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/* 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>
This library is free software; you can redistribute it and/or
modify it under the terms of the GNU Lesser General Public
License as published by the Free Software Foundation; either
version 2.1 of the License, or (at your option) any later version.
This library is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
Lesser General Public License for more details.
You should have received a copy of the GNU Lesser General Public
License along with this library; if not, see
<http://www.gnu.org/licenses/>. */
#include <float.h>
#include <math.h>
#include <math_private.h>
#include <math-underflow.h>
#include <libm-alias-ldouble.h>
/* arctan(k/8), k = 0, ..., 82 */
static const _Float128 atantbl[84] = {
L(0.0000000000000000000000000000000000000000E0),
L(1.2435499454676143503135484916387102557317E-1), /* arctan(0.125) */
L(2.4497866312686415417208248121127581091414E-1),
L(3.5877067027057222039592006392646049977698E-1),
L(4.6364760900080611621425623146121440202854E-1),
L(5.5859931534356243597150821640166127034645E-1),
L(6.4350110879328438680280922871732263804151E-1),
L(7.1882999962162450541701415152590465395142E-1),
L(7.8539816339744830961566084581987572104929E-1),
L(8.4415398611317100251784414827164750652594E-1),
L(8.9605538457134395617480071802993782702458E-1),
L(9.4200004037946366473793717053459358607166E-1),
L(9.8279372324732906798571061101466601449688E-1),
L(1.0191413442663497346383429170230636487744E0),
L(1.0516502125483736674598673120862998296302E0),
L(1.0808390005411683108871567292171998202703E0),
L(1.1071487177940905030170654601785370400700E0),
L(1.1309537439791604464709335155363278047493E0),
L(1.1525719972156675180401498626127513797495E0),
L(1.1722738811284763866005949441337046149712E0),
L(1.1902899496825317329277337748293183376012E0),
L(1.2068173702852525303955115800565576303133E0),
L(1.2220253232109896370417417439225704908830E0),
L(1.2360594894780819419094519711090786987027E0),
L(1.2490457723982544258299170772810901230778E0),
L(1.2610933822524404193139408812473357720101E0),
L(1.2722973952087173412961937498224804940684E0),
L(1.2827408797442707473628852511364955306249E0),
L(1.2924966677897852679030914214070816845853E0),
L(1.3016288340091961438047858503666855921414E0),
L(1.3101939350475556342564376891719053122733E0),
L(1.3182420510168370498593302023271362531155E0),
L(1.3258176636680324650592392104284756311844E0),
L(1.3329603993374458675538498697331558093700E0),
L(1.3397056595989995393283037525895557411039E0),
L(1.3460851583802539310489409282517796256512E0),
L(1.3521273809209546571891479413898128509842E0),
L(1.3578579772154994751124898859640585287459E0),
L(1.3633001003596939542892985278250991189943E0),
L(1.3684746984165928776366381936948529556191E0),
L(1.3734007669450158608612719264449611486510E0),
L(1.3780955681325110444536609641291551522494E0),
L(1.3825748214901258580599674177685685125566E0),
L(1.3868528702577214543289381097042486034883E0),
L(1.3909428270024183486427686943836432060856E0),
L(1.3948567013423687823948122092044222644895E0),
L(1.3986055122719575950126700816114282335732E0),
L(1.4021993871854670105330304794336492676944E0),
L(1.4056476493802697809521934019958079881002E0),
L(1.4089588955564736949699075250792569287156E0),
L(1.4121410646084952153676136718584891599630E0),
L(1.4152014988178669079462550975833894394929E0),
L(1.4181469983996314594038603039700989523716E0),
L(1.4209838702219992566633046424614466661176E0),
L(1.4237179714064941189018190466107297503086E0),
L(1.4263547484202526397918060597281265695725E0),
L(1.4288992721907326964184700745371983590908E0),
L(1.4313562697035588982240194668401779312122E0),
L(1.4337301524847089866404719096698873648610E0),
L(1.4360250423171655234964275337155008780675E0),
L(1.4382447944982225979614042479354815855386E0),
L(1.4403930189057632173997301031392126865694E0),
L(1.4424730991091018200252920599377292525125E0),
L(1.4444882097316563655148453598508037025938E0),
L(1.4464413322481351841999668424758804165254E0),
L(1.4483352693775551917970437843145232637695E0),
L(1.4501726582147939000905940595923466567576E0),
L(1.4519559822271314199339700039142990228105E0),
L(1.4536875822280323362423034480994649820285E0),
L(1.4553696664279718992423082296859928222270E0),
L(1.4570043196511885530074841089245667532358E0),
L(1.4585935117976422128825857356750737658039E0),
L(1.4601391056210009726721818194296893361233E0),
L(1.4616428638860188872060496086383008594310E0),
L(1.4631064559620759326975975316301202111560E0),
L(1.4645314639038178118428450961503371619177E0),
L(1.4659193880646627234129855241049975398470E0),
L(1.4672716522843522691530527207287398276197E0),
L(1.4685896086876430842559640450619880951144E0),
L(1.4698745421276027686510391411132998919794E0),
L(1.4711276743037345918528755717617308518553E0),
L(1.4723501675822635384916444186631899205983E0),
L(1.4735431285433308455179928682541563973416E0), /* arctan(10.25) */
L(1.5707963267948966192313216916397514420986E0) /* pi/2 */
};
/* arctan t = t + t^3 p(t^2) / q(t^2)
|t| <= 0.09375
peak relative error 5.3e-37 */
static const _Float128
p0 = L(-4.283708356338736809269381409828726405572E1),
p1 = L(-8.636132499244548540964557273544599863825E1),
p2 = L(-5.713554848244551350855604111031839613216E1),
p3 = L(-1.371405711877433266573835355036413750118E1),
p4 = L(-8.638214309119210906997318946650189640184E-1),
q0 = L(1.285112506901621042780814422948906537959E2),
q1 = L(3.361907253914337187957855834229672347089E2),
q2 = L(3.180448303864130128268191635189365331680E2),
q3 = L(1.307244136980865800160844625025280344686E2),
q4 = L(2.173623741810414221251136181221172551416E1);
/* q5 = 1.000000000000000000000000000000000000000E0 */
static const _Float128 huge = L(1.0e4930);
_Float128
__atanl (_Float128 x)
{
int k, sign;
_Float128 t, u, p, q;
ieee854_long_double_shape_type s;
s.value = x;
k = s.parts32.w0;
if (k & 0x80000000)
sign = 1;
else
sign = 0;
/* Check for IEEE special cases. */
k &= 0x7fffffff;
if (k >= 0x7fff0000)
{
/* NaN. */
if ((k & 0xffff) | s.parts32.w1 | s.parts32.w2 | s.parts32.w3)
return (x + x);
/* Infinity. */
if (sign)
return -atantbl[83];
else
return atantbl[83];
}
if (k <= 0x3fc50000) /* |x| < 2**-58 */
{
math_check_force_underflow (x);
/* Raise inexact. */
if (huge + x > 0.0)
return x;
}
if (k >= 0x40720000) /* |x| > 2**115 */
{
/* Saturate result to {-,+}pi/2 */
if (sign)
return -atantbl[83];
else
return atantbl[83];
}
if (sign)
x = -x;
if (k >= 0x40024800) /* 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 = L(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 */
u = atantbl[k] + u;
if (sign)
return (-u);
else
return u;
}
libm_alias_ldouble (__atan, atan)
|
