1 /* Copyright 2001 by Stephen L. Moshier <moshier@na-net.ornl.gov>
3 This program is free software: you can redistribute it and/or modify
4 it under the terms of the GNU General Public License as published by
5 the Free Software Foundation; either version 3 of the License, or
6 (at your option) any later version.
8 This program is distributed in the hope that it will be useful,
9 but WITHOUT ANY WARRANTY; without even the implied warranty of
10 MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
11 GNU General Public License for more details.
13 You should have received a copy of the GNU General Public License
14 along with this program. If not, see <http://www.gnu.org/licenses/>. */
21 #if HAVE_SAME_LONG_DOUBLE_AS_DOUBLE
31 /* Code based on glibc/sysdeps/ieee754/ldbl-128/s_atanl.c. */
35 * Inverse circular tangent for 128-bit long double precision
42 * long double x, y, atanl();
50 * Returns radian angle between -pi/2 and +pi/2 whose tangent is x.
52 * The function uses a rational approximation of the form
53 * t + t^3 P(t^2)/Q(t^2), optimized for |t| < 0.09375.
55 * The argument is reduced using the identity
56 * arctan x - arctan u = arctan ((x-u)/(1 + ux))
57 * and an 83-entry lookup table for arctan u, with u = 0, 1/8, ..., 10.25.
58 * Use of the table improves the execution speed of the routine.
65 * arithmetic domain # trials peak rms
66 * IEEE -19, 19 4e5 1.7e-34 5.4e-35
71 * This program uses integer operations on bit fields of floating-point
72 * numbers. It does not work with data structures other than the
77 /* arctan(k/8), k = 0, ..., 82 */
78 static const long double atantbl[84] = {
79 0.0000000000000000000000000000000000000000E0L,
80 1.2435499454676143503135484916387102557317E-1L, /* arctan(0.125) */
81 2.4497866312686415417208248121127581091414E-1L,
82 3.5877067027057222039592006392646049977698E-1L,
83 4.6364760900080611621425623146121440202854E-1L,
84 5.5859931534356243597150821640166127034645E-1L,
85 6.4350110879328438680280922871732263804151E-1L,
86 7.1882999962162450541701415152590465395142E-1L,
87 7.8539816339744830961566084581987572104929E-1L,
88 8.4415398611317100251784414827164750652594E-1L,
89 8.9605538457134395617480071802993782702458E-1L,
90 9.4200004037946366473793717053459358607166E-1L,
91 9.8279372324732906798571061101466601449688E-1L,
92 1.0191413442663497346383429170230636487744E0L,
93 1.0516502125483736674598673120862998296302E0L,
94 1.0808390005411683108871567292171998202703E0L,
95 1.1071487177940905030170654601785370400700E0L,
96 1.1309537439791604464709335155363278047493E0L,
97 1.1525719972156675180401498626127513797495E0L,
98 1.1722738811284763866005949441337046149712E0L,
99 1.1902899496825317329277337748293183376012E0L,
100 1.2068173702852525303955115800565576303133E0L,
101 1.2220253232109896370417417439225704908830E0L,
102 1.2360594894780819419094519711090786987027E0L,
103 1.2490457723982544258299170772810901230778E0L,
104 1.2610933822524404193139408812473357720101E0L,
105 1.2722973952087173412961937498224804940684E0L,
106 1.2827408797442707473628852511364955306249E0L,
107 1.2924966677897852679030914214070816845853E0L,
108 1.3016288340091961438047858503666855921414E0L,
109 1.3101939350475556342564376891719053122733E0L,
110 1.3182420510168370498593302023271362531155E0L,
111 1.3258176636680324650592392104284756311844E0L,
112 1.3329603993374458675538498697331558093700E0L,
113 1.3397056595989995393283037525895557411039E0L,
114 1.3460851583802539310489409282517796256512E0L,
115 1.3521273809209546571891479413898128509842E0L,
116 1.3578579772154994751124898859640585287459E0L,
117 1.3633001003596939542892985278250991189943E0L,
118 1.3684746984165928776366381936948529556191E0L,
119 1.3734007669450158608612719264449611486510E0L,
120 1.3780955681325110444536609641291551522494E0L,
121 1.3825748214901258580599674177685685125566E0L,
122 1.3868528702577214543289381097042486034883E0L,
123 1.3909428270024183486427686943836432060856E0L,
124 1.3948567013423687823948122092044222644895E0L,
125 1.3986055122719575950126700816114282335732E0L,
126 1.4021993871854670105330304794336492676944E0L,
127 1.4056476493802697809521934019958079881002E0L,
128 1.4089588955564736949699075250792569287156E0L,
129 1.4121410646084952153676136718584891599630E0L,
130 1.4152014988178669079462550975833894394929E0L,
131 1.4181469983996314594038603039700989523716E0L,
132 1.4209838702219992566633046424614466661176E0L,
133 1.4237179714064941189018190466107297503086E0L,
134 1.4263547484202526397918060597281265695725E0L,
135 1.4288992721907326964184700745371983590908E0L,
136 1.4313562697035588982240194668401779312122E0L,
137 1.4337301524847089866404719096698873648610E0L,
138 1.4360250423171655234964275337155008780675E0L,
139 1.4382447944982225979614042479354815855386E0L,
140 1.4403930189057632173997301031392126865694E0L,
141 1.4424730991091018200252920599377292525125E0L,
142 1.4444882097316563655148453598508037025938E0L,
143 1.4464413322481351841999668424758804165254E0L,
144 1.4483352693775551917970437843145232637695E0L,
145 1.4501726582147939000905940595923466567576E0L,
146 1.4519559822271314199339700039142990228105E0L,
147 1.4536875822280323362423034480994649820285E0L,
148 1.4553696664279718992423082296859928222270E0L,
149 1.4570043196511885530074841089245667532358E0L,
150 1.4585935117976422128825857356750737658039E0L,
151 1.4601391056210009726721818194296893361233E0L,
152 1.4616428638860188872060496086383008594310E0L,
153 1.4631064559620759326975975316301202111560E0L,
154 1.4645314639038178118428450961503371619177E0L,
155 1.4659193880646627234129855241049975398470E0L,
156 1.4672716522843522691530527207287398276197E0L,
157 1.4685896086876430842559640450619880951144E0L,
158 1.4698745421276027686510391411132998919794E0L,
159 1.4711276743037345918528755717617308518553E0L,
160 1.4723501675822635384916444186631899205983E0L,
161 1.4735431285433308455179928682541563973416E0L, /* arctan(10.25) */
162 1.5707963267948966192313216916397514420986E0L /* pi/2 */
166 /* arctan t = t + t^3 p(t^2) / q(t^2)
168 peak relative error 5.3e-37 */
170 static const long double
171 p0 = -4.283708356338736809269381409828726405572E1L,
172 p1 = -8.636132499244548540964557273544599863825E1L,
173 p2 = -5.713554848244551350855604111031839613216E1L,
174 p3 = -1.371405711877433266573835355036413750118E1L,
175 p4 = -8.638214309119210906997318946650189640184E-1L,
176 q0 = 1.285112506901621042780814422948906537959E2L,
177 q1 = 3.361907253914337187957855834229672347089E2L,
178 q2 = 3.180448303864130128268191635189365331680E2L,
179 q3 = 1.307244136980865800160844625025280344686E2L,
180 q4 = 2.173623741810414221251136181221172551416E1L;
181 /* q5 = 1.000000000000000000000000000000000000000E0 */
185 atanl (long double x)
188 long double t, u, p, q;
190 /* Check for zero or NaN. */
191 if (isnanl (x) || x == 0.0)
215 /* Index of nearest table element.
216 Roundoff to integer is asymmetrical to avoid cancellation when t < 0
220 /* Small arctan argument. */
221 t = (x - u) / (1.0 + x * u);
224 /* Arctan of small argument t. */
226 p = ((((p4 * u) + p3) * u + p2) * u + p1) * u + p0;
227 q = ((((u + q4) * u + q3) * u + q2) * u + q1) * u + q0;
228 u = t * u * p / q + t;
230 /* arctan x = arctan u + arctan t */