libquadmath/math/atanq.c

   1 /*                                                      s_atanl.c
   2  *
   3  *      Inverse circular tangent for 128-bit __float128 precision
   4  *      (arctangent)
   5  *
   6  *
   7  *
   8  * SYNOPSIS:
   9  *
  10  * __float128 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, write to the Free Software
  59     Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307  USA */
  60
  61
  62 #include "quadmath-imp.h"
  63
  64 /* arctan(k/8), k = 0, ..., 82 */
  65 static const __float128 atantbl[84] = {
  66   0.0000000000000000000000000000000000000000E0Q,
  67   1.2435499454676143503135484916387102557317E-1Q, /* arctan(0.125)  */
  68   2.4497866312686415417208248121127581091414E-1Q,
  69   3.5877067027057222039592006392646049977698E-1Q,
  70   4.6364760900080611621425623146121440202854E-1Q,
  71   5.5859931534356243597150821640166127034645E-1Q,
  72   6.4350110879328438680280922871732263804151E-1Q,
  73   7.1882999962162450541701415152590465395142E-1Q,
  74   7.8539816339744830961566084581987572104929E-1Q,
  75   8.4415398611317100251784414827164750652594E-1Q,
  76   8.9605538457134395617480071802993782702458E-1Q,
  77   9.4200004037946366473793717053459358607166E-1Q,
  78   9.8279372324732906798571061101466601449688E-1Q,
  79   1.0191413442663497346383429170230636487744E0Q,
  80   1.0516502125483736674598673120862998296302E0Q,
  81   1.0808390005411683108871567292171998202703E0Q,
  82   1.1071487177940905030170654601785370400700E0Q,
  83   1.1309537439791604464709335155363278047493E0Q,
  84   1.1525719972156675180401498626127513797495E0Q,
  85   1.1722738811284763866005949441337046149712E0Q,
  86   1.1902899496825317329277337748293183376012E0Q,
  87   1.2068173702852525303955115800565576303133E0Q,
  88   1.2220253232109896370417417439225704908830E0Q,
  89   1.2360594894780819419094519711090786987027E0Q,
  90   1.2490457723982544258299170772810901230778E0Q,
  91   1.2610933822524404193139408812473357720101E0Q,
  92   1.2722973952087173412961937498224804940684E0Q,
  93   1.2827408797442707473628852511364955306249E0Q,
  94   1.2924966677897852679030914214070816845853E0Q,
  95   1.3016288340091961438047858503666855921414E0Q,
  96   1.3101939350475556342564376891719053122733E0Q,
  97   1.3182420510168370498593302023271362531155E0Q,
  98   1.3258176636680324650592392104284756311844E0Q,
  99   1.3329603993374458675538498697331558093700E0Q,
 100   1.3397056595989995393283037525895557411039E0Q,
 101   1.3460851583802539310489409282517796256512E0Q,
 102   1.3521273809209546571891479413898128509842E0Q,
 103   1.3578579772154994751124898859640585287459E0Q,
 104   1.3633001003596939542892985278250991189943E0Q,
 105   1.3684746984165928776366381936948529556191E0Q,
 106   1.3734007669450158608612719264449611486510E0Q,
 107   1.3780955681325110444536609641291551522494E0Q,
 108   1.3825748214901258580599674177685685125566E0Q,
 109   1.3868528702577214543289381097042486034883E0Q,
 110   1.3909428270024183486427686943836432060856E0Q,
 111   1.3948567013423687823948122092044222644895E0Q,
 112   1.3986055122719575950126700816114282335732E0Q,
 113   1.4021993871854670105330304794336492676944E0Q,
 114   1.4056476493802697809521934019958079881002E0Q,
 115   1.4089588955564736949699075250792569287156E0Q,
 116   1.4121410646084952153676136718584891599630E0Q,
 117   1.4152014988178669079462550975833894394929E0Q,
 118   1.4181469983996314594038603039700989523716E0Q,
 119   1.4209838702219992566633046424614466661176E0Q,
 120   1.4237179714064941189018190466107297503086E0Q,
 121   1.4263547484202526397918060597281265695725E0Q,
 122   1.4288992721907326964184700745371983590908E0Q,
 123   1.4313562697035588982240194668401779312122E0Q,
 124   1.4337301524847089866404719096698873648610E0Q,
 125   1.4360250423171655234964275337155008780675E0Q,
 126   1.4382447944982225979614042479354815855386E0Q,
 127   1.4403930189057632173997301031392126865694E0Q,
 128   1.4424730991091018200252920599377292525125E0Q,
 129   1.4444882097316563655148453598508037025938E0Q,
 130   1.4464413322481351841999668424758804165254E0Q,
 131   1.4483352693775551917970437843145232637695E0Q,
 132   1.4501726582147939000905940595923466567576E0Q,
 133   1.4519559822271314199339700039142990228105E0Q,
 134   1.4536875822280323362423034480994649820285E0Q,
 135   1.4553696664279718992423082296859928222270E0Q,
 136   1.4570043196511885530074841089245667532358E0Q,
 137   1.4585935117976422128825857356750737658039E0Q,
 138   1.4601391056210009726721818194296893361233E0Q,
 139   1.4616428638860188872060496086383008594310E0Q,
 140   1.4631064559620759326975975316301202111560E0Q,
 141   1.4645314639038178118428450961503371619177E0Q,
 142   1.4659193880646627234129855241049975398470E0Q,
 143   1.4672716522843522691530527207287398276197E0Q,
 144   1.4685896086876430842559640450619880951144E0Q,
 145   1.4698745421276027686510391411132998919794E0Q,
 146   1.4711276743037345918528755717617308518553E0Q,
 147   1.4723501675822635384916444186631899205983E0Q,
 148   1.4735431285433308455179928682541563973416E0Q, /* arctan(10.25) */
 149   1.5707963267948966192313216916397514420986E0Q  /* pi/2 */
 150 };
 151
 152
 153 /* arctan t = t + t^3 p(t^2) / q(t^2)
 154    |t| <= 0.09375
 155    peak relative error 5.3e-37 */
 156
 157 static const __float128
 158   p0 = -4.283708356338736809269381409828726405572E1Q,
 159   p1 = -8.636132499244548540964557273544599863825E1Q,
 160   p2 = -5.713554848244551350855604111031839613216E1Q,
 161   p3 = -1.371405711877433266573835355036413750118E1Q,
 162   p4 = -8.638214309119210906997318946650189640184E-1Q,
 163   q0 = 1.285112506901621042780814422948906537959E2Q,
 164   q1 = 3.361907253914337187957855834229672347089E2Q,
 165   q2 = 3.180448303864130128268191635189365331680E2Q,
 166   q3 = 1.307244136980865800160844625025280344686E2Q,
 167   q4 = 2.173623741810414221251136181221172551416E1Q;
 168   /* q5 = 1.000000000000000000000000000000000000000E0 */
 169
 170 static const __float128 huge = 1.0e4930Q;
 171
 172 __float128
 173 atanq (__float128 x)
 174 {
 175   int k, sign;
 176   __float128 t, u, p, q;
 177   ieee854_float128 s;
 178
 179   s.value = x;
 180   k = s.words32.w0;
 181   if (k & 0x80000000)
 182     sign = 1;
 183   else
 184     sign = 0;
 185
 186   /* Check for IEEE special cases.  */
 187   k &= 0x7fffffff;
 188   if (k >= 0x7fff0000)
 189     {
 190       /* NaN. */
 191       if ((k & 0xffff) | s.words32.w1 | s.words32.w2 | s.words32.w3)
 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 <= 0x3fc50000) /* |x| < 2**-58 */
 202     {
 203       math_check_force_underflow (x);
 204       /* Raise inexact. */
 205       if (huge + x > 0.0)
 206         return x;
 207     }
 208
 209   if (k >= 0x40720000) /* |x| > 2**115 */
 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 >= 0x40024800) /* 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 = 8.0Q * x + 0.25Q;
 232       u = 0.125Q * 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 }