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
69 /* arctan(k/8), k = 0, ..., 82 */
70 static const long double atantbl[84] = {
71 0.0000000000000000000000000000000000000000E0L,
72 1.2435499454676143503135484916387102557317E-1L, /* arctan(0.125) */
73 2.4497866312686415417208248121127581091414E-1L,
74 3.5877067027057222039592006392646049977698E-1L,
75 4.6364760900080611621425623146121440202854E-1L,
76 5.5859931534356243597150821640166127034645E-1L,
77 6.4350110879328438680280922871732263804151E-1L,
78 7.1882999962162450541701415152590465395142E-1L,
79 7.8539816339744830961566084581987572104929E-1L,
80 8.4415398611317100251784414827164750652594E-1L,
81 8.9605538457134395617480071802993782702458E-1L,
82 9.4200004037946366473793717053459358607166E-1L,
83 9.8279372324732906798571061101466601449688E-1L,
84 1.0191413442663497346383429170230636487744E0L,
85 1.0516502125483736674598673120862998296302E0L,
86 1.0808390005411683108871567292171998202703E0L,
87 1.1071487177940905030170654601785370400700E0L,
88 1.1309537439791604464709335155363278047493E0L,
89 1.1525719972156675180401498626127513797495E0L,
90 1.1722738811284763866005949441337046149712E0L,
91 1.1902899496825317329277337748293183376012E0L,
92 1.2068173702852525303955115800565576303133E0L,
93 1.2220253232109896370417417439225704908830E0L,
94 1.2360594894780819419094519711090786987027E0L,
95 1.2490457723982544258299170772810901230778E0L,
96 1.2610933822524404193139408812473357720101E0L,
97 1.2722973952087173412961937498224804940684E0L,
98 1.2827408797442707473628852511364955306249E0L,
99 1.2924966677897852679030914214070816845853E0L,
100 1.3016288340091961438047858503666855921414E0L,
101 1.3101939350475556342564376891719053122733E0L,
102 1.3182420510168370498593302023271362531155E0L,
103 1.3258176636680324650592392104284756311844E0L,
104 1.3329603993374458675538498697331558093700E0L,
105 1.3397056595989995393283037525895557411039E0L,
106 1.3460851583802539310489409282517796256512E0L,
107 1.3521273809209546571891479413898128509842E0L,
108 1.3578579772154994751124898859640585287459E0L,
109 1.3633001003596939542892985278250991189943E0L,
110 1.3684746984165928776366381936948529556191E0L,
111 1.3734007669450158608612719264449611486510E0L,
112 1.3780955681325110444536609641291551522494E0L,
113 1.3825748214901258580599674177685685125566E0L,
114 1.3868528702577214543289381097042486034883E0L,
115 1.3909428270024183486427686943836432060856E0L,
116 1.3948567013423687823948122092044222644895E0L,
117 1.3986055122719575950126700816114282335732E0L,
118 1.4021993871854670105330304794336492676944E0L,
119 1.4056476493802697809521934019958079881002E0L,
120 1.4089588955564736949699075250792569287156E0L,
121 1.4121410646084952153676136718584891599630E0L,
122 1.4152014988178669079462550975833894394929E0L,
123 1.4181469983996314594038603039700989523716E0L,
124 1.4209838702219992566633046424614466661176E0L,
125 1.4237179714064941189018190466107297503086E0L,
126 1.4263547484202526397918060597281265695725E0L,
127 1.4288992721907326964184700745371983590908E0L,
128 1.4313562697035588982240194668401779312122E0L,
129 1.4337301524847089866404719096698873648610E0L,
130 1.4360250423171655234964275337155008780675E0L,
131 1.4382447944982225979614042479354815855386E0L,
132 1.4403930189057632173997301031392126865694E0L,
133 1.4424730991091018200252920599377292525125E0L,
134 1.4444882097316563655148453598508037025938E0L,
135 1.4464413322481351841999668424758804165254E0L,
136 1.4483352693775551917970437843145232637695E0L,
137 1.4501726582147939000905940595923466567576E0L,
138 1.4519559822271314199339700039142990228105E0L,
139 1.4536875822280323362423034480994649820285E0L,
140 1.4553696664279718992423082296859928222270E0L,
141 1.4570043196511885530074841089245667532358E0L,
142 1.4585935117976422128825857356750737658039E0L,
143 1.4601391056210009726721818194296893361233E0L,
144 1.4616428638860188872060496086383008594310E0L,
145 1.4631064559620759326975975316301202111560E0L,
146 1.4645314639038178118428450961503371619177E0L,
147 1.4659193880646627234129855241049975398470E0L,
148 1.4672716522843522691530527207287398276197E0L,
149 1.4685896086876430842559640450619880951144E0L,
150 1.4698745421276027686510391411132998919794E0L,
151 1.4711276743037345918528755717617308518553E0L,
152 1.4723501675822635384916444186631899205983E0L,
153 1.4735431285433308455179928682541563973416E0L, /* arctan(10.25) */
154 1.5707963267948966192313216916397514420986E0L /* pi/2 */
158 /* arctan t = t + t^3 p(t^2) / q(t^2)
160 peak relative error 5.3e-37 */
162 static const long double
163 p0 = -4.283708356338736809269381409828726405572E1L,
164 p1 = -8.636132499244548540964557273544599863825E1L,
165 p2 = -5.713554848244551350855604111031839613216E1L,
166 p3 = -1.371405711877433266573835355036413750118E1L,
167 p4 = -8.638214309119210906997318946650189640184E-1L,
168 q0 = 1.285112506901621042780814422948906537959E2L,
169 q1 = 3.361907253914337187957855834229672347089E2L,
170 q2 = 3.180448303864130128268191635189365331680E2L,
171 q3 = 1.307244136980865800160844625025280344686E2L,
172 q4 = 2.173623741810414221251136181221172551416E1L;
173 /* q5 = 1.000000000000000000000000000000000000000E0 */
177 atanl (long double x)
180 long double t, u, p, q;
182 /* Check for zero or NaN. */
183 if (isnanl (x) || x == 0.0)
207 /* Index of nearest table element.
208 Roundoff to integer is asymmetrical to avoid cancellation when t < 0
212 /* Small arctan argument. */
213 t = (x - u) / (1.0 + x * u);
216 /* Arctan of small argument t. */
218 p = ((((p4 * u) + p3) * u + p2) * u + p1) * u + p0;
219 q = ((((u + q4) * u + q3) * u + q2) * u + q1) * u + q0;
220 u = t * u * p / q + t;
222 /* arctan x = arctan u + arctan t */