1 /*- 2 * Copyright (c) 2007-2013 Bruce D. Evans 3 * All rights reserved. 4 * 5 * Redistribution and use in source and binary forms, with or without 6 * modification, are permitted provided that the following conditions 7 * are met: 8 * 1. Redistributions of source code must retain the above copyright 9 * notice unmodified, this list of conditions, and the following 10 * disclaimer. 11 * 2. Redistributions in binary form must reproduce the above copyright 12 * notice, this list of conditions and the following disclaimer in the 13 * documentation and/or other materials provided with the distribution. 14 * 15 * THIS SOFTWARE IS PROVIDED BY THE AUTHOR ``AS IS'' AND ANY EXPRESS OR 16 * IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES 17 * OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. 18 * IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY DIRECT, INDIRECT, 19 * INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT 20 * NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, 21 * DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY 22 * THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT 23 * (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF 24 * THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. 25 */ 26 27 #include <sys/cdefs.h> 28 __FBSDID("$FreeBSD$"); 29 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 <assert.h> 79 #include <fenv.h> 80 #endif 81 82 #include "fpmath.h" 83 #include "math.h" 84 #ifndef NO_STRUCT_RETURN 85 #define STRUCT_RETURN 86 #endif 87 #include "math_private.h" 88 89 #if !defined(NO_UTAB) && !defined(NO_UTABL) 90 #define USE_UTAB 91 #endif 92 93 /* 94 * Domain [-0.005280, 0.004838], range ~[-1.1577e-37, 1.1582e-37]: 95 * |log(1 + d)/d - p(d)| < 2**-122.7 96 */ 97 static const long double 98 P2 = -0.5L, 99 P3 = 3.33333333333333333333333333333233795e-1L, /* 0x15555555555555555555555554d42.0p-114L */ 100 P4 = -2.49999999999999999999999999941139296e-1L, /* -0x1ffffffffffffffffffffffdab14e.0p-115L */ 101 P5 = 2.00000000000000000000000085468039943e-1L, /* 0x19999999999999999999a6d3567f4.0p-115L */ 102 P6 = -1.66666666666666666666696142372698408e-1L, /* -0x15555555555555555567267a58e13.0p-115L */ 103 P7 = 1.42857142857142857119522943477166120e-1L, /* 0x1249249249249248ed79a0ae434de.0p-115L */ 104 P8 = -1.24999999999999994863289015033581301e-1L; /* -0x1fffffffffffffa13e91765e46140.0p-116L */ 105 /* Double precision gives ~ 53 + log2(P9 * max(|d|)**8) ~= 120 bits. */ 106 static const double 107 P9 = 1.1111111111111401e-1, /* 0x1c71c71c71c7ed.0p-56 */ 108 P10 = -1.0000000000040135e-1, /* -0x199999999a0a92.0p-56 */ 109 P11 = 9.0909090728136258e-2, /* 0x1745d173962111.0p-56 */ 110 P12 = -8.3333318851855284e-2, /* -0x1555551722c7a3.0p-56 */ 111 P13 = 7.6928634666404178e-2, /* 0x13b1985204a4ae.0p-56 */ 112 P14 = -7.1626810078462499e-2; /* -0x12562276cdc5d0.0p-56 */ 113 114 static volatile const double zero = 0; 115 116 #define INTERVALS 128 117 #define LOG2_INTERVALS 7 118 #define TSIZE (INTERVALS + 1) 119 #define G(i) (T[(i)].G) 120 #define F_hi(i) (T[(i)].F_hi) 121 #define F_lo(i) (T[(i)].F_lo) 122 #define ln2_hi F_hi(TSIZE - 1) 123 #define ln2_lo F_lo(TSIZE - 1) 124 #define E(i) (U[(i)].E) 125 #define H(i) (U[(i)].H) 126 127 static const struct { 128 float G; /* 1/(1 + i/128) rounded to 8/9 bits */ 129 float F_hi; /* log(1 / G_i) rounded (see below) */ 130 /* The compiler will insert 8 bytes of padding here. */ 131 long double F_lo; /* next 113 bits for log(1 / G_i) */ 132 } T[TSIZE] = { 133 /* 134 * ln2_hi and each F_hi(i) are rounded to a number of bits that 135 * makes F_hi(i) + dk*ln2_hi exact for all i and all dk. 136 * 137 * The last entry (for X just below 2) is used to define ln2_hi 138 * and ln2_lo, to ensure that F_hi(i) and F_lo(i) cancel exactly 139 * with dk*ln2_hi and dk*ln2_lo, respectively, when dk = -1. 140 * This is needed for accuracy when x is just below 1. (To avoid 141 * special cases, such x are "reduced" strangely to X just below 142 * 2 and dk = -1, and then the exact cancellation is needed 143 * because any the error from any non-exactness would be too 144 * large). 145 * 146 * The relevant range of dk is [-16445, 16383]. The maximum number 147 * of bits in F_hi(i) that works is very dependent on i but has 148 * a minimum of 93. We only need about 12 bits in F_hi(i) for 149 * it to provide enough extra precision. 150 * 151 * We round F_hi(i) to 24 bits so that it can have type float, 152 * mainly to minimize the size of the table. Using all 24 bits 153 * in a float for it automatically satisfies the above constraints. 154 */ 155 0x800000.0p-23, 0, 0, 156 0xfe0000.0p-24, 0x8080ac.0p-30, -0x14ee431dae6674afa0c4bfe16e8fd.0p-144L, 157 0xfc0000.0p-24, 0x8102b3.0p-29, -0x1db29ee2d83717be918e1119642ab.0p-144L, 158 0xfa0000.0p-24, 0xc24929.0p-29, 0x1191957d173697cf302cc9476f561.0p-143L, 159 0xf80000.0p-24, 0x820aec.0p-28, 0x13ce8888e02e78eba9b1113bc1c18.0p-142L, 160 0xf60000.0p-24, 0xa33577.0p-28, -0x17a4382ce6eb7bfa509bec8da5f22.0p-142L, 161 0xf48000.0p-24, 0xbc42cb.0p-28, -0x172a21161a107674986dcdca6709c.0p-143L, 162 0xf30000.0p-24, 0xd57797.0p-28, -0x1e09de07cb958897a3ea46e84abb3.0p-142L, 163 0xf10000.0p-24, 0xf7518e.0p-28, 0x1ae1eec1b036c484993c549c4bf40.0p-151L, 164 0xef0000.0p-24, 0x8cb9df.0p-27, -0x1d7355325d560d9e9ab3d6ebab580.0p-141L, 165 0xed8000.0p-24, 0x999ec0.0p-27, -0x1f9f02d256d5037108f4ec21e48cd.0p-142L, 166 0xec0000.0p-24, 0xa6988b.0p-27, -0x16fc0a9d12c17a70f7a684c596b12.0p-143L, 167 0xea0000.0p-24, 0xb80698.0p-27, 0x15d581c1e8da99ded322fb08b8462.0p-141L, 168 0xe80000.0p-24, 0xc99af3.0p-27, -0x1535b3ba8f150ae09996d7bb4653e.0p-143L, 169 0xe70000.0p-24, 0xd273b2.0p-27, 0x163786f5251aefe0ded34c8318f52.0p-145L, 170 0xe50000.0p-24, 0xe442c0.0p-27, 0x1bc4b2368e32d56699c1799a244d4.0p-144L, 171 0xe38000.0p-24, 0xf1b83f.0p-27, 0x1c6090f684e6766abceccab1d7174.0p-141L, 172 0xe20000.0p-24, 0xff448a.0p-27, -0x1890aa69ac9f4215f93936b709efb.0p-142L, 173 0xe08000.0p-24, 0x8673f6.0p-26, 0x1b9985194b6affd511b534b72a28e.0p-140L, 174 0xdf0000.0p-24, 0x8d515c.0p-26, -0x1dc08d61c6ef1d9b2ef7e68680598.0p-143L, 175 0xdd8000.0p-24, 0x943a9e.0p-26, -0x1f72a2dac729b3f46662238a9425a.0p-142L, 176 0xdc0000.0p-24, 0x9b2fe6.0p-26, -0x1fd4dfd3a0afb9691aed4d5e3df94.0p-140L, 177 0xda8000.0p-24, 0xa2315d.0p-26, -0x11b26121629c46c186384993e1c93.0p-142L, 178 0xd90000.0p-24, 0xa93f2f.0p-26, 0x1286d633e8e5697dc6a402a56fce1.0p-141L, 179 0xd78000.0p-24, 0xb05988.0p-26, 0x16128eba9367707ebfa540e45350c.0p-144L, 180 0xd60000.0p-24, 0xb78094.0p-26, 0x16ead577390d31ef0f4c9d43f79b2.0p-140L, 181 0xd50000.0p-24, 0xbc4c6c.0p-26, 0x151131ccf7c7b75e7d900b521c48d.0p-141L, 182 0xd38000.0p-24, 0xc3890a.0p-26, -0x115e2cd714bd06508aeb00d2ae3e9.0p-140L, 183 0xd20000.0p-24, 0xcad2d7.0p-26, -0x1847f406ebd3af80485c2f409633c.0p-142L, 184 0xd10000.0p-24, 0xcfb620.0p-26, 0x1c2259904d686581799fbce0b5f19.0p-141L, 185 0xcf8000.0p-24, 0xd71653.0p-26, 0x1ece57a8d5ae54f550444ecf8b995.0p-140L, 186 0xce0000.0p-24, 0xde843a.0p-26, -0x1f109d4bc4595412b5d2517aaac13.0p-141L, 187 0xcd0000.0p-24, 0xe37fde.0p-26, 0x1bc03dc271a74d3a85b5b43c0e727.0p-141L, 188 0xcb8000.0p-24, 0xeb050c.0p-26, -0x1bf2badc0df841a71b79dd5645b46.0p-145L, 189 0xca0000.0p-24, 0xf29878.0p-26, -0x18efededd89fbe0bcfbe6d6db9f66.0p-147L, 190 0xc90000.0p-24, 0xf7ad6f.0p-26, 0x1373ff977baa6911c7bafcb4d84fb.0p-141L, 191 0xc80000.0p-24, 0xfcc8e3.0p-26, 0x196766f2fb328337cc050c6d83b22.0p-140L, 192 0xc68000.0p-24, 0x823f30.0p-25, 0x19bd076f7c434e5fcf1a212e2a91e.0p-139L, 193 0xc58000.0p-24, 0x84d52c.0p-25, -0x1a327257af0f465e5ecab5f2a6f81.0p-139L, 194 0xc40000.0p-24, 0x88bc74.0p-25, 0x113f23def19c5a0fe396f40f1dda9.0p-141L, 195 0xc30000.0p-24, 0x8b5ae6.0p-25, 0x1759f6e6b37de945a049a962e66c6.0p-139L, 196 0xc20000.0p-24, 0x8dfccb.0p-25, 0x1ad35ca6ed5147bdb6ddcaf59c425.0p-141L, 197 0xc10000.0p-24, 0x90a22b.0p-25, 0x1a1d71a87deba46bae9827221dc98.0p-139L, 198 0xbf8000.0p-24, 0x94a0d8.0p-25, -0x139e5210c2b730e28aba001a9b5e0.0p-140L, 199 0xbe8000.0p-24, 0x974f16.0p-25, -0x18f6ebcff3ed72e23e13431adc4a5.0p-141L, 200 0xbd8000.0p-24, 0x9a00f1.0p-25, -0x1aa268be39aab7148e8d80caa10b7.0p-139L, 201 0xbc8000.0p-24, 0x9cb672.0p-25, -0x14c8815839c5663663d15faed7771.0p-139L, 202 0xbb0000.0p-24, 0xa0cda1.0p-25, 0x1eaf46390dbb2438273918db7df5c.0p-141L, 203 0xba0000.0p-24, 0xa38c6e.0p-25, 0x138e20d831f698298adddd7f32686.0p-141L, 204 0xb90000.0p-24, 0xa64f05.0p-25, -0x1e8d3c41123615b147a5d47bc208f.0p-142L, 205 0xb80000.0p-24, 0xa91570.0p-25, 0x1ce28f5f3840b263acb4351104631.0p-140L, 206 0xb70000.0p-24, 0xabdfbb.0p-25, -0x186e5c0a42423457e22d8c650b355.0p-139L, 207 0xb60000.0p-24, 0xaeadef.0p-25, -0x14d41a0b2a08a465dc513b13f567d.0p-143L, 208 0xb50000.0p-24, 0xb18018.0p-25, 0x16755892770633947ffe651e7352f.0p-139L, 209 0xb40000.0p-24, 0xb45642.0p-25, -0x16395ebe59b15228bfe8798d10ff0.0p-142L, 210 0xb30000.0p-24, 0xb73077.0p-25, 0x1abc65c8595f088b61a335f5b688c.0p-140L, 211 0xb20000.0p-24, 0xba0ec4.0p-25, -0x1273089d3dad88e7d353e9967d548.0p-139L, 212 0xb10000.0p-24, 0xbcf133.0p-25, 0x10f9f67b1f4bbf45de06ecebfaf6d.0p-139L, 213 0xb00000.0p-24, 0xbfd7d2.0p-25, -0x109fab904864092b34edda19a831e.0p-140L, 214 0xaf0000.0p-24, 0xc2c2ac.0p-25, -0x1124680aa43333221d8a9b475a6ba.0p-139L, 215 0xae8000.0p-24, 0xc439b3.0p-25, -0x1f360cc4710fbfe24b633f4e8d84d.0p-140L, 216 0xad8000.0p-24, 0xc72afd.0p-25, -0x132d91f21d89c89c45003fc5d7807.0p-140L, 217 0xac8000.0p-24, 0xca20a2.0p-25, -0x16bf9b4d1f8da8002f2449e174504.0p-139L, 218 0xab8000.0p-24, 0xcd1aae.0p-25, 0x19deb5ce6a6a8717d5626e16acc7d.0p-141L, 219 0xaa8000.0p-24, 0xd0192f.0p-25, 0x1a29fb48f7d3ca87dabf351aa41f4.0p-139L, 220 0xaa0000.0p-24, 0xd19a20.0p-25, 0x1127d3c6457f9d79f51dcc73014c9.0p-141L, 221 0xa90000.0p-24, 0xd49f6a.0p-25, -0x1ba930e486a0ac42d1bf9199188e7.0p-141L, 222 0xa80000.0p-24, 0xd7a94b.0p-25, -0x1b6e645f31549dd1160bcc45c7e2c.0p-139L, 223 0xa70000.0p-24, 0xdab7d0.0p-25, 0x1118a425494b610665377f15625b6.0p-140L, 224 0xa68000.0p-24, 0xdc40d5.0p-25, 0x1966f24d29d3a2d1b2176010478be.0p-140L, 225 0xa58000.0p-24, 0xdf566d.0p-25, -0x1d8e52eb2248f0c95dd83626d7333.0p-142L, 226 0xa48000.0p-24, 0xe270ce.0p-25, -0x1ee370f96e6b67ccb006a5b9890ea.0p-140L, 227 0xa40000.0p-24, 0xe3ffce.0p-25, 0x1d155324911f56db28da4d629d00a.0p-140L, 228 0xa30000.0p-24, 0xe72179.0p-25, -0x1fe6e2f2f867d8f4d60c713346641.0p-140L, 229 0xa20000.0p-24, 0xea4812.0p-25, 0x1b7be9add7f4d3b3d406b6cbf3ce5.0p-140L, 230 0xa18000.0p-24, 0xebdd3d.0p-25, 0x1b3cfb3f7511dd73692609040ccc2.0p-139L, 231 0xa08000.0p-24, 0xef0b5b.0p-25, -0x1220de1f7301901b8ad85c25afd09.0p-139L, 232 0xa00000.0p-24, 0xf0a451.0p-25, -0x176364c9ac81cc8a4dfb804de6867.0p-140L, 233 0x9f0000.0p-24, 0xf3da16.0p-25, 0x1eed6b9aafac8d42f78d3e65d3727.0p-141L, 234 0x9e8000.0p-24, 0xf576e9.0p-25, 0x1d593218675af269647b783d88999.0p-139L, 235 0x9d8000.0p-24, 0xf8b47c.0p-25, -0x13e8eb7da053e063714615f7cc91d.0p-144L, 236 0x9d0000.0p-24, 0xfa553f.0p-25, 0x1c063259bcade02951686d5373aec.0p-139L, 237 0x9c0000.0p-24, 0xfd9ac5.0p-25, 0x1ef491085fa3c1649349630531502.0p-139L, 238 0x9b8000.0p-24, 0xff3f8c.0p-25, 0x1d607a7c2b8c5320619fb9433d841.0p-139L, 239 0x9a8000.0p-24, 0x814697.0p-24, -0x12ad3817004f3f0bdff99f932b273.0p-138L, 240 0x9a0000.0p-24, 0x821b06.0p-24, -0x189fc53117f9e54e78103a2bc1767.0p-141L, 241 0x990000.0p-24, 0x83c5f8.0p-24, 0x14cf15a048907b7d7f47ddb45c5a3.0p-139L, 242 0x988000.0p-24, 0x849c7d.0p-24, 0x1cbb1d35fb82873b04a9af1dd692c.0p-138L, 243 0x978000.0p-24, 0x864ba6.0p-24, 0x1128639b814f9b9770d8cb6573540.0p-138L, 244 0x970000.0p-24, 0x87244c.0p-24, 0x184733853300f002e836dfd47bd41.0p-139L, 245 0x968000.0p-24, 0x87fdaa.0p-24, 0x109d23aef77dd5cd7cc94306fb3ff.0p-140L, 246 0x958000.0p-24, 0x89b293.0p-24, -0x1a81ef367a59de2b41eeebd550702.0p-138L, 247 0x950000.0p-24, 0x8a8e20.0p-24, -0x121ad3dbb2f45275c917a30df4ac9.0p-138L, 248 0x948000.0p-24, 0x8b6a6a.0p-24, -0x1cfb981628af71a89df4e6df2e93b.0p-139L, 249 0x938000.0p-24, 0x8d253a.0p-24, -0x1d21730ea76cfdec367828734cae5.0p-139L, 250 0x930000.0p-24, 0x8e03c2.0p-24, 0x135cc00e566f76b87333891e0dec4.0p-138L, 251 0x928000.0p-24, 0x8ee30d.0p-24, -0x10fcb5df257a263e3bf446c6e3f69.0p-140L, 252 0x918000.0p-24, 0x90a3ee.0p-24, -0x16e171b15433d723a4c7380a448d8.0p-139L, 253 0x910000.0p-24, 0x918587.0p-24, -0x1d050da07f3236f330972da2a7a87.0p-139L, 254 0x908000.0p-24, 0x9267e7.0p-24, 0x1be03669a5268d21148c6002becd3.0p-139L, 255 0x8f8000.0p-24, 0x942f04.0p-24, 0x10b28e0e26c336af90e00533323ba.0p-139L, 256 0x8f0000.0p-24, 0x9513c3.0p-24, 0x1a1d820da57cf2f105a89060046aa.0p-138L, 257 0x8e8000.0p-24, 0x95f950.0p-24, -0x19ef8f13ae3cf162409d8ea99d4c0.0p-139L, 258 0x8e0000.0p-24, 0x96dfab.0p-24, -0x109e417a6e507b9dc10dac743ad7a.0p-138L, 259 0x8d0000.0p-24, 0x98aed2.0p-24, 0x10d01a2c5b0e97c4990b23d9ac1f5.0p-139L, 260 0x8c8000.0p-24, 0x9997a2.0p-24, -0x1d6a50d4b61ea74540bdd2aa99a42.0p-138L, 261 0x8c0000.0p-24, 0x9a8145.0p-24, 0x1b3b190b83f9527e6aba8f2d783c1.0p-138L, 262 0x8b8000.0p-24, 0x9b6bbf.0p-24, 0x13a69fad7e7abe7ba81c664c107e0.0p-138L, 263 0x8b0000.0p-24, 0x9c5711.0p-24, -0x11cd12316f576aad348ae79867223.0p-138L, 264 0x8a8000.0p-24, 0x9d433b.0p-24, 0x1c95c444b807a246726b304ccae56.0p-139L, 265 0x898000.0p-24, 0x9f1e22.0p-24, -0x1b9c224ea698c2f9b47466d6123fe.0p-139L, 266 0x890000.0p-24, 0xa00ce1.0p-24, 0x125ca93186cf0f38b4619a2483399.0p-141L, 267 0x888000.0p-24, 0xa0fc80.0p-24, -0x1ee38a7bc228b3597043be78eaf49.0p-139L, 268 0x880000.0p-24, 0xa1ed00.0p-24, -0x1a0db876613d204147dc69a07a649.0p-138L, 269 0x878000.0p-24, 0xa2de62.0p-24, 0x193224e8516c008d3602a7b41c6e8.0p-139L, 270 0x870000.0p-24, 0xa3d0a9.0p-24, 0x1fa28b4d2541aca7d5844606b2421.0p-139L, 271 0x868000.0p-24, 0xa4c3d6.0p-24, 0x1c1b5760fb4571acbcfb03f16daf4.0p-138L, 272 0x858000.0p-24, 0xa6acea.0p-24, 0x1fed5d0f65949c0a345ad743ae1ae.0p-140L, 273 0x850000.0p-24, 0xa7a2d4.0p-24, 0x1ad270c9d749362382a7688479e24.0p-140L, 274 0x848000.0p-24, 0xa899ab.0p-24, 0x199ff15ce532661ea9643a3a2d378.0p-139L, 275 0x840000.0p-24, 0xa99171.0p-24, 0x1a19e15ccc45d257530a682b80490.0p-139L, 276 0x838000.0p-24, 0xaa8a28.0p-24, -0x121a14ec532b35ba3e1f868fd0b5e.0p-140L, 277 0x830000.0p-24, 0xab83d1.0p-24, 0x1aee319980bff3303dd481779df69.0p-139L, 278 0x828000.0p-24, 0xac7e6f.0p-24, -0x18ffd9e3900345a85d2d86161742e.0p-140L, 279 0x820000.0p-24, 0xad7a03.0p-24, -0x1e4db102ce29f79b026b64b42caa1.0p-140L, 280 0x818000.0p-24, 0xae768f.0p-24, 0x17c35c55a04a82ab19f77652d977a.0p-141L, 281 0x810000.0p-24, 0xaf7415.0p-24, 0x1448324047019b48d7b98c1cf7234.0p-138L, 282 0x808000.0p-24, 0xb07298.0p-24, -0x1750ee3915a197e9c7359dd94152f.0p-138L, 283 0x800000.0p-24, 0xb17218.0p-24, -0x105c610ca86c3898cff81a12a17e2.0p-141L, 284 }; 285 286 #ifdef USE_UTAB 287 static const struct { 288 float H; /* 1 + i/INTERVALS (exact) */ 289 float E; /* H(i) * G(i) - 1 (exact) */ 290 } U[TSIZE] = { 291 0x800000.0p-23, 0, 292 0x810000.0p-23, -0x800000.0p-37, 293 0x820000.0p-23, -0x800000.0p-35, 294 0x830000.0p-23, -0x900000.0p-34, 295 0x840000.0p-23, -0x800000.0p-33, 296 0x850000.0p-23, -0xc80000.0p-33, 297 0x860000.0p-23, -0xa00000.0p-36, 298 0x870000.0p-23, 0x940000.0p-33, 299 0x880000.0p-23, 0x800000.0p-35, 300 0x890000.0p-23, -0xc80000.0p-34, 301 0x8a0000.0p-23, 0xe00000.0p-36, 302 0x8b0000.0p-23, 0x900000.0p-33, 303 0x8c0000.0p-23, -0x800000.0p-35, 304 0x8d0000.0p-23, -0xe00000.0p-33, 305 0x8e0000.0p-23, 0x880000.0p-33, 306 0x8f0000.0p-23, -0xa80000.0p-34, 307 0x900000.0p-23, -0x800000.0p-35, 308 0x910000.0p-23, 0x800000.0p-37, 309 0x920000.0p-23, 0x900000.0p-35, 310 0x930000.0p-23, 0xd00000.0p-35, 311 0x940000.0p-23, 0xe00000.0p-35, 312 0x950000.0p-23, 0xc00000.0p-35, 313 0x960000.0p-23, 0xe00000.0p-36, 314 0x970000.0p-23, -0x800000.0p-38, 315 0x980000.0p-23, -0xc00000.0p-35, 316 0x990000.0p-23, -0xd00000.0p-34, 317 0x9a0000.0p-23, 0x880000.0p-33, 318 0x9b0000.0p-23, 0xe80000.0p-35, 319 0x9c0000.0p-23, -0x800000.0p-35, 320 0x9d0000.0p-23, 0xb40000.0p-33, 321 0x9e0000.0p-23, 0x880000.0p-34, 322 0x9f0000.0p-23, -0xe00000.0p-35, 323 0xa00000.0p-23, 0x800000.0p-33, 324 0xa10000.0p-23, -0x900000.0p-36, 325 0xa20000.0p-23, -0xb00000.0p-33, 326 0xa30000.0p-23, -0xa00000.0p-36, 327 0xa40000.0p-23, 0x800000.0p-33, 328 0xa50000.0p-23, -0xf80000.0p-35, 329 0xa60000.0p-23, 0x880000.0p-34, 330 0xa70000.0p-23, -0x900000.0p-33, 331 0xa80000.0p-23, -0x800000.0p-35, 332 0xa90000.0p-23, 0x900000.0p-34, 333 0xaa0000.0p-23, 0xa80000.0p-33, 334 0xab0000.0p-23, -0xac0000.0p-34, 335 0xac0000.0p-23, -0x800000.0p-37, 336 0xad0000.0p-23, 0xf80000.0p-35, 337 0xae0000.0p-23, 0xf80000.0p-34, 338 0xaf0000.0p-23, -0xac0000.0p-33, 339 0xb00000.0p-23, -0x800000.0p-33, 340 0xb10000.0p-23, -0xb80000.0p-34, 341 0xb20000.0p-23, -0x800000.0p-34, 342 0xb30000.0p-23, -0xb00000.0p-35, 343 0xb40000.0p-23, -0x800000.0p-35, 344 0xb50000.0p-23, -0xe00000.0p-36, 345 0xb60000.0p-23, -0x800000.0p-35, 346 0xb70000.0p-23, -0xb00000.0p-35, 347 0xb80000.0p-23, -0x800000.0p-34, 348 0xb90000.0p-23, -0xb80000.0p-34, 349 0xba0000.0p-23, -0x800000.0p-33, 350 0xbb0000.0p-23, -0xac0000.0p-33, 351 0xbc0000.0p-23, 0x980000.0p-33, 352 0xbd0000.0p-23, 0xbc0000.0p-34, 353 0xbe0000.0p-23, 0xe00000.0p-36, 354 0xbf0000.0p-23, -0xb80000.0p-35, 355 0xc00000.0p-23, -0x800000.0p-33, 356 0xc10000.0p-23, 0xa80000.0p-33, 357 0xc20000.0p-23, 0x900000.0p-34, 358 0xc30000.0p-23, -0x800000.0p-35, 359 0xc40000.0p-23, -0x900000.0p-33, 360 0xc50000.0p-23, 0x820000.0p-33, 361 0xc60000.0p-23, 0x800000.0p-38, 362 0xc70000.0p-23, -0x820000.0p-33, 363 0xc80000.0p-23, 0x800000.0p-33, 364 0xc90000.0p-23, -0xa00000.0p-36, 365 0xca0000.0p-23, -0xb00000.0p-33, 366 0xcb0000.0p-23, 0x840000.0p-34, 367 0xcc0000.0p-23, -0xd00000.0p-34, 368 0xcd0000.0p-23, 0x800000.0p-33, 369 0xce0000.0p-23, -0xe00000.0p-35, 370 0xcf0000.0p-23, 0xa60000.0p-33, 371 0xd00000.0p-23, -0x800000.0p-35, 372 0xd10000.0p-23, 0xb40000.0p-33, 373 0xd20000.0p-23, -0x800000.0p-35, 374 0xd30000.0p-23, 0xaa0000.0p-33, 375 0xd40000.0p-23, -0xe00000.0p-35, 376 0xd50000.0p-23, 0x880000.0p-33, 377 0xd60000.0p-23, -0xd00000.0p-34, 378 0xd70000.0p-23, 0x9c0000.0p-34, 379 0xd80000.0p-23, -0xb00000.0p-33, 380 0xd90000.0p-23, -0x800000.0p-38, 381 0xda0000.0p-23, 0xa40000.0p-33, 382 0xdb0000.0p-23, -0xdc0000.0p-34, 383 0xdc0000.0p-23, 0xc00000.0p-35, 384 0xdd0000.0p-23, 0xca0000.0p-33, 385 0xde0000.0p-23, -0xb80000.0p-34, 386 0xdf0000.0p-23, 0xd00000.0p-35, 387 0xe00000.0p-23, 0xc00000.0p-33, 388 0xe10000.0p-23, -0xf40000.0p-34, 389 0xe20000.0p-23, 0x800000.0p-37, 390 0xe30000.0p-23, 0x860000.0p-33, 391 0xe40000.0p-23, -0xc80000.0p-33, 392 0xe50000.0p-23, -0xa80000.0p-34, 393 0xe60000.0p-23, 0xe00000.0p-36, 394 0xe70000.0p-23, 0x880000.0p-33, 395 0xe80000.0p-23, -0xe00000.0p-33, 396 0xe90000.0p-23, -0xfc0000.0p-34, 397 0xea0000.0p-23, -0x800000.0p-35, 398 0xeb0000.0p-23, 0xe80000.0p-35, 399 0xec0000.0p-23, 0x900000.0p-33, 400 0xed0000.0p-23, 0xe20000.0p-33, 401 0xee0000.0p-23, -0xac0000.0p-33, 402 0xef0000.0p-23, -0xc80000.0p-34, 403 0xf00000.0p-23, -0x800000.0p-35, 404 0xf10000.0p-23, 0x800000.0p-35, 405 0xf20000.0p-23, 0xb80000.0p-34, 406 0xf30000.0p-23, 0x940000.0p-33, 407 0xf40000.0p-23, 0xc80000.0p-33, 408 0xf50000.0p-23, -0xf20000.0p-33, 409 0xf60000.0p-23, -0xc80000.0p-33, 410 0xf70000.0p-23, -0xa20000.0p-33, 411 0xf80000.0p-23, -0x800000.0p-33, 412 0xf90000.0p-23, -0xc40000.0p-34, 413 0xfa0000.0p-23, -0x900000.0p-34, 414 0xfb0000.0p-23, -0xc80000.0p-35, 415 0xfc0000.0p-23, -0x800000.0p-35, 416 0xfd0000.0p-23, -0x900000.0p-36, 417 0xfe0000.0p-23, -0x800000.0p-37, 418 0xff0000.0p-23, -0x800000.0p-39, 419 0x800000.0p-22, 0, 420 }; 421 #endif /* USE_UTAB */ 422 423 #ifdef STRUCT_RETURN 424 #define RETURN1(rp, v) do { \ 425 (rp)->hi = (v); \ 426 (rp)->lo_set = 0; \ 427 return; \ 428 } while (0) 429 430 #define RETURN2(rp, h, l) do { \ 431 (rp)->hi = (h); \ 432 (rp)->lo = (l); \ 433 (rp)->lo_set = 1; \ 434 return; \ 435 } while (0) 436 437 struct ld { 438 long double hi; 439 long double lo; 440 int lo_set; 441 }; 442 #else 443 #define RETURN1(rp, v) RETURNF(v) 444 #define RETURN2(rp, h, l) RETURNI((h) + (l)) 445 #endif 446 447 #ifdef STRUCT_RETURN 448 static inline __always_inline void 449 k_logl(long double x, struct ld *rp) 450 #else 451 long double 452 logl(long double x) 453 #endif 454 { 455 long double d, val_hi, val_lo; 456 double dd, dk; 457 uint64_t lx, llx; 458 int i, k; 459 uint16_t hx; 460 461 EXTRACT_LDBL128_WORDS(hx, lx, llx, x); 462 k = -16383; 463 #if 0 /* Hard to do efficiently. Don't do it until we support all modes. */ 464 if (x == 1) 465 RETURN1(rp, 0); /* log(1) = +0 in all rounding modes */ 466 #endif 467 if (hx == 0 || hx >= 0x8000) { /* zero, negative or subnormal? */ 468 if (((hx & 0x7fff) | lx | llx) == 0) 469 RETURN1(rp, -1 / zero); /* log(+-0) = -Inf */ 470 if (hx != 0) 471 /* log(neg or NaN) = qNaN: */ 472 RETURN1(rp, (x - x) / zero); 473 x *= 0x1.0p113; /* subnormal; scale up x */ 474 EXTRACT_LDBL128_WORDS(hx, lx, llx, x); 475 k = -16383 - 113; 476 } else if (hx >= 0x7fff) 477 RETURN1(rp, x + x); /* log(Inf or NaN) = Inf or qNaN */ 478 #ifndef STRUCT_RETURN 479 ENTERI(); 480 #endif 481 k += hx; 482 dk = k; 483 484 /* Scale x to be in [1, 2). */ 485 SET_LDBL_EXPSIGN(x, 0x3fff); 486 487 /* 0 <= i <= INTERVALS: */ 488 #define L2I (49 - LOG2_INTERVALS) 489 i = (lx + (1LL << (L2I - 2))) >> (L2I - 1); 490 491 /* 492 * -0.005280 < d < 0.004838. In particular, the infinite- 493 * precision |d| is <= 2**-7. Rounding of G(i) to 8 bits 494 * ensures that d is representable without extra precision for 495 * this bound on |d| (since when this calculation is expressed 496 * as x*G(i)-1, the multiplication needs as many extra bits as 497 * G(i) has and the subtraction cancels 8 bits). But for 498 * most i (107 cases out of 129), the infinite-precision |d| 499 * is <= 2**-8. G(i) is rounded to 9 bits for such i to give 500 * better accuracy (this works by improving the bound on |d|, 501 * which in turn allows rounding to 9 bits in more cases). 502 * This is only important when the original x is near 1 -- it 503 * lets us avoid using a special method to give the desired 504 * accuracy for such x. 505 */ 506 if (0) 507 d = x * G(i) - 1; 508 else { 509 #ifdef USE_UTAB 510 d = (x - H(i)) * G(i) + E(i); 511 #else 512 long double x_hi; 513 double x_lo; 514 515 /* 516 * Split x into x_hi + x_lo to calculate x*G(i)-1 exactly. 517 * G(i) has at most 9 bits, so the splitting point is not 518 * critical. 519 */ 520 INSERT_LDBL128_WORDS(x_hi, 0x3fff, lx, 521 llx & 0xffffffffff000000ULL); 522 x_lo = x - x_hi; 523 d = x_hi * G(i) - 1 + x_lo * G(i); 524 #endif 525 } 526 527 /* 528 * Our algorithm depends on exact cancellation of F_lo(i) and 529 * F_hi(i) with dk*ln_2_lo and dk*ln2_hi when k is -1 and i is 530 * at the end of the table. This and other technical complications 531 * make it difficult to avoid the double scaling in (dk*ln2) * 532 * log(base) for base != e without losing more accuracy and/or 533 * efficiency than is gained. 534 */ 535 /* 536 * Use double precision operations wherever possible, since long 537 * double operations are emulated and are very slow on the only 538 * known machines that support ld128 (sparc64). Also, don't try 539 * to improve parallelism by increasing the number of operations, 540 * since any parallelism on such machines is needed for the 541 * emulation. Horner's method is good for this, and is also good 542 * for accuracy. Horner's method doesn't handle the `lo' term 543 * well, either for efficiency or accuracy. However, for accuracy 544 * we evaluate d * d * P2 separately to take advantage of 545 * by P2 being exact, and this gives a good place to sum the 'lo' 546 * term too. 547 */ 548 dd = (double)d; 549 val_lo = d * d * d * (P3 + 550 d * (P4 + d * (P5 + d * (P6 + d * (P7 + d * (P8 + 551 dd * (P9 + dd * (P10 + dd * (P11 + dd * (P12 + dd * (P13 + 552 dd * P14))))))))))) + (F_lo(i) + dk * ln2_lo) + d * d * P2; 553 val_hi = d; 554 #ifdef DEBUG 555 if (fetestexcept(FE_UNDERFLOW)) 556 breakpoint(); 557 #endif 558 559 _3sumF(val_hi, val_lo, F_hi(i) + dk * ln2_hi); 560 RETURN2(rp, val_hi, val_lo); 561 } 562 563 long double 564 log1pl(long double x) 565 { 566 long double d, d_hi, f_lo, val_hi, val_lo; 567 long double f_hi, twopminusk; 568 double d_lo, dd, dk; 569 uint64_t lx, llx; 570 int i, k; 571 int16_t ax, hx; 572 573 DOPRINT_START(&x); 574 EXTRACT_LDBL128_WORDS(hx, lx, llx, x); 575 if (hx < 0x3fff) { /* x < 1, or x neg NaN */ 576 ax = hx & 0x7fff; 577 if (ax >= 0x3fff) { /* x <= -1, or x neg NaN */ 578 if (ax == 0x3fff && (lx | llx) == 0) 579 RETURNP(-1 / zero); /* log1p(-1) = -Inf */ 580 /* log1p(x < 1, or x NaN) = qNaN: */ 581 RETURNP((x - x) / (x - x)); 582 } 583 if (ax <= 0x3f8d) { /* |x| < 2**-113 */ 584 if ((int)x == 0) 585 RETURNP(x); /* x with inexact if x != 0 */ 586 } 587 f_hi = 1; 588 f_lo = x; 589 } else if (hx >= 0x7fff) { /* x +Inf or non-neg NaN */ 590 RETURNP(x + x); /* log1p(Inf or NaN) = Inf or qNaN */ 591 } else if (hx < 0x40e1) { /* 1 <= x < 2**226 */ 592 f_hi = x; 593 f_lo = 1; 594 } else { /* 2**226 <= x < +Inf */ 595 f_hi = x; 596 f_lo = 0; /* avoid underflow of the P3 term */ 597 } 598 ENTERI(); 599 x = f_hi + f_lo; 600 f_lo = (f_hi - x) + f_lo; 601 602 EXTRACT_LDBL128_WORDS(hx, lx, llx, x); 603 k = -16383; 604 605 k += hx; 606 dk = k; 607 608 SET_LDBL_EXPSIGN(x, 0x3fff); 609 twopminusk = 1; 610 SET_LDBL_EXPSIGN(twopminusk, 0x7ffe - (hx & 0x7fff)); 611 f_lo *= twopminusk; 612 613 i = (lx + (1LL << (L2I - 2))) >> (L2I - 1); 614 615 /* 616 * x*G(i)-1 (with a reduced x) can be represented exactly, as 617 * above, but now we need to evaluate the polynomial on d = 618 * (x+f_lo)*G(i)-1 and extra precision is needed for that. 619 * Since x+x_lo is a hi+lo decomposition and subtracting 1 620 * doesn't lose too many bits, an inexact calculation for 621 * f_lo*G(i) is good enough. 622 */ 623 if (0) 624 d_hi = x * G(i) - 1; 625 else { 626 #ifdef USE_UTAB 627 d_hi = (x - H(i)) * G(i) + E(i); 628 #else 629 long double x_hi; 630 double x_lo; 631 632 INSERT_LDBL128_WORDS(x_hi, 0x3fff, lx, 633 llx & 0xffffffffff000000ULL); 634 x_lo = x - x_hi; 635 d_hi = x_hi * G(i) - 1 + x_lo * G(i); 636 #endif 637 } 638 d_lo = f_lo * G(i); 639 640 /* 641 * This is _2sumF(d_hi, d_lo) inlined. The condition 642 * (d_hi == 0 || |d_hi| >= |d_lo|) for using _2sumF() is not 643 * always satisifed, so it is not clear that this works, but 644 * it works in practice. It works even if it gives a wrong 645 * normalized d_lo, since |d_lo| > |d_hi| implies that i is 646 * nonzero and d is tiny, so the F(i) term dominates d_lo. 647 * In float precision: 648 * (By exhaustive testing, the worst case is d_hi = 0x1.bp-25. 649 * And if d is only a little tinier than that, we would have 650 * another underflow problem for the P3 term; this is also ruled 651 * out by exhaustive testing.) 652 */ 653 d = d_hi + d_lo; 654 d_lo = d_hi - d + d_lo; 655 d_hi = d; 656 657 dd = (double)d; 658 val_lo = d * d * d * (P3 + 659 d * (P4 + d * (P5 + d * (P6 + d * (P7 + d * (P8 + 660 dd * (P9 + dd * (P10 + dd * (P11 + dd * (P12 + dd * (P13 + 661 dd * P14))))))))))) + (F_lo(i) + dk * ln2_lo + d_lo) + d * d * P2; 662 val_hi = d_hi; 663 #ifdef DEBUG 664 if (fetestexcept(FE_UNDERFLOW)) 665 breakpoint(); 666 #endif 667 668 _3sumF(val_hi, val_lo, F_hi(i) + dk * ln2_hi); 669 RETURN2PI(val_hi, val_lo); 670 } 671 672 #ifdef STRUCT_RETURN 673 674 long double 675 logl(long double x) 676 { 677 struct ld r; 678 679 ENTERI(); 680 DOPRINT_START(&x); 681 k_logl(x, &r); 682 RETURNSPI(&r); 683 } 684 685 /* 686 * 29+113 bit decompositions. The bits are distributed so that the products 687 * of the hi terms are exact in double precision. The types are chosen so 688 * that the products of the hi terms are done in at least double precision, 689 * without any explicit conversions. More natural choices would require a 690 * slow long double precision multiplication. 691 */ 692 static const double 693 invln10_hi = 4.3429448176175356e-1, /* 0x1bcb7b15000000.0p-54 */ 694 invln2_hi = 1.4426950402557850e0; /* 0x17154765000000.0p-52 */ 695 static const long double 696 invln10_lo = 1.41498268538580090791605082294397000e-10L, /* 0x137287195355baaafad33dc323ee3.0p-145L */ 697 invln2_lo = 6.33178418956604368501892137426645911e-10L; /* 0x15c17f0bbbe87fed0691d3e88eb57.0p-143L */ 698 699 long double 700 log10l(long double x) 701 { 702 struct ld r; 703 long double lo; 704 float hi; 705 706 ENTERI(); 707 DOPRINT_START(&x); 708 k_logl(x, &r); 709 if (!r.lo_set) 710 RETURNPI(r.hi); 711 _2sumF(r.hi, r.lo); 712 hi = r.hi; 713 lo = r.lo + (r.hi - hi); 714 RETURN2PI(invln10_hi * hi, 715 (invln10_lo + invln10_hi) * lo + invln10_lo * hi); 716 } 717 718 long double 719 log2l(long double x) 720 { 721 struct ld r; 722 long double lo; 723 float hi; 724 725 ENTERI(); 726 DOPRINT_START(&x); 727 k_logl(x, &r); 728 if (!r.lo_set) 729 RETURNPI(r.hi); 730 _2sumF(r.hi, r.lo); 731 hi = r.hi; 732 lo = r.lo + (r.hi - hi); 733 RETURN2PI(invln2_hi * hi, 734 (invln2_lo + invln2_hi) * lo + invln2_lo * hi); 735 } 736 737 #endif /* STRUCT_RETURN */ 738
