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, atanq();
11 *
12 * y = atanq( 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 <http://www.gnu.org/licenses/>. */
60
61 #include "quadmath-imp.h"
62
63 /* arctan(k/8), k = 0, ..., 82 */
64 static const __float128 atantbl[84] = {
65 0.0000000000000000000000000000000000000000E0Q,
66 1.2435499454676143503135484916387102557317E-1Q, /* arctan(0.125) */
67 2.4497866312686415417208248121127581091414E-1Q,
68 3.5877067027057222039592006392646049977698E-1Q,
69 4.6364760900080611621425623146121440202854E-1Q,
70 5.5859931534356243597150821640166127034645E-1Q,
71 6.4350110879328438680280922871732263804151E-1Q,
72 7.1882999962162450541701415152590465395142E-1Q,
73 7.8539816339744830961566084581987572104929E-1Q,
74 8.4415398611317100251784414827164750652594E-1Q,
75 8.9605538457134395617480071802993782702458E-1Q,
76 9.4200004037946366473793717053459358607166E-1Q,
77 9.8279372324732906798571061101466601449688E-1Q,
78 1.0191413442663497346383429170230636487744E0Q,
79 1.0516502125483736674598673120862998296302E0Q,
80 1.0808390005411683108871567292171998202703E0Q,
81 1.1071487177940905030170654601785370400700E0Q,
82 1.1309537439791604464709335155363278047493E0Q,
83 1.1525719972156675180401498626127513797495E0Q,
84 1.1722738811284763866005949441337046149712E0Q,
85 1.1902899496825317329277337748293183376012E0Q,
86 1.2068173702852525303955115800565576303133E0Q,
87 1.2220253232109896370417417439225704908830E0Q,
88 1.2360594894780819419094519711090786987027E0Q,
89 1.2490457723982544258299170772810901230778E0Q,
90 1.2610933822524404193139408812473357720101E0Q,
91 1.2722973952087173412961937498224804940684E0Q,
92 1.2827408797442707473628852511364955306249E0Q,
93 1.2924966677897852679030914214070816845853E0Q,
94 1.3016288340091961438047858503666855921414E0Q,
95 1.3101939350475556342564376891719053122733E0Q,
96 1.3182420510168370498593302023271362531155E0Q,
97 1.3258176636680324650592392104284756311844E0Q,
98 1.3329603993374458675538498697331558093700E0Q,
99 1.3397056595989995393283037525895557411039E0Q,
100 1.3460851583802539310489409282517796256512E0Q,
101 1.3521273809209546571891479413898128509842E0Q,
102 1.3578579772154994751124898859640585287459E0Q,
103 1.3633001003596939542892985278250991189943E0Q,
104 1.3684746984165928776366381936948529556191E0Q,
105 1.3734007669450158608612719264449611486510E0Q,
106 1.3780955681325110444536609641291551522494E0Q,
107 1.3825748214901258580599674177685685125566E0Q,
108 1.3868528702577214543289381097042486034883E0Q,
109 1.3909428270024183486427686943836432060856E0Q,
110 1.3948567013423687823948122092044222644895E0Q,
111 1.3986055122719575950126700816114282335732E0Q,
112 1.4021993871854670105330304794336492676944E0Q,
113 1.4056476493802697809521934019958079881002E0Q,
114 1.4089588955564736949699075250792569287156E0Q,
115 1.4121410646084952153676136718584891599630E0Q,
116 1.4152014988178669079462550975833894394929E0Q,
117 1.4181469983996314594038603039700989523716E0Q,
118 1.4209838702219992566633046424614466661176E0Q,
119 1.4237179714064941189018190466107297503086E0Q,
120 1.4263547484202526397918060597281265695725E0Q,
121 1.4288992721907326964184700745371983590908E0Q,
122 1.4313562697035588982240194668401779312122E0Q,
123 1.4337301524847089866404719096698873648610E0Q,
124 1.4360250423171655234964275337155008780675E0Q,
125 1.4382447944982225979614042479354815855386E0Q,
126 1.4403930189057632173997301031392126865694E0Q,
127 1.4424730991091018200252920599377292525125E0Q,
128 1.4444882097316563655148453598508037025938E0Q,
129 1.4464413322481351841999668424758804165254E0Q,
130 1.4483352693775551917970437843145232637695E0Q,
131 1.4501726582147939000905940595923466567576E0Q,
132 1.4519559822271314199339700039142990228105E0Q,
133 1.4536875822280323362423034480994649820285E0Q,
134 1.4553696664279718992423082296859928222270E0Q,
135 1.4570043196511885530074841089245667532358E0Q,
136 1.4585935117976422128825857356750737658039E0Q,
137 1.4601391056210009726721818194296893361233E0Q,
138 1.4616428638860188872060496086383008594310E0Q,
139 1.4631064559620759326975975316301202111560E0Q,
140 1.4645314639038178118428450961503371619177E0Q,
141 1.4659193880646627234129855241049975398470E0Q,
142 1.4672716522843522691530527207287398276197E0Q,
143 1.4685896086876430842559640450619880951144E0Q,
144 1.4698745421276027686510391411132998919794E0Q,
145 1.4711276743037345918528755717617308518553E0Q,
146 1.4723501675822635384916444186631899205983E0Q,
147 1.4735431285433308455179928682541563973416E0Q, /* arctan(10.25) */
148 1.5707963267948966192313216916397514420986E0Q /* pi/2 */
149 };
150
151
152 /* arctan t = t + t^3 p(t^2) / q(t^2)
153 |t| <= 0.09375
154 peak relative error 5.3e-37 */
155
156 static const __float128
157 p0 = -4.283708356338736809269381409828726405572E1Q,
158 p1 = -8.636132499244548540964557273544599863825E1Q,
159 p2 = -5.713554848244551350855604111031839613216E1Q,
160 p3 = -1.371405711877433266573835355036413750118E1Q,
161 p4 = -8.638214309119210906997318946650189640184E-1Q,
162 q0 = 1.285112506901621042780814422948906537959E2Q,
163 q1 = 3.361907253914337187957855834229672347089E2Q,
164 q2 = 3.180448303864130128268191635189365331680E2Q,
165 q3 = 1.307244136980865800160844625025280344686E2Q,
166 q4 = 2.173623741810414221251136181221172551416E1Q;
167 /* q5 = 1.000000000000000000000000000000000000000E0 */
168
169 static const __float128 huge = 1.0e4930Q;
170
171 __float128
172 atanq (__float128 x)
173 {
174 int k, sign;
175 __float128 t, u, p, q;
176 ieee854_float128 s;
177
178 s.value = x;
179 k = s.words32.w0;
180 if (k & 0x80000000)
181 sign = 1;
182 else
183 sign = 0;
184
185 /* Check for IEEE special cases. */
186 k &= 0x7fffffff;
187 if (k >= 0x7fff0000)
188 {
189 /* NaN. */
190 if ((k & 0xffff) | s.words32.w1 | s.words32.w2 | s.words32.w3)
191 return (x + x);
192
193 /* Infinity. */
194 if (sign)
195 return -atantbl[83];
196 else
197 return atantbl[83];
198 }
199
200 if (k <= 0x3fc50000) /* |x| < 2**-58 */
201 {
202 math_check_force_underflow (x);
203 /* Raise inexact. */
204 if (huge + x > 0.0)
205 return x;
206 }
207
208 if (k >= 0x40720000) /* |x| > 2**115 */
209 {
210 /* Saturate result to {-,+}pi/2 */
211 if (sign)
212 return -atantbl[83];
213 else
214 return atantbl[83];
215 }
216
217 if (sign)
218 x = -x;
219
220 if (k >= 0x40024800) /* 10.25 */
221 {
222 k = 83;
223 t = -1.0/x;
224 }
225 else
226 {
227 /* Index of nearest table element.
228 Roundoff to integer is asymmetrical to avoid cancellation when t < 0
229 (cf. fdlibm). */
230 k = 8.0 * x + 0.25;
231 u = 0.125Q * k;
232 /* Small arctan argument. */
233 t = (x - u) / (1.0 + x * u);
234 }
235
236 /* Arctan of small argument t. */
237 u = t * t;
238 p = ((((p4 * u) + p3) * u + p2) * u + p1) * u + p0;
239 q = ((((u + q4) * u + q3) * u + q2) * u + q1) * u + q0;
240 u = t * u * p / q + t;
241
242 /* arctan x = arctan u + arctan t */
243 u = atantbl[k] + u;
244 if (sign)
245 return (-u);
246 else
247 return u;
248 }