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); |
252 | |