sysdeps/ieee754/ldbl-128ibm/s_atanl.c

   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     <http://www.gnu.org/licenses/>.  */
  60
  61
  62 #include <math.h>
  63 #include <math_private.h>
  64 #include <math_ldbl_opt.h>
  65
  66 /* arctan(k/8), k = 0, ..., 82 */
  67 static const long double atantbl[84] = {
  68   0.0000000000000000000000000000000000000000E0L,
  69   1.2435499454676143503135484916387102557317E-1L, /* arctan(0.125)  */
  70   2.4497866312686415417208248121127581091414E-1L,
  71   3.5877067027057222039592006392646049977698E-1L,
  72   4.6364760900080611621425623146121440202854E-1L,
  73   5.5859931534356243597150821640166127034645E-1L,
  74   6.4350110879328438680280922871732263804151E-1L,
  75   7.1882999962162450541701415152590465395142E-1L,
  76   7.8539816339744830961566084581987572104929E-1L,
  77   8.4415398611317100251784414827164750652594E-1L,
  78   8.9605538457134395617480071802993782702458E-1L,
  79   9.4200004037946366473793717053459358607166E-1L,
  80   9.8279372324732906798571061101466601449688E-1L,
  81   1.0191413442663497346383429170230636487744E0L,
  82   1.0516502125483736674598673120862998296302E0L,
  83   1.0808390005411683108871567292171998202703E0L,
  84   1.1071487177940905030170654601785370400700E0L,
  85   1.1309537439791604464709335155363278047493E0L,
  86   1.1525719972156675180401498626127513797495E0L,
  87   1.1722738811284763866005949441337046149712E0L,
  88   1.1902899496825317329277337748293183376012E0L,
  89   1.2068173702852525303955115800565576303133E0L,
  90   1.2220253232109896370417417439225704908830E0L,
  91   1.2360594894780819419094519711090786987027E0L,
  92   1.2490457723982544258299170772810901230778E0L,
  93   1.2610933822524404193139408812473357720101E0L,
  94   1.2722973952087173412961937498224804940684E0L,
  95   1.2827408797442707473628852511364955306249E0L,
  96   1.2924966677897852679030914214070816845853E0L,
  97   1.3016288340091961438047858503666855921414E0L,
  98   1.3101939350475556342564376891719053122733E0L,
  99   1.3182420510168370498593302023271362531155E0L,
 100   1.3258176636680324650592392104284756311844E0L,
 101   1.3329603993374458675538498697331558093700E0L,
 102   1.3397056595989995393283037525895557411039E0L,
 103   1.3460851583802539310489409282517796256512E0L,
 104   1.3521273809209546571891479413898128509842E0L,
 105   1.3578579772154994751124898859640585287459E0L,
 106   1.3633001003596939542892985278250991189943E0L,
 107   1.3684746984165928776366381936948529556191E0L,
 108   1.3734007669450158608612719264449611486510E0L,
 109   1.3780955681325110444536609641291551522494E0L,
 110   1.3825748214901258580599674177685685125566E0L,
 111   1.3868528702577214543289381097042486034883E0L,
 112   1.3909428270024183486427686943836432060856E0L,
 113   1.3948567013423687823948122092044222644895E0L,
 114   1.3986055122719575950126700816114282335732E0L,
 115   1.4021993871854670105330304794336492676944E0L,
 116   1.4056476493802697809521934019958079881002E0L,
 117   1.4089588955564736949699075250792569287156E0L,
 118   1.4121410646084952153676136718584891599630E0L,
 119   1.4152014988178669079462550975833894394929E0L,
 120   1.4181469983996314594038603039700989523716E0L,
 121   1.4209838702219992566633046424614466661176E0L,
 122   1.4237179714064941189018190466107297503086E0L,
 123   1.4263547484202526397918060597281265695725E0L,
 124   1.4288992721907326964184700745371983590908E0L,
 125   1.4313562697035588982240194668401779312122E0L,
 126   1.4337301524847089866404719096698873648610E0L,
 127   1.4360250423171655234964275337155008780675E0L,
 128   1.4382447944982225979614042479354815855386E0L,
 129   1.4403930189057632173997301031392126865694E0L,
 130   1.4424730991091018200252920599377292525125E0L,
 131   1.4444882097316563655148453598508037025938E0L,
 132   1.4464413322481351841999668424758804165254E0L,
 133   1.4483352693775551917970437843145232637695E0L,
 134   1.4501726582147939000905940595923466567576E0L,
 135   1.4519559822271314199339700039142990228105E0L,
 136   1.4536875822280323362423034480994649820285E0L,
 137   1.4553696664279718992423082296859928222270E0L,
 138   1.4570043196511885530074841089245667532358E0L,
 139   1.4585935117976422128825857356750737658039E0L,
 140   1.4601391056210009726721818194296893361233E0L,
 141   1.4616428638860188872060496086383008594310E0L,
 142   1.4631064559620759326975975316301202111560E0L,
 143   1.4645314639038178118428450961503371619177E0L,
 144   1.4659193880646627234129855241049975398470E0L,
 145   1.4672716522843522691530527207287398276197E0L,
 146   1.4685896086876430842559640450619880951144E0L,
 147   1.4698745421276027686510391411132998919794E0L,
 148   1.4711276743037345918528755717617308518553E0L,
 149   1.4723501675822635384916444186631899205983E0L,
 150   1.4735431285433308455179928682541563973416E0L, /* arctan(10.25) */
 151   1.5707963267948966192313216916397514420986E0L  /* pi/2 */
 152 };
 153
 154
 155 /* arctan t = t + t^3 p(t^2) / q(t^2)
 156    |t| <= 0.09375
 157    peak relative error 5.3e-37 */
 158
 159 static const long double
 160   p0 = -4.283708356338736809269381409828726405572E1L,
 161   p1 = -8.636132499244548540964557273544599863825E1L,
 162   p2 = -5.713554848244551350855604111031839613216E1L,
 163   p3 = -1.371405711877433266573835355036413750118E1L,
 164   p4 = -8.638214309119210906997318946650189640184E-1L,
 165   q0 = 1.285112506901621042780814422948906537959E2L,
 166   q1 = 3.361907253914337187957855834229672347089E2L,
 167   q2 = 3.180448303864130128268191635189365331680E2L,
 168   q3 = 1.307244136980865800160844625025280344686E2L,
 169   q4 = 2.173623741810414221251136181221172551416E1L;
 170   /* q5 = 1.000000000000000000000000000000000000000E0 */
 171
 172
 173 long double
 174 __atanl (long double x)
 175 {
 176   int32_t k, sign, lx;
 177   long double t, u, p, q;
 178   double xhi;
 179
 180   xhi = ldbl_high (x);
 181   EXTRACT_WORDS (k, lx, xhi);
 182   sign = k & 0x80000000;
 183
 184   /* Check for IEEE special cases.  */
 185   k &= 0x7fffffff;
 186   if (k >= 0x7ff00000)
 187     {
 188       /* NaN. */
 189       if (((k - 0x7ff00000) | lx) != 0)
 190         return (x + x);
 191
 192       /* Infinity. */
 193       if (sign)
 194         return -atantbl[83];
 195       else
 196         return atantbl[83];
 197     }
 198
 199   if (k <= 0x3c800000) /* |x| <= 2**-55.  */
 200     {
 201       /* Raise inexact.  */
 202       if (1e300L + x > 0.0)
 203         return x;
 204     }
 205
 206   if (k >= 0x46c00000) /* |x| >= 2**109.  */
 207     {
 208       /* Saturate result to {-,+}pi/2.  */
 209       if (sign)
 210         return -atantbl[83];
 211       else
 212         return atantbl[83];
 213     }
 214
 215   if (sign)
 216       x = -x;
 217
 218   if (k >= 0x40248000) /* 10.25 */
 219     {
 220       k = 83;
 221       t = -1.0/x;
 222     }
 223   else
 224     {
 225       /* Index of nearest table element.
 226          Roundoff to integer is asymmetrical to avoid cancellation when t < 0
 227          (cf. fdlibm). */
 228       k = 8.0 * x + 0.25;
 229       u = 0.125 * k;
 230       /* Small arctan argument.  */
 231       t = (x - u) / (1.0 + x * u);
 232     }
 233
 234   /* Arctan of small argument t.  */
 235   u = t * t;
 236   p =     ((((p4 * u) + p3) * u + p2) * u + p1) * u + p0;
 237   q = ((((u + q4) * u + q3) * u + q2) * u + q1) * u + q0;
 238   u = t * u * p / q  +  t;
 239
 240   /* arctan x = arctan u  +  arctan t */
 241   u = atantbl[k] + u;
 242   if (sign)
 243     return (-u);
 244   else
 245     return u;
 246 }
 247
 248 long_double_symbol (libm, __atanl, atanl);