git: d47b524eadb5 - stable/13 - * lib/msun/Makefile b/lib/msun/Makefile: . Disconnect imprecise.c from the build. This file can be deleted. . Add b_tgammal.c to the build for ld80 and ld128 targets. The ld128 is a 'git mv' of imprecise.c to ld128/b_tgammal.c.
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Date: Wed, 26 Jul 2023 23:51:24 UTC
The branch stable/13 has been updated by emaste:
URL: https://cgit.FreeBSD.org/src/commit/?id=d47b524eadb5a87506fe6ae03febbf72fad2ee06
commit d47b524eadb5a87506fe6ae03febbf72fad2ee06
Author: Mark Murray <markm@FreeBSD.org>
AuthorDate: 2021-12-14 09:08:57 +0000
Commit: Ed Maste <emaste@FreeBSD.org>
CommitDate: 2023-07-26 23:07:30 +0000
* lib/msun/Makefile b/lib/msun/Makefile:
. Disconnect imprecise.c from the build. This file can be deleted.
. Add b_tgammal.c to the build for ld80 and ld128 targets. The ld128
is a 'git mv' of imprecise.c to ld128/b_tgammal.c.
* lib/msun/ld80/b_expl.c:
. New file. Implement __exp__D for ld80 targets. This is based on
bsdsrc/b_exp.c.
* lib/msun/ld80/b_logl.c:
. New file. Implement __log__D for ld80 targets. This is based on
bsdsrc/b_log.c.
* lib/msun/ld80/b_tgammal.c b/lib/msun/ld80/b_tgammal.c
. New file. Implement tgammal(x) for ld80 targets.
Submitted by: Steve Kargl
Differential Revision: https://reviews.freebsd.org/D33444
Reviewed by: pfg
(cherry picked from commit 03a88e3de9c68182d21df94b1c8c7ced930dbd1f)
---
lib/msun/Makefile | 3 +-
lib/msun/{src/imprecise.c => ld128/b_tgammal.c} | 0
lib/msun/ld80/b_expl.c | 113 +++++++
lib/msun/ld80/b_logl.c | 375 +++++++++++++++++++++
lib/msun/ld80/b_tgammal.c | 419 ++++++++++++++++++++++++
5 files changed, 908 insertions(+), 2 deletions(-)
diff --git a/lib/msun/Makefile b/lib/msun/Makefile
index 24d740d761d5..8cb67f3700b9 100644
--- a/lib/msun/Makefile
+++ b/lib/msun/Makefile
@@ -69,7 +69,6 @@ COMMON_SRCS= b_tgamma.c \
e_pow.c e_powf.c e_rem_pio2.c \
e_rem_pio2f.c e_remainder.c e_remainderf.c e_scalb.c e_scalbf.c \
e_sinh.c e_sinhf.c e_sqrt.c e_sqrtf.c fenv.c \
- imprecise.c \
k_cos.c k_cosf.c k_exp.c k_expf.c k_rem_pio2.c k_sin.c k_sinf.c \
k_tan.c k_tanf.c \
s_asinh.c s_asinhf.c s_atan.c s_atanf.c s_carg.c s_cargf.c s_cargl.c \
@@ -116,7 +115,7 @@ SYMBOL_MAPS= ${SYM_MAPS}
COMMON_SRCS+= s_copysignl.c s_fabsl.c s_llrintl.c s_lrintl.c s_modfl.c
.if ${LDBL_PREC} != 53
# If long double != double use these; otherwise, we alias the double versions.
-COMMON_SRCS+= catrigl.c \
+COMMON_SRCS+= b_tgammal.c catrigl.c \
e_acoshl.c e_acosl.c e_asinl.c e_atan2l.c e_atanhl.c \
e_coshl.c e_fmodl.c e_hypotl.c \
e_lgammal.c e_lgammal_r.c e_powl.c \
diff --git a/lib/msun/src/imprecise.c b/lib/msun/ld128/b_tgammal.c
similarity index 100%
rename from lib/msun/src/imprecise.c
rename to lib/msun/ld128/b_tgammal.c
diff --git a/lib/msun/ld80/b_expl.c b/lib/msun/ld80/b_expl.c
new file mode 100644
index 000000000000..21bacf4497a3
--- /dev/null
+++ b/lib/msun/ld80/b_expl.c
@@ -0,0 +1,113 @@
+/*-
+ * SPDX-License-Identifier: BSD-3-Clause
+ *
+ * Copyright (c) 1985, 1993
+ * The Regents of the University of California. All rights reserved.
+ *
+ * Redistribution and use in source and binary forms, with or without
+ * modification, are permitted provided that the following conditions
+ * are met:
+ * 1. Redistributions of source code must retain the above copyright
+ * notice, this list of conditions and the following disclaimer.
+ * 2. Redistributions in binary form must reproduce the above copyright
+ * notice, this list of conditions and the following disclaimer in the
+ * documentation and/or other materials provided with the distribution.
+ * 3. Neither the name of the University nor the names of its contributors
+ * may be used to endorse or promote products derived from this software
+ * without specific prior written permission.
+ *
+ * THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND
+ * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
+ * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
+ * ARE DISCLAIMED. IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE
+ * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
+ * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
+ * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
+ * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
+ * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
+ * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
+ * SUCH DAMAGE.
+ */
+
+/*
+ * See bsdsrc/b_exp.c for implementation details.
+ *
+ * bsdrc/b_exp.c converted to long double by Steven G. Kargl.
+ */
+
+#include "fpmath.h"
+#include "math_private.h"
+
+static const union IEEEl2bits
+ p0u = LD80C(0xaaaaaaaaaaaaaaab, -3, 1.66666666666666666671e-01L),
+ p1u = LD80C(0xb60b60b60b60b59a, -9, -2.77777777777777775377e-03L),
+ p2u = LD80C(0x8ab355e008a3cfce, -14, 6.61375661375629297465e-05L),
+ p3u = LD80C(0xddebbc994b0c1376, -20, -1.65343915327882529784e-06L),
+ p4u = LD80C(0xb354784cb4ef4c41, -25, 4.17535101591534118469e-08L),
+ p5u = LD80C(0x913e8a718382ce75, -30, -1.05679137034774806475e-09L),
+ p6u = LD80C(0xe8f0042aa134502e, -36, 2.64819349895429516863e-11L);
+#define p1 (p0u.e)
+#define p2 (p1u.e)
+#define p3 (p2u.e)
+#define p4 (p3u.e)
+#define p5 (p4u.e)
+#define p6 (p5u.e)
+#define p7 (p6u.e)
+
+/*
+ * lnhuge = (LDBL_MAX_EXP + 9) * log(2.)
+ * lntiny = (LDBL_MIN_EXP - 64 - 10) * log(2.)
+ * invln2 = 1 / log(2.)
+ */
+static const union IEEEl2bits
+ln2hiu = LD80C(0xb17217f700000000, -1, 6.93147180369123816490e-01L),
+ln2lou = LD80C(0xd1cf79abc9e3b398, -33, 1.90821492927058781614e-10L),
+lnhugeu = LD80C(0xb18b0c0330a8fad9, 13, 1.13627617309191834574e+04L),
+lntinyu = LD80C(0xb236f28a68bc3bd7, 13, -1.14057368561139000667e+04L),
+invln2u = LD80C(0xb8aa3b295c17f0bc, 0, 1.44269504088896340739e+00L);
+#define ln2hi (ln2hiu.e)
+#define ln2lo (ln2lou.e)
+#define lnhuge (lnhugeu.e)
+#define lntiny (lntinyu.e)
+#define invln2 (invln2u.e)
+
+/* returns exp(r = x + c) for |c| < |x| with no overlap. */
+
+static long double
+__exp__D(long double x, long double c)
+{
+ long double hi, lo, z;
+ int k;
+
+ if (x != x) /* x is NaN. */
+ return(x);
+
+ if (x <= lnhuge) {
+ if (x >= lntiny) {
+ /* argument reduction: x --> x - k*ln2 */
+ z = invln2 * x;
+ k = z + copysignl(0.5L, x);
+
+ /*
+ * Express (x + c) - k * ln2 as hi - lo.
+ * Let x = hi - lo rounded.
+ */
+ hi = x - k * ln2hi; /* Exact. */
+ lo = k * ln2lo - c;
+ x = hi - lo;
+
+ /* Return 2^k*[1+x+x*c/(2+c)] */
+ z = x * x;
+ c = x - z * (p1 + z * (p2 + z * (p3 + z * (p4 +
+ z * (p5 + z * (p6 + z * p7))))));
+ c = (x * c) / (2 - c);
+
+ return (ldexpl(1 + (hi - (lo - c)), k));
+ } else {
+ /* exp(-INF) is 0. exp(-big) underflows to 0. */
+ return (isfinite(x) ? ldexpl(1., -5000) : 0);
+ }
+ } else
+ /* exp(INF) is INF, exp(+big#) overflows to INF */
+ return (isfinite(x) ? ldexpl(1., 5000) : x);
+}
diff --git a/lib/msun/ld80/b_logl.c b/lib/msun/ld80/b_logl.c
new file mode 100644
index 000000000000..b11eacbe17a1
--- /dev/null
+++ b/lib/msun/ld80/b_logl.c
@@ -0,0 +1,375 @@
+/*-
+ * SPDX-License-Identifier: BSD-3-Clause
+ *
+ * Copyright (c) 1992, 1993
+ * The Regents of the University of California. All rights reserved.
+ *
+ * Redistribution and use in source and binary forms, with or without
+ * modification, are permitted provided that the following conditions
+ * are met:
+ * 1. Redistributions of source code must retain the above copyright
+ * notice, this list of conditions and the following disclaimer.
+ * 2. Redistributions in binary form must reproduce the above copyright
+ * notice, this list of conditions and the following disclaimer in the
+ * documentation and/or other materials provided with the distribution.
+ * 3. Neither the name of the University nor the names of its contributors
+ * may be used to endorse or promote products derived from this software
+ * without specific prior written permission.
+ *
+ * THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND
+ * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
+ * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
+ * ARE DISCLAIMED. IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE
+ * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
+ * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
+ * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
+ * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
+ * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
+ * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
+ * SUCH DAMAGE.
+ */
+
+/*
+ * See bsdsrc/b_log.c for implementation details.
+ *
+ * bsdrc/b_log.c converted to long double by Steven G. Kargl.
+ */
+
+#define N 128
+
+/*
+ * Coefficients in the polynomial approximation of log(1+f/F).
+ * Domain of x is [0,1./256] with 2**(-84.48) precision.
+ */
+static const union IEEEl2bits
+ a1u = LD80C(0xaaaaaaaaaaaaaaab, -4, 8.33333333333333333356e-02L),
+ a2u = LD80C(0xcccccccccccccd29, -7, 1.25000000000000000781e-02L),
+ a3u = LD80C(0x9249249241ed3764, -9, 2.23214285711721994134e-03L),
+ a4u = LD80C(0xe38e959e1e7e01cf, -12, 4.34030476540000360640e-04L);
+#define A1 (a1u.e)
+#define A2 (a2u.e)
+#define A3 (a3u.e)
+#define A4 (a4u.e)
+
+/*
+ * Table of log(Fj) = logF_head[j] + logF_tail[j], for Fj = 1+j/128.
+ * Used for generation of extend precision logarithms.
+ * The constant 35184372088832 is 2^45, so the divide is exact.
+ * It ensures correct reading of logF_head, even for inaccurate
+ * decimal-to-binary conversion routines. (Everybody gets the
+ * right answer for integers less than 2^53.)
+ * Values for log(F) were generated using error < 10^-57 absolute
+ * with the bc -l package.
+ */
+
+static double logF_head[N+1] = {
+ 0.,
+ .007782140442060381246,
+ .015504186535963526694,
+ .023167059281547608406,
+ .030771658666765233647,
+ .038318864302141264488,
+ .045809536031242714670,
+ .053244514518837604555,
+ .060624621816486978786,
+ .067950661908525944454,
+ .075223421237524235039,
+ .082443669210988446138,
+ .089612158689760690322,
+ .096729626458454731618,
+ .103796793681567578460,
+ .110814366340264314203,
+ .117783035656430001836,
+ .124703478501032805070,
+ .131576357788617315236,
+ .138402322859292326029,
+ .145182009844575077295,
+ .151916042025732167530,
+ .158605030176659056451,
+ .165249572895390883786,
+ .171850256926518341060,
+ .178407657472689606947,
+ .184922338493834104156,
+ .191394852999565046047,
+ .197825743329758552135,
+ .204215541428766300668,
+ .210564769107350002741,
+ .216873938300523150246,
+ .223143551314024080056,
+ .229374101064877322642,
+ .235566071312860003672,
+ .241719936886966024758,
+ .247836163904594286577,
+ .253915209980732470285,
+ .259957524436686071567,
+ .265963548496984003577,
+ .271933715484010463114,
+ .277868451003087102435,
+ .283768173130738432519,
+ .289633292582948342896,
+ .295464212893421063199,
+ .301261330578199704177,
+ .307025035294827830512,
+ .312755710004239517729,
+ .318453731118097493890,
+ .324119468654316733591,
+ .329753286372579168528,
+ .335355541920762334484,
+ .340926586970454081892,
+ .346466767346100823488,
+ .351976423156884266063,
+ .357455888922231679316,
+ .362905493689140712376,
+ .368325561158599157352,
+ .373716409793814818840,
+ .379078352934811846353,
+ .384411698910298582632,
+ .389716751140440464951,
+ .394993808240542421117,
+ .400243164127459749579,
+ .405465108107819105498,
+ .410659924985338875558,
+ .415827895143593195825,
+ .420969294644237379543,
+ .426084395310681429691,
+ .431173464818130014464,
+ .436236766774527495726,
+ .441274560805140936281,
+ .446287102628048160113,
+ .451274644139630254358,
+ .456237433481874177232,
+ .461175715122408291790,
+ .466089729924533457960,
+ .470979715219073113985,
+ .475845904869856894947,
+ .480688529345570714212,
+ .485507815781602403149,
+ .490303988045525329653,
+ .495077266798034543171,
+ .499827869556611403822,
+ .504556010751912253908,
+ .509261901790523552335,
+ .513945751101346104405,
+ .518607764208354637958,
+ .523248143765158602036,
+ .527867089620485785417,
+ .532464798869114019908,
+ .537041465897345915436,
+ .541597282432121573947,
+ .546132437597407260909,
+ .550647117952394182793,
+ .555141507540611200965,
+ .559615787935399566777,
+ .564070138285387656651,
+ .568504735352689749561,
+ .572919753562018740922,
+ .577315365035246941260,
+ .581691739635061821900,
+ .586049045003164792433,
+ .590387446602107957005,
+ .594707107746216934174,
+ .599008189645246602594,
+ .603290851438941899687,
+ .607555250224322662688,
+ .611801541106615331955,
+ .616029877215623855590,
+ .620240409751204424537,
+ .624433288012369303032,
+ .628608659422752680256,
+ .632766669570628437213,
+ .636907462236194987781,
+ .641031179420679109171,
+ .645137961373620782978,
+ .649227946625615004450,
+ .653301272011958644725,
+ .657358072709030238911,
+ .661398482245203922502,
+ .665422632544505177065,
+ .669430653942981734871,
+ .673422675212350441142,
+ .677398823590920073911,
+ .681359224807238206267,
+ .685304003098281100392,
+ .689233281238557538017,
+ .693147180560117703862
+};
+
+static double logF_tail[N+1] = {
+ 0.,
+ -.00000000000000543229938420049,
+ .00000000000000172745674997061,
+ -.00000000000001323017818229233,
+ -.00000000000001154527628289872,
+ -.00000000000000466529469958300,
+ .00000000000005148849572685810,
+ -.00000000000002532168943117445,
+ -.00000000000005213620639136504,
+ -.00000000000001819506003016881,
+ .00000000000006329065958724544,
+ .00000000000008614512936087814,
+ -.00000000000007355770219435028,
+ .00000000000009638067658552277,
+ .00000000000007598636597194141,
+ .00000000000002579999128306990,
+ -.00000000000004654729747598444,
+ -.00000000000007556920687451336,
+ .00000000000010195735223708472,
+ -.00000000000017319034406422306,
+ -.00000000000007718001336828098,
+ .00000000000010980754099855238,
+ -.00000000000002047235780046195,
+ -.00000000000008372091099235912,
+ .00000000000014088127937111135,
+ .00000000000012869017157588257,
+ .00000000000017788850778198106,
+ .00000000000006440856150696891,
+ .00000000000016132822667240822,
+ -.00000000000007540916511956188,
+ -.00000000000000036507188831790,
+ .00000000000009120937249914984,
+ .00000000000018567570959796010,
+ -.00000000000003149265065191483,
+ -.00000000000009309459495196889,
+ .00000000000017914338601329117,
+ -.00000000000001302979717330866,
+ .00000000000023097385217586939,
+ .00000000000023999540484211737,
+ .00000000000015393776174455408,
+ -.00000000000036870428315837678,
+ .00000000000036920375082080089,
+ -.00000000000009383417223663699,
+ .00000000000009433398189512690,
+ .00000000000041481318704258568,
+ -.00000000000003792316480209314,
+ .00000000000008403156304792424,
+ -.00000000000034262934348285429,
+ .00000000000043712191957429145,
+ -.00000000000010475750058776541,
+ -.00000000000011118671389559323,
+ .00000000000037549577257259853,
+ .00000000000013912841212197565,
+ .00000000000010775743037572640,
+ .00000000000029391859187648000,
+ -.00000000000042790509060060774,
+ .00000000000022774076114039555,
+ .00000000000010849569622967912,
+ -.00000000000023073801945705758,
+ .00000000000015761203773969435,
+ .00000000000003345710269544082,
+ -.00000000000041525158063436123,
+ .00000000000032655698896907146,
+ -.00000000000044704265010452446,
+ .00000000000034527647952039772,
+ -.00000000000007048962392109746,
+ .00000000000011776978751369214,
+ -.00000000000010774341461609578,
+ .00000000000021863343293215910,
+ .00000000000024132639491333131,
+ .00000000000039057462209830700,
+ -.00000000000026570679203560751,
+ .00000000000037135141919592021,
+ -.00000000000017166921336082431,
+ -.00000000000028658285157914353,
+ -.00000000000023812542263446809,
+ .00000000000006576659768580062,
+ -.00000000000028210143846181267,
+ .00000000000010701931762114254,
+ .00000000000018119346366441110,
+ .00000000000009840465278232627,
+ -.00000000000033149150282752542,
+ -.00000000000018302857356041668,
+ -.00000000000016207400156744949,
+ .00000000000048303314949553201,
+ -.00000000000071560553172382115,
+ .00000000000088821239518571855,
+ -.00000000000030900580513238244,
+ -.00000000000061076551972851496,
+ .00000000000035659969663347830,
+ .00000000000035782396591276383,
+ -.00000000000046226087001544578,
+ .00000000000062279762917225156,
+ .00000000000072838947272065741,
+ .00000000000026809646615211673,
+ -.00000000000010960825046059278,
+ .00000000000002311949383800537,
+ -.00000000000058469058005299247,
+ -.00000000000002103748251144494,
+ -.00000000000023323182945587408,
+ -.00000000000042333694288141916,
+ -.00000000000043933937969737844,
+ .00000000000041341647073835565,
+ .00000000000006841763641591466,
+ .00000000000047585534004430641,
+ .00000000000083679678674757695,
+ -.00000000000085763734646658640,
+ .00000000000021913281229340092,
+ -.00000000000062242842536431148,
+ -.00000000000010983594325438430,
+ .00000000000065310431377633651,
+ -.00000000000047580199021710769,
+ -.00000000000037854251265457040,
+ .00000000000040939233218678664,
+ .00000000000087424383914858291,
+ .00000000000025218188456842882,
+ -.00000000000003608131360422557,
+ -.00000000000050518555924280902,
+ .00000000000078699403323355317,
+ -.00000000000067020876961949060,
+ .00000000000016108575753932458,
+ .00000000000058527188436251509,
+ -.00000000000035246757297904791,
+ -.00000000000018372084495629058,
+ .00000000000088606689813494916,
+ .00000000000066486268071468700,
+ .00000000000063831615170646519,
+ .00000000000025144230728376072,
+ -.00000000000017239444525614834
+};
+/*
+ * Extra precision variant, returning struct {double a, b;};
+ * log(x) = a + b to 63 bits, with 'a' rounded to 24 bits.
+ */
+static struct Double
+__log__D(long double x)
+{
+ int m, j;
+ long double F, f, g, q, u, v, u1, u2;
+ struct Double r;
+
+ /*
+ * Argument reduction: 1 <= g < 2; x/2^m = g;
+ * y = F*(1 + f/F) for |f| <= 2^-8
+ */
+ g = frexpl(x, &m);
+ g *= 2;
+ m--;
+ if (m == DBL_MIN_EXP - 1) {
+ j = ilogbl(g);
+ m += j;
+ g = ldexpl(g, -j);
+ }
+ j = N * (g - 1) + 0.5L;
+ F = (1.L / N) * j + 1;
+ f = g - F;
+
+ g = 1 / (2 * F + f);
+ u = 2 * f * g;
+ v = u * u;
+ q = u * v * (A1 + v * (A2 + v * (A3 + v * A4)));
+ if (m | j) {
+ u1 = u + 513;
+ u1 -= 513;
+ } else {
+ u1 = (float)u;
+ }
+ u2 = (2 * (f - F * u1) - u1 * f) * g;
+
+ u1 += m * (long double)logF_head[N] + logF_head[j];
+
+ u2 += logF_tail[j];
+ u2 += q;
+ u2 += logF_tail[N] * m;
+ r.a = (float)(u1 + u2); /* Only difference is here. */
+ r.b = (u1 - r.a) + u2;
+ return (r);
+}
diff --git a/lib/msun/ld80/b_tgammal.c b/lib/msun/ld80/b_tgammal.c
new file mode 100644
index 000000000000..121248adb1b2
--- /dev/null
+++ b/lib/msun/ld80/b_tgammal.c
@@ -0,0 +1,419 @@
+/*-
+ * SPDX-License-Identifier: BSD-3-Clause
+ *
+ * Copyright (c) 1992, 1993
+ * The Regents of the University of California. All rights reserved.
+ *
+ * Redistribution and use in source and binary forms, with or without
+ * modification, are permitted provided that the following conditions
+ * are met:
+ * 1. Redistributions of source code must retain the above copyright
+ * notice, this list of conditions and the following disclaimer.
+ * 2. Redistributions in binary form must reproduce the above copyright
+ * notice, this list of conditions and the following disclaimer in the
+ * documentation and/or other materials provided with the distribution.
+ * 3. Neither the name of the University nor the names of its contributors
+ * may be used to endorse or promote products derived from this software
+ * without specific prior written permission.
+ *
+ * THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND
+ * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
+ * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
+ * ARE DISCLAIMED. IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE
+ * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
+ * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
+ * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
+ * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
+ * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
+ * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
+ * SUCH DAMAGE.
+ */
+
+/*
+ * The original code, FreeBSD's old svn r93211, contain the following
+ * attribution:
+ *
+ * This code by P. McIlroy, Oct 1992;
+ *
+ * The financial support of UUNET Communications Services is greatfully
+ * acknowledged.
+ *
+ * bsdrc/b_tgamma.c converted to long double by Steven G. Kargl.
+ */
+
+/*
+ * See bsdsrc/t_tgamma.c for implementation details.
+ */
+
+#include <float.h>
+
+#if LDBL_MAX_EXP != 0x4000
+#error "Unsupported long double format"
+#endif
+
+#ifdef __i386__
+#include <ieeefp.h>
+#endif
+
+#include "fpmath.h"
+#include "math.h"
+#include "math_private.h"
+
+/* Used in b_log.c and below. */
+struct Double {
+ long double a;
+ long double b;
+};
+
+#include "b_logl.c"
+#include "b_expl.c"
+
+static const double zero = 0.;
+static const volatile double tiny = 1e-300;
+/*
+ * x >= 6
+ *
+ * Use the asymptotic approximation (Stirling's formula) adjusted for
+ * equal-ripples:
+ *
+ * log(G(x)) ~= (x-0.5)*(log(x)-1) + 0.5(log(2*pi)-1) + 1/x*P(1/(x*x))
+ *
+ * Keep extra precision in multiplying (x-.5)(log(x)-1), to avoid
+ * premature round-off.
+ *
+ * Accurate to max(ulp(1/128) absolute, 2^-66 relative) error.
+ */
+
+/*
+ * The following is a decomposition of 0.5 * (log(2*pi) - 1) into the
+ * first 12 bits in ln2pi_hi and the trailing 64 bits in ln2pi_lo. The
+ * variables are clearly misnamed.
+ */
+static const union IEEEl2bits
+ln2pi_hiu = LD80C(0xd680000000000000, -2, 4.18945312500000000000e-01L),
+ln2pi_lou = LD80C(0xe379b414b596d687, -18, -6.77929532725821967032e-06L);
+#define ln2pi_hi (ln2pi_hiu.e)
+#define ln2pi_lo (ln2pi_lou.e)
+
+static const union IEEEl2bits
+ Pa0u = LD80C(0xaaaaaaaaaaaaaaaa, -4, 8.33333333333333333288e-02L),
+ Pa1u = LD80C(0xb60b60b60b5fcd59, -9, -2.77777777777776516326e-03L),
+ Pa2u = LD80C(0xd00d00cffbb47014, -11, 7.93650793635429639018e-04L),
+ Pa3u = LD80C(0x9c09c07c0805343e, -11, -5.95238087960599252215e-04L),
+ Pa4u = LD80C(0xdca8d31f8e6e5e8f, -11, 8.41749082509607342883e-04L),
+ Pa5u = LD80C(0xfb4d4289632f1638, -10, -1.91728055205541624556e-03L),
+ Pa6u = LD80C(0xd15a4ba04078d3f8, -8, 6.38893788027752396194e-03L),
+ Pa7u = LD80C(0xe877283110bcad95, -6, -2.83771309846297590312e-02L),
+ Pa8u = LD80C(0x8da97eed13717af8, -3, 1.38341887683837576925e-01L),
+ Pa9u = LD80C(0xf093b1c1584e30ce, -2, -4.69876818515470146031e-01L);
+#define Pa0 (Pa0u.e)
+#define Pa1 (Pa1u.e)
+#define Pa2 (Pa2u.e)
+#define Pa3 (Pa3u.e)
+#define Pa4 (Pa4u.e)
+#define Pa5 (Pa5u.e)
+#define Pa6 (Pa6u.e)
+#define Pa7 (Pa7u.e)
+#define Pa8 (Pa8u.e)
+#define Pa9 (Pa9u.e)
+
+static struct Double
+large_gam(long double x)
+{
+ long double p, z, thi, tlo, xhi, xlo;
+ long double logx;
+ struct Double u;
+
+ z = 1 / (x * x);
+ p = Pa0 + z * (Pa1 + z * (Pa2 + z * (Pa3 + z * (Pa4 + z * (Pa5 +
+ z * (Pa6 + z * (Pa7 + z * (Pa8 + z * Pa9))))))));
+ p = p / x;
+
+ u = __log__D(x);
+ u.a -= 1;
+
+ /* Split (x - 0.5) in high and low parts. */
+ x -= 0.5L;
+ xhi = (float)x;
+ xlo = x - xhi;
+
+ /* Compute t = (x-.5)*(log(x)-1) in extra precision. */
+ thi = xhi * u.a;
+ tlo = xlo * u.a + x * u.b;
+
+ /* Compute thi + tlo + ln2pi_hi + ln2pi_lo + p. */
+ tlo += ln2pi_lo;
+ tlo += p;
+ u.a = ln2pi_hi + tlo;
+ u.a += thi;
+ u.b = thi - u.a;
+ u.b += ln2pi_hi;
+ u.b += tlo;
+ return (u);
+}
+/*
+ * Rational approximation, A0 + x * x * P(x) / Q(x), on the interval
+ * [1.066.., 2.066..] accurate to 4.25e-19.
+ *
+ * Returns r.a + r.b = a0 + (z + c)^2 * p / q, with r.a truncated.
+ */
+static const union IEEEl2bits
+ a0_hiu = LD80C(0xe2b6e4153a57746c, -1, 8.85603194410888700265e-01L),
+ a0_lou = LD80C(0x851566d40f32c76d, -66, 1.40907742727049706207e-20L);
+#define a0_hi (a0_hiu.e)
+#define a0_lo (a0_lou.e)
+
+static const union IEEEl2bits
+P0u = LD80C(0xdb629fb9bbdc1c1d, -2, 4.28486815855585429733e-01L),
+P1u = LD80C(0xe6f4f9f5641aa6be, -3, 2.25543885805587730552e-01L),
+P2u = LD80C(0xead1bd99fdaf7cc1, -6, 2.86644652514293482381e-02L),
+P3u = LD80C(0x9ccc8b25838ab1e0, -8, 4.78512567772456362048e-03L),
+P4u = LD80C(0x8f0c4383ef9ce72a, -9, 2.18273781132301146458e-03L),
+P5u = LD80C(0xe732ab2c0a2778da, -13, 2.20487522485636008928e-04L),
+P6u = LD80C(0xce70b27ca822b297, -16, 2.46095923774929264284e-05L),
+P7u = LD80C(0xa309e2e16fb63663, -19, 2.42946473022376182921e-06L),
+P8u = LD80C(0xaf9c110efb2c633d, -23, 1.63549217667765869987e-07L),
+Q1u = LD80C(0xd4d7422719f48f15, -1, 8.31409582658993993626e-01L),
+Q2u = LD80C(0xe13138ea404f1268, -5, -5.49785826915643198508e-02L),
+Q3u = LD80C(0xd1c6cc91989352c0, -4, -1.02429960435139887683e-01L),
+Q4u = LD80C(0xa7e9435a84445579, -7, 1.02484853505908820524e-02L),
+Q5u = LD80C(0x83c7c34db89b7bda, -8, 4.02161632832052872697e-03L),
+Q6u = LD80C(0xbed06bf6e1c14e5b, -11, -7.27898206351223022157e-04L),
+Q7u = LD80C(0xef05bf841d4504c0, -18, 7.12342421869453515194e-06L),
+Q8u = LD80C(0xf348d08a1ff53cb1, -19, 3.62522053809474067060e-06L);
+#define P0 (P0u.e)
+#define P1 (P1u.e)
+#define P2 (P2u.e)
+#define P3 (P3u.e)
+#define P4 (P4u.e)
+#define P5 (P5u.e)
+#define P6 (P6u.e)
+#define P7 (P7u.e)
+#define P8 (P8u.e)
+#define Q1 (Q1u.e)
+#define Q2 (Q2u.e)
+#define Q3 (Q3u.e)
+#define Q4 (Q4u.e)
+#define Q5 (Q5u.e)
+#define Q6 (Q6u.e)
+#define Q7 (Q7u.e)
+#define Q8 (Q8u.e)
+
+static struct Double
+ratfun_gam(long double z, long double c)
+{
+ long double p, q, thi, tlo;
+ struct Double r;
+
+ q = 1 + z * (Q1 + z * (Q2 + z * (Q3 + z * (Q4 + z * (Q5 +
+ z * (Q6 + z * (Q7 + z * Q8)))))));
+ p = P0 + z * (P1 + z * (P2 + z * (P3 + z * (P4 + z * (P5 +
+ z * (P6 + z * (P7 + z * P8)))))));
+ p = p / q;
+
+ /* Split z into high and low parts. */
+ thi = (float)z;
+ tlo = (z - thi) + c;
+ tlo *= (thi + z);
+
+ /* Split (z+c)^2 into high and low parts. */
+ thi *= thi;
+ q = thi;
+ thi = (float)thi;
+ tlo += (q - thi);
+
+ /* Split p/q into high and low parts. */
+ r.a = (float)p;
+ r.b = p - r.a;
+
+ tlo = tlo * p + thi * r.b + a0_lo;
+ thi *= r.a; /* t = (z+c)^2*(P/Q) */
+ r.a = (float)(thi + a0_hi);
+ r.b = ((a0_hi - r.a) + thi) + tlo;
+ return (r); /* r = a0 + t */
+}
+/*
+ * x < 6
+ *
+ * Use argument reduction G(x+1) = xG(x) to reach the range [1.066124,
+ * 2.066124]. Use a rational approximation centered at the minimum
+ * (x0+1) to ensure monotonicity.
+ *
+ * Good to < 1 ulp. (provably .90 ulp; .87 ulp on 1,000,000 runs.)
+ * It also has correct monotonicity.
+ */
+static const union IEEEl2bits
+ xm1u = LD80C(0xec5b0c6ad7c7edc3, -2, 4.61632144968362341254e-01L);
+#define x0 (xm1u.e)
+
+static const double
+ left = -0.3955078125; /* left boundary for rat. approx */
+
+static long double
+small_gam(long double x)
+{
+ long double t, y, ym1;
+ struct Double yy, r;
+
+ y = x - 1;
+
+ if (y <= 1 + (left + x0)) {
+ yy = ratfun_gam(y - x0, 0);
+ return (yy.a + yy.b);
+ }
+
+ r.a = (float)y;
+ yy.a = r.a - 1;
+ y = y - 1 ;
+ r.b = yy.b = y - yy.a;
+
+ /* Argument reduction: G(x+1) = x*G(x) */
+ for (ym1 = y - 1; ym1 > left + x0; y = ym1--, yy.a--) {
+ t = r.a * yy.a;
+ r.b = r.a * yy.b + y * r.b;
+ r.a = (float)t;
+ r.b += (t - r.a);
+ }
+
+ /* Return r*tgamma(y). */
+ yy = ratfun_gam(y - x0, 0);
+ y = r.b * (yy.a + yy.b) + r.a * yy.b;
+ y += yy.a * r.a;
+ return (y);
+}
+/*
+ * Good on (0, 1+x0+left]. Accurate to 1 ulp.
+ */
+static long double
+smaller_gam(long double x)
+{
+ long double d, rhi, rlo, t, xhi, xlo;
+ struct Double r;
+
+ if (x < x0 + left) {
+ t = (float)x;
+ d = (t + x) * (x - t);
+ t *= t;
+ xhi = (float)(t + x);
+ xlo = x - xhi;
+ xlo += t;
+ xlo += d;
+ t = 1 - x0;
+ t += x;
+ d = 1 - x0;
+ d -= t;
+ d += x;
+ x = xhi + xlo;
+ } else {
+ xhi = (float)x;
+ xlo = x - xhi;
+ t = x - x0;
+ d = - x0 - t;
+ d += x;
+ }
+
+ r = ratfun_gam(t, d);
+ d = (float)(r.a / x);
+ r.a -= d * xhi;
+ r.a -= d * xlo;
+ r.a += r.b;
+
+ return (d + r.a / x);
+}
+/*
+ * x < 0
+ *
+ * Use reflection formula, G(x) = pi/(sin(pi*x)*x*G(x)).
+ * At negative integers, return NaN and raise invalid.
+ */
+static const union IEEEl2bits
+piu = LD80C(0xc90fdaa22168c235, 1, 3.14159265358979323851e+00L);
+#define pi (piu.e)
+
+static long double
+neg_gam(long double x)
+{
+ int sgn = 1;
+ struct Double lg, lsine;
+ long double y, z;
+
+ y = ceill(x);
+ if (y == x) /* Negative integer. */
+ return ((x - x) / zero);
+
+ z = y - x;
+ if (z > 0.5)
+ z = 1 - z;
+
+ y = y / 2;
+ if (y == ceill(y))
+ sgn = -1;
+
+ if (z < 0.25)
+ z = sinpil(z);
+ else
+ z = cospil(0.5 - z);
+
+ /* Special case: G(1-x) = Inf; G(x) may be nonzero. */
+ if (x < -1753) {
+
+ if (x < -1760)
+ return (sgn * tiny * tiny);
+ y = expl(lgammal(x) / 2);
+ y *= y;
+ return (sgn < 0 ? -y : y);
+ }
+
+
+ y = 1 - x;
+ if (1 - y == x)
+ y = tgammal(y);
+ else /* 1-x is inexact */
+ y = - x * tgammal(-x);
+
+ if (sgn < 0) y = -y;
+ return (pi / (y * z));
+}
+/*
+ * xmax comes from lgamma(xmax) - emax * log(2) = 0.
+ * static const float xmax = 35.040095f
+ * static const double xmax = 171.624376956302725;
+ * ld80: LD80C(0xdb718c066b352e20, 10, 1.75554834290446291689e+03L),
+ * ld128: 1.75554834290446291700388921607020320e+03L,
+ *
+ * iota is a sloppy threshold to isolate x = 0.
+ */
+static const double xmax = 1755.54834290446291689;
+static const double iota = 0x1p-116;
+
+long double
+tgammal(long double x)
+{
+ struct Double u;
+
+ ENTERI();
+
+ if (x >= 6) {
+ if (x > xmax)
+ RETURNI(x / zero);
+ u = large_gam(x);
+ RETURNI(__exp__D(u.a, u.b));
+ }
+
+ if (x >= 1 + left + x0)
+ RETURNI(small_gam(x));
+
+ if (x > iota)
+ RETURNI(smaller_gam(x));
+
+ if (x > -iota) {
+ if (x != 0)
+ u.a = 1 - tiny; /* raise inexact */
+ RETURNI(1 / x);
+ }
+
+ if (!isfinite(x))
+ RETURNI(x - x); /* x is NaN or -Inf */
+
*** 2 LINES SKIPPED ***