]>
git.ipfire.org Git - thirdparty/glibc.git/blob - sysdeps/ieee754/ldbl-128ibm/s_atanl.c
0e16e74fd652b60cfba1233014c1934bc3d4de5f
3 * Inverse circular tangent for 128-bit long double precision
10 * long double x, y, atanl();
18 * Returns radian angle between -pi/2 and +pi/2 whose tangent is x.
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.
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.
33 * arithmetic domain # trials peak rms
34 * IEEE -19, 19 4e5 1.7e-34 5.4e-35
39 * This program uses integer operations on bit fields of floating-point
40 * numbers. It does not work with data structures other than the
45 /* Copyright 2001 by Stephen L. Moshier <moshier@na-net.ornl.gov>
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.
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.
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/>. */
64 #include <math_private.h>
65 #include <math-underflow.h>
66 #include <math_ldbl_opt.h>
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 */
157 /* arctan t = t + t^3 p(t^2) / q(t^2)
159 peak relative error 5.3e-37 */
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 */
176 __atanl (long double x
)
179 long double t
, u
, p
, q
;
183 EXTRACT_WORDS (k
, lx
, xhi
);
184 sign
= k
& 0x80000000;
186 /* Check for IEEE special cases. */
191 if (((k
- 0x7ff00000) | lx
) != 0)
201 if (k
<= 0x3c800000) /* |x| <= 2**-55. */
203 math_check_force_underflow (x
);
205 if (1e300L
+ x
> 0.0)
209 if (k
>= 0x46c00000) /* |x| >= 2**109. */
211 /* Saturate result to {-,+}pi/2. */
221 if (k
>= 0x40248000) /* 10.25 */
228 /* Index of nearest table element.
229 Roundoff to integer is asymmetrical to avoid cancellation when t < 0
233 /* Small arctan argument. */
234 t
= (x
- u
) / (1.0 + x
* u
);
237 /* Arctan of small argument 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
;
243 /* arctan x = arctan u + arctan t */
251 long_double_symbol (libm
, __atanl
, atanl
);