xref: /illumos-gate/usr/src/lib/libm/common/complex/k_atan2l.c (revision 25c28e83)

/*
 * CDDL HEADER START
 *
 * The contents of this file are subject to the terms of the
 * Common Development and Distribution License (the "License").
 * You may not use this file except in compliance with the License.
 *
 * You can obtain a copy of the license at usr/src/OPENSOLARIS.LICENSE
 * or http://www.opensolaris.org/os/licensing.
 * See the License for the specific language governing permissions
 * and limitations under the License.
 *
 * When distributing Covered Code, include this CDDL HEADER in each
 * file and include the License file at usr/src/OPENSOLARIS.LICENSE.
 * If applicable, add the following below this CDDL HEADER, with the
 * fields enclosed by brackets "[]" replaced with your own identifying
 * information: Portions Copyright [yyyy] [name of copyright owner]
 *
 * CDDL HEADER END
 */
/*
 * Copyright 2011 Nexenta Systems, Inc.  All rights reserved.
 */
/*
 * Copyright 2005 Sun Microsystems, Inc.  All rights reserved.
 * Use is subject to license terms.
 */

#include "libm.h"		/* __k_atan2l */
#include "complex_wrapper.h"

#if defined(__sparc)
#define	HALF(x)  ((int *) &x)[3] = 0; ((int *) &x)[2] &= 0xfe000000
#elif defined(__x86)
#define	HALF(x)  ((int *) &x)[0] = 0
#endif

/*
 * long double __k_atan2l(long double y, long double x, long double *e)
 *
 * Compute atan2l with error terms.
 *
 * Important formula:
 *                  3       5
 *                 x       x
 * atan(x) = x - ----- + ----- - ...	(for x <= 1)
 *                 3       5
 *
 *           pi     1     1
 *         = --- - --- + --- - ...	(for x > 1)
 *                         3
 *            2     x    3x
 *
 * Arg(x + y i) = sign(y) * atan2(|y|, x)
 *              = sign(y) * atan(|y|/x)		  (for x > 0)
 *                sign(y) * (PI - atan(|y|/|x|))  (for x < 0)
 * Thus if x >> y (IEEE double: EXP(x) - EXP(y) >= 60):
 *	1. (x > 0): atan2(y,x) ~ y/x
 *	2. (x < 0): atan2(y,x) ~ sign(y) (PI - |y/x|))
 * Otherwise if x << y:
 *	atan2(y,x) ~ sign(y)*PI/2 - x/y
 *
 * __k_atan2l call static functions mx_polyl, mx_atanl
 */


/*
 * (void) mx_polyl (long double *z, long double *a, long double *e, int n)
 * return
 *	e = a  + z*(a  + z*(a  + ... z*(a  + e)...))
 *	     0       2       4           2n
 * Note:
 * 1.	e and coefficient ai are represented by two long double numbers.
 *	For e, the first one contain the leading 53 bits (30 for x86 exteneded)
 *      and the second one contain the remaining 113 bits (64 for x86 extended).
 *	For ai, the first one contian the leading 53 bits (or 30 for x86)
 *      rounded, and the second is the remaining 113 bits (or 64 for x86).
 * 2.	z is an array of three doubles.
 * 	z[0] :	the rounded value of Z (the intended value of z)
 * 	z[1] :	the leading 32 (or 56) bits of Z rounded
 * 	z[2] :	the remaining 113 (or 64) bits of Z
 *		Note that z[0] = z[1]+z[2] rounded.
 *
 */

static void
mx_polyl(const long double *z, const long double *a, long double *e, int n) {
	long double r, s, t, p_h, p_l, z_h, z_l, p, w;
	int i;
	n = n + n;
	p = e[0] + a[n];
	p_l = a[n + 1];
	w = p; HALF(w);
	p_h = w;
	p   = a[n - 2] + z[0] * p;
	z_h = z[1]; z_l = z[2];
	p_l += e[0] - (p_h - a[n]);

	for (i = n - 2; i >= 2; i -= 2) {

	/* compute p = ai + z * p */
		t = z_h * p_h;
		s = z[0] * p_l + p_h * z_l;
		w = p;  HALF(w);
		p_h = w;
		s  += a[i + 1];
		r   = t - (p_h - a[i]);
		p   = a[i - 2] + z[0] * p;
		p_l = r + s;
	}
	w = p; HALF(w);
	e[0] = w;
	t   = z_h * p_h;
	s   = z[0] * p_l + p_h * z_l;
	r   = t - (e[0] - a[0]);
	e[1] = r + s;
}

/*
 * Table of constants for atan from 0.125 to 8
 *	0.125 -- 0x3ffc0000  --- (increment at bit 12)
 *		 0x3ffc1000
 *		 0x3ffc2000
 *	...	...
 *		 0x4001f000
 *	8.000 -- 0x40020000	 (total: 97)
 */

static const long double TBL_atan_hil[] = {
#if defined(__sparc)
1.2435499454676143503135484916387102416568e-01L,
1.3203976161463874927468440652656953226250e-01L,
1.3970887428916364518336777673909505681607e-01L,
1.4736148108865163560980276039684551821066e-01L,
1.5499674192394098230371437493349219133371e-01L,
1.6261382859794857537364156376155780062019e-01L,
1.7021192528547440449049660709976171369543e-01L,
1.7779022899267607079662479921582468899456e-01L,
1.8534794999569476488602596122854464667261e-01L,
1.9288431225797466419705871069022730349878e-01L,
2.0039855382587851465394578503437838446153e-01L,
2.0788992720226299360533498310299432475629e-01L,
2.1535769969773804802445962716648964165745e-01L,
2.2280115375939451577103212214043255525024e-01L,
2.3021958727684373024017095967980299065551e-01L,
2.3761231386547125247388363432563777919892e-01L,
2.4497866312686415417208248121127580641959e-01L,
2.5962962940825753102994644318397190560106e-01L,
2.7416745111965879759937189834217578592444e-01L,
2.8858736189407739562361141995821834504332e-01L,
3.0288486837497140556055609450555821812277e-01L,
3.1705575320914700980901557667446732975852e-01L,
3.3109607670413209494433878775694455421259e-01L,
3.4500217720710510886768128690005168408290e-01L,
3.5877067027057222039592006392646052215363e-01L,
3.7239844667675422192365503828370182641413e-01L,
3.8588266939807377589769548460723139638186e-01L,
3.9922076957525256561471669615886476491104e-01L,
4.1241044159738730689979128966712694260920e-01L,
4.2544963737004228954226360518079233013817e-01L,
4.3833655985795780544561604921477130895882e-01L,
4.5106965598852347637563925728219344073798e-01L,
4.6364760900080611621425623146121439713344e-01L,
4.8833395105640552386716496074706484459644e-01L,
5.1238946031073770666660102058425923805558e-01L,
5.3581123796046370026908506870769144698471e-01L,
5.5859931534356243597150821640166122875873e-01L,
5.8075635356767039920327447500150082375122e-01L,
6.0228734613496418168212269420423291922459e-01L,
6.2319932993406593099247534906037459367793e-01L,
6.4350110879328438680280922871732260447265e-01L,
6.6320299270609325536325431023827583417226e-01L,
6.8231655487474807825642998171115298784729e-01L,
7.0085440788445017245795128178675127318623e-01L,
7.1882999962162450541701415152590469891043e-01L,
7.3625742898142813174283527108914662479274e-01L,
7.5315128096219438952473937026902888600575e-01L,
7.6952648040565826040682003598565401726598e-01L,
7.8539816339744830961566084581987569936977e-01L,
8.1569192331622341102146083874564582672284e-01L,
8.4415398611317100251784414827164746738632e-01L,
8.7090345707565295314017311259781407291650e-01L,
8.9605538457134395617480071802993779546602e-01L,
9.1971960535041681722860345482108940969311e-01L,
9.4200004037946366473793717053459362115891e-01L,
9.6299433068093620181519583599709989677298e-01L,
9.8279372324732906798571061101466603762572e-01L,
1.0014831356942347329183295953014374896343e+00L,
1.0191413442663497346383429170230636212354e+00L,
1.0358412530088001765846944703254440735476e+00L,
1.0516502125483736674598673120862999026920e+00L,
1.0666303653157435630791763474202799086015e+00L,
1.0808390005411683108871567292171997859003e+00L,
1.0943289073211899198927883146102352763033e+00L,
1.1071487177940905030170654601785370497543e+00L,
1.1309537439791604464709335155363277560026e+00L,
1.1525719972156675180401498626127514672834e+00L,
1.1722738811284763866005949441337046006865e+00L,
1.1902899496825317329277337748293182803384e+00L,
1.2068173702852525303955115800565576625682e+00L,
1.2220253232109896370417417439225704120294e+00L,
1.2360594894780819419094519711090786146210e+00L,
1.2490457723982544258299170772810900483550e+00L,
1.2610933822524404193139408812473357640124e+00L,
1.2722973952087173412961937498224805746463e+00L,
1.2827408797442707473628852511364955164072e+00L,
1.2924966677897852679030914214070816723528e+00L,
1.3016288340091961438047858503666855024453e+00L,
1.3101939350475556342564376891719053437537e+00L,
1.3182420510168370498593302023271363040427e+00L,
1.3258176636680324650592392104284756886164e+00L,
1.3397056595989995393283037525895557850243e+00L,
1.3521273809209546571891479413898127598774e+00L,
1.3633001003596939542892985278250991560269e+00L,
1.3734007669450158608612719264449610604836e+00L,
1.3825748214901258580599674177685685163955e+00L,
1.3909428270024183486427686943836432395486e+00L,
1.3986055122719575950126700816114282727858e+00L,
1.4056476493802697809521934019958080664406e+00L,
1.4121410646084952153676136718584890852820e+00L,
1.4181469983996314594038603039700988632607e+00L,
1.4237179714064941189018190466107297108905e+00L,
1.4288992721907326964184700745371984001389e+00L,
1.4337301524847089866404719096698873880264e+00L,
1.4382447944982225979614042479354816039669e+00L,
1.4424730991091018200252920599377291810352e+00L,
1.4464413322481351841999668424758803866109e+00L,
#elif defined(__x86)
1.243549945356789976358413696289e-01L,  1.320397615781985223293304443359e-01L,
1.397088742814958095550537109375e-01L,  1.473614810383878648281097412109e-01L,
1.549967419123277068138122558594e-01L,  1.626138285500928759574890136719e-01L,
1.702119252295233309268951416016e-01L,  1.777902289759367704391479492188e-01L,
1.853479499695822596549987792969e-01L,  1.928843122441321611404418945312e-01L,
2.003985538030974566936492919922e-01L,  2.078899272019043564796447753906e-01L,
2.153576996643096208572387695312e-01L,  2.228011537226848304271697998047e-01L,
2.302195872762240469455718994141e-01L,  2.376123138237744569778442382812e-01L,
2.449786631041206419467926025391e-01L,  2.596296293195337057113647460938e-01L,
2.741674510762095451354980468750e-01L,  2.885873618070036172866821289062e-01L,
3.028848683461546897888183593750e-01L,  3.170557531993836164474487304688e-01L,
3.310960766393691301345825195312e-01L,  3.450021771714091300964355468750e-01L,
3.587706702528521418571472167969e-01L,  3.723984466632828116416931152344e-01L,
3.858826693613082170486450195312e-01L,  3.992207695264369249343872070312e-01L,
4.124104415532201528549194335938e-01L,  4.254496373469009995460510253906e-01L,
4.383365598041564226150512695312e-01L,  4.510696559445932507514953613281e-01L,
4.636476089945062994956970214844e-01L,  4.883339509833604097366333007812e-01L,
5.123894601128995418548583984375e-01L,  5.358112377580255270004272460938e-01L,
5.585993151180446147918701171875e-01L,  5.807563534472137689590454101562e-01L,
6.022873460315167903900146484375e-01L,  6.231993297114968299865722656250e-01L,
6.435011087451130151748657226562e-01L,  6.632029926404356956481933593750e-01L,
6.823165547102689743041992187500e-01L,  7.008544078562408685684204101562e-01L,
7.188299994450062513351440429688e-01L,  7.362574287690222263336181640625e-01L,
7.531512808054685592651367187500e-01L,  7.695264802314341068267822265625e-01L,
7.853981633670628070831298828125e-01L,  8.156919232569634914398193359375e-01L,
8.441539860796183347702026367188e-01L,  8.709034570492804050445556640625e-01L,
8.960553845390677452087402343750e-01L,  9.197196052409708499908447265625e-01L,
9.420000403188169002532958984375e-01L,  9.629943305626511573791503906250e-01L,
9.827937232330441474914550781250e-01L,  1.001483135391026735305786132812e+00L,
1.019141343887895345687866210938e+00L,  1.035841252654790878295898437500e+00L,
1.051650212146341800689697265625e+00L,  1.066630364861339330673217773438e+00L,
1.080839000176638364791870117188e+00L,  1.094328907318413257598876953125e+00L,
1.107148717623203992843627929688e+00L,  1.130953743588179349899291992188e+00L,
1.152571997139602899551391601562e+00L,  1.172273880802094936370849609375e+00L,
1.190289949532598257064819335938e+00L,  1.206817369908094406127929687500e+00L,
1.222025323193520307540893554688e+00L,  1.236059489194303750991821289062e+00L,
1.249045772012323141098022460938e+00L,  1.261093381792306900024414062500e+00L,
1.272297394927591085433959960938e+00L,  1.282740879338234663009643554688e+00L,
1.292496667709201574325561523438e+00L,  1.301628833636641502380371093750e+00L,
1.310193934943526983261108398438e+00L,  1.318242050707340240478515625000e+00L,
1.325817663222551345825195312500e+00L,  1.339705659542232751846313476562e+00L,
1.352127380669116973876953125000e+00L,  1.363300099968910217285156250000e+00L,
1.373400766868144273757934570312e+00L,  1.382574821356683969497680664062e+00L,
1.390942826867103576660156250000e+00L,  1.398605511989444494247436523438e+00L,
1.405647648964077234268188476562e+00L,  1.412141064181923866271972656250e+00L,
1.418146998155862092971801757812e+00L,  1.423717970959842205047607421875e+00L,
1.428899271879345178604125976562e+00L,  1.433730152435600757598876953125e+00L,
1.438244794495403766632080078125e+00L,  1.442473099101334810256958007812e+00L,
1.446441331878304481506347656250e+00L,
#endif
};
static const long double TBL_atan_lol[] = {
#if defined(__sparc)
1.4074869197628063802317202820414310039556e-36L,
-4.9596961594739925555730439437999675295505e-36L,
8.9527745625194648873931213446361849472788e-36L,
1.1880437423207895718180765843544965589427e-35L,
-2.7810278112045145378425375128234365381448e-37L,
1.4797220377023800327295536234315147262387e-36L,
-4.2169561400548198732870384801849639863829e-36L,
7.2431229666913484649930323656316023494680e-36L,
-2.1573430089839170299895679353790663182462e-36L,
-9.9515745405126723554452367298128605186305e-36L,
-3.9065558992324838181617569730397882363067e-36L,
5.5260292271793726813211980664661124518807e-36L,
8.8415722215914321807682254318036452043689e-36L,
-8.1767728791586179254193323628285599800711e-36L,
-1.3344123034656142243797113823028330070762e-36L,
-4.4927331207813382908930733924681325892188e-36L,
4.4945511471812490393201824336762495687730e-36L,
-1.6688081504279223555776724459648440567274e-35L,
1.5629757586107955769461086568937329684113e-35L,
-2.2389835563308078552507970385331510848109e-35L,
-4.8312321745547311551870450671182151367050e-36L,
-1.4336172352905832876958926610980698844309e-35L,
-8.7440181998899932802989174170960593316080e-36L,
5.9284636008529837445780360785464550143016e-36L,
-2.2376651248436241276061055295043514993630e-35L,
6.0745837599336105414280310756677442136480e-36L,
1.5372187110451949677792344762029967023093e-35L,
2.0976068056751156241657121582478790247159e-35L,
-5.5623956405495438060726862202622807523700e-36L,
1.9697366707832471841858411934897351901523e-35L,
2.1070311964479488509034733639424887543697e-35L,
-2.3027356362982001602256518510854229844561e-35L,
4.8950964225733349266861843522029764772843e-36L,
-7.2380143477794458213872723050820253166391e-36L,
1.6365648865703614031637443396049568858105e-35L,
-3.9885811958234530793729129919803234197399e-35L,
4.1587722120912613510417783923227421336929e-35L,
3.8347421454556472153684687377337135027394e-35L,
-9.2251178933638721723515896465489002497864e-36L,
1.4094619690455989526175736741854656192178e-36L,
3.3568857805472235270612851425810803679451e-35L,
3.9090991055522552395018106803232118803401e-35L,
5.2956416979654208140521862707297033857956e-36L,
-5.0960846819945514367847063923662507136721e-36L,
-4.4959014425277615858329680393918315204998e-35L,
3.8039226544551634266566857615962609653834e-35L,
-4.4056522872895512108308642196611689657618e-36L,
1.6025024192482161076223807753425619076948e-36L,
2.1679525325309452561992610065108380635264e-35L,
1.9844038013515422125715362925736754104066e-35L,
3.9139619471799746834505227353568432457241e-35L,
2.1113443807975453505518453436799561854730e-35L,
3.1558557277444692755039816944392770185432e-35L,
1.6295044520355461408265585619500238335614e-35L,
-3.5087245209270305856151230356171213582305e-35L,
2.9041041864282855679591055270946117300088e-35L,
-2.3128843453818356590931995209806627233282e-35L,
-7.7124923181471578439967973820714857839953e-35L,
2.7539027829886922429092063590445808781462e-35L,
-9.4500899453181308951084545990839335972452e-35L,
-7.3061755302032092337594946001641651543473e-35L,
-4.1736144813953752193952770157406952602798e-35L,
3.4369948356256407045344855262863733571105e-35L,
-6.3790243492298090907302084924276831116460e-35L,
-9.6842943816353261291004127866079538980649e-36L,
4.8746757539138870909275958326700072821615e-35L,
-8.7533886477084190884511601368582548254655e-35L,
1.4284743992327918892692551138086727754845e-35L,
5.7262776211073389542565625693479173445042e-35L,
-3.2254883148780411245594822270747948565684e-35L,
7.8853548190609877325965525252380833808405e-35L,
8.4081736739037194097515038365370730251333e-35L,
7.4722870357563683815078242981933587273670e-35L,
7.9977202825793435289434813600890494256112e-36L,
-8.0577840773362139054848492346292673645405e-35L,
1.4217746753670583065490040209048757624336e-35L,
1.2232486914221205004109743560319090913328e-35L,
8.9696055070830036447361957217943988339065e-35L,
-3.1480394435081884410686066739846269858951e-35L,
-5.0927146040715345013240642517608928352977e-35L,
-5.7431997715924136568133859432702789493569e-35L,
-4.3920451405083770279099766080476485439987e-35L,
9.1106753984907715563018666776308759323326e-35L,
-3.7032569014272841009512400773061537538358e-35L,
8.8167419429746714276909825405131416764489e-35L,
-3.8389341696028352503752312861740895209678e-36L,
-3.3462959341960891546340895508017603408404e-35L,
-3.9212626776786074383916188498955828634947e-35L,
-7.8340397396377867255864494568594088378648e-35L,
7.4681018632456986520600640340627309824469e-35L,
8.9110918618956918451135594876165314884113e-35L,
3.9418160632271890530431797145664308529115e-35L,
-4.1048114088580104820193435638327617443913e-35L,
-2.3165419451582153326383944756220900454330e-35L,
-1.8428312581525319409399330203703211113843e-35L,
7.1477316546709482345411712017906842769961e-35L,
2.9914501578435874662153637707016094237004e-35L,
#elif defined(__x86)
1.108243739551347953496477557317e-11L,  3.644022694535396219063202730280e-11L,
7.667835628314065801595065768845e-12L,  5.026377078169301918590803009109e-11L,
1.161327548990211907411719105561e-11L,  4.785569941615255008968280209991e-11L,
5.595107356360146549819920947848e-11L,  1.673930035747684999707469623769e-11L,
2.611250523102718193166964451527e-11L,  1.384250305661681615897729354721e-11L,
2.278105796029649304219088055497e-11L,  3.586371256902077123693302823191e-13L,
3.342842716722085763523965049902e-11L,  3.670968534386232233574504707347e-11L,
6.196832945990602657404893210974e-13L,  4.169679549603939604438777470618e-11L,
2.274351222528987867221331091414e-11L,  8.872382531858169709022188891298e-11L,
4.344925246387385146717580155420e-11L,  8.707377833692929105196832265348e-11L,
2.881671577173773513055821329154e-11L,  9.763393361566846205717315422347e-12L,
6.476296480975626822569454546857e-11L,  3.569597877124574002505169001136e-11L,
1.772007853877284712958549977698e-11L,  1.347141028196192304932683248872e-11L,
3.676555884905046507598141175404e-11L,  4.881564068032948912761478588710e-11L,
4.416715404487185607337693704681e-11L,  2.314128999621257979016734983553e-11L,
5.380138283056477968352133002913e-11L,  4.393022562414389595406841771063e-11L,
6.299816718559209976839402028537e-12L,  7.304511413053165996581483735843e-11L,
1.978381648117426221467592544212e-10L,  2.024381732686578226139414070989e-10L,
2.255178211796380992141612703464e-10L,  1.204566302442290648452508620986e-10L,
1.034473912921080457667329099995e-10L,  2.225691010059030834353745950874e-10L,
4.817137162794350606107263804151e-11L,  6.565755971506095086327587326326e-11L,
1.644791039522307629611529931429e-10L,  2.820930388953087163050126809014e-11L,
1.766182540818701085571546539514e-10L,  2.124059054092171070266466628320e-10L,
1.567258302596026515190288816001e-10L,  1.742241535800378094231540188685e-10L,
3.038550253253096300737572104929e-11L,  5.925991958164150280814584656688e-11L,
3.355266774764151155289750652594e-11L,  2.637254809561744853531409402995e-11L,
3.227621096606048365493782702458e-11L,  1.094459672377587282585894259882e-10L,
6.064676448464127209709358607166e-11L,  1.182850444360454453720999258140e-10L,
1.428492049425553288966601449688e-11L,  3.032079976125434624889374125094e-10L,
3.784543889504767060855636487744e-10L,  3.540092982887960328254439790467e-10L,
4.020318667701700464612998296302e-10L,  4.544042324059585739827798668654e-10L,
3.645299460952866120296998202703e-10L,  2.776662293911361485235212513020e-12L,
1.708865101734375304910370400700e-10L,  3.909810965716415233488278047493e-10L,
7.606461848875826105025137974947e-11L,  3.263814502297453347587046149712e-10L,
1.499334758629144388918183376012e-10L,  3.771581242675818925565576303133e-10L,
1.746932950084818923507049088298e-11L,  2.837781909176306820465786987027e-10L,
3.859312847318946163435901230778e-10L,  4.601335192895268187473357720101e-10L,
2.811262558622337888849804940684e-10L,  4.060360843532416964489955306249e-10L,
8.058369357752989796958168458531e-11L,  3.725546414244147566166855921414e-10L,
1.040286509953292907344053122733e-10L,  3.094968093808145773271362531155e-10L,
4.454811192340438979284756311844e-10L,  5.676678748199027602705574110388e-11L,
2.518376833121948163898128509842e-10L,  3.907837370041422778250991189943e-10L,
7.687158710333735613246114865100e-11L,  1.334418885622867537060685125566e-10L,
1.353147719826124443836432060856e-10L,  2.825131007652335581739282335732e-10L,
4.161925466840049254333079881002e-10L,  4.265713490956410156084891599630e-10L,
2.437693664320585461575989523716e-10L,  4.466519138542116247357297503086e-10L,
3.113875178143440979746983590908e-10L,  4.910822904159495654488736486097e-11L,
2.818831329324169810481585538618e-12L,  7.767009768334052125229252512543e-12L,
3.698307026936191862258804165254e-10L,
#endif
};

/*
 * mx_atanl(x, err)
 * Table look-up algorithm
 * By K.C. Ng, March 9, 1989
 *
 * Algorithm.
 *
 * The algorithm is based on atan(x)=atan(y)+atan((x-y)/(1+x*y)).
 * We use poly1(x) to approximate atan(x) for x in [0,1/8] with
 * error (relative)
 * 	|(atan(x)-poly1(x))/x|<= 2^-140
 *
 * and use poly2(x) to approximate atan(x) for x in [0,1/65] with
 * error
 *	|atan(x)-poly2(x)|<= 2^-143.7
 *
 * Here poly1 and poly2 are odd polynomial with the following form:
 *		x + x^3*(a1+x^2*(a2+...))
 *
 * (0). Purge off Inf and NaN and 0
 * (1). Reduce x to positive by atan(x) = -atan(-x).
 * (2). For x <= 1/8, use
 *	(2.1) if x < 2^(-prec/2), atan(x) =  x  with inexact flag raised
 *	(2.2) Otherwise
 *		atan(x) = poly1(x)
 * (3). For x >= 8 then (prec = 78)
 *	(3.1) if x >= 2^prec,     atan(x) = atan(inf) - pio2_lo
 *	(3.2) if x >= 2^(prec/3), atan(x) = atan(inf) - 1/x
 *	(3.3) if x >  65,         atan(x) = atan(inf) - poly2(1/x)
 *	(3.4) Otherwise,	  atan(x) = atan(inf) - poly1(1/x)
 *
 * (4). Now x is in (0.125, 8)
 *      Find y that match x to 4.5 bit after binary (easy).
 *	If iy is the high word of y, then
 *		single : j = (iy - 0x3e000000) >> 19
 *		double : j = (iy - 0x3fc00000) >> 16
 *		quad   : j = (iy - 0x3ffc0000) >> 12
 *
 *	Let s = (x-y)/(1+x*y). Then
 *	atan(x) = atan(y) + poly1(s)
 *		= _TBL_atan_hi[j] + (_TBL_atan_lo[j] + poly2(s) )
 *
 *	Note. |s| <= 1.5384615385e-02 = 1/65. Maxium occurs at x = 1.03125
 *
 */

/*
 * p[0] - p[16] for atan(x) =
 *		x + x^3*(p1+x^2*(p2+...))
 */
static const long double pe[] = {
	1.0L,
	0.0L,
#if defined(__sparc)
	-0.33333333333333332870740406406184774823L,
	-4.62592926927148558508441072595508240609e-18L,
	0.19999999999999999722444243843710864894L,
	2.77555756156289124602047010782090464486e-18L,
	-0.14285714285714285615158658515611023176L,
	-9.91270557700756738621231719241800559409e-19L,
#elif defined(__x86)
	-0.33333333325572311878204345703125L,
	-7.76102145512898763020833333192787755766644373e-11L,
	0.19999999995343387126922607421875L,
	4.65661287307739257812498949613909375938538636e-11L,
	-0.142857142840512096881866455078125L,
	-1.66307602609906877787419703858463013035681375e-11L,
#endif
};

static const long double p[] = { /* p[0] - p[16] */
	1.0L,
	-3.33333333333333333333333333333333333319278775586e-0001L,
	1.99999999999999999999999999999999894961390937601e-0001L,
	-1.42857142857142857142857142856866970385846301312e-0001L,
	1.11111111111111111111111110742899094415954427738e-0001L,
	-9.09090909090909090909087972707015549231951421806e-0002L,
	7.69230769230769230767699003016385628597359717046e-0002L,
	-6.66666666666666666113842763495291228025226575259e-0002L,
	5.88235294117646915706902204947653640091126695962e-0002L,
	-5.26315789473657016886225044679594035524579379810e-0002L,
	4.76190476186633969331771169790375592681525481267e-0002L,
	-4.34782608290146274616081389793141896576997370161e-0002L,
	3.99999968161267722260103962788865225205057218988e-0002L,
	-3.70368536844778256320786172745225703228683638328e-0002L,
	3.44752320396524479494062858284036892703898522150e-0002L,
	-3.20491216046653214683721787776813360591233428081e-0002L,
	2.67632651033434456758550618122802167256870856514e-0002L,
};

/* q[0] - q[9] */
static const long double qe[] = {
	1.0L,
	0.0L,
#if defined(__sparc)
	-0.33333333333333332870740406406184774823486804962158203125L,
	-4.625929269271485585069345465471207312531868714634217630e-18L,
	0.19999999999999999722444243843710864894092082977294921875L,
	2.7755575615628864268260553912956813621977220359134667560e-18L,
#elif defined(__x86)
	-0.33333333325572311878204345703125L,
	-7.76102145512898763020833333042135150927893e-11L,
	0.19999999995343387126922607421875L,
	4.656612873077392578124507576697622106863058e-11L,
#endif
};

static const long double q[] = {	/* q[0] - q[9] */
	-3.33333333333333333333333333333333333304213515094e-0001L,
	1.99999999999999999999999999999995075766976221077e-0001L,
	-1.42857142857142857142857142570379604317921113079e-0001L,
	1.11111111111111111111102923861900979127978214077e-0001L,
	-9.09090909090909089586854075816999506863320031460e-0002L,
	7.69230769230756334929213246003824644696974730368e-0002L,
	-6.66666666589192433974402013508912138168133579856e-0002L,
	5.88235013696778007696800252045588307023299350858e-0002L,
	-5.25754959898164576495303840687699583228444695685e-0002L,
};

static const long double
two8700  = 9.140338438955067659002088492701e+2618L, /* 2^8700 */
twom8700 = 1.094051392821643668051436593760e-2619L, /* 2^-8700 */
one   = 1.0L,
zero  = 0.0L,
pi    = 3.1415926535897932384626433832795028841971693993751L,
pio2  = 1.57079632679489661923132169163975144209858469968755L,
pio4  = 0.785398163397448309615660845819875721049292349843776L,
pi3o4 = 2.356194490192344928846982537459627163147877049531329L,
#if defined(__sparc)
pi_lo    = 8.67181013012378102479704402604335196876232e-35L,
pio2_lo  = 4.33590506506189051239852201302167598438116e-35L,
pio4_lo  = 2.16795253253094525619926100651083799219058e-35L,
pi3o4_lo = 6.50385759759283576859778301953251397657174e-35L;
#elif defined(__x86)
pi_lo    = -5.01655761266833202355732708e-20L,
pio2_lo  = -2.50827880633416601177866354e-20L,
pio4_lo  = -1.25413940316708300588933177e-20L,
pi3o4_lo = -9.18342907192877118770525931e-20L;
#endif

static long double
mx_atanl(long double x, long double *err) {
	long double y, z, r, s, t, w, s_h, s_l, x_h, x_l, zz[3], ee[2], z_h,
		z_l, r_h, r_l, u, v;
	int ix, iy, hx, i, j;
	float fx;

	hx = HI_XWORD(x);
	ix = hx & (~0x80000000);

	/* for |x| < 1/8 */
	if (ix < 0x3ffc0000) {
		if (ix < 0x3ff30000) {	/* when |x| < 2**-12 */
			if (ix < 0x3fc60000) {	/* if |x| < 2**-prec/2 */
				*err = (long double) ((int) x);
				return (x);
			}
			z = x * x;
			t = q[8];
			for (i = 7; i >= 0; i--) t = q[i] + z * t;
			t *= x * z;
			r = x + t;
			*err = t - (r - x);
			return (r);
		}
		z = x * x;

		/* use long double precision at p4 and on */
		t = p[16];
		for (i = 15; i >= 4; i--) t = p[i] + z * t;
		ee[0] = z * t;

		x_h = x;  HALF(x_h);
		z_h = z;  HALF(z_h);
		x_l = x - x_h;
		z_l = (x_h * x_h - z_h);
		zz[0] = z;
		zz[1] = z_h;
		zz[2] = z_l + x_l * (x + x_h);

		/* compute (1+z*(p1+z*(p2+z*(p3+e)))) */

		mx_polyl(zz, pe, ee, 3);

		/* finally x*(1+z*(p1+...)) */
		r = x_h * ee[0];
		t = x * ee[1] + x_l * ee[0];
		s = t + r;
		*err = t - (s - r);
		return (s);
	}
	/* for |x| >= 8.0 */
	if (ix >= 0x40020000) {	/* x >=  8 */
		x = fabsl(x);
		if (ix >= 0x402e0000) {	/* x >=  2**47 */
			if (ix >= 0x408b0000) {	/* x >=  2**140 */
				y = -pio2_lo;
			} else
				y = one / x - pio2_lo;
			if (hx >= 0) {
				t = pio2 - y;
				*err = -(y - (pio2 - t));
			} else {
				t = y - pio2;
				*err = y - (pio2 + t);
			}
			return (t);
		} else {
			/* compute r = 1/x */
			r = one / x;
			z = r * r;
			x_h = x; HALF(x_h);
			r_h = r; HALF(r_h);
			z_h = z; HALF(z_h);
			r_l = r * ((x_h - x) * r_h - (x_h * r_h - one));
			z_l = (r_h * r_h - z_h);
			zz[0] = z;
			zz[1] = z_h;
			zz[2] = z_l + r_l * (r + r_h);
			if (ix < 0x40050400) {	/* 8 <  x <  65 */
				/* use double precision at p4 and on */
				t = p[16];
				for (i = 15; i >= 4; i--) t = p[i] + z * t;
				ee[0] = z * t;
				/* compute (1+z*(p1+z*(p2+z*(p3+e)))) */
				mx_polyl(zz, pe, ee, 3);
			} else { /* x < 65 < 2**47 */
				/* use long double at q3 and on */
				t = q[8];
				for (i = 7; i >= 2; i--) t = q[i] + z * t;
				ee[0] = z * t;
				/* compute (1+z*(q1+z*(q2+e))) */
				mx_polyl(zz, qe, ee, 2);
			}
			/* pio2 - r*(1+...) */
			v = r_h * ee[0];
			t = pio2_lo - (r * ee[1] + r_l * ee[0]);
			if (hx >= 0) {
				s = pio2 - v;
				t -= (v - (pio2 - s));
			} else {
				s = v - pio2;
				t = -(t - (v - (s + pio2)));
			}
			w = s + t;
			*err = t - (w - s);
			return (w);
		}
	}
	/* now x is between 1/8 and 8 */
	iy = (ix + 0x00000800) & 0x7ffff000;
	j = (iy - 0x3ffc0000) >> 12;
	((int *) &fx)[0] = 0x3e000000 + (j << 19);
	y = (long double) fx;
	x = fabsl(x);

	w = (x - y);
	v = 1.0L / (one + x * y);
	s = w * v;
	z = s * s;
	/* use long double precision at q3 and on */
	t = q[8];
	for (i = 7; i >= 2; i--) t = q[i] + z * t;
	ee[0] = z * t;
	s_h = s; HALF(s_h);
	z_h = z; HALF(z_h);
	x_h = x; HALF(x_h);
	t = one + x * y; HALF(t);
	r = -((x_h - x) * y - (x_h * y - (t - one)));
	s_l = -v * (s_h * r - (w - s_h * t));
	z_l = (s_h * s_h - z_h);
	zz[0] = z;
	zz[1] = z_h;
	zz[2] = z_l + s_l * (s + s_h);
	/* compute (1+z*(q1+z*(q2+e))) by call mx_poly */
	mx_polyl(zz, qe, ee, 2);
	v = s_h * ee[0];
	t = TBL_atan_lol[j] + (s * ee[1] + s_l * ee[0]);
	u = TBL_atan_hil[j];
	s = u + v;
	t += (v - (s - u));
	w = s + t;
	*err = t - (w - s);
	if (hx < 0) {
		w = -w;
		*err = -*err;
	}
	return (w);
}

long double
__k_atan2l(long double y, long double x, long double *w) {
	long double t, xh, th, t1, t2, w1, w2;
	int ix, iy, hx, hy;

	hy = HI_XWORD(y);
	hx = HI_XWORD(x);
	iy = hy & ~0x80000000;
	ix = hx & ~0x80000000;

	*w = 0.0;
	if (ix >= 0x7fff0000 || iy >= 0x7fff0000) {	/* ignore inexact */
		if (isnanl(x) || isnanl(y))
			return (x * y);
		else if (iy < 0x7fff0000) {
			if (hx >= 0) {	/* ATAN2(+-finite, +inf) is +-0 */
				*w *= y;
				return (*w);
			} else {	/* ATAN2(+-finite, -inf) is +-pi */
				*w = copysignl(pi_lo, y);
				return (copysignl(pi, y));
			}
		} else if (ix < 0x7fff0000) {
					/* ATAN2(+-inf, finite) is +-pi/2 */
			*w = (hy >= 0)? pio2_lo : -pio2_lo;
			return ((hy >= 0)? pio2 : -pio2);
		} else if (hx > 0) {	/* ATAN2(+-INF,+INF) = +-pi/4 */
			*w = (hy >= 0)? pio4_lo : -pio4_lo;
			return ((hy >= 0)? pio4 : -pio4);
		} else {		/* ATAN2(+-INF,-INF) = +-3pi/4 */
			*w = (hy >= 0)? pi3o4_lo : -pi3o4_lo;
			return ((hy >= 0)? pi3o4 : -pi3o4);
		}
	} else if (x == zero || y == zero) {
		if (y == zero) {
			if (hx >= 0) /* ATAN2(+-0, +(0 <= x <= inf)) is +-0 */
				return (y);
			else { 	/* ATAN2(+-0, -(0 <= x <= inf)) is +-pi */
				*w = (hy >= 0)? pi_lo : -pi_lo;
				return ((hy >= 0)? pi : -pi);
			}
		} else { /* ATAN2(+-(anything but 0 and NaN), 0) is +-pi/2 */
			*w = (hy >= 0)? pio2_lo : -pio2_lo;
			return ((hy >= 0)? pio2 : -pio2);
		}
	} else if (iy - ix > 0x00640000) { /* |x/y| < 2 ** -100 */
		*w = (hy >= 0)? pio2_lo : -pio2_lo;
		return ((hy >= 0)? pio2 : -pio2);
	} else if (ix - iy > 0x00640000) { /* |y/x| < 2 ** -100 */
		if (hx < 0) {
			*w = (hy >= 0)? pi_lo : -pi_lo;
			return ((hy >= 0)? pi : -pi);
		} else {
			t = y / x;
			th = t; HALF(th);
			xh = x; HALF(xh);
			t1 = (x - xh) * t + xh * (t - th);
			t2 = y - xh * th;
			*w = (t2 - t1) / x;
			return (t);
		}
	} else {
		if (ix >= 0x5fff3000) {
			x *= twom8700;
			y *= twom8700;
		} else if (ix < 0x203d0000) {
			x *= two8700;
			y *= two8700;
		}
		y = fabsl(y);
		x = fabsl(x);
		t = y / x;
		th = t; HALF(th);
		xh = x; HALF(xh);
		t1 = (x - xh) * t + xh * (t - th);
		t2 = y - xh * th;
		w1 = mx_atanl(t, &w2);
		w2 += (t2 - t1) / (x + y * t);
		if (hx < 0) {
			t1 = pi - w1;
			t2 = pi - t1;
			w2 = (pi_lo - w2) - (w1 - t2);
			w1 = t1;
		}
		*w = (hy >= 0)? w2 : -w2;
		return ((hy >= 0)? w1 : -w1);
	}
}