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 Intel 80-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 #ifdef __i386__ 83 #include <ieeefp.h> 84 #endif 85 86 #include "fpmath.h" 87 #include "math.h" 88 #define i386_SSE_GOOD 89 #ifndef NO_STRUCT_RETURN 90 #define STRUCT_RETURN 91 #endif 92 #include "math_private.h" 93 94 #if !defined(NO_UTAB) && !defined(NO_UTABL) 95 #define USE_UTAB 96 #endif 97 98 /* 99 * Domain [-0.005280, 0.004838], range ~[-5.1736e-22, 5.1738e-22]: 100 * |log(1 + d)/d - p(d)| < 2**-70.7 101 */ 102 static const double 103 P2 = -0.5, 104 P3 = 3.3333333333333359e-1, /* 0x1555555555555a.0p-54 */ 105 P4 = -2.5000000000004424e-1, /* -0x1000000000031d.0p-54 */ 106 P5 = 1.9999999992970016e-1, /* 0x1999999972f3c7.0p-55 */ 107 P6 = -1.6666666072191585e-1, /* -0x15555548912c09.0p-55 */ 108 P7 = 1.4286227413310518e-1, /* 0x12494f9d9def91.0p-55 */ 109 P8 = -1.2518388626763144e-1; /* -0x1006068cc0b97c.0p-55 */ 110 111 static volatile const double zero = 0; 112 113 #define INTERVALS 128 114 #define LOG2_INTERVALS 7 115 #define TSIZE (INTERVALS + 1) 116 #define G(i) (T[(i)].G) 117 #define F_hi(i) (T[(i)].F_hi) 118 #define F_lo(i) (T[(i)].F_lo) 119 #define ln2_hi F_hi(TSIZE - 1) 120 #define ln2_lo F_lo(TSIZE - 1) 121 #define E(i) (U[(i)].E) 122 #define H(i) (U[(i)].H) 123 124 static const struct { 125 float G; /* 1/(1 + i/128) rounded to 8/9 bits */ 126 float F_hi; /* log(1 / G_i) rounded (see below) */ 127 double F_lo; /* next 53 bits for log(1 / G_i) */ 128 } T[TSIZE] = { 129 /* 130 * ln2_hi and each F_hi(i) are rounded to a number of bits that 131 * makes F_hi(i) + dk*ln2_hi exact for all i and all dk. 132 * 133 * The last entry (for X just below 2) is used to define ln2_hi 134 * and ln2_lo, to ensure that F_hi(i) and F_lo(i) cancel exactly 135 * with dk*ln2_hi and dk*ln2_lo, respectively, when dk = -1. 136 * This is needed for accuracy when x is just below 1. (To avoid 137 * special cases, such x are "reduced" strangely to X just below 138 * 2 and dk = -1, and then the exact cancellation is needed 139 * because any the error from any non-exactness would be too 140 * large). 141 * 142 * We want to share this table between double precision and ld80, 143 * so the relevant range of dk is the larger one of ld80 144 * ([-16445, 16383]) and the relevant exactness requirement is 145 * the stricter one of double precision. The maximum number of 146 * bits in F_hi(i) that works is very dependent on i but has 147 * a minimum of 33. We only need about 12 bits in F_hi(i) for 148 * it to provide enough extra precision in double precision (11 149 * more than that are required for ld80). 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, -0x14ee431dae6675.0p-84, 157 0xfc0000.0p-24, 0x8102b3.0p-29, -0x1db29ee2d83718.0p-84, 158 0xfa0000.0p-24, 0xc24929.0p-29, 0x1191957d173698.0p-83, 159 0xf80000.0p-24, 0x820aec.0p-28, 0x13ce8888e02e79.0p-82, 160 0xf60000.0p-24, 0xa33577.0p-28, -0x17a4382ce6eb7c.0p-82, 161 0xf48000.0p-24, 0xbc42cb.0p-28, -0x172a21161a1076.0p-83, 162 0xf30000.0p-24, 0xd57797.0p-28, -0x1e09de07cb9589.0p-82, 163 0xf10000.0p-24, 0xf7518e.0p-28, 0x1ae1eec1b036c5.0p-91, 164 0xef0000.0p-24, 0x8cb9df.0p-27, -0x1d7355325d560e.0p-81, 165 0xed8000.0p-24, 0x999ec0.0p-27, -0x1f9f02d256d503.0p-82, 166 0xec0000.0p-24, 0xa6988b.0p-27, -0x16fc0a9d12c17a.0p-83, 167 0xea0000.0p-24, 0xb80698.0p-27, 0x15d581c1e8da9a.0p-81, 168 0xe80000.0p-24, 0xc99af3.0p-27, -0x1535b3ba8f150b.0p-83, 169 0xe70000.0p-24, 0xd273b2.0p-27, 0x163786f5251af0.0p-85, 170 0xe50000.0p-24, 0xe442c0.0p-27, 0x1bc4b2368e32d5.0p-84, 171 0xe38000.0p-24, 0xf1b83f.0p-27, 0x1c6090f684e676.0p-81, 172 0xe20000.0p-24, 0xff448a.0p-27, -0x1890aa69ac9f42.0p-82, 173 0xe08000.0p-24, 0x8673f6.0p-26, 0x1b9985194b6b00.0p-80, 174 0xdf0000.0p-24, 0x8d515c.0p-26, -0x1dc08d61c6ef1e.0p-83, 175 0xdd8000.0p-24, 0x943a9e.0p-26, -0x1f72a2dac729b4.0p-82, 176 0xdc0000.0p-24, 0x9b2fe6.0p-26, -0x1fd4dfd3a0afb9.0p-80, 177 0xda8000.0p-24, 0xa2315d.0p-26, -0x11b26121629c47.0p-82, 178 0xd90000.0p-24, 0xa93f2f.0p-26, 0x1286d633e8e569.0p-81, 179 0xd78000.0p-24, 0xb05988.0p-26, 0x16128eba936770.0p-84, 180 0xd60000.0p-24, 0xb78094.0p-26, 0x16ead577390d32.0p-80, 181 0xd50000.0p-24, 0xbc4c6c.0p-26, 0x151131ccf7c7b7.0p-81, 182 0xd38000.0p-24, 0xc3890a.0p-26, -0x115e2cd714bd06.0p-80, 183 0xd20000.0p-24, 0xcad2d7.0p-26, -0x1847f406ebd3b0.0p-82, 184 0xd10000.0p-24, 0xcfb620.0p-26, 0x1c2259904d6866.0p-81, 185 0xcf8000.0p-24, 0xd71653.0p-26, 0x1ece57a8d5ae55.0p-80, 186 0xce0000.0p-24, 0xde843a.0p-26, -0x1f109d4bc45954.0p-81, 187 0xcd0000.0p-24, 0xe37fde.0p-26, 0x1bc03dc271a74d.0p-81, 188 0xcb8000.0p-24, 0xeb050c.0p-26, -0x1bf2badc0df842.0p-85, 189 0xca0000.0p-24, 0xf29878.0p-26, -0x18efededd89fbe.0p-87, 190 0xc90000.0p-24, 0xf7ad6f.0p-26, 0x1373ff977baa69.0p-81, 191 0xc80000.0p-24, 0xfcc8e3.0p-26, 0x196766f2fb3283.0p-80, 192 0xc68000.0p-24, 0x823f30.0p-25, 0x19bd076f7c434e.0p-79, 193 0xc58000.0p-24, 0x84d52c.0p-25, -0x1a327257af0f46.0p-79, 194 0xc40000.0p-24, 0x88bc74.0p-25, 0x113f23def19c5a.0p-81, 195 0xc30000.0p-24, 0x8b5ae6.0p-25, 0x1759f6e6b37de9.0p-79, 196 0xc20000.0p-24, 0x8dfccb.0p-25, 0x1ad35ca6ed5148.0p-81, 197 0xc10000.0p-24, 0x90a22b.0p-25, 0x1a1d71a87deba4.0p-79, 198 0xbf8000.0p-24, 0x94a0d8.0p-25, -0x139e5210c2b731.0p-80, 199 0xbe8000.0p-24, 0x974f16.0p-25, -0x18f6ebcff3ed73.0p-81, 200 0xbd8000.0p-24, 0x9a00f1.0p-25, -0x1aa268be39aab7.0p-79, 201 0xbc8000.0p-24, 0x9cb672.0p-25, -0x14c8815839c566.0p-79, 202 0xbb0000.0p-24, 0xa0cda1.0p-25, 0x1eaf46390dbb24.0p-81, 203 0xba0000.0p-24, 0xa38c6e.0p-25, 0x138e20d831f698.0p-81, 204 0xb90000.0p-24, 0xa64f05.0p-25, -0x1e8d3c41123616.0p-82, 205 0xb80000.0p-24, 0xa91570.0p-25, 0x1ce28f5f3840b2.0p-80, 206 0xb70000.0p-24, 0xabdfbb.0p-25, -0x186e5c0a424234.0p-79, 207 0xb60000.0p-24, 0xaeadef.0p-25, -0x14d41a0b2a08a4.0p-83, 208 0xb50000.0p-24, 0xb18018.0p-25, 0x16755892770634.0p-79, 209 0xb40000.0p-24, 0xb45642.0p-25, -0x16395ebe59b152.0p-82, 210 0xb30000.0p-24, 0xb73077.0p-25, 0x1abc65c8595f09.0p-80, 211 0xb20000.0p-24, 0xba0ec4.0p-25, -0x1273089d3dad89.0p-79, 212 0xb10000.0p-24, 0xbcf133.0p-25, 0x10f9f67b1f4bbf.0p-79, 213 0xb00000.0p-24, 0xbfd7d2.0p-25, -0x109fab90486409.0p-80, 214 0xaf0000.0p-24, 0xc2c2ac.0p-25, -0x1124680aa43333.0p-79, 215 0xae8000.0p-24, 0xc439b3.0p-25, -0x1f360cc4710fc0.0p-80, 216 0xad8000.0p-24, 0xc72afd.0p-25, -0x132d91f21d89c9.0p-80, 217 0xac8000.0p-24, 0xca20a2.0p-25, -0x16bf9b4d1f8da8.0p-79, 218 0xab8000.0p-24, 0xcd1aae.0p-25, 0x19deb5ce6a6a87.0p-81, 219 0xaa8000.0p-24, 0xd0192f.0p-25, 0x1a29fb48f7d3cb.0p-79, 220 0xaa0000.0p-24, 0xd19a20.0p-25, 0x1127d3c6457f9d.0p-81, 221 0xa90000.0p-24, 0xd49f6a.0p-25, -0x1ba930e486a0ac.0p-81, 222 0xa80000.0p-24, 0xd7a94b.0p-25, -0x1b6e645f31549e.0p-79, 223 0xa70000.0p-24, 0xdab7d0.0p-25, 0x1118a425494b61.0p-80, 224 0xa68000.0p-24, 0xdc40d5.0p-25, 0x1966f24d29d3a3.0p-80, 225 0xa58000.0p-24, 0xdf566d.0p-25, -0x1d8e52eb2248f1.0p-82, 226 0xa48000.0p-24, 0xe270ce.0p-25, -0x1ee370f96e6b68.0p-80, 227 0xa40000.0p-24, 0xe3ffce.0p-25, 0x1d155324911f57.0p-80, 228 0xa30000.0p-24, 0xe72179.0p-25, -0x1fe6e2f2f867d9.0p-80, 229 0xa20000.0p-24, 0xea4812.0p-25, 0x1b7be9add7f4d4.0p-80, 230 0xa18000.0p-24, 0xebdd3d.0p-25, 0x1b3cfb3f7511dd.0p-79, 231 0xa08000.0p-24, 0xef0b5b.0p-25, -0x1220de1f730190.0p-79, 232 0xa00000.0p-24, 0xf0a451.0p-25, -0x176364c9ac81cd.0p-80, 233 0x9f0000.0p-24, 0xf3da16.0p-25, 0x1eed6b9aafac8d.0p-81, 234 0x9e8000.0p-24, 0xf576e9.0p-25, 0x1d593218675af2.0p-79, 235 0x9d8000.0p-24, 0xf8b47c.0p-25, -0x13e8eb7da053e0.0p-84, 236 0x9d0000.0p-24, 0xfa553f.0p-25, 0x1c063259bcade0.0p-79, 237 0x9c0000.0p-24, 0xfd9ac5.0p-25, 0x1ef491085fa3c1.0p-79, 238 0x9b8000.0p-24, 0xff3f8c.0p-25, 0x1d607a7c2b8c53.0p-79, 239 0x9a8000.0p-24, 0x814697.0p-24, -0x12ad3817004f3f.0p-78, 240 0x9a0000.0p-24, 0x821b06.0p-24, -0x189fc53117f9e5.0p-81, 241 0x990000.0p-24, 0x83c5f8.0p-24, 0x14cf15a048907b.0p-79, 242 0x988000.0p-24, 0x849c7d.0p-24, 0x1cbb1d35fb8287.0p-78, 243 0x978000.0p-24, 0x864ba6.0p-24, 0x1128639b814f9c.0p-78, 244 0x970000.0p-24, 0x87244c.0p-24, 0x184733853300f0.0p-79, 245 0x968000.0p-24, 0x87fdaa.0p-24, 0x109d23aef77dd6.0p-80, 246 0x958000.0p-24, 0x89b293.0p-24, -0x1a81ef367a59de.0p-78, 247 0x950000.0p-24, 0x8a8e20.0p-24, -0x121ad3dbb2f452.0p-78, 248 0x948000.0p-24, 0x8b6a6a.0p-24, -0x1cfb981628af72.0p-79, 249 0x938000.0p-24, 0x8d253a.0p-24, -0x1d21730ea76cfe.0p-79, 250 0x930000.0p-24, 0x8e03c2.0p-24, 0x135cc00e566f77.0p-78, 251 0x928000.0p-24, 0x8ee30d.0p-24, -0x10fcb5df257a26.0p-80, 252 0x918000.0p-24, 0x90a3ee.0p-24, -0x16e171b15433d7.0p-79, 253 0x910000.0p-24, 0x918587.0p-24, -0x1d050da07f3237.0p-79, 254 0x908000.0p-24, 0x9267e7.0p-24, 0x1be03669a5268d.0p-79, 255 0x8f8000.0p-24, 0x942f04.0p-24, 0x10b28e0e26c337.0p-79, 256 0x8f0000.0p-24, 0x9513c3.0p-24, 0x1a1d820da57cf3.0p-78, 257 0x8e8000.0p-24, 0x95f950.0p-24, -0x19ef8f13ae3cf1.0p-79, 258 0x8e0000.0p-24, 0x96dfab.0p-24, -0x109e417a6e507c.0p-78, 259 0x8d0000.0p-24, 0x98aed2.0p-24, 0x10d01a2c5b0e98.0p-79, 260 0x8c8000.0p-24, 0x9997a2.0p-24, -0x1d6a50d4b61ea7.0p-78, 261 0x8c0000.0p-24, 0x9a8145.0p-24, 0x1b3b190b83f952.0p-78, 262 0x8b8000.0p-24, 0x9b6bbf.0p-24, 0x13a69fad7e7abe.0p-78, 263 0x8b0000.0p-24, 0x9c5711.0p-24, -0x11cd12316f576b.0p-78, 264 0x8a8000.0p-24, 0x9d433b.0p-24, 0x1c95c444b807a2.0p-79, 265 0x898000.0p-24, 0x9f1e22.0p-24, -0x1b9c224ea698c3.0p-79, 266 0x890000.0p-24, 0xa00ce1.0p-24, 0x125ca93186cf0f.0p-81, 267 0x888000.0p-24, 0xa0fc80.0p-24, -0x1ee38a7bc228b3.0p-79, 268 0x880000.0p-24, 0xa1ed00.0p-24, -0x1a0db876613d20.0p-78, 269 0x878000.0p-24, 0xa2de62.0p-24, 0x193224e8516c01.0p-79, 270 0x870000.0p-24, 0xa3d0a9.0p-24, 0x1fa28b4d2541ad.0p-79, 271 0x868000.0p-24, 0xa4c3d6.0p-24, 0x1c1b5760fb4572.0p-78, 272 0x858000.0p-24, 0xa6acea.0p-24, 0x1fed5d0f65949c.0p-80, 273 0x850000.0p-24, 0xa7a2d4.0p-24, 0x1ad270c9d74936.0p-80, 274 0x848000.0p-24, 0xa899ab.0p-24, 0x199ff15ce53266.0p-79, 275 0x840000.0p-24, 0xa99171.0p-24, 0x1a19e15ccc45d2.0p-79, 276 0x838000.0p-24, 0xaa8a28.0p-24, -0x121a14ec532b36.0p-80, 277 0x830000.0p-24, 0xab83d1.0p-24, 0x1aee319980bff3.0p-79, 278 0x828000.0p-24, 0xac7e6f.0p-24, -0x18ffd9e3900346.0p-80, 279 0x820000.0p-24, 0xad7a03.0p-24, -0x1e4db102ce29f8.0p-80, 280 0x818000.0p-24, 0xae768f.0p-24, 0x17c35c55a04a83.0p-81, 281 0x810000.0p-24, 0xaf7415.0p-24, 0x1448324047019b.0p-78, 282 0x808000.0p-24, 0xb07298.0p-24, -0x1750ee3915a198.0p-78, 283 0x800000.0p-24, 0xb17218.0p-24, -0x105c610ca86c39.0p-81, 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, dk, val_hi, val_lo, z; 456 uint64_t ix, lx; 457 int i, k; 458 uint16_t hx; 459 460 EXTRACT_LDBL80_WORDS(hx, lx, x); 461 k = -16383; 462 #if 0 /* Hard to do efficiently. Don't do it until we support all modes. */ 463 if (x == 1) 464 RETURN1(rp, 0); /* log(1) = +0 in all rounding modes */ 465 #endif 466 if (hx == 0 || hx >= 0x8000) { /* zero, negative or subnormal? */ 467 if (((hx & 0x7fff) | lx) == 0) 468 RETURN1(rp, -1 / zero); /* log(+-0) = -Inf */ 469 if (hx != 0) 470 /* log(neg or [pseudo-]NaN) = qNaN: */ 471 RETURN1(rp, (x - x) / zero); 472 x *= 0x1.0p65; /* subnormal; scale up x */ 473 /* including pseudo-subnormals */ 474 EXTRACT_LDBL80_WORDS(hx, lx, x); 475 k = -16383 - 65; 476 } else if (hx >= 0x7fff || (lx & 0x8000000000000000ULL) == 0) 477 RETURN1(rp, x + x); /* log(Inf or NaN) = Inf or qNaN */ 478 /* log(pseudo-Inf) = qNaN */ 479 /* log(pseudo-NaN) = qNaN */ 480 /* log(unnormal) = qNaN */ 481 #ifndef STRUCT_RETURN 482 ENTERI(); 483 #endif 484 k += hx; 485 ix = lx & 0x7fffffffffffffffULL; 486 dk = k; 487 488 /* Scale x to be in [1, 2). */ 489 SET_LDBL_EXPSIGN(x, 0x3fff); 490 491 /* 0 <= i <= INTERVALS: */ 492 #define L2I (64 - LOG2_INTERVALS) 493 i = (ix + (1LL << (L2I - 2))) >> (L2I - 1); 494 495 /* 496 * -0.005280 < d < 0.004838. In particular, the infinite- 497 * precision |d| is <= 2**-7. Rounding of G(i) to 8 bits 498 * ensures that d is representable without extra precision for 499 * this bound on |d| (since when this calculation is expressed 500 * as x*G(i)-1, the multiplication needs as many extra bits as 501 * G(i) has and the subtraction cancels 8 bits). But for 502 * most i (107 cases out of 129), the infinite-precision |d| 503 * is <= 2**-8. G(i) is rounded to 9 bits for such i to give 504 * better accuracy (this works by improving the bound on |d|, 505 * which in turn allows rounding to 9 bits in more cases). 506 * This is only important when the original x is near 1 -- it 507 * lets us avoid using a special method to give the desired 508 * accuracy for such x. 509 */ 510 if (0) 511 d = x * G(i) - 1; 512 else { 513 #ifdef USE_UTAB 514 d = (x - H(i)) * G(i) + E(i); 515 #else 516 long double x_hi, x_lo; 517 float fx_hi; 518 519 /* 520 * Split x into x_hi + x_lo to calculate x*G(i)-1 exactly. 521 * G(i) has at most 9 bits, so the splitting point is not 522 * critical. 523 */ 524 SET_FLOAT_WORD(fx_hi, (lx >> 40) | 0x3f800000); 525 x_hi = fx_hi; 526 x_lo = x - x_hi; 527 d = x_hi * G(i) - 1 + x_lo * G(i); 528 #endif 529 } 530 531 /* 532 * Our algorithm depends on exact cancellation of F_lo(i) and 533 * F_hi(i) with dk*ln_2_lo and dk*ln2_hi when k is -1 and i is 534 * at the end of the table. This and other technical complications 535 * make it difficult to avoid the double scaling in (dk*ln2) * 536 * log(base) for base != e without losing more accuracy and/or 537 * efficiency than is gained. 538 */ 539 z = d * d; 540 val_lo = z * d * z * (z * (d * P8 + P7) + (d * P6 + P5)) + 541 (F_lo(i) + dk * ln2_lo + z * d * (d * P4 + P3)) + z * P2; 542 val_hi = d; 543 #ifdef DEBUG 544 if (fetestexcept(FE_UNDERFLOW)) 545 breakpoint(); 546 #endif 547 548 _3sumF(val_hi, val_lo, F_hi(i) + dk * ln2_hi); 549 RETURN2(rp, val_hi, val_lo); 550 } 551 552 long double 553 log1pl(long double x) 554 { 555 long double d, d_hi, d_lo, dk, f_lo, val_hi, val_lo, z; 556 long double f_hi, twopminusk; 557 uint64_t ix, lx; 558 int i, k; 559 int16_t ax, hx; 560 561 DOPRINT_START(&x); 562 EXTRACT_LDBL80_WORDS(hx, lx, x); 563 if (hx < 0x3fff) { /* x < 1, or x neg NaN */ 564 ax = hx & 0x7fff; 565 if (ax >= 0x3fff) { /* x <= -1, or x neg NaN */ 566 if (ax == 0x3fff && lx == 0x8000000000000000ULL) 567 RETURNP(-1 / zero); /* log1p(-1) = -Inf */ 568 /* log1p(x < 1, or x [pseudo-]NaN) = qNaN: */ 569 RETURNP((x - x) / (x - x)); 570 } 571 if (ax <= 0x3fbe) { /* |x| < 2**-64 */ 572 if ((int)x == 0) 573 RETURNP(x); /* x with inexact if x != 0 */ 574 } 575 f_hi = 1; 576 f_lo = x; 577 } else if (hx >= 0x7fff) { /* x +Inf or non-neg NaN */ 578 RETURNP(x + x); /* log1p(Inf or NaN) = Inf or qNaN */ 579 /* log1p(pseudo-Inf) = qNaN */ 580 /* log1p(pseudo-NaN) = qNaN */ 581 /* log1p(unnormal) = qNaN */ 582 } else if (hx < 0x407f) { /* 1 <= x < 2**128 */ 583 f_hi = x; 584 f_lo = 1; 585 } else { /* 2**128 <= x < +Inf */ 586 f_hi = x; 587 f_lo = 0; /* avoid underflow of the P5 term */ 588 } 589 ENTERI(); 590 x = f_hi + f_lo; 591 f_lo = (f_hi - x) + f_lo; 592 593 EXTRACT_LDBL80_WORDS(hx, lx, x); 594 k = -16383; 595 596 k += hx; 597 ix = lx & 0x7fffffffffffffffULL; 598 dk = k; 599 600 SET_LDBL_EXPSIGN(x, 0x3fff); 601 twopminusk = 1; 602 SET_LDBL_EXPSIGN(twopminusk, 0x7ffe - (hx & 0x7fff)); 603 f_lo *= twopminusk; 604 605 i = (ix + (1LL << (L2I - 2))) >> (L2I - 1); 606 607 /* 608 * x*G(i)-1 (with a reduced x) can be represented exactly, as 609 * above, but now we need to evaluate the polynomial on d = 610 * (x+f_lo)*G(i)-1 and extra precision is needed for that. 611 * Since x+x_lo is a hi+lo decomposition and subtracting 1 612 * doesn't lose too many bits, an inexact calculation for 613 * f_lo*G(i) is good enough. 614 */ 615 if (0) 616 d_hi = x * G(i) - 1; 617 else { 618 #ifdef USE_UTAB 619 d_hi = (x - H(i)) * G(i) + E(i); 620 #else 621 long double x_hi, x_lo; 622 float fx_hi; 623 624 SET_FLOAT_WORD(fx_hi, (lx >> 40) | 0x3f800000); 625 x_hi = fx_hi; 626 x_lo = x - x_hi; 627 d_hi = x_hi * G(i) - 1 + x_lo * G(i); 628 #endif 629 } 630 d_lo = f_lo * G(i); 631 632 /* 633 * This is _2sumF(d_hi, d_lo) inlined. The condition 634 * (d_hi == 0 || |d_hi| >= |d_lo|) for using _2sumF() is not 635 * always satisifed, so it is not clear that this works, but 636 * it works in practice. It works even if it gives a wrong 637 * normalized d_lo, since |d_lo| > |d_hi| implies that i is 638 * nonzero and d is tiny, so the F(i) term dominates d_lo. 639 * In float precision: 640 * (By exhaustive testing, the worst case is d_hi = 0x1.bp-25. 641 * And if d is only a little tinier than that, we would have 642 * another underflow problem for the P3 term; this is also ruled 643 * out by exhaustive testing.) 644 */ 645 d = d_hi + d_lo; 646 d_lo = d_hi - d + d_lo; 647 d_hi = d; 648 649 z = d * d; 650 val_lo = z * d * z * (z * (d * P8 + P7) + (d * P6 + P5)) + 651 (F_lo(i) + dk * ln2_lo + d_lo + z * d * (d * P4 + P3)) + z * P2; 652 val_hi = d_hi; 653 #ifdef DEBUG 654 if (fetestexcept(FE_UNDERFLOW)) 655 breakpoint(); 656 #endif 657 658 _3sumF(val_hi, val_lo, F_hi(i) + dk * ln2_hi); 659 RETURN2PI(val_hi, val_lo); 660 } 661 662 #ifdef STRUCT_RETURN 663 664 long double 665 logl(long double x) 666 { 667 struct ld r; 668 669 ENTERI(); 670 DOPRINT_START(&x); 671 k_logl(x, &r); 672 RETURNSPI(&r); 673 } 674 675 static const double 676 invln10_hi = 4.3429448190317999e-1, /* 0x1bcb7b1526e000.0p-54 */ 677 invln10_lo = 7.1842412889749798e-14, /* 0x1438ca9aadd558.0p-96 */ 678 invln2_hi = 1.4426950408887933e0, /* 0x171547652b8000.0p-52 */ 679 invln2_lo = 1.7010652264631490e-13; /* 0x17f0bbbe87fed0.0p-95 */ 680 681 long double 682 log10l(long double x) 683 { 684 struct ld r; 685 long double hi, lo; 686 687 ENTERI(); 688 DOPRINT_START(&x); 689 k_logl(x, &r); 690 if (!r.lo_set) 691 RETURNPI(r.hi); 692 _2sumF(r.hi, r.lo); 693 hi = (float)r.hi; 694 lo = r.lo + (r.hi - hi); 695 RETURN2PI(invln10_hi * hi, 696 (invln10_lo + invln10_hi) * lo + invln10_lo * hi); 697 } 698 699 long double 700 log2l(long double x) 701 { 702 struct ld r; 703 long double hi, lo; 704 705 ENTERI(); 706 DOPRINT_START(&x); 707 k_logl(x, &r); 708 if (!r.lo_set) 709 RETURNPI(r.hi); 710 _2sumF(r.hi, r.lo); 711 hi = (float)r.hi; 712 lo = r.lo + (r.hi - hi); 713 RETURN2PI(invln2_hi * hi, 714 (invln2_lo + invln2_hi) * lo + invln2_lo * hi); 715 } 716 717 #endif /* STRUCT_RETURN */ 718