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
65 /* arctan(k/8), k = 0, ..., 82 */
66 static const long double atantbl[84] = {
67 0.0000000000000000000000000000000000000000E0L,
68 1.2435499454676143503135484916387102557317E-1L, /* arctan(0.125) */
69 2.4497866312686415417208248121127581091414E-1L,
70 3.5877067027057222039592006392646049977698E-1L,
71 4.6364760900080611621425623146121440202854E-1L,
72 5.5859931534356243597150821640166127034645E-1L,
73 6.4350110879328438680280922871732263804151E-1L,
74 7.1882999962162450541701415152590465395142E-1L,
75 7.8539816339744830961566084581987572104929E-1L,
76 8.4415398611317100251784414827164750652594E-1L,
77 8.9605538457134395617480071802993782702458E-1L,
78 9.4200004037946366473793717053459358607166E-1L,
79 9.8279372324732906798571061101466601449688E-1L,
80 1.0191413442663497346383429170230636487744E0L,
81 1.0516502125483736674598673120862998296302E0L,
82 1.0808390005411683108871567292171998202703E0L,
83 1.1071487177940905030170654601785370400700E0L,
84 1.1309537439791604464709335155363278047493E0L,
85 1.1525719972156675180401498626127513797495E0L,
86 1.1722738811284763866005949441337046149712E0L,
87 1.1902899496825317329277337748293183376012E0L,
88 1.2068173702852525303955115800565576303133E0L,
89 1.2220253232109896370417417439225704908830E0L,
90 1.2360594894780819419094519711090786987027E0L,
91 1.2490457723982544258299170772810901230778E0L,
92 1.2610933822524404193139408812473357720101E0L,
93 1.2722973952087173412961937498224804940684E0L,
94 1.2827408797442707473628852511364955306249E0L,
95 1.2924966677897852679030914214070816845853E0L,
96 1.3016288340091961438047858503666855921414E0L,
97 1.3101939350475556342564376891719053122733E0L,
98 1.3182420510168370498593302023271362531155E0L,
99 1.3258176636680324650592392104284756311844E0L,
100 1.3329603993374458675538498697331558093700E0L,
101 1.3397056595989995393283037525895557411039E0L,
102 1.3460851583802539310489409282517796256512E0L,
103 1.3521273809209546571891479413898128509842E0L,
104 1.3578579772154994751124898859640585287459E0L,
105 1.3633001003596939542892985278250991189943E0L,
106 1.3684746984165928776366381936948529556191E0L,
107 1.3734007669450158608612719264449611486510E0L,
108 1.3780955681325110444536609641291551522494E0L,
109 1.3825748214901258580599674177685685125566E0L,
110 1.3868528702577214543289381097042486034883E0L,
111 1.3909428270024183486427686943836432060856E0L,
112 1.3948567013423687823948122092044222644895E0L,
113 1.3986055122719575950126700816114282335732E0L,
114 1.4021993871854670105330304794336492676944E0L,
115 1.4056476493802697809521934019958079881002E0L,
116 1.4089588955564736949699075250792569287156E0L,
117 1.4121410646084952153676136718584891599630E0L,
118 1.4152014988178669079462550975833894394929E0L,
119 1.4181469983996314594038603039700989523716E0L,
120 1.4209838702219992566633046424614466661176E0L,
121 1.4237179714064941189018190466107297503086E0L,
122 1.4263547484202526397918060597281265695725E0L,
123 1.4288992721907326964184700745371983590908E0L,
124 1.4313562697035588982240194668401779312122E0L,
125 1.4337301524847089866404719096698873648610E0L,
126 1.4360250423171655234964275337155008780675E0L,
127 1.4382447944982225979614042479354815855386E0L,
128 1.4403930189057632173997301031392126865694E0L,
129 1.4424730991091018200252920599377292525125E0L,
130 1.4444882097316563655148453598508037025938E0L,
131 1.4464413322481351841999668424758804165254E0L,
132 1.4483352693775551917970437843145232637695E0L,
133 1.4501726582147939000905940595923466567576E0L,
134 1.4519559822271314199339700039142990228105E0L,
135 1.4536875822280323362423034480994649820285E0L,
136 1.4553696664279718992423082296859928222270E0L,
137 1.4570043196511885530074841089245667532358E0L,
138 1.4585935117976422128825857356750737658039E0L,
139 1.4601391056210009726721818194296893361233E0L,
140 1.4616428638860188872060496086383008594310E0L,
141 1.4631064559620759326975975316301202111560E0L,
142 1.4645314639038178118428450961503371619177E0L,
143 1.4659193880646627234129855241049975398470E0L,
144 1.4672716522843522691530527207287398276197E0L,
145 1.4685896086876430842559640450619880951144E0L,
146 1.4698745421276027686510391411132998919794E0L,
147 1.4711276743037345918528755717617308518553E0L,
148 1.4723501675822635384916444186631899205983E0L,
149 1.4735431285433308455179928682541563973416E0L, /* arctan(10.25) */
150 1.5707963267948966192313216916397514420986E0L /* pi/2 */
154 /* arctan t = t + t^3 p(t^2) / q(t^2)
156 peak relative error 5.3e-37 */
158 static const long double
159 p0 = -4.283708356338736809269381409828726405572E1L,
160 p1 = -8.636132499244548540964557273544599863825E1L,
161 p2 = -5.713554848244551350855604111031839613216E1L,
162 p3 = -1.371405711877433266573835355036413750118E1L,
163 p4 = -8.638214309119210906997318946650189640184E-1L,
164 q0 = 1.285112506901621042780814422948906537959E2L,
165 q1 = 3.361907253914337187957855834229672347089E2L,
166 q2 = 3.180448303864130128268191635189365331680E2L,
167 q3 = 1.307244136980865800160844625025280344686E2L,
168 q4 = 2.173623741810414221251136181221172551416E1L;
169 /* q5 = 1.000000000000000000000000000000000000000E0 */
173 atanl (long double x)
176 long double t, u, p, q;
178 /* Check for zero or NaN. */
179 if (isnanl (x) || x == 0.0)
203 /* Index of nearest table element.
204 Roundoff to integer is asymmetrical to avoid cancellation when t < 0
208 /* Small arctan argument. */
209 t = (x - u) / (1.0 + x * u);
212 /* Arctan of small argument t. */
214 p = ((((p4 * u) + p3) * u + p2) * u + p1) * u + p0;
215 q = ((((u + q4) * u + q3) * u + q2) * u + q1) * u + q0;
216 u = t * u * p / q + t;
218 /* arctan x = arctan u + arctan t */