FE49096581B // 55data8 0x3FC981634011AA75 // 56data8 0x3FC9F6C407089664 // 57data8 0x3FCA58E729348F43 // 58data8 0x3FCABB55C31693AC // 59data8 0x3FCB1E104919EFD0 // 60data8 0x3FCB94EE93E367CA // 61data8 0x3FCBF851C067555E // 62data8 0x3FCC5C0254BF23A5 // 63data8 0x3FCCC000C9DB3C52 // 64data8 0x3FCD244D99C85673 // 65data8 0x3FCD88E93FB2F450 // 66data8 0x3FCDEDD437EAEF00 // 67data8 0x3FCE530EFFE71012 // 68data8 0x3FCEB89A1648B971 // 69data8 0x3FCF1E75FADF9BDE // 70data8 0x3FCF84A32EAD7C35 // 71data8 0x3FCFEB2233EA07CD // 72data8 0x3FD028F9C7035C1C // 73data8 0x3FD05C8BE0D9635A // 74data8 0x3FD085EB8F8AE797 // 75data8 0x3FD0B9C8E32D1911 // 76data8 0x3FD0EDD060B78080 // 77data8 0x3FD122024CF0063F // 78data8 0x3FD14BE2927AECD4 // 79data8 0x3FD180618EF18ADF // 80data8 0x3FD1B50BBE2FC63B // 81data8 0x3FD1DF4CC7CF242D // 82data8 0x3FD214456D0EB8D4 // 83data8 0x3FD23EC5991EBA49 // 84data8 0x3FD2740D9F870AFB // 85data8 0x3FD29ECDABCDFA03 // 86data8 0x3FD2D46602ADCCEE // 87data8 0x3FD2FF66B04EA9D4 // 88data8 0x3FD335504B355A37 // 89data8 0x3FD360925EC44F5C // 90data8 0x3FD38BF1C3337E74 // 91data8 0x3FD3C25277333183 // 92data8 0x3FD3EDF463C1683E // 93data8 0x3FD419B423D5E8C7 // 94data8 0x3FD44591E0539F48 // 95data8 0x3FD47C9175B6F0AD // 96data8 0x3FD4A8B341552B09 // 97data8 0x3FD4D4F39089019F // 98data8 0x3FD501528DA1F967 // 99data8 0x3FD52DD06347D4F6 // 100data8 0x3FD55A6D3C7B8A89 // 101data8 0x3FD5925D2B112A59 // 102data8 0x3FD5BF406B543DB1 // 103data8 0x3FD5EC433D5C35AD // 104data8 0x3FD61965CDB02C1E // 105data8 0x3FD646A84935B2A1 // 106data8 0x3FD6740ADD31DE94 // 107data8 0x3FD6A18DB74A58C5 // 108data8 0x3FD6CF31058670EC // 109data8 0x3FD6F180E852F0B9 // 110data8 0x3FD71F5D71B894EF // 111data8 0x3FD74D5AEFD66D5C // 112data8 0x3FD77B79922BD37D // 113data8 0x3FD7A9B9889F19E2 // 114data8 0x3FD7D81B037EB6A6 // 115data8 0x3FD8069E33827230 // 116data8 0x3FD82996D3EF8BCA // 117data8 0x3FD85855776DCBFA // 118data8 0x3FD8873658327CCE // 119data8 0x3FD8AA75973AB8CE // 120data8 0x3FD8D992DC8824E4 // 121data8 0x3FD908D2EA7D9511 // 122data8 0x3FD92C59E79C0E56 // 123data8 0x3FD95BD750EE3ED2 // 124data8 0x3FD98B7811A3EE5B // 125data8 0x3FD9AF47F33D406B // 126data8 0x3FD9DF270C1914A7 // 127data8 0x3FDA0325ED14FDA4 // 128data8 0x3FDA33440224FA78 // 129data8 0x3FDA57725E80C382 // 130data8 0x3FDA87D0165DD199 // 131data8 0x3FDAAC2E6C03F895 // 132data8 0x3FDADCCC6FDF6A81 // 133data8 0x3FDB015B3EB1E790 // 134data8 0x3FDB323A3A635948 // 135data8 0x3FDB56FA04462909 // 136data8 0x3FDB881AA659BC93 // 137data8 0x3FDBAD0BEF3DB164 // 138data8 0x3FDBD21297781C2F // 139data8 0x3FDC039236F08818 // 140data8 0x3FDC28CB1E4D32FC // 141data8 0x3FDC4E19B84723C1 // 142data8 0x3FDC7FF9C74554C9 // 143data8 0x3FDCA57B64E9DB05 // 144data8 0x3FDCCB130A5CEBAF // 145data8 0x3FDCF0C0D18F326F // 146data8 0x3FDD232075B5A201 // 147data8 0x3FDD490246DEFA6B // 148data8 0x3FDD6EFA918D25CD // 149data8 0x3FDD9509707AE52F // 150data8 0x3FDDBB2EFE92C554 // 151data8 0x3FDDEE2F3445E4AE // 152data8 0x3FDE148A1A2726CD // 153data8 0x3FDE3AFC0A49FF3F // 154data8 0x3FDE6185206D516D // 155data8 0x3FDE882578823D51 // 156data8 0x3FDEAEDD2EAC990C // 157data8 0x3FDED5AC5F436BE2 // 158data8 0x3FDEFC9326D16AB8 // 159data8 0x3FDF2391A21575FF // 160data8 0x3FDF4AA7EE03192C // 161data8 0x3FDF71D627C30BB0 // 162data8 0x3FDF991C6CB3B379 // 163data8 0x3FDFC07ADA69A90F // 164data8 0x3FDFE7F18EB03D3E // 165data8 0x3FE007C053C5002E // 166data8 0x3FE01B942198A5A0 // 167data8 0x3FE02F74400C64EA // 168data8 0x3FE04360BE7603AC // 169data8 0x3FE05759AC47FE33 // 170data8 0x3FE06B5F1911CF51 // 171data8 0x3FE078BF0533C568 // 172data8 0x3FE08CD9687E7B0E // 173data8 0x3FE0A10074CF9019 // 174data8 0x3FE0B5343A234476 // 175data8 0x3FE0C974C89431CD // 176data8 0x3FE0DDC2305B9886 // 177data8 0x3FE0EB524BAFC918 // 178data8 0x3FE0FFB54213A475 // 179data8 0x3FE114253DA97D9F // 180data8 0x3FE128A24F1D9AFF // 181data8 0x3FE1365252BF0864 // 182data8 0x3FE14AE558B4A92D // 183data8 0x3FE15F85A19C765B // 184data8 0x3FE16D4D38C119FA // 185data8 0x3FE18203C20DD133 // 186data8 0x3FE196C7BC4B1F3A // 187data8 0x3FE1A4A738B7A33C // 188data8 0x3FE1B981C0C9653C // 189data8 0x3FE1CE69E8BB106A // 190data8 0x3FE1DC619DE06944 // 191data8 0x3FE1F160A2AD0DA3 // 192data8 0x3FE2066D7740737E // 193data8 0x3FE2147DBA47A393 // 194data8 0x3FE229A1BC5EBAC3 // 195data8 0x3FE237C1841A502E // 196data8 0x3FE24CFCE6F80D9A // 197data8 0x3FE25B2C55CD5762 // 198data8 0x3FE2707F4D5F7C40 // 199data8 0x3FE285E0842CA383 // 200data8 0x3FE294294708B773 // 201data8 0x3FE2A9A2670AFF0C // 202data8 0x3FE2B7FB2C8D1CC0 // 203data8 0x3FE2C65A6395F5F5 // 204data8 0x3FE2DBF557B0DF42 // 205data8 0x3FE2EA64C3F97654 // 206data8 0x3FE3001823684D73 // 207data8 0x3FE30E97E9A8B5CC // 208data8 0x3FE32463EBDD34E9 // 209data8 0x3FE332F4314AD795 // 210data8 0x3FE348D90E7464CF // 211data8 0x3FE35779F8C43D6D // 212data8 0x3FE36621961A6A99 // 213data8 0x3FE37C299F3C366A // 214data8 0x3FE38AE2171976E7 // 215data8 0x3FE399A157A603E7 // 216data8 0x3FE3AFCCFE77B9D1 // 217data8 0x3FE3BE9D503533B5 // 218data8 0x3FE3CD7480B4A8A2 // 219data8 0x3FE3E3C43918F76C // 220data8 0x3FE3F2ACB27ED6C6 // 221data8 0x3FE4019C2125CA93 // 222data8 0x3FE4181061389722 // 223data8 0x3FE42711518DF545 // 224data8 0x3FE436194E12B6BF // 225data8 0x3FE445285D68EA69 // 226data8 0x3FE45BCC464C893A // 227data8 0x3FE46AED21F117FC // 228data8 0x3FE47A1527E8A2D3 // 229data8 0x3FE489445EFFFCCB // 230data8 0x3FE4A018BCB69835 // 231data8 0x3FE4AF5A0C9D65D7 // 232data8 0x3FE4BEA2A5BDBE87 // 233data8 0x3FE4CDF28F10AC46 // 234data8 0x3FE4DD49CF994058 // 235data8 0x3FE4ECA86E64A683 // 236data8 0x3FE503C43CD8EB68 // 237data8 0x3FE513356667FC57 // 238data8 0x3FE522AE0738A3D7 // 239data8 0x3FE5322E26867857 // 240data8 0x3FE541B5CB979809 // 241data8 0x3FE55144FDBCBD62 // 242data8 0x3FE560DBC45153C6 // 243data8 0x3FE5707A26BB8C66 // 244data8 0x3FE587F60ED5B8FF // 245data8 0x3FE597A7977C8F31 // 246data8 0x3FE5A760D634BB8A // 247data8 0x3FE5B721D295F10E // 248data8 0x3FE5C6EA94431EF9 // 249data8 0x3FE5D6BB22EA86F5 // 250data8 0x3FE5E6938645D38F // 251data8 0x3FE5F673C61A2ED1 // 252data8 0x3FE6065BEA385926 // 253data8 0x3FE6164BFA7CC06B // 254data8 0x3FE62643FECF9742 // 255//// two parts of ln(2)data8 0x3FE62E42FEF00000,0x3DD473DE6AF278ED//// lo parts of ln(1/frcpa(1+i/256)), i=0...255data4 0x20E70672 // 0data4 0x1F60A5D0 // 1data4 0x218EABA0 // 2data4 0x21403104 // 3data4 0x20E9B54E // 4data4 0x21EE1382 // 5data4 0x226014E3 // 6data4 0x2095E5C9 // 7data4 0x228BA9D4 // 8data4 0x22932B86 // 9data4 0x22608A57 // 10data4 0x220209F3 // 11data4 0x212882CC // 12data4 0x220D46E2 // 13data4 0x21FA4C28 // 14data4 0x229E5BD9 // 15data4 0x228C9838 // 16data4 0x2311F954 // 17data4 0x221365DF // 18data4 0x22BD0CB3 // 19data4 0x223D4BB7 // 20data4 0x22A71BBE // 21data4 0x237DB2FA // 22data4 0x23194C9D // 23data4 0x22EC639E // 24data4 0x2367E669 // 25data4 0x232E1D5F // 26data4 0x234A639B // 27data4 0x2365C0E0 // 28data4 0x234646C1 // 29data4 0x220CBF9C // 30data4 0x22A00FD4 // 31data4 0x2306A3F2 // 32data4 0x23745A9B // 33data4 0x2398D756 // 34data4 0x23DD0B6A // 35data4 0x23DE338B // 36data4 0x23A222DF // 37data4 0x223164F8 // 38data4 0x23B4E87B // 39data4 0x23D6CCB8 // 40data4 0x220C2099 // 41data4 0x21B86B67 // 42