4 * Copyright (c) 2018, Arm Limited.
5 * SPDX-License-Identifier: MIT
10 #define N (1 << LOG_TABLE_BITS)
12 const struct log_data __log_data
= {
13 .ln2hi
= 0x1.62e42fefa3800p
-1,
14 .ln2lo
= 0x1.ef35793c76730p
-45,
16 // relative error: 0x1.c04d76cp-63
17 // in -0x1p-4 0x1.09p-4 (|log(1+x)| > 0x1p-4 outside the interval)
20 -0x1.ffffffffffdcbp
-3,
22 -0x1.55555556745a7p
-3,
24 -0x1.fffffa4423d65p
-4,
26 -0x1.999eb43b068ffp
-4,
28 -0x1.5521375d145cdp
-4,
31 // relative error: 0x1.926199e8p-56
32 // abs error: 0x1.882ff33p-65
33 // in -0x1.fp-9 0x1.fp-9
34 -0x1.0000000000001p
-1,
38 -0x1.55575e506c89fp
-3,
43 log(x) = k ln2 + log(c) + log(z/c)
44 log(z/c) = poly(z/c - 1)
46 where z is in [1.6p-1; 1.6p0] which is split into N subintervals and z falls
47 into the ith one, then table entries are computed as
50 tab[i].logc = (double)log(c)
51 tab2[i].chi = (double)c
52 tab2[i].clo = (double)(c - (double)c)
54 where c is near the center of the subinterval and is chosen by trying +-2^29
55 floating point invc candidates around 1/center and selecting one for which
57 1) the rounding error in 0x1.8p9 + logc is 0,
58 2) the rounding error in z - chi - clo is < 0x1p-66 and
59 3) the rounding error in (double)log(c) is minimized (< 0x1p-66).
61 Note: 1) ensures that k*ln2hi + logc can be computed without rounding error,
62 2) ensures that z/c - 1 can be computed as (z - chi - clo)*invc with close to
63 a single rounding error when there is no fast fma for z*invc - 1, 3) ensures
64 that logc + poly(z/c - 1) has small error, however near x == 1 when
65 |log(x)| < 0x1p-4, this is not enough so that is special cased. */
67 {0x1.734f0c3e0de9fp
+0, -0x1.7cc7f79e69000p
-2},
68 {0x1.713786a2ce91fp
+0, -0x1.76feec20d0000p
-2},
69 {0x1.6f26008fab5a0p
+0, -0x1.713e31351e000p
-2},
70 {0x1.6d1a61f138c7dp
+0, -0x1.6b85b38287800p
-2},
71 {0x1.6b1490bc5b4d1p
+0, -0x1.65d5590807800p
-2},
72 {0x1.69147332f0cbap
+0, -0x1.602d076180000p
-2},
73 {0x1.6719f18224223p
+0, -0x1.5a8ca86909000p
-2},
74 {0x1.6524f99a51ed9p
+0, -0x1.54f4356035000p
-2},
75 {0x1.63356aa8f24c4p
+0, -0x1.4f637c36b4000p
-2},
76 {0x1.614b36b9ddc14p
+0, -0x1.49da7fda85000p
-2},
77 {0x1.5f66452c65c4cp
+0, -0x1.445923989a800p
-2},
78 {0x1.5d867b5912c4fp
+0, -0x1.3edf439b0b800p
-2},
79 {0x1.5babccb5b90dep
+0, -0x1.396ce448f7000p
-2},
80 {0x1.59d61f2d91a78p
+0, -0x1.3401e17bda000p
-2},
81 {0x1.5805612465687p
+0, -0x1.2e9e2ef468000p
-2},
82 {0x1.56397cee76bd3p
+0, -0x1.2941b3830e000p
-2},
83 {0x1.54725e2a77f93p
+0, -0x1.23ec58cda8800p
-2},
84 {0x1.52aff42064583p
+0, -0x1.1e9e129279000p
-2},
85 {0x1.50f22dbb2bddfp
+0, -0x1.1956d2b48f800p
-2},
86 {0x1.4f38f4734ded7p
+0, -0x1.141679ab9f800p
-2},
87 {0x1.4d843cfde2840p
+0, -0x1.0edd094ef9800p
-2},
88 {0x1.4bd3ec078a3c8p
+0, -0x1.09aa518db1000p
-2},
89 {0x1.4a27fc3e0258ap
+0, -0x1.047e65263b800p
-2},
90 {0x1.4880524d48434p
+0, -0x1.feb224586f000p
-3},
91 {0x1.46dce1b192d0bp
+0, -0x1.f474a7517b000p
-3},
92 {0x1.453d9d3391854p
+0, -0x1.ea4443d103000p
-3},
93 {0x1.43a2744b4845ap
+0, -0x1.e020d44e9b000p
-3},
94 {0x1.420b54115f8fbp
+0, -0x1.d60a22977f000p
-3},
95 {0x1.40782da3ef4b1p
+0, -0x1.cc00104959000p
-3},
96 {0x1.3ee8f5d57fe8fp
+0, -0x1.c202956891000p
-3},
97 {0x1.3d5d9a00b4ce9p
+0, -0x1.b81178d811000p
-3},
98 {0x1.3bd60c010c12bp
+0, -0x1.ae2c9ccd3d000p
-3},
99 {0x1.3a5242b75dab8p
+0, -0x1.a45402e129000p
-3},
100 {0x1.38d22cd9fd002p
+0, -0x1.9a877681df000p
-3},
101 {0x1.3755bc5847a1cp
+0, -0x1.90c6d69483000p
-3},
102 {0x1.35dce49ad36e2p
+0, -0x1.87120a645c000p
-3},
103 {0x1.34679984dd440p
+0, -0x1.7d68fb4143000p
-3},
104 {0x1.32f5cceffcb24p
+0, -0x1.73cb83c627000p
-3},
105 {0x1.3187775a10d49p
+0, -0x1.6a39a9b376000p
-3},
106 {0x1.301c8373e3990p
+0, -0x1.60b3154b7a000p
-3},
107 {0x1.2eb4ebb95f841p
+0, -0x1.5737d76243000p
-3},
108 {0x1.2d50a0219a9d1p
+0, -0x1.4dc7b8fc23000p
-3},
109 {0x1.2bef9a8b7fd2ap
+0, -0x1.4462c51d20000p
-3},
110 {0x1.2a91c7a0c1babp
+0, -0x1.3b08abc830000p
-3},
111 {0x1.293726014b530p
+0, -0x1.31b996b490000p
-3},
112 {0x1.27dfa5757a1f5p
+0, -0x1.2875490a44000p
-3},
113 {0x1.268b39b1d3bbfp
+0, -0x1.1f3b9f879a000p
-3},
114 {0x1.2539d838ff5bdp
+0, -0x1.160c8252ca000p
-3},
115 {0x1.23eb7aac9083bp
+0, -0x1.0ce7f57f72000p
-3},
116 {0x1.22a012ba940b6p
+0, -0x1.03cdc49fea000p
-3},
117 {0x1.2157996cc4132p
+0, -0x1.f57bdbc4b8000p
-4},
118 {0x1.201201dd2fc9bp
+0, -0x1.e370896404000p
-4},
119 {0x1.1ecf4494d480bp
+0, -0x1.d17983ef94000p
-4},
120 {0x1.1d8f5528f6569p
+0, -0x1.bf9674ed8a000p
-4},
121 {0x1.1c52311577e7cp
+0, -0x1.adc79202f6000p
-4},
122 {0x1.1b17c74cb26e9p
+0, -0x1.9c0c3e7288000p
-4},
123 {0x1.19e010c2c1ab6p
+0, -0x1.8a646b372c000p
-4},
124 {0x1.18ab07bb670bdp
+0, -0x1.78d01b3ac0000p
-4},
125 {0x1.1778a25efbcb6p
+0, -0x1.674f145380000p
-4},
126 {0x1.1648d354c31dap
+0, -0x1.55e0e6d878000p
-4},
127 {0x1.151b990275fddp
+0, -0x1.4485cdea1e000p
-4},
128 {0x1.13f0ea432d24cp
+0, -0x1.333d94d6aa000p
-4},
129 {0x1.12c8b7210f9dap
+0, -0x1.22079f8c56000p
-4},
130 {0x1.11a3028ecb531p
+0, -0x1.10e4698622000p
-4},
131 {0x1.107fbda8434afp
+0, -0x1.ffa6c6ad20000p
-5},
132 {0x1.0f5ee0f4e6bb3p
+0, -0x1.dda8d4a774000p
-5},
133 {0x1.0e4065d2a9fcep
+0, -0x1.bbcece4850000p
-5},
134 {0x1.0d244632ca521p
+0, -0x1.9a1894012c000p
-5},
135 {0x1.0c0a77ce2981ap
+0, -0x1.788583302c000p
-5},
136 {0x1.0af2f83c636d1p
+0, -0x1.5715e67d68000p
-5},
137 {0x1.09ddb98a01339p
+0, -0x1.35c8a49658000p
-5},
138 {0x1.08cabaf52e7dfp
+0, -0x1.149e364154000p
-5},
139 {0x1.07b9f2f4e28fbp
+0, -0x1.e72c082eb8000p
-6},
140 {0x1.06ab58c358f19p
+0, -0x1.a55f152528000p
-6},
141 {0x1.059eea5ecf92cp
+0, -0x1.63d62cf818000p
-6},
142 {0x1.04949cdd12c90p
+0, -0x1.228fb8caa0000p
-6},
143 {0x1.038c6c6f0ada9p
+0, -0x1.c317b20f90000p
-7},
144 {0x1.02865137932a9p
+0, -0x1.419355daa0000p
-7},
145 {0x1.0182427ea7348p
+0, -0x1.81203c2ec0000p
-8},
146 {0x1.008040614b195p
+0, -0x1.0040979240000p
-9},
147 {0x1.fe01ff726fa1ap
-1, 0x1.feff384900000p
-9},
148 {0x1.fa11cc261ea74p
-1, 0x1.7dc41353d0000p
-7},
149 {0x1.f6310b081992ep
-1, 0x1.3cea3c4c28000p
-6},
150 {0x1.f25f63ceeadcdp
-1, 0x1.b9fc114890000p
-6},
151 {0x1.ee9c8039113e7p
-1, 0x1.1b0d8ce110000p
-5},
152 {0x1.eae8078cbb1abp
-1, 0x1.58a5bd001c000p
-5},
153 {0x1.e741aa29d0c9bp
-1, 0x1.95c8340d88000p
-5},
154 {0x1.e3a91830a99b5p
-1, 0x1.d276aef578000p
-5},
155 {0x1.e01e009609a56p
-1, 0x1.07598e598c000p
-4},
156 {0x1.dca01e577bb98p
-1, 0x1.253f5e30d2000p
-4},
157 {0x1.d92f20b7c9103p
-1, 0x1.42edd8b380000p
-4},
158 {0x1.d5cac66fb5ccep
-1, 0x1.606598757c000p
-4},
159 {0x1.d272caa5ede9dp
-1, 0x1.7da76356a0000p
-4},
160 {0x1.cf26e3e6b2ccdp
-1, 0x1.9ab434e1c6000p
-4},
161 {0x1.cbe6da2a77902p
-1, 0x1.b78c7bb0d6000p
-4},
162 {0x1.c8b266d37086dp
-1, 0x1.d431332e72000p
-4},
163 {0x1.c5894bd5d5804p
-1, 0x1.f0a3171de6000p
-4},
164 {0x1.c26b533bb9f8cp
-1, 0x1.067152b914000p
-3},
165 {0x1.bf583eeece73fp
-1, 0x1.147858292b000p
-3},
166 {0x1.bc4fd75db96c1p
-1, 0x1.2266ecdca3000p
-3},
167 {0x1.b951e0c864a28p
-1, 0x1.303d7a6c55000p
-3},
168 {0x1.b65e2c5ef3e2cp
-1, 0x1.3dfc33c331000p
-3},
169 {0x1.b374867c9888bp
-1, 0x1.4ba366b7a8000p
-3},
170 {0x1.b094b211d304ap
-1, 0x1.5933928d1f000p
-3},
171 {0x1.adbe885f2ef7ep
-1, 0x1.66acd2418f000p
-3},
172 {0x1.aaf1d31603da2p
-1, 0x1.740f8ec669000p
-3},
173 {0x1.a82e63fd358a7p
-1, 0x1.815c0f51af000p
-3},
174 {0x1.a5740ef09738bp
-1, 0x1.8e92954f68000p
-3},
175 {0x1.a2c2a90ab4b27p
-1, 0x1.9bb3602f84000p
-3},
176 {0x1.a01a01393f2d1p
-1, 0x1.a8bed1c2c0000p
-3},
177 {0x1.9d79f24db3c1bp
-1, 0x1.b5b515c01d000p
-3},
178 {0x1.9ae2505c7b190p
-1, 0x1.c2967ccbcc000p
-3},
179 {0x1.9852ef297ce2fp
-1, 0x1.cf635d5486000p
-3},
180 {0x1.95cbaeea44b75p
-1, 0x1.dc1bd3446c000p
-3},
181 {0x1.934c69de74838p
-1, 0x1.e8c01b8cfe000p
-3},
182 {0x1.90d4f2f6752e6p
-1, 0x1.f5509c0179000p
-3},
183 {0x1.8e6528effd79dp
-1, 0x1.00e6c121fb800p
-2},
184 {0x1.8bfce9fcc007cp
-1, 0x1.071b80e93d000p
-2},
185 {0x1.899c0dabec30ep
-1, 0x1.0d46b9e867000p
-2},
186 {0x1.87427aa2317fbp
-1, 0x1.13687334bd000p
-2},
187 {0x1.84f00acb39a08p
-1, 0x1.1980d67234800p
-2},
188 {0x1.82a49e8653e55p
-1, 0x1.1f8ffe0cc8000p
-2},
189 {0x1.8060195f40260p
-1, 0x1.2595fd7636800p
-2},
190 {0x1.7e22563e0a329p
-1, 0x1.2b9300914a800p
-2},
191 {0x1.7beb377dcb5adp
-1, 0x1.3187210436000p
-2},
192 {0x1.79baa679725c2p
-1, 0x1.377266dec1800p
-2},
193 {0x1.77907f2170657p
-1, 0x1.3d54ffbaf3000p
-2},
194 {0x1.756cadbd6130cp
-1, 0x1.432eee32fe000p
-2},
198 {0x1.61000014fb66bp
-1, 0x1.e026c91425b3cp
-56},
199 {0x1.63000034db495p
-1, 0x1.dbfea48005d41p
-55},
200 {0x1.650000d94d478p
-1, 0x1.e7fa786d6a5b7p
-55},
201 {0x1.67000074e6fadp
-1, 0x1.1fcea6b54254cp
-57},
202 {0x1.68ffffedf0faep
-1, -0x1.c7e274c590efdp
-56},
203 {0x1.6b0000763c5bcp
-1, -0x1.ac16848dcda01p
-55},
204 {0x1.6d0001e5cc1f6p
-1, 0x1.33f1c9d499311p
-55},
205 {0x1.6efffeb05f63ep
-1, -0x1.e80041ae22d53p
-56},
206 {0x1.710000e86978p
-1, 0x1.bff6671097952p
-56},
207 {0x1.72ffffc67e912p
-1, 0x1.c00e226bd8724p
-55},
208 {0x1.74fffdf81116ap
-1, -0x1.e02916ef101d2p
-57},
209 {0x1.770000f679c9p
-1, -0x1.7fc71cd549c74p
-57},
210 {0x1.78ffffa7ec835p
-1, 0x1.1bec19ef50483p
-55},
211 {0x1.7affffe20c2e6p
-1, -0x1.07e1729cc6465p
-56},
212 {0x1.7cfffed3fc9p
-1, -0x1.08072087b8b1cp
-55},
213 {0x1.7efffe9261a76p
-1, 0x1.dc0286d9df9aep
-55},
214 {0x1.81000049ca3e8p
-1, 0x1.97fd251e54c33p
-55},
215 {0x1.8300017932c8fp
-1, -0x1.afee9b630f381p
-55},
216 {0x1.850000633739cp
-1, 0x1.9bfbf6b6535bcp
-55},
217 {0x1.87000204289c6p
-1, -0x1.bbf65f3117b75p
-55},
218 {0x1.88fffebf57904p
-1, -0x1.9006ea23dcb57p
-55},
219 {0x1.8b00022bc04dfp
-1, -0x1.d00df38e04b0ap
-56},
220 {0x1.8cfffe50c1b8ap
-1, -0x1.8007146ff9f05p
-55},
221 {0x1.8effffc918e43p
-1, 0x1.3817bd07a7038p
-55},
222 {0x1.910001efa5fc7p
-1, 0x1.93e9176dfb403p
-55},
223 {0x1.9300013467bb9p
-1, 0x1.f804e4b980276p
-56},
224 {0x1.94fffe6ee076fp
-1, -0x1.f7ef0d9ff622ep
-55},
225 {0x1.96fffde3c12d1p
-1, -0x1.082aa962638bap
-56},
226 {0x1.98ffff4458a0dp
-1, -0x1.7801b9164a8efp
-55},
227 {0x1.9afffdd982e3ep
-1, -0x1.740e08a5a9337p
-55},
228 {0x1.9cfffed49fb66p
-1, 0x1.fce08c19bep
-60},
229 {0x1.9f00020f19c51p
-1, -0x1.a3faa27885b0ap
-55},
230 {0x1.a10001145b006p
-1, 0x1.4ff489958da56p
-56},
231 {0x1.a300007bbf6fap
-1, 0x1.cbeab8a2b6d18p
-55},
232 {0x1.a500010971d79p
-1, 0x1.8fecadd78793p
-55},
233 {0x1.a70001df52e48p
-1, -0x1.f41763dd8abdbp
-55},
234 {0x1.a90001c593352p
-1, -0x1.ebf0284c27612p
-55},
235 {0x1.ab0002a4f3e4bp
-1, -0x1.9fd043cff3f5fp
-57},
236 {0x1.acfffd7ae1ed1p
-1, -0x1.23ee7129070b4p
-55},
237 {0x1.aefffee510478p
-1, 0x1.a063ee00edea3p
-57},
238 {0x1.b0fffdb650d5bp
-1, 0x1.a06c8381f0ab9p
-58},
239 {0x1.b2ffffeaaca57p
-1, -0x1.9011e74233c1dp
-56},
240 {0x1.b4fffd995badcp
-1, -0x1.9ff1068862a9fp
-56},
241 {0x1.b7000249e659cp
-1, 0x1.aff45d0864f3ep
-55},
242 {0x1.b8ffff987164p
-1, 0x1.cfe7796c2c3f9p
-56},
243 {0x1.bafffd204cb4fp
-1, -0x1.3ff27eef22bc4p
-57},
244 {0x1.bcfffd2415c45p
-1, -0x1.cffb7ee3bea21p
-57},
245 {0x1.beffff86309dfp
-1, -0x1.14103972e0b5cp
-55},
246 {0x1.c0fffe1b57653p
-1, 0x1.bc16494b76a19p
-55},
247 {0x1.c2ffff1fa57e3p
-1, -0x1.4feef8d30c6edp
-57},
248 {0x1.c4fffdcbfe424p
-1, -0x1.43f68bcec4775p
-55},
249 {0x1.c6fffed54b9f7p
-1, 0x1.47ea3f053e0ecp
-55},
250 {0x1.c8fffeb998fd5p
-1, 0x1.383068df992f1p
-56},
251 {0x1.cb0002125219ap
-1, -0x1.8fd8e64180e04p
-57},
252 {0x1.ccfffdd94469cp
-1, 0x1.e7ebe1cc7ea72p
-55},
253 {0x1.cefffeafdc476p
-1, 0x1.ebe39ad9f88fep
-55},
254 {0x1.d1000169af82bp
-1, 0x1.57d91a8b95a71p
-56},
255 {0x1.d30000d0ff71dp
-1, 0x1.9c1906970c7dap
-55},
256 {0x1.d4fffea790fc4p
-1, -0x1.80e37c558fe0cp
-58},
257 {0x1.d70002edc87e5p
-1, -0x1.f80d64dc10f44p
-56},
258 {0x1.d900021dc82aap
-1, -0x1.47c8f94fd5c5cp
-56},
259 {0x1.dafffd86b0283p
-1, 0x1.c7f1dc521617ep
-55},
260 {0x1.dd000296c4739p
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