newlib/libm/common/log2_data.c

   1 /* Data for log2.
   2    Copyright (c) 2018 Arm Ltd.  All rights reserved.
   3
   4    SPDX-License-Identifier: BSD-3-Clause
   5
   6    Redistribution and use in source and binary forms, with or without
   7    modification, are permitted provided that the following conditions
   8    are met:
   9    1. Redistributions of source code must retain the above copyright
  10       notice, this list of conditions and the following 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    3. The name of the company may not be used to endorse or promote
  15       products derived from this software without specific prior written
  16       permission.
  17
  18    THIS SOFTWARE IS PROVIDED BY ARM LTD ``AS IS'' AND ANY EXPRESS OR IMPLIED
  19    WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF
  20    MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED.
  21    IN NO EVENT SHALL ARM LTD BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
  22    SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED
  23    TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR
  24    PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF
  25    LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING
  26    NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
  27    SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. */
  28
  29 #include "fdlibm.h"
  30 #if !__OBSOLETE_MATH
  31
  32 #include "math_config.h"
  33
  34 #define N (1 << LOG2_TABLE_BITS)
  35
  36 const struct log2_data __log2_data = {
  37 // First coefficient: 0x1.71547652b82fe1777d0ffda0d24p0
  38 .invln2hi = 0x1.7154765200000p+0,
  39 .invln2lo = 0x1.705fc2eefa200p-33,
  40 .poly1 = {
  41 #if LOG2_POLY1_ORDER == 11
  42 // relative error: 0x1.2fad8188p-63
  43 // in -0x1.5b51p-5 0x1.6ab2p-5
  44 -0x1.71547652b82fep-1,
  45 0x1.ec709dc3a03f7p-2,
  46 -0x1.71547652b7c3fp-2,
  47 0x1.2776c50f05be4p-2,
  48 -0x1.ec709dd768fe5p-3,
  49 0x1.a61761ec4e736p-3,
  50 -0x1.7153fbc64a79bp-3,
  51 0x1.484d154f01b4ap-3,
  52 -0x1.289e4a72c383cp-3,
  53 0x1.0b32f285aee66p-3,
  54 #endif
  55 },
  56 .poly = {
  57 #if N == 64 && LOG2_POLY_ORDER == 7
  58 // relative error: 0x1.a72c2bf8p-58
  59 // abs error: 0x1.67a552c8p-66
  60 // in -0x1.f45p-8 0x1.f45p-8
  61 -0x1.71547652b8339p-1,
  62 0x1.ec709dc3a04bep-2,
  63 -0x1.7154764702ffbp-2,
  64 0x1.2776c50034c48p-2,
  65 -0x1.ec7b328ea92bcp-3,
  66 0x1.a6225e117f92ep-3,
  67 #endif
  68 },
  69 /* Algorithm:
  70
  71         x = 2^k z
  72         log2(x) = k + log2(c) + log2(z/c)
  73         log2(z/c) = poly(z/c - 1)
  74
  75 where z is in [1.6p-1; 1.6p0] which is split into N subintervals and z falls
  76 into the ith one, then table entries are computed as
  77
  78         tab[i].invc = 1/c
  79         tab[i].logc = (double)log2(c)
  80         tab2[i].chi = (double)c
  81         tab2[i].clo = (double)(c - (double)c)
  82
  83 where c is near the center of the subinterval and is chosen by trying +-2^29
  84 floating point invc candidates around 1/center and selecting one for which
  85
  86         1) the rounding error in 0x1.8p10 + logc is 0,
  87         2) the rounding error in z - chi - clo is < 0x1p-64 and
  88         3) the rounding error in (double)log2(c) is minimized (< 0x1p-68).
  89
  90 Note: 1) ensures that k + logc can be computed without rounding error, 2)
  91 ensures that z/c - 1 can be computed as (z - chi - clo)*invc with close to a
  92 single rounding error when there is no fast fma for z*invc - 1, 3) ensures
  93 that logc + poly(z/c - 1) has small error, however near x == 1 when
  94 |log2(x)| < 0x1p-4, this is not enough so that is special cased.  */
  95 .tab = {
  96 #if N == 64
  97 {0x1.724286bb1acf8p+0, -0x1.1095feecdb000p-1},
  98 {0x1.6e1f766d2cca1p+0, -0x1.08494bd76d000p-1},
  99 {0x1.6a13d0e30d48ap+0, -0x1.00143aee8f800p-1},
 100 {0x1.661ec32d06c85p+0, -0x1.efec5360b4000p-2},
 101 {0x1.623fa951198f8p+0, -0x1.dfdd91ab7e000p-2},
 102 {0x1.5e75ba4cf026cp+0, -0x1.cffae0cc79000p-2},
 103 {0x1.5ac055a214fb8p+0, -0x1.c043811fda000p-2},
 104 {0x1.571ed0f166e1ep+0, -0x1.b0b67323ae000p-2},
 105 {0x1.53909590bf835p+0, -0x1.a152f5a2db000p-2},
 106 {0x1.5014fed61adddp+0, -0x1.9217f5af86000p-2},
 107 {0x1.4cab88e487bd0p+0, -0x1.8304db0719000p-2},
 108 {0x1.49539b4334feep+0, -0x1.74189f9a9e000p-2},
 109 {0x1.460cbdfafd569p+0, -0x1.6552bb5199000p-2},
 110 {0x1.42d664ee4b953p+0, -0x1.56b23a29b1000p-2},
 111 {0x1.3fb01111dd8a6p+0, -0x1.483650f5fa000p-2},
 112 {0x1.3c995b70c5836p+0, -0x1.39de937f6a000p-2},
 113 {0x1.3991c4ab6fd4ap+0, -0x1.2baa1538d6000p-2},
 114 {0x1.3698e0ce099b5p+0, -0x1.1d98340ca4000p-2},
 115 {0x1.33ae48213e7b2p+0, -0x1.0fa853a40e000p-2},
 116 {0x1.30d191985bdb1p+0, -0x1.01d9c32e73000p-2},
 117 {0x1.2e025cab271d7p+0, -0x1.e857da2fa6000p-3},
 118 {0x1.2b404cf13cd82p+0, -0x1.cd3c8633d8000p-3},
 119 {0x1.288b02c7ccb50p+0, -0x1.b26034c14a000p-3},
 120 {0x1.25e2263944de5p+0, -0x1.97c1c2f4fe000p-3},
 121 {0x1.234563d8615b1p+0, -0x1.7d6023f800000p-3},
 122 {0x1.20b46e33eaf38p+0, -0x1.633a71a05e000p-3},
 123 {0x1.1e2eefdcda3ddp+0, -0x1.494f5e9570000p-3},
 124 {0x1.1bb4a580b3930p+0, -0x1.2f9e424e0a000p-3},
 125 {0x1.19453847f2200p+0, -0x1.162595afdc000p-3},
 126 {0x1.16e06c0d5d73cp+0, -0x1.f9c9a75bd8000p-4},
 127 {0x1.1485f47b7e4c2p+0, -0x1.c7b575bf9c000p-4},
 128 {0x1.12358ad0085d1p+0, -0x1.960c60ff48000p-4},
 129 {0x1.0fef00f532227p+0, -0x1.64ce247b60000p-4},
 130 {0x1.0db2077d03a8fp+0, -0x1.33f78b2014000p-4},
 131 {0x1.0b7e6d65980d9p+0, -0x1.0387d1a42c000p-4},
 132 {0x1.0953efe7b408dp+0, -0x1.a6f9208b50000p-5},
 133 {0x1.07325cac53b83p+0, -0x1.47a954f770000p-5},
 134 {0x1.05197e40d1b5cp+0, -0x1.d23a8c50c0000p-6},
 135 {0x1.03091c1208ea2p+0, -0x1.16a2629780000p-6},
 136 {0x1.0101025b37e21p+0, -0x1.720f8d8e80000p-8},
 137 {0x1.fc07ef9caa76bp-1, 0x1.6fe53b1500000p-7},
 138 {0x1.f4465d3f6f184p-1, 0x1.11ccce10f8000p-5},
 139 {0x1.ecc079f84107fp-1, 0x1.c4dfc8c8b8000p-5},
 140 {0x1.e573a99975ae8p-1, 0x1.3aa321e574000p-4},
 141 {0x1.de5d6f0bd3de6p-1, 0x1.918a0d08b8000p-4},
 142 {0x1.d77b681ff38b3p-1, 0x1.e72e9da044000p-4},
 143 {0x1.d0cb5724de943p-1, 0x1.1dcd2507f6000p-3},
 144 {0x1.ca4b2dc0e7563p-1, 0x1.476ab03dea000p-3},
 145 {0x1.c3f8ee8d6cb51p-1, 0x1.7074377e22000p-3},
 146 {0x1.bdd2b4f020c4cp-1, 0x1.98ede8ba94000p-3},
 147 {0x1.b7d6c006015cap-1, 0x1.c0db86ad2e000p-3},
 148 {0x1.b20366e2e338fp-1, 0x1.e840aafcee000p-3},
 149 {0x1.ac57026295039p-1, 0x1.0790ab4678000p-2},
 150 {0x1.a6d01bc2731ddp-1, 0x1.1ac056801c000p-2},
 151 {0x1.a16d3bc3ff18bp-1, 0x1.2db11d4fee000p-2},
 152 {0x1.9c2d14967feadp-1, 0x1.406464ec58000p-2},
 153 {0x1.970e4f47c9902p-1, 0x1.52dbe093af000p-2},
 154 {0x1.920fb3982bcf2p-1, 0x1.651902050d000p-2},
 155 {0x1.8d30187f759f1p-1, 0x1.771d2cdeaf000p-2},
 156 {0x1.886e5ebb9f66dp-1, 0x1.88e9c857d9000p-2},
 157 {0x1.83c97b658b994p-1, 0x1.9a80155e16000p-2},
 158 {0x1.7f405ffc61022p-1, 0x1.abe186ed3d000p-2},
 159 {0x1.7ad22181415cap-1, 0x1.bd0f2aea0e000p-2},
 160 {0x1.767dcf99eff8cp-1, 0x1.ce0a43dbf4000p-2},
 161 #endif
 162 },
 163 #if !HAVE_FAST_FMA
 164 .tab2 = {
 165 # if N == 64
 166 {0x1.6200012b90a8ep-1, 0x1.904ab0644b605p-55},
 167 {0x1.66000045734a6p-1, 0x1.1ff9bea62f7a9p-57},
 168 {0x1.69fffc325f2c5p-1, 0x1.27ecfcb3c90bap-55},
 169 {0x1.6e00038b95a04p-1, 0x1.8ff8856739326p-55},
 170 {0x1.71fffe09994e3p-1, 0x1.afd40275f82b1p-55},
 171 {0x1.7600015590e1p-1, -0x1.2fd75b4238341p-56},
 172 {0x1.7a00012655bd5p-1, 0x1.808e67c242b76p-56},
 173 {0x1.7e0003259e9a6p-1, -0x1.208e426f622b7p-57},
 174 {0x1.81fffedb4b2d2p-1, -0x1.402461ea5c92fp-55},
 175 {0x1.860002dfafcc3p-1, 0x1.df7f4a2f29a1fp-57},
 176 {0x1.89ffff78c6b5p-1, -0x1.e0453094995fdp-55},
 177 {0x1.8e00039671566p-1, -0x1.a04f3bec77b45p-55},
 178 {0x1.91fffe2bf1745p-1, -0x1.7fa34400e203cp-56},
 179 {0x1.95fffcc5c9fd1p-1, -0x1.6ff8005a0695dp-56},
 180 {0x1.9a0003bba4767p-1, 0x1.0f8c4c4ec7e03p-56},
 181 {0x1.9dfffe7b92da5p-1, 0x1.e7fd9478c4602p-55},
 182 {0x1.a1fffd72efdafp-1, -0x1.a0c554dcdae7ep-57},
 183 {0x1.a5fffde04ff95p-1, 0x1.67da98ce9b26bp-55},
 184 {0x1.a9fffca5e8d2bp-1, -0x1.284c9b54c13dep-55},
 185 {0x1.adfffddad03eap-1, 0x1.812c8ea602e3cp-58},
 186 {0x1.b1ffff10d3d4dp-1, -0x1.efaddad27789cp-55},
 187 {0x1.b5fffce21165ap-1, 0x1.3cb1719c61237p-58},
 188 {0x1.b9fffd950e674p-1, 0x1.3f7d94194cep-56},
 189 {0x1.be000139ca8afp-1, 0x1.50ac4215d9bcp-56},
 190 {0x1.c20005b46df99p-1, 0x1.beea653e9c1c9p-57},
 191 {0x1.c600040b9f7aep-1, -0x1.c079f274a70d6p-56},
 192 {0x1.ca0006255fd8ap-1, -0x1.a0b4076e84c1fp-56},
 193 {0x1.cdfffd94c095dp-1, 0x1.8f933f99ab5d7p-55},
 194 {0x1.d1ffff975d6cfp-1, -0x1.82c08665fe1bep-58},
 195 {0x1.d5fffa2561c93p-1, -0x1.b04289bd295f3p-56},
 196 {0x1.d9fff9d228b0cp-1, 0x1.70251340fa236p-55},
 197 {0x1.de00065bc7e16p-1, -0x1.5011e16a4d80cp-56},
 198 {0x1.e200002f64791p-1, 0x1.9802f09ef62ep-55},
 199 {0x1.e600057d7a6d8p-1, -0x1.e0b75580cf7fap-56},
 200 {0x1.ea00027edc00cp-1, -0x1.c848309459811p-55},
 201 {0x1.ee0006cf5cb7cp-1, -0x1.f8027951576f4p-55},
 202 {0x1.f2000782b7dccp-1, -0x1.f81d97274538fp-55},
 203 {0x1.f6000260c450ap-1, -0x1.071002727ffdcp-59},
 204 {0x1.f9fffe88cd533p-1, -0x1.81bdce1fda8bp-58},
 205 {0x1.fdfffd50f8689p-1, 0x1.7f91acb918e6ep-55},
 206 {0x1.0200004292367p+0, 0x1.b7ff365324681p-54},
 207 {0x1.05fffe3e3d668p+0, 0x1.6fa08ddae957bp-55},
 208 {0x1.0a0000a85a757p+0, -0x1.7e2de80d3fb91p-58},
 209 {0x1.0e0001a5f3fccp+0, -0x1.1823305c5f014p-54},
 210 {0x1.11ffff8afbaf5p+0, -0x1.bfabb6680bac2p-55},
 211 {0x1.15fffe54d91adp+0, -0x1.d7f121737e7efp-54},
 212 {0x1.1a00011ac36e1p+0, 0x1.c000a0516f5ffp-54},
 213 {0x1.1e00019c84248p+0, -0x1.082fbe4da5dap-54},
 214 {0x1.220000ffe5e6ep+0, -0x1.8fdd04c9cfb43p-55},
 215 {0x1.26000269fd891p+0, 0x1.cfe2a7994d182p-55},
 216 {0x1.2a00029a6e6dap+0, -0x1.00273715e8bc5p-56},
 217 {0x1.2dfffe0293e39p+0, 0x1.b7c39dab2a6f9p-54},
 218 {0x1.31ffff7dcf082p+0, 0x1.df1336edc5254p-56},
 219 {0x1.35ffff05a8b6p+0, -0x1.e03564ccd31ebp-54},
 220 {0x1.3a0002e0eaeccp+0, 0x1.5f0e74bd3a477p-56},
 221 {0x1.3e000043bb236p+0, 0x1.c7dcb149d8833p-54},
 222 {0x1.4200002d187ffp+0, 0x1.e08afcf2d3d28p-56},
 223 {0x1.460000d387cb1p+0, 0x1.20837856599a6p-55},
 224 {0x1.4a00004569f89p+0, -0x1.9fa5c904fbcd2p-55},
 225 {0x1.4e000043543f3p+0, -0x1.81125ed175329p-56},
 226 {0x1.51fffcc027f0fp+0, 0x1.883d8847754dcp-54},
 227 {0x1.55ffffd87b36fp+0, -0x1.709e731d02807p-55},
 228 {0x1.59ffff21df7bap+0, 0x1.7f79f68727b02p-55},
 229 {0x1.5dfffebfc3481p+0, -0x1.180902e30e93ep-54},
 230 # endif
 231 },
 232 #endif /* !HAVE_FAST_FMA */
 233 };
 234 #endif /* __OBSOLETE_MATH */