1 /*- 2 * SPDX-License-Identifier: BSD-2-Clause 3 * 4 * Copyright (c) 2007-2013 Bruce D. Evans 5 * All rights reserved. 6 * 7 * Redistribution and use in source and binary forms, with or without 8 * modification, are permitted provided that the following conditions 9 * are met: 10 * 1. Redistributions of source code must retain the above copyright 11 * notice unmodified, this list of conditions, and the following 12 * disclaimer. 13 * 2. Redistributions in binary form must reproduce the above copyright 14 * notice, this list of conditions and the following disclaimer in the 15 * documentation and/or other materials provided with the distribution. 16 * 17 * THIS SOFTWARE IS PROVIDED BY THE AUTHOR ``AS IS'' AND ANY EXPRESS OR 18 * IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES 19 * OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. 20 * IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY DIRECT, INDIRECT, 21 * INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT 22 * NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, 23 * DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY 24 * THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT 25 * (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF 26 * THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. 27 */ 28 29 #include <sys/cdefs.h> 30 /** 31 * Implementation of the natural logarithm of x for 128-bit format. 32 * 33 * First decompose x into its base 2 representation: 34 * 35 * log(x) = log(X * 2**k), where X is in [1, 2) 36 * = log(X) + k * log(2). 37 * 38 * Let X = X_i + e, where X_i is the center of one of the intervals 39 * [-1.0/256, 1.0/256), [1.0/256, 3.0/256), .... [2.0-1.0/256, 2.0+1.0/256) 40 * and X is in this interval. Then 41 * 42 * log(X) = log(X_i + e) 43 * = log(X_i * (1 + e / X_i)) 44 * = log(X_i) + log(1 + e / X_i). 45 * 46 * The values log(X_i) are tabulated below. Let d = e / X_i and use 47 * 48 * log(1 + d) = p(d) 49 * 50 * where p(d) = d - 0.5*d*d + ... is a special minimax polynomial of 51 * suitably high degree. 52 * 53 * To get sufficiently small roundoff errors, k * log(2), log(X_i), and 54 * sometimes (if |k| is not large) the first term in p(d) must be evaluated 55 * and added up in extra precision. Extra precision is not needed for the 56 * rest of p(d). In the worst case when k = 0 and log(X_i) is 0, the final 57 * error is controlled mainly by the error in the second term in p(d). The 58 * error in this term itself is at most 0.5 ulps from the d*d operation in 59 * it. The error in this term relative to the first term is thus at most 60 * 0.5 * |-0.5| * |d| < 1.0/1024 ulps. We aim for an accumulated error of 61 * at most twice this at the point of the final rounding step. Thus the 62 * final error should be at most 0.5 + 1.0/512 = 0.5020 ulps. Exhaustive 63 * testing of a float variant of this function showed a maximum final error 64 * of 0.5008 ulps. Non-exhaustive testing of a double variant of this 65 * function showed a maximum final error of 0.5078 ulps (near 1+1.0/256). 66 * 67 * We made the maximum of |d| (and thus the total relative error and the 68 * degree of p(d)) small by using a large number of intervals. Using 69 * centers of intervals instead of endpoints reduces this maximum by a 70 * factor of 2 for a given number of intervals. p(d) is special only 71 * in beginning with the Taylor coefficients 0 + 1*d, which tends to happen 72 * naturally. The most accurate minimax polynomial of a given degree might 73 * be different, but then we wouldn't want it since we would have to do 74 * extra work to avoid roundoff error (especially for P0*d instead of d). 75 */ 76 77 #ifdef DEBUG 78 #include <fenv.h> 79 #endif 80 81 #include "math.h" 82 #ifndef NO_STRUCT_RETURN 83 #define STRUCT_RETURN 84 #endif 85 #include "math_private.h" 86 87 #if !defined(NO_UTAB) && !defined(NO_UTABL) 88 #define USE_UTAB 89 #endif 90 91 /* 92 * Domain [-0.005280, 0.004838], range ~[-1.1577e-37, 1.1582e-37]: 93 * |log(1 + d)/d - p(d)| < 2**-122.7 94 */ 95 static const long double 96 P2 = -0.5L, 97 P3 = 3.33333333333333333333333333333233795e-1L, /* 0x15555555555555555555555554d42.0p-114L */ 98 P4 = -2.49999999999999999999999999941139296e-1L, /* -0x1ffffffffffffffffffffffdab14e.0p-115L */ 99 P5 = 2.00000000000000000000000085468039943e-1L, /* 0x19999999999999999999a6d3567f4.0p-115L */ 100 P6 = -1.66666666666666666666696142372698408e-1L, /* -0x15555555555555555567267a58e13.0p-115L */ 101 P7 = 1.42857142857142857119522943477166120e-1L, /* 0x1249249249249248ed79a0ae434de.0p-115L */ 102 P8 = -1.24999999999999994863289015033581301e-1L; /* -0x1fffffffffffffa13e91765e46140.0p-116L */ 103 /* Double precision gives ~ 53 + log2(P9 * max(|d|)**8) ~= 120 bits. */ 104 static const double 105 P9 = 1.1111111111111401e-1, /* 0x1c71c71c71c7ed.0p-56 */ 106 P10 = -1.0000000000040135e-1, /* -0x199999999a0a92.0p-56 */ 107 P11 = 9.0909090728136258e-2, /* 0x1745d173962111.0p-56 */ 108 P12 = -8.3333318851855284e-2, /* -0x1555551722c7a3.0p-56 */ 109 P13 = 7.6928634666404178e-2, /* 0x13b1985204a4ae.0p-56 */ 110 P14 = -7.1626810078462499e-2; /* -0x12562276cdc5d0.0p-56 */ 111 112 static volatile const double zero = 0; 113 114 #define INTERVALS 128 115 #define LOG2_INTERVALS 7 116 #define TSIZE (INTERVALS + 1) 117 #define G(i) (T[(i)].G) 118 #define F_hi(i) (T[(i)].F_hi) 119 #define F_lo(i) (T[(i)].F_lo) 120 #define ln2_hi F_hi(TSIZE - 1) 121 #define ln2_lo F_lo(TSIZE - 1) 122 #define E(i) (U[(i)].E) 123 #define H(i) (U[(i)].H) 124 125 static const struct { 126 float G; /* 1/(1 + i/128) rounded to 8/9 bits */ 127 float F_hi; /* log(1 / G_i) rounded (see below) */ 128 /* The compiler will insert 8 bytes of padding here. */ 129 long double F_lo; /* next 113 bits for log(1 / G_i) */ 130 } T[TSIZE] = { 131 /* 132 * ln2_hi and each F_hi(i) are rounded to a number of bits that 133 * makes F_hi(i) + dk*ln2_hi exact for all i and all dk. 134 * 135 * The last entry (for X just below 2) is used to define ln2_hi 136 * and ln2_lo, to ensure that F_hi(i) and F_lo(i) cancel exactly 137 * with dk*ln2_hi and dk*ln2_lo, respectively, when dk = -1. 138 * This is needed for accuracy when x is just below 1. (To avoid 139 * special cases, such x are "reduced" strangely to X just below 140 * 2 and dk = -1, and then the exact cancellation is needed 141 * because any the error from any non-exactness would be too 142 * large). 143 * 144 * The relevant range of dk is [-16445, 16383]. The maximum number 145 * of bits in F_hi(i) that works is very dependent on i but has 146 * a minimum of 93. We only need about 12 bits in F_hi(i) for 147 * it to provide enough extra precision. 148 * 149 * We round F_hi(i) to 24 bits so that it can have type float, 150 * mainly to minimize the size of the table. Using all 24 bits 151 * in a float for it automatically satisfies the above constraints. 152 */ 153 { 0x800000.0p-23, 0, 0 }, 154 { 0xfe0000.0p-24, 0x8080ac.0p-30, -0x14ee431dae6674afa0c4bfe16e8fd.0p-144L }, 155 { 0xfc0000.0p-24, 0x8102b3.0p-29, -0x1db29ee2d83717be918e1119642ab.0p-144L }, 156 { 0xfa0000.0p-24, 0xc24929.0p-29, 0x1191957d173697cf302cc9476f561.0p-143L }, 157 { 0xf80000.0p-24, 0x820aec.0p-28, 0x13ce8888e02e78eba9b1113bc1c18.0p-142L }, 158 { 0xf60000.0p-24, 0xa33577.0p-28, -0x17a4382ce6eb7bfa509bec8da5f22.0p-142L }, 159 { 0xf48000.0p-24, 0xbc42cb.0p-28, -0x172a21161a107674986dcdca6709c.0p-143L }, 160 { 0xf30000.0p-24, 0xd57797.0p-28, -0x1e09de07cb958897a3ea46e84abb3.0p-142L }, 161 { 0xf10000.0p-24, 0xf7518e.0p-28, 0x1ae1eec1b036c484993c549c4bf40.0p-151L }, 162 { 0xef0000.0p-24, 0x8cb9df.0p-27, -0x1d7355325d560d9e9ab3d6ebab580.0p-141L }, 163 { 0xed8000.0p-24, 0x999ec0.0p-27, -0x1f9f02d256d5037108f4ec21e48cd.0p-142L }, 164 { 0xec0000.0p-24, 0xa6988b.0p-27, -0x16fc0a9d12c17a70f7a684c596b12.0p-143L }, 165 { 0xea0000.0p-24, 0xb80698.0p-27, 0x15d581c1e8da99ded322fb08b8462.0p-141L }, 166 { 0xe80000.0p-24, 0xc99af3.0p-27, -0x1535b3ba8f150ae09996d7bb4653e.0p-143L }, 167 { 0xe70000.0p-24, 0xd273b2.0p-27, 0x163786f5251aefe0ded34c8318f52.0p-145L }, 168 { 0xe50000.0p-24, 0xe442c0.0p-27, 0x1bc4b2368e32d56699c1799a244d4.0p-144L }, 169 { 0xe38000.0p-24, 0xf1b83f.0p-27, 0x1c6090f684e6766abceccab1d7174.0p-141L }, 170 { 0xe20000.0p-24, 0xff448a.0p-27, -0x1890aa69ac9f4215f93936b709efb.0p-142L }, 171 { 0xe08000.0p-24, 0x8673f6.0p-26, 0x1b9985194b6affd511b534b72a28e.0p-140L }, 172 { 0xdf0000.0p-24, 0x8d515c.0p-26, -0x1dc08d61c6ef1d9b2ef7e68680598.0p-143L }, 173 { 0xdd8000.0p-24, 0x943a9e.0p-26, -0x1f72a2dac729b3f46662238a9425a.0p-142L }, 174 { 0xdc0000.0p-24, 0x9b2fe6.0p-26, -0x1fd4dfd3a0afb9691aed4d5e3df94.0p-140L }, 175 { 0xda8000.0p-24, 0xa2315d.0p-26, -0x11b26121629c46c186384993e1c93.0p-142L }, 176 { 0xd90000.0p-24, 0xa93f2f.0p-26, 0x1286d633e8e5697dc6a402a56fce1.0p-141L }, 177 { 0xd78000.0p-24, 0xb05988.0p-26, 0x16128eba9367707ebfa540e45350c.0p-144L }, 178 { 0xd60000.0p-24, 0xb78094.0p-26, 0x16ead577390d31ef0f4c9d43f79b2.0p-140L }, 179 { 0xd50000.0p-24, 0xbc4c6c.0p-26, 0x151131ccf7c7b75e7d900b521c48d.0p-141L }, 180 { 0xd38000.0p-24, 0xc3890a.0p-26, -0x115e2cd714bd06508aeb00d2ae3e9.0p-140L }, 181 { 0xd20000.0p-24, 0xcad2d7.0p-26, -0x1847f406ebd3af80485c2f409633c.0p-142L }, 182 { 0xd10000.0p-24, 0xcfb620.0p-26, 0x1c2259904d686581799fbce0b5f19.0p-141L }, 183 { 0xcf8000.0p-24, 0xd71653.0p-26, 0x1ece57a8d5ae54f550444ecf8b995.0p-140L }, 184 { 0xce0000.0p-24, 0xde843a.0p-26, -0x1f109d4bc4595412b5d2517aaac13.0p-141L }, 185 { 0xcd0000.0p-24, 0xe37fde.0p-26, 0x1bc03dc271a74d3a85b5b43c0e727.0p-141L }, 186 { 0xcb8000.0p-24, 0xeb050c.0p-26, -0x1bf2badc0df841a71b79dd5645b46.0p-145L }, 187 { 0xca0000.0p-24, 0xf29878.0p-26, -0x18efededd89fbe0bcfbe6d6db9f66.0p-147L }, 188 { 0xc90000.0p-24, 0xf7ad6f.0p-26, 0x1373ff977baa6911c7bafcb4d84fb.0p-141L }, 189 { 0xc80000.0p-24, 0xfcc8e3.0p-26, 0x196766f2fb328337cc050c6d83b22.0p-140L }, 190 { 0xc68000.0p-24, 0x823f30.0p-25, 0x19bd076f7c434e5fcf1a212e2a91e.0p-139L }, 191 { 0xc58000.0p-24, 0x84d52c.0p-25, -0x1a327257af0f465e5ecab5f2a6f81.0p-139L }, 192 { 0xc40000.0p-24, 0x88bc74.0p-25, 0x113f23def19c5a0fe396f40f1dda9.0p-141L }, 193 { 0xc30000.0p-24, 0x8b5ae6.0p-25, 0x1759f6e6b37de945a049a962e66c6.0p-139L }, 194 { 0xc20000.0p-24, 0x8dfccb.0p-25, 0x1ad35ca6ed5147bdb6ddcaf59c425.0p-141L }, 195 { 0xc10000.0p-24, 0x90a22b.0p-25, 0x1a1d71a87deba46bae9827221dc98.0p-139L }, 196 { 0xbf8000.0p-24, 0x94a0d8.0p-25, -0x139e5210c2b730e28aba001a9b5e0.0p-140L }, 197 { 0xbe8000.0p-24, 0x974f16.0p-25, -0x18f6ebcff3ed72e23e13431adc4a5.0p-141L }, 198 { 0xbd8000.0p-24, 0x9a00f1.0p-25, -0x1aa268be39aab7148e8d80caa10b7.0p-139L }, 199 { 0xbc8000.0p-24, 0x9cb672.0p-25, -0x14c8815839c5663663d15faed7771.0p-139L }, 200 { 0xbb0000.0p-24, 0xa0cda1.0p-25, 0x1eaf46390dbb2438273918db7df5c.0p-141L }, 201 { 0xba0000.0p-24, 0xa38c6e.0p-25, 0x138e20d831f698298adddd7f32686.0p-141L }, 202 { 0xb90000.0p-24, 0xa64f05.0p-25, -0x1e8d3c41123615b147a5d47bc208f.0p-142L }, 203 { 0xb80000.0p-24, 0xa91570.0p-25, 0x1ce28f5f3840b263acb4351104631.0p-140L }, 204 { 0xb70000.0p-24, 0xabdfbb.0p-25, -0x186e5c0a42423457e22d8c650b355.0p-139L }, 205 { 0xb60000.0p-24, 0xaeadef.0p-25, -0x14d41a0b2a08a465dc513b13f567d.0p-143L }, 206 { 0xb50000.0p-24, 0xb18018.0p-25, 0x16755892770633947ffe651e7352f.0p-139L }, 207 { 0xb40000.0p-24, 0xb45642.0p-25, -0x16395ebe59b15228bfe8798d10ff0.0p-142L }, 208 { 0xb30000.0p-24, 0xb73077.0p-25, 0x1abc65c8595f088b61a335f5b688c.0p-140L }, 209 { 0xb20000.0p-24, 0xba0ec4.0p-25, -0x1273089d3dad88e7d353e9967d548.0p-139L }, 210 { 0xb10000.0p-24, 0xbcf133.0p-25, 0x10f9f67b1f4bbf45de06ecebfaf6d.0p-139L }, 211 { 0xb00000.0p-24, 0xbfd7d2.0p-25, -0x109fab904864092b34edda19a831e.0p-140L }, 212 { 0xaf0000.0p-24, 0xc2c2ac.0p-25, -0x1124680aa43333221d8a9b475a6ba.0p-139L }, 213 { 0xae8000.0p-24, 0xc439b3.0p-25, -0x1f360cc4710fbfe24b633f4e8d84d.0p-140L }, 214 { 0xad8000.0p-24, 0xc72afd.0p-25, -0x132d91f21d89c89c45003fc5d7807.0p-140L }, 215 { 0xac8000.0p-24, 0xca20a2.0p-25, -0x16bf9b4d1f8da8002f2449e174504.0p-139L }, 216 { 0xab8000.0p-24, 0xcd1aae.0p-25, 0x19deb5ce6a6a8717d5626e16acc7d.0p-141L }, 217 { 0xaa8000.0p-24, 0xd0192f.0p-25, 0x1a29fb48f7d3ca87dabf351aa41f4.0p-139L }, 218 { 0xaa0000.0p-24, 0xd19a20.0p-25, 0x1127d3c6457f9d79f51dcc73014c9.0p-141L }, 219 { 0xa90000.0p-24, 0xd49f6a.0p-25, -0x1ba930e486a0ac42d1bf9199188e7.0p-141L }, 220 { 0xa80000.0p-24, 0xd7a94b.0p-25, -0x1b6e645f31549dd1160bcc45c7e2c.0p-139L }, 221 { 0xa70000.0p-24, 0xdab7d0.0p-25, 0x1118a425494b610665377f15625b6.0p-140L }, 222 { 0xa68000.0p-24, 0xdc40d5.0p-25, 0x1966f24d29d3a2d1b2176010478be.0p-140L }, 223 { 0xa58000.0p-24, 0xdf566d.0p-25, -0x1d8e52eb2248f0c95dd83626d7333.0p-142L }, 224 { 0xa48000.0p-24, 0xe270ce.0p-25, -0x1ee370f96e6b67ccb006a5b9890ea.0p-140L }, 225 { 0xa40000.0p-24, 0xe3ffce.0p-25, 0x1d155324911f56db28da4d629d00a.0p-140L }, 226 { 0xa30000.0p-24, 0xe72179.0p-25, -0x1fe6e2f2f867d8f4d60c713346641.0p-140L }, 227 { 0xa20000.0p-24, 0xea4812.0p-25, 0x1b7be9add7f4d3b3d406b6cbf3ce5.0p-140L }, 228 { 0xa18000.0p-24, 0xebdd3d.0p-25, 0x1b3cfb3f7511dd73692609040ccc2.0p-139L }, 229 { 0xa08000.0p-24, 0xef0b5b.0p-25, -0x1220de1f7301901b8ad85c25afd09.0p-139L }, 230 { 0xa00000.0p-24, 0xf0a451.0p-25, -0x176364c9ac81cc8a4dfb804de6867.0p-140L }, 231 { 0x9f0000.0p-24, 0xf3da16.0p-25, 0x1eed6b9aafac8d42f78d3e65d3727.0p-141L }, 232 { 0x9e8000.0p-24, 0xf576e9.0p-25, 0x1d593218675af269647b783d88999.0p-139L }, 233 { 0x9d8000.0p-24, 0xf8b47c.0p-25, -0x13e8eb7da053e063714615f7cc91d.0p-144L }, 234 { 0x9d0000.0p-24, 0xfa553f.0p-25, 0x1c063259bcade02951686d5373aec.0p-139L }, 235 { 0x9c0000.0p-24, 0xfd9ac5.0p-25, 0x1ef491085fa3c1649349630531502.0p-139L }, 236 { 0x9b8000.0p-24, 0xff3f8c.0p-25, 0x1d607a7c2b8c5320619fb9433d841.0p-139L }, 237 { 0x9a8000.0p-24, 0x814697.0p-24, -0x12ad3817004f3f0bdff99f932b273.0p-138L }, 238 { 0x9a0000.0p-24, 0x821b06.0p-24, -0x189fc53117f9e54e78103a2bc1767.0p-141L }, 239 { 0x990000.0p-24, 0x83c5f8.0p-24, 0x14cf15a048907b7d7f47ddb45c5a3.0p-139L }, 240 { 0x988000.0p-24, 0x849c7d.0p-24, 0x1cbb1d35fb82873b04a9af1dd692c.0p-138L }, 241 { 0x978000.0p-24, 0x864ba6.0p-24, 0x1128639b814f9b9770d8cb6573540.0p-138L }, 242 { 0x970000.0p-24, 0x87244c.0p-24, 0x184733853300f002e836dfd47bd41.0p-139L }, 243 { 0x968000.0p-24, 0x87fdaa.0p-24, 0x109d23aef77dd5cd7cc94306fb3ff.0p-140L }, 244 { 0x958000.0p-24, 0x89b293.0p-24, -0x1a81ef367a59de2b41eeebd550702.0p-138L }, 245 { 0x950000.0p-24, 0x8a8e20.0p-24, -0x121ad3dbb2f45275c917a30df4ac9.0p-138L }, 246 { 0x948000.0p-24, 0x8b6a6a.0p-24, -0x1cfb981628af71a89df4e6df2e93b.0p-139L }, 247 { 0x938000.0p-24, 0x8d253a.0p-24, -0x1d21730ea76cfdec367828734cae5.0p-139L }, 248 { 0x930000.0p-24, 0x8e03c2.0p-24, 0x135cc00e566f76b87333891e0dec4.0p-138L }, 249 { 0x928000.0p-24, 0x8ee30d.0p-24, -0x10fcb5df257a263e3bf446c6e3f69.0p-140L }, 250 { 0x918000.0p-24, 0x90a3ee.0p-24, -0x16e171b15433d723a4c7380a448d8.0p-139L }, 251 { 0x910000.0p-24, 0x918587.0p-24, -0x1d050da07f3236f330972da2a7a87.0p-139L }, 252 { 0x908000.0p-24, 0x9267e7.0p-24, 0x1be03669a5268d21148c6002becd3.0p-139L }, 253 { 0x8f8000.0p-24, 0x942f04.0p-24, 0x10b28e0e26c336af90e00533323ba.0p-139L }, 254 { 0x8f0000.0p-24, 0x9513c3.0p-24, 0x1a1d820da57cf2f105a89060046aa.0p-138L }, 255 { 0x8e8000.0p-24, 0x95f950.0p-24, -0x19ef8f13ae3cf162409d8ea99d4c0.0p-139L }, 256 { 0x8e0000.0p-24, 0x96dfab.0p-24, -0x109e417a6e507b9dc10dac743ad7a.0p-138L }, 257 { 0x8d0000.0p-24, 0x98aed2.0p-24, 0x10d01a2c5b0e97c4990b23d9ac1f5.0p-139L }, 258 { 0x8c8000.0p-24, 0x9997a2.0p-24, -0x1d6a50d4b61ea74540bdd2aa99a42.0p-138L }, 259 { 0x8c0000.0p-24, 0x9a8145.0p-24, 0x1b3b190b83f9527e6aba8f2d783c1.0p-138L }, 260 { 0x8b8000.0p-24, 0x9b6bbf.0p-24, 0x13a69fad7e7abe7ba81c664c107e0.0p-138L }, 261 { 0x8b0000.0p-24, 0x9c5711.0p-24, -0x11cd12316f576aad348ae79867223.0p-138L }, 262 { 0x8a8000.0p-24, 0x9d433b.0p-24, 0x1c95c444b807a246726b304ccae56.0p-139L }, 263 { 0x898000.0p-24, 0x9f1e22.0p-24, -0x1b9c224ea698c2f9b47466d6123fe.0p-139L }, 264 { 0x890000.0p-24, 0xa00ce1.0p-24, 0x125ca93186cf0f38b4619a2483399.0p-141L }, 265 { 0x888000.0p-24, 0xa0fc80.0p-24, -0x1ee38a7bc228b3597043be78eaf49.0p-139L }, 266 { 0x880000.0p-24, 0xa1ed00.0p-24, -0x1a0db876613d204147dc69a07a649.0p-138L }, 267 { 0x878000.0p-24, 0xa2de62.0p-24, 0x193224e8516c008d3602a7b41c6e8.0p-139L }, 268 { 0x870000.0p-24, 0xa3d0a9.0p-24, 0x1fa28b4d2541aca7d5844606b2421.0p-139L }, 269 { 0x868000.0p-24, 0xa4c3d6.0p-24, 0x1c1b5760fb4571acbcfb03f16daf4.0p-138L }, 270 { 0x858000.0p-24, 0xa6acea.0p-24, 0x1fed5d0f65949c0a345ad743ae1ae.0p-140L }, 271 { 0x850000.0p-24, 0xa7a2d4.0p-24, 0x1ad270c9d749362382a7688479e24.0p-140L }, 272 { 0x848000.0p-24, 0xa899ab.0p-24, 0x199ff15ce532661ea9643a3a2d378.0p-139L }, 273 { 0x840000.0p-24, 0xa99171.0p-24, 0x1a19e15ccc45d257530a682b80490.0p-139L }, 274 { 0x838000.0p-24, 0xaa8a28.0p-24, -0x121a14ec532b35ba3e1f868fd0b5e.0p-140L }, 275 { 0x830000.0p-24, 0xab83d1.0p-24, 0x1aee319980bff3303dd481779df69.0p-139L }, 276 { 0x828000.0p-24, 0xac7e6f.0p-24, -0x18ffd9e3900345a85d2d86161742e.0p-140L }, 277 { 0x820000.0p-24, 0xad7a03.0p-24, -0x1e4db102ce29f79b026b64b42caa1.0p-140L }, 278 { 0x818000.0p-24, 0xae768f.0p-24, 0x17c35c55a04a82ab19f77652d977a.0p-141L }, 279 { 0x810000.0p-24, 0xaf7415.0p-24, 0x1448324047019b48d7b98c1cf7234.0p-138L }, 280 { 0x808000.0p-24, 0xb07298.0p-24, -0x1750ee3915a197e9c7359dd94152f.0p-138L }, 281 { 0x800000.0p-24, 0xb17218.0p-24, -0x105c610ca86c3898cff81a12a17e2.0p-141L }, 282 }; 283 284 #ifdef USE_UTAB 285 static const struct { 286 float H; /* 1 + i/INTERVALS (exact) */ 287 float E; /* H(i) * G(i) - 1 (exact) */ 288 } U[TSIZE] = { 289 { 0x800000.0p-23, 0 }, 290 { 0x810000.0p-23, -0x800000.0p-37 }, 291 { 0x820000.0p-23, -0x800000.0p-35 }, 292 { 0x830000.0p-23, -0x900000.0p-34 }, 293 { 0x840000.0p-23, -0x800000.0p-33 }, 294 { 0x850000.0p-23, -0xc80000.0p-33 }, 295 { 0x860000.0p-23, -0xa00000.0p-36 }, 296 { 0x870000.0p-23, 0x940000.0p-33 }, 297 { 0x880000.0p-23, 0x800000.0p-35 }, 298 { 0x890000.0p-23, -0xc80000.0p-34 }, 299 { 0x8a0000.0p-23, 0xe00000.0p-36 }, 300 { 0x8b0000.0p-23, 0x900000.0p-33 }, 301 { 0x8c0000.0p-23, -0x800000.0p-35 }, 302 { 0x8d0000.0p-23, -0xe00000.0p-33 }, 303 { 0x8e0000.0p-23, 0x880000.0p-33 }, 304 { 0x8f0000.0p-23, -0xa80000.0p-34 }, 305 { 0x900000.0p-23, -0x800000.0p-35 }, 306 { 0x910000.0p-23, 0x800000.0p-37 }, 307 { 0x920000.0p-23, 0x900000.0p-35 }, 308 { 0x930000.0p-23, 0xd00000.0p-35 }, 309 { 0x940000.0p-23, 0xe00000.0p-35 }, 310 { 0x950000.0p-23, 0xc00000.0p-35 }, 311 { 0x960000.0p-23, 0xe00000.0p-36 }, 312 { 0x970000.0p-23, -0x800000.0p-38 }, 313 { 0x980000.0p-23, -0xc00000.0p-35 }, 314 { 0x990000.0p-23, -0xd00000.0p-34 }, 315 { 0x9a0000.0p-23, 0x880000.0p-33 }, 316 { 0x9b0000.0p-23, 0xe80000.0p-35 }, 317 { 0x9c0000.0p-23, -0x800000.0p-35 }, 318 { 0x9d0000.0p-23, 0xb40000.0p-33 }, 319 { 0x9e0000.0p-23, 0x880000.0p-34 }, 320 { 0x9f0000.0p-23, -0xe00000.0p-35 }, 321 { 0xa00000.0p-23, 0x800000.0p-33 }, 322 { 0xa10000.0p-23, -0x900000.0p-36 }, 323 { 0xa20000.0p-23, -0xb00000.0p-33 }, 324 { 0xa30000.0p-23, -0xa00000.0p-36 }, 325 { 0xa40000.0p-23, 0x800000.0p-33 }, 326 { 0xa50000.0p-23, -0xf80000.0p-35 }, 327 { 0xa60000.0p-23, 0x880000.0p-34 }, 328 { 0xa70000.0p-23, -0x900000.0p-33 }, 329 { 0xa80000.0p-23, -0x800000.0p-35 }, 330 { 0xa90000.0p-23, 0x900000.0p-34 }, 331 { 0xaa0000.0p-23, 0xa80000.0p-33 }, 332 { 0xab0000.0p-23, -0xac0000.0p-34 }, 333 { 0xac0000.0p-23, -0x800000.0p-37 }, 334 { 0xad0000.0p-23, 0xf80000.0p-35 }, 335 { 0xae0000.0p-23, 0xf80000.0p-34 }, 336 { 0xaf0000.0p-23, -0xac0000.0p-33 }, 337 { 0xb00000.0p-23, -0x800000.0p-33 }, 338 { 0xb10000.0p-23, -0xb80000.0p-34 }, 339 { 0xb20000.0p-23, -0x800000.0p-34 }, 340 { 0xb30000.0p-23, -0xb00000.0p-35 }, 341 { 0xb40000.0p-23, -0x800000.0p-35 }, 342 { 0xb50000.0p-23, -0xe00000.0p-36 }, 343 { 0xb60000.0p-23, -0x800000.0p-35 }, 344 { 0xb70000.0p-23, -0xb00000.0p-35 }, 345 { 0xb80000.0p-23, -0x800000.0p-34 }, 346 { 0xb90000.0p-23, -0xb80000.0p-34 }, 347 { 0xba0000.0p-23, -0x800000.0p-33 }, 348 { 0xbb0000.0p-23, -0xac0000.0p-33 }, 349 { 0xbc0000.0p-23, 0x980000.0p-33 }, 350 { 0xbd0000.0p-23, 0xbc0000.0p-34 }, 351 { 0xbe0000.0p-23, 0xe00000.0p-36 }, 352 { 0xbf0000.0p-23, -0xb80000.0p-35 }, 353 { 0xc00000.0p-23, -0x800000.0p-33 }, 354 { 0xc10000.0p-23, 0xa80000.0p-33 }, 355 { 0xc20000.0p-23, 0x900000.0p-34 }, 356 { 0xc30000.0p-23, -0x800000.0p-35 }, 357 { 0xc40000.0p-23, -0x900000.0p-33 }, 358 { 0xc50000.0p-23, 0x820000.0p-33 }, 359 { 0xc60000.0p-23, 0x800000.0p-38 }, 360 { 0xc70000.0p-23, -0x820000.0p-33 }, 361 { 0xc80000.0p-23, 0x800000.0p-33 }, 362 { 0xc90000.0p-23, -0xa00000.0p-36 }, 363 { 0xca0000.0p-23, -0xb00000.0p-33 }, 364 { 0xcb0000.0p-23, 0x840000.0p-34 }, 365 { 0xcc0000.0p-23, -0xd00000.0p-34 }, 366 { 0xcd0000.0p-23, 0x800000.0p-33 }, 367 { 0xce0000.0p-23, -0xe00000.0p-35 }, 368 { 0xcf0000.0p-23, 0xa60000.0p-33 }, 369 { 0xd00000.0p-23, -0x800000.0p-35 }, 370 { 0xd10000.0p-23, 0xb40000.0p-33 }, 371 { 0xd20000.0p-23, -0x800000.0p-35 }, 372 { 0xd30000.0p-23, 0xaa0000.0p-33 }, 373 { 0xd40000.0p-23, -0xe00000.0p-35 }, 374 { 0xd50000.0p-23, 0x880000.0p-33 }, 375 { 0xd60000.0p-23, -0xd00000.0p-34 }, 376 { 0xd70000.0p-23, 0x9c0000.0p-34 }, 377 { 0xd80000.0p-23, -0xb00000.0p-33 }, 378 { 0xd90000.0p-23, -0x800000.0p-38 }, 379 { 0xda0000.0p-23, 0xa40000.0p-33 }, 380 { 0xdb0000.0p-23, -0xdc0000.0p-34 }, 381 { 0xdc0000.0p-23, 0xc00000.0p-35 }, 382 { 0xdd0000.0p-23, 0xca0000.0p-33 }, 383 { 0xde0000.0p-23, -0xb80000.0p-34 }, 384 { 0xdf0000.0p-23, 0xd00000.0p-35 }, 385 { 0xe00000.0p-23, 0xc00000.0p-33 }, 386 { 0xe10000.0p-23, -0xf40000.0p-34 }, 387 { 0xe20000.0p-23, 0x800000.0p-37 }, 388 { 0xe30000.0p-23, 0x860000.0p-33 }, 389 { 0xe40000.0p-23, -0xc80000.0p-33 }, 390 { 0xe50000.0p-23, -0xa80000.0p-34 }, 391 { 0xe60000.0p-23, 0xe00000.0p-36 }, 392 { 0xe70000.0p-23, 0x880000.0p-33 }, 393 { 0xe80000.0p-23, -0xe00000.0p-33 }, 394 { 0xe90000.0p-23, -0xfc0000.0p-34 }, 395 { 0xea0000.0p-23, -0x800000.0p-35 }, 396 { 0xeb0000.0p-23, 0xe80000.0p-35 }, 397 { 0xec0000.0p-23, 0x900000.0p-33 }, 398 { 0xed0000.0p-23, 0xe20000.0p-33 }, 399 { 0xee0000.0p-23, -0xac0000.0p-33 }, 400 { 0xef0000.0p-23, -0xc80000.0p-34 }, 401 { 0xf00000.0p-23, -0x800000.0p-35 }, 402 { 0xf10000.0p-23, 0x800000.0p-35 }, 403 { 0xf20000.0p-23, 0xb80000.0p-34 }, 404 { 0xf30000.0p-23, 0x940000.0p-33 }, 405 { 0xf40000.0p-23, 0xc80000.0p-33 }, 406 { 0xf50000.0p-23, -0xf20000.0p-33 }, 407 { 0xf60000.0p-23, -0xc80000.0p-33 }, 408 { 0xf70000.0p-23, -0xa20000.0p-33 }, 409 { 0xf80000.0p-23, -0x800000.0p-33 }, 410 { 0xf90000.0p-23, -0xc40000.0p-34 }, 411 { 0xfa0000.0p-23, -0x900000.0p-34 }, 412 { 0xfb0000.0p-23, -0xc80000.0p-35 }, 413 { 0xfc0000.0p-23, -0x800000.0p-35 }, 414 { 0xfd0000.0p-23, -0x900000.0p-36 }, 415 { 0xfe0000.0p-23, -0x800000.0p-37 }, 416 { 0xff0000.0p-23, -0x800000.0p-39 }, 417 { 0x800000.0p-22, 0 }, 418 }; 419 #endif /* USE_UTAB */ 420 421 #ifdef STRUCT_RETURN 422 #define RETURN1(rp, v) do { \ 423 (rp)->hi = (v); \ 424 (rp)->lo_set = 0; \ 425 return; \ 426 } while (0) 427 428 #define RETURN2(rp, h, l) do { \ 429 (rp)->hi = (h); \ 430 (rp)->lo = (l); \ 431 (rp)->lo_set = 1; \ 432 return; \ 433 } while (0) 434 435 struct ld { 436 long double hi; 437 long double lo; 438 int lo_set; 439 }; 440 #else 441 #define RETURN1(rp, v) RETURNF(v) 442 #define RETURN2(rp, h, l) RETURNI((h) + (l)) 443 #endif 444 445 #ifdef STRUCT_RETURN 446 static inline __always_inline void 447 k_logl(long double x, struct ld *rp) 448 #else 449 long double 450 logl(long double x) 451 #endif 452 { 453 long double d, val_hi, val_lo; 454 double dd, dk; 455 uint64_t lx, llx; 456 int i, k; 457 uint16_t hx; 458 459 EXTRACT_LDBL128_WORDS(hx, lx, llx, x); 460 k = -16383; 461 #if 0 /* Hard to do efficiently. Don't do it until we support all modes. */ 462 if (x == 1) 463 RETURN1(rp, 0); /* log(1) = +0 in all rounding modes */ 464 #endif 465 if (hx == 0 || hx >= 0x8000) { /* zero, negative or subnormal? */ 466 if (((hx & 0x7fff) | lx | llx) == 0) 467 RETURN1(rp, -1 / zero); /* log(+-0) = -Inf */ 468 if (hx != 0) 469 /* log(neg or NaN) = qNaN: */ 470 RETURN1(rp, (x - x) / zero); 471 x *= 0x1.0p113; /* subnormal; scale up x */ 472 EXTRACT_LDBL128_WORDS(hx, lx, llx, x); 473 k = -16383 - 113; 474 } else if (hx >= 0x7fff) 475 RETURN1(rp, x + x); /* log(Inf or NaN) = Inf or qNaN */ 476 #ifndef STRUCT_RETURN 477 ENTERI(); 478 #endif 479 k += hx; 480 dk = k; 481 482 /* Scale x to be in [1, 2). */ 483 SET_LDBL_EXPSIGN(x, 0x3fff); 484 485 /* 0 <= i <= INTERVALS: */ 486 #define L2I (49 - LOG2_INTERVALS) 487 i = (lx + (1LL << (L2I - 2))) >> (L2I - 1); 488 489 /* 490 * -0.005280 < d < 0.004838. In particular, the infinite- 491 * precision |d| is <= 2**-7. Rounding of G(i) to 8 bits 492 * ensures that d is representable without extra precision for 493 * this bound on |d| (since when this calculation is expressed 494 * as x*G(i)-1, the multiplication needs as many extra bits as 495 * G(i) has and the subtraction cancels 8 bits). But for 496 * most i (107 cases out of 129), the infinite-precision |d| 497 * is <= 2**-8. G(i) is rounded to 9 bits for such i to give 498 * better accuracy (this works by improving the bound on |d|, 499 * which in turn allows rounding to 9 bits in more cases). 500 * This is only important when the original x is near 1 -- it 501 * lets us avoid using a special method to give the desired 502 * accuracy for such x. 503 */ 504 if (0) 505 d = x * G(i) - 1; 506 else { 507 #ifdef USE_UTAB 508 d = (x - H(i)) * G(i) + E(i); 509 #else 510 long double x_hi; 511 double x_lo; 512 513 /* 514 * Split x into x_hi + x_lo to calculate x*G(i)-1 exactly. 515 * G(i) has at most 9 bits, so the splitting point is not 516 * critical. 517 */ 518 INSERT_LDBL128_WORDS(x_hi, 0x3fff, lx, 519 llx & 0xffffffffff000000ULL); 520 x_lo = x - x_hi; 521 d = x_hi * G(i) - 1 + x_lo * G(i); 522 #endif 523 } 524 525 /* 526 * Our algorithm depends on exact cancellation of F_lo(i) and 527 * F_hi(i) with dk*ln_2_lo and dk*ln2_hi when k is -1 and i is 528 * at the end of the table. This and other technical complications 529 * make it difficult to avoid the double scaling in (dk*ln2) * 530 * log(base) for base != e without losing more accuracy and/or 531 * efficiency than is gained. 532 */ 533 /* 534 * Use double precision operations wherever possible, since 535 * long double operations are emulated and were very slow on 536 * the old sparc64 and unknown on the newer aarch64 and riscv 537 * machines. Also, don't try to improve parallelism by 538 * increasing the number of operations, since any parallelism 539 * on such machines is needed for the emulation. Horner's 540 * method is good for this, and is also good for accuracy. 541 * Horner's method doesn't handle the `lo' term well, either 542 * for efficiency or accuracy. However, for accuracy we 543 * evaluate d * d * P2 separately to take advantage of by P2 544 * being exact, and this gives a good place to sum the 'lo' 545 * term too. 546 */ 547 dd = (double)d; 548 val_lo = d * d * d * (P3 + 549 d * (P4 + d * (P5 + d * (P6 + d * (P7 + d * (P8 + 550 dd * (P9 + dd * (P10 + dd * (P11 + dd * (P12 + dd * (P13 + 551 dd * P14))))))))))) + (F_lo(i) + dk * ln2_lo) + d * d * P2; 552 val_hi = d; 553 #ifdef DEBUG 554 if (fetestexcept(FE_UNDERFLOW)) 555 breakpoint(); 556 #endif 557 558 _3sumF(val_hi, val_lo, F_hi(i) + dk * ln2_hi); 559 RETURN2(rp, val_hi, val_lo); 560 } 561 562 long double 563 log1pl(long double x) 564 { 565 long double d, d_hi, f_lo, val_hi, val_lo; 566 long double f_hi, twopminusk; 567 double d_lo, dd, dk; 568 uint64_t lx, llx; 569 int i, k; 570 int16_t ax, hx; 571 572 EXTRACT_LDBL128_WORDS(hx, lx, llx, x); 573 if (hx < 0x3fff) { /* x < 1, or x neg NaN */ 574 ax = hx & 0x7fff; 575 if (ax >= 0x3fff) { /* x <= -1, or x neg NaN */ 576 if (ax == 0x3fff && (lx | llx) == 0) 577 RETURNF(-1 / zero); /* log1p(-1) = -Inf */ 578 /* log1p(x < 1, or x NaN) = qNaN: */ 579 RETURNF((x - x) / (x - x)); 580 } 581 if (ax <= 0x3f8d) { /* |x| < 2**-113 */ 582 if ((int)x == 0) 583 RETURNF(x); /* x with inexact if x != 0 */ 584 } 585 f_hi = 1; 586 f_lo = x; 587 } else if (hx >= 0x7fff) { /* x +Inf or non-neg NaN */ 588 RETURNF(x + x); /* log1p(Inf or NaN) = Inf or qNaN */ 589 } else if (hx < 0x40e1) { /* 1 <= x < 2**226 */ 590 f_hi = x; 591 f_lo = 1; 592 } else { /* 2**226 <= x < +Inf */ 593 f_hi = x; 594 f_lo = 0; /* avoid underflow of the P3 term */ 595 } 596 ENTERI(); 597 x = f_hi + f_lo; 598 f_lo = (f_hi - x) + f_lo; 599 600 EXTRACT_LDBL128_WORDS(hx, lx, llx, x); 601 k = -16383; 602 603 k += hx; 604 dk = k; 605 606 SET_LDBL_EXPSIGN(x, 0x3fff); 607 twopminusk = 1; 608 SET_LDBL_EXPSIGN(twopminusk, 0x7ffe - (hx & 0x7fff)); 609 f_lo *= twopminusk; 610 611 i = (lx + (1LL << (L2I - 2))) >> (L2I - 1); 612 613 /* 614 * x*G(i)-1 (with a reduced x) can be represented exactly, as 615 * above, but now we need to evaluate the polynomial on d = 616 * (x+f_lo)*G(i)-1 and extra precision is needed for that. 617 * Since x+x_lo is a hi+lo decomposition and subtracting 1 618 * doesn't lose too many bits, an inexact calculation for 619 * f_lo*G(i) is good enough. 620 */ 621 if (0) 622 d_hi = x * G(i) - 1; 623 else { 624 #ifdef USE_UTAB 625 d_hi = (x - H(i)) * G(i) + E(i); 626 #else 627 long double x_hi; 628 double x_lo; 629 630 INSERT_LDBL128_WORDS(x_hi, 0x3fff, lx, 631 llx & 0xffffffffff000000ULL); 632 x_lo = x - x_hi; 633 d_hi = x_hi * G(i) - 1 + x_lo * G(i); 634 #endif 635 } 636 d_lo = f_lo * G(i); 637 638 /* 639 * This is _2sumF(d_hi, d_lo) inlined. The condition 640 * (d_hi == 0 || |d_hi| >= |d_lo|) for using _2sumF() is not 641 * always satisifed, so it is not clear that this works, but 642 * it works in practice. It works even if it gives a wrong 643 * normalized d_lo, since |d_lo| > |d_hi| implies that i is 644 * nonzero and d is tiny, so the F(i) term dominates d_lo. 645 * In float precision: 646 * (By exhaustive testing, the worst case is d_hi = 0x1.bp-25. 647 * And if d is only a little tinier than that, we would have 648 * another underflow problem for the P3 term; this is also ruled 649 * out by exhaustive testing.) 650 */ 651 d = d_hi + d_lo; 652 d_lo = d_hi - d + d_lo; 653 d_hi = d; 654 655 dd = (double)d; 656 val_lo = d * d * d * (P3 + 657 d * (P4 + d * (P5 + d * (P6 + d * (P7 + d * (P8 + 658 dd * (P9 + dd * (P10 + dd * (P11 + dd * (P12 + dd * (P13 + 659 dd * P14))))))))))) + (F_lo(i) + dk * ln2_lo + d_lo) + d * d * P2; 660 val_hi = d_hi; 661 #ifdef DEBUG 662 if (fetestexcept(FE_UNDERFLOW)) 663 breakpoint(); 664 #endif 665 666 _3sumF(val_hi, val_lo, F_hi(i) + dk * ln2_hi); 667 RETURNI(val_hi + val_lo); 668 } 669 670 #ifdef STRUCT_RETURN 671 672 long double 673 logl(long double x) 674 { 675 struct ld r; 676 677 ENTERI(); 678 k_logl(x, &r); 679 RETURNSPI(&r); 680 } 681 682 /* 683 * 29+113 bit decompositions. The bits are distributed so that the products 684 * of the hi terms are exact in double precision. The types are chosen so 685 * that the products of the hi terms are done in at least double precision, 686 * without any explicit conversions. More natural choices would require a 687 * slow long double precision multiplication. 688 */ 689 static const double 690 invln10_hi = 4.3429448176175356e-1, /* 0x1bcb7b15000000.0p-54 */ 691 invln2_hi = 1.4426950402557850e0; /* 0x17154765000000.0p-52 */ 692 static const long double 693 invln10_lo = 1.41498268538580090791605082294397000e-10L, /* 0x137287195355baaafad33dc323ee3.0p-145L */ 694 invln2_lo = 6.33178418956604368501892137426645911e-10L, /* 0x15c17f0bbbe87fed0691d3e88eb57.0p-143L */ 695 invln10_lo_plus_hi = invln10_lo + invln10_hi, 696 invln2_lo_plus_hi = invln2_lo + invln2_hi; 697 698 long double 699 log10l(long double x) 700 { 701 struct ld r; 702 long double hi, lo; 703 704 ENTERI(); 705 k_logl(x, &r); 706 if (!r.lo_set) 707 RETURNI(r.hi); 708 _2sumF(r.hi, r.lo); 709 hi = (float)r.hi; 710 lo = r.lo + (r.hi - hi); 711 RETURNI(invln10_hi * hi + (invln10_lo_plus_hi * lo + invln10_lo * hi)); 712 } 713 714 long double 715 log2l(long double x) 716 { 717 struct ld r; 718 long double hi, lo; 719 720 ENTERI(); 721 k_logl(x, &r); 722 if (!r.lo_set) 723 RETURNI(r.hi); 724 _2sumF(r.hi, r.lo); 725 hi = (float)r.hi; 726 lo = r.lo + (r.hi - hi); 727 RETURNI(invln2_hi * hi + (invln2_lo_plus_hi * lo + invln2_lo * hi)); 728 } 729 730 #endif /* STRUCT_RETURN */ 731