libquadmath/math/atanq.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, atanq();
  11  *
  12  * y = atanq( 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 #include "quadmath-imp.h"
  62
  63 /* arctan(k/8), k = 0, ..., 82 */
  64 static const __float128 atantbl[84] = {
  65   0.0000000000000000000000000000000000000000E0Q,
  66   1.2435499454676143503135484916387102557317E-1Q, /* arctan(0.125)  */
  67   2.4497866312686415417208248121127581091414E-1Q,
  68   3.5877067027057222039592006392646049977698E-1Q,
  69   4.6364760900080611621425623146121440202854E-1Q,
  70   5.5859931534356243597150821640166127034645E-1Q,
  71   6.4350110879328438680280922871732263804151E-1Q,
  72   7.1882999962162450541701415152590465395142E-1Q,
  73   7.8539816339744830961566084581987572104929E-1Q,
  74   8.4415398611317100251784414827164750652594E-1Q,
  75   8.9605538457134395617480071802993782702458E-1Q,
  76   9.4200004037946366473793717053459358607166E-1Q,
  77   9.8279372324732906798571061101466601449688E-1Q,
  78   1.0191413442663497346383429170230636487744E0Q,
  79   1.0516502125483736674598673120862998296302E0Q,
  80   1.0808390005411683108871567292171998202703E0Q,
  81   1.1071487177940905030170654601785370400700E0Q,
  82   1.1309537439791604464709335155363278047493E0Q,
  83   1.1525719972156675180401498626127513797495E0Q,
  84   1.1722738811284763866005949441337046149712E0Q,
  85   1.1902899496825317329277337748293183376012E0Q,
  86   1.2068173702852525303955115800565576303133E0Q,
  87   1.2220253232109896370417417439225704908830E0Q,
  88   1.2360594894780819419094519711090786987027E0Q,
  89   1.2490457723982544258299170772810901230778E0Q,
  90   1.2610933822524404193139408812473357720101E0Q,
  91   1.2722973952087173412961937498224804940684E0Q,
  92   1.2827408797442707473628852511364955306249E0Q,
  93   1.2924966677897852679030914214070816845853E0Q,
  94   1.3016288340091961438047858503666855921414E0Q,
  95   1.3101939350475556342564376891719053122733E0Q,
  96   1.3182420510168370498593302023271362531155E0Q,
  97   1.3258176636680324650592392104284756311844E0Q,
  98   1.3329603993374458675538498697331558093700E0Q,
  99   1.3397056595989995393283037525895557411039E0Q,
 100   1.3460851583802539310489409282517796256512E0Q,
 101   1.3521273809209546571891479413898128509842E0Q,
 102   1.3578579772154994751124898859640585287459E0Q,
 103   1.3633001003596939542892985278250991189943E0Q,
 104   1.3684746984165928776366381936948529556191E0Q,
 105   1.3734007669450158608612719264449611486510E0Q,
 106   1.3780955681325110444536609641291551522494E0Q,
 107   1.3825748214901258580599674177685685125566E0Q,
 108   1.3868528702577214543289381097042486034883E0Q,
 109   1.3909428270024183486427686943836432060856E0Q,
 110   1.3948567013423687823948122092044222644895E0Q,
 111   1.3986055122719575950126700816114282335732E0Q,
 112   1.4021993871854670105330304794336492676944E0Q,
 113   1.4056476493802697809521934019958079881002E0Q,
 114   1.4089588955564736949699075250792569287156E0Q,
 115   1.4121410646084952153676136718584891599630E0Q,
 116   1.4152014988178669079462550975833894394929E0Q,
 117   1.4181469983996314594038603039700989523716E0Q,
 118   1.4209838702219992566633046424614466661176E0Q,
 119   1.4237179714064941189018190466107297503086E0Q,
 120   1.4263547484202526397918060597281265695725E0Q,
 121   1.4288992721907326964184700745371983590908E0Q,
 122   1.4313562697035588982240194668401779312122E0Q,
 123   1.4337301524847089866404719096698873648610E0Q,
 124   1.4360250423171655234964275337155008780675E0Q,
 125   1.4382447944982225979614042479354815855386E0Q,
 126   1.4403930189057632173997301031392126865694E0Q,
 127   1.4424730991091018200252920599377292525125E0Q,
 128   1.4444882097316563655148453598508037025938E0Q,
 129   1.4464413322481351841999668424758804165254E0Q,
 130   1.4483352693775551917970437843145232637695E0Q,
 131   1.4501726582147939000905940595923466567576E0Q,
 132   1.4519559822271314199339700039142990228105E0Q,
 133   1.4536875822280323362423034480994649820285E0Q,
 134   1.4553696664279718992423082296859928222270E0Q,
 135   1.4570043196511885530074841089245667532358E0Q,
 136   1.4585935117976422128825857356750737658039E0Q,
 137   1.4601391056210009726721818194296893361233E0Q,
 138   1.4616428638860188872060496086383008594310E0Q,
 139   1.4631064559620759326975975316301202111560E0Q,
 140   1.4645314639038178118428450961503371619177E0Q,
 141   1.4659193880646627234129855241049975398470E0Q,
 142   1.4672716522843522691530527207287398276197E0Q,
 143   1.4685896086876430842559640450619880951144E0Q,
 144   1.4698745421276027686510391411132998919794E0Q,
 145   1.4711276743037345918528755717617308518553E0Q,
 146   1.4723501675822635384916444186631899205983E0Q,
 147   1.4735431285433308455179928682541563973416E0Q, /* arctan(10.25) */
 148   1.5707963267948966192313216916397514420986E0Q  /* pi/2 */
 149 };
 150
 151
 152 /* arctan t = t + t^3 p(t^2) / q(t^2)
 153    |t| <= 0.09375
 154    peak relative error 5.3e-37 */
 155
 156 static const __float128
 157   p0 = -4.283708356338736809269381409828726405572E1Q,
 158   p1 = -8.636132499244548540964557273544599863825E1Q,
 159   p2 = -5.713554848244551350855604111031839613216E1Q,
 160   p3 = -1.371405711877433266573835355036413750118E1Q,
 161   p4 = -8.638214309119210906997318946650189640184E-1Q,
 162   q0 = 1.285112506901621042780814422948906537959E2Q,
 163   q1 = 3.361907253914337187957855834229672347089E2Q,
 164   q2 = 3.180448303864130128268191635189365331680E2Q,
 165   q3 = 1.307244136980865800160844625025280344686E2Q,
 166   q4 = 2.173623741810414221251136181221172551416E1Q;
 167   /* q5 = 1.000000000000000000000000000000000000000E0 */
 168
 169 static const __float128 huge = 1.0e4930Q;
 170
 171 __float128
 172 atanq (__float128 x)
 173 {
 174   int k, sign;
 175   __float128 t, u, p, q;
 176   ieee854_float128 s;
 177
 178   s.value = x;
 179   k = s.words32.w0;
 180   if (k & 0x80000000)
 181     sign = 1;
 182   else
 183     sign = 0;
 184
 185   /* Check for IEEE special cases.  */
 186   k &= 0x7fffffff;
 187   if (k >= 0x7fff0000)
 188     {
 189       /* NaN. */
 190       if ((k & 0xffff) | s.words32.w1 | s.words32.w2 | s.words32.w3)
 191         return (x + x);
 192
 193       /* Infinity. */
 194       if (sign)
 195         return -atantbl[83];
 196       else
 197         return atantbl[83];
 198     }
 199
 200   if (k <= 0x3fc50000) /* |x| < 2**-58 */
 201     {
 202       math_check_force_underflow (x);
 203       /* Raise inexact. */
 204       if (huge + x > 0.0)
 205         return x;
 206     }
 207
 208   if (k >= 0x40720000) /* |x| > 2**115 */
 209     {
 210       /* Saturate result to {-,+}pi/2 */
 211       if (sign)
 212         return -atantbl[83];
 213       else
 214         return atantbl[83];
 215     }
 216
 217   if (sign)
 218       x = -x;
 219
 220   if (k >= 0x40024800) /* 10.25 */
 221     {
 222       k = 83;
 223       t = -1.0/x;
 224     }
 225   else
 226     {
 227       /* Index of nearest table element.
 228          Roundoff to integer is asymmetrical to avoid cancellation when t < 0
 229          (cf. fdlibm). */
 230       k = 8.0 * x + 0.25;
 231       u = 0.125Q * k;
 232       /* Small arctan argument.  */
 233       t = (x - u) / (1.0 + x * u);
 234     }
 235
 236   /* Arctan of small argument t.  */
 237   u = t * t;
 238   p =     ((((p4 * u) + p3) * u + p2) * u + p1) * u + p0;
 239   q = ((((u + q4) * u + q3) * u + q2) * u + q1) * u + q0;
 240   u = t * u * p / q  +  t;
 241
 242   /* arctan x = arctan u  +  arctan t */
 243   u = atantbl[k] + u;
 244   if (sign)
 245     return (-u);
 246   else
 247     return u;
 248 }