1 /*
2 * CDDL HEADER START
3 *
4 * The contents of this file are subject to the terms of the
5 * Common Development and Distribution License (the "License").
6 * You may not use this file except in compliance with the License.
7 *
8 * You can obtain a copy of the license at usr/src/OPENSOLARIS.LICENSE
9 * or http://www.opensolaris.org/os/licensing.
10 * See the License for the specific language governing permissions
11 * and limitations under the License.
12 *
13 * When distributing Covered Code, include this CDDL HEADER in each
14 * file and include the License file at usr/src/OPENSOLARIS.LICENSE.
15 * If applicable, add the following below this CDDL HEADER, with the
16 * fields enclosed by brackets "[]" replaced with your own identifying
17 * information: Portions Copyright [yyyy] [name of copyright owner]
18 *
19 * CDDL HEADER END
20 */
21 /*
22 * Copyright 2011 Nexenta Systems, Inc. All rights reserved.
23 */
24 /*
25 * Copyright 2005 Sun Microsystems, Inc. All rights reserved.
26 * Use is subject to license terms.
27 */
28
29 #include "libm.h" /* __k_atan2l */
30 #include "complex_wrapper.h"
31
32 #if defined(__sparc)
33 #define HALF(x) ((int *) &x)[3] = 0; ((int *) &x)[2] &= 0xfe000000
34 #elif defined(__x86)
35 #define HALF(x) ((int *) &x)[0] = 0
36 #endif
37
38 /*
39 * long double __k_atan2l(long double y, long double x, long double *e)
40 *
41 * Compute atan2l with error terms.
42 *
43 * Important formula:
44 * 3 5
45 * x x
46 * atan(x) = x - ----- + ----- - ... (for x <= 1)
47 * 3 5
48 *
49 * pi 1 1
50 * = --- - --- + --- - ... (for x > 1)
51 * 3
52 * 2 x 3x
53 *
54 * Arg(x + y i) = sign(y) * atan2(|y|, x)
55 * = sign(y) * atan(|y|/x) (for x > 0)
56 * sign(y) * (PI - atan(|y|/|x|)) (for x < 0)
57 * Thus if x >> y (IEEE double: EXP(x) - EXP(y) >= 60):
58 * 1. (x > 0): atan2(y,x) ~ y/x
59 * 2. (x < 0): atan2(y,x) ~ sign(y) (PI - |y/x|))
60 * Otherwise if x << y:
61 * atan2(y,x) ~ sign(y)*PI/2 - x/y
62 *
63 * __k_atan2l call static functions mx_polyl, mx_atanl
64 */
65
66
67 /*
68 * (void) mx_polyl (long double *z, long double *a, long double *e, int n)
69 * return
70 * e = a + z*(a + z*(a + ... z*(a + e)...))
71 * 0 2 4 2n
72 * Note:
73 * 1. e and coefficient ai are represented by two long double numbers.
74 * For e, the first one contain the leading 53 bits (30 for x86 exteneded)
75 * and the second one contain the remaining 113 bits (64 for x86 extended).
76 * For ai, the first one contian the leading 53 bits (or 30 for x86)
77 * rounded, and the second is the remaining 113 bits (or 64 for x86).
78 * 2. z is an array of three doubles.
79 * z[0] : the rounded value of Z (the intended value of z)
80 * z[1] : the leading 32 (or 56) bits of Z rounded
81 * z[2] : the remaining 113 (or 64) bits of Z
82 * Note that z[0] = z[1]+z[2] rounded.
83 *
84 */
85
86 static void
mx_polyl(const long double * z,const long double * a,long double * e,int n)87 mx_polyl(const long double *z, const long double *a, long double *e, int n) {
88 long double r, s, t, p_h, p_l, z_h, z_l, p, w;
89 int i;
90 n = n + n;
91 p = e[0] + a[n];
92 p_l = a[n + 1];
93 w = p; HALF(w);
94 p_h = w;
95 p = a[n - 2] + z[0] * p;
96 z_h = z[1]; z_l = z[2];
97 p_l += e[0] - (p_h - a[n]);
98
99 for (i = n - 2; i >= 2; i -= 2) {
100
101 /* compute p = ai + z * p */
102 t = z_h * p_h;
103 s = z[0] * p_l + p_h * z_l;
104 w = p; HALF(w);
105 p_h = w;
106 s += a[i + 1];
107 r = t - (p_h - a[i]);
108 p = a[i - 2] + z[0] * p;
109 p_l = r + s;
110 }
111 w = p; HALF(w);
112 e[0] = w;
113 t = z_h * p_h;
114 s = z[0] * p_l + p_h * z_l;
115 r = t - (e[0] - a[0]);
116 e[1] = r + s;
117 }
118
119 /*
120 * Table of constants for atan from 0.125 to 8
121 * 0.125 -- 0x3ffc0000 --- (increment at bit 12)
122 * 0x3ffc1000
123 * 0x3ffc2000
124 * ... ...
125 * 0x4001f000
126 * 8.000 -- 0x40020000 (total: 97)
127 */
128
129 static const long double TBL_atan_hil[] = {
130 #if defined(__sparc)
131 1.2435499454676143503135484916387102416568e-01L,
132 1.3203976161463874927468440652656953226250e-01L,
133 1.3970887428916364518336777673909505681607e-01L,
134 1.4736148108865163560980276039684551821066e-01L,
135 1.5499674192394098230371437493349219133371e-01L,
136 1.6261382859794857537364156376155780062019e-01L,
137 1.7021192528547440449049660709976171369543e-01L,
138 1.7779022899267607079662479921582468899456e-01L,
139 1.8534794999569476488602596122854464667261e-01L,
140 1.9288431225797466419705871069022730349878e-01L,
141 2.0039855382587851465394578503437838446153e-01L,
142 2.0788992720226299360533498310299432475629e-01L,
143 2.1535769969773804802445962716648964165745e-01L,
144 2.2280115375939451577103212214043255525024e-01L,
145 2.3021958727684373024017095967980299065551e-01L,
146 2.3761231386547125247388363432563777919892e-01L,
147 2.4497866312686415417208248121127580641959e-01L,
148 2.5962962940825753102994644318397190560106e-01L,
149 2.7416745111965879759937189834217578592444e-01L,
150 2.8858736189407739562361141995821834504332e-01L,
151 3.0288486837497140556055609450555821812277e-01L,
152 3.1705575320914700980901557667446732975852e-01L,
153 3.3109607670413209494433878775694455421259e-01L,
154 3.4500217720710510886768128690005168408290e-01L,
155 3.5877067027057222039592006392646052215363e-01L,
156 3.7239844667675422192365503828370182641413e-01L,
157 3.8588266939807377589769548460723139638186e-01L,
158 3.9922076957525256561471669615886476491104e-01L,
159 4.1241044159738730689979128966712694260920e-01L,
160 4.2544963737004228954226360518079233013817e-01L,
161 4.3833655985795780544561604921477130895882e-01L,
162 4.5106965598852347637563925728219344073798e-01L,
163 4.6364760900080611621425623146121439713344e-01L,
164 4.8833395105640552386716496074706484459644e-01L,
165 5.1238946031073770666660102058425923805558e-01L,
166 5.3581123796046370026908506870769144698471e-01L,
167 5.5859931534356243597150821640166122875873e-01L,
168 5.8075635356767039920327447500150082375122e-01L,
169 6.0228734613496418168212269420423291922459e-01L,
170 6.2319932993406593099247534906037459367793e-01L,
171 6.4350110879328438680280922871732260447265e-01L,
172 6.6320299270609325536325431023827583417226e-01L,
173 6.8231655487474807825642998171115298784729e-01L,
174 7.0085440788445017245795128178675127318623e-01L,
175 7.1882999962162450541701415152590469891043e-01L,
176 7.3625742898142813174283527108914662479274e-01L,
177 7.5315128096219438952473937026902888600575e-01L,
178 7.6952648040565826040682003598565401726598e-01L,
179 7.8539816339744830961566084581987569936977e-01L,
180 8.1569192331622341102146083874564582672284e-01L,
181 8.4415398611317100251784414827164746738632e-01L,
182 8.7090345707565295314017311259781407291650e-01L,
183 8.9605538457134395617480071802993779546602e-01L,
184 9.1971960535041681722860345482108940969311e-01L,
185 9.4200004037946366473793717053459362115891e-01L,
186 9.6299433068093620181519583599709989677298e-01L,
187 9.8279372324732906798571061101466603762572e-01L,
188 1.0014831356942347329183295953014374896343e+00L,
189 1.0191413442663497346383429170230636212354e+00L,
190 1.0358412530088001765846944703254440735476e+00L,
191 1.0516502125483736674598673120862999026920e+00L,
192 1.0666303653157435630791763474202799086015e+00L,
193 1.0808390005411683108871567292171997859003e+00L,
194 1.0943289073211899198927883146102352763033e+00L,
195 1.1071487177940905030170654601785370497543e+00L,
196 1.1309537439791604464709335155363277560026e+00L,
197 1.1525719972156675180401498626127514672834e+00L,
198 1.1722738811284763866005949441337046006865e+00L,
199 1.1902899496825317329277337748293182803384e+00L,
200 1.2068173702852525303955115800565576625682e+00L,
201 1.2220253232109896370417417439225704120294e+00L,
202 1.2360594894780819419094519711090786146210e+00L,
203 1.2490457723982544258299170772810900483550e+00L,
204 1.2610933822524404193139408812473357640124e+00L,
205 1.2722973952087173412961937498224805746463e+00L,
206 1.2827408797442707473628852511364955164072e+00L,
207 1.2924966677897852679030914214070816723528e+00L,
208 1.3016288340091961438047858503666855024453e+00L,
209 1.3101939350475556342564376891719053437537e+00L,
210 1.3182420510168370498593302023271363040427e+00L,
211 1.3258176636680324650592392104284756886164e+00L,
212 1.3397056595989995393283037525895557850243e+00L,
213 1.3521273809209546571891479413898127598774e+00L,
214 1.3633001003596939542892985278250991560269e+00L,
215 1.3734007669450158608612719264449610604836e+00L,
216 1.3825748214901258580599674177685685163955e+00L,
217 1.3909428270024183486427686943836432395486e+00L,
218 1.3986055122719575950126700816114282727858e+00L,
219 1.4056476493802697809521934019958080664406e+00L,
220 1.4121410646084952153676136718584890852820e+00L,
221 1.4181469983996314594038603039700988632607e+00L,
222 1.4237179714064941189018190466107297108905e+00L,
223 1.4288992721907326964184700745371984001389e+00L,
224 1.4337301524847089866404719096698873880264e+00L,
225 1.4382447944982225979614042479354816039669e+00L,
226 1.4424730991091018200252920599377291810352e+00L,
227 1.4464413322481351841999668424758803866109e+00L,
228 #elif defined(__x86)
229 1.243549945356789976358413696289e-01L, 1.320397615781985223293304443359e-01L,
230 1.397088742814958095550537109375e-01L, 1.473614810383878648281097412109e-01L,
231 1.549967419123277068138122558594e-01L, 1.626138285500928759574890136719e-01L,
232 1.702119252295233309268951416016e-01L, 1.777902289759367704391479492188e-01L,
233 1.853479499695822596549987792969e-01L, 1.928843122441321611404418945312e-01L,
234 2.003985538030974566936492919922e-01L, 2.078899272019043564796447753906e-01L,
235 2.153576996643096208572387695312e-01L, 2.228011537226848304271697998047e-01L,
236 2.302195872762240469455718994141e-01L, 2.376123138237744569778442382812e-01L,
237 2.449786631041206419467926025391e-01L, 2.596296293195337057113647460938e-01L,
238 2.741674510762095451354980468750e-01L, 2.885873618070036172866821289062e-01L,
239 3.028848683461546897888183593750e-01L, 3.170557531993836164474487304688e-01L,
240 3.310960766393691301345825195312e-01L, 3.450021771714091300964355468750e-01L,
241 3.587706702528521418571472167969e-01L, 3.723984466632828116416931152344e-01L,
242 3.858826693613082170486450195312e-01L, 3.992207695264369249343872070312e-01L,
243 4.124104415532201528549194335938e-01L, 4.254496373469009995460510253906e-01L,
244 4.383365598041564226150512695312e-01L, 4.510696559445932507514953613281e-01L,
245 4.636476089945062994956970214844e-01L, 4.883339509833604097366333007812e-01L,
246 5.123894601128995418548583984375e-01L, 5.358112377580255270004272460938e-01L,
247 5.585993151180446147918701171875e-01L, 5.807563534472137689590454101562e-01L,
248 6.022873460315167903900146484375e-01L, 6.231993297114968299865722656250e-01L,
249 6.435011087451130151748657226562e-01L, 6.632029926404356956481933593750e-01L,
250 6.823165547102689743041992187500e-01L, 7.008544078562408685684204101562e-01L,
251 7.188299994450062513351440429688e-01L, 7.362574287690222263336181640625e-01L,
252 7.531512808054685592651367187500e-01L, 7.695264802314341068267822265625e-01L,
253 7.853981633670628070831298828125e-01L, 8.156919232569634914398193359375e-01L,
254 8.441539860796183347702026367188e-01L, 8.709034570492804050445556640625e-01L,
255 8.960553845390677452087402343750e-01L, 9.197196052409708499908447265625e-01L,
256 9.420000403188169002532958984375e-01L, 9.629943305626511573791503906250e-01L,
257 9.827937232330441474914550781250e-01L, 1.001483135391026735305786132812e+00L,
258 1.019141343887895345687866210938e+00L, 1.035841252654790878295898437500e+00L,
259 1.051650212146341800689697265625e+00L, 1.066630364861339330673217773438e+00L,
260 1.080839000176638364791870117188e+00L, 1.094328907318413257598876953125e+00L,
261 1.107148717623203992843627929688e+00L, 1.130953743588179349899291992188e+00L,
262 1.152571997139602899551391601562e+00L, 1.172273880802094936370849609375e+00L,
263 1.190289949532598257064819335938e+00L, 1.206817369908094406127929687500e+00L,
264 1.222025323193520307540893554688e+00L, 1.236059489194303750991821289062e+00L,
265 1.249045772012323141098022460938e+00L, 1.261093381792306900024414062500e+00L,
266 1.272297394927591085433959960938e+00L, 1.282740879338234663009643554688e+00L,
267 1.292496667709201574325561523438e+00L, 1.301628833636641502380371093750e+00L,
268 1.310193934943526983261108398438e+00L, 1.318242050707340240478515625000e+00L,
269 1.325817663222551345825195312500e+00L, 1.339705659542232751846313476562e+00L,
270 1.352127380669116973876953125000e+00L, 1.363300099968910217285156250000e+00L,
271 1.373400766868144273757934570312e+00L, 1.382574821356683969497680664062e+00L,
272 1.390942826867103576660156250000e+00L, 1.398605511989444494247436523438e+00L,
273 1.405647648964077234268188476562e+00L, 1.412141064181923866271972656250e+00L,
274 1.418146998155862092971801757812e+00L, 1.423717970959842205047607421875e+00L,
275 1.428899271879345178604125976562e+00L, 1.433730152435600757598876953125e+00L,
276 1.438244794495403766632080078125e+00L, 1.442473099101334810256958007812e+00L,
277 1.446441331878304481506347656250e+00L,
278 #endif
279 };
280 static const long double TBL_atan_lol[] = {
281 #if defined(__sparc)
282 1.4074869197628063802317202820414310039556e-36L,
283 -4.9596961594739925555730439437999675295505e-36L,
284 8.9527745625194648873931213446361849472788e-36L,
285 1.1880437423207895718180765843544965589427e-35L,
286 -2.7810278112045145378425375128234365381448e-37L,
287 1.4797220377023800327295536234315147262387e-36L,
288 -4.2169561400548198732870384801849639863829e-36L,
289 7.2431229666913484649930323656316023494680e-36L,
290 -2.1573430089839170299895679353790663182462e-36L,
291 -9.9515745405126723554452367298128605186305e-36L,
292 -3.9065558992324838181617569730397882363067e-36L,
293 5.5260292271793726813211980664661124518807e-36L,
294 8.8415722215914321807682254318036452043689e-36L,
295 -8.1767728791586179254193323628285599800711e-36L,
296 -1.3344123034656142243797113823028330070762e-36L,
297 -4.4927331207813382908930733924681325892188e-36L,
298 4.4945511471812490393201824336762495687730e-36L,
299 -1.6688081504279223555776724459648440567274e-35L,
300 1.5629757586107955769461086568937329684113e-35L,
301 -2.2389835563308078552507970385331510848109e-35L,
302 -4.8312321745547311551870450671182151367050e-36L,
303 -1.4336172352905832876958926610980698844309e-35L,
304 -8.7440181998899932802989174170960593316080e-36L,
305 5.9284636008529837445780360785464550143016e-36L,
306 -2.2376651248436241276061055295043514993630e-35L,
307 6.0745837599336105414280310756677442136480e-36L,
308 1.5372187110451949677792344762029967023093e-35L,
309 2.0976068056751156241657121582478790247159e-35L,
310 -5.5623956405495438060726862202622807523700e-36L,
311 1.9697366707832471841858411934897351901523e-35L,
312 2.1070311964479488509034733639424887543697e-35L,
313 -2.3027356362982001602256518510854229844561e-35L,
314 4.8950964225733349266861843522029764772843e-36L,
315 -7.2380143477794458213872723050820253166391e-36L,
316 1.6365648865703614031637443396049568858105e-35L,
317 -3.9885811958234530793729129919803234197399e-35L,
318 4.1587722120912613510417783923227421336929e-35L,
319 3.8347421454556472153684687377337135027394e-35L,
320 -9.2251178933638721723515896465489002497864e-36L,
321 1.4094619690455989526175736741854656192178e-36L,
322 3.3568857805472235270612851425810803679451e-35L,
323 3.9090991055522552395018106803232118803401e-35L,
324 5.2956416979654208140521862707297033857956e-36L,
325 -5.0960846819945514367847063923662507136721e-36L,
326 -4.4959014425277615858329680393918315204998e-35L,
327 3.8039226544551634266566857615962609653834e-35L,
328 -4.4056522872895512108308642196611689657618e-36L,
329 1.6025024192482161076223807753425619076948e-36L,
330 2.1679525325309452561992610065108380635264e-35L,
331 1.9844038013515422125715362925736754104066e-35L,
332 3.9139619471799746834505227353568432457241e-35L,
333 2.1113443807975453505518453436799561854730e-35L,
334 3.1558557277444692755039816944392770185432e-35L,
335 1.6295044520355461408265585619500238335614e-35L,
336 -3.5087245209270305856151230356171213582305e-35L,
337 2.9041041864282855679591055270946117300088e-35L,
338 -2.3128843453818356590931995209806627233282e-35L,
339 -7.7124923181471578439967973820714857839953e-35L,
340 2.7539027829886922429092063590445808781462e-35L,
341 -9.4500899453181308951084545990839335972452e-35L,
342 -7.3061755302032092337594946001641651543473e-35L,
343 -4.1736144813953752193952770157406952602798e-35L,
344 3.4369948356256407045344855262863733571105e-35L,
345 -6.3790243492298090907302084924276831116460e-35L,
346 -9.6842943816353261291004127866079538980649e-36L,
347 4.8746757539138870909275958326700072821615e-35L,
348 -8.7533886477084190884511601368582548254655e-35L,
349 1.4284743992327918892692551138086727754845e-35L,
350 5.7262776211073389542565625693479173445042e-35L,
351 -3.2254883148780411245594822270747948565684e-35L,
352 7.8853548190609877325965525252380833808405e-35L,
353 8.4081736739037194097515038365370730251333e-35L,
354 7.4722870357563683815078242981933587273670e-35L,
355 7.9977202825793435289434813600890494256112e-36L,
356 -8.0577840773362139054848492346292673645405e-35L,
357 1.4217746753670583065490040209048757624336e-35L,
358 1.2232486914221205004109743560319090913328e-35L,
359 8.9696055070830036447361957217943988339065e-35L,
360 -3.1480394435081884410686066739846269858951e-35L,
361 -5.0927146040715345013240642517608928352977e-35L,
362 -5.7431997715924136568133859432702789493569e-35L,
363 -4.3920451405083770279099766080476485439987e-35L,
364 9.1106753984907715563018666776308759323326e-35L,
365 -3.7032569014272841009512400773061537538358e-35L,
366 8.8167419429746714276909825405131416764489e-35L,
367 -3.8389341696028352503752312861740895209678e-36L,
368 -3.3462959341960891546340895508017603408404e-35L,
369 -3.9212626776786074383916188498955828634947e-35L,
370 -7.8340397396377867255864494568594088378648e-35L,
371 7.4681018632456986520600640340627309824469e-35L,
372 8.9110918618956918451135594876165314884113e-35L,
373 3.9418160632271890530431797145664308529115e-35L,
374 -4.1048114088580104820193435638327617443913e-35L,
375 -2.3165419451582153326383944756220900454330e-35L,
376 -1.8428312581525319409399330203703211113843e-35L,
377 7.1477316546709482345411712017906842769961e-35L,
378 2.9914501578435874662153637707016094237004e-35L,
379 #elif defined(__x86)
380 1.108243739551347953496477557317e-11L, 3.644022694535396219063202730280e-11L,
381 7.667835628314065801595065768845e-12L, 5.026377078169301918590803009109e-11L,
382 1.161327548990211907411719105561e-11L, 4.785569941615255008968280209991e-11L,
383 5.595107356360146549819920947848e-11L, 1.673930035747684999707469623769e-11L,
384 2.611250523102718193166964451527e-11L, 1.384250305661681615897729354721e-11L,
385 2.278105796029649304219088055497e-11L, 3.586371256902077123693302823191e-13L,
386 3.342842716722085763523965049902e-11L, 3.670968534386232233574504707347e-11L,
387 6.196832945990602657404893210974e-13L, 4.169679549603939604438777470618e-11L,
388 2.274351222528987867221331091414e-11L, 8.872382531858169709022188891298e-11L,
389 4.344925246387385146717580155420e-11L, 8.707377833692929105196832265348e-11L,
390 2.881671577173773513055821329154e-11L, 9.763393361566846205717315422347e-12L,
391 6.476296480975626822569454546857e-11L, 3.569597877124574002505169001136e-11L,
392 1.772007853877284712958549977698e-11L, 1.347141028196192304932683248872e-11L,
393 3.676555884905046507598141175404e-11L, 4.881564068032948912761478588710e-11L,
394 4.416715404487185607337693704681e-11L, 2.314128999621257979016734983553e-11L,
395 5.380138283056477968352133002913e-11L, 4.393022562414389595406841771063e-11L,
396 6.299816718559209976839402028537e-12L, 7.304511413053165996581483735843e-11L,
397 1.978381648117426221467592544212e-10L, 2.024381732686578226139414070989e-10L,
398 2.255178211796380992141612703464e-10L, 1.204566302442290648452508620986e-10L,
399 1.034473912921080457667329099995e-10L, 2.225691010059030834353745950874e-10L,
400 4.817137162794350606107263804151e-11L, 6.565755971506095086327587326326e-11L,
401 1.644791039522307629611529931429e-10L, 2.820930388953087163050126809014e-11L,
402 1.766182540818701085571546539514e-10L, 2.124059054092171070266466628320e-10L,
403 1.567258302596026515190288816001e-10L, 1.742241535800378094231540188685e-10L,
404 3.038550253253096300737572104929e-11L, 5.925991958164150280814584656688e-11L,
405 3.355266774764151155289750652594e-11L, 2.637254809561744853531409402995e-11L,
406 3.227621096606048365493782702458e-11L, 1.094459672377587282585894259882e-10L,
407 6.064676448464127209709358607166e-11L, 1.182850444360454453720999258140e-10L,
408 1.428492049425553288966601449688e-11L, 3.032079976125434624889374125094e-10L,
409 3.784543889504767060855636487744e-10L, 3.540092982887960328254439790467e-10L,
410 4.020318667701700464612998296302e-10L, 4.544042324059585739827798668654e-10L,
411 3.645299460952866120296998202703e-10L, 2.776662293911361485235212513020e-12L,
412 1.708865101734375304910370400700e-10L, 3.909810965716415233488278047493e-10L,
413 7.606461848875826105025137974947e-11L, 3.263814502297453347587046149712e-10L,
414 1.499334758629144388918183376012e-10L, 3.771581242675818925565576303133e-10L,
415 1.746932950084818923507049088298e-11L, 2.837781909176306820465786987027e-10L,
416 3.859312847318946163435901230778e-10L, 4.601335192895268187473357720101e-10L,
417 2.811262558622337888849804940684e-10L, 4.060360843532416964489955306249e-10L,
418 8.058369357752989796958168458531e-11L, 3.725546414244147566166855921414e-10L,
419 1.040286509953292907344053122733e-10L, 3.094968093808145773271362531155e-10L,
420 4.454811192340438979284756311844e-10L, 5.676678748199027602705574110388e-11L,
421 2.518376833121948163898128509842e-10L, 3.907837370041422778250991189943e-10L,
422 7.687158710333735613246114865100e-11L, 1.334418885622867537060685125566e-10L,
423 1.353147719826124443836432060856e-10L, 2.825131007652335581739282335732e-10L,
424 4.161925466840049254333079881002e-10L, 4.265713490956410156084891599630e-10L,
425 2.437693664320585461575989523716e-10L, 4.466519138542116247357297503086e-10L,
426 3.113875178143440979746983590908e-10L, 4.910822904159495654488736486097e-11L,
427 2.818831329324169810481585538618e-12L, 7.767009768334052125229252512543e-12L,
428 3.698307026936191862258804165254e-10L,
429 #endif
430 };
431
432 /*
433 * mx_atanl(x, err)
434 * Table look-up algorithm
435 * By K.C. Ng, March 9, 1989
436 *
437 * Algorithm.
438 *
439 * The algorithm is based on atan(x)=atan(y)+atan((x-y)/(1+x*y)).
440 * We use poly1(x) to approximate atan(x) for x in [0,1/8] with
441 * error (relative)
442 * |(atan(x)-poly1(x))/x|<= 2^-140
443 *
444 * and use poly2(x) to approximate atan(x) for x in [0,1/65] with
445 * error
446 * |atan(x)-poly2(x)|<= 2^-143.7
447 *
448 * Here poly1 and poly2 are odd polynomial with the following form:
449 * x + x^3*(a1+x^2*(a2+...))
450 *
451 * (0). Purge off Inf and NaN and 0
452 * (1). Reduce x to positive by atan(x) = -atan(-x).
453 * (2). For x <= 1/8, use
454 * (2.1) if x < 2^(-prec/2), atan(x) = x with inexact flag raised
455 * (2.2) Otherwise
456 * atan(x) = poly1(x)
457 * (3). For x >= 8 then (prec = 78)
458 * (3.1) if x >= 2^prec, atan(x) = atan(inf) - pio2_lo
459 * (3.2) if x >= 2^(prec/3), atan(x) = atan(inf) - 1/x
460 * (3.3) if x > 65, atan(x) = atan(inf) - poly2(1/x)
461 * (3.4) Otherwise, atan(x) = atan(inf) - poly1(1/x)
462 *
463 * (4). Now x is in (0.125, 8)
464 * Find y that match x to 4.5 bit after binary (easy).
465 * If iy is the high word of y, then
466 * single : j = (iy - 0x3e000000) >> 19
467 * double : j = (iy - 0x3fc00000) >> 16
468 * quad : j = (iy - 0x3ffc0000) >> 12
469 *
470 * Let s = (x-y)/(1+x*y). Then
471 * atan(x) = atan(y) + poly1(s)
472 * = _TBL_atan_hi[j] + (_TBL_atan_lo[j] + poly2(s) )
473 *
474 * Note. |s| <= 1.5384615385e-02 = 1/65. Maxium occurs at x = 1.03125
475 *
476 */
477
478 /*
479 * p[0] - p[16] for atan(x) =
480 * x + x^3*(p1+x^2*(p2+...))
481 */
482 static const long double pe[] = {
483 1.0L,
484 0.0L,
485 #if defined(__sparc)
486 -0.33333333333333332870740406406184774823L,
487 -4.62592926927148558508441072595508240609e-18L,
488 0.19999999999999999722444243843710864894L,
489 2.77555756156289124602047010782090464486e-18L,
490 -0.14285714285714285615158658515611023176L,
491 -9.91270557700756738621231719241800559409e-19L,
492 #elif defined(__x86)
493 -0.33333333325572311878204345703125L,
494 -7.76102145512898763020833333192787755766644373e-11L,
495 0.19999999995343387126922607421875L,
496 4.65661287307739257812498949613909375938538636e-11L,
497 -0.142857142840512096881866455078125L,
498 -1.66307602609906877787419703858463013035681375e-11L,
499 #endif
500 };
501
502 static const long double p[] = { /* p[0] - p[16] */
503 1.0L,
504 -3.33333333333333333333333333333333333319278775586e-0001L,
505 1.99999999999999999999999999999999894961390937601e-0001L,
506 -1.42857142857142857142857142856866970385846301312e-0001L,
507 1.11111111111111111111111110742899094415954427738e-0001L,
508 -9.09090909090909090909087972707015549231951421806e-0002L,
509 7.69230769230769230767699003016385628597359717046e-0002L,
510 -6.66666666666666666113842763495291228025226575259e-0002L,
511 5.88235294117646915706902204947653640091126695962e-0002L,
512 -5.26315789473657016886225044679594035524579379810e-0002L,
513 4.76190476186633969331771169790375592681525481267e-0002L,
514 -4.34782608290146274616081389793141896576997370161e-0002L,
515 3.99999968161267722260103962788865225205057218988e-0002L,
516 -3.70368536844778256320786172745225703228683638328e-0002L,
517 3.44752320396524479494062858284036892703898522150e-0002L,
518 -3.20491216046653214683721787776813360591233428081e-0002L,
519 2.67632651033434456758550618122802167256870856514e-0002L,
520 };
521
522 /* q[0] - q[9] */
523 static const long double qe[] = {
524 1.0L,
525 0.0L,
526 #if defined(__sparc)
527 -0.33333333333333332870740406406184774823486804962158203125L,
528 -4.625929269271485585069345465471207312531868714634217630e-18L,
529 0.19999999999999999722444243843710864894092082977294921875L,
530 2.7755575615628864268260553912956813621977220359134667560e-18L,
531 #elif defined(__x86)
532 -0.33333333325572311878204345703125L,
533 -7.76102145512898763020833333042135150927893e-11L,
534 0.19999999995343387126922607421875L,
535 4.656612873077392578124507576697622106863058e-11L,
536 #endif
537 };
538
539 static const long double q[] = { /* q[0] - q[9] */
540 -3.33333333333333333333333333333333333304213515094e-0001L,
541 1.99999999999999999999999999999995075766976221077e-0001L,
542 -1.42857142857142857142857142570379604317921113079e-0001L,
543 1.11111111111111111111102923861900979127978214077e-0001L,
544 -9.09090909090909089586854075816999506863320031460e-0002L,
545 7.69230769230756334929213246003824644696974730368e-0002L,
546 -6.66666666589192433974402013508912138168133579856e-0002L,
547 5.88235013696778007696800252045588307023299350858e-0002L,
548 -5.25754959898164576495303840687699583228444695685e-0002L,
549 };
550
551 static const long double
552 two8700 = 9.140338438955067659002088492701e+2618L, /* 2^8700 */
553 twom8700 = 1.094051392821643668051436593760e-2619L, /* 2^-8700 */
554 one = 1.0L,
555 zero = 0.0L,
556 pi = 3.1415926535897932384626433832795028841971693993751L,
557 pio2 = 1.57079632679489661923132169163975144209858469968755L,
558 pio4 = 0.785398163397448309615660845819875721049292349843776L,
559 pi3o4 = 2.356194490192344928846982537459627163147877049531329L,
560 #if defined(__sparc)
561 pi_lo = 8.67181013012378102479704402604335196876232e-35L,
562 pio2_lo = 4.33590506506189051239852201302167598438116e-35L,
563 pio4_lo = 2.16795253253094525619926100651083799219058e-35L,
564 pi3o4_lo = 6.50385759759283576859778301953251397657174e-35L;
565 #elif defined(__x86)
566 pi_lo = -5.01655761266833202355732708e-20L,
567 pio2_lo = -2.50827880633416601177866354e-20L,
568 pio4_lo = -1.25413940316708300588933177e-20L,
569 pi3o4_lo = -9.18342907192877118770525931e-20L;
570 #endif
571
572 static long double
mx_atanl(long double x,long double * err)573 mx_atanl(long double x, long double *err) {
574 long double y, z, r, s, t, w, s_h, s_l, x_h, x_l, zz[3], ee[2], z_h,
575 z_l, r_h, r_l, u, v;
576 int ix, iy, hx, i, j;
577 float fx;
578
579 hx = HI_XWORD(x);
580 ix = hx & (~0x80000000);
581
582 /* for |x| < 1/8 */
583 if (ix < 0x3ffc0000) {
584 if (ix < 0x3ff30000) { /* when |x| < 2**-12 */
585 if (ix < 0x3fc60000) { /* if |x| < 2**-prec/2 */
586 *err = (long double) ((int) x);
587 return (x);
588 }
589 z = x * x;
590 t = q[8];
591 for (i = 7; i >= 0; i--) t = q[i] + z * t;
592 t *= x * z;
593 r = x + t;
594 *err = t - (r - x);
595 return (r);
596 }
597 z = x * x;
598
599 /* use long double precision at p4 and on */
600 t = p[16];
601 for (i = 15; i >= 4; i--) t = p[i] + z * t;
602 ee[0] = z * t;
603
604 x_h = x; HALF(x_h);
605 z_h = z; HALF(z_h);
606 x_l = x - x_h;
607 z_l = (x_h * x_h - z_h);
608 zz[0] = z;
609 zz[1] = z_h;
610 zz[2] = z_l + x_l * (x + x_h);
611
612 /* compute (1+z*(p1+z*(p2+z*(p3+e)))) */
613
614 mx_polyl(zz, pe, ee, 3);
615
616 /* finally x*(1+z*(p1+...)) */
617 r = x_h * ee[0];
618 t = x * ee[1] + x_l * ee[0];
619 s = t + r;
620 *err = t - (s - r);
621 return (s);
622 }
623 /* for |x| >= 8.0 */
624 if (ix >= 0x40020000) { /* x >= 8 */
625 x = fabsl(x);
626 if (ix >= 0x402e0000) { /* x >= 2**47 */
627 if (ix >= 0x408b0000) { /* x >= 2**140 */
628 y = -pio2_lo;
629 } else
630 y = one / x - pio2_lo;
631 if (hx >= 0) {
632 t = pio2 - y;
633 *err = -(y - (pio2 - t));
634 } else {
635 t = y - pio2;
636 *err = y - (pio2 + t);
637 }
638 return (t);
639 } else {
640 /* compute r = 1/x */
641 r = one / x;
642 z = r * r;
643 x_h = x; HALF(x_h);
644 r_h = r; HALF(r_h);
645 z_h = z; HALF(z_h);
646 r_l = r * ((x_h - x) * r_h - (x_h * r_h - one));
647 z_l = (r_h * r_h - z_h);
648 zz[0] = z;
649 zz[1] = z_h;
650 zz[2] = z_l + r_l * (r + r_h);
651 if (ix < 0x40050400) { /* 8 < x < 65 */
652 /* use double precision at p4 and on */
653 t = p[16];
654 for (i = 15; i >= 4; i--) t = p[i] + z * t;
655 ee[0] = z * t;
656 /* compute (1+z*(p1+z*(p2+z*(p3+e)))) */
657 mx_polyl(zz, pe, ee, 3);
658 } else { /* x < 65 < 2**47 */
659 /* use long double at q3 and on */
660 t = q[8];
661 for (i = 7; i >= 2; i--) t = q[i] + z * t;
662 ee[0] = z * t;
663 /* compute (1+z*(q1+z*(q2+e))) */
664 mx_polyl(zz, qe, ee, 2);
665 }
666 /* pio2 - r*(1+...) */
667 v = r_h * ee[0];
668 t = pio2_lo - (r * ee[1] + r_l * ee[0]);
669 if (hx >= 0) {
670 s = pio2 - v;
671 t -= (v - (pio2 - s));
672 } else {
673 s = v - pio2;
674 t = -(t - (v - (s + pio2)));
675 }
676 w = s + t;
677 *err = t - (w - s);
678 return (w);
679 }
680 }
681 /* now x is between 1/8 and 8 */
682 iy = (ix + 0x00000800) & 0x7ffff000;
683 j = (iy - 0x3ffc0000) >> 12;
684 ((int *) &fx)[0] = 0x3e000000 + (j << 19);
685 y = (long double) fx;
686 x = fabsl(x);
687
688 w = (x - y);
689 v = 1.0L / (one + x * y);
690 s = w * v;
691 z = s * s;
692 /* use long double precision at q3 and on */
693 t = q[8];
694 for (i = 7; i >= 2; i--) t = q[i] + z * t;
695 ee[0] = z * t;
696 s_h = s; HALF(s_h);
697 z_h = z; HALF(z_h);
698 x_h = x; HALF(x_h);
699 t = one + x * y; HALF(t);
700 r = -((x_h - x) * y - (x_h * y - (t - one)));
701 s_l = -v * (s_h * r - (w - s_h * t));
702 z_l = (s_h * s_h - z_h);
703 zz[0] = z;
704 zz[1] = z_h;
705 zz[2] = z_l + s_l * (s + s_h);
706 /* compute (1+z*(q1+z*(q2+e))) by call mx_poly */
707 mx_polyl(zz, qe, ee, 2);
708 v = s_h * ee[0];
709 t = TBL_atan_lol[j] + (s * ee[1] + s_l * ee[0]);
710 u = TBL_atan_hil[j];
711 s = u + v;
712 t += (v - (s - u));
713 w = s + t;
714 *err = t - (w - s);
715 if (hx < 0) {
716 w = -w;
717 *err = -*err;
718 }
719 return (w);
720 }
721
722 long double
__k_atan2l(long double y,long double x,long double * w)723 __k_atan2l(long double y, long double x, long double *w) {
724 long double t, xh, th, t1, t2, w1, w2;
725 int ix, iy, hx, hy;
726
727 hy = HI_XWORD(y);
728 hx = HI_XWORD(x);
729 iy = hy & ~0x80000000;
730 ix = hx & ~0x80000000;
731
732 *w = 0.0;
733 if (ix >= 0x7fff0000 || iy >= 0x7fff0000) { /* ignore inexact */
734 if (isnanl(x) || isnanl(y))
735 return (x * y);
736 else if (iy < 0x7fff0000) {
737 if (hx >= 0) { /* ATAN2(+-finite, +inf) is +-0 */
738 *w *= y;
739 return (*w);
740 } else { /* ATAN2(+-finite, -inf) is +-pi */
741 *w = copysignl(pi_lo, y);
742 return (copysignl(pi, y));
743 }
744 } else if (ix < 0x7fff0000) {
745 /* ATAN2(+-inf, finite) is +-pi/2 */
746 *w = (hy >= 0)? pio2_lo : -pio2_lo;
747 return ((hy >= 0)? pio2 : -pio2);
748 } else if (hx > 0) { /* ATAN2(+-INF,+INF) = +-pi/4 */
749 *w = (hy >= 0)? pio4_lo : -pio4_lo;
750 return ((hy >= 0)? pio4 : -pio4);
751 } else { /* ATAN2(+-INF,-INF) = +-3pi/4 */
752 *w = (hy >= 0)? pi3o4_lo : -pi3o4_lo;
753 return ((hy >= 0)? pi3o4 : -pi3o4);
754 }
755 } else if (x == zero || y == zero) {
756 if (y == zero) {
757 if (hx >= 0) /* ATAN2(+-0, +(0 <= x <= inf)) is +-0 */
758 return (y);
759 else { /* ATAN2(+-0, -(0 <= x <= inf)) is +-pi */
760 *w = (hy >= 0)? pi_lo : -pi_lo;
761 return ((hy >= 0)? pi : -pi);
762 }
763 } else { /* ATAN2(+-(anything but 0 and NaN), 0) is +-pi/2 */
764 *w = (hy >= 0)? pio2_lo : -pio2_lo;
765 return ((hy >= 0)? pio2 : -pio2);
766 }
767 } else if (iy - ix > 0x00640000) { /* |x/y| < 2 ** -100 */
768 *w = (hy >= 0)? pio2_lo : -pio2_lo;
769 return ((hy >= 0)? pio2 : -pio2);
770 } else if (ix - iy > 0x00640000) { /* |y/x| < 2 ** -100 */
771 if (hx < 0) {
772 *w = (hy >= 0)? pi_lo : -pi_lo;
773 return ((hy >= 0)? pi : -pi);
774 } else {
775 t = y / x;
776 th = t; HALF(th);
777 xh = x; HALF(xh);
778 t1 = (x - xh) * t + xh * (t - th);
779 t2 = y - xh * th;
780 *w = (t2 - t1) / x;
781 return (t);
782 }
783 } else {
784 if (ix >= 0x5fff3000) {
785 x *= twom8700;
786 y *= twom8700;
787 } else if (ix < 0x203d0000) {
788 x *= two8700;
789 y *= two8700;
790 }
791 y = fabsl(y);
792 x = fabsl(x);
793 t = y / x;
794 th = t; HALF(th);
795 xh = x; HALF(xh);
796 t1 = (x - xh) * t + xh * (t - th);
797 t2 = y - xh * th;
798 w1 = mx_atanl(t, &w2);
799 w2 += (t2 - t1) / (x + y * t);
800 if (hx < 0) {
801 t1 = pi - w1;
802 t2 = pi - t1;
803 w2 = (pi_lo - w2) - (w1 - t2);
804 w1 = t1;
805 }
806 *w = (hy >= 0)? w2 : -w2;
807 return ((hy >= 0)? w1 : -w1);
808 }
809 }
810