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4239f144ce
libquadmath sources are mostly based on glibc sources at present, but derived from them by a manual editing / substitution process and with subsequent manual merges. The manual effort involved in merges means they are sometimes incomplete and long-delayed. Since libquadmath was first created, glibc's support for this format has undergone significant changes so that it can also be used in glibc to provide *f128 functions for the _Float128 type from TS 18661-3. This makes it significantly easier to use it for libquadmath in a more automated fashion, since glibc has a float128_private.h header that redefines many identifiers as macros as needed for building *f128 functions. Simply using float128_private.h directly in libquadmath, with unmodified glibc sources except for changing function names in that one header to be *q instead of *f128, would be tricky, given its dependence on lots of other glibc-internal headers (whereas libquadmath supports non-glibc systems), and also given how some libm functions in glibc are built from type-generic templates using a further set of macros rather than from separate function implementations for each type. So instead this patch adds a script update-quadmath.py to convert glibc sources into libquadmath ones, and the script reads float128_private.h to identify many of the substitutions it should make. quadmath-imp.h is updated with various new internal definitions, taken from glibc as needed; this is the main place expected to need updating manually when subsequent merges from glibc are done using the script. No attempt is made to make the script output match the details of existing formatting, although the differences are of a size that makes a rough comparison (ignoring whitespace) possible. Two new public interfaces are added to libquadmath, exp2q and issignalingq, at a new QUADMATH_1.2 symbol version, since those interfaces are used internally by some of the glibc sources being merged into libquadmath; although there is a new symbol version, no change however is made to the libtool version in the libtool-version file. Although there are various other interfaces now in glibc libm but not in libquadmath, this patch does nothing to add such interfaces (although adding many of them would in fact be easy to do, given the script). One internal file (not providing any public interfaces), math/isinf_nsq.c, is removed, as no longer used by anything in libquadmath after the merge. Conditionals in individual source files on <fenv.h> availability or features are moved into quadmath-imp.h (providing trivial macro versions of the functions if real implementations aren't available), to simplify the substitutions in individual source files. Note however that I haven't tested for any configurations lacking <fenv.h>, so further changes could well be needed there. Two files in libquadmath/math/ are based on glibc sources but not updated in this patch: fmaq.c and rem_pio2q.c. Both could be updated after further changes to the script (and quadmath-imp.h as needed); in the case of rem_pio2q.c, based on two separate glibc source files, those separate files would naturally be split out into separate libquadmath source files in the process (as done in this patch with expq_table.h and tanq_kernel.c, where previously two glibc source files had been merged into one libquadmath source file). complex.c, nanq.c and sqrtq.c are not based on glibc sources (though four of the (trivial) functions in complex.c could readily be replaced by instead using the four corresponding files from glibc, if desired). libquadmath also has printf/ and strtod/ sources based on glibc, also mostly not updated for a long time. Again the script could no doubt be made to generate those automatically, although that would be a larger change (effectively some completely separate logic in the script, not sharing much if anything with the existing code). Bootstrapped with no regressions on x86_64-pc-linux-gnu. PR libquadmath/68686 * Makefile.am: (libquadmath_la_SOURCES): Remove math/isinf_nsq.c. Add math/exp2q.c math/issignalingq.c math/lgammaq_neg.c math/lgammaq_product.c math/tanq_kernel.c math/tgammaq_product.c math/casinhq_kernel.c. * Makefile.in: Regenerate. * libquadmath.texi (exp2q, issignalingq): Document. * quadmath-imp.h: Include <errno.h>, <limits.h>, <stdbool.h> and <fenv.h>. (HIGH_ORDER_BIT_IS_SET_FOR_SNAN, FIX_FLT128_LONG_CONVERT_OVERFLOW) (FIX_FLT128_LLONG_CONVERT_OVERFLOW, __quadmath_kernel_tanq) (__quadmath_gamma_productq, __quadmath_gammaq_r) (__quadmath_lgamma_negq, __quadmath_lgamma_productq) (__quadmath_lgammaq_r, __quadmath_kernel_casinhq, mul_splitq) (math_check_force_underflow_complex, __glibc_likely) (__glibc_unlikely, struct rm_ctx, SET_RESTORE_ROUNDF128) (libc_feholdsetround_ctx, libc_feresetround_ctx): New. (feraiseexcept, fenv_t, feholdexcept, fesetround, feupdateenv) (fesetenv, fetestexcept, feclearexcept): Define if not supported through <fenv.h>. (__quadmath_isinf_nsq): Remove. * quadmath.h (exp2q, issignalingq): New. * quadmath.map (QUADMATH_1.2): New. * quadmath_weak.h (exp2q, issignalingq): New. * update-quadmath.py: New file. * math/isinf_nsq.c: Remove file. * math/casinhq_kernel.c, math/exp2q.c, math/expq_table.h, math/issignalingq.c, math/lgammaq_neg.c, math/lgammaq_product.c, math/tanq_kernel.c, math/tgammaq_product.c: New files. Generated from glibc sources with update-quadmath.py. * math/acoshq.c, math/acosq.c, math/asinhq.c, math/asinq.c, math/atan2q.c, math/atanhq.c, math/atanq.c, math/cacoshq.c, math/cacosq.c, math/casinhq.c, math/casinq.c, math/catanhq.c, math/catanq.c, math/cbrtq.c, math/ccoshq.c, math/ceilq.c, math/cexpq.c, math/cimagq.c, math/clog10q.c, math/clogq.c, math/conjq.c, math/copysignq.c, math/coshq.c, math/cosq.c, math/cosq_kernel.c, math/cprojq.c, math/crealq.c, math/csinhq.c, math/csinq.c, math/csqrtq.c, math/ctanhq.c, math/ctanq.c, math/erfq.c, math/expm1q.c, math/expq.c, math/fabsq.c, math/fdimq.c, math/finiteq.c, math/floorq.c, math/fmaxq.c, math/fminq.c, math/fmodq.c, math/frexpq.c, math/hypotq.c, math/ilogbq.c, math/isinfq.c, math/isnanq.c, math/j0q.c, math/j1q.c, math/jnq.c, math/ldexpq.c, math/lgammaq.c, math/llrintq.c, math/llroundq.c, math/log10q.c, math/log1pq.c, math/log2q.c, math/logbq.c, math/logq.c, math/lrintq.c, math/lroundq.c, math/modfq.c, math/nearbyintq.c, math/nextafterq.c, math/powq.c, math/remainderq.c, math/remquoq.c, math/rintq.c, math/roundq.c, math/scalblnq.c, math/scalbnq.c, math/signbitq.c, math/sincos_table.c, math/sincosq.c, math/sincosq_kernel.c, math/sinhq.c, math/sinq.c, math/sinq_kernel.c, math/tanhq.c, math/tanq.c, math/tgammaq.c, math/truncq.c, math/x2y2m1q.c: Regenerate from glibc sources with update-quadmath.py. From-SVN: r265822
550 lines
23 KiB
C
550 lines
23 KiB
C
/* lgammal expanding around zeros.
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Copyright (C) 2015-2018 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 "quadmath-imp.h"
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static const __float128 lgamma_zeros[][2] =
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{
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{ -0x2.74ff92c01f0d82abec9f315f1a08p+0Q, 0xe.d3ccb7fb2658634a2b9f6b2ba81p-116Q },
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{ -0x2.bf6821437b20197995a4b4641eaep+0Q, -0xb.f4b00b4829f961e428533e6ad048p-116Q },
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{ -0x3.24c1b793cb35efb8be699ad3d9bap+0Q, -0x6.5454cb7fac60e3f16d9d7840c2ep-116Q },
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{ -0x3.f48e2a8f85fca170d4561291236cp+0Q, -0xc.320a4887d1cb4c711828a75d5758p-116Q },
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{ -0x4.0a139e16656030c39f0b0de18114p+0Q, 0x1.53e84029416e1242006b2b3d1cfp-112Q },
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{ -0x4.fdd5de9bbabf3510d0aa40769884p+0Q, -0x1.01d7d78125286f78d1e501f14966p-112Q },
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{ -0x5.021a95fc2db6432a4c56e595394cp+0Q, -0x1.ecc6af0430d4fe5746fa7233356fp-112Q },
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{ -0x5.ffa4bd647d0357dd4ed62cbd31ecp+0Q, -0x1.f8e3f8e5deba2d67dbd70dd96ce1p-112Q },
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{ -0x6.005ac9625f233b607c2d96d16384p+0Q, -0x1.cb86ac569340cf1e5f24df7aab7bp-112Q },
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{ -0x6.fff2fddae1bbff3d626b65c23fd4p+0Q, 0x1.e0bfcff5c457ebcf4d3ad9674167p-112Q },
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{ -0x7.000cff7b7f87adf4482dcdb98784p+0Q, 0x1.54d99e35a74d6407b80292df199fp-112Q },
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{ -0x7.fffe5fe05673c3ca9e82b522b0ccp+0Q, 0x1.62d177c832e0eb42c2faffd1b145p-112Q },
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{ -0x8.0001a01459fc9f60cb3cec1cec88p+0Q, 0x2.8998835ac7277f7bcef67c47f188p-112Q },
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{ -0x8.ffffd1c425e80ffc864e95749258p+0Q, -0x1.e7e20210e7f81cf781b44e9d2b02p-112Q },
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{ -0x9.00002e3bb47d86d6d843fedc352p+0Q, 0x2.14852f613a16291751d2ab751f7ep-112Q },
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{ -0x9.fffffb606bdfdcd062ae77a50548p+0Q, 0x3.962d1490cc2e8f031c7007eaa1ap-116Q },
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{ -0xa.0000049f93bb9927b45d95e1544p+0Q, -0x1.e03086db9146a9287bd4f2172d5ap-112Q },
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{ -0xa.ffffff9466e9f1b36dacd2adbd18p+0Q, -0xd.05a4e458062f3f95345a4d9c9b6p-116Q },
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{ -0xb.0000006b9915315d965a6ffea41p+0Q, 0x1.b415c6fff233e7b7fdc3a094246fp-112Q },
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{ -0xb.fffffff7089387387de41acc3d4p+0Q, 0x3.687427c6373bd74a10306e10a28ep-112Q },
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{ -0xc.00000008f76c7731567c0f0250fp+0Q, -0x3.87920df5675833859190eb128ef6p-112Q },
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{ -0xc.ffffffff4f6dcf617f97a5ffc758p+0Q, 0x2.ab72d76f32eaee2d1a42ed515d3ap-116Q },
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{ -0xd.00000000b092309c06683dd1b9p+0Q, -0x3.e3700857a15c19ac5a611de9688ap-112Q },
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{ -0xd.fffffffff36345ab9e184a3e09dp+0Q, -0x1.176dc48e47f62d917973dd44e553p-112Q },
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{ -0xe.000000000c9cba545e94e75ec57p+0Q, -0x1.8f753e2501e757a17cf2ecbeeb89p-112Q },
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{ -0xe.ffffffffff28c060c6604ef3037p+0Q, -0x1.f89d37357c9e3dc17c6c6e63becap-112Q },
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{ -0xf.0000000000d73f9f399bd0e420f8p+0Q, -0x5.e9ee31b0b890744fc0e3fbc01048p-116Q },
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{ -0xf.fffffffffff28c060c6621f512e8p+0Q, 0xd.1b2eec9d960bd9adc5be5f5fa5p-116Q },
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{ -0x1.000000000000d73f9f399da1424cp+4Q, 0x6.c46e0e88305d2800f0e414c506a8p-116Q },
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{ -0x1.0ffffffffffff3569c47e7a93e1cp+4Q, -0x4.6a08a2e008a998ebabb8087efa2cp-112Q },
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{ -0x1.1000000000000ca963b818568887p+4Q, -0x6.ca5a3a64ec15db0a95caf2c9ffb4p-112Q },
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{ -0x1.1fffffffffffff4bec3ce234132dp+4Q, -0x8.b2b726187c841cb92cd5221e444p-116Q },
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{ -0x1.20000000000000b413c31dcbeca5p+4Q, 0x3.c4d005344b6cd0e7231120294abcp-112Q },
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{ -0x1.2ffffffffffffff685b25cbf5f54p+4Q, -0x5.ced932e38485f7dd296b8fa41448p-112Q },
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{ -0x1.30000000000000097a4da340a0acp+4Q, 0x7.e484e0e0ffe38d406ebebe112f88p-112Q },
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{ -0x1.3fffffffffffffff86af516ff7f7p+4Q, -0x6.bd67e720d57854502b7db75e1718p-112Q },
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{ -0x1.40000000000000007950ae900809p+4Q, 0x6.bec33375cac025d9c073168c5d9p-112Q },
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{ -0x1.4ffffffffffffffffa391c4248c3p+4Q, 0x5.c63022b62b5484ba346524db607p-112Q },
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{ -0x1.500000000000000005c6e3bdb73dp+4Q, -0x5.c62f55ed5322b2685c5e9a51e6a8p-112Q },
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{ -0x1.5fffffffffffffffffbcc71a492p+4Q, -0x1.eb5aeb96c74d7ad25e060528fb5p-112Q },
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{ -0x1.6000000000000000004338e5b6ep+4Q, 0x1.eb5aec04b2f2eb663e4e3d8a018cp-112Q },
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{ -0x1.6ffffffffffffffffffd13c97d9dp+4Q, -0x3.8fcc4d08d6fe5aa56ab04307ce7ep-112Q },
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{ -0x1.70000000000000000002ec368263p+4Q, 0x3.8fcc4d090cee2f5d0b69a99c353cp-112Q },
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{ -0x1.7fffffffffffffffffffe0d30fe7p+4Q, 0x7.2f577cca4b4c8cb1dc14001ac5ecp-112Q },
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{ -0x1.800000000000000000001f2cf019p+4Q, -0x7.2f577cca4b3442e35f0040b3b9e8p-112Q },
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{ -0x1.8ffffffffffffffffffffec0c332p+4Q, -0x2.e9a0572b1bb5b95f346a92d67a6p-112Q },
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{ -0x1.90000000000000000000013f3ccep+4Q, 0x2.e9a0572b1bb5c371ddb3561705ap-112Q },
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{ -0x1.9ffffffffffffffffffffff3b8bdp+4Q, -0x1.cad8d32e386fd783e97296d63dcbp-116Q },
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{ -0x1.a0000000000000000000000c4743p+4Q, 0x1.cad8d32e386fd7c1ab8c1fe34c0ep-116Q },
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{ -0x1.afffffffffffffffffffffff8b95p+4Q, -0x3.8f48cc5737d5979c39db806c5406p-112Q },
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{ -0x1.b00000000000000000000000746bp+4Q, 0x3.8f48cc5737d5979c3b3a6bda06f6p-112Q },
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{ -0x1.bffffffffffffffffffffffffbd8p+4Q, 0x6.2898d42174dcf171470d8c8c6028p-112Q },
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{ -0x1.c000000000000000000000000428p+4Q, -0x6.2898d42174dcf171470d18ba412cp-112Q },
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{ -0x1.cfffffffffffffffffffffffffdbp+4Q, -0x4.c0ce9794ea50a839e311320bde94p-112Q },
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{ -0x1.d000000000000000000000000025p+4Q, 0x4.c0ce9794ea50a839e311322f7cf8p-112Q },
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{ -0x1.dfffffffffffffffffffffffffffp+4Q, 0x3.932c5047d60e60caded4c298a174p-112Q },
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{ -0x1.e000000000000000000000000001p+4Q, -0x3.932c5047d60e60caded4c298973ap-112Q },
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{ -0x1.fp+4Q, 0xa.1a6973c1fade2170f7237d35fe3p-116Q },
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{ -0x1.fp+4Q, -0xa.1a6973c1fade2170f7237d35fe08p-116Q },
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{ -0x2p+4Q, 0x5.0d34b9e0fd6f10b87b91be9aff1p-120Q },
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{ -0x2p+4Q, -0x5.0d34b9e0fd6f10b87b91be9aff0cp-120Q },
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{ -0x2.1p+4Q, 0x2.73024a9ba1aa36a7059bff52e844p-124Q },
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{ -0x2.1p+4Q, -0x2.73024a9ba1aa36a7059bff52e844p-124Q },
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{ -0x2.2p+4Q, 0x1.2710231c0fd7a13f8a2b4af9d6b7p-128Q },
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{ -0x2.2p+4Q, -0x1.2710231c0fd7a13f8a2b4af9d6b7p-128Q },
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{ -0x2.3p+4Q, 0x8.6e2ce38b6c8f9419e3fad3f0312p-136Q },
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{ -0x2.3p+4Q, -0x8.6e2ce38b6c8f9419e3fad3f0312p-136Q },
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{ -0x2.4p+4Q, 0x3.bf30652185952560d71a254e4eb8p-140Q },
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{ -0x2.4p+4Q, -0x3.bf30652185952560d71a254e4eb8p-140Q },
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{ -0x2.5p+4Q, 0x1.9ec8d1c94e85af4c78b15c3d89d3p-144Q },
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{ -0x2.5p+4Q, -0x1.9ec8d1c94e85af4c78b15c3d89d3p-144Q },
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{ -0x2.6p+4Q, 0xa.ea565ce061d57489e9b85276274p-152Q },
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{ -0x2.6p+4Q, -0xa.ea565ce061d57489e9b85276274p-152Q },
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{ -0x2.7p+4Q, 0x4.7a6512692eb37804111dabad30ecp-156Q },
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{ -0x2.7p+4Q, -0x4.7a6512692eb37804111dabad30ecp-156Q },
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{ -0x2.8p+4Q, 0x1.ca8ed42a12ae3001a07244abad2bp-160Q },
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{ -0x2.8p+4Q, -0x1.ca8ed42a12ae3001a07244abad2bp-160Q },
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{ -0x2.9p+4Q, 0xb.2f30e1ce812063f12e7e8d8d96e8p-168Q },
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{ -0x2.9p+4Q, -0xb.2f30e1ce812063f12e7e8d8d96e8p-168Q },
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{ -0x2.ap+4Q, 0x4.42bd49d4c37a0db136489772e428p-172Q },
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{ -0x2.ap+4Q, -0x4.42bd49d4c37a0db136489772e428p-172Q },
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{ -0x2.bp+4Q, 0x1.95db45257e5122dcbae56def372p-176Q },
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{ -0x2.bp+4Q, -0x1.95db45257e5122dcbae56def372p-176Q },
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{ -0x2.cp+4Q, 0x9.3958d81ff63527ecf993f3fb6f48p-184Q },
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{ -0x2.cp+4Q, -0x9.3958d81ff63527ecf993f3fb6f48p-184Q },
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{ -0x2.dp+4Q, 0x3.47970e4440c8f1c058bd238c9958p-188Q },
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{ -0x2.dp+4Q, -0x3.47970e4440c8f1c058bd238c9958p-188Q },
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{ -0x2.ep+4Q, 0x1.240804f65951062ca46e4f25c608p-192Q },
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{ -0x2.ep+4Q, -0x1.240804f65951062ca46e4f25c608p-192Q },
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{ -0x2.fp+4Q, 0x6.36a382849fae6de2d15362d8a394p-200Q },
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{ -0x2.fp+4Q, -0x6.36a382849fae6de2d15362d8a394p-200Q },
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{ -0x3p+4Q, 0x2.123680d6dfe4cf4b9b1bcb9d8bdcp-204Q },
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{ -0x3p+4Q, -0x2.123680d6dfe4cf4b9b1bcb9d8bdcp-204Q },
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{ -0x3.1p+4Q, 0xa.d21786ff5842eca51fea0870919p-212Q },
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{ -0x3.1p+4Q, -0xa.d21786ff5842eca51fea0870919p-212Q },
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{ -0x3.2p+4Q, 0x3.766dedc259af040be140a68a6c04p-216Q },
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};
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static const __float128 e_hi = 0x2.b7e151628aed2a6abf7158809cf4p+0Q;
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static const __float128 e_lo = 0xf.3c762e7160f38b4da56a784d9048p-116Q;
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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 __float128 lgamma_coeff[] =
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{
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0x1.5555555555555555555555555555p-4Q,
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-0xb.60b60b60b60b60b60b60b60b60b8p-12Q,
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0x3.4034034034034034034034034034p-12Q,
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-0x2.7027027027027027027027027028p-12Q,
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0x3.72a3c5631fe46ae1d4e700dca8f2p-12Q,
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-0x7.daac36664f1f207daac36664f1f4p-12Q,
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0x1.a41a41a41a41a41a41a41a41a41ap-8Q,
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-0x7.90a1b2c3d4e5f708192a3b4c5d7p-8Q,
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0x2.dfd2c703c0cfff430edfd2c703cp-4Q,
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-0x1.6476701181f39edbdb9ce625987dp+0Q,
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0xd.672219167002d3a7a9c886459cp+0Q,
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-0x9.cd9292e6660d55b3f712eb9e07c8p+4Q,
|
|
0x8.911a740da740da740da740da741p+8Q,
|
|
-0x8.d0cc570e255bf59ff6eec24b49p+12Q,
|
|
0xa.8d1044d3708d1c219ee4fdc446ap+16Q,
|
|
-0xe.8844d8a169abbc406169abbc406p+20Q,
|
|
0x1.6d29a0f6433b79890cede62433b8p+28Q,
|
|
-0x2.88a233b3c8cddaba9809357125d8p+32Q,
|
|
0x5.0dde6f27500939a85c40939a85c4p+36Q,
|
|
-0xb.4005bde03d4642a243581714af68p+40Q,
|
|
0x1.bc8cd6f8f1f755c78753cdb5d5c9p+48Q,
|
|
-0x4.bbebb143bb94de5a0284fa7ec424p+52Q,
|
|
0xe.2e1337f5af0bed90b6b0a352d4fp+56Q,
|
|
-0x2.e78250162b62405ad3e4bfe61b38p+64Q,
|
|
0xa.5f7eef9e71ac7c80326ab4cc8bfp+68Q,
|
|
-0x2.83be0395e550213369924971b21ap+76Q,
|
|
0xa.8ebfe48da17dd999790760b0cep+80Q,
|
|
};
|
|
|
|
#define NCOEFF (sizeof (lgamma_coeff) / sizeof (lgamma_coeff[0]))
|
|
|
|
/* Polynomial approximations to (|gamma(x)|-1)(x-n)/(x-x0), where n is
|
|
the integer end-point of the half-integer interval containing x and
|
|
x0 is the zero of lgamma in that half-integer interval. Each
|
|
polynomial is expressed in terms of x-xm, where xm is the midpoint
|
|
of the interval for which the polynomial applies. */
|
|
|
|
static const __float128 poly_coeff[] =
|
|
{
|
|
/* Interval [-2.125, -2] (polynomial degree 23). */
|
|
-0x1.0b71c5c54d42eb6c17f30b7aa8f5p+0Q,
|
|
-0xc.73a1dc05f34951602554c6d7506p-4Q,
|
|
-0x1.ec841408528b51473e6c425ee5ffp-4Q,
|
|
-0xe.37c9da26fc3c9a3c1844c8c7f1cp-4Q,
|
|
-0x1.03cd87c519305703b021fa33f827p-4Q,
|
|
-0xe.ae9ada65e09aa7f1c75216128f58p-4Q,
|
|
0x9.b11855a4864b5731cf85736015a8p-8Q,
|
|
-0xe.f28c133e697a95c28607c9701dep-4Q,
|
|
0x2.6ec14a1c586a72a7cc33ee569d6ap-4Q,
|
|
-0xf.57cab973e14464a262fc24723c38p-4Q,
|
|
0x4.5b0fc25f16e52997b2886bbae808p-4Q,
|
|
-0xf.f50e59f1a9b56e76e988dac9ccf8p-4Q,
|
|
0x6.5f5eae15e9a93369e1d85146c6fcp-4Q,
|
|
-0x1.0d2422daac459e33e0994325ed23p+0Q,
|
|
0x8.82000a0e7401fb1117a0e6606928p-4Q,
|
|
-0x1.1f492f178a3f1b19f58a2ca68e55p+0Q,
|
|
0xa.cb545f949899a04c160b19389abp-4Q,
|
|
-0x1.36165a1b155ba3db3d1b77caf498p+0Q,
|
|
0xd.44c5d5576f74302e5cf79e183eep-4Q,
|
|
-0x1.51f22e0cdd33d3d481e326c02f3ep+0Q,
|
|
0xf.f73a349c08244ac389c007779bfp-4Q,
|
|
-0x1.73317bf626156ba716747c4ca866p+0Q,
|
|
0x1.379c3c97b9bc71e1c1c4802dd657p+0Q,
|
|
-0x1.a72a351c54f902d483052000f5dfp+0Q,
|
|
/* Interval [-2.25, -2.125] (polynomial degree 24). */
|
|
-0xf.2930890d7d675a80c36afb0fd5e8p-4Q,
|
|
-0xc.a5cfde054eab5c6770daeca577f8p-4Q,
|
|
0x3.9c9e0fdebb07cdf89c61d41c9238p-4Q,
|
|
-0x1.02a5ad35605fcf4af65a6dbacb84p+0Q,
|
|
0x9.6e9b1185bb48be9de1918e00a2e8p-4Q,
|
|
-0x1.4d8332f3cfbfa116fd611e9ce90dp+0Q,
|
|
0x1.1c0c8cb4d9f4b1d490e1a41fae4dp+0Q,
|
|
-0x1.c9a6f5ae9130cd0299e293a42714p+0Q,
|
|
0x1.d7e9307fd58a2ea997f29573a112p+0Q,
|
|
-0x2.921cb3473d96178ca2a11d2a8d46p+0Q,
|
|
0x2.e8d59113b6f3409ff8db226e9988p+0Q,
|
|
-0x3.cbab931625a1ae2b26756817f264p+0Q,
|
|
0x4.7d9f0f05d5296d18663ca003912p+0Q,
|
|
-0x5.ade9cba12a14ea485667b7135bbp+0Q,
|
|
0x6.dc983a5da74fb48e767b7fec0a3p+0Q,
|
|
-0x8.8d9ed454ae31d9e138dd8ee0d1a8p+0Q,
|
|
0xa.6fa099d4e7c202e0c0fd6ed8492p+0Q,
|
|
-0xc.ebc552a8090a0f0115e92d4ebbc8p+0Q,
|
|
0xf.d695e4772c0d829b53fba9ca5568p+0Q,
|
|
-0x1.38c32ae38e5e9eb79b2a4c5570a9p+4Q,
|
|
0x1.8035145646cfab49306d0999a51bp+4Q,
|
|
-0x1.d930adbb03dd342a4c2a8c4e1af6p+4Q,
|
|
0x2.45c2edb1b4943ddb3686cd9c6524p+4Q,
|
|
-0x2.e818ebbfafe2f916fa21abf7756p+4Q,
|
|
0x3.9804ce51d0fb9a430a711fd7307p+4Q,
|
|
/* Interval [-2.375, -2.25] (polynomial degree 25). */
|
|
-0xd.7d28d505d6181218a25f31d5e45p-4Q,
|
|
-0xe.69649a3040985140cdf946829fap-4Q,
|
|
0xb.0d74a2827d053a8d44595012484p-4Q,
|
|
-0x1.924b0922853617cac181afbc08ddp+0Q,
|
|
0x1.d49b12bccf0a568582e2d3c410f3p+0Q,
|
|
-0x3.0898bb7d8c4093e636279c791244p+0Q,
|
|
0x4.207a6cac711cb53868e8a5057eep+0Q,
|
|
-0x6.39ee63ea4fb1dcab0c9144bf3ddcp+0Q,
|
|
0x8.e2e2556a797b649bf3f53bd26718p+0Q,
|
|
-0xd.0e83ac82552ef12af508589e7a8p+0Q,
|
|
0x1.2e4525e0ce6670563c6484a82b05p+4Q,
|
|
-0x1.b8e350d6a8f2b222fa390a57c23dp+4Q,
|
|
0x2.805cd69b919087d8a80295892c2cp+4Q,
|
|
-0x3.a42585424a1b7e64c71743ab014p+4Q,
|
|
0x5.4b4f409f98de49f7bfb03c05f984p+4Q,
|
|
-0x7.b3c5827fbe934bc820d6832fb9fcp+4Q,
|
|
0xb.33b7b90cc96c425526e0d0866e7p+4Q,
|
|
-0x1.04b77047ac4f59ee3775ca10df0dp+8Q,
|
|
0x1.7b366f5e94a34f41386eac086313p+8Q,
|
|
-0x2.2797338429385c9849ca6355bfc2p+8Q,
|
|
0x3.225273cf92a27c9aac1b35511256p+8Q,
|
|
-0x4.8f078aa48afe6cb3a4e89690f898p+8Q,
|
|
0x6.9f311d7b6654fc1d0b5195141d04p+8Q,
|
|
-0x9.a0c297b6b4621619ca9bacc48ed8p+8Q,
|
|
0xe.ce1f06b6f90d92138232a76e4cap+8Q,
|
|
-0x1.5b0e6806fa064daf011613e43b17p+12Q,
|
|
/* Interval [-2.5, -2.375] (polynomial degree 27). */
|
|
-0xb.74ea1bcfff94b2c01afba9daa7d8p-4Q,
|
|
-0x1.2a82bd590c37538cab143308de4dp+0Q,
|
|
0x1.88020f828b966fec66b8649fd6fcp+0Q,
|
|
-0x3.32279f040eb694970e9db24863dcp+0Q,
|
|
0x5.57ac82517767e68a721005853864p+0Q,
|
|
-0x9.c2aedcfe22833de43834a0a6cc4p+0Q,
|
|
0x1.12c132f1f5577f99e1a0ed3538e1p+4Q,
|
|
-0x1.ea94e26628a3de3597f7bb55a948p+4Q,
|
|
0x3.66b4ac4fa582f58b59f96b2f7c7p+4Q,
|
|
-0x6.0cf746a9cf4cba8c39afcc73fc84p+4Q,
|
|
0xa.c102ef2c20d75a342197df7fedf8p+4Q,
|
|
-0x1.31ebff06e8f14626782df58db3b6p+8Q,
|
|
0x2.1fd6f0c0e710994e059b9dbdb1fep+8Q,
|
|
-0x3.c6d76040407f447f8b5074f07706p+8Q,
|
|
0x6.b6d18e0d8feb4c2ef5af6a40ed18p+8Q,
|
|
-0xb.efaf542c529f91e34217f24ae6a8p+8Q,
|
|
0x1.53852d873210e7070f5d9eb2296p+12Q,
|
|
-0x2.5b977c0ddc6d540717173ac29fc8p+12Q,
|
|
0x4.310d452ae05100eff1e02343a724p+12Q,
|
|
-0x7.73a5d8f20c4f986a7dd1912b2968p+12Q,
|
|
0xd.3f5ea2484f3fca15eab1f4d1a218p+12Q,
|
|
-0x1.78d18aac156d1d93a2ffe7e08d3fp+16Q,
|
|
0x2.9df49ca75e5b567f5ea3e47106cp+16Q,
|
|
-0x4.a7149af8961a08aa7c3233b5bb94p+16Q,
|
|
0x8.3db10ffa742c707c25197d989798p+16Q,
|
|
-0xe.a26d6dd023cadd02041a049ec368p+16Q,
|
|
0x1.c825d90514e7c57c7fa5316f947cp+20Q,
|
|
-0x3.34bb81e5a0952df8ca1abdc6684cp+20Q,
|
|
/* Interval [-2.625, -2.5] (polynomial degree 28). */
|
|
-0x3.d10108c27ebafad533c20eac32bp-4Q,
|
|
0x1.cd557caff7d2b2085f41dbec5106p+0Q,
|
|
0x3.819b4856d399520dad9776ea2cacp+0Q,
|
|
0x6.8505cbad03dc34c5e42e8b12eb78p+0Q,
|
|
0xb.c1b2e653a9e38f82b399c94e7f08p+0Q,
|
|
0x1.50a53a38f148138105124df65419p+4Q,
|
|
0x2.57ae00cbe5232cbeeed34d89727ap+4Q,
|
|
0x4.2b156301b8604db85a601544bfp+4Q,
|
|
0x7.6989ed23ca3ca7579b3462592b5cp+4Q,
|
|
0xd.2dd2976557939517f831f5552cc8p+4Q,
|
|
0x1.76e1c3430eb860969bce40cd494p+8Q,
|
|
0x2.9a77bf5488742466db3a2c7c1ec6p+8Q,
|
|
0x4.a0d62ed7266e8eb36f725a8ebcep+8Q,
|
|
0x8.3a6184dd3021067df2f8b91e99c8p+8Q,
|
|
0xe.a0ade1538245bf55d39d7e436b1p+8Q,
|
|
0x1.a01359fae8617b5826dd74428e9p+12Q,
|
|
0x2.e3b0a32caae77251169acaca1ad4p+12Q,
|
|
0x5.2301257c81589f62b38fb5993ee8p+12Q,
|
|
0x9.21c9275db253d4e719b73b18cb9p+12Q,
|
|
0x1.03c104bc96141cda3f3fa4b112bcp+16Q,
|
|
0x1.cdc8ed65119196a08b0c78f1445p+16Q,
|
|
0x3.34f31d2eaacf34382cdb0073572ap+16Q,
|
|
0x5.b37628cadf12bf0000907d0ef294p+16Q,
|
|
0xa.22d8b332c0b1e6a616f425dfe5ap+16Q,
|
|
0x1.205b01444804c3ff922cd78b4c42p+20Q,
|
|
0x1.fe8f0cea9d1e0ff25be2470b4318p+20Q,
|
|
0x3.8872aebeb368399aee02b39340aep+20Q,
|
|
0x6.ebd560d351e84e26a4381f5b293cp+20Q,
|
|
0xc.c3644d094b0dae2fbcbf682cd428p+20Q,
|
|
/* Interval [-2.75, -2.625] (polynomial degree 26). */
|
|
-0x6.b5d252a56e8a75458a27ed1c2dd4p-4Q,
|
|
0x1.28d60383da3ac721aed3c5794da9p+0Q,
|
|
0x1.db6513ada8a66ea77d87d9a8827bp+0Q,
|
|
0x2.e217118f9d348a27f7506a707e6ep+0Q,
|
|
0x4.450112c5cbf725a0fb9802396c9p+0Q,
|
|
0x6.4af99151eae7810a75df2a0303c4p+0Q,
|
|
0x9.2db598b4a97a7f69aeef32aec758p+0Q,
|
|
0xd.62bef9c22471f5ee47ea1b9c0b5p+0Q,
|
|
0x1.379f294e412bd62328326d4222f9p+4Q,
|
|
0x1.c5827349d8865f1e8825c37c31c6p+4Q,
|
|
0x2.93a7e7a75b7568cc8cbe8c016c12p+4Q,
|
|
0x3.bf9bb882afe57edb383d41879d3ap+4Q,
|
|
0x5.73c737828cee095c43a5566731c8p+4Q,
|
|
0x7.ee4653493a7f81e0442062b3823cp+4Q,
|
|
0xb.891c6b83fc8b55bd973b5d962d6p+4Q,
|
|
0x1.0c775d7de3bf9b246c0208e0207ep+8Q,
|
|
0x1.867ee43ec4bd4f4fd56abc05110ap+8Q,
|
|
0x2.37fe9ba6695821e9822d8c8af0a6p+8Q,
|
|
0x3.3a2c667e37c942f182cd3223a936p+8Q,
|
|
0x4.b1b500eb59f3f782c7ccec88754p+8Q,
|
|
0x6.d3efd3b65b3d0d8488d30b79fa4cp+8Q,
|
|
0x9.ee8224e65bed5ced8b75eaec609p+8Q,
|
|
0xe.72416e510cca77d53fc615c1f3dp+8Q,
|
|
0x1.4fb538b0a2dfe567a8904b7e0445p+12Q,
|
|
0x1.e7f56a9266cf525a5b8cf4cb76cep+12Q,
|
|
0x2.f0365c983f68c597ee49d099cce8p+12Q,
|
|
0x4.53aa229e1b9f5b5e59625265951p+12Q,
|
|
/* Interval [-2.875, -2.75] (polynomial degree 24). */
|
|
-0x8.a41b1e4f36ff88dc820815607d68p-4Q,
|
|
0xc.da87d3b69dc0f2f9c6f368b8ca1p-4Q,
|
|
0x1.1474ad5c36158a7bea04fd2f98c6p+0Q,
|
|
0x1.761ecb90c555df6555b7dba955b6p+0Q,
|
|
0x1.d279bff9ae291caf6c4b4bcb3202p+0Q,
|
|
0x2.4e5d00559a6e2b9b5d7fe1f6689cp+0Q,
|
|
0x2.d57545a75cee8743ae2b17bc8d24p+0Q,
|
|
0x3.8514eee3aac88b89bec2307021bap+0Q,
|
|
0x4.5235e3b6e1891ffeb87fed9f8a24p+0Q,
|
|
0x5.562acdb10eef3c9a773b3e27a864p+0Q,
|
|
0x6.8ec8965c76efe03c26bff60b1194p+0Q,
|
|
0x8.15251aca144877af32658399f9b8p+0Q,
|
|
0x9.f08d56aba174d844138af782c0f8p+0Q,
|
|
0xc.3dbbeda2679e8a1346ccc3f6da88p+0Q,
|
|
0xf.0f5bfd5eacc26db308ffa0556fa8p+0Q,
|
|
0x1.28a6ccd84476fbc713d6bab49ac9p+4Q,
|
|
0x1.6d0a3ae2a3b1c8ff400641a3a21fp+4Q,
|
|
0x1.c15701b28637f87acfb6a91d33b5p+4Q,
|
|
0x2.28fbe0eccf472089b017651ca55ep+4Q,
|
|
0x2.a8a453004f6e8ffaacd1603bc3dp+4Q,
|
|
0x3.45ae4d9e1e7cd1a5dba0e4ec7f6cp+4Q,
|
|
0x4.065fbfacb7fad3e473cb577a61e8p+4Q,
|
|
0x4.f3d1473020927acac1944734a39p+4Q,
|
|
0x6.54bb091245815a36fb74e314dd18p+4Q,
|
|
0x7.d7f445129f7fb6c055e582d3f6ep+4Q,
|
|
/* Interval [-3, -2.875] (polynomial degree 23). */
|
|
-0xa.046d667e468f3e44dcae1afcc648p-4Q,
|
|
0x9.70b88dcc006c214d8d996fdf5ccp-4Q,
|
|
0xa.a8a39421c86d3ff24931a0929fp-4Q,
|
|
0xd.2f4d1363f324da2b357c8b6ec94p-4Q,
|
|
0xd.ca9aa1a3a5c00de11bf60499a97p-4Q,
|
|
0xf.cf09c31eeb52a45dfa7ebe3778dp-4Q,
|
|
0x1.04b133a39ed8a09691205660468bp+0Q,
|
|
0x1.22b547a06edda944fcb12fd9b5ecp+0Q,
|
|
0x1.2c57fce7db86a91df09602d344b3p+0Q,
|
|
0x1.4aade4894708f84795212fe257eep+0Q,
|
|
0x1.579c8b7b67ec4afed5b28c8bf787p+0Q,
|
|
0x1.776820e7fc80ae5284239733078ap+0Q,
|
|
0x1.883ab28c7301fde4ca6b8ec26ec8p+0Q,
|
|
0x1.aa2ef6e1ae52eb42c9ee83b206e3p+0Q,
|
|
0x1.bf4ad50f0a9a9311300cf0c51ee7p+0Q,
|
|
0x1.e40206e0e96b1da463814dde0d09p+0Q,
|
|
0x1.fdcbcffef3a21b29719c2bd9feb1p+0Q,
|
|
0x2.25e2e8948939c4d42cf108fae4bep+0Q,
|
|
0x2.44ce14d2b59c1c0e6bf2cfa81018p+0Q,
|
|
0x2.70ee80bbd0387162be4861c43622p+0Q,
|
|
0x2.954b64d2c2ebf3489b949c74476p+0Q,
|
|
0x2.c616e133a811c1c9446105208656p+0Q,
|
|
0x3.05a69dfe1a9ba1079f90fcf26bd4p+0Q,
|
|
0x3.410d2ad16a0506de29736e6aafdap+0Q,
|
|
};
|
|
|
|
static const size_t poly_deg[] =
|
|
{
|
|
23,
|
|
24,
|
|
25,
|
|
27,
|
|
28,
|
|
26,
|
|
24,
|
|
23,
|
|
};
|
|
|
|
static const size_t poly_end[] =
|
|
{
|
|
23,
|
|
48,
|
|
74,
|
|
102,
|
|
131,
|
|
158,
|
|
183,
|
|
207,
|
|
};
|
|
|
|
/* Compute sin (pi * X) for -0.25 <= X <= 0.5. */
|
|
|
|
static __float128
|
|
lg_sinpi (__float128 x)
|
|
{
|
|
if (x <= 0.25Q)
|
|
return sinq (M_PIq * x);
|
|
else
|
|
return cosq (M_PIq * (0.5Q - x));
|
|
}
|
|
|
|
/* Compute cos (pi * X) for -0.25 <= X <= 0.5. */
|
|
|
|
static __float128
|
|
lg_cospi (__float128 x)
|
|
{
|
|
if (x <= 0.25Q)
|
|
return cosq (M_PIq * x);
|
|
else
|
|
return sinq (M_PIq * (0.5Q - x));
|
|
}
|
|
|
|
/* Compute cot (pi * X) for -0.25 <= X <= 0.5. */
|
|
|
|
static __float128
|
|
lg_cotpi (__float128 x)
|
|
{
|
|
return lg_cospi (x) / lg_sinpi (x);
|
|
}
|
|
|
|
/* Compute lgamma of a negative argument -50 < X < -2, setting
|
|
*SIGNGAMP accordingly. */
|
|
|
|
__float128
|
|
__quadmath_lgamma_negq (__float128 x, int *signgamp)
|
|
{
|
|
/* Determine the half-integer region X lies in, handle exact
|
|
integers and determine the sign of the result. */
|
|
int i = floorq (-2 * x);
|
|
if ((i & 1) == 0 && i == -2 * x)
|
|
return 1.0Q / 0.0Q;
|
|
__float128 xn = ((i & 1) == 0 ? -i / 2 : (-i - 1) / 2);
|
|
i -= 4;
|
|
*signgamp = ((i & 2) == 0 ? -1 : 1);
|
|
|
|
SET_RESTORE_ROUNDF128 (FE_TONEAREST);
|
|
|
|
/* Expand around the zero X0 = X0_HI + X0_LO. */
|
|
__float128 x0_hi = lgamma_zeros[i][0], x0_lo = lgamma_zeros[i][1];
|
|
__float128 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 = floorq (-8 * x) - 16;
|
|
__float128 xm = (-33 - 2 * j) * 0.0625Q;
|
|
__float128 x_adj = x - xm;
|
|
size_t deg = poly_deg[j];
|
|
size_t end = poly_end[j];
|
|
__float128 g = poly_coeff[end];
|
|
for (size_t j = 1; j <= deg; j++)
|
|
g = g * x_adj + poly_coeff[end - j];
|
|
return log1pq (g * xdiff / (x - xn));
|
|
}
|
|
|
|
/* The result we want is log (sinpi (X0) / sinpi (X))
|
|
+ log (gamma (1 - X0) / gamma (1 - X)). */
|
|
__float128 x_idiff = fabsq (xn - x), x0_idiff = fabsq (xn - x0_hi - x0_lo);
|
|
__float128 log_sinpi_ratio;
|
|
if (x0_idiff < x_idiff * 0.5Q)
|
|
/* Use log not log1p to avoid inaccuracy from log1p of arguments
|
|
close to -1. */
|
|
log_sinpi_ratio = logq (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. */
|
|
__float128 x0diff2 = ((i & 1) == 0 ? xdiff : -xdiff) * 0.5Q;
|
|
__float128 sx0d2 = lg_sinpi (x0diff2);
|
|
__float128 cx0d2 = lg_cospi (x0diff2);
|
|
log_sinpi_ratio = log1pq (2 * sx0d2
|
|
* (-sx0d2 + cx0d2 * lg_cotpi (x_idiff)));
|
|
}
|
|
|
|
__float128 log_gamma_ratio;
|
|
__float128 y0 = 1 - x0_hi;
|
|
__float128 y0_eps = -x0_hi + (1 - y0) - x0_lo;
|
|
__float128 y = 1 - x;
|
|
__float128 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. */
|
|
__float128 log_gamma_adj = 0;
|
|
if (i < 20)
|
|
{
|
|
int n_up = (21 - i) / 2;
|
|
__float128 ny0, ny0_eps, ny, ny_eps;
|
|
ny0 = y0 + n_up;
|
|
ny0_eps = y0 - (ny0 - n_up) + y0_eps;
|
|
y0 = ny0;
|
|
y0_eps = ny0_eps;
|
|
ny = y + n_up;
|
|
ny_eps = y - (ny - n_up) + y_eps;
|
|
y = ny;
|
|
y_eps = ny_eps;
|
|
__float128 prodm1 = __quadmath_lgamma_productq (xdiff, y - n_up, y_eps, n_up);
|
|
log_gamma_adj = -log1pq (prodm1);
|
|
}
|
|
__float128 log_gamma_high
|
|
= (xdiff * log1pq ((y0 - e_hi - e_lo + y0_eps) / e_hi)
|
|
+ (y - 0.5Q + y_eps) * log1pq (xdiff / y) + log_gamma_adj);
|
|
/* Compute the sum of (B_2k / 2k(2k-1))(Y0^-(2k-1) - Y^-(2k-1)). */
|
|
__float128 y0r = 1 / y0, yr = 1 / y;
|
|
__float128 y0r2 = y0r * y0r, yr2 = yr * yr;
|
|
__float128 rdiff = -xdiff / (y * y0);
|
|
__float128 bterm[NCOEFF];
|
|
__float128 dlast = rdiff, elast = rdiff * yr * (yr + y0r);
|
|
bterm[0] = dlast * lgamma_coeff[0];
|
|
for (size_t j = 1; j < NCOEFF; j++)
|
|
{
|
|
__float128 dnext = dlast * y0r2 + elast;
|
|
__float128 enext = elast * yr2;
|
|
bterm[j] = dnext * lgamma_coeff[j];
|
|
dlast = dnext;
|
|
elast = enext;
|
|
}
|
|
__float128 log_gamma_low = 0;
|
|
for (size_t j = 0; j < NCOEFF; j++)
|
|
log_gamma_low += bterm[NCOEFF - 1 - j];
|
|
log_gamma_ratio = log_gamma_high + log_gamma_low;
|
|
|
|
return log_sinpi_ratio + log_gamma_ratio;
|
|
}
|