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[llvm-project.git] / libc / src / math / generic / log10.cpp
blob47569b4758a4bbc1723544f9ee7b56d588d898f9
1 //===-- Double-precision log10(x) function --------------------------------===//
2 //
3 // Part of the LLVM Project, under the Apache License v2.0 with LLVM Exceptions.
4 // See https://llvm.org/LICENSE.txt for license information.
5 // SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception
6 //
7 //===----------------------------------------------------------------------===//
8
9 #include "src/math/log10.h"
10 #include "src/__support/FPUtil/FEnvImpl.h"
11 #include "src/__support/FPUtil/FPBits.h"
12 #include "src/__support/FPUtil/double_double.h"
13 #include "src/__support/FPUtil/dyadic_float.h"
14 #include "src/__support/FPUtil/multiply_add.h"
15 #include "src/__support/common.h"
16 #include "src/__support/macros/optimization.h" // LIBC_UNLIKELY
17
18 namespace __llvm_libc {
19
20 // 192-bit precision dyadic floating point numbers.
21 using Float192 = typename fputil::DyadicFloat<192>;
22 using MType = typename Float192::MantissaType;
23
24 namespace {
25
26 // log10(2) generated by Sollya with:
27 // > a = round(log10(2), 43, RN);
28 constexpr double LOG10_2_HI = 0x1.34413509f78p-2; // LSB = 2^-43
29 // > b = round(log10(2) - a, D, RN);
30 constexpr double LOG10_2_LO = 0x1.fef311f12b358p-46; // LSB = 2^-98
31 constexpr double LOG10_2_ULP[2] = {0x1.0p-97, 0.0};
32
33 // Generated by Sollya with:
34 // > for i from 0 to 127 do {
35 // r = 2^-8 * nearestint( 2^8 * (1 + (i + 0.5)*2^(-7) - 2^(-15)) /
36 // ((1 + i*2^-7)*(1 + (i + 1)*2^-7)) );
37 // print(r, ",");
38 // };
39 // To improve the accuracy with inputs close to 1, we replace R[0] with 1.0.
40 constexpr double R[128] = {
41 1.0, 0x1.fap-1, 0x1.f6p-1, 0x1.f2p-1, 0x1.eep-1, 0x1.eap-1, 0x1.e8p-1,
42 0x1.e4p-1, 0x1.ep-1, 0x1.dcp-1, 0x1.dap-1, 0x1.d6p-1, 0x1.d2p-1, 0x1.dp-1,
43 0x1.ccp-1, 0x1.c8p-1, 0x1.c6p-1, 0x1.c2p-1, 0x1.cp-1, 0x1.bcp-1, 0x1.bap-1,
44 0x1.b6p-1, 0x1.b4p-1, 0x1.bp-1, 0x1.aep-1, 0x1.aap-1, 0x1.a8p-1, 0x1.a6p-1,
45 0x1.a2p-1, 0x1.ap-1, 0x1.9ep-1, 0x1.9ap-1, 0x1.98p-1, 0x1.96p-1, 0x1.94p-1,
46 0x1.9p-1, 0x1.8ep-1, 0x1.8cp-1, 0x1.8ap-1, 0x1.88p-1, 0x1.84p-1, 0x1.82p-1,
47 0x1.8p-1, 0x1.7ep-1, 0x1.7cp-1, 0x1.7ap-1, 0x1.78p-1, 0x1.76p-1, 0x1.74p-1,
48 0x1.72p-1, 0x1.7p-1, 0x1.6ep-1, 0x1.6cp-1, 0x1.6ap-1, 0x1.68p-1, 0x1.66p-1,
49 0x1.64p-1, 0x1.62p-1, 0x1.6p-1, 0x1.5ep-1, 0x1.5cp-1, 0x1.5ap-1, 0x1.58p-1,
50 0x1.56p-1, 0x1.54p-1, 0x1.52p-1, 0x1.5p-1, 0x1.5p-1, 0x1.4ep-1, 0x1.4cp-1,
51 0x1.4ap-1, 0x1.48p-1, 0x1.46p-1, 0x1.46p-1, 0x1.44p-1, 0x1.42p-1, 0x1.4p-1,
52 0x1.3ep-1, 0x1.3ep-1, 0x1.3cp-1, 0x1.3ap-1, 0x1.38p-1, 0x1.38p-1, 0x1.36p-1,
53 0x1.34p-1, 0x1.32p-1, 0x1.32p-1, 0x1.3p-1, 0x1.2ep-1, 0x1.2ep-1, 0x1.2cp-1,
54 0x1.2ap-1, 0x1.2ap-1, 0x1.28p-1, 0x1.26p-1, 0x1.26p-1, 0x1.24p-1, 0x1.22p-1,
55 0x1.22p-1, 0x1.2p-1, 0x1.1ep-1, 0x1.1ep-1, 0x1.1cp-1, 0x1.1cp-1, 0x1.1ap-1,
56 0x1.18p-1, 0x1.18p-1, 0x1.16p-1, 0x1.16p-1, 0x1.14p-1, 0x1.12p-1, 0x1.12p-1,
57 0x1.1p-1, 0x1.1p-1, 0x1.0ep-1, 0x1.0ep-1, 0x1.0cp-1, 0x1.0ap-1, 0x1.0ap-1,
58 0x1.08p-1, 0x1.08p-1, 0x1.06p-1, 0x1.06p-1, 0x1.04p-1, 0x1.04p-1, 0x1.02p-1,
59 0x1.02p-1, 0x1p-1,
60 };
61
62 // Generated by Sollya with:
63 // for i from 0 to 127 do {
64 // r = 2^-8 * nearestint( 2^8 * (1 + (i + 0.5)*2^(-7) - 2^(-15)) /
65 // ((1 + i*2^-7)*(1 + (i + 1)*2^-7)) );
66 // b = nearestint(log10(r)*2^43) * 2^-43;
67 // c = round(log10(r) - b, D, RN);
68 // print("{", -c, ",", -b, "},");
69 // };
70 // We replace LOG10_R[0] with log10(1.0) == 0.0
71 constexpr fputil::DoubleDouble LOG10_R[128] = {
72 {0.0, 0.0},
73 {-0x1.dfa6d47e47379p-45, 0x1.4f8205236p-8},
74 {-0x1.11bc6f2b9a3acp-45, 0x1.18b2dc8d3p-7},
75 {-0x1.685fc114e61bfp-46, 0x1.8a8c06bb2p-7},
76 {0x1.3c757d5b7376ap-45, 0x1.fd503c39p-7},
77 {0x1.f3cbd9c5111b1p-47, 0x1.3881a7b818p-6},
78 {-0x1.e22fab794a816p-45, 0x1.559bd2407p-6},
79 {0x1.421bab5f034a4p-45, 0x1.902c31d628p-6},
80 {0x1.fedb4b594a31bp-45, 0x1.cb38fccd88p-6},
81 {-0x1.c4e824b246c7fp-47, 0x1.0362241e64p-5},
82 {-0x1.df23ed2ebe477p-45, 0x1.125d0432ecp-5},
83 {-0x1.00c12f7a1b586p-47, 0x1.30838cdc3p-5},
84 {0x1.e5ff3439d368dp-46, 0x1.4eec0e2458p-5},
85 {0x1.2951bb9cd2fb7p-45, 0x1.5e3966b7e8p-5},
86 {0x1.fae96a708581ep-46, 0x1.7d070145f4p-5},
87 {0x1.badcf3d6e4566p-46, 0x1.9c197abfp-5},
88 {-0x1.b0197d2cb982ep-48, 0x1.abbcebd85p-5},
89 {-0x1.24b4a6b5ce4d4p-52, 0x1.cb38fccd8cp-5},
90 {-0x1.40bcd23c3e44cp-45, 0x1.db11ed766cp-5},
91 {-0x1.024e9d08ce301p-45, 0x1.fafa6d398p-5},
92 {0x1.77c3e779b9fcfp-48, 0x1.0585283764p-4},
93 {0x1.a9a57734f2038p-48, 0x1.15b11a094ap-4},
94 {-0x1.d227d61f9e88dp-45, 0x1.1dd5460c8cp-4},
95 {0x1.d288560689912p-53, 0x1.2e3a740b78p-4},
96 {-0x1.df5de49ddb16p-46, 0x1.367ba3aaa2p-4},
97 {0x1.5b873a39e56dcp-47, 0x1.471ba8a7dep-4},
98 {0x1.75da8a5871b9ap-45, 0x1.4f7aad9bbcp-4},
99 {0x1.ef5f3c10990f4p-45, 0x1.57e3d47c3ap-4},
100 {-0x1.02359c584ac3cp-45, 0x1.68d4eaf26ep-4},
101 {-0x1.41149840eaa65p-46, 0x1.715d0ce368p-4},
102 {-0x1.ff281b9601ce6p-46, 0x1.79efb57b1p-4},
103 {0x1.2b9da13d5c8cbp-47, 0x1.8b350364c6p-4},
104 {-0x1.80743406505e6p-48, 0x1.93e7de0fc4p-4},
105 {-0x1.76b169f6b4949p-49, 0x1.9ca5aa172ap-4},
106 {-0x1.bc8889d0fe1ap-47, 0x1.a56e8325f6p-4},
107 {-0x1.03ad4133e8c4cp-45, 0x1.b721cd1716p-4},
108 {0x1.72a4e1d198491p-46, 0x1.c00c776722p-4},
109 {-0x1.ddd18dedb6656p-45, 0x1.c902a19e66p-4},
110 {0x1.5e533080ecf32p-47, 0x1.d204698cb4p-4},
111 {0x1.7e865b8783768p-45, 0x1.db11ed766ap-4},
112 {0x1.5f7ef576ada0cp-45, 0x1.ed50a4a26ep-4},
113 {0x1.c9a3bd0891bccp-46, 0x1.f68216c9ccp-4},
114 {-0x1.ff229f20ed3d2p-46, 0x1.ffbfc2bbc8p-4},
115 {-0x1.6f30673aae7efp-45, 0x1.0484e4942bp-3},
116 {0x1.dae5ed5e3f34cp-45, 0x1.093025a199p-3},
117 {-0x1.3eea49e637bb3p-45, 0x1.0de1b56357p-3},
118 {0x1.82c6326f70b35p-46, 0x1.1299a4fb3ep-3},
119 {0x1.f04d633b79054p-45, 0x1.175805d158p-3},
120 {0x1.8b891b6d05a73p-48, 0x1.1c1ce9955cp-3},
121 {-0x1.35ca658049a0ap-51, 0x1.20e8624039p-3},
122 {0x1.ff081a4e81f0bp-45, 0x1.25ba8215afp-3},
123 {0x1.1e3f04f63ee01p-45, 0x1.2a935ba5f1p-3},
124 {-0x1.e1471e5cb397ep-45, 0x1.2f7301cf4fp-3},
125 {-0x1.5bd54fd6eb7d9p-45, 0x1.345987bfefp-3},
126 {0x1.fe93f791a7264p-46, 0x1.394700f795p-3},
127 {-0x1.8aebce3ec8738p-45, 0x1.3e3b814974p-3},
128 {0x1.b0722aa2559f2p-45, 0x1.43371cde07p-3},
129 {-0x1.bcb784188a058p-46, 0x1.4839e83507p-3},
130 {0x1.20ca9a2bcc728p-45, 0x1.4d43f8275ap-3},
131 {0x1.bb95ec8fd8f68p-45, 0x1.525561e925p-3},
132 {0x1.4e036062e2e73p-48, 0x1.576e3b0bdep-3},
133 {-0x1.e560b5e7a02b4p-51, 0x1.5c8e998073p-3},
134 {0x1.1e472c751a4cp-49, 0x1.61b6939983p-3},
135 {-0x1.1116ad28239bp-48, 0x1.66e6400da4p-3},
136 {0x1.9ad1d9e405fb9p-46, 0x1.6c1db5f9bbp-3},
137 {-0x1.41149840eaa65p-45, 0x1.715d0ce368p-3},
138 {0x1.bfb1de334b1cbp-45, 0x1.76a45cbb7ep-3},
139 {0x1.bfb1de334b1cbp-45, 0x1.76a45cbb7ep-3},
140 {-0x1.9fc01708d86e7p-48, 0x1.7bf3bde09ap-3},
141 {0x1.4adaf7fe992b6p-45, 0x1.814b4921bdp-3},
142 {-0x1.c0b434f5d2b7ap-46, 0x1.86ab17c10cp-3},
143 {0x1.4c7acd659652fp-45, 0x1.8c13437695p-3},
144 {0x1.3e99da1b485cfp-45, 0x1.9183e67339p-3},
145 {0x1.3e99da1b485cfp-45, 0x1.9183e67339p-3},
146 {-0x1.fb7d8e74e02ap-46, 0x1.96fd1b63ap-3},
147 {0x1.7c9690248757ep-46, 0x1.9c7efd734ap-3},
148 {-0x1.0cee0ed4ca7e9p-52, 0x1.a209a84fbdp-3},
149 {0x1.d924eb1fec6c5p-47, 0x1.a79d382bc2p-3},
150 {0x1.d924eb1fec6c5p-47, 0x1.a79d382bc2p-3},
151 {0x1.fe6eb5a4df1dfp-49, 0x1.ad39c9c2c6p-3},
152 {0x1.4c9cce6dd603bp-46, 0x1.b2df7a5c5p-3},
153 {-0x1.a01b9bb0ebf1bp-45, 0x1.b88e67cf98p-3},
154 {-0x1.a01b9bb0ebf1bp-45, 0x1.b88e67cf98p-3},
155 {0x1.2ee896e06dbe8p-45, 0x1.be46b08735p-3},
156 {-0x1.ff2381071d23ep-49, 0x1.c4087384f5p-3},
157 {-0x1.2f9fd61140aa6p-45, 0x1.c9d3d065c6p-3},
158 {-0x1.2f9fd61140aa6p-45, 0x1.c9d3d065c6p-3},
159 {-0x1.2386ad8b819b8p-45, 0x1.cfa8e765ccp-3},
160 {0x1.d63da3ac9f0d5p-45, 0x1.d587d96494p-3},
161 {0x1.d63da3ac9f0d5p-45, 0x1.d587d96494p-3},
162 {0x1.fcc16f3ba09cbp-45, 0x1.db70c7e96ep-3},
163 {-0x1.cc52c1ba2d838p-45, 0x1.e163d527e7p-3},
164 {-0x1.cc52c1ba2d838p-45, 0x1.e163d527e7p-3},
165 {0x1.f97877007c127p-46, 0x1.e76124046bp-3},
166 {0x1.fbd42fdc335fdp-47, 0x1.ed68d81919p-3},
167 {0x1.fbd42fdc335fdp-47, 0x1.ed68d81919p-3},
168 {-0x1.cbc0789055864p-45, 0x1.f37b15bab1p-3},
169 {0x1.2737df7f29668p-45, 0x1.f99801fdb7p-3},
170 {0x1.2737df7f29668p-45, 0x1.f99801fdb7p-3},
171 {-0x1.ff229f20ed3d2p-45, 0x1.ffbfc2bbc8p-3},
172 {0x1.00809c16ae3ecp-46, 0x1.02f93f4c87p-2},
173 {0x1.00809c16ae3ecp-46, 0x1.02f93f4c87p-2},
174 {-0x1.aa1358e87a6b4p-45, 0x1.06182e84fd8p-2},
175 {-0x1.aa1358e87a6b4p-45, 0x1.06182e84fd8p-2},
176 {-0x1.f171b2c5f2274p-48, 0x1.093cc32c91p-2},
177 {-0x1.42d6ae59e7d9cp-45, 0x1.0c6711d6acp-2},
178 {-0x1.42d6ae59e7d9cp-45, 0x1.0c6711d6acp-2},
179 {0x1.eac1871dbdbbfp-45, 0x1.0f972f87ffp-2},
180 {0x1.eac1871dbdbbfp-45, 0x1.0f972f87ffp-2},
181 {0x1.feed957c16a44p-46, 0x1.12cd31b9c98p-2},
182 {0x1.a6023a51d15b6p-46, 0x1.16092e5d3a8p-2},
183 {0x1.a6023a51d15b6p-46, 0x1.16092e5d3a8p-2},
184 {0x1.cefc5208422d8p-45, 0x1.194b3bdef68p-2},
185 {0x1.cefc5208422d8p-45, 0x1.194b3bdef68p-2},
186 {-0x1.1d4f1e8c2daffp-55, 0x1.1c93712abc8p-2},
187 {-0x1.1d4f1e8c2daffp-55, 0x1.1c93712abc8p-2},
188 {0x1.40eb9b53054c3p-46, 0x1.1fe1e5af2cp-2},
189 {0x1.9bbc8038401fcp-45, 0x1.2336b161b3p-2},
190 {0x1.9bbc8038401fcp-45, 0x1.2336b161b3p-2},
191 {0x1.09ca54daae9f9p-48, 0x1.2691ecc29fp-2},
192 {0x1.09ca54daae9f9p-48, 0x1.2691ecc29fp-2},
193 {0x1.2b528446968a4p-48, 0x1.29f3b0e1558p-2},
194 {0x1.2b528446968a4p-48, 0x1.29f3b0e1558p-2},
195 {-0x1.4548507c3dd04p-46, 0x1.2d5c1760b88p-2},
196 {-0x1.4548507c3dd04p-46, 0x1.2d5c1760b88p-2},
197 {-0x1.db59b99249f3ap-46, 0x1.30cb3a7bb38p-2},
198 {-0x1.db59b99249f3ap-46, 0x1.30cb3a7bb38p-2},
199 {0x1.fef311f12b358p-46, 0x1.34413509f78p-2},
200 };
201
202 constexpr double LOG10_R_ULP[2] = {0x1.0p-96, 0.0};
203
204 // Generated with Sollya:
205 // > P = fpminimax(log10(1 + x)/x, 6, [|D...|], [-2^-7; 2^-7], absolute);
206 // > dirtyinfnorm(log10(1 + x)/x - P, [-2^-7, 2^-7]);
207 // 0x1.9535684fb3064623001de9b13e6adf06355b5d75bp-57
208 constexpr double COEFFS[7] = {0x1.bcb7b1526e50ep-2, -0x1.bcb7b1526e53fp-3,
209 0x1.287a763700e4p-3, -0x1.bcb7b14641063p-4,
210 0x1.63c61abdf033fp-4, -0x1.28808b8a217fcp-4,
211 0x1.ffe99fc1908c6p-5};
212
213 constexpr double P_ERR = 0x1.0p-52;
214
215 // Number of extra range reduction steps.
216 constexpr size_t R_STEPS = 5;
217 constexpr size_t R_BITS = 4;
218 constexpr size_t R_SIZES = 1 << (R_BITS + 1);
219
220 // Generated by Sollya with:
221 // for i from 0 to 4 do {
222 // N = 11 + 4*i;
223 // print ("{");
224 // for j from -2^4 to 2^4 - 1 do {
225 // r = 2^(-N) * nearestint(2^(N) * ( 1 + (j + 0.5)*2^(-N) - 2^(-2*N-1)) /
226 // ((1 + j * 2^(-N)) * (1 + (j + 1)*2^(-N))));
227 // print(r, ",");
228 // };
229 // print("},");
230 // };
231 constexpr double RR[R_STEPS][R_SIZES] = {
232 {
233 0x1.02p0, 0x1.01ep0, 0x1.01cp0, 0x1.01ap0, 0x1.018p0, 0x1.016p0,
234 0x1.014p0, 0x1.012p0, 0x1.01p0, 0x1.00ep0, 0x1.00cp0, 0x1.00ap0,
235 0x1.008p0, 0x1.006p0, 0x1.004p0, 0x1p0, 0x1p0, 0x1.ffcp-1,
236 0x1.ff8p-1, 0x1.ff4p-1, 0x1.ffp-1, 0x1.fecp-1, 0x1.fe8p-1, 0x1.fe4p-1,
237 0x1.fep-1, 0x1.fdcp-1, 0x1.fd8p-1, 0x1.fd4p-1, 0x1.fdp-1, 0x1.fccp-1,
238 0x1.fc8p-1, 0x1.fc4p-1,
239 },
240 {
241 0x1.002p0, 0x1.001ep0, 0x1.001cp0, 0x1.001ap0, 0x1.0018p0,
242 0x1.0016p0, 0x1.0014p0, 0x1.0012p0, 0x1.001p0, 0x1.000ep0,
243 0x1.000cp0, 0x1.000ap0, 0x1.0008p0, 0x1.0006p0, 0x1.0004p0,
244 0x1p0, 0x1p0, 0x1.fffcp-1, 0x1.fff8p-1, 0x1.fff4p-1,
245 0x1.fffp-1, 0x1.ffecp-1, 0x1.ffe8p-1, 0x1.ffe4p-1, 0x1.ffep-1,
246 0x1.ffdcp-1, 0x1.ffd8p-1, 0x1.ffd4p-1, 0x1.ffdp-1, 0x1.ffccp-1,
247 0x1.ffc8p-1, 0x1.ffc4p-1,
248 },
249 {
250 0x1.0002p0, 0x1.0001ep0, 0x1.0001cp0, 0x1.0001ap0, 0x1.00018p0,
251 0x1.00016p0, 0x1.00014p0, 0x1.00012p0, 0x1.0001p0, 0x1.0000ep0,
252 0x1.0000cp0, 0x1.0000ap0, 0x1.00008p0, 0x1.00006p0, 0x1.00004p0,
253 0x1p0, 0x1p0, 0x1.ffffcp-1, 0x1.ffff8p-1, 0x1.ffff4p-1,
254 0x1.ffffp-1, 0x1.fffecp-1, 0x1.fffe8p-1, 0x1.fffe4p-1, 0x1.fffep-1,
255 0x1.fffdcp-1, 0x1.fffd8p-1, 0x1.fffd4p-1, 0x1.fffdp-1, 0x1.fffccp-1,
256 0x1.fffc8p-1, 0x1.fffc4p-1,
257 },
258 {
259 0x1.00002p0, 0x1.00001ep0, 0x1.00001cp0, 0x1.00001ap0,
260 0x1.000018p0, 0x1.000016p0, 0x1.000014p0, 0x1.000012p0,
261 0x1.00001p0, 0x1.00000ep0, 0x1.00000cp0, 0x1.00000ap0,
262 0x1.000008p0, 0x1.000006p0, 0x1.000004p0, 0x1p0,
263 0x1p0, 0x1.fffffcp-1, 0x1.fffff8p-1, 0x1.fffff4p-1,
264 0x1.fffffp-1, 0x1.ffffecp-1, 0x1.ffffe8p-1, 0x1.ffffe4p-1,
265 0x1.ffffep-1, 0x1.ffffdcp-1, 0x1.ffffd8p-1, 0x1.ffffd4p-1,
266 0x1.ffffdp-1, 0x1.ffffccp-1, 0x1.ffffc8p-1, 0x1.ffffc4p-1,
267 },
268 {
269 0x1.000002p0, 0x1.000001ep0, 0x1.000001cp0, 0x1.000001ap0,
270 0x1.0000018p0, 0x1.0000016p0, 0x1.0000014p0, 0x1.0000012p0,
271 0x1.000001p0, 0x1.000000ep0, 0x1.000000cp0, 0x1.000000ap0,
272 0x1.0000008p0, 0x1.0000006p0, 0x1.0000004p0, 0x1p0,
273 0x1p0, 0x1.ffffffcp-1, 0x1.ffffff8p-1, 0x1.ffffff4p-1,
274 0x1.ffffffp-1, 0x1.fffffecp-1, 0x1.fffffe8p-1, 0x1.fffffe4p-1,
275 0x1.fffffep-1, 0x1.fffffdcp-1, 0x1.fffffd8p-1, 0x1.fffffd4p-1,
276 0x1.fffffdp-1, 0x1.fffffccp-1, 0x1.fffffc8p-1, 0x1.fffffc4p-1,
277 },
278 };
279
280 // log10(2) with 192-bit prepcision generated by SageMath with:
281 // sage: (s, m, e) = RealField(192)(2).log10().sign_exponent_mantissa();
282 // sage: print("MType({", hex(m % 2^64), ",", hex((m >> 64) % 2^64), ",",
283 // hex((m >> 128) % 2^64), "})");
284 const Float192 LOG10_2(/*sign=*/false, /*exponent=*/-193, /*mantissa=*/
285 MType({0x26ad30c543d1f34a, 0x8f8959ac0b7c9178,
286 0x9a209a84fbcff798}));
287
288 // -log10(r) with 192-bit precision generated by SageMath with:
289 //
290 // for i in range(128):
291 // r = 2^-8 * round( 2^8 * (1 + (i + 1/2)*2^(-7) - 2^(-15)) / ((1 + i*2^-7)*(1
292 // + (i + 1)*2^-7)) ); s, m, e = RR(r).log10().sign_mantissa_exponent();
293 // print("{false,", e, ", MType({", hex(m % 2^64), ",", hex((m >> 64) % 2^64),
294 // ",", hex((m >> 128) % 2^64), "})},");
295 const Float192 LOG10_R_F192[128] = {
296 {false, 0, MType(0)},
297 {false, -199,
298 MType({0x3b7bb8a51a78c8bd, 0x6e321c6a7cdecc4, 0xa7c10291a88164ae})},
299 {false, -198,
300 MType({0xf4ba30d77042aac0, 0xa8cb8a86f6040a23, 0x8c596e4695dc8721})},
301 {false, -198,
302 MType({0x49fd078256b3ba8b, 0xeb19e4156cfa1bb7, 0xc546035d8e97a03e})},
303 {false, -198,
304 MType({0x3d964e8c5b12cf00, 0xb6e6ed36c9800088, 0xfea81e1c8278eafa})},
305 {false, -197,
306 MType({0x9c97a4a794669437, 0x714446c4f6a91d15, 0x9c40d3dc0c7cf2f6})},
307 {false, -197,
308 MType({0x1fc57458d58421ab, 0x86b57ea610c7db33, 0xaacde920361dd054})},
309 {false, -197,
310 MType({0x89be5385cc62fb89, 0x5f034a40e6a2f09c, 0xc81618eb15421bab})},
311 {false, -197,
312 MType({0x8b3be744561e443e, 0x594a31b2c5cc891b, 0xe59c7e66c5fedb4b})},
313 {false, -196,
314 MType({0xb1afe1d0add7d035, 0x69b7270237094316, 0x81b1120f31c762fb})},
315 {false, -196,
316 MType({0x6821aa1f743fb5b9, 0x68a0dc47567691c9, 0x892e821975106e09})},
317 {false, -196,
318 MType({0x1935f7436a3ea32c, 0x10bc94f44d216b49, 0x9841c66e17dfe7da})},
319 {false, -196,
320 MType({0x3dfba019afa885f9, 0xe74da327d8d80cb, 0xa77607122c797fcd})},
321 {false, -196,
322 MType({0x7d002c9ce2b6dff3, 0xce697dbaa00d4c7d, 0xaf1cb35bf494a8dd})},
323 {false, -196,
324 MType({0x9ff137c69b2cf640, 0x9c216079dcf0ea95, 0xbe8380a2fa7eba5a})},
325 {false, -196,
326 MType({0x94d5d9c17fa207c5, 0xf5b91598205b8144, 0xce0cbd5f806eb73c})},
327 {false, -196,
328 MType({0x8f995b90cbedc22d, 0x2d3467d253e2d1fb, 0xd5de75ec27e4fe68})},
329 {false, -196,
330 MType({0x8b3be744561e443e, 0x594a31b2c5cc891b, 0xe59c7e66c5fedb4b})},
331 {false, -196,
332 MType({0x97079feab7423d9b, 0xe1e0dda0b3d375a3, 0xed88f6bb355fa196})},
333 {false, -196,
334 MType({0xe86f834386695343, 0x7b98e7f593daf19e, 0xfd7d369cbf7ed8b1})},
335 {false, -195,
336 MType({0x5d4cae75ae8c48e7, 0x3bcdcfe7b23976cd, 0x82c2941bb20bbe1f})},
337 {false, -195,
338 MType({0x67047204db2989f7, 0xb9a7901be7521304, 0x8ad88d04a50d4d2b})},
339 {false, -195,
340 MType({0x829acbf3088a2849, 0x78185dcc37fda019, 0x8eeaa306458b760a})},
341 {false, -195,
342 MType({0x9d5d4ea0a7c8afb1, 0x1581a26448e2ac0e, 0x971d3a05bc0074a2})},
343 {false, -195,
344 MType({0xa078ffeb008a04d8, 0x6c449d409f883fe2, 0x9b3dd1d550c41443})},
345 {false, -195,
346 MType({0x2ede0aa760148d65, 0xa39e56dbb661c829, 0xa38dd453ef15b873})},
347 {false, -195,
348 MType({0x1716e6f9b3225217, 0x961c6e690d8879b4, 0xa7bd56cdde5d76a2})},
349 {false, -195,
350 MType({0x5bcb3818decefa22, 0x42643ced81ec14a, 0xabf1ea3e1d7bd7cf})},
351 {false, -195,
352 MType({0x164c684a1951204a, 0xe9ed4f101c5ef101, 0xb46a757936bf7298})},
353 {false, -195,
354 MType({0xcff88d82b3417f9b, 0xf7e2ab36f09e9013, 0xb8ae8671b3d7dd6c})},
355 {false, -195,
356 MType({0x8a86f0b2dfd799c2, 0x8d3fc63485e7ff12, 0xbcf7dabd87c01afc})},
357 {false, -195,
358 MType({0xf0e9563d52c1d533, 0x13d5c8cb231a53bd, 0xc59a81b26312b9da})},
359 {false, -195,
360 MType({0xa004d184e2faf2ea, 0x5fcd7d0ce937375e, 0xc9f3ef07e1f3fc5e})},
361 {false, -195,
362 MType({0x8a03e643ccfc8ec7, 0x58252dada9f06110, 0xce52d50b94fa253a})},
363 {false, -195,
364 MType({0xed0ad59454d36575, 0x62f01e5ff43708aa, 0xd2b74192fae43777})},
365 {false, -195,
366 MType({0x3af155936af69c0b, 0xb305ced1419fe924, 0xdb90e68b8abf14af})},
367 {false, -195,
368 MType({0x77f157444abbe2fc, 0x3a330921681e2481, 0xe0063bb3912e549c})},
369 {false, -195,
370 MType({0x693eca0a0148cc18, 0x849266a85513dc6d, 0xe48150cf32888b9c})},
371 {false, -195,
372 MType({0x73dac2c5ab4c8eda, 0x80ecf3266b4dcf4, 0xe90234c65a15e533})},
373 {false, -195,
374 MType({0x97079feab7423d9b, 0xe1e0dda0b3d375a3, 0xed88f6bb355fa196})},
375 {false, -195,
376 MType({0x22b519828722ce77, 0x5dab68307fedefcd, 0xf6a852513757dfbd})},
377 {false, -195,
378 MType({0x66278c914a763228, 0xa112379749fd91b8, 0xfb410b64e6393477})},
379 {false, -195,
380 MType({0x12b0b091e7b0299d, 0x1be2585c279c50a5, 0xffdfe15de3c01bac})},
381 {false, -194,
382 MType({0x3f42ceed6e05646f, 0x18aa302171017dcb, 0x8242724a155219f3})},
383 {false, -194,
384 MType({0xd146a4518aa0076f, 0xabc7e698502d43bf, 0x849812d0ccbb5cbd})},
385 {false, -194,
386 MType({0x34708f4b01b4e73f, 0xc339089a51663370, 0x86f0dab1ab5822b6})},
387 {false, -194,
388 MType({0x5e3e593143a3f512, 0x26f70b34ce5cf201, 0x894cd27d9f182c63})},
389 {false, -194,
390 MType({0xb5f326771e3d87a9, 0x676f20a87ab433de, 0x8bac02e8ac3e09ac})},
391 {false, -194,
392 MType({0x633dc078567bc8d9, 0x6db4169cc4b83bc3, 0x8e0e74caae062e24})},
393 {false, -194,
394 MType({0x662d0eb20c51da4f, 0xcd3fdb2fad0d1fd6, 0x907431201c7f651a})},
395 {false, -194,
396 MType({0x7fed29520fdf89d7, 0x49d03e163250d1d4, 0x92dd410ad7bfe103})},
397 {false, -194,
398 MType({0xed5106d40e889904, 0x9ec7dc02d5e723b8, 0x9549add2f8a3c7e0})},
399 {false, -194,
400 MType({0xcdbe2ebc07714f34, 0x34698d03a5442572, 0x97b980e7a743d71c})},
401 {false, -194,
402 MType({0x1e9782e43af2d3d7, 0x522904d1e47f3de, 0x9a2cc3dff7548556})},
403 {false, -194,
404 MType({0xfabfd5317e9bdd56, 0x791a72646c87b975, 0x9ca3807bca9fe93f})},
405 {false, -194,
406 MType({0x2dc99e57977305af, 0x3826f190d655d736, 0x9f1dc0a4b9cea286})},
407 {false, -194,
408 MType({0x5a4dfeb5b0db1f24, 0x544ab3e48199b299, 0xa19b8e6f03b60e45})},
409 {false, -194,
410 MType({0x2888ece2bf37b7e9, 0xbe775fa82961114e, 0xa41cf41a83643487})},
411 {false, -194,
412 MType({0xef4915fda0982302, 0x45798e5019e6c081, 0xa6a1fc13ad241953})},
413 {false, -194,
414 MType({0x33a92c4634bdd6c, 0x91fb1ed0cdc4d1fb, 0xa92ab0f492b772bd})},
415 {false, -194,
416 MType({0xe89863702c85cccb, 0x818b8b9cbbd17b71, 0xabb71d85ef05380d})},
417 {false, -194,
418 MType({0x482ab24ae8fdf5a4, 0xa50c2fea60c5b3b2, 0xae474cc0397f0d4f})},
419 {false, -194,
420 MType({0x682d5b55e9594eb3, 0x58ea34980ad8b720, 0xb0db49ccc1823c8e})},
421 {false, -194,
422 MType({0xae8dceb953c096c0, 0x4b5f71941be508a3, 0xb3732006d1fbbba5})},
423 {false, -194,
424 MType({0xfc1396a39c34fef3, 0x9e405fb8bcb1ff1d, 0xb60edafcdd99ad1d})},
425 {false, -194,
426 MType({0xcff88d82b3417f9b, 0xf7e2ab36f09e9013, 0xb8ae8671b3d7dd6c})},
427 {false, -194,
428 MType({0x6f1a4043a1a8a435, 0xc669639640c305bb, 0xbb522e5dbf37f63b})},
429 {false, -194,
430 MType({0x6f1a4043a1a8a435, 0xc669639640c305bb, 0xbb522e5dbf37f63b})},
431 {false, -194,
432 MType({0x6c4433809b0babe7, 0xa3dc9e464e98764b, 0xbdf9def04cf980ff})},
433 {false, -194,
434 MType({0x38a1d9b9b9370d6d, 0xffd3256b59fa9c59, 0xc0a5a490dea95b5e})},
435 {false, -194,
436 MType({0x60b092e62b5beb8a, 0xb0a2d48672a051a5, 0xc3558be085e3f4bc})},
437 {false, -194,
438 MType({0x1065867f127536e0, 0xacb2ca5d4ca1c10e, 0xc609a1bb4aa98f59})},
439 {false, -194,
440 MType({0x8030cfce4646062f, 0x43690b9e3cde0d01, 0xc8c1f3399ca7d33b})},
441 {false, -194,
442 MType({0x8030cfce4646062f, 0x43690b9e3cde0d01, 0xc8c1f3399ca7d33b})},
443 {false, -194,
444 MType({0xd806ff9947bc6ca7, 0x18b1fd60383f7e59, 0xcb7e8db1cfe04827})},
445 {false, -194,
446 MType({0x65af114cbdb0193e, 0x248757e5f45af3d, 0xce3f7eb9a517c969})},
447 {false, -194,
448 MType({0x3569862a1e8f9a4c, 0x7c4acd605be48bc1, 0xd104d427de7fbcc4})},
449 {false, -194,
450 MType({0x94e3cbc5cd693dda, 0x58ff63629a92652c, 0xd3ce9c15e10ec927})},
451 {false, -194,
452 MType({0x94e3cbc5cd693dda, 0x58ff63629a92652c, 0xd3ce9c15e10ec927})},
453 {false, -194,
454 MType({0x1e0a73c970dbbf33, 0x6b49be3bd8c89f10, 0xd69ce4e16303fcdd})},
455 {false, -194,
456 MType({0x59899d5040e21558, 0xe6dd603a881e9060, 0xd96fbd2e2814c9cc})},
457 {false, -194,
458 MType({0x71549f0a4d78d49c, 0x89e281c98c1d705c, 0xdc4733e7cbcbfc8c})},
459 {false, -194,
460 MType({0x71549f0a4d78d49c, 0x89e281c98c1d705c, 0xdc4733e7cbcbfc8c})},
461 {false, -194,
462 MType({0xf4eee5981334e57, 0xdc0db7cf0cce9f32, 0xdf2358439aa5dd12})},
463 {false, -194,
464 MType({0xd50afdf84e68b269, 0xfdf1c5b846db9dea, 0xe20439c27a7c01b8})},
465 {false, -194,
466 MType({0xd95ac10b65558e44, 0x3dd7eab48869c401, 0xe4e9e832e2da0c05})},
467 {false, -194,
468 MType({0xd95ac10b65558e44, 0x3dd7eab48869c401, 0xe4e9e832e2da0c05})},
469 {false, -194,
470 MType({0xe9377fb1f3b453b6, 0x4e8fcc900b41daee, 0xe7d473b2e5db8f2a})},
471 {false, -194,
472 MType({0xe772954c39d07f3b, 0x7593e1a9e9173599, 0xeac3ecb24a3ac7b4})},
473 {false, -194,
474 MType({0xe772954c39d07f3b, 0x7593e1a9e9173599, 0xeac3ecb24a3ac7b4})},
475 {false, -194,
476 MType({0xa6d10312a95362d4, 0xe7741396b49e1ce4, 0xedb863f4b73f982d})},
477 {false, -194,
478 MType({0xd0d55aebaeb5abfc, 0xc8ba4f8f47b85a5b, 0xf0b1ea93f34675a7})},
479 {false, -194,
480 MType({0xd0d55aebaeb5abfc, 0xc8ba4f8f47b85a5b, 0xf0b1ea93f34675a7})},
481 {false, -194,
482 MType({0x7e1dea1275662695, 0x7007c1276821b705, 0xf3b09202359f9787})},
483 {false, -194,
484 MType({0x54dc283e4f79339c, 0x7ee19afe6db7e324, 0xf6b46c0c8c8fdea1})},
485 {false, -194,
486 MType({0x54dc283e4f79339c, 0x7ee19afe6db7e324, 0xf6b46c0c8c8fdea1})},
487 {false, -194,
488 MType({0xf7844244016096c0, 0xedf54f37f6d4041f, 0xf9bd8add584687f0})},
489 {false, -194,
490 MType({0x94a99151573d5249, 0xefe52ccf03e7dee0, 0xfccc00fedba4e6fb})},
491 {false, -194,
492 MType({0x94a99151573d5249, 0xefe52ccf03e7dee0, 0xfccc00fedba4e6fb})},
493 {false, -194,
494 MType({0x12b0b091e7b0299d, 0x1be2585c279c50a5, 0xffdfe15de3c01bac})},
495 {false, -193,
496 MType({0xeba2ae5f7d1c7168, 0xe0b571f5c91b0445, 0x817c9fa643880404})},
497 {false, -193,
498 MType({0xeba2ae5f7d1c7168, 0xe0b571f5c91b0445, 0x817c9fa643880404})},
499 {false, -193,
500 MType({0x2db8874aa1eb0c3c, 0x7178594bef2def59, 0x830c17427ea55eca})},
501 {false, -193,
502 MType({0x2db8874aa1eb0c3c, 0x7178594bef2def59, 0x830c17427ea55eca})},
503 {false, -193,
504 MType({0xf3cb58bd1bbe04f0, 0x9a741bb171158d29, 0x849e6196487c1d1c})},
505 {false, -193,
506 MType({0xd95b712663014da, 0x1a618264446cb495, 0x863388eb55ebd295})},
507 {false, -193,
508 MType({0xd95b712663014da, 0x1a618264446cb495, 0x863388eb55ebd295})},
509 {false, -193,
510 MType({0x2e66a10dfdce751, 0x71dbdbbec51d7657, 0x87cb97c3ff9eac18})},
511 {false, -193,
512 MType({0x2e66a10dfdce751, 0x71dbdbbec51d7657, 0x87cb97c3ff9eac18})},
513 {false, -193,
514 MType({0x84a2c0cd81dbcf53, 0xabe0b522230f7d13, 0x896698dce4cff76c})},
515 {false, -193,
516 MType({0xae1f8a1def2acf5a, 0xd28e8adafea703b3, 0x8b04972e9d4d3011})},
517 {false, -193,
518 MType({0xae1f8a1def2acf5a, 0xd28e8adafea703b3, 0x8b04972e9d4d3011})},
519 {false, -193,
520 MType({0x8a02390202a4a59d, 0x208422d83be34b26, 0x8ca59def7b5cefc5})},
521 {false, -193,
522 MType({0x8a02390202a4a59d, 0x208422d83be34b26, 0x8ca59def7b5cefc5})},
523 {false, -193,
524 MType({0x420c16bd3939f912, 0xc385cf49402af0e4, 0x8e49b8955e3ffb8a})},
525 {false, -193,
526 MType({0x420c16bd3939f912, 0xc385cf49402af0e4, 0x8e49b8955e3ffb8a})},
527 {false, -193,
528 MType({0x8a1754a1ee7c990, 0xda982a614e12c6dd, 0x8ff0f2d7960a075c})},
529 {false, -193,
530 MType({0xe77cb3d650c2718e, 0x38401fc1c1b5c2b, 0x919b58b0d999bbc8})},
531 {false, -193,
532 MType({0xe77cb3d650c2718e, 0x38401fc1c1b5c2b, 0x919b58b0d999bbc8})},
533 {false, -193,
534 MType({0x3c50b7234a381be8, 0xa9b55d3f16da746a, 0x9348f6614f821394})},
535 {false, -193,
536 MType({0x3c50b7234a381be8, 0xa9b55d3f16da746a, 0x9348f6614f821394})},
537 {false, -193,
538 MType({0xd31cd763e50a0231, 0x88d2d1473d4f7f4, 0x94f9d870aac256a5})},
539 {false, -193,
540 MType({0xd31cd763e50a0231, 0x88d2d1473d4f7f4, 0x94f9d870aac256a5})},
541 {false, -193,
542 MType({0x8eb2e675bc182d0d, 0x7c1e117dea19e9e5, 0x96ae0bb05c35d5bd})},
543 {false, -193,
544 MType({0x8eb2e675bc182d0d, 0x7c1e117dea19e9e5, 0x96ae0bb05c35d5bd})},
545 {false, -193,
546 MType({0x78c2d9cf6e98b1c1, 0x336db0630f536fb9, 0x98659d3dd9b12532})},
547 {false, -193,
548 MType({0x78c2d9cf6e98b1c1, 0x336db0630f536fb9, 0x98659d3dd9b12532})},
549 {false, -193,
550 MType({0x26ad30c543d1f34a, 0x8f8959ac0b7c9178, 0x9a209a84fbcff798})},
551 };
552
553 // -log10(r) for further range reduction steps, generated by SageMath with:
554 //
555 // RR = RealField(192);
556 // for i in range(5):
557 // N = 11 + 4*i;
558 // print("{");
559 // for j in range(-2^4, 2^4):
560 // r = 2^(-N) * round(2^(N) * ( 1 + (j + 0.5)*2^(-N) - 2^(-2*N-1) ) / ((1 +
561 // j * 2^(-N)) * (1 + (j + 1)*2^(-N)))); a = -RR(r).log10(); if j in [0,
562 // -1]:
563 // r = 1; a = RR(0);
564 // s, m, e = a.sign_mantissa_exponent()
565 // sgn = "{false," if s == 1 else "{true,";
566 // print(sgn, e, ", MType({", hex(m % 2^64), ",", hex((m >> 64) % 2^64),
567 // ",", hex((m >> 128) % 2^64), "})},");
568 // print("},");
569 const Float192 LOG10_RR[R_STEPS][R_SIZES] = {
570 {
571 {true, -200,
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881 {false, -216,
882 MType({0xb70a87a69080a500, 0x99308918494a4a1f, 0xa6c4e2fbfd08b172})},
883 {false, -216,
884 MType({0x8995c0b3074c289d, 0xd5579f970cf000a8, 0xb4aaa09c47738304})},
885 {false, -216,
886 MType({0x72650f85eed7af38, 0xd4ddab9f8032bbab, 0xc2905e3e4e960b8e})},
887 {false, -216,
888 MType({0xeb04ee2cfc701a37, 0xc5b134a5bb25cf2d, 0xd0761be212704b7f})},
889 },
890 };
891
892 // > P = fpminimax(log10(1 + x)/x, 4, [|192...|], [-2^-27, 2^-27]);
893 // > P;
894 // > dirtyinfnorm(log10(1 + x)/x - P, [-2^-27, 2^-27]);
895 // 0x1.287a7...p-143
896 const Float192 BIG_COEFFS[5]{
897 {false, -194,
898 MType({0x1fc14ee0c0158d73, 0x762ec7601912bf70, 0x58f189dd49436234})},
899 {true, -195,
900 MType({0x8f42a80e947f6357, 0x67836140b941fe04, 0xde5bd8a93728747a})},
901 {false, -194,
902 MType({0xf6f00690b1fa8ba9, 0x78e7c71fc8ca2d3a, 0x943d3b1b7a1af663})},
903 {true, -191,
904 MType({0x255a69358264a2d1, 0xa6ab7555f5a64d33, 0x1bcb7b1526e50e32})},
905 {false, -193,
906 MType({0x3ee3460246fdf301, 0x355baaafad33dc32, 0xde5bd8a937287195})},
907 };
908
909 // Reuse the output of the fast pass range reduction.
910 // |m_x| < 2^-7
911 double log10_accurate(int e_x, int index, double m_x) {
912 Float192 e_x_f192(static_cast<float>(e_x));
913 Float192 sum = fputil::quick_add(LOG10_R_F192[index],
914 fputil::quick_mul(LOG10_2, e_x_f192));
915
916 fputil::DoubleDouble mx{/*lo*/ 0.0, /*hi*/ m_x};
917
918 // Further range reductions.
919 double scale = 0x1.0p+7;
920 for (size_t i = 0; i < R_STEPS; ++i) {
921 scale *= 0x1.0p+4;
922 int id = static_cast<int>(fputil::multiply_add(mx.hi, scale, 0x1.0p+4));
923 double r = RR[i][id];
924 fputil::DoubleDouble rm = fputil::exact_mult(r, mx.hi);
925 rm.hi += r - 1.0;
926 rm.lo = fputil::multiply_add(r, mx.lo, rm.lo);
927 mx = fputil::exact_add(rm.hi, rm.lo);
928 sum = fputil::quick_add(sum, LOG10_RR[i][id]);
929 }
930 // Now |m_x| <= 2^-27
931 Float192 m_hi(mx.hi);
932 Float192 m_lo(mx.lo);
933 Float192 m = fputil::quick_add(m_hi, m_lo);
934 Float192 p = fputil::quick_mul(m, BIG_COEFFS[0]);
935
936 for (size_t i = 1; i < 5; ++i) {
937 auto aa = fputil::quick_add(p, BIG_COEFFS[i]);
938 p = fputil::quick_mul(m, aa);
939 };
940
941 return static_cast<double>(fputil::quick_add(sum, p));
942 }
943
944 } // namespace
945
946 // TODO(lntue): Make the implementation correctly rounded for non-FMA targets.
947 LLVM_LIBC_FUNCTION(double, log10, (double x)) {
948 using FPBits_t = typename fputil::FPBits<double>;
949 FPBits_t xbits(x);
950 int x_e = -1023;
951
952 if (LIBC_UNLIKELY(xbits.uintval() < FPBits_t::MIN_NORMAL ||
953 xbits.uintval() > FPBits_t::MAX_NORMAL)) {
954 if (xbits.is_zero()) {
955 // return -Inf and raise FE_DIVBYZERO.
956 fputil::set_errno_if_required(ERANGE);
957 fputil::raise_except_if_required(FE_DIVBYZERO);
958 return static_cast<double>(FPBits_t::neg_inf());
959 }
960 if (xbits.get_sign() && !xbits.is_nan()) {
961 fputil::set_errno_if_required(EDOM);
962 fputil::raise_except_if_required(FE_INVALID);
963 return FPBits_t::build_quiet_nan(0);
964 }
965 if (xbits.is_inf_or_nan()) {
966 return x;
967 }
968 // Normalize denormal inputs.
969 xbits.set_val(x * 0x1.0p52);
970 x_e -= 52;
971 }
972
973 // log10(x) = log10(2^x_e * x_m)
974 // = x_e * log10(2) + log10(x_m)
975
976 // Range reduction for log10(x_m):
977 // For each x_m, we would like to find R such that:
978 // |R * x_m - 1| < C
979 uint64_t x_u = xbits.uintval();
980 int shifted = x_u >> 45;
981 int index = shifted & 0x7F;
982 double r = R[index];
983
984 x_e += (x_u >> 52) & 0x7FF;
985 double e_x = static_cast<double>(x_e);
986
987 int e_err = (e_x == -1) && (index == 0x7F);
988 int logr_err = (index == 0);
989
990 double err =
991 fputil::multiply_add(e_x, LOG10_2_ULP[e_err], LOG10_R_ULP[logr_err]);
992
993 // hi is exact
994 double hi = fputil::multiply_add(e_x, LOG10_2_HI, LOG10_R[index].hi);
995 // lo errors ~ e_x * LSB(LOG10_2_LO) + LSB(LOG10_R[index].lo) + rounding err
996 // <= 2 * (e_x * LSB(LOG10_2_LO) + LSB(LOG10_R[index].lo))
997 double lo = fputil::multiply_add(e_x, LOG10_2_LO, LOG10_R[index].lo);
998 // A bound on the error is given
999 // in "Note on FastTwoSum with Directed Roundings"
1000 // by Paul Zimmermann, https://hal.inria.fr/hal-03798376, 2022.
1001 // Theorem 1 says that
1002 // the difference between a+b and hi+lo is bounded by 2u^2|a+b|
1003 // and also by 2u^2|hi|. Here u=2^-53, thus we get:
1004 // |(a+b)-(hi+lo)| <= 2^-105 min(|a+b|,|hi|)
1005 // So the overall errors <= 2^-105 min(|a+b|, |hi|) + 2*(...)
1006 fputil::DoubleDouble rr = fputil::exact_add(hi, lo);
1007
1008 uint64_t x_m = (x_u & 0x000F'FFFF'FFFF'FFFFULL) | 0x3FF0'0000'0000'0000ULL;
1009 double m = FPBits_t(x_m).get_val();
1010
1011 double u = fputil::multiply_add(r, m, -1.0); // exact
1012 err = fputil::multiply_add(u, P_ERR, err);
1013
1014 // Degree-7 minimax polynomial
1015 double u_sq = u * u;
1016 double p0 = u * COEFFS[0];
1017 double p1 = fputil::multiply_add(u, COEFFS[2], COEFFS[1]);
1018 double p2 = fputil::multiply_add(u, COEFFS[4], COEFFS[3]);
1019 double p3 = fputil::multiply_add(u, COEFFS[6], COEFFS[5]);
1020 double p01 = fputil::multiply_add(u_sq, p1, p0);
1021 double p23 = fputil::multiply_add(u_sq, p3, p2);
1022 double u4 = u_sq * u_sq;
1023 fputil::DoubleDouble re = fputil::exact_add(rr.hi, p01);
1024 double ll = fputil::multiply_add(u4, p23, re.lo + rr.lo);
1025 // Lower bound from the result
1026 double left = re.hi + (ll - err);
1027 // Upper bound from the result
1028 double right = re.hi + (ll + err);
1029
1030 // Ziv's test if fast pass is accurate enough.
1031 if (left == right)
1032 return left;
1033
1034 // Exact cases:
1035 switch (x_u) {
1036 case 0x3ff0000000000000: // x = 1.0
1037 return 0.0;
1038 case 0x4024000000000000: // x = 10.0
1039 return 1.0;
1040 case 0x4059000000000000: // x = 10^2
1041 return 2.0;
1042 case 0x408f400000000000: // x = 10^3
1043 return 3.0;
1044 case 0x40c3880000000000: // x = 10^4
1045 return 4.0;
1046 case 0x40f86a0000000000: // x = 10^5
1047 return 5.0;
1048 case 0x412e848000000000: // x = 10^6
1049 return 6.0;
1050 case 0x416312d000000000: // x = 10^7
1051 return 7.0;
1052 case 0x4197d78400000000: // x = 10^8
1053 return 8.0;
1054 case 0x41cdcd6500000000: // x = 10^9
1055 return 9.0;
1056 case 0x4202a05f20000000: // x = 10^10
1057 return 10.0;
1058 case 0x42374876e8000000: // x = 10^11
1059 return 11.0;
1060 case 0x426d1a94a2000000: // x = 10^12
1061 return 12.0;
1062 case 0x42a2309ce5400000: // x = 10^13
1063 return 13.0;
1064 case 0x42d6bcc41e900000: // x = 10^14
1065 return 14.0;
1066 case 0x430c6bf526340000: // x = 10^15
1067 return 15.0;
1068 case 0x4341c37937e08000: // x = 10^16
1069 return 16.0;
1070 case 0x4376345785d8a000: // x = 10^17
1071 return 17.0;
1072 case 0x43abc16d674ec800: // x = 10^18
1073 return 18.0;
1074 case 0x43e158e460913d00: // x = 10^19
1075 return 19.0;
1076 case 0x4415af1d78b58c40: // x = 10^20
1077 return 20.0;
1078 case 0x444b1ae4d6e2ef50: // x = 10^21
1079 return 21.0;
1080 case 0x4480f0cf064dd592: // x = 10^22
1081 return 22.0;
1082 }
1083
1084 return log10_accurate(x_e, index, u);
1085 }
1086
1087 } // namespace __llvm_libc
LLVM monorepo