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     <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);