1 /* s_atanl.c
2 *
3 * Inverse circular tangent for 128-bit long double precision
4 * (arctangent)
5 *
6 *
7 *
8 * SYNOPSIS:
9 *
10 * long double x, y, atanl();
11 *
12 * y = atanl( x );
13 *
14 *
15 *
16 * DESCRIPTION:
17 *
18 * Returns radian angle between -pi/2 and +pi/2 whose tangent is x.
19 *
20 * The function uses a rational approximation of the form
21 * t + t^3 P(t^2)/Q(t^2), optimized for |t| < 0.09375.
22 *
23 * The argument is reduced using the identity
24 * arctan x - arctan u = arctan ((x-u)/(1 + ux))
25 * and an 83-entry lookup table for arctan u, with u = 0, 1/8, ..., 10.25.
26 * Use of the table improves the execution speed of the routine.
27 *
28 *
29 *
30 * ACCURACY:
31 *
32 * Relative error:
33 * arithmetic domain # trials peak rms
34 * IEEE -19, 19 4e5 1.7e-34 5.4e-35
35 *
36 *
37 * WARNING:
38 *
39 * This program uses integer operations on bit fields of floating-point
40 * numbers. It does not work with data structures other than the
41 * structure assumed.
42 *
43 */
44
45 /* Copyright 2001 by Stephen L. Moshier <moshier@na-net.ornl.gov>
46
47 This library is free software; you can redistribute it and/or
48 modify it under the terms of the GNU Lesser General Public
49 License as published by the Free Software Foundation; either
50 version 2.1 of the License, or (at your option) any later version.
51
52 This library is distributed in the hope that it will be useful,
53 but WITHOUT ANY WARRANTY; without even the implied warranty of
54 MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
55 Lesser General Public License for more details.
56
57 You should have received a copy of the GNU Lesser General Public
58 License along with this library; if not, see
59 <https://www.gnu.org/licenses/>. */
60
61
62 #include <float.h>
63 #include <math.h>
64 #include <math_private.h>
65 #include <math-underflow.h>
66 #include <math_ldbl_opt.h>
67
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 */
154 };
155
156
157 /* arctan t = t + t^3 p(t^2) / q(t^2)
158 |t| <= 0.09375
159 peak relative error 5.3e-37 */
160
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 */
173
174
175 long double
176 __atanl (long double x)
177 {
178 int32_t k, sign, lx;
179 long double t, u, p, q;
180 double xhi;
181
182 xhi = ldbl_high (x);
183 EXTRACT_WORDS (k, lx, xhi);
184 sign = k & 0x80000000;
185
186 /* Check for IEEE special cases. */
187 k &= 0x7fffffff;
188 if (k >= 0x7ff00000)
189 {
190 /* NaN. */
191 if (((k - 0x7ff00000) | lx) != 0)
192 return (x + x);
193
194 /* Infinity. */
195 if (sign)
196 return -atantbl[83];
197 else
198 return atantbl[83];
199 }
200
201 if (k <= 0x3c800000) /* |x| <= 2**-55. */
202 {
203 math_check_force_underflow (x);
204 /* Raise inexact. */
205 if (1e300L + x > 0.0)
206 return x;
207 }
208
209 if (k >= 0x46c00000) /* |x| >= 2**109. */
210 {
211 /* Saturate result to {-,+}pi/2. */
212 if (sign)
213 return -atantbl[83];
214 else
215 return atantbl[83];
216 }
217
218 if (sign)
219 x = -x;
220
221 if (k >= 0x40248000) /* 10.25 */
222 {
223 k = 83;
224 t = -1.0/x;
225 }
226 else
227 {
228 /* Index of nearest table element.
229 Roundoff to integer is asymmetrical to avoid cancellation when t < 0
230 (cf. fdlibm). */
231 k = 8.0 * x + 0.25;
232 u = 0.125 * k;
233 /* Small arctan argument. */
234 t = (x - u) / (1.0 + x * u);
235 }
236
237 /* Arctan of small argument t. */
238 u = t * t;
239 p = ((((p4 * u) + p3) * u + p2) * u + p1) * u + p0;
240 q = ((((u + q4) * u + q3) * u + q2) * u + q1) * u + q0;
241 u = t * u * p / q + t;
242
243 /* arctan x = arctan u + arctan t */
244 u = atantbl[k] + u;
245 if (sign)
246 return (-u);
247 else
248 return u;
249 }
250
251 long_double_symbol (libm, __atanl, atanl);