atanl.c

/*							s_atanl.c
 *
 *	Inverse circular tangent for 128-bit long double precision
 *      (arctangent)
 *
 *
 *
 * SYNOPSIS:
 *
 * long double x, y, atanl();
 *
 * y = atanl( x );
 *
 *
 *
 * DESCRIPTION:
 *
 * Returns radian angle between -pi/2 and +pi/2 whose tangent is x.
 *
 * The function uses a rational approximation of the form
 * t + t^3 P(t^2)/Q(t^2), optimized for |t| < 0.09375.
 *
 * The argument is reduced using the identity
 *    arctan x - arctan u  =  arctan ((x-u)/(1 + ux))
 * and an 83-entry lookup table for arctan u, with u = 0, 1/8, ..., 10.25.
 * Use of the table improves the execution speed of the routine.
 *
 *
 *
 * ACCURACY:
 *
 *                      Relative error:
 * arithmetic   domain     # trials      peak         rms
 *    IEEE      -19, 19       4e5       1.7e-34     5.4e-35
 *
 *
 * WARNING:
 *
 * This program uses integer operations on bit fields of floating-point
 * numbers.  It does not work with data structures other than the
 * structure assumed.
 *
 */

/* Copyright 2001 by Stephen L. Moshier <moshier@na-net.ornl.gov>  */

#include "mathl.h"

#include <math.h>

/* arctan(k/8), k = 0, ..., 82 */
static const long double atantbl[84] = {
  0.0000000000000000000000000000000000000000E0L,
  1.2435499454676143503135484916387102557317E-1L, /* arctan(0.125)  */
  2.4497866312686415417208248121127581091414E-1L,
  3.5877067027057222039592006392646049977698E-1L,
  4.6364760900080611621425623146121440202854E-1L,
  5.5859931534356243597150821640166127034645E-1L,
  6.4350110879328438680280922871732263804151E-1L,
  7.1882999962162450541701415152590465395142E-1L,
  7.8539816339744830961566084581987572104929E-1L,
  8.4415398611317100251784414827164750652594E-1L,
  8.9605538457134395617480071802993782702458E-1L,
  9.4200004037946366473793717053459358607166E-1L,
  9.8279372324732906798571061101466601449688E-1L,
  1.0191413442663497346383429170230636487744E0L,
  1.0516502125483736674598673120862998296302E0L,
  1.0808390005411683108871567292171998202703E0L,
  1.1071487177940905030170654601785370400700E0L,
  1.1309537439791604464709335155363278047493E0L,
  1.1525719972156675180401498626127513797495E0L,
  1.1722738811284763866005949441337046149712E0L,
  1.1902899496825317329277337748293183376012E0L,
  1.2068173702852525303955115800565576303133E0L,
  1.2220253232109896370417417439225704908830E0L,
  1.2360594894780819419094519711090786987027E0L,
  1.2490457723982544258299170772810901230778E0L,
  1.2610933822524404193139408812473357720101E0L,
  1.2722973952087173412961937498224804940684E0L,
  1.2827408797442707473628852511364955306249E0L,
  1.2924966677897852679030914214070816845853E0L,
  1.3016288340091961438047858503666855921414E0L,
  1.3101939350475556342564376891719053122733E0L,
  1.3182420510168370498593302023271362531155E0L,
  1.3258176636680324650592392104284756311844E0L,
  1.3329603993374458675538498697331558093700E0L,
  1.3397056595989995393283037525895557411039E0L,
  1.3460851583802539310489409282517796256512E0L,
  1.3521273809209546571891479413898128509842E0L,
  1.3578579772154994751124898859640585287459E0L,
  1.3633001003596939542892985278250991189943E0L,
  1.3684746984165928776366381936948529556191E0L,
  1.3734007669450158608612719264449611486510E0L,
  1.3780955681325110444536609641291551522494E0L,
  1.3825748214901258580599674177685685125566E0L,
  1.3868528702577214543289381097042486034883E0L,
  1.3909428270024183486427686943836432060856E0L,
  1.3948567013423687823948122092044222644895E0L,
  1.3986055122719575950126700816114282335732E0L,
  1.4021993871854670105330304794336492676944E0L,
  1.4056476493802697809521934019958079881002E0L,
  1.4089588955564736949699075250792569287156E0L,
  1.4121410646084952153676136718584891599630E0L,
  1.4152014988178669079462550975833894394929E0L,
  1.4181469983996314594038603039700989523716E0L,
  1.4209838702219992566633046424614466661176E0L,
  1.4237179714064941189018190466107297503086E0L,
  1.4263547484202526397918060597281265695725E0L,
  1.4288992721907326964184700745371983590908E0L,
  1.4313562697035588982240194668401779312122E0L,
  1.4337301524847089866404719096698873648610E0L,
  1.4360250423171655234964275337155008780675E0L,
  1.4382447944982225979614042479354815855386E0L,
  1.4403930189057632173997301031392126865694E0L,
  1.4424730991091018200252920599377292525125E0L,
  1.4444882097316563655148453598508037025938E0L,
  1.4464413322481351841999668424758804165254E0L,
  1.4483352693775551917970437843145232637695E0L,
  1.4501726582147939000905940595923466567576E0L,
  1.4519559822271314199339700039142990228105E0L,
  1.4536875822280323362423034480994649820285E0L,
  1.4553696664279718992423082296859928222270E0L,
  1.4570043196511885530074841089245667532358E0L,
  1.4585935117976422128825857356750737658039E0L,
  1.4601391056210009726721818194296893361233E0L,
  1.4616428638860188872060496086383008594310E0L,
  1.4631064559620759326975975316301202111560E0L,
  1.4645314639038178118428450961503371619177E0L,
  1.4659193880646627234129855241049975398470E0L,
  1.4672716522843522691530527207287398276197E0L,
  1.4685896086876430842559640450619880951144E0L,
  1.4698745421276027686510391411132998919794E0L,
  1.4711276743037345918528755717617308518553E0L,
  1.4723501675822635384916444186631899205983E0L,
  1.4735431285433308455179928682541563973416E0L, /* arctan(10.25) */
  1.5707963267948966192313216916397514420986E0L  /* pi/2 */
};


/* arctan t = t + t^3 p(t^2) / q(t^2)
   |t| <= 0.09375
   peak relative error 5.3e-37 */

static const long double
  p0 = -4.283708356338736809269381409828726405572E1L,
  p1 = -8.636132499244548540964557273544599863825E1L,
  p2 = -5.713554848244551350855604111031839613216E1L,
  p3 = -1.371405711877433266573835355036413750118E1L,
  p4 = -8.638214309119210906997318946650189640184E-1L,
  q0 = 1.285112506901621042780814422948906537959E2L,
  q1 = 3.361907253914337187957855834229672347089E2L,
  q2 = 3.180448303864130128268191635189365331680E2L,
  q3 = 1.307244136980865800160844625025280344686E2L,
  q4 = 2.173623741810414221251136181221172551416E1L;
  /* q5 = 1.000000000000000000000000000000000000000E0 */


long double
atanl (long double x)
{
  int k, sign;
  long double t, u, p, q;

  sign = (x < 0) ? -1 : 1;

  /* Check for zero or NaN.  */
  if (x != x || x == 0.0)
    return x + x;

  /* Check for infinity.  */
  if (x + x == x)
    return sign * atantbl[83];

  x *= sign;

  if (x >= 10.25) /* 10.25 */
    {
      k = 83;
      t = -1.0/x;
    }
  else
    {
      /* Index of nearest table element.
	 Roundoff to integer is asymmetrical to avoid cancellation when t < 0
         (cf. fdlibm). */
      k = 8.0 * x + 0.25;
      u = 0.125 * k;
      /* Small arctan argument.  */
      t = (x - u) / (1.0 + x * u);
    }

  /* Arctan of small argument t.  */
  u = t * t;
  p =     ((((p4 * u) + p3) * u + p2) * u + p1) * u + p0;
  q = ((((u + q4) * u + q3) * u + q2) * u + q1) * u + q0;
  u = t * u * p / q  +  t;

  /* arctan x = arctan u  +  arctan t */
  return sign * (atantbl[k] + u);
}