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