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 2, or (at your option)
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; see the file COPYING.
15 If not, write to the Free Software Foundation,
16 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301, USA. */
25 * Inverse circular tangent for 128-bit long double precision
32 * long double x, y, atanl();
40 * Returns radian angle between -pi/2 and +pi/2 whose tangent is x.
42 * The function uses a rational approximation of the form
43 * t + t^3 P(t^2)/Q(t^2), optimized for |t| < 0.09375.
45 * The argument is reduced using the identity
46 * arctan x - arctan u = arctan ((x-u)/(1 + ux))
47 * and an 83-entry lookup table for arctan u, with u = 0, 1/8, ..., 10.25.
48 * Use of the table improves the execution speed of the routine.
55 * arithmetic domain # trials peak rms
56 * IEEE -19, 19 4e5 1.7e-34 5.4e-35
61 * This program uses integer operations on bit fields of floating-point
62 * numbers. It does not work with data structures other than the
68 /* arctan(k/8), k = 0, ..., 82 */
69 static const long double atantbl[84] = {
70 0.0000000000000000000000000000000000000000E0L,
71 1.2435499454676143503135484916387102557317E-1L, /* arctan(0.125) */
72 2.4497866312686415417208248121127581091414E-1L,
73 3.5877067027057222039592006392646049977698E-1L,
74 4.6364760900080611621425623146121440202854E-1L,
75 5.5859931534356243597150821640166127034645E-1L,
76 6.4350110879328438680280922871732263804151E-1L,
77 7.1882999962162450541701415152590465395142E-1L,
78 7.8539816339744830961566084581987572104929E-1L,
79 8.4415398611317100251784414827164750652594E-1L,
80 8.9605538457134395617480071802993782702458E-1L,
81 9.4200004037946366473793717053459358607166E-1L,
82 9.8279372324732906798571061101466601449688E-1L,
83 1.0191413442663497346383429170230636487744E0L,
84 1.0516502125483736674598673120862998296302E0L,
85 1.0808390005411683108871567292171998202703E0L,
86 1.1071487177940905030170654601785370400700E0L,
87 1.1309537439791604464709335155363278047493E0L,
88 1.1525719972156675180401498626127513797495E0L,
89 1.1722738811284763866005949441337046149712E0L,
90 1.1902899496825317329277337748293183376012E0L,
91 1.2068173702852525303955115800565576303133E0L,
92 1.2220253232109896370417417439225704908830E0L,
93 1.2360594894780819419094519711090786987027E0L,
94 1.2490457723982544258299170772810901230778E0L,
95 1.2610933822524404193139408812473357720101E0L,
96 1.2722973952087173412961937498224804940684E0L,
97 1.2827408797442707473628852511364955306249E0L,
98 1.2924966677897852679030914214070816845853E0L,
99 1.3016288340091961438047858503666855921414E0L,
100 1.3101939350475556342564376891719053122733E0L,
101 1.3182420510168370498593302023271362531155E0L,
102 1.3258176636680324650592392104284756311844E0L,
103 1.3329603993374458675538498697331558093700E0L,
104 1.3397056595989995393283037525895557411039E0L,
105 1.3460851583802539310489409282517796256512E0L,
106 1.3521273809209546571891479413898128509842E0L,
107 1.3578579772154994751124898859640585287459E0L,
108 1.3633001003596939542892985278250991189943E0L,
109 1.3684746984165928776366381936948529556191E0L,
110 1.3734007669450158608612719264449611486510E0L,
111 1.3780955681325110444536609641291551522494E0L,
112 1.3825748214901258580599674177685685125566E0L,
113 1.3868528702577214543289381097042486034883E0L,
114 1.3909428270024183486427686943836432060856E0L,
115 1.3948567013423687823948122092044222644895E0L,
116 1.3986055122719575950126700816114282335732E0L,
117 1.4021993871854670105330304794336492676944E0L,
118 1.4056476493802697809521934019958079881002E0L,
119 1.4089588955564736949699075250792569287156E0L,
120 1.4121410646084952153676136718584891599630E0L,
121 1.4152014988178669079462550975833894394929E0L,
122 1.4181469983996314594038603039700989523716E0L,
123 1.4209838702219992566633046424614466661176E0L,
124 1.4237179714064941189018190466107297503086E0L,
125 1.4263547484202526397918060597281265695725E0L,
126 1.4288992721907326964184700745371983590908E0L,
127 1.4313562697035588982240194668401779312122E0L,
128 1.4337301524847089866404719096698873648610E0L,
129 1.4360250423171655234964275337155008780675E0L,
130 1.4382447944982225979614042479354815855386E0L,
131 1.4403930189057632173997301031392126865694E0L,
132 1.4424730991091018200252920599377292525125E0L,
133 1.4444882097316563655148453598508037025938E0L,
134 1.4464413322481351841999668424758804165254E0L,
135 1.4483352693775551917970437843145232637695E0L,
136 1.4501726582147939000905940595923466567576E0L,
137 1.4519559822271314199339700039142990228105E0L,
138 1.4536875822280323362423034480994649820285E0L,
139 1.4553696664279718992423082296859928222270E0L,
140 1.4570043196511885530074841089245667532358E0L,
141 1.4585935117976422128825857356750737658039E0L,
142 1.4601391056210009726721818194296893361233E0L,
143 1.4616428638860188872060496086383008594310E0L,
144 1.4631064559620759326975975316301202111560E0L,
145 1.4645314639038178118428450961503371619177E0L,
146 1.4659193880646627234129855241049975398470E0L,
147 1.4672716522843522691530527207287398276197E0L,
148 1.4685896086876430842559640450619880951144E0L,
149 1.4698745421276027686510391411132998919794E0L,
150 1.4711276743037345918528755717617308518553E0L,
151 1.4723501675822635384916444186631899205983E0L,
152 1.4735431285433308455179928682541563973416E0L, /* arctan(10.25) */
153 1.5707963267948966192313216916397514420986E0L /* pi/2 */
157 /* arctan t = t + t^3 p(t^2) / q(t^2)
159 peak relative error 5.3e-37 */
161 static const long double
162 p0 = -4.283708356338736809269381409828726405572E1L,
163 p1 = -8.636132499244548540964557273544599863825E1L,
164 p2 = -5.713554848244551350855604111031839613216E1L,
165 p3 = -1.371405711877433266573835355036413750118E1L,
166 p4 = -8.638214309119210906997318946650189640184E-1L,
167 q0 = 1.285112506901621042780814422948906537959E2L,
168 q1 = 3.361907253914337187957855834229672347089E2L,
169 q2 = 3.180448303864130128268191635189365331680E2L,
170 q3 = 1.307244136980865800160844625025280344686E2L,
171 q4 = 2.173623741810414221251136181221172551416E1L;
172 /* q5 = 1.000000000000000000000000000000000000000E0 */
176 atanl (long double x)
179 long double t, u, p, q;
183 /* Check for zero or NaN. */
184 if (x != x || x == 0.0)
206 /* Index of nearest table element.
207 Roundoff to integer is asymmetrical to avoid cancellation when t < 0
211 /* Small arctan argument. */
212 t = (x - u) / (1.0 + x * u);
215 /* Arctan of small argument t. */
217 p = ((((p4 * u) + p3) * u + p2) * u + p1) * u + p0;
218 q = ((((u + q4) * u + q3) * u + q2) * u + q1) * u + q0;
219 u = t * u * p / q + t;
221 /* arctan x = arctan u + arctan t */