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050f29c188
The existing implementations of lgamma functions (except for the ia64 versions) use the reflection formula for negative arguments. This suffers large inaccuracy from cancellation near zeros of lgamma (near where the gamma function is +/- 1). This patch fixes this inaccuracy. For arguments above -2, there are no zeros and no large cancellation, while for sufficiently large negative arguments the zeros are so close to integers that even for integers +/- 1ulp the log(gamma(1-x)) term dominates and cancellation is not significant. Thus, it is only necessary to take special care about cancellation for arguments around a limited number of zeros. Accordingly, this patch uses precomputed tables of relevant zeros, expressed as the sum of two floating-point values. The log of the ratio of two sines can be computed accurately using log1p in cases where log would lose accuracy. The log of the ratio of two gamma(1-x) values can be computed using Stirling's approximation (the difference between two values of that approximation to lgamma being computable without computing the two values and then subtracting), with appropriate adjustments (which don't reduce accuracy too much) in cases where 1-x is too small to use Stirling's approximation directly. In the interval from -3 to -2, using the ratios of sines and of gamma(1-x) can still produce too much cancellation between those two parts of the computation (and that interval is also the worst interval for computing the ratio between gamma(1-x) values, which computation becomes more accurate, while being less critical for the final result, for larger 1-x). Because this can result in errors slightly above those accepted in glibc, this interval is instead dealt with by polynomial approximations. Separate polynomial approximations to (|gamma(x)|-1)(x-n)/(x-x0) are used for each interval of length 1/8 from -3 to -2, where n (-3 or -2) is the nearest integer to the 1/8-interval and x0 is the zero of lgamma in the relevant half-integer interval (-3 to -2.5 or -2.5 to -2). Together, the two approaches are intended to give sufficient accuracy for all negative arguments in the problem range. Outside that range, the previous implementation continues to be used. Tested for x86_64, x86, mips64 and powerpc. The mips64 and powerpc testing shows up pre-existing problems for ldbl-128 and ldbl-128ibm with large negative arguments giving spurious "invalid" exceptions (exposed by newly added tests for cases this patch doesn't affect the logic for); I'll address those problems separately. [BZ #2542] [BZ #2543] [BZ #2558] * sysdeps/ieee754/dbl-64/e_lgamma_r.c (__ieee754_lgamma_r): Call __lgamma_neg for arguments from -28.0 to -2.0. * sysdeps/ieee754/flt-32/e_lgammaf_r.c (__ieee754_lgammaf_r): Call __lgamma_negf for arguments from -15.0 to -2.0. * sysdeps/ieee754/ldbl-128/e_lgammal_r.c (__ieee754_lgammal_r): Call __lgamma_negl for arguments from -48.0 or -50.0 to -2.0. * sysdeps/ieee754/ldbl-96/e_lgammal_r.c (__ieee754_lgammal_r): Call __lgamma_negl for arguments from -33.0 to -2.0. * sysdeps/ieee754/dbl-64/lgamma_neg.c: New file. * sysdeps/ieee754/dbl-64/lgamma_product.c: Likewise. * sysdeps/ieee754/flt-32/lgamma_negf.c: Likewise. * sysdeps/ieee754/flt-32/lgamma_productf.c: Likewise. * sysdeps/ieee754/ldbl-128/lgamma_negl.c: Likewise. * sysdeps/ieee754/ldbl-128/lgamma_productl.c: Likewise. * sysdeps/ieee754/ldbl-128ibm/lgamma_negl.c: Likewise. * sysdeps/ieee754/ldbl-128ibm/lgamma_productl.c: Likewise. * sysdeps/ieee754/ldbl-96/lgamma_negl.c: Likewise. * sysdeps/ieee754/ldbl-96/lgamma_product.c: Likewise. * sysdeps/ieee754/ldbl-96/lgamma_productl.c: Likewise. * sysdeps/generic/math_private.h (__lgamma_negf): New prototype. (__lgamma_neg): Likewise. (__lgamma_negl): Likewise. (__lgamma_product): Likewise. (__lgamma_productl): Likewise. * math/Makefile (libm-calls): Add lgamma_neg and lgamma_product. * math/auto-libm-test-in: Add more tests of lgamma. * math/auto-libm-test-out: Regenerated. * sysdeps/i386/fpu/libm-test-ulps: Update. * sysdeps/x86_64/fpu/libm-test-ulps: Likewise.
533 lines
22 KiB
C
533 lines
22 KiB
C
/* lgammal expanding around zeros.
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Copyright (C) 2015 Free Software Foundation, Inc.
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This file is part of the GNU C Library.
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The GNU C Library is free software; you can redistribute it and/or
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modify it under the terms of the GNU Lesser General Public
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License as published by the Free Software Foundation; either
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version 2.1 of the License, or (at your option) any later version.
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The GNU C Library is distributed in the hope that it will be useful,
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but WITHOUT ANY WARRANTY; without even the implied warranty of
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MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
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Lesser General Public License for more details.
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You should have received a copy of the GNU Lesser General Public
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License along with the GNU C Library; if not, see
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<http://www.gnu.org/licenses/>. */
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#include <float.h>
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#include <math.h>
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#include <math_private.h>
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static const long double lgamma_zeros[][2] =
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{
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{ -0x2.74ff92c01f0d82abec9f315f1ap+0L, -0x7.12c334804d9a79cb5d46094d46p-112L },
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{ -0x2.bf6821437b20197995a4b4641fp+0L, 0x5.140b4ff4b7d6069e1bd7acc196p-108L },
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{ -0x3.24c1b793cb35efb8be699ad3dap+0L, 0x4.59abab3480539f1c0e926287cp-108L },
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{ -0x3.f48e2a8f85fca170d456129123p+0L, -0x6.cc320a4887d1cb4c711828a75ep-108L },
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{ -0x4.0a139e16656030c39f0b0de182p+0L, 0xe.d53e84029416e1242006b2b3dp-108L },
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{ -0x4.fdd5de9bbabf3510d0aa407698p+0L, -0x8.501d7d78125286f78d1e501f14p-108L },
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{ -0x5.021a95fc2db6432a4c56e5953ap+0L, 0xb.2133950fbcf2b01a8b9058dcccp-108L },
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{ -0x5.ffa4bd647d0357dd4ed62cbd32p+0L, 0x1.2071c071a2145d2982428f2269p-108L },
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{ -0x6.005ac9625f233b607c2d96d164p+0L, 0x7.a347953a96cbf30e1a0db20856p-108L },
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{ -0x6.fff2fddae1bbff3d626b65c24p+0L, 0x2.de0bfcff5c457ebcf4d3ad9674p-108L },
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{ -0x7.000cff7b7f87adf4482dcdb988p+0L, 0x7.d54d99e35a74d6407b80292df2p-108L },
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{ -0x7.fffe5fe05673c3ca9e82b522bp+0L, -0xc.a9d2e8837cd1f14bd3d05002e4p-108L },
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{ -0x8.0001a01459fc9f60cb3cec1cecp+0L, -0x8.576677ca538d88084310983b8p-108L },
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{ -0x8.ffffd1c425e80ffc864e957494p+0L, 0x1.a6181dfdef1807e3087e4bb163p-104L },
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{ -0x9.00002e3bb47d86d6d843fedc34p+0L, -0x1.1deb7ad09ec5e9d6e8ae2d548bp-104L },
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{ -0x9.fffffb606bdfdcd062ae77a504p+0L, -0x1.47c69d2eb6f33d170fce38ff818p-104L },
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{ -0xa.0000049f93bb9927b45d95e154p+0L, -0x4.1e03086db9146a9287bd4f2172p-108L },
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{ -0xa.ffffff9466e9f1b36dacd2adbcp+0L, -0x1.18d05a4e458062f3f95345a4dap-104L },
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{ -0xb.0000006b9915315d965a6ffea4p+0L, -0xe.4bea39000dcc1848023c5f6bdcp-112L },
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{ -0xb.fffffff7089387387de41acc3cp+0L, -0x1.3c978bd839c8c428b5efcf91ef8p-104L },
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{ -0xc.00000008f76c7731567c0f025p+0L, -0xf.387920df5675833859190eb128p-108L },
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{ -0xc.ffffffff4f6dcf617f97a5ffc8p+0L, 0xa.82ab72d76f32eaee2d1a42ed5p-108L },
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{ -0xd.00000000b092309c06683dd1b8p+0L, -0x1.03e3700857a15c19ac5a611de98p-104L },
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{ -0xd.fffffffff36345ab9e184a3e08p+0L, -0x1.d1176dc48e47f62d917973dd45p-104L },
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{ -0xe.000000000c9cba545e94e75ec4p+0L, -0x1.718f753e2501e757a17cf2ecbfp-104L },
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{ -0xe.ffffffffff28c060c6604ef304p+0L, 0x8.e0762c8ca8361c23e8393919c4p-108L },
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{ -0xf.0000000000d73f9f399bd0e42p+0L, -0xf.85e9ee31b0b890744fc0e3fbcp-108L },
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{ -0xf.fffffffffff28c060c6621f514p+0L, 0x1.18d1b2eec9d960bd9adc5be5f6p-104L },
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{ -0x1.000000000000d73f9f399da1428p+4L, 0x3.406c46e0e88305d2800f0e414cp-104L },
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{ -0x1.0ffffffffffff3569c47e7a93ep+4L, -0x1.c46a08a2e008a998ebabb8087fp-104L },
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{ -0x1.1000000000000ca963b81856888p+4L, -0x7.6ca5a3a64ec15db0a95caf2cap-108L },
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{ -0x1.1fffffffffffff4bec3ce23413p+4L, -0x2.d08b2b726187c841cb92cd5222p-104L },
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{ -0x1.20000000000000b413c31dcbec8p+4L, -0x2.4c3b2ffacbb4932f18dceedfd7p-104L },
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{ -0x1.2ffffffffffffff685b25cbf5f8p+4L, 0x2.ba3126cd1c7b7a0822d694705cp-104L },
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{ -0x1.30000000000000097a4da340a08p+4L, -0x2.b81b7b1f1f001c72bf914141efp-104L },
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{ -0x1.3fffffffffffffff86af516ff8p+4L, 0x8.9429818df2a87abafd48248a2p-108L },
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{ -0x1.40000000000000007950ae9008p+4L, -0x8.9413ccc8a353fda263f8ce973cp-108L },
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{ -0x1.4ffffffffffffffffa391c4249p+4L, 0x3.d5c63022b62b5484ba346524dbp-104L },
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{ -0x1.500000000000000005c6e3bdb7p+4L, -0x3.d5c62f55ed5322b2685c5e9a52p-104L },
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{ -0x1.5fffffffffffffffffbcc71a49p+4L, -0x2.01eb5aeb96c74d7ad25e060529p-104L },
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{ -0x1.6000000000000000004338e5b7p+4L, 0x2.01eb5aec04b2f2eb663e4e3d8ap-104L },
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{ -0x1.6ffffffffffffffffffd13c97d8p+4L, -0x1.d38fcc4d08d6fe5aa56ab04308p-104L },
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{ -0x1.70000000000000000002ec36828p+4L, 0x1.d38fcc4d090cee2f5d0b69a99cp-104L },
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{ -0x1.7fffffffffffffffffffe0d31p+4L, 0x1.972f577cca4b4c8cb1dc14001bp-104L },
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{ -0x1.800000000000000000001f2cfp+4L, -0x1.972f577cca4b3442e35f0040b38p-104L },
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{ -0x1.8ffffffffffffffffffffec0c3p+4L, -0x3.22e9a0572b1bb5b95f346a92d6p-104L },
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{ -0x1.90000000000000000000013f3dp+4L, 0x3.22e9a0572b1bb5c371ddb35617p-104L },
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{ -0x1.9ffffffffffffffffffffff3b88p+4L, -0x3.d01cad8d32e386fd783e97296dp-104L },
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{ -0x1.a0000000000000000000000c478p+4L, 0x3.d01cad8d32e386fd7c1ab8c1fep-104L },
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{ -0x1.afffffffffffffffffffffff8b8p+4L, -0x1.538f48cc5737d5979c39db806c8p-104L },
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{ -0x1.b00000000000000000000000748p+4L, 0x1.538f48cc5737d5979c3b3a6bdap-104L },
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{ -0x1.bffffffffffffffffffffffffcp+4L, 0x2.862898d42174dcf171470d8c8cp-104L },
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{ -0x1.c0000000000000000000000004p+4L, -0x2.862898d42174dcf171470d18bap-104L },
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{ -0x1.dp+4L, 0x2.4b3f31686b15af57c61ceecdf4p-104L },
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{ -0x1.dp+4L, -0x2.4b3f31686b15af57c61ceecdd1p-104L },
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{ -0x1.ep+4L, 0x1.3932c5047d60e60caded4c298ap-108L },
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{ -0x1.ep+4L, -0x1.3932c5047d60e60caded4c29898p-108L },
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{ -0x1.fp+4L, 0xa.1a6973c1fade2170f7237d36p-116L },
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{ -0x1.fp+4L, -0xa.1a6973c1fade2170f7237d36p-116L },
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{ -0x2p+4L, 0x5.0d34b9e0fd6f10b87b91be9bp-120L },
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{ -0x2p+4L, -0x5.0d34b9e0fd6f10b87b91be9bp-120L },
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{ -0x2.1p+4L, 0x2.73024a9ba1aa36a7059bff52e8p-124L },
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{ -0x2.1p+4L, -0x2.73024a9ba1aa36a7059bff52e8p-124L },
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{ -0x2.2p+4L, 0x1.2710231c0fd7a13f8a2b4af9d68p-128L },
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{ -0x2.2p+4L, -0x1.2710231c0fd7a13f8a2b4af9d68p-128L },
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{ -0x2.3p+4L, 0x8.6e2ce38b6c8f9419e3fad3f03p-136L },
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{ -0x2.3p+4L, -0x8.6e2ce38b6c8f9419e3fad3f03p-136L },
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{ -0x2.4p+4L, 0x3.bf30652185952560d71a254e4fp-140L },
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{ -0x2.4p+4L, -0x3.bf30652185952560d71a254e4fp-140L },
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{ -0x2.5p+4L, 0x1.9ec8d1c94e85af4c78b15c3d8ap-144L },
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{ -0x2.5p+4L, -0x1.9ec8d1c94e85af4c78b15c3d8ap-144L },
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{ -0x2.6p+4L, 0xa.ea565ce061d57489e9b8527628p-152L },
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{ -0x2.6p+4L, -0xa.ea565ce061d57489e9b8527628p-152L },
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{ -0x2.7p+4L, 0x4.7a6512692eb37804111dabad3p-156L },
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{ -0x2.7p+4L, -0x4.7a6512692eb37804111dabad3p-156L },
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{ -0x2.8p+4L, 0x1.ca8ed42a12ae3001a07244abadp-160L },
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{ -0x2.8p+4L, -0x1.ca8ed42a12ae3001a07244abadp-160L },
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{ -0x2.9p+4L, 0xb.2f30e1ce812063f12e7e8d8d98p-168L },
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{ -0x2.9p+4L, -0xb.2f30e1ce812063f12e7e8d8d98p-168L },
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{ -0x2.ap+4L, 0x4.42bd49d4c37a0db136489772e4p-172L },
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{ -0x2.ap+4L, -0x4.42bd49d4c37a0db136489772e4p-172L },
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{ -0x2.bp+4L, 0x1.95db45257e5122dcbae56def37p-176L },
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{ -0x2.bp+4L, -0x1.95db45257e5122dcbae56def37p-176L },
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{ -0x2.cp+4L, 0x9.3958d81ff63527ecf993f3fb7p-184L },
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{ -0x2.cp+4L, -0x9.3958d81ff63527ecf993f3fb7p-184L },
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{ -0x2.dp+4L, 0x3.47970e4440c8f1c058bd238c99p-188L },
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{ -0x2.dp+4L, -0x3.47970e4440c8f1c058bd238c99p-188L },
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{ -0x2.ep+4L, 0x1.240804f65951062ca46e4f25c6p-192L },
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{ -0x2.ep+4L, -0x1.240804f65951062ca46e4f25c6p-192L },
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{ -0x2.fp+4L, 0x6.36a382849fae6de2d15362d8a4p-200L },
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{ -0x2.fp+4L, -0x6.36a382849fae6de2d15362d8a4p-200L },
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{ -0x3p+4L, 0x2.123680d6dfe4cf4b9b1bcb9d8cp-204L },
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};
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static const long double e_hi = 0x2.b7e151628aed2a6abf7158809dp+0L;
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static const long double e_lo = -0xb.0c389d18e9f0c74b25a9587b28p-112L;
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/* Coefficients B_2k / 2k(2k-1) of x^-(2k-1) in Stirling's
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approximation to lgamma function. */
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static const long double lgamma_coeff[] =
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{
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0x1.555555555555555555555555558p-4L,
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-0xb.60b60b60b60b60b60b60b60b6p-12L,
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0x3.4034034034034034034034034p-12L,
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-0x2.7027027027027027027027027p-12L,
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0x3.72a3c5631fe46ae1d4e700dca9p-12L,
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-0x7.daac36664f1f207daac36664f2p-12L,
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0x1.a41a41a41a41a41a41a41a41a4p-8L,
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-0x7.90a1b2c3d4e5f708192a3b4c5ep-8L,
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0x2.dfd2c703c0cfff430edfd2c704p-4L,
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-0x1.6476701181f39edbdb9ce625988p+0L,
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0xd.672219167002d3a7a9c886459cp+0L,
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-0x9.cd9292e6660d55b3f712eb9e08p+4L,
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0x8.911a740da740da740da740da74p+8L,
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-0x8.d0cc570e255bf59ff6eec24b48p+12L,
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0xa.8d1044d3708d1c219ee4fdc448p+16L,
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-0xe.8844d8a169abbc406169abbc4p+20L,
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0x1.6d29a0f6433b79890cede624338p+28L,
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-0x2.88a233b3c8cddaba9809357126p+32L,
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0x5.0dde6f27500939a85c40939a86p+36L,
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-0xb.4005bde03d4642a243581714bp+40L,
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0x1.bc8cd6f8f1f755c78753cdb5d6p+48L,
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-0x4.bbebb143bb94de5a0284fa7ec4p+52L,
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0xe.2e1337f5af0bed90b6b0a352d4p+56L,
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-0x2.e78250162b62405ad3e4bfe61bp+64L,
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0xa.5f7eef9e71ac7c80326ab4cc8cp+68L,
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-0x2.83be0395e550213369924971b2p+76L,
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};
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#define NCOEFF (sizeof (lgamma_coeff) / sizeof (lgamma_coeff[0]))
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/* Polynomial approximations to (|gamma(x)|-1)(x-n)/(x-x0), where n is
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the integer end-point of the half-integer interval containing x and
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x0 is the zero of lgamma in that half-integer interval. Each
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polynomial is expressed in terms of x-xm, where xm is the midpoint
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of the interval for which the polynomial applies. */
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static const long double poly_coeff[] =
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{
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/* Interval [-2.125, -2] (polynomial degree 21). */
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-0x1.0b71c5c54d42eb6c17f30b7aa9p+0L,
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-0xc.73a1dc05f34951602554c6d76cp-4L,
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-0x1.ec841408528b51473e6c42f1c58p-4L,
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-0xe.37c9da26fc3c9a3c1844c04b84p-4L,
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-0x1.03cd87c519305703b00b046ce4p-4L,
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-0xe.ae9ada65e09aa7f1c817c91048p-4L,
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0x9.b11855a4864b571b6a4f571c88p-8L,
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-0xe.f28c133e697a95ba2dabb97584p-4L,
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0x2.6ec14a1c586a7ddb6c4be90fe1p-4L,
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-0xf.57cab973e14496f0900851c0d4p-4L,
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0x4.5b0fc25f16b0df37175495c70cp-4L,
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-0xf.f50e59f1a8fb8c402091e3cd3cp-4L,
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0x6.5f5eae1681d1e50e575c3d4d36p-4L,
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-0x1.0d2422dac7ea8a52db6bf0d14fp+0L,
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0x8.820008f221eae5a36e15913bacp-4L,
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-0x1.1f492eec53b9481ea23a7e944ep+0L,
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0xa.cb55b4d662945e8cf1f81ee5b4p-4L,
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-0x1.3616863983e131d7935700ccd48p+0L,
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0xd.43c783ebab66074d18709d5cap-4L,
|
|
-0x1.51d5dbc56bc85976871c6e51f78p+0L,
|
|
0x1.06253af656eb6b2ed998387aabp+0L,
|
|
-0x1.7d910a0aadc63d7a1ef7690dbb8p+0L,
|
|
/* Interval [-2.25, -2.125] (polynomial degree 22). */
|
|
-0xf.2930890d7d675a80c36afb0fd4p-4L,
|
|
-0xc.a5cfde054eab5c6770daeca684p-4L,
|
|
0x3.9c9e0fdebb07cdf89c61d434adp-4L,
|
|
-0x1.02a5ad35605fcf4af65a67fe8a8p+0L,
|
|
0x9.6e9b1185bb48be9de18d8bbeb8p-4L,
|
|
-0x1.4d8332f3cfbfa116fdf648372cp+0L,
|
|
0x1.1c0c8cb4d9f4b1d495142b53ebp+0L,
|
|
-0x1.c9a6f5ae9130ccfb9b7e39136f8p+0L,
|
|
0x1.d7e9307fd58a2e85209d0e83eap+0L,
|
|
-0x2.921cb3473d96462f22c171712fp+0L,
|
|
0x2.e8d59113b6f3fc1ed3b556b62cp+0L,
|
|
-0x3.cbab931624e3b6cf299cea1213p+0L,
|
|
0x4.7d9f0f05d2c4cf91e41ea1f048p+0L,
|
|
-0x5.ade9cba31affa276fe516135eep+0L,
|
|
0x6.dc983a62cf6ddc935ae3c5b9ap+0L,
|
|
-0x8.8d9ed100b2a7813f82cbd83e3cp+0L,
|
|
0xa.6fa0926892835a9a29c9b8db8p+0L,
|
|
-0xc.ebc90aff4ffe319d70bef0d61p+0L,
|
|
0xf.d69cf50ab226bacece014c0b44p+0L,
|
|
-0x1.389964ac7cfef4578eec028e5c8p+4L,
|
|
0x1.7ff0d2090164e25901f97cab3bp+4L,
|
|
-0x1.e9e6d282da6bd004619d073071p+4L,
|
|
0x2.5d719ab6ad4be8b5c32b0fba2ap+4L,
|
|
/* Interval [-2.375, -2.25] (polynomial degree 24). */
|
|
-0xd.7d28d505d6181218a25f31d5e4p-4L,
|
|
-0xe.69649a3040985140cdf946827cp-4L,
|
|
0xb.0d74a2827d053a8d4459500f88p-4L,
|
|
-0x1.924b0922853617cac181b097e48p+0L,
|
|
0x1.d49b12bccf0a568582e2dbf8ep+0L,
|
|
-0x3.0898bb7d8c4093e6360d26bbc5p+0L,
|
|
0x4.207a6cac711cb538684f74619ep+0L,
|
|
-0x6.39ee63ea4fb1dcac86ab337e3cp+0L,
|
|
0x8.e2e2556a797b64a1b9328a3978p+0L,
|
|
-0xd.0e83ac82552ee5596df1706ff4p+0L,
|
|
0x1.2e4525e0ce666e48fac68ddcdep+4L,
|
|
-0x1.b8e350d6a8f6597ed2eb3c2eff8p+4L,
|
|
0x2.805cd69b9197ee0089dd1b1c46p+4L,
|
|
-0x3.a42585423e4d00db075f2d687ep+4L,
|
|
0x5.4b4f409f874e2a7dcd8aa4a62ap+4L,
|
|
-0x7.b3c5829962ca1b95535db9cc4ep+4L,
|
|
0xb.33b7b928986ec6b219e2e15a98p+4L,
|
|
-0x1.04b76dec4115106bb16316d9cd8p+8L,
|
|
0x1.7b366d8d46f179d5c5302d6534p+8L,
|
|
-0x2.2799846ddc54813d40da622b99p+8L,
|
|
0x3.2253a862c1078a3ccabac65bebp+8L,
|
|
-0x4.8d92cebc90a4a29816f4952f4ep+8L,
|
|
0x6.9ebb8f9d72c66c80c4f4492e7ap+8L,
|
|
-0xa.2850a483f9ba0e43f5848b5cd8p+8L,
|
|
0xe.e1b6bdce83b27944edab8c428p+8L,
|
|
/* Interval [-2.5, -2.375] (polynomial degree 25). */
|
|
-0xb.74ea1bcfff94b2c01afba9daa8p-4L,
|
|
-0x1.2a82bd590c37538cab143308e3p+0L,
|
|
0x1.88020f828b966fec66b8648d16p+0L,
|
|
-0x3.32279f040eb694970e9db0308bp+0L,
|
|
0x5.57ac82517767e68a72142041b4p+0L,
|
|
-0x9.c2aedcfe22833de438786dc658p+0L,
|
|
0x1.12c132f1f5577f99dbfb7ecb408p+4L,
|
|
-0x1.ea94e26628a3de3557dc349db8p+4L,
|
|
0x3.66b4ac4fa582f5cbe7e19d10c6p+4L,
|
|
-0x6.0cf746a9cf4cbcb0004cb01f66p+4L,
|
|
0xa.c102ef2c20d5a313cbfd37f5b8p+4L,
|
|
-0x1.31ebff06e8f08f58d1c35eacfdp+8L,
|
|
0x2.1fd6f0c0e788660ba1f1573722p+8L,
|
|
-0x3.c6d760404305e75356a86a11d6p+8L,
|
|
0x6.b6d18e0c31a2ba4d5b5ac78676p+8L,
|
|
-0xb.efaf5426343e6b41a823ed6c44p+8L,
|
|
0x1.53852db2fe01305b9f336d132d8p+12L,
|
|
-0x2.5b977cb2b568382e71ca93a36bp+12L,
|
|
0x4.310d090a6119c7d85a2786a616p+12L,
|
|
-0x7.73a518387ef1d4d04917dfb25cp+12L,
|
|
0xd.3f965798601aabd24bdaa6e68cp+12L,
|
|
-0x1.78db20b0b166480c93cf0031198p+16L,
|
|
0x2.9be0068b65cf13bd1cf71f0eccp+16L,
|
|
-0x4.a221230466b9cd51d5b811d6b6p+16L,
|
|
0x8.f6f8c13e2b52aa3e30a4ce6898p+16L,
|
|
-0x1.02145337ff16b44fa7c2adf7f28p+20L,
|
|
/* Interval [-2.625, -2.5] (polynomial degree 26). */
|
|
-0x3.d10108c27ebafad533c20eac33p-4L,
|
|
0x1.cd557caff7d2b2085f41dbec538p+0L,
|
|
0x3.819b4856d399520dad9776ebb9p+0L,
|
|
0x6.8505cbad03dc34c5e42e89c4b4p+0L,
|
|
0xb.c1b2e653a9e38f82b3997134a8p+0L,
|
|
0x1.50a53a38f1481381051544750ep+4L,
|
|
0x2.57ae00cbe5232cbeef4e94eb2cp+4L,
|
|
0x4.2b156301b8604db82856d5767p+4L,
|
|
0x7.6989ed23ca3ca751fc9c32eb88p+4L,
|
|
0xd.2dd29765579396f3a456772c44p+4L,
|
|
0x1.76e1c3430eb8630991d1aa8a248p+8L,
|
|
0x2.9a77bf548873743fe65d025f56p+8L,
|
|
0x4.a0d62ed7266389753842d7be74p+8L,
|
|
0x8.3a6184dd32d31ec73fc6f2d37cp+8L,
|
|
0xe.a0ade153a3bf0247db49e11ae8p+8L,
|
|
0x1.a01359fa74d4eaf8858bbc35f68p+12L,
|
|
0x2.e3b0a32845cbc135bae4a5216cp+12L,
|
|
0x5.23012653815fe88456170a7dc6p+12L,
|
|
0x9.21c92dcde748ec199bc9c65738p+12L,
|
|
0x1.03c0f3621b4c67d2d86e5e813d8p+16L,
|
|
0x1.cdc884edcc9f5404f2708551cb8p+16L,
|
|
0x3.35025f0b1624d1ffc86688bf03p+16L,
|
|
0x5.b3bd9562ebf2409c5ce99929ep+16L,
|
|
0xa.1a229b1986d9f89cb80abccfdp+16L,
|
|
0x1.1e69136ebd520146d51837f3308p+20L,
|
|
0x2.2d2738c72449db2524171b9271p+20L,
|
|
0x4.036e80cc6621b836f94f426834p+20L,
|
|
/* Interval [-2.75, -2.625] (polynomial degree 24). */
|
|
-0x6.b5d252a56e8a75458a27ed1c2ep-4L,
|
|
0x1.28d60383da3ac721aed3c57949p+0L,
|
|
0x1.db6513ada8a66ea77d87d9a796p+0L,
|
|
0x2.e217118f9d348a27f7506c4b4fp+0L,
|
|
0x4.450112c5cbf725a0fb982fc44cp+0L,
|
|
0x6.4af99151eae7810a75a5fceac8p+0L,
|
|
0x9.2db598b4a97a7f69ab7be31128p+0L,
|
|
0xd.62bef9c22471f5f17955733c6p+0L,
|
|
0x1.379f294e412bd6255506135f4a8p+4L,
|
|
0x1.c5827349d8865d858d4f85f3c38p+4L,
|
|
0x2.93a7e7a75b755bbea1785a1349p+4L,
|
|
0x3.bf9bb882afed66a08b22ed7a45p+4L,
|
|
0x5.73c737828d2044aca95fdef33ep+4L,
|
|
0x7.ee46534920f1c81574db260f0ep+4L,
|
|
0xb.891c6b837b513eaf1592fe78ccp+4L,
|
|
0x1.0c775d815bf741526a3dd66ded8p+8L,
|
|
0x1.867ee44cf11f26455a8924a56bp+8L,
|
|
0x2.37fe968baa1018e55cae680f1dp+8L,
|
|
0x3.3a2c557f686679eb5d8e960fd1p+8L,
|
|
0x4.b1ba0539d4d80cc9174738b992p+8L,
|
|
0x6.d3fd80155b6d2211956cb6bc5ap+8L,
|
|
0x9.eb5a96b0ee3d9ca523f5fbc1fp+8L,
|
|
0xe.6b37429c1acc7dc19ef312dda4p+8L,
|
|
0x1.621132d6aa138b203a28e4792fp+12L,
|
|
0x2.09610219270e2ce11a985d4d36p+12L,
|
|
/* Interval [-2.875, -2.75] (polynomial degree 23). */
|
|
-0x8.a41b1e4f36ff88dc820815607cp-4L,
|
|
0xc.da87d3b69dc0f2f9c6f368b8c8p-4L,
|
|
0x1.1474ad5c36158a7bea04fd30b28p+0L,
|
|
0x1.761ecb90c555df6555b7dbb9ce8p+0L,
|
|
0x1.d279bff9ae291caf6c4b17497f8p+0L,
|
|
0x2.4e5d00559a6e2b9b5d7e35b575p+0L,
|
|
0x2.d57545a75cee8743b1ff6e22b8p+0L,
|
|
0x3.8514eee3aac88b89d2d4ddef4ep+0L,
|
|
0x4.5235e3b6e1891fd9c975383318p+0L,
|
|
0x5.562acdb10eef3c14a780490e3cp+0L,
|
|
0x6.8ec8965c76f0b261bc41b5e532p+0L,
|
|
0x8.15251aca144a98a1e1c0981388p+0L,
|
|
0x9.f08d56ab9e7eee9515a457214cp+0L,
|
|
0xc.3dbbeda2620d5be4fe8621ce6p+0L,
|
|
0xf.0f5bfd65b3feb6d745a2cdbf9cp+0L,
|
|
0x1.28a6ccd8dd27fb90fcaa31d37dp+4L,
|
|
0x1.6d0a3a3091c3d64cfd1a3c5769p+4L,
|
|
0x1.c1570107e02d5ab0b8bea6d6c98p+4L,
|
|
0x2.28fc9b295b583fa469de7acceap+4L,
|
|
0x2.a8a4cac0217026bbdbce34f4adp+4L,
|
|
0x3.4532c98bce75262ac0ede53edep+4L,
|
|
0x4.062fd9ba18e00e55c25a4f0688p+4L,
|
|
0x5.22e00e6d9846a3451fad5587f8p+4L,
|
|
0x6.5d0f7ce92a0bf928d4a30e92c6p+4L,
|
|
/* Interval [-3, -2.875] (polynomial degree 22). */
|
|
-0xa.046d667e468f3e44dcae1afcc8p-4L,
|
|
0x9.70b88dcc006c214d8d996fdf7p-4L,
|
|
0xa.a8a39421c86d3ff24931a093c4p-4L,
|
|
0xd.2f4d1363f324da2b357c850124p-4L,
|
|
0xd.ca9aa1a3a5c00de11bf5d7047p-4L,
|
|
0xf.cf09c31eeb52a45dfb25e50ebcp-4L,
|
|
0x1.04b133a39ed8a096914cc78812p+0L,
|
|
0x1.22b547a06edda9447f516a2ee7p+0L,
|
|
0x1.2c57fce7db86a91c8d0f12077b8p+0L,
|
|
0x1.4aade4894708fb8b78365e9bf88p+0L,
|
|
0x1.579c8b7b67ec5179ecc4e9c7dp+0L,
|
|
0x1.776820e7fc7361c50e7ef40a88p+0L,
|
|
0x1.883ab28c72ef238ada6c480ab18p+0L,
|
|
0x1.aa2ef6e1d11b9fcea06a1dcab1p+0L,
|
|
0x1.bf4ad50f2dd2aeb02395ea08648p+0L,
|
|
0x1.e40206a5477615838e02279dfc8p+0L,
|
|
0x1.fdcbcfd4b0777fb173b85d5b398p+0L,
|
|
0x2.25e32b3b3c89e833029169a17bp+0L,
|
|
0x2.44ce344ff0bda6570fe3d0a76dp+0L,
|
|
0x2.70bfba6fa079faf2dbf31d2216p+0L,
|
|
0x2.953e22a97725cc179ad21024fap+0L,
|
|
0x2.d8ccc51524659a499eee0f267p+0L,
|
|
0x3.080fbb09c14936c2171c8a51bcp+0L,
|
|
};
|
|
|
|
static const size_t poly_deg[] =
|
|
{
|
|
21,
|
|
22,
|
|
24,
|
|
25,
|
|
26,
|
|
24,
|
|
23,
|
|
22,
|
|
};
|
|
|
|
static const size_t poly_end[] =
|
|
{
|
|
21,
|
|
44,
|
|
69,
|
|
95,
|
|
122,
|
|
147,
|
|
171,
|
|
194,
|
|
};
|
|
|
|
/* Compute sin (pi * X) for -0.25 <= X <= 0.5. */
|
|
|
|
static long double
|
|
lg_sinpi (long double x)
|
|
{
|
|
if (x <= 0.25L)
|
|
return __sinl (M_PIl * x);
|
|
else
|
|
return __cosl (M_PIl * (0.5L - x));
|
|
}
|
|
|
|
/* Compute cos (pi * X) for -0.25 <= X <= 0.5. */
|
|
|
|
static long double
|
|
lg_cospi (long double x)
|
|
{
|
|
if (x <= 0.25L)
|
|
return __cosl (M_PIl * x);
|
|
else
|
|
return __sinl (M_PIl * (0.5L - x));
|
|
}
|
|
|
|
/* Compute cot (pi * X) for -0.25 <= X <= 0.5. */
|
|
|
|
static long double
|
|
lg_cotpi (long double x)
|
|
{
|
|
return lg_cospi (x) / lg_sinpi (x);
|
|
}
|
|
|
|
/* Compute lgamma of a negative argument -48 < X < -2, setting
|
|
*SIGNGAMP accordingly. */
|
|
|
|
long double
|
|
__lgamma_negl (long double x, int *signgamp)
|
|
{
|
|
/* Determine the half-integer region X lies in, handle exact
|
|
integers and determine the sign of the result. */
|
|
int i = __floorl (-2 * x);
|
|
if ((i & 1) == 0 && i == -2 * x)
|
|
return 1.0L / 0.0L;
|
|
long double xn = ((i & 1) == 0 ? -i / 2 : (-i - 1) / 2);
|
|
i -= 4;
|
|
*signgamp = ((i & 2) == 0 ? -1 : 1);
|
|
|
|
SET_RESTORE_ROUNDL (FE_TONEAREST);
|
|
|
|
/* Expand around the zero X0 = X0_HI + X0_LO. */
|
|
long double x0_hi = lgamma_zeros[i][0], x0_lo = lgamma_zeros[i][1];
|
|
long double xdiff = x - x0_hi - x0_lo;
|
|
|
|
/* For arguments in the range -3 to -2, use polynomial
|
|
approximations to an adjusted version of the gamma function. */
|
|
if (i < 2)
|
|
{
|
|
int j = __floorl (-8 * x) - 16;
|
|
long double xm = (-33 - 2 * j) * 0.0625L;
|
|
long double x_adj = x - xm;
|
|
size_t deg = poly_deg[j];
|
|
size_t end = poly_end[j];
|
|
long double g = poly_coeff[end];
|
|
for (size_t j = 1; j <= deg; j++)
|
|
g = g * x_adj + poly_coeff[end - j];
|
|
return __log1pl (g * xdiff / (x - xn));
|
|
}
|
|
|
|
/* The result we want is log (sinpi (X0) / sinpi (X))
|
|
+ log (gamma (1 - X0) / gamma (1 - X)). */
|
|
long double x_idiff = fabsl (xn - x), x0_idiff = fabsl (xn - x0_hi - x0_lo);
|
|
long double log_sinpi_ratio;
|
|
if (x0_idiff < x_idiff * 0.5L)
|
|
/* Use log not log1p to avoid inaccuracy from log1p of arguments
|
|
close to -1. */
|
|
log_sinpi_ratio = __ieee754_logl (lg_sinpi (x0_idiff)
|
|
/ lg_sinpi (x_idiff));
|
|
else
|
|
{
|
|
/* Use log1p not log to avoid inaccuracy from log of arguments
|
|
close to 1. X0DIFF2 has positive sign if X0 is further from
|
|
XN than X is from XN, negative sign otherwise. */
|
|
long double x0diff2 = ((i & 1) == 0 ? xdiff : -xdiff) * 0.5L;
|
|
long double sx0d2 = lg_sinpi (x0diff2);
|
|
long double cx0d2 = lg_cospi (x0diff2);
|
|
log_sinpi_ratio = __log1pl (2 * sx0d2
|
|
* (-sx0d2 + cx0d2 * lg_cotpi (x_idiff)));
|
|
}
|
|
|
|
long double log_gamma_ratio;
|
|
long double y0 = 1 - x0_hi;
|
|
long double y0_eps = -x0_hi + (1 - y0) - x0_lo;
|
|
long double y = 1 - x;
|
|
long double y_eps = -x + (1 - y);
|
|
/* We now wish to compute LOG_GAMMA_RATIO
|
|
= log (gamma (Y0 + Y0_EPS) / gamma (Y + Y_EPS)). XDIFF
|
|
accurately approximates the difference Y0 + Y0_EPS - Y -
|
|
Y_EPS. Use Stirling's approximation. First, we may need to
|
|
adjust into the range where Stirling's approximation is
|
|
sufficiently accurate. */
|
|
long double log_gamma_adj = 0;
|
|
if (i < 18)
|
|
{
|
|
int n_up = (19 - i) / 2;
|
|
long double ny0, ny0_eps, ny, ny_eps;
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ny0 = y0 + n_up;
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ny0_eps = y0 - (ny0 - n_up) + y0_eps;
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y0 = ny0;
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y0_eps = ny0_eps;
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ny = y + n_up;
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ny_eps = y - (ny - n_up) + y_eps;
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y = ny;
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y_eps = ny_eps;
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long double prodm1 = __lgamma_productl (xdiff, y - n_up, y_eps, n_up);
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log_gamma_adj = -__log1pl (prodm1);
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}
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long double log_gamma_high
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= (xdiff * __log1pl ((y0 - e_hi - e_lo + y0_eps) / e_hi)
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+ (y - 0.5L + y_eps) * __log1pl (xdiff / y) + log_gamma_adj);
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/* Compute the sum of (B_2k / 2k(2k-1))(Y0^-(2k-1) - Y^-(2k-1)). */
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long double y0r = 1 / y0, yr = 1 / y;
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long double y0r2 = y0r * y0r, yr2 = yr * yr;
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long double rdiff = -xdiff / (y * y0);
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long double bterm[NCOEFF];
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long double dlast = rdiff, elast = rdiff * yr * (yr + y0r);
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bterm[0] = dlast * lgamma_coeff[0];
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for (size_t j = 1; j < NCOEFF; j++)
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|
{
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long double dnext = dlast * y0r2 + elast;
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long double enext = elast * yr2;
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bterm[j] = dnext * lgamma_coeff[j];
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dlast = dnext;
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elast = enext;
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}
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long double log_gamma_low = 0;
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for (size_t j = 0; j < NCOEFF; j++)
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log_gamma_low += bterm[NCOEFF - 1 - j];
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log_gamma_ratio = log_gamma_high + log_gamma_low;
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|
|
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return log_sinpi_ratio + log_gamma_ratio;
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}
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