1 /*- 2 * SPDX-License-Identifier: BSD-2-Clause-FreeBSD 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 __FBSDID("$FreeBSD$"); 31 32 /** 33 * Implementation of the natural logarithm of x for Intel 80-bit format. 34 * 35 * First decompose x into its base 2 representation: 36 * 37 * log(x) = log(X * 2**k), where X is in [1, 2) 38 * = log(X) + k * log(2). 39 * 40 * Let X = X_i + e, where X_i is the center of one of the intervals 41 * [-1.0/256, 1.0/256), [1.0/256, 3.0/256), .... [2.0-1.0/256, 2.0+1.0/256) 42 * and X is in this interval. Then 43 * 44 * log(X) = log(X_i + e) 45 * = log(X_i * (1 + e / X_i)) 46 * = log(X_i) + log(1 + e / X_i). 47 * 48 * The values log(X_i) are tabulated below. Let d = e / X_i and use 49 * 50 * log(1 + d) = p(d) 51 * 52 * where p(d) = d - 0.5*d*d + ... is a special minimax polynomial of 53 * suitably high degree. 54 * 55 * To get sufficiently small roundoff errors, k * log(2), log(X_i), and 56 * sometimes (if |k| is not large) the first term in p(d) must be evaluated 57 * and added up in extra precision. Extra precision is not needed for the 58 * rest of p(d). In the worst case when k = 0 and log(X_i) is 0, the final 59 * error is controlled mainly by the error in the second term in p(d). The 60 * error in this term itself is at most 0.5 ulps from the d*d operation in 61 * it. The error in this term relative to the first term is thus at most 62 * 0.5 * |-0.5| * |d| < 1.0/1024 ulps. We aim for an accumulated error of 63 * at most twice this at the point of the final rounding step. Thus the 64 * final error should be at most 0.5 + 1.0/512 = 0.5020 ulps. Exhaustive 65 * testing of a float variant of this function showed a maximum final error 66 * of 0.5008 ulps. Non-exhaustive testing of a double variant of this 67 * function showed a maximum final error of 0.5078 ulps (near 1+1.0/256). 68 * 69 * We made the maximum of |d| (and thus the total relative error and the 70 * degree of p(d)) small by using a large number of intervals. Using 71 * centers of intervals instead of endpoints reduces this maximum by a 72 * factor of 2 for a given number of intervals. p(d) is special only 73 * in beginning with the Taylor coefficients 0 + 1*d, which tends to happen 74 * naturally. The most accurate minimax polynomial of a given degree might 75 * be different, but then we wouldn't want it since we would have to do 76 * extra work to avoid roundoff error (especially for P0*d instead of d). 77 */ 78 79 #ifdef DEBUG 80 #include <assert.h> 81 #include <fenv.h> 82 #endif 83 84 #ifdef __i386__ 85 #include <ieeefp.h> 86 #endif 87 88 #include "fpmath.h" 89 #include "math.h" 90 #define i386_SSE_GOOD 91 #ifndef NO_STRUCT_RETURN 92 #define STRUCT_RETURN 93 #endif 94 #include "math_private.h" 95 96 #if !defined(NO_UTAB) && !defined(NO_UTABL) 97 #define USE_UTAB 98 #endif 99 100 /* 101 * Domain [-0.005280, 0.004838], range ~[-5.1736e-22, 5.1738e-22]: 102 * |log(1 + d)/d - p(d)| < 2**-70.7 103 */ 104 static const double 105 P2 = -0.5, 106 P3 = 3.3333333333333359e-1, /* 0x1555555555555a.0p-54 */ 107 P4 = -2.5000000000004424e-1, /* -0x1000000000031d.0p-54 */ 108 P5 = 1.9999999992970016e-1, /* 0x1999999972f3c7.0p-55 */ 109 P6 = -1.6666666072191585e-1, /* -0x15555548912c09.0p-55 */ 110 P7 = 1.4286227413310518e-1, /* 0x12494f9d9def91.0p-55 */ 111 P8 = -1.2518388626763144e-1; /* -0x1006068cc0b97c.0p-55 */ 112 113 static volatile const double zero = 0; 114 115 #define INTERVALS 128 116 #define LOG2_INTERVALS 7 117 #define TSIZE (INTERVALS + 1) 118 #define G(i) (T[(i)].G) 119 #define F_hi(i) (T[(i)].F_hi) 120 #define F_lo(i) (T[(i)].F_lo) 121 #define ln2_hi F_hi(TSIZE - 1) 122 #define ln2_lo F_lo(TSIZE - 1) 123 #define E(i) (U[(i)].E) 124 #define H(i) (U[(i)].H) 125 126 static const struct { 127 float G; /* 1/(1 + i/128) rounded to 8/9 bits */ 128 float F_hi; /* log(1 / G_i) rounded (see below) */ 129 double F_lo; /* next 53 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 * We want to share this table between double precision and ld80, 145 * so the relevant range of dk is the larger one of ld80 146 * ([-16445, 16383]) and the relevant exactness requirement is 147 * the stricter one of double precision. The maximum number of 148 * bits in F_hi(i) that works is very dependent on i but has 149 * a minimum of 33. We only need about 12 bits in F_hi(i) for 150 * it to provide enough extra precision in double precision (11 151 * more than that are required for ld80). 152 * 153 * We round F_hi(i) to 24 bits so that it can have type float, 154 * mainly to minimize the size of the table. Using all 24 bits 155 * in a float for it automatically satisfies the above constraints. 156 */ 157 { 0x800000.0p-23, 0, 0 }, 158 { 0xfe0000.0p-24, 0x8080ac.0p-30, -0x14ee431dae6675.0p-84 }, 159 { 0xfc0000.0p-24, 0x8102b3.0p-29, -0x1db29ee2d83718.0p-84 }, 160 { 0xfa0000.0p-24, 0xc24929.0p-29, 0x1191957d173698.0p-83 }, 161 { 0xf80000.0p-24, 0x820aec.0p-28, 0x13ce8888e02e79.0p-82 }, 162 { 0xf60000.0p-24, 0xa33577.0p-28, -0x17a4382ce6eb7c.0p-82 }, 163 { 0xf48000.0p-24, 0xbc42cb.0p-28, -0x172a21161a1076.0p-83 }, 164 { 0xf30000.0p-24, 0xd57797.0p-28, -0x1e09de07cb9589.0p-82 }, 165 { 0xf10000.0p-24, 0xf7518e.0p-28, 0x1ae1eec1b036c5.0p-91 }, 166 { 0xef0000.0p-24, 0x8cb9df.0p-27, -0x1d7355325d560e.0p-81 }, 167 { 0xed8000.0p-24, 0x999ec0.0p-27, -0x1f9f02d256d503.0p-82 }, 168 { 0xec0000.0p-24, 0xa6988b.0p-27, -0x16fc0a9d12c17a.0p-83 }, 169 { 0xea0000.0p-24, 0xb80698.0p-27, 0x15d581c1e8da9a.0p-81 }, 170 { 0xe80000.0p-24, 0xc99af3.0p-27, -0x1535b3ba8f150b.0p-83 }, 171 { 0xe70000.0p-24, 0xd273b2.0p-27, 0x163786f5251af0.0p-85 }, 172 { 0xe50000.0p-24, 0xe442c0.0p-27, 0x1bc4b2368e32d5.0p-84 }, 173 { 0xe38000.0p-24, 0xf1b83f.0p-27, 0x1c6090f684e676.0p-81 }, 174 { 0xe20000.0p-24, 0xff448a.0p-27, -0x1890aa69ac9f42.0p-82 }, 175 { 0xe08000.0p-24, 0x8673f6.0p-26, 0x1b9985194b6b00.0p-80 }, 176 { 0xdf0000.0p-24, 0x8d515c.0p-26, -0x1dc08d61c6ef1e.0p-83 }, 177 { 0xdd8000.0p-24, 0x943a9e.0p-26, -0x1f72a2dac729b4.0p-82 }, 178 { 0xdc0000.0p-24, 0x9b2fe6.0p-26, -0x1fd4dfd3a0afb9.0p-80 }, 179 { 0xda8000.0p-24, 0xa2315d.0p-26, -0x11b26121629c47.0p-82 }, 180 { 0xd90000.0p-24, 0xa93f2f.0p-26, 0x1286d633e8e569.0p-81 }, 181 { 0xd78000.0p-24, 0xb05988.0p-26, 0x16128eba936770.0p-84 }, 182 { 0xd60000.0p-24, 0xb78094.0p-26, 0x16ead577390d32.0p-80 }, 183 { 0xd50000.0p-24, 0xbc4c6c.0p-26, 0x151131ccf7c7b7.0p-81 }, 184 { 0xd38000.0p-24, 0xc3890a.0p-26, -0x115e2cd714bd06.0p-80 }, 185 { 0xd20000.0p-24, 0xcad2d7.0p-26, -0x1847f406ebd3b0.0p-82 }, 186 { 0xd10000.0p-24, 0xcfb620.0p-26, 0x1c2259904d6866.0p-81 }, 187 { 0xcf8000.0p-24, 0xd71653.0p-26, 0x1ece57a8d5ae55.0p-80 }, 188 { 0xce0000.0p-24, 0xde843a.0p-26, -0x1f109d4bc45954.0p-81 }, 189 { 0xcd0000.0p-24, 0xe37fde.0p-26, 0x1bc03dc271a74d.0p-81 }, 190 { 0xcb8000.0p-24, 0xeb050c.0p-26, -0x1bf2badc0df842.0p-85 }, 191 { 0xca0000.0p-24, 0xf29878.0p-26, -0x18efededd89fbe.0p-87 }, 192 { 0xc90000.0p-24, 0xf7ad6f.0p-26, 0x1373ff977baa69.0p-81 }, 193 { 0xc80000.0p-24, 0xfcc8e3.0p-26, 0x196766f2fb3283.0p-80 }, 194 { 0xc68000.0p-24, 0x823f30.0p-25, 0x19bd076f7c434e.0p-79 }, 195 { 0xc58000.0p-24, 0x84d52c.0p-25, -0x1a327257af0f46.0p-79 }, 196 { 0xc40000.0p-24, 0x88bc74.0p-25, 0x113f23def19c5a.0p-81 }, 197 { 0xc30000.0p-24, 0x8b5ae6.0p-25, 0x1759f6e6b37de9.0p-79 }, 198 { 0xc20000.0p-24, 0x8dfccb.0p-25, 0x1ad35ca6ed5148.0p-81 }, 199 { 0xc10000.0p-24, 0x90a22b.0p-25, 0x1a1d71a87deba4.0p-79 }, 200 { 0xbf8000.0p-24, 0x94a0d8.0p-25, -0x139e5210c2b731.0p-80 }, 201 { 0xbe8000.0p-24, 0x974f16.0p-25, -0x18f6ebcff3ed73.0p-81 }, 202 { 0xbd8000.0p-24, 0x9a00f1.0p-25, -0x1aa268be39aab7.0p-79 }, 203 { 0xbc8000.0p-24, 0x9cb672.0p-25, -0x14c8815839c566.0p-79 }, 204 { 0xbb0000.0p-24, 0xa0cda1.0p-25, 0x1eaf46390dbb24.0p-81 }, 205 { 0xba0000.0p-24, 0xa38c6e.0p-25, 0x138e20d831f698.0p-81 }, 206 { 0xb90000.0p-24, 0xa64f05.0p-25, -0x1e8d3c41123616.0p-82 }, 207 { 0xb80000.0p-24, 0xa91570.0p-25, 0x1ce28f5f3840b2.0p-80 }, 208 { 0xb70000.0p-24, 0xabdfbb.0p-25, -0x186e5c0a424234.0p-79 }, 209 { 0xb60000.0p-24, 0xaeadef.0p-25, -0x14d41a0b2a08a4.0p-83 }, 210 { 0xb50000.0p-24, 0xb18018.0p-25, 0x16755892770634.0p-79 }, 211 { 0xb40000.0p-24, 0xb45642.0p-25, -0x16395ebe59b152.0p-82 }, 212 { 0xb30000.0p-24, 0xb73077.0p-25, 0x1abc65c8595f09.0p-80 }, 213 { 0xb20000.0p-24, 0xba0ec4.0p-25, -0x1273089d3dad89.0p-79 }, 214 { 0xb10000.0p-24, 0xbcf133.0p-25, 0x10f9f67b1f4bbf.0p-79 }, 215 { 0xb00000.0p-24, 0xbfd7d2.0p-25, -0x109fab90486409.0p-80 }, 216 { 0xaf0000.0p-24, 0xc2c2ac.0p-25, -0x1124680aa43333.0p-79 }, 217 { 0xae8000.0p-24, 0xc439b3.0p-25, -0x1f360cc4710fc0.0p-80 }, 218 { 0xad8000.0p-24, 0xc72afd.0p-25, -0x132d91f21d89c9.0p-80 }, 219 { 0xac8000.0p-24, 0xca20a2.0p-25, -0x16bf9b4d1f8da8.0p-79 }, 220 { 0xab8000.0p-24, 0xcd1aae.0p-25, 0x19deb5ce6a6a87.0p-81 }, 221 { 0xaa8000.0p-24, 0xd0192f.0p-25, 0x1a29fb48f7d3cb.0p-79 }, 222 { 0xaa0000.0p-24, 0xd19a20.0p-25, 0x1127d3c6457f9d.0p-81 }, 223 { 0xa90000.0p-24, 0xd49f6a.0p-25, -0x1ba930e486a0ac.0p-81 }, 224 { 0xa80000.0p-24, 0xd7a94b.0p-25, -0x1b6e645f31549e.0p-79 }, 225 { 0xa70000.0p-24, 0xdab7d0.0p-25, 0x1118a425494b61.0p-80 }, 226 { 0xa68000.0p-24, 0xdc40d5.0p-25, 0x1966f24d29d3a3.0p-80 }, 227 { 0xa58000.0p-24, 0xdf566d.0p-25, -0x1d8e52eb2248f1.0p-82 }, 228 { 0xa48000.0p-24, 0xe270ce.0p-25, -0x1ee370f96e6b68.0p-80 }, 229 { 0xa40000.0p-24, 0xe3ffce.0p-25, 0x1d155324911f57.0p-80 }, 230 { 0xa30000.0p-24, 0xe72179.0p-25, -0x1fe6e2f2f867d9.0p-80 }, 231 { 0xa20000.0p-24, 0xea4812.0p-25, 0x1b7be9add7f4d4.0p-80 }, 232 { 0xa18000.0p-24, 0xebdd3d.0p-25, 0x1b3cfb3f7511dd.0p-79 }, 233 { 0xa08000.0p-24, 0xef0b5b.0p-25, -0x1220de1f730190.0p-79 }, 234 { 0xa00000.0p-24, 0xf0a451.0p-25, -0x176364c9ac81cd.0p-80 }, 235 { 0x9f0000.0p-24, 0xf3da16.0p-25, 0x1eed6b9aafac8d.0p-81 }, 236 { 0x9e8000.0p-24, 0xf576e9.0p-25, 0x1d593218675af2.0p-79 }, 237 { 0x9d8000.0p-24, 0xf8b47c.0p-25, -0x13e8eb7da053e0.0p-84 }, 238 { 0x9d0000.0p-24, 0xfa553f.0p-25, 0x1c063259bcade0.0p-79 }, 239 { 0x9c0000.0p-24, 0xfd9ac5.0p-25, 0x1ef491085fa3c1.0p-79 }, 240 { 0x9b8000.0p-24, 0xff3f8c.0p-25, 0x1d607a7c2b8c53.0p-79 }, 241 { 0x9a8000.0p-24, 0x814697.0p-24, -0x12ad3817004f3f.0p-78 }, 242 { 0x9a0000.0p-24, 0x821b06.0p-24, -0x189fc53117f9e5.0p-81 }, 243 { 0x990000.0p-24, 0x83c5f8.0p-24, 0x14cf15a048907b.0p-79 }, 244 { 0x988000.0p-24, 0x849c7d.0p-24, 0x1cbb1d35fb8287.0p-78 }, 245 { 0x978000.0p-24, 0x864ba6.0p-24, 0x1128639b814f9c.0p-78 }, 246 { 0x970000.0p-24, 0x87244c.0p-24, 0x184733853300f0.0p-79 }, 247 { 0x968000.0p-24, 0x87fdaa.0p-24, 0x109d23aef77dd6.0p-80 }, 248 { 0x958000.0p-24, 0x89b293.0p-24, -0x1a81ef367a59de.0p-78 }, 249 { 0x950000.0p-24, 0x8a8e20.0p-24, -0x121ad3dbb2f452.0p-78 }, 250 { 0x948000.0p-24, 0x8b6a6a.0p-24, -0x1cfb981628af72.0p-79 }, 251 { 0x938000.0p-24, 0x8d253a.0p-24, -0x1d21730ea76cfe.0p-79 }, 252 { 0x930000.0p-24, 0x8e03c2.0p-24, 0x135cc00e566f77.0p-78 }, 253 { 0x928000.0p-24, 0x8ee30d.0p-24, -0x10fcb5df257a26.0p-80 }, 254 { 0x918000.0p-24, 0x90a3ee.0p-24, -0x16e171b15433d7.0p-79 }, 255 { 0x910000.0p-24, 0x918587.0p-24, -0x1d050da07f3237.0p-79 }, 256 { 0x908000.0p-24, 0x9267e7.0p-24, 0x1be03669a5268d.0p-79 }, 257 { 0x8f8000.0p-24, 0x942f04.0p-24, 0x10b28e0e26c337.0p-79 }, 258 { 0x8f0000.0p-24, 0x9513c3.0p-24, 0x1a1d820da57cf3.0p-78 }, 259 { 0x8e8000.0p-24, 0x95f950.0p-24, -0x19ef8f13ae3cf1.0p-79 }, 260 { 0x8e0000.0p-24, 0x96dfab.0p-24, -0x109e417a6e507c.0p-78 }, 261 { 0x8d0000.0p-24, 0x98aed2.0p-24, 0x10d01a2c5b0e98.0p-79 }, 262 { 0x8c8000.0p-24, 0x9997a2.0p-24, -0x1d6a50d4b61ea7.0p-78 }, 263 { 0x8c0000.0p-24, 0x9a8145.0p-24, 0x1b3b190b83f952.0p-78 }, 264 { 0x8b8000.0p-24, 0x9b6bbf.0p-24, 0x13a69fad7e7abe.0p-78 }, 265 { 0x8b0000.0p-24, 0x9c5711.0p-24, -0x11cd12316f576b.0p-78 }, 266 { 0x8a8000.0p-24, 0x9d433b.0p-24, 0x1c95c444b807a2.0p-79 }, 267 { 0x898000.0p-24, 0x9f1e22.0p-24, -0x1b9c224ea698c3.0p-79 }, 268 { 0x890000.0p-24, 0xa00ce1.0p-24, 0x125ca93186cf0f.0p-81 }, 269 { 0x888000.0p-24, 0xa0fc80.0p-24, -0x1ee38a7bc228b3.0p-79 }, 270 { 0x880000.0p-24, 0xa1ed00.0p-24, -0x1a0db876613d20.0p-78 }, 271 { 0x878000.0p-24, 0xa2de62.0p-24, 0x193224e8516c01.0p-79 }, 272 { 0x870000.0p-24, 0xa3d0a9.0p-24, 0x1fa28b4d2541ad.0p-79 }, 273 { 0x868000.0p-24, 0xa4c3d6.0p-24, 0x1c1b5760fb4572.0p-78 }, 274 { 0x858000.0p-24, 0xa6acea.0p-24, 0x1fed5d0f65949c.0p-80 }, 275 { 0x850000.0p-24, 0xa7a2d4.0p-24, 0x1ad270c9d74936.0p-80 }, 276 { 0x848000.0p-24, 0xa899ab.0p-24, 0x199ff15ce53266.0p-79 }, 277 { 0x840000.0p-24, 0xa99171.0p-24, 0x1a19e15ccc45d2.0p-79 }, 278 { 0x838000.0p-24, 0xaa8a28.0p-24, -0x121a14ec532b36.0p-80 }, 279 { 0x830000.0p-24, 0xab83d1.0p-24, 0x1aee319980bff3.0p-79 }, 280 { 0x828000.0p-24, 0xac7e6f.0p-24, -0x18ffd9e3900346.0p-80 }, 281 { 0x820000.0p-24, 0xad7a03.0p-24, -0x1e4db102ce29f8.0p-80 }, 282 { 0x818000.0p-24, 0xae768f.0p-24, 0x17c35c55a04a83.0p-81 }, 283 { 0x810000.0p-24, 0xaf7415.0p-24, 0x1448324047019b.0p-78 }, 284 { 0x808000.0p-24, 0xb07298.0p-24, -0x1750ee3915a198.0p-78 }, 285 { 0x800000.0p-24, 0xb17218.0p-24, -0x105c610ca86c39.0p-81 }, 286 }; 287 288 #ifdef USE_UTAB 289 static const struct { 290 float H; /* 1 + i/INTERVALS (exact) */ 291 float E; /* H(i) * G(i) - 1 (exact) */ 292 } U[TSIZE] = { 293 { 0x800000.0p-23, 0 }, 294 { 0x810000.0p-23, -0x800000.0p-37 }, 295 { 0x820000.0p-23, -0x800000.0p-35 }, 296 { 0x830000.0p-23, -0x900000.0p-34 }, 297 { 0x840000.0p-23, -0x800000.0p-33 }, 298 { 0x850000.0p-23, -0xc80000.0p-33 }, 299 { 0x860000.0p-23, -0xa00000.0p-36 }, 300 { 0x870000.0p-23, 0x940000.0p-33 }, 301 { 0x880000.0p-23, 0x800000.0p-35 }, 302 { 0x890000.0p-23, -0xc80000.0p-34 }, 303 { 0x8a0000.0p-23, 0xe00000.0p-36 }, 304 { 0x8b0000.0p-23, 0x900000.0p-33 }, 305 { 0x8c0000.0p-23, -0x800000.0p-35 }, 306 { 0x8d0000.0p-23, -0xe00000.0p-33 }, 307 { 0x8e0000.0p-23, 0x880000.0p-33 }, 308 { 0x8f0000.0p-23, -0xa80000.0p-34 }, 309 { 0x900000.0p-23, -0x800000.0p-35 }, 310 { 0x910000.0p-23, 0x800000.0p-37 }, 311 { 0x920000.0p-23, 0x900000.0p-35 }, 312 { 0x930000.0p-23, 0xd00000.0p-35 }, 313 { 0x940000.0p-23, 0xe00000.0p-35 }, 314 { 0x950000.0p-23, 0xc00000.0p-35 }, 315 { 0x960000.0p-23, 0xe00000.0p-36 }, 316 { 0x970000.0p-23, -0x800000.0p-38 }, 317 { 0x980000.0p-23, -0xc00000.0p-35 }, 318 { 0x990000.0p-23, -0xd00000.0p-34 }, 319 { 0x9a0000.0p-23, 0x880000.0p-33 }, 320 { 0x9b0000.0p-23, 0xe80000.0p-35 }, 321 { 0x9c0000.0p-23, -0x800000.0p-35 }, 322 { 0x9d0000.0p-23, 0xb40000.0p-33 }, 323 { 0x9e0000.0p-23, 0x880000.0p-34 }, 324 { 0x9f0000.0p-23, -0xe00000.0p-35 }, 325 { 0xa00000.0p-23, 0x800000.0p-33 }, 326 { 0xa10000.0p-23, -0x900000.0p-36 }, 327 { 0xa20000.0p-23, -0xb00000.0p-33 }, 328 { 0xa30000.0p-23, -0xa00000.0p-36 }, 329 { 0xa40000.0p-23, 0x800000.0p-33 }, 330 { 0xa50000.0p-23, -0xf80000.0p-35 }, 331 { 0xa60000.0p-23, 0x880000.0p-34 }, 332 { 0xa70000.0p-23, -0x900000.0p-33 }, 333 { 0xa80000.0p-23, -0x800000.0p-35 }, 334 { 0xa90000.0p-23, 0x900000.0p-34 }, 335 { 0xaa0000.0p-23, 0xa80000.0p-33 }, 336 { 0xab0000.0p-23, -0xac0000.0p-34 }, 337 { 0xac0000.0p-23, -0x800000.0p-37 }, 338 { 0xad0000.0p-23, 0xf80000.0p-35 }, 339 { 0xae0000.0p-23, 0xf80000.0p-34 }, 340 { 0xaf0000.0p-23, -0xac0000.0p-33 }, 341 { 0xb00000.0p-23, -0x800000.0p-33 }, 342 { 0xb10000.0p-23, -0xb80000.0p-34 }, 343 { 0xb20000.0p-23, -0x800000.0p-34 }, 344 { 0xb30000.0p-23, -0xb00000.0p-35 }, 345 { 0xb40000.0p-23, -0x800000.0p-35 }, 346 { 0xb50000.0p-23, -0xe00000.0p-36 }, 347 { 0xb60000.0p-23, -0x800000.0p-35 }, 348 { 0xb70000.0p-23, -0xb00000.0p-35 }, 349 { 0xb80000.0p-23, -0x800000.0p-34 }, 350 { 0xb90000.0p-23, -0xb80000.0p-34 }, 351 { 0xba0000.0p-23, -0x800000.0p-33 }, 352 { 0xbb0000.0p-23, -0xac0000.0p-33 }, 353 { 0xbc0000.0p-23, 0x980000.0p-33 }, 354 { 0xbd0000.0p-23, 0xbc0000.0p-34 }, 355 { 0xbe0000.0p-23, 0xe00000.0p-36 }, 356 { 0xbf0000.0p-23, -0xb80000.0p-35 }, 357 { 0xc00000.0p-23, -0x800000.0p-33 }, 358 { 0xc10000.0p-23, 0xa80000.0p-33 }, 359 { 0xc20000.0p-23, 0x900000.0p-34 }, 360 { 0xc30000.0p-23, -0x800000.0p-35 }, 361 { 0xc40000.0p-23, -0x900000.0p-33 }, 362 { 0xc50000.0p-23, 0x820000.0p-33 }, 363 { 0xc60000.0p-23, 0x800000.0p-38 }, 364 { 0xc70000.0p-23, -0x820000.0p-33 }, 365 { 0xc80000.0p-23, 0x800000.0p-33 }, 366 { 0xc90000.0p-23, -0xa00000.0p-36 }, 367 { 0xca0000.0p-23, -0xb00000.0p-33 }, 368 { 0xcb0000.0p-23, 0x840000.0p-34 }, 369 { 0xcc0000.0p-23, -0xd00000.0p-34 }, 370 { 0xcd0000.0p-23, 0x800000.0p-33 }, 371 { 0xce0000.0p-23, -0xe00000.0p-35 }, 372 { 0xcf0000.0p-23, 0xa60000.0p-33 }, 373 { 0xd00000.0p-23, -0x800000.0p-35 }, 374 { 0xd10000.0p-23, 0xb40000.0p-33 }, 375 { 0xd20000.0p-23, -0x800000.0p-35 }, 376 { 0xd30000.0p-23, 0xaa0000.0p-33 }, 377 { 0xd40000.0p-23, -0xe00000.0p-35 }, 378 { 0xd50000.0p-23, 0x880000.0p-33 }, 379 { 0xd60000.0p-23, -0xd00000.0p-34 }, 380 { 0xd70000.0p-23, 0x9c0000.0p-34 }, 381 { 0xd80000.0p-23, -0xb00000.0p-33 }, 382 { 0xd90000.0p-23, -0x800000.0p-38 }, 383 { 0xda0000.0p-23, 0xa40000.0p-33 }, 384 { 0xdb0000.0p-23, -0xdc0000.0p-34 }, 385 { 0xdc0000.0p-23, 0xc00000.0p-35 }, 386 { 0xdd0000.0p-23, 0xca0000.0p-33 }, 387 { 0xde0000.0p-23, -0xb80000.0p-34 }, 388 { 0xdf0000.0p-23, 0xd00000.0p-35 }, 389 { 0xe00000.0p-23, 0xc00000.0p-33 }, 390 { 0xe10000.0p-23, -0xf40000.0p-34 }, 391 { 0xe20000.0p-23, 0x800000.0p-37 }, 392 { 0xe30000.0p-23, 0x860000.0p-33 }, 393 { 0xe40000.0p-23, -0xc80000.0p-33 }, 394 { 0xe50000.0p-23, -0xa80000.0p-34 }, 395 { 0xe60000.0p-23, 0xe00000.0p-36 }, 396 { 0xe70000.0p-23, 0x880000.0p-33 }, 397 { 0xe80000.0p-23, -0xe00000.0p-33 }, 398 { 0xe90000.0p-23, -0xfc0000.0p-34 }, 399 { 0xea0000.0p-23, -0x800000.0p-35 }, 400 { 0xeb0000.0p-23, 0xe80000.0p-35 }, 401 { 0xec0000.0p-23, 0x900000.0p-33 }, 402 { 0xed0000.0p-23, 0xe20000.0p-33 }, 403 { 0xee0000.0p-23, -0xac0000.0p-33 }, 404 { 0xef0000.0p-23, -0xc80000.0p-34 }, 405 { 0xf00000.0p-23, -0x800000.0p-35 }, 406 { 0xf10000.0p-23, 0x800000.0p-35 }, 407 { 0xf20000.0p-23, 0xb80000.0p-34 }, 408 { 0xf30000.0p-23, 0x940000.0p-33 }, 409 { 0xf40000.0p-23, 0xc80000.0p-33 }, 410 { 0xf50000.0p-23, -0xf20000.0p-33 }, 411 { 0xf60000.0p-23, -0xc80000.0p-33 }, 412 { 0xf70000.0p-23, -0xa20000.0p-33 }, 413 { 0xf80000.0p-23, -0x800000.0p-33 }, 414 { 0xf90000.0p-23, -0xc40000.0p-34 }, 415 { 0xfa0000.0p-23, -0x900000.0p-34 }, 416 { 0xfb0000.0p-23, -0xc80000.0p-35 }, 417 { 0xfc0000.0p-23, -0x800000.0p-35 }, 418 { 0xfd0000.0p-23, -0x900000.0p-36 }, 419 { 0xfe0000.0p-23, -0x800000.0p-37 }, 420 { 0xff0000.0p-23, -0x800000.0p-39 }, 421 { 0x800000.0p-22, 0 }, 422 }; 423 #endif /* USE_UTAB */ 424 425 #ifdef STRUCT_RETURN 426 #define RETURN1(rp, v) do { \ 427 (rp)->hi = (v); \ 428 (rp)->lo_set = 0; \ 429 return; \ 430 } while (0) 431 432 #define RETURN2(rp, h, l) do { \ 433 (rp)->hi = (h); \ 434 (rp)->lo = (l); \ 435 (rp)->lo_set = 1; \ 436 return; \ 437 } while (0) 438 439 struct ld { 440 long double hi; 441 long double lo; 442 int lo_set; 443 }; 444 #else 445 #define RETURN1(rp, v) RETURNF(v) 446 #define RETURN2(rp, h, l) RETURNI((h) + (l)) 447 #endif 448 449 #ifdef STRUCT_RETURN 450 static inline __always_inline void 451 k_logl(long double x, struct ld *rp) 452 #else 453 long double 454 logl(long double x) 455 #endif 456 { 457 long double d, dk, val_hi, val_lo, z; 458 uint64_t ix, lx; 459 int i, k; 460 uint16_t hx; 461 462 EXTRACT_LDBL80_WORDS(hx, lx, x); 463 k = -16383; 464 #if 0 /* Hard to do efficiently. Don't do it until we support all modes. */ 465 if (x == 1) 466 RETURN1(rp, 0); /* log(1) = +0 in all rounding modes */ 467 #endif 468 if (hx == 0 || hx >= 0x8000) { /* zero, negative or subnormal? */ 469 if (((hx & 0x7fff) | lx) == 0) 470 RETURN1(rp, -1 / zero); /* log(+-0) = -Inf */ 471 if (hx != 0) 472 /* log(neg or [pseudo-]NaN) = qNaN: */ 473 RETURN1(rp, (x - x) / zero); 474 x *= 0x1.0p65; /* subnormal; scale up x */ 475 /* including pseudo-subnormals */ 476 EXTRACT_LDBL80_WORDS(hx, lx, x); 477 k = -16383 - 65; 478 } else if (hx >= 0x7fff || (lx & 0x8000000000000000ULL) == 0) 479 RETURN1(rp, x + x); /* log(Inf or NaN) = Inf or qNaN */ 480 /* log(pseudo-Inf) = qNaN */ 481 /* log(pseudo-NaN) = qNaN */ 482 /* log(unnormal) = qNaN */ 483 #ifndef STRUCT_RETURN 484 ENTERI(); 485 #endif 486 k += hx; 487 ix = lx & 0x7fffffffffffffffULL; 488 dk = k; 489 490 /* Scale x to be in [1, 2). */ 491 SET_LDBL_EXPSIGN(x, 0x3fff); 492 493 /* 0 <= i <= INTERVALS: */ 494 #define L2I (64 - LOG2_INTERVALS) 495 i = (ix + (1LL << (L2I - 2))) >> (L2I - 1); 496 497 /* 498 * -0.005280 < d < 0.004838. In particular, the infinite- 499 * precision |d| is <= 2**-7. Rounding of G(i) to 8 bits 500 * ensures that d is representable without extra precision for 501 * this bound on |d| (since when this calculation is expressed 502 * as x*G(i)-1, the multiplication needs as many extra bits as 503 * G(i) has and the subtraction cancels 8 bits). But for 504 * most i (107 cases out of 129), the infinite-precision |d| 505 * is <= 2**-8. G(i) is rounded to 9 bits for such i to give 506 * better accuracy (this works by improving the bound on |d|, 507 * which in turn allows rounding to 9 bits in more cases). 508 * This is only important when the original x is near 1 -- it 509 * lets us avoid using a special method to give the desired 510 * accuracy for such x. 511 */ 512 if (0) 513 d = x * G(i) - 1; 514 else { 515 #ifdef USE_UTAB 516 d = (x - H(i)) * G(i) + E(i); 517 #else 518 long double x_hi, x_lo; 519 float fx_hi; 520 521 /* 522 * Split x into x_hi + x_lo to calculate x*G(i)-1 exactly. 523 * G(i) has at most 9 bits, so the splitting point is not 524 * critical. 525 */ 526 SET_FLOAT_WORD(fx_hi, (lx >> 40) | 0x3f800000); 527 x_hi = fx_hi; 528 x_lo = x - x_hi; 529 d = x_hi * G(i) - 1 + x_lo * G(i); 530 #endif 531 } 532 533 /* 534 * Our algorithm depends on exact cancellation of F_lo(i) and 535 * F_hi(i) with dk*ln_2_lo and dk*ln2_hi when k is -1 and i is 536 * at the end of the table. This and other technical complications 537 * make it difficult to avoid the double scaling in (dk*ln2) * 538 * log(base) for base != e without losing more accuracy and/or 539 * efficiency than is gained. 540 */ 541 z = d * d; 542 val_lo = z * d * z * (z * (d * P8 + P7) + (d * P6 + P5)) + 543 (F_lo(i) + dk * ln2_lo + z * d * (d * P4 + P3)) + z * P2; 544 val_hi = d; 545 #ifdef DEBUG 546 if (fetestexcept(FE_UNDERFLOW)) 547 breakpoint(); 548 #endif 549 550 _3sumF(val_hi, val_lo, F_hi(i) + dk * ln2_hi); 551 RETURN2(rp, val_hi, val_lo); 552 } 553 554 long double 555 log1pl(long double x) 556 { 557 long double d, d_hi, d_lo, dk, f_lo, val_hi, val_lo, z; 558 long double f_hi, twopminusk; 559 uint64_t ix, lx; 560 int i, k; 561 int16_t ax, hx; 562 563 DOPRINT_START(&x); 564 EXTRACT_LDBL80_WORDS(hx, lx, x); 565 if (hx < 0x3fff) { /* x < 1, or x neg NaN */ 566 ax = hx & 0x7fff; 567 if (ax >= 0x3fff) { /* x <= -1, or x neg NaN */ 568 if (ax == 0x3fff && lx == 0x8000000000000000ULL) 569 RETURNP(-1 / zero); /* log1p(-1) = -Inf */ 570 /* log1p(x < 1, or x [pseudo-]NaN) = qNaN: */ 571 RETURNP((x - x) / (x - x)); 572 } 573 if (ax <= 0x3fbe) { /* |x| < 2**-64 */ 574 if ((int)x == 0) 575 RETURNP(x); /* x with inexact if x != 0 */ 576 } 577 f_hi = 1; 578 f_lo = x; 579 } else if (hx >= 0x7fff) { /* x +Inf or non-neg NaN */ 580 RETURNP(x + x); /* log1p(Inf or NaN) = Inf or qNaN */ 581 /* log1p(pseudo-Inf) = qNaN */ 582 /* log1p(pseudo-NaN) = qNaN */ 583 /* log1p(unnormal) = qNaN */ 584 } else if (hx < 0x407f) { /* 1 <= x < 2**128 */ 585 f_hi = x; 586 f_lo = 1; 587 } else { /* 2**128 <= x < +Inf */ 588 f_hi = x; 589 f_lo = 0; /* avoid underflow of the P5 term */ 590 } 591 ENTERI(); 592 x = f_hi + f_lo; 593 f_lo = (f_hi - x) + f_lo; 594 595 EXTRACT_LDBL80_WORDS(hx, lx, x); 596 k = -16383; 597 598 k += hx; 599 ix = lx & 0x7fffffffffffffffULL; 600 dk = k; 601 602 SET_LDBL_EXPSIGN(x, 0x3fff); 603 twopminusk = 1; 604 SET_LDBL_EXPSIGN(twopminusk, 0x7ffe - (hx & 0x7fff)); 605 f_lo *= twopminusk; 606 607 i = (ix + (1LL << (L2I - 2))) >> (L2I - 1); 608 609 /* 610 * x*G(i)-1 (with a reduced x) can be represented exactly, as 611 * above, but now we need to evaluate the polynomial on d = 612 * (x+f_lo)*G(i)-1 and extra precision is needed for that. 613 * Since x+x_lo is a hi+lo decomposition and subtracting 1 614 * doesn't lose too many bits, an inexact calculation for 615 * f_lo*G(i) is good enough. 616 */ 617 if (0) 618 d_hi = x * G(i) - 1; 619 else { 620 #ifdef USE_UTAB 621 d_hi = (x - H(i)) * G(i) + E(i); 622 #else 623 long double x_hi, x_lo; 624 float fx_hi; 625 626 SET_FLOAT_WORD(fx_hi, (lx >> 40) | 0x3f800000); 627 x_hi = fx_hi; 628 x_lo = x - x_hi; 629 d_hi = x_hi * G(i) - 1 + x_lo * G(i); 630 #endif 631 } 632 d_lo = f_lo * G(i); 633 634 /* 635 * This is _2sumF(d_hi, d_lo) inlined. The condition 636 * (d_hi == 0 || |d_hi| >= |d_lo|) for using _2sumF() is not 637 * always satisifed, so it is not clear that this works, but 638 * it works in practice. It works even if it gives a wrong 639 * normalized d_lo, since |d_lo| > |d_hi| implies that i is 640 * nonzero and d is tiny, so the F(i) term dominates d_lo. 641 * In float precision: 642 * (By exhaustive testing, the worst case is d_hi = 0x1.bp-25. 643 * And if d is only a little tinier than that, we would have 644 * another underflow problem for the P3 term; this is also ruled 645 * out by exhaustive testing.) 646 */ 647 d = d_hi + d_lo; 648 d_lo = d_hi - d + d_lo; 649 d_hi = d; 650 651 z = d * d; 652 val_lo = z * d * z * (z * (d * P8 + P7) + (d * P6 + P5)) + 653 (F_lo(i) + dk * ln2_lo + d_lo + z * d * (d * P4 + P3)) + z * P2; 654 val_hi = d_hi; 655 #ifdef DEBUG 656 if (fetestexcept(FE_UNDERFLOW)) 657 breakpoint(); 658 #endif 659 660 _3sumF(val_hi, val_lo, F_hi(i) + dk * ln2_hi); 661 RETURN2PI(val_hi, val_lo); 662 } 663 664 #ifdef STRUCT_RETURN 665 666 long double 667 logl(long double x) 668 { 669 struct ld r; 670 671 ENTERI(); 672 DOPRINT_START(&x); 673 k_logl(x, &r); 674 RETURNSPI(&r); 675 } 676 677 static const double 678 invln10_hi = 4.3429448190317999e-1, /* 0x1bcb7b1526e000.0p-54 */ 679 invln10_lo = 7.1842412889749798e-14, /* 0x1438ca9aadd558.0p-96 */ 680 invln2_hi = 1.4426950408887933e0, /* 0x171547652b8000.0p-52 */ 681 invln2_lo = 1.7010652264631490e-13; /* 0x17f0bbbe87fed0.0p-95 */ 682 683 long double 684 log10l(long double x) 685 { 686 struct ld r; 687 long double hi, lo; 688 689 ENTERI(); 690 DOPRINT_START(&x); 691 k_logl(x, &r); 692 if (!r.lo_set) 693 RETURNPI(r.hi); 694 _2sumF(r.hi, r.lo); 695 hi = (float)r.hi; 696 lo = r.lo + (r.hi - hi); 697 RETURN2PI(invln10_hi * hi, 698 (invln10_lo + invln10_hi) * lo + invln10_lo * hi); 699 } 700 701 long double 702 log2l(long double x) 703 { 704 struct ld r; 705 long double hi, lo; 706 707 ENTERI(); 708 DOPRINT_START(&x); 709 k_logl(x, &r); 710 if (!r.lo_set) 711 RETURNPI(r.hi); 712 _2sumF(r.hi, r.lo); 713 hi = (float)r.hi; 714 lo = r.lo + (r.hi - hi); 715 RETURN2PI(invln2_hi * hi, 716 (invln2_lo + invln2_hi) * lo + invln2_lo * hi); 717 } 718 719 #endif /* STRUCT_RETURN */ 720