1 /* lgammal expanding around zeros.
2 Copyright (C) 2015-2018 Free Software Foundation, Inc.
3 This file is part of the GNU C Library.
5 The GNU C Library is free software; you can redistribute it and/or
6 modify it under the terms of the GNU Lesser General Public
7 License as published by the Free Software Foundation; either
8 version 2.1 of the License, or (at your option) any later version.
10 The GNU C Library is distributed in the hope that it will be useful,
11 but WITHOUT ANY WARRANTY; without even the implied warranty of
12 MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
13 Lesser General Public License for more details.
15 You should have received a copy of the GNU Lesser General Public
16 License along with the GNU C Library; if not, see
17 <http://www.gnu.org/licenses/>. */
19 #include "quadmath-imp.h"
21 static const __float128 lgamma_zeros
[][2] =
23 { -0x2.74ff92c01f0d82abec9f315f1a08p
+0Q
, 0xe.d3ccb7fb2658634a2b9f6b2ba81p
-116Q
},
24 { -0x2.bf6821437b20197995a4b4641eaep
+0Q
, -0xb.f4b00b4829f961e428533e6ad048p
-116Q
},
25 { -0x3.24c1b793cb35efb8be699ad3d9bap
+0Q
, -0x6.5454cb7fac60e3f16d9d7840c2ep
-116Q
},
26 { -0x3.f48e2a8f85fca170d4561291236cp
+0Q
, -0xc.320a4887d1cb4c711828a75d5758p
-116Q
},
27 { -0x4.0a139e16656030c39f0b0de18114p
+0Q
, 0x1.53e84029416e1242006b2b3d1cfp
-112Q
},
28 { -0x4.fdd5de9bbabf3510d0aa40769884p
+0Q
, -0x1.01d7d78125286f78d1e501f14966p
-112Q
},
29 { -0x5.021a95fc2db6432a4c56e595394cp
+0Q
, -0x1.ecc6af0430d4fe5746fa7233356fp
-112Q
},
30 { -0x5.ffa4bd647d0357dd4ed62cbd31ecp
+0Q
, -0x1.f8e3f8e5deba2d67dbd70dd96ce1p
-112Q
},
31 { -0x6.005ac9625f233b607c2d96d16384p
+0Q
, -0x1.cb86ac569340cf1e5f24df7aab7bp
-112Q
},
32 { -0x6.fff2fddae1bbff3d626b65c23fd4p
+0Q
, 0x1.e0bfcff5c457ebcf4d3ad9674167p
-112Q
},
33 { -0x7.000cff7b7f87adf4482dcdb98784p
+0Q
, 0x1.54d99e35a74d6407b80292df199fp
-112Q
},
34 { -0x7.fffe5fe05673c3ca9e82b522b0ccp
+0Q
, 0x1.62d177c832e0eb42c2faffd1b145p
-112Q
},
35 { -0x8.0001a01459fc9f60cb3cec1cec88p
+0Q
, 0x2.8998835ac7277f7bcef67c47f188p
-112Q
},
36 { -0x8.ffffd1c425e80ffc864e95749258p
+0Q
, -0x1.e7e20210e7f81cf781b44e9d2b02p
-112Q
},
37 { -0x9.00002e3bb47d86d6d843fedc352p
+0Q
, 0x2.14852f613a16291751d2ab751f7ep
-112Q
},
38 { -0x9.fffffb606bdfdcd062ae77a50548p
+0Q
, 0x3.962d1490cc2e8f031c7007eaa1ap
-116Q
},
39 { -0xa.0000049f93bb9927b45d95e1544p
+0Q
, -0x1.e03086db9146a9287bd4f2172d5ap
-112Q
},
40 { -0xa.ffffff9466e9f1b36dacd2adbd18p
+0Q
, -0xd.05a4e458062f3f95345a4d9c9b6p
-116Q
},
41 { -0xb.0000006b9915315d965a6ffea41p
+0Q
, 0x1.b415c6fff233e7b7fdc3a094246fp
-112Q
},
42 { -0xb.fffffff7089387387de41acc3d4p
+0Q
, 0x3.687427c6373bd74a10306e10a28ep
-112Q
},
43 { -0xc.00000008f76c7731567c0f0250fp
+0Q
, -0x3.87920df5675833859190eb128ef6p
-112Q
},
44 { -0xc.ffffffff4f6dcf617f97a5ffc758p
+0Q
, 0x2.ab72d76f32eaee2d1a42ed515d3ap
-116Q
},
45 { -0xd.00000000b092309c06683dd1b9p
+0Q
, -0x3.e3700857a15c19ac5a611de9688ap
-112Q
},
46 { -0xd.fffffffff36345ab9e184a3e09dp
+0Q
, -0x1.176dc48e47f62d917973dd44e553p
-112Q
},
47 { -0xe.000000000c9cba545e94e75ec57p
+0Q
, -0x1.8f753e2501e757a17cf2ecbeeb89p
-112Q
},
48 { -0xe.ffffffffff28c060c6604ef3037p
+0Q
, -0x1.f89d37357c9e3dc17c6c6e63becap
-112Q
},
49 { -0xf.0000000000d73f9f399bd0e420f8p
+0Q
, -0x5.e9ee31b0b890744fc0e3fbc01048p
-116Q
},
50 { -0xf.fffffffffff28c060c6621f512e8p
+0Q
, 0xd.1b2eec9d960bd9adc5be5f5fa5p
-116Q
},
51 { -0x1.000000000000d73f9f399da1424cp
+4Q
, 0x6.c46e0e88305d2800f0e414c506a8p
-116Q
},
52 { -0x1.0ffffffffffff3569c47e7a93e1cp
+4Q
, -0x4.6a08a2e008a998ebabb8087efa2cp
-112Q
},
53 { -0x1.1000000000000ca963b818568887p
+4Q
, -0x6.ca5a3a64ec15db0a95caf2c9ffb4p
-112Q
},
54 { -0x1.1fffffffffffff4bec3ce234132dp
+4Q
, -0x8.b2b726187c841cb92cd5221e444p
-116Q
},
55 { -0x1.20000000000000b413c31dcbeca5p
+4Q
, 0x3.c4d005344b6cd0e7231120294abcp
-112Q
},
56 { -0x1.2ffffffffffffff685b25cbf5f54p
+4Q
, -0x5.ced932e38485f7dd296b8fa41448p
-112Q
},
57 { -0x1.30000000000000097a4da340a0acp
+4Q
, 0x7.e484e0e0ffe38d406ebebe112f88p
-112Q
},
58 { -0x1.3fffffffffffffff86af516ff7f7p
+4Q
, -0x6.bd67e720d57854502b7db75e1718p
-112Q
},
59 { -0x1.40000000000000007950ae900809p
+4Q
, 0x6.bec33375cac025d9c073168c5d9p
-112Q
},
60 { -0x1.4ffffffffffffffffa391c4248c3p
+4Q
, 0x5.c63022b62b5484ba346524db607p
-112Q
},
61 { -0x1.500000000000000005c6e3bdb73dp
+4Q
, -0x5.c62f55ed5322b2685c5e9a51e6a8p
-112Q
},
62 { -0x1.5fffffffffffffffffbcc71a492p
+4Q
, -0x1.eb5aeb96c74d7ad25e060528fb5p
-112Q
},
63 { -0x1.6000000000000000004338e5b6ep
+4Q
, 0x1.eb5aec04b2f2eb663e4e3d8a018cp
-112Q
},
64 { -0x1.6ffffffffffffffffffd13c97d9dp
+4Q
, -0x3.8fcc4d08d6fe5aa56ab04307ce7ep
-112Q
},
65 { -0x1.70000000000000000002ec368263p
+4Q
, 0x3.8fcc4d090cee2f5d0b69a99c353cp
-112Q
},
66 { -0x1.7fffffffffffffffffffe0d30fe7p
+4Q
, 0x7.2f577cca4b4c8cb1dc14001ac5ecp
-112Q
},
67 { -0x1.800000000000000000001f2cf019p
+4Q
, -0x7.2f577cca4b3442e35f0040b3b9e8p
-112Q
},
68 { -0x1.8ffffffffffffffffffffec0c332p
+4Q
, -0x2.e9a0572b1bb5b95f346a92d67a6p
-112Q
},
69 { -0x1.90000000000000000000013f3ccep
+4Q
, 0x2.e9a0572b1bb5c371ddb3561705ap
-112Q
},
70 { -0x1.9ffffffffffffffffffffff3b8bdp
+4Q
, -0x1.cad8d32e386fd783e97296d63dcbp
-116Q
},
71 { -0x1.a0000000000000000000000c4743p
+4Q
, 0x1.cad8d32e386fd7c1ab8c1fe34c0ep
-116Q
},
72 { -0x1.afffffffffffffffffffffff8b95p
+4Q
, -0x3.8f48cc5737d5979c39db806c5406p
-112Q
},
73 { -0x1.b00000000000000000000000746bp
+4Q
, 0x3.8f48cc5737d5979c3b3a6bda06f6p
-112Q
},
74 { -0x1.bffffffffffffffffffffffffbd8p
+4Q
, 0x6.2898d42174dcf171470d8c8c6028p
-112Q
},
75 { -0x1.c000000000000000000000000428p
+4Q
, -0x6.2898d42174dcf171470d18ba412cp
-112Q
},
76 { -0x1.cfffffffffffffffffffffffffdbp
+4Q
, -0x4.c0ce9794ea50a839e311320bde94p
-112Q
},
77 { -0x1.d000000000000000000000000025p
+4Q
, 0x4.c0ce9794ea50a839e311322f7cf8p
-112Q
},
78 { -0x1.dfffffffffffffffffffffffffffp
+4Q
, 0x3.932c5047d60e60caded4c298a174p
-112Q
},
79 { -0x1.e000000000000000000000000001p
+4Q
, -0x3.932c5047d60e60caded4c298973ap
-112Q
},
80 { -0x1.fp
+4Q
, 0xa.1a6973c1fade2170f7237d35fe3p
-116Q
},
81 { -0x1.fp
+4Q
, -0xa.1a6973c1fade2170f7237d35fe08p
-116Q
},
82 { -0x2p
+4Q
, 0x5.0d34b9e0fd6f10b87b91be9aff1p
-120Q
},
83 { -0x2p
+4Q
, -0x5.0d34b9e0fd6f10b87b91be9aff0cp
-120Q
},
84 { -0x2.1p
+4Q
, 0x2.73024a9ba1aa36a7059bff52e844p
-124Q
},
85 { -0x2.1p
+4Q
, -0x2.73024a9ba1aa36a7059bff52e844p
-124Q
},
86 { -0x2.2p
+4Q
, 0x1.2710231c0fd7a13f8a2b4af9d6b7p
-128Q
},
87 { -0x2.2p
+4Q
, -0x1.2710231c0fd7a13f8a2b4af9d6b7p
-128Q
},
88 { -0x2.3p
+4Q
, 0x8.6e2ce38b6c8f9419e3fad3f0312p
-136Q
},
89 { -0x2.3p
+4Q
, -0x8.6e2ce38b6c8f9419e3fad3f0312p
-136Q
},
90 { -0x2.4p
+4Q
, 0x3.bf30652185952560d71a254e4eb8p
-140Q
},
91 { -0x2.4p
+4Q
, -0x3.bf30652185952560d71a254e4eb8p
-140Q
},
92 { -0x2.5p
+4Q
, 0x1.9ec8d1c94e85af4c78b15c3d89d3p
-144Q
},
93 { -0x2.5p
+4Q
, -0x1.9ec8d1c94e85af4c78b15c3d89d3p
-144Q
},
94 { -0x2.6p
+4Q
, 0xa.ea565ce061d57489e9b85276274p
-152Q
},
95 { -0x2.6p
+4Q
, -0xa.ea565ce061d57489e9b85276274p
-152Q
},
96 { -0x2.7p
+4Q
, 0x4.7a6512692eb37804111dabad30ecp
-156Q
},
97 { -0x2.7p
+4Q
, -0x4.7a6512692eb37804111dabad30ecp
-156Q
},
98 { -0x2.8p
+4Q
, 0x1.ca8ed42a12ae3001a07244abad2bp
-160Q
},
99 { -0x2.8p
+4Q
, -0x1.ca8ed42a12ae3001a07244abad2bp
-160Q
},
100 { -0x2.9p
+4Q
, 0xb.2f30e1ce812063f12e7e8d8d96e8p
-168Q
},
101 { -0x2.9p
+4Q
, -0xb.2f30e1ce812063f12e7e8d8d96e8p
-168Q
},
102 { -0x2.ap
+4Q
, 0x4.42bd49d4c37a0db136489772e428p
-172Q
},
103 { -0x2.ap
+4Q
, -0x4.42bd49d4c37a0db136489772e428p
-172Q
},
104 { -0x2.bp
+4Q
, 0x1.95db45257e5122dcbae56def372p
-176Q
},
105 { -0x2.bp
+4Q
, -0x1.95db45257e5122dcbae56def372p
-176Q
},
106 { -0x2.cp
+4Q
, 0x9.3958d81ff63527ecf993f3fb6f48p
-184Q
},
107 { -0x2.cp
+4Q
, -0x9.3958d81ff63527ecf993f3fb6f48p
-184Q
},
108 { -0x2.dp
+4Q
, 0x3.47970e4440c8f1c058bd238c9958p
-188Q
},
109 { -0x2.dp
+4Q
, -0x3.47970e4440c8f1c058bd238c9958p
-188Q
},
110 { -0x2.ep
+4Q
, 0x1.240804f65951062ca46e4f25c608p
-192Q
},
111 { -0x2.ep
+4Q
, -0x1.240804f65951062ca46e4f25c608p
-192Q
},
112 { -0x2.fp
+4Q
, 0x6.36a382849fae6de2d15362d8a394p
-200Q
},
113 { -0x2.fp
+4Q
, -0x6.36a382849fae6de2d15362d8a394p
-200Q
},
114 { -0x3p
+4Q
, 0x2.123680d6dfe4cf4b9b1bcb9d8bdcp
-204Q
},
115 { -0x3p
+4Q
, -0x2.123680d6dfe4cf4b9b1bcb9d8bdcp
-204Q
},
116 { -0x3.1p
+4Q
, 0xa.d21786ff5842eca51fea0870919p
-212Q
},
117 { -0x3.1p
+4Q
, -0xa.d21786ff5842eca51fea0870919p
-212Q
},
118 { -0x3.2p
+4Q
, 0x3.766dedc259af040be140a68a6c04p
-216Q
},
121 static const __float128 e_hi
= 0x2.b7e151628aed2a6abf7158809cf4p
+0Q
;
122 static const __float128 e_lo
= 0xf.3c762e7160f38b4da56a784d9048p
-116Q
;
125 /* Coefficients B_2k / 2k(2k-1) of x^-(2k-1) in Stirling's
126 approximation to lgamma function. */
128 static const __float128 lgamma_coeff
[] =
130 0x1.5555555555555555555555555555p
-4Q
,
131 -0xb.60b60b60b60b60b60b60b60b60b8p
-12Q
,
132 0x3.4034034034034034034034034034p
-12Q
,
133 -0x2.7027027027027027027027027028p
-12Q
,
134 0x3.72a3c5631fe46ae1d4e700dca8f2p
-12Q
,
135 -0x7.daac36664f1f207daac36664f1f4p
-12Q
,
136 0x1.a41a41a41a41a41a41a41a41a41ap
-8Q
,
137 -0x7.90a1b2c3d4e5f708192a3b4c5d7p
-8Q
,
138 0x2.dfd2c703c0cfff430edfd2c703cp
-4Q
,
139 -0x1.6476701181f39edbdb9ce625987dp
+0Q
,
140 0xd.672219167002d3a7a9c886459cp
+0Q
,
141 -0x9.cd9292e6660d55b3f712eb9e07c8p
+4Q
,
142 0x8.911a740da740da740da740da741p
+8Q
,
143 -0x8.d0cc570e255bf59ff6eec24b49p
+12Q
,
144 0xa.8d1044d3708d1c219ee4fdc446ap
+16Q
,
145 -0xe.8844d8a169abbc406169abbc406p
+20Q
,
146 0x1.6d29a0f6433b79890cede62433b8p
+28Q
,
147 -0x2.88a233b3c8cddaba9809357125d8p
+32Q
,
148 0x5.0dde6f27500939a85c40939a85c4p
+36Q
,
149 -0xb.4005bde03d4642a243581714af68p
+40Q
,
150 0x1.bc8cd6f8f1f755c78753cdb5d5c9p
+48Q
,
151 -0x4.bbebb143bb94de5a0284fa7ec424p
+52Q
,
152 0xe.2e1337f5af0bed90b6b0a352d4fp
+56Q
,
153 -0x2.e78250162b62405ad3e4bfe61b38p
+64Q
,
154 0xa.5f7eef9e71ac7c80326ab4cc8bfp
+68Q
,
155 -0x2.83be0395e550213369924971b21ap
+76Q
,
156 0xa.8ebfe48da17dd999790760b0cep
+80Q
,
159 #define NCOEFF (sizeof (lgamma_coeff) / sizeof (lgamma_coeff[0]))
161 /* Polynomial approximations to (|gamma(x)|-1)(x-n)/(x-x0), where n is
162 the integer end-point of the half-integer interval containing x and
163 x0 is the zero of lgamma in that half-integer interval. Each
164 polynomial is expressed in terms of x-xm, where xm is the midpoint
165 of the interval for which the polynomial applies. */
167 static const __float128 poly_coeff
[] =
169 /* Interval [-2.125, -2] (polynomial degree 23). */
170 -0x1.0b71c5c54d42eb6c17f30b7aa8f5p
+0Q
,
171 -0xc.73a1dc05f34951602554c6d7506p
-4Q
,
172 -0x1.ec841408528b51473e6c425ee5ffp
-4Q
,
173 -0xe.37c9da26fc3c9a3c1844c8c7f1cp
-4Q
,
174 -0x1.03cd87c519305703b021fa33f827p
-4Q
,
175 -0xe.ae9ada65e09aa7f1c75216128f58p
-4Q
,
176 0x9.b11855a4864b5731cf85736015a8p
-8Q
,
177 -0xe.f28c133e697a95c28607c9701dep
-4Q
,
178 0x2.6ec14a1c586a72a7cc33ee569d6ap
-4Q
,
179 -0xf.57cab973e14464a262fc24723c38p
-4Q
,
180 0x4.5b0fc25f16e52997b2886bbae808p
-4Q
,
181 -0xf.f50e59f1a9b56e76e988dac9ccf8p
-4Q
,
182 0x6.5f5eae15e9a93369e1d85146c6fcp
-4Q
,
183 -0x1.0d2422daac459e33e0994325ed23p
+0Q
,
184 0x8.82000a0e7401fb1117a0e6606928p
-4Q
,
185 -0x1.1f492f178a3f1b19f58a2ca68e55p
+0Q
,
186 0xa.cb545f949899a04c160b19389abp
-4Q
,
187 -0x1.36165a1b155ba3db3d1b77caf498p
+0Q
,
188 0xd.44c5d5576f74302e5cf79e183eep
-4Q
,
189 -0x1.51f22e0cdd33d3d481e326c02f3ep
+0Q
,
190 0xf.f73a349c08244ac389c007779bfp
-4Q
,
191 -0x1.73317bf626156ba716747c4ca866p
+0Q
,
192 0x1.379c3c97b9bc71e1c1c4802dd657p
+0Q
,
193 -0x1.a72a351c54f902d483052000f5dfp
+0Q
,
194 /* Interval [-2.25, -2.125] (polynomial degree 24). */
195 -0xf.2930890d7d675a80c36afb0fd5e8p
-4Q
,
196 -0xc.a5cfde054eab5c6770daeca577f8p
-4Q
,
197 0x3.9c9e0fdebb07cdf89c61d41c9238p
-4Q
,
198 -0x1.02a5ad35605fcf4af65a6dbacb84p
+0Q
,
199 0x9.6e9b1185bb48be9de1918e00a2e8p
-4Q
,
200 -0x1.4d8332f3cfbfa116fd611e9ce90dp
+0Q
,
201 0x1.1c0c8cb4d9f4b1d490e1a41fae4dp
+0Q
,
202 -0x1.c9a6f5ae9130cd0299e293a42714p
+0Q
,
203 0x1.d7e9307fd58a2ea997f29573a112p
+0Q
,
204 -0x2.921cb3473d96178ca2a11d2a8d46p
+0Q
,
205 0x2.e8d59113b6f3409ff8db226e9988p
+0Q
,
206 -0x3.cbab931625a1ae2b26756817f264p
+0Q
,
207 0x4.7d9f0f05d5296d18663ca003912p
+0Q
,
208 -0x5.ade9cba12a14ea485667b7135bbp
+0Q
,
209 0x6.dc983a5da74fb48e767b7fec0a3p
+0Q
,
210 -0x8.8d9ed454ae31d9e138dd8ee0d1a8p
+0Q
,
211 0xa.6fa099d4e7c202e0c0fd6ed8492p
+0Q
,
212 -0xc.ebc552a8090a0f0115e92d4ebbc8p
+0Q
,
213 0xf.d695e4772c0d829b53fba9ca5568p
+0Q
,
214 -0x1.38c32ae38e5e9eb79b2a4c5570a9p
+4Q
,
215 0x1.8035145646cfab49306d0999a51bp
+4Q
,
216 -0x1.d930adbb03dd342a4c2a8c4e1af6p
+4Q
,
217 0x2.45c2edb1b4943ddb3686cd9c6524p
+4Q
,
218 -0x2.e818ebbfafe2f916fa21abf7756p
+4Q
,
219 0x3.9804ce51d0fb9a430a711fd7307p
+4Q
,
220 /* Interval [-2.375, -2.25] (polynomial degree 25). */
221 -0xd.7d28d505d6181218a25f31d5e45p
-4Q
,
222 -0xe.69649a3040985140cdf946829fap
-4Q
,
223 0xb.0d74a2827d053a8d44595012484p
-4Q
,
224 -0x1.924b0922853617cac181afbc08ddp
+0Q
,
225 0x1.d49b12bccf0a568582e2d3c410f3p
+0Q
,
226 -0x3.0898bb7d8c4093e636279c791244p
+0Q
,
227 0x4.207a6cac711cb53868e8a5057eep
+0Q
,
228 -0x6.39ee63ea4fb1dcab0c9144bf3ddcp
+0Q
,
229 0x8.e2e2556a797b649bf3f53bd26718p
+0Q
,
230 -0xd.0e83ac82552ef12af508589e7a8p
+0Q
,
231 0x1.2e4525e0ce6670563c6484a82b05p
+4Q
,
232 -0x1.b8e350d6a8f2b222fa390a57c23dp
+4Q
,
233 0x2.805cd69b919087d8a80295892c2cp
+4Q
,
234 -0x3.a42585424a1b7e64c71743ab014p
+4Q
,
235 0x5.4b4f409f98de49f7bfb03c05f984p
+4Q
,
236 -0x7.b3c5827fbe934bc820d6832fb9fcp
+4Q
,
237 0xb.33b7b90cc96c425526e0d0866e7p
+4Q
,
238 -0x1.04b77047ac4f59ee3775ca10df0dp
+8Q
,
239 0x1.7b366f5e94a34f41386eac086313p
+8Q
,
240 -0x2.2797338429385c9849ca6355bfc2p
+8Q
,
241 0x3.225273cf92a27c9aac1b35511256p
+8Q
,
242 -0x4.8f078aa48afe6cb3a4e89690f898p
+8Q
,
243 0x6.9f311d7b6654fc1d0b5195141d04p
+8Q
,
244 -0x9.a0c297b6b4621619ca9bacc48ed8p
+8Q
,
245 0xe.ce1f06b6f90d92138232a76e4cap
+8Q
,
246 -0x1.5b0e6806fa064daf011613e43b17p
+12Q
,
247 /* Interval [-2.5, -2.375] (polynomial degree 27). */
248 -0xb.74ea1bcfff94b2c01afba9daa7d8p
-4Q
,
249 -0x1.2a82bd590c37538cab143308de4dp
+0Q
,
250 0x1.88020f828b966fec66b8649fd6fcp
+0Q
,
251 -0x3.32279f040eb694970e9db24863dcp
+0Q
,
252 0x5.57ac82517767e68a721005853864p
+0Q
,
253 -0x9.c2aedcfe22833de43834a0a6cc4p
+0Q
,
254 0x1.12c132f1f5577f99e1a0ed3538e1p
+4Q
,
255 -0x1.ea94e26628a3de3597f7bb55a948p
+4Q
,
256 0x3.66b4ac4fa582f58b59f96b2f7c7p
+4Q
,
257 -0x6.0cf746a9cf4cba8c39afcc73fc84p
+4Q
,
258 0xa.c102ef2c20d75a342197df7fedf8p
+4Q
,
259 -0x1.31ebff06e8f14626782df58db3b6p
+8Q
,
260 0x2.1fd6f0c0e710994e059b9dbdb1fep
+8Q
,
261 -0x3.c6d76040407f447f8b5074f07706p
+8Q
,
262 0x6.b6d18e0d8feb4c2ef5af6a40ed18p
+8Q
,
263 -0xb.efaf542c529f91e34217f24ae6a8p
+8Q
,
264 0x1.53852d873210e7070f5d9eb2296p
+12Q
,
265 -0x2.5b977c0ddc6d540717173ac29fc8p
+12Q
,
266 0x4.310d452ae05100eff1e02343a724p
+12Q
,
267 -0x7.73a5d8f20c4f986a7dd1912b2968p
+12Q
,
268 0xd.3f5ea2484f3fca15eab1f4d1a218p
+12Q
,
269 -0x1.78d18aac156d1d93a2ffe7e08d3fp
+16Q
,
270 0x2.9df49ca75e5b567f5ea3e47106cp
+16Q
,
271 -0x4.a7149af8961a08aa7c3233b5bb94p
+16Q
,
272 0x8.3db10ffa742c707c25197d989798p
+16Q
,
273 -0xe.a26d6dd023cadd02041a049ec368p
+16Q
,
274 0x1.c825d90514e7c57c7fa5316f947cp
+20Q
,
275 -0x3.34bb81e5a0952df8ca1abdc6684cp
+20Q
,
276 /* Interval [-2.625, -2.5] (polynomial degree 28). */
277 -0x3.d10108c27ebafad533c20eac32bp
-4Q
,
278 0x1.cd557caff7d2b2085f41dbec5106p
+0Q
,
279 0x3.819b4856d399520dad9776ea2cacp
+0Q
,
280 0x6.8505cbad03dc34c5e42e8b12eb78p
+0Q
,
281 0xb.c1b2e653a9e38f82b399c94e7f08p
+0Q
,
282 0x1.50a53a38f148138105124df65419p
+4Q
,
283 0x2.57ae00cbe5232cbeeed34d89727ap
+4Q
,
284 0x4.2b156301b8604db85a601544bfp
+4Q
,
285 0x7.6989ed23ca3ca7579b3462592b5cp
+4Q
,
286 0xd.2dd2976557939517f831f5552cc8p
+4Q
,
287 0x1.76e1c3430eb860969bce40cd494p
+8Q
,
288 0x2.9a77bf5488742466db3a2c7c1ec6p
+8Q
,
289 0x4.a0d62ed7266e8eb36f725a8ebcep
+8Q
,
290 0x8.3a6184dd3021067df2f8b91e99c8p
+8Q
,
291 0xe.a0ade1538245bf55d39d7e436b1p
+8Q
,
292 0x1.a01359fae8617b5826dd74428e9p
+12Q
,
293 0x2.e3b0a32caae77251169acaca1ad4p
+12Q
,
294 0x5.2301257c81589f62b38fb5993ee8p
+12Q
,
295 0x9.21c9275db253d4e719b73b18cb9p
+12Q
,
296 0x1.03c104bc96141cda3f3fa4b112bcp
+16Q
,
297 0x1.cdc8ed65119196a08b0c78f1445p
+16Q
,
298 0x3.34f31d2eaacf34382cdb0073572ap
+16Q
,
299 0x5.b37628cadf12bf0000907d0ef294p
+16Q
,
300 0xa.22d8b332c0b1e6a616f425dfe5ap
+16Q
,
301 0x1.205b01444804c3ff922cd78b4c42p
+20Q
,
302 0x1.fe8f0cea9d1e0ff25be2470b4318p
+20Q
,
303 0x3.8872aebeb368399aee02b39340aep
+20Q
,
304 0x6.ebd560d351e84e26a4381f5b293cp
+20Q
,
305 0xc.c3644d094b0dae2fbcbf682cd428p
+20Q
,
306 /* Interval [-2.75, -2.625] (polynomial degree 26). */
307 -0x6.b5d252a56e8a75458a27ed1c2dd4p
-4Q
,
308 0x1.28d60383da3ac721aed3c5794da9p
+0Q
,
309 0x1.db6513ada8a66ea77d87d9a8827bp
+0Q
,
310 0x2.e217118f9d348a27f7506a707e6ep
+0Q
,
311 0x4.450112c5cbf725a0fb9802396c9p
+0Q
,
312 0x6.4af99151eae7810a75df2a0303c4p
+0Q
,
313 0x9.2db598b4a97a7f69aeef32aec758p
+0Q
,
314 0xd.62bef9c22471f5ee47ea1b9c0b5p
+0Q
,
315 0x1.379f294e412bd62328326d4222f9p
+4Q
,
316 0x1.c5827349d8865f1e8825c37c31c6p
+4Q
,
317 0x2.93a7e7a75b7568cc8cbe8c016c12p
+4Q
,
318 0x3.bf9bb882afe57edb383d41879d3ap
+4Q
,
319 0x5.73c737828cee095c43a5566731c8p
+4Q
,
320 0x7.ee4653493a7f81e0442062b3823cp
+4Q
,
321 0xb.891c6b83fc8b55bd973b5d962d6p
+4Q
,
322 0x1.0c775d7de3bf9b246c0208e0207ep
+8Q
,
323 0x1.867ee43ec4bd4f4fd56abc05110ap
+8Q
,
324 0x2.37fe9ba6695821e9822d8c8af0a6p
+8Q
,
325 0x3.3a2c667e37c942f182cd3223a936p
+8Q
,
326 0x4.b1b500eb59f3f782c7ccec88754p
+8Q
,
327 0x6.d3efd3b65b3d0d8488d30b79fa4cp
+8Q
,
328 0x9.ee8224e65bed5ced8b75eaec609p
+8Q
,
329 0xe.72416e510cca77d53fc615c1f3dp
+8Q
,
330 0x1.4fb538b0a2dfe567a8904b7e0445p
+12Q
,
331 0x1.e7f56a9266cf525a5b8cf4cb76cep
+12Q
,
332 0x2.f0365c983f68c597ee49d099cce8p
+12Q
,
333 0x4.53aa229e1b9f5b5e59625265951p
+12Q
,
334 /* Interval [-2.875, -2.75] (polynomial degree 24). */
335 -0x8.a41b1e4f36ff88dc820815607d68p
-4Q
,
336 0xc.da87d3b69dc0f2f9c6f368b8ca1p
-4Q
,
337 0x1.1474ad5c36158a7bea04fd2f98c6p
+0Q
,
338 0x1.761ecb90c555df6555b7dba955b6p
+0Q
,
339 0x1.d279bff9ae291caf6c4b4bcb3202p
+0Q
,
340 0x2.4e5d00559a6e2b9b5d7fe1f6689cp
+0Q
,
341 0x2.d57545a75cee8743ae2b17bc8d24p
+0Q
,
342 0x3.8514eee3aac88b89bec2307021bap
+0Q
,
343 0x4.5235e3b6e1891ffeb87fed9f8a24p
+0Q
,
344 0x5.562acdb10eef3c9a773b3e27a864p
+0Q
,
345 0x6.8ec8965c76efe03c26bff60b1194p
+0Q
,
346 0x8.15251aca144877af32658399f9b8p
+0Q
,
347 0x9.f08d56aba174d844138af782c0f8p
+0Q
,
348 0xc.3dbbeda2679e8a1346ccc3f6da88p
+0Q
,
349 0xf.0f5bfd5eacc26db308ffa0556fa8p
+0Q
,
350 0x1.28a6ccd84476fbc713d6bab49ac9p
+4Q
,
351 0x1.6d0a3ae2a3b1c8ff400641a3a21fp
+4Q
,
352 0x1.c15701b28637f87acfb6a91d33b5p
+4Q
,
353 0x2.28fbe0eccf472089b017651ca55ep
+4Q
,
354 0x2.a8a453004f6e8ffaacd1603bc3dp
+4Q
,
355 0x3.45ae4d9e1e7cd1a5dba0e4ec7f6cp
+4Q
,
356 0x4.065fbfacb7fad3e473cb577a61e8p
+4Q
,
357 0x4.f3d1473020927acac1944734a39p
+4Q
,
358 0x6.54bb091245815a36fb74e314dd18p
+4Q
,
359 0x7.d7f445129f7fb6c055e582d3f6ep
+4Q
,
360 /* Interval [-3, -2.875] (polynomial degree 23). */
361 -0xa.046d667e468f3e44dcae1afcc648p
-4Q
,
362 0x9.70b88dcc006c214d8d996fdf5ccp
-4Q
,
363 0xa.a8a39421c86d3ff24931a0929fp
-4Q
,
364 0xd.2f4d1363f324da2b357c8b6ec94p
-4Q
,
365 0xd.ca9aa1a3a5c00de11bf60499a97p
-4Q
,
366 0xf.cf09c31eeb52a45dfa7ebe3778dp
-4Q
,
367 0x1.04b133a39ed8a09691205660468bp
+0Q
,
368 0x1.22b547a06edda944fcb12fd9b5ecp
+0Q
,
369 0x1.2c57fce7db86a91df09602d344b3p
+0Q
,
370 0x1.4aade4894708f84795212fe257eep
+0Q
,
371 0x1.579c8b7b67ec4afed5b28c8bf787p
+0Q
,
372 0x1.776820e7fc80ae5284239733078ap
+0Q
,
373 0x1.883ab28c7301fde4ca6b8ec26ec8p
+0Q
,
374 0x1.aa2ef6e1ae52eb42c9ee83b206e3p
+0Q
,
375 0x1.bf4ad50f0a9a9311300cf0c51ee7p
+0Q
,
376 0x1.e40206e0e96b1da463814dde0d09p
+0Q
,
377 0x1.fdcbcffef3a21b29719c2bd9feb1p
+0Q
,
378 0x2.25e2e8948939c4d42cf108fae4bep
+0Q
,
379 0x2.44ce14d2b59c1c0e6bf2cfa81018p
+0Q
,
380 0x2.70ee80bbd0387162be4861c43622p
+0Q
,
381 0x2.954b64d2c2ebf3489b949c74476p
+0Q
,
382 0x2.c616e133a811c1c9446105208656p
+0Q
,
383 0x3.05a69dfe1a9ba1079f90fcf26bd4p
+0Q
,
384 0x3.410d2ad16a0506de29736e6aafdap
+0Q
,
387 static const size_t poly_deg
[] =
399 static const size_t poly_end
[] =
411 /* Compute sin (pi * X) for -0.25 <= X <= 0.5. */
414 lg_sinpi (__float128 x
)
417 return sinq (M_PIq
* x
);
419 return cosq (M_PIq
* (0.5Q
- x
));
422 /* Compute cos (pi * X) for -0.25 <= X <= 0.5. */
425 lg_cospi (__float128 x
)
428 return cosq (M_PIq
* x
);
430 return sinq (M_PIq
* (0.5Q
- x
));
433 /* Compute cot (pi * X) for -0.25 <= X <= 0.5. */
436 lg_cotpi (__float128 x
)
438 return lg_cospi (x
) / lg_sinpi (x
);
441 /* Compute lgamma of a negative argument -50 < X < -2, setting
442 *SIGNGAMP accordingly. */
445 __quadmath_lgamma_negq (__float128 x
, int *signgamp
)
447 /* Determine the half-integer region X lies in, handle exact
448 integers and determine the sign of the result. */
449 int i
= floorq (-2 * x
);
450 if ((i
& 1) == 0 && i
== -2 * x
)
452 __float128 xn
= ((i
& 1) == 0 ? -i
/ 2 : (-i
- 1) / 2);
454 *signgamp
= ((i
& 2) == 0 ? -1 : 1);
456 SET_RESTORE_ROUNDF128 (FE_TONEAREST
);
458 /* Expand around the zero X0 = X0_HI + X0_LO. */
459 __float128 x0_hi
= lgamma_zeros
[i
][0], x0_lo
= lgamma_zeros
[i
][1];
460 __float128 xdiff
= x
- x0_hi
- x0_lo
;
462 /* For arguments in the range -3 to -2, use polynomial
463 approximations to an adjusted version of the gamma function. */
466 int j
= floorq (-8 * x
) - 16;
467 __float128 xm
= (-33 - 2 * j
) * 0.0625Q
;
468 __float128 x_adj
= x
- xm
;
469 size_t deg
= poly_deg
[j
];
470 size_t end
= poly_end
[j
];
471 __float128 g
= poly_coeff
[end
];
472 for (size_t j
= 1; j
<= deg
; j
++)
473 g
= g
* x_adj
+ poly_coeff
[end
- j
];
474 return log1pq (g
* xdiff
/ (x
- xn
));
477 /* The result we want is log (sinpi (X0) / sinpi (X))
478 + log (gamma (1 - X0) / gamma (1 - X)). */
479 __float128 x_idiff
= fabsq (xn
- x
), x0_idiff
= fabsq (xn
- x0_hi
- x0_lo
);
480 __float128 log_sinpi_ratio
;
481 if (x0_idiff
< x_idiff
* 0.5Q
)
482 /* Use log not log1p to avoid inaccuracy from log1p of arguments
484 log_sinpi_ratio
= logq (lg_sinpi (x0_idiff
)
485 / lg_sinpi (x_idiff
));
488 /* Use log1p not log to avoid inaccuracy from log of arguments
489 close to 1. X0DIFF2 has positive sign if X0 is further from
490 XN than X is from XN, negative sign otherwise. */
491 __float128 x0diff2
= ((i
& 1) == 0 ? xdiff
: -xdiff
) * 0.5Q
;
492 __float128 sx0d2
= lg_sinpi (x0diff2
);
493 __float128 cx0d2
= lg_cospi (x0diff2
);
494 log_sinpi_ratio
= log1pq (2 * sx0d2
495 * (-sx0d2
+ cx0d2
* lg_cotpi (x_idiff
)));
498 __float128 log_gamma_ratio
;
499 __float128 y0
= 1 - x0_hi
;
500 __float128 y0_eps
= -x0_hi
+ (1 - y0
) - x0_lo
;
501 __float128 y
= 1 - x
;
502 __float128 y_eps
= -x
+ (1 - y
);
503 /* We now wish to compute LOG_GAMMA_RATIO
504 = log (gamma (Y0 + Y0_EPS) / gamma (Y + Y_EPS)). XDIFF
505 accurately approximates the difference Y0 + Y0_EPS - Y -
506 Y_EPS. Use Stirling's approximation. First, we may need to
507 adjust into the range where Stirling's approximation is
508 sufficiently accurate. */
509 __float128 log_gamma_adj
= 0;
512 int n_up
= (21 - i
) / 2;
513 __float128 ny0
, ny0_eps
, ny
, ny_eps
;
515 ny0_eps
= y0
- (ny0
- n_up
) + y0_eps
;
519 ny_eps
= y
- (ny
- n_up
) + y_eps
;
522 __float128 prodm1
= __quadmath_lgamma_productq (xdiff
, y
- n_up
, y_eps
, n_up
);
523 log_gamma_adj
= -log1pq (prodm1
);
525 __float128 log_gamma_high
526 = (xdiff
* log1pq ((y0
- e_hi
- e_lo
+ y0_eps
) / e_hi
)
527 + (y
- 0.5Q
+ y_eps
) * log1pq (xdiff
/ y
) + log_gamma_adj
);
528 /* Compute the sum of (B_2k / 2k(2k-1))(Y0^-(2k-1) - Y^-(2k-1)). */
529 __float128 y0r
= 1 / y0
, yr
= 1 / y
;
530 __float128 y0r2
= y0r
* y0r
, yr2
= yr
* yr
;
531 __float128 rdiff
= -xdiff
/ (y
* y0
);
532 __float128 bterm
[NCOEFF
];
533 __float128 dlast
= rdiff
, elast
= rdiff
* yr
* (yr
+ y0r
);
534 bterm
[0] = dlast
* lgamma_coeff
[0];
535 for (size_t j
= 1; j
< NCOEFF
; j
++)
537 __float128 dnext
= dlast
* y0r2
+ elast
;
538 __float128 enext
= elast
* yr2
;
539 bterm
[j
] = dnext
* lgamma_coeff
[j
];
543 __float128 log_gamma_low
= 0;
544 for (size_t j
= 0; j
< NCOEFF
; j
++)
545 log_gamma_low
+= bterm
[NCOEFF
- 1 - j
];
546 log_gamma_ratio
= log_gamma_high
+ log_gamma_low
;
548 return log_sinpi_ratio
+ log_gamma_ratio
;