.file "log1p.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 // 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 // // 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 = r8 GR_ad_1 = r8 GR_ad_2 = r9 GR_Exp = r10 GR_N = r11 GR_signexp_x = r14 GR_exp_mask = r15 GR_exp_bias = r16 GR_05 = r17 GR_A3 = r18 GR_Sig = r19 GR_Ind = r19 GR_exp_x = r20 GR_SAVE_B0 = r33 GR_SAVE_PFS = r34 GR_SAVE_GP = r35 GR_SAVE_SP = r36 GR_Parameter_X = r37 GR_Parameter_Y = r38 GR_Parameter_RESULT = r39 GR_Parameter_TAG = r40 FR_NormX = f7 FR_RcpX = f9 FR_r = f10 FR_r2 = f11 FR_r4 = f12 FR_N = f13 FR_Ln2hi = f14 FR_Ln2lo = f15 FR_A7 = f32 FR_A6 = f33 FR_A5 = f34 FR_A4 = f35 FR_A3 = f36 FR_A2 = f37 FR_Thi = f38 FR_NxLn2hipThi = f38 FR_NxLn2pT = f38 FR_Tlo = f39 FR_NxLn2lopTlo = f39 FR_Xp1 = f40 FR_Y = f1 FR_X = f10 FR_RESULT = f8 // Data //============================================================== RODATA .align 16 LOCAL_OBJECT_START(log_data) // coefficients of polynomial approximation data8 0x3FC2494104381A8E // A7 data8 0xBFC5556D556BBB69 // A6 data8 0x3FC999999988B5E9 // A5 data8 0xBFCFFFFFFFF6FFF5 // A4 // // hi parts of ln(1/frcpa(1+i/256)), i=0...255 data8 0x3F60040155D5889D // 0 data8 0x3F78121214586B54 // 1 data8 0x3F841929F96832EF // 2 data8 0x3F8C317384C75F06 // 3 data8 0x3F91A6B91AC73386 // 4 data8 0x3F95BA9A5D9AC039 // 5 data8 0x3F99D2A8074325F3 // 6 data8 0x3F9D6B2725979802 // 7 data8 0x3FA0C58FA19DFAA9 // 8 data8 0x3FA2954C78CBCE1A // 9 data8 0x3FA4A94D2DA96C56 // 10 data8 0x3FA67C94F2D4BB58 // 11 data8 0x3FA85188B630F068 // 12 data8 0x3FAA6B8ABE73AF4C // 13 data8 0x3FAC441E06F72A9E // 14 data8 0x3FAE1E6713606D06 // 15 data8 0x3FAFFA6911AB9300 // 16 data8 0x3FB0EC139C5DA600 // 17 data8 0x3FB1DBD2643D190B // 18 data8 0x3FB2CC7284FE5F1C // 19 data8 0x3FB3BDF5A7D1EE64 // 20 data8 0x3FB4B05D7AA012E0 // 21 data8 0x3FB580DB7CEB5701 // 22 data8 0x3FB674F089365A79 // 23 data8 0x3FB769EF2C6B568D // 24 data8 0x3FB85FD927506A47 // 25 data8 0x3FB9335E5D594988 // 26 data8 0x3FBA2B0220C8E5F4 // 27 data8 0x3FBB0004AC1A86AB // 28 data8 0x3FBBF968769FCA10 // 29 data8 0x3FBCCFEDBFEE13A8 // 30 data8 0x3FBDA727638446A2 // 31 data8 0x3FBEA3257FE10F79 // 32 data8 0x3FBF7BE9FEDBFDE5 // 33 data8 0x3FC02AB352FF25F3 // 34 data8 0x3FC097CE579D204C // 35 data8 0x3FC1178E8227E47B // 36 data8 0x3FC185747DBECF33 // 37 data8 0x3FC1F3B925F25D41 // 38 data8 0x3FC2625D1E6DDF56 // 39 data8 0x3FC2D1610C868139 // 40 data8 0x3FC340C59741142E // 41 data8 0x3FC3B08B6757F2A9 // 42 data8 0x3FC40DFB08378003 // 43 data8 0x3FC47E74E8CA5F7C // 44 data8 0x3FC4EF51F6466DE4 // 45 data8 0x3FC56092E02BA516 // 46 data8 0x3FC5D23857CD74D4 // 47 data8 0x3FC6313A37335D76 // 48 data8 0x3FC6A399DABBD383 // 49 data8 0x3FC70337DD3CE41A // 50 data8 0x3FC77654128F6127 // 51 data8 0x3FC7E9D82A0B022D // 52 data8 0x3FC84A6B759F512E // 53 data8 0x3FC8AB47D5F5A30F // 54 data8 0x3FC91FE49096581B // 55 data8 0x3FC981634011AA75 // 56 data8 0x3FC9F6C407089664 // 57 data8 0x3FCA58E729348F43 // 58 data8 0x3FCABB55C31693AC // 59 data8 0x3FCB1E104919EFD0 // 60 data8 0x3FCB94EE93E367CA // 61 data8 0x3FCBF851C067555E // 62 data8 0x3FCC5C0254BF23A5 // 63 data8 0x3FCCC000C9DB3C52 // 64 data8 0x3FCD244D99C85673 // 65 data8 0x3FCD88E93FB2F450 // 66 data8 0x3FCDEDD437EAEF00 // 67 data8 0x3FCE530EFFE71012 // 68 data8 0x3FCEB89A1648B971 // 69 data8 0x3FCF1E75FADF9BDE // 70 data8 0x3FCF84A32EAD7C35 // 71 data8 0x3FCFEB2233EA07CD // 72 data8 0x3FD028F9C7035C1C // 73 data8 0x3FD05C8BE0D9635A // 74 data8 0x3FD085EB8F8AE797 // 75 data8 0x3FD0B9C8E32D1911 // 76 data8 0x3FD0EDD060B78080 // 77 data8 0x3FD122024CF0063F // 78 data8 0x3FD14BE2927AECD4 // 79 data8 0x3FD180618EF18ADF // 80 data8 0x3FD1B50BBE2FC63B // 81 data8 0x3FD1DF4CC7CF242D // 82 data8 0x3FD214456D0EB8D4 // 83 data8 0x3FD23EC5991EBA49 // 84 data8 0x3FD2740D9F870AFB // 85 data8 0x3FD29ECDABCDFA03 // 86 data8 0x3FD2D46602ADCCEE // 87 data8 0x3FD2FF66B04EA9D4 // 88 data8 0x3FD335504B355A37 // 89 data8 0x3FD360925EC44F5C // 90 data8 0x3FD38BF1C3337E74 // 91 data8 0x3FD3C25277333183 // 92 data8 0x3FD3EDF463C1683E // 93 data8 0x3FD419B423D5E8C7 // 94 data8 0x3FD44591E0539F48 // 95 data8 0x3FD47C9175B6F0AD // 96 data8 0x3FD4A8B341552B09 // 97 data8 0x3FD4D4F39089019F // 98 data8 0x3FD501528DA1F967 // 99 data8 0x3FD52DD06347D4F6 // 100 data8 0x3FD55A6D3C7B8A89 // 101 data8 0x3FD5925D2B112A59 // 102 data8 0x3FD5BF406B543DB1 // 103 data8 0x3FD5EC433D5C35AD // 104 data8 0x3FD61965CDB02C1E // 105 data8 0x3FD646A84935B2A1 // 106 data8 0x3FD6740ADD31DE94 // 107 data8 0x3FD6A18DB74A58C5 // 108 data8 0x3FD6CF31058670EC // 109 data8 0x3FD6F180E852F0B9 // 110 data8 0x3FD71F5D71B894EF // 111 data8 0x3FD74D5AEFD66D5C // 112 data8 0x3FD77B79922BD37D // 113 data8 0x3FD7A9B9889F19E2 // 114 data8 0x3FD7D81B037EB6A6 // 115 data8 0x3FD8069E33827230 // 116 data8 0x3FD82996D3EF8BCA // 117 data8 0x3FD85855776DCBFA // 118 data8 0x3FD8873658327CCE // 119 data8 0x3FD8AA75973AB8CE // 120 data8 0x3FD8D992DC8824E4 // 121 data8 0x3FD908D2EA7D9511 // 122 data8 0x3FD92C59E79C0E56 // 123 data8 0x3FD95BD750EE3ED2 // 124 data8 0x3FD98B7811A3EE5B // 125 data8 0x3FD9AF47F33D406B // 126 data8 0x3FD9DF270C1914A7 // 127 data8 0x3FDA0325ED14FDA4 // 128 data8 0x3FDA33440224FA78 // 129 data8 0x3FDA57725E80C382 // 130 data8 0x3FDA87D0165DD199 // 131 data8 0x3FDAAC2E6C03F895 // 132 data8 0x3FDADCCC6FDF6A81 // 133 data8 0x3FDB015B3EB1E790 // 134 data8 0x3FDB323A3A635948 // 135 data8 0x3FDB56FA04462909 // 136 data8 0x3FDB881AA659BC93 // 137 data8 0x3FDBAD0BEF3DB164 // 138 data8 0x3FDBD21297781C2F // 139 data8 0x3FDC039236F08818 // 140 data8 0x3FDC28CB1E4D32FC // 141 data8 0x3FDC4E19B84723C1 // 142 data8 0x3FDC7FF9C74554C9 // 143 data8 0x3FDCA57B64E9DB05 // 144 data8 0x3FDCCB130A5CEBAF // 145 data8 0x3FDCF0C0D18F326F // 146 data8 0x3FDD232075B5A201 // 147 data8 0x3FDD490246DEFA6B // 148 data8 0x3FDD6EFA918D25CD // 149 data8 0x3FDD9509707AE52F // 150 data8 0x3FDDBB2EFE92C554 // 151 data8 0x3FDDEE2F3445E4AE // 152 data8 0x3FDE148A1A2726CD // 153 data8 0x3FDE3AFC0A49FF3F // 154 data8 0x3FDE6185206D516D // 155 data8 0x3FDE882578823D51 // 156 data8 0x3FDEAEDD2EAC990C // 157 data8 0x3FDED5AC5F436BE2 // 158 data8 0x3FDEFC9326D16AB8 // 159 data8 0x3FDF2391A21575FF // 160 data8 0x3FDF4AA7EE03192C // 161 data8 0x3FDF71D627C30BB0 // 162 data8 0x3FDF991C6CB3B379 // 163 data8 0x3FDFC07ADA69A90F // 164 data8 0x3FDFE7F18EB03D3E // 165 data8 0x3FE007C053C5002E // 166 data8 0x3FE01B942198A5A0 // 167 data8 0x3FE02F74400C64EA // 168 data8 0x3FE04360BE7603AC // 169 data8 0x3FE05759AC47FE33 // 170 data8 0x3FE06B5F1911CF51 // 171 data8 0x3FE078BF0533C568 // 172 data8 0x3FE08CD9687E7B0E // 173 data8 0x3FE0A10074CF9019 // 174 data8 0x3FE0B5343A234476 // 175 data8 0x3FE0C974C89431CD // 176 data8 0x3FE0DDC2305B9886 // 177 data8 0x3FE0EB524BAFC918 // 178 data8 0x3FE0FFB54213A475 // 179 data8 0x3FE114253DA97D9F // 180 data8 0x3FE128A24F1D9AFF // 181 data8 0x3FE1365252BF0864 // 182 data8 0x3FE14AE558B4A92D // 183 data8 0x3FE15F85A19C765B // 184 data8 0x3FE16D4D38C119FA // 185 data8 0x3FE18203C20DD133 // 186 data8 0x3FE196C7BC4B1F3A // 187 data8 0x3FE1A4A738B7A33C // 188 data8 0x3FE1B981C0C9653C // 189 data8 0x3FE1CE69E8BB106A // 190 data8 0x3FE1DC619DE06944 // 191 data8 0x3FE1F160A2AD0DA3 // 192 data8 0x3FE2066D7740737E // 193 data8 0x3FE2147DBA47A393 // 194 data8 0x3FE229A1BC5EBAC3 // 195 data8 0x3FE237C1841A502E // 196 data8 0x3FE24CFCE6F80D9A // 197 data8 0x3FE25B2C55CD5762 // 198 data8 0x3FE2707F4D5F7C40 // 199 data8 0x3FE285E0842CA383 // 200 data8 0x3FE294294708B773 // 201 data8 0x3FE2A9A2670AFF0C // 202 data8 0x3FE2B7FB2C8D1CC0 // 203 data8 0x3FE2C65A6395F5F5 // 204 data8 0x3FE2DBF557B0DF42 // 205 data8 0x3FE2EA64C3F97654 // 206 data8 0x3FE3001823684D73 // 207 data8 0x3FE30E97E9A8B5CC // 208 data8 0x3FE32463EBDD34E9 // 209 data8 0x3FE332F4314AD795 // 210 data8 0x3FE348D90E7464CF // 211 data8 0x3FE35779F8C43D6D // 212 data8 0x3FE36621961A6A99 // 213 data8 0x3FE37C299F3C366A // 214 data8 0x3FE38AE2171976E7 // 215 data8 0x3FE399A157A603E7 // 216 data8 0x3FE3AFCCFE77B9D1 // 217 data8 0x3FE3BE9D503533B5 // 218 data8 0x3FE3CD7480B4A8A2 // 219 data8 0x3FE3E3C43918F76C // 220 data8 0x3FE3F2ACB27ED6C6 // 221 data8 0x3FE4019C2125CA93 // 222 data8 0x3FE4181061389722 // 223 data8 0x3FE42711518DF545 // 224 data8 0x3FE436194E12B6BF // 225 data8 0x3FE445285D68EA69 // 226 data8 0x3FE45BCC464C893A // 227 data8 0x3FE46AED21F117FC // 228 data8 0x3FE47A1527E8A2D3 // 229 data8 0x3FE489445EFFFCCB // 230 data8 0x3FE4A018BCB69835 // 231 data8 0x3FE4AF5A0C9D65D7 // 232 data8 0x3FE4BEA2A5BDBE87 // 233 data8 0x3FE4CDF28F10AC46 // 234 data8 0x3FE4DD49CF994058 // 235 data8 0x3FE4ECA86E64A683 // 236 data8 0x3FE503C43CD8EB68 // 237 data8 0x3FE513356667FC57 // 238 data8 0x3FE522AE0738A3D7 // 239 data8 0x3FE5322E26867857 // 240 data8 0x3FE541B5CB979809 // 241 data8 0x3FE55144FDBCBD62 // 242 data8 0x3FE560DBC45153C6 // 243 data8 0x3FE5707A26BB8C66 // 244 data8 0x3FE587F60ED5B8FF // 245 data8 0x3FE597A7977C8F31 // 246 data8 0x3FE5A760D634BB8A // 247 data8 0x3FE5B721D295F10E // 248 data8 0x3FE5C6EA94431EF9 // 249 data8 0x3FE5D6BB22EA86F5 // 250 data8 0x3FE5E6938645D38F // 251 data8 0x3FE5F673C61A2ED1 // 252 data8 0x3FE6065BEA385926 // 253 data8 0x3FE6164BFA7CC06B // 254 data8 0x3FE62643FECF9742 // 255 // // two parts of ln(2) data8 0x3FE62E42FEF00000,0x3DD473DE6AF278ED // // lo parts of ln(1/frcpa(1+i/256)), i=0...255 data4 0x20E70672 // 0 data4 0x1F60A5D0 // 1 data4 0x218EABA0 // 2 data4 0x21403104 // 3 data4 0x20E9B54E // 4 data4 0x21EE1382 // 5 data4 0x226014E3 // 6 data4 0x2095E5C9 // 7 data4 0x228BA9D4 // 8 data4 0x22932B86 // 9 data4 0x22608A57 // 10 data4 0x220209F3 // 11 data4 0x212882CC // 12 data4 0x220D46E2 // 13 data4 0x21FA4C28 // 14 data4 0x229E5BD9 // 15 data4 0x228C9838 // 16 data4 0x2311F954 // 17 data4 0x221365DF // 18 data4 0x22BD0CB3 // 19 data4 0x223D4BB7 // 20 data4 0x22A71BBE // 21 data4 0x237DB2FA // 22 data4 0x23194C9D // 23 data4 0x22EC639E // 24 data4 0x2367E669 // 25 data4 0x232E1D5F // 26 data4 0x234A639B // 27 data4 0x2365C0E0 // 28 data4 0x234646C1 // 29 data4 0x220CBF9C // 30 data4 0x22A00FD4 // 31 data4 0x2306A3F2 // 32 data4 0x23745A9B // 33 data4 0x2398D756 // 34 data4 0x23DD0B6A // 35 data4 0x23DE338B // 36 data4 0x23A222DF // 37 data4 0x223164F8 // 38 data4 0x23B4E87B // 39 data4 0x23D6CCB8 // 40 data4 0x220C2099 // 41 data4 0x21B86B67 // 42 data4 0x236D14F1 // 43 data4 0x225A923F // 44 data4 0x22748723 // 45 data4 0x22200D13 // 46 data4 0x23C296EA // 47 data4 0x2302AC38 // 48 data4 0x234B1996 // 49 data4 0x2385E298 // 50 data4 0x23175BE5 // 51 data4 0x2193F482 // 52 data4 0x23BFEA90 // 53 data4 0x23D70A0C // 54 data4 0x231CF30A // 55 data4 0x235D9E90 // 56 data4 0x221AD0CB // 57 data4 0x22FAA08B // 58 data4 0x23D29A87 // 59 data4 0x20C4B2FE // 60 data4 0x2381B8B7 // 61 data4 0x23F8D9FC // 62 data4 0x23EAAE7B // 63 data4 0x2329E8AA // 64 data4 0x23EC0322 // 65 data4 0x2357FDCB // 66 data4 0x2392A9AD // 67 data4 0x22113B02 // 68 data4 0x22DEE901 // 69 data4 0x236A6D14 // 70 data4 0x2371D33E // 71 data4 0x2146F005 // 72 data4 0x23230B06 // 73 data4 0x22F1C77D // 74 data4 0x23A89FA3 // 75 data4 0x231D1241 // 76 data4 0x244DA96C // 77 data4 0x23ECBB7D // 78 data4 0x223E42B4 // 79 data4 0x23801BC9 // 80 data4 0x23573263 // 81 data4 0x227C1158 // 82 data4 0x237BD749 // 83 data4 0x21DDBAE9 // 84 data4 0x23401735 // 85 data4 0x241D9DEE // 86 data4 0x23BC88CB // 87 data4 0x2396D5F1 // 88 data4 0x23FC89CF // 89 data4 0x2414F9A2 // 90 data4 0x2474A0F5 // 91 data4 0x24354B60 // 92 data4 0x23C1EB40 // 93 data4 0x2306DD92 // 94 data4 0x24353B6B // 95 data4 0x23CD1701 // 96 data4 0x237C7A1C // 97 data4 0x245793AA // 98 data4 0x24563695 // 99 data4 0x23C51467 // 100 data4 0x24476B68 // 101 data4 0x212585A9 // 102 data4 0x247B8293 // 103 data4 0x2446848A // 104 data4 0x246A53F8 // 105 data4 0x246E496D // 106 data4 0x23ED1D36 // 107 data4 0x2314C258 // 108 data4 0x233244A7 // 109 data4 0x245B7AF0 // 110 data4 0x24247130 // 111 data4 0x22D67B38 // 112 data4 0x2449F620 // 113 data4 0x23BBC8B8 // 114 data4 0x237D3BA0 // 115 data4 0x245E8F13 // 116 data4 0x2435573F // 117 data4 0x242DE666 // 118 data4 0x2463BC10 // 119 data4 0x2466587D // 120 data4 0x2408144B // 121 data4 0x2405F0E5 // 122 data4 0x22381CFF // 123 data4 0x24154F9B // 124 data4 0x23A4E96E // 125 data4 0x24052967 // 126 data4 0x2406963F // 127 data4 0x23F7D3CB // 128 data4 0x2448AFF4 // 129 data4 0x24657A21 // 130 data4 0x22FBC230 // 131 data4 0x243C8DEA // 132 data4 0x225DC4B7 // 133 data4 0x23496EBF // 134 data4 0x237C2B2B // 135 data4 0x23A4A5B1 // 136 data4 0x2394E9D1 // 137 data4 0x244BC950 // 138 data4 0x23C7448F // 139 data4 0x2404A1AD // 140 data4 0x246511D5 // 141 data4 0x24246526 // 142 data4 0x23111F57 // 143 data4 0x22868951 // 144 data4 0x243EB77F // 145 data4 0x239F3DFF // 146 data4 0x23089666 // 147 data4 0x23EBFA6A // 148 data4 0x23C51312 // 149 data4 0x23E1DD5E // 150 data4 0x232C0944 // 151 data4 0x246A741F // 152 data4 0x2414DF8D // 153 data4 0x247B5546 // 154 data4 0x2415C980 // 155 data4 0x24324ABD // 156 data4 0x234EB5E5 // 157 data4 0x2465E43E // 158 data4 0x242840D1 // 159 data4 0x24444057 // 160 data4 0x245E56F0 // 161 data4 0x21AE30F8 // 162 data4 0x23FB3283 // 163 data4 0x247A4D07 // 164 data4 0x22AE314D // 165 data4 0x246B7727 // 166 data4 0x24EAD526 // 167 data4 0x24B41DC9 // 168 data4 0x24EE8062 // 169 data4 0x24A0C7C4 // 170 data4 0x24E8DA67 // 171 data4 0x231120F7 // 172 data4 0x24401FFB // 173 data4 0x2412DD09 // 174 data4 0x248C131A // 175 data4 0x24C0A7CE // 176 data4 0x243DD4C8 // 177 data4 0x24457FEB // 178 data4 0x24DEEFBB // 179 data4 0x243C70AE // 180 data4 0x23E7A6FA // 181 data4 0x24C2D311 // 182 data4 0x23026255 // 183 data4 0x2437C9B9 // 184 data4 0x246BA847 // 185 data4 0x2420B448 // 186 data4 0x24C4CF5A // 187 data4 0x242C4981 // 188 data4 0x24DE1525 // 189 data4 0x24F5CC33 // 190 data4 0x235A85DA // 191 data4 0x24A0B64F // 192 data4 0x244BA0A4 // 193 data4 0x24AAF30A // 194 data4 0x244C86F9 // 195 data4 0x246D5B82 // 196 data4 0x24529347 // 197 data4 0x240DD008 // 198 data4 0x24E98790 // 199 data4 0x2489B0CE // 200 data4 0x22BC29AC // 201 data4 0x23F37C7A // 202 data4 0x24987FE8 // 203 data4 0x22AFE20B // 204 data4 0x24C8D7C2 // 205 data4 0x24B28B7D // 206 data4 0x23B6B271 // 207 data4 0x24C77CB6 // 208 data4 0x24EF1DCA // 209 data4 0x24A4F0AC // 210 data4 0x24CF113E // 211 data4 0x2496BBAB // 212 data4 0x23C7CC8A // 213 data4 0x23AE3961 // 214 data4 0x2410A895 // 215 data4 0x23CE3114 // 216 data4 0x2308247D // 217 data4 0x240045E9 // 218 data4 0x24974F60 // 219 data4 0x242CB39F // 220 data4 0x24AB8D69 // 221 data4 0x23436788 // 222 data4 0x24305E9E // 223 data4 0x243E71A9 // 224 data4 0x23C2A6B3 // 225 data4 0x23FFE6CF // 226 data4 0x2322D801 // 227 data4 0x24515F21 // 228 data4 0x2412A0D6 // 229 data4 0x24E60D44 // 230 data4 0x240D9251 // 231 data4 0x247076E2 // 232 data4 0x229B101B // 233 data4 0x247B12DE // 234 data4 0x244B9127 // 235 data4 0x2499EC42 // 236 data4 0x21FC3963 // 237 data4 0x23E53266 // 238 data4 0x24CE102D // 239 data4 0x23CC45D2 // 240 data4 0x2333171D // 241 data4 0x246B3533 // 242 data4 0x24931129 // 243 data4 0x24405FFA // 244 data4 0x24CF464D // 245 data4 0x237095CD // 246 data4 0x24F86CBD // 247 data4 0x24E2D84B // 248 data4 0x21ACBB44 // 249 data4 0x24F43A8C // 250 data4 0x249DB931 // 251 data4 0x24A385EF // 252 data4 0x238B1279 // 253 data4 0x2436213E // 254 data4 0x24F18A3B // 255 LOCAL_OBJECT_END(log_data) // Code //============================================================== .section .text GLOBAL_IEEE754_ENTRY(log1p) { .mfi getf.exp GR_signexp_x = f8 // if x is unorm then must recompute fadd.s1 FR_Xp1 = f8, f1 // Form 1+x mov GR_05 = 0xfffe } { .mlx addl GR_ad_1 = @ltoff(log_data),gp movl GR_A3 = 0x3fd5555555555557 // double precision memory // representation of A3 } ;; { .mfi ld8 GR_ad_1 = [GR_ad_1] fclass.m p8,p0 = f8,0xb // Is x unorm? mov GR_exp_mask = 0x1ffff } { .mfi nop.m 0 fnorm.s1 FR_NormX = f8 // Normalize x mov GR_exp_bias = 0xffff } ;; { .mfi setf.exp FR_A2 = GR_05 // create A2 = 0.5 fclass.m p9,p0 = f8,0x1E1 // is x NaN, NaT or +Inf? nop.i 0 } { .mib setf.d FR_A3 = GR_A3 // create A3 add GR_ad_2 = 16,GR_ad_1 // address of A5,A4 (p8) br.cond.spnt log1p_unorm // Branch if x=unorm } ;; log1p_common: { .mfi nop.m 0 frcpa.s1 FR_RcpX,p0 = f1,FR_Xp1 nop.i 0 } { .mfb nop.m 0 (p9) fma.d.s0 f8 = f8,f1,f0 // set V-flag (p9) br.ret.spnt b0 // exit for NaN, NaT and +Inf } ;; { .mfi getf.exp GR_Exp = FR_Xp1 // signexp of x+1 fclass.m p10,p0 = FR_Xp1,0x3A // is 1+x < 0? and GR_exp_x = GR_exp_mask, GR_signexp_x // biased exponent of x } { .mfi ldfpd FR_A7,FR_A6 = [GR_ad_1] nop.f 0 nop.i 0 } ;; { .mfi getf.sig GR_Sig = FR_Xp1 // get significand to calculate index // for Thi,Tlo if |x| >= 2^-8 fcmp.eq.s1 p12,p0 = f8,f0 // is x equal to 0? sub GR_exp_x = GR_exp_x, GR_exp_bias // true exponent of x } ;; { .mfi sub GR_N = GR_Exp,GR_exp_bias // true exponent of x+1 fcmp.eq.s1 p11,p0 = FR_Xp1,f0 // is x = -1? cmp.gt p6,p7 = -8, GR_exp_x // Is |x| < 2^-8 } { .mfb ldfpd FR_A5,FR_A4 = [GR_ad_2],16 nop.f 0 (p10) br.cond.spnt log1p_lt_minus_1 // jump if x < -1 } ;; // p6 is true if |x| < 1/256 // p7 is true if |x| >= 1/256 .pred.rel "mutex",p6,p7 { .mfi (p7) add GR_ad_1 = 0x820,GR_ad_1 // address of log(2) parts (p6) fms.s1 FR_r = f8,f1,f0 // range reduction for |x|<1/256 (p6) cmp.gt.unc p10,p0 = -80, GR_exp_x // Is |x| < 2^-80 } { .mfb (p7) setf.sig FR_N = GR_N // copy unbiased exponent of x to the // significand field of FR_N (p7) fms.s1 FR_r = FR_RcpX,FR_Xp1,f1 // range reduction for |x|>=1/256 (p12) br.ret.spnt b0 // exit for x=0, return x } ;; { .mib (p7) ldfpd FR_Ln2hi,FR_Ln2lo = [GR_ad_1],16 (p7) extr.u GR_Ind = GR_Sig,55,8 // get bits from 55 to 62 as index (p11) br.cond.spnt log1p_eq_minus_1 // jump if x = -1 } ;; { .mmf (p7) shladd GR_ad_2 = GR_Ind,3,GR_ad_2 // address of Thi (p7) shladd GR_ad_1 = GR_Ind,2,GR_ad_1 // address of Tlo (p10) fnma.d.s0 f8 = f8,f8,f8 // If |x| very small, result=x-x*x } ;; { .mmb (p7) ldfd FR_Thi = [GR_ad_2] (p7) ldfs FR_Tlo = [GR_ad_1] (p10) br.ret.spnt b0 // Exit if |x| < 2^(-80) } ;; { .mfi nop.m 0 fma.s1 FR_r2 = FR_r,FR_r,f0 // r^2 nop.i 0 } { .mfi nop.m 0 fms.s1 FR_A2 = FR_A3,FR_r,FR_A2 // A3*r+A2 nop.i 0 } ;; { .mfi nop.m 0 fma.s1 FR_A6 = FR_A7,FR_r,FR_A6 // A7*r+A6 nop.i 0 } { .mfi nop.m 0 fma.s1 FR_A4 = FR_A5,FR_r,FR_A4 // A5*r+A4 nop.i 0 } ;; { .mfi nop.m 0 (p7) fcvt.xf FR_N = FR_N nop.i 0 } ;; { .mfi nop.m 0 fma.s1 FR_r4 = FR_r2,FR_r2,f0 // r^4 nop.i 0 } { .mfi nop.m 0 // (A3*r+A2)*r^2+r fma.s1 FR_A2 = FR_A2,FR_r2,FR_r nop.i 0 } ;; { .mfi nop.m 0 // (A7*r+A6)*r^2+(A5*r+A4) fma.s1 FR_A4 = FR_A6,FR_r2,FR_A4 nop.i 0 } ;; { .mfi nop.m 0 // N*Ln2hi+Thi (p7) fma.s1 FR_NxLn2hipThi = FR_N,FR_Ln2hi,FR_Thi nop.i 0 } { .mfi nop.m 0 // N*Ln2lo+Tlo (p7) fma.s1 FR_NxLn2lopTlo = FR_N,FR_Ln2lo,FR_Tlo nop.i 0 } ;; { .mfi nop.m 0 (p7) fma.s1 f8 = FR_A4,FR_r4,FR_A2 // P(r) if |x| >= 1/256 nop.i 0 } { .mfi nop.m 0 // (N*Ln2hi+Thi) + (N*Ln2lo+Tlo) (p7) fma.s1 FR_NxLn2pT = FR_NxLn2hipThi,f1,FR_NxLn2lopTlo nop.i 0 } ;; .pred.rel "mutex",p6,p7 { .mfi nop.m 0 (p6) fma.d.s0 f8 = FR_A4,FR_r4,FR_A2 // result if 2^(-80) <= |x| < 1/256 nop.i 0 } { .mfb nop.m 0 (p7) fma.d.s0 f8 = f8,f1,FR_NxLn2pT // result if |x| >= 1/256 br.ret.sptk b0 // Exit if |x| >= 2^(-80) } ;; .align 32 log1p_unorm: // Here if x=unorm { .mfb getf.exp GR_signexp_x = FR_NormX // recompute biased exponent nop.f 0 br.cond.sptk log1p_common } ;; .align 32 log1p_eq_minus_1: // Here if x=-1 { .mfi nop.m 0 fmerge.s FR_X = f8,f8 // keep input argument for subsequent // call of __libm_error_support# nop.i 0 } ;; { .mfi mov GR_TAG = 140 // set libm error in case of log1p(-1). frcpa.s0 f8,p0 = f8,f0 // log1p(-1) should be equal to -INF. // We can get it using frcpa because it // sets result to the IEEE-754 mandated // quotient of f8/f0. nop.i 0 } { .mib nop.m 0 nop.i 0 br.cond.sptk log_libm_err } ;; .align 32 log1p_lt_minus_1: // Here if x < -1 { .mfi nop.m 0 fmerge.s FR_X = f8,f8 nop.i 0 } ;; { .mfi mov GR_TAG = 141 // set libm error in case of x < -1. frcpa.s0 f8,p0 = f0,f0 // log1p(x) x < -1 should be equal to NaN. // We can get it using frcpa because it // sets result to the IEEE-754 mandated // quotient of f0/f0 i.e. NaN. nop.i 0 } ;; .align 32 log_libm_err: { .mmi alloc r32 = ar.pfs,1,4,4,0 mov GR_Parameter_TAG = GR_TAG nop.i 0 } ;; GLOBAL_IEEE754_END(log1p) LOCAL_LIBM_ENTRY(__libm_error_region) .prologue { .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 };; { .mmi 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 { .mib 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] = 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 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 };; { .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#