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 /*
23 * Copyright 2011 Nexenta Systems, Inc. All rights reserved.
24 */
25
26 /*
27 * Copyright 2005 Sun Microsystems, Inc. All rights reserved.
28 * Use is subject to license terms.
29 */
30
31 #include "libm.h" /* __k_atan2l */
32 #include "complex_wrapper.h"
33
34 #if defined(__sparc)
35 #define HALF(x) ((int *)&x)[3] = 0; ((int *)&x)[2] &= 0xfe000000
36 #elif defined(__x86)
37 #define HALF(x) ((int *)&x)[0] = 0
38 #endif
39
40 /*
41 * long double __k_atan2l(long double y, long double x, long double *e)
42 *
43 * Compute atan2l with error terms.
44 *
45 * Important formula:
46 * 3 5
47 * x x
48 * atan(x) = x - ----- + ----- - ... (for x <= 1)
49 * 3 5
50 *
51 * pi 1 1
52 * = --- - --- + --- - ... (for x > 1)
53 * 3
54 * 2 x 3x
55 *
56 * Arg(x + y i) = sign(y) * atan2(|y|, x)
57 * = sign(y) * atan(|y|/x) (for x > 0)
58 * sign(y) * (PI - atan(|y|/|x|)) (for x < 0)
59 * Thus if x >> y (IEEE double: EXP(x) - EXP(y) >= 60):
60 * 1. (x > 0): atan2(y,x) ~ y/x
61 * 2. (x < 0): atan2(y,x) ~ sign(y) (PI - |y/x|))
62 * Otherwise if x << y:
63 * atan2(y,x) ~ sign(y)*PI/2 - x/y
64 *
65 * __k_atan2l call static functions mx_polyl, mx_atanl
66 */
67
68 /*
69 * (void) mx_polyl (long double *z, long double *a, long double *e, int n)
70 * return
71 * e = a + z*(a + z*(a + ... z*(a + e)...))
72 * 0 2 4 2n
73 * Note:
74 * 1. e and coefficient ai are represented by two long double numbers.
75 * For e, the first one contain the leading 53 bits (30 for x86 exteneded)
76 * and the second one contain the remaining 113 bits (64 for x86 extended).
77 * For ai, the first one contian the leading 53 bits (or 30 for x86)
78 * rounded, and the second is the remaining 113 bits (or 64 for x86).
79 * 2. z is an array of three doubles.
80 * z[0] : the rounded value of Z (the intended value of z)
81 * z[1] : the leading 32 (or 56) bits of Z rounded
82 * z[2] : the remaining 113 (or 64) bits of Z
83 * Note that z[0] = z[1]+z[2] rounded.
84 *
85 */
86 static void
87 mx_polyl(const long double *z, const long double *a, long double *e, int n)
88 {
89 long double r, s, t, p_h, p_l, z_h, z_l, p, w;
90 int i;
91
92 n = n + n;
93 p = e[0] + a[n];
94 p_l = a[n + 1];
95 w = p;
96 HALF(w);
97 p_h = w;
98 p = a[n - 2] + z[0] * p;
99 z_h = z[1];
100 z_l = z[2];
101 p_l += e[0] - (p_h - a[n]);
102
103 for (i = n - 2; i >= 2; i -= 2) {
104 /* compute p = ai + z * p */
105 t = z_h * p_h;
106 s = z[0] * p_l + p_h * z_l;
107 w = p;
108 HALF(w);
109 p_h = w;
110 s += a[i + 1];
111 r = t - (p_h - a[i]);
112 p = a[i - 2] + z[0] * p;
113 p_l = r + s;
114 }
115
116 w = p;
117 HALF(w);
118 e[0] = w;
119 t = z_h * p_h;
120 s = z[0] * p_l + p_h * z_l;
121 r = t - (e[0] - a[0]);
122 e[1] = r + s;
123 }
124
125 /*
126 * Table of constants for atan from 0.125 to 8
127 * 0.125 -- 0x3ffc0000 --- (increment at bit 12)
128 * 0x3ffc1000
129 * 0x3ffc2000
130 * ... ...
131 * 0x4001f000
132 * 8.000 -- 0x40020000 (total: 97)
133 */
134
135 static const long double TBL_atan_hil[] = {
136 #if defined(__sparc)
137 1.2435499454676143503135484916387102416568e-01L,
138 1.3203976161463874927468440652656953226250e-01L,
139 1.3970887428916364518336777673909505681607e-01L,
140 1.4736148108865163560980276039684551821066e-01L,
141 1.5499674192394098230371437493349219133371e-01L,
142 1.6261382859794857537364156376155780062019e-01L,
143 1.7021192528547440449049660709976171369543e-01L,
144 1.7779022899267607079662479921582468899456e-01L,
145 1.8534794999569476488602596122854464667261e-01L,
146 1.9288431225797466419705871069022730349878e-01L,
147 2.0039855382587851465394578503437838446153e-01L,
148 2.0788992720226299360533498310299432475629e-01L,
149 2.1535769969773804802445962716648964165745e-01L,
150 2.2280115375939451577103212214043255525024e-01L,
151 2.3021958727684373024017095967980299065551e-01L,
152 2.3761231386547125247388363432563777919892e-01L,
153 2.4497866312686415417208248121127580641959e-01L,
154 2.5962962940825753102994644318397190560106e-01L,
155 2.7416745111965879759937189834217578592444e-01L,
156 2.8858736189407739562361141995821834504332e-01L,
157 3.0288486837497140556055609450555821812277e-01L,
158 3.1705575320914700980901557667446732975852e-01L,
159 3.3109607670413209494433878775694455421259e-01L,
160 3.4500217720710510886768128690005168408290e-01L,
161 3.5877067027057222039592006392646052215363e-01L,
162 3.7239844667675422192365503828370182641413e-01L,
163 3.8588266939807377589769548460723139638186e-01L,
164 3.9922076957525256561471669615886476491104e-01L,
165 4.1241044159738730689979128966712694260920e-01L,
166 4.2544963737004228954226360518079233013817e-01L,
167 4.3833655985795780544561604921477130895882e-01L,
168 4.5106965598852347637563925728219344073798e-01L,
169 4.6364760900080611621425623146121439713344e-01L,
170 4.8833395105640552386716496074706484459644e-01L,
171 5.1238946031073770666660102058425923805558e-01L,
172 5.3581123796046370026908506870769144698471e-01L,
173 5.5859931534356243597150821640166122875873e-01L,
174 5.8075635356767039920327447500150082375122e-01L,
175 6.0228734613496418168212269420423291922459e-01L,
176 6.2319932993406593099247534906037459367793e-01L,
177 6.4350110879328438680280922871732260447265e-01L,
178 6.6320299270609325536325431023827583417226e-01L,
179 6.8231655487474807825642998171115298784729e-01L,
180 7.0085440788445017245795128178675127318623e-01L,
181 7.1882999962162450541701415152590469891043e-01L,
182 7.3625742898142813174283527108914662479274e-01L,
183 7.5315128096219438952473937026902888600575e-01L,
184 7.6952648040565826040682003598565401726598e-01L,
185 7.8539816339744830961566084581987569936977e-01L,
186 8.1569192331622341102146083874564582672284e-01L,
187 8.4415398611317100251784414827164746738632e-01L,
188 8.7090345707565295314017311259781407291650e-01L,
189 8.9605538457134395617480071802993779546602e-01L,
190 9.1971960535041681722860345482108940969311e-01L,
191 9.4200004037946366473793717053459362115891e-01L,
192 9.6299433068093620181519583599709989677298e-01L,
193 9.8279372324732906798571061101466603762572e-01L,
194 1.0014831356942347329183295953014374896343e+00L,
195 1.0191413442663497346383429170230636212354e+00L,
196 1.0358412530088001765846944703254440735476e+00L,
197 1.0516502125483736674598673120862999026920e+00L,
198 1.0666303653157435630791763474202799086015e+00L,
199 1.0808390005411683108871567292171997859003e+00L,
200 1.0943289073211899198927883146102352763033e+00L,
201 1.1071487177940905030170654601785370497543e+00L,
202 1.1309537439791604464709335155363277560026e+00L,
203 1.1525719972156675180401498626127514672834e+00L,
204 1.1722738811284763866005949441337046006865e+00L,
205 1.1902899496825317329277337748293182803384e+00L,
206 1.2068173702852525303955115800565576625682e+00L,
207 1.2220253232109896370417417439225704120294e+00L,
208 1.2360594894780819419094519711090786146210e+00L,
209 1.2490457723982544258299170772810900483550e+00L,
210 1.2610933822524404193139408812473357640124e+00L,
211 1.2722973952087173412961937498224805746463e+00L,
212 1.2827408797442707473628852511364955164072e+00L,
213 1.2924966677897852679030914214070816723528e+00L,
214 1.3016288340091961438047858503666855024453e+00L,
215 1.3101939350475556342564376891719053437537e+00L,
216 1.3182420510168370498593302023271363040427e+00L,
217 1.3258176636680324650592392104284756886164e+00L,
218 1.3397056595989995393283037525895557850243e+00L,
219 1.3521273809209546571891479413898127598774e+00L,
220 1.3633001003596939542892985278250991560269e+00L,
221 1.3734007669450158608612719264449610604836e+00L,
222 1.3825748214901258580599674177685685163955e+00L,
223 1.3909428270024183486427686943836432395486e+00L,
224 1.3986055122719575950126700816114282727858e+00L,
225 1.4056476493802697809521934019958080664406e+00L,
226 1.4121410646084952153676136718584890852820e+00L,
227 1.4181469983996314594038603039700988632607e+00L,
228 1.4237179714064941189018190466107297108905e+00L,
229 1.4288992721907326964184700745371984001389e+00L,
230 1.4337301524847089866404719096698873880264e+00L,
231 1.4382447944982225979614042479354816039669e+00L,
232 1.4424730991091018200252920599377291810352e+00L,
233 1.4464413322481351841999668424758803866109e+00L,
234 #elif defined(__x86)
235 1.243549945356789976358413696289e-01L,
236 1.320397615781985223293304443359e-01L,
237 1.397088742814958095550537109375e-01L,
238 1.473614810383878648281097412109e-01L,
239 1.549967419123277068138122558594e-01L,
240 1.626138285500928759574890136719e-01L,
241 1.702119252295233309268951416016e-01L,
242 1.777902289759367704391479492188e-01L,
243 1.853479499695822596549987792969e-01L,
244 1.928843122441321611404418945312e-01L,
245 2.003985538030974566936492919922e-01L,
246 2.078899272019043564796447753906e-01L,
247 2.153576996643096208572387695312e-01L,
248 2.228011537226848304271697998047e-01L,
249 2.302195872762240469455718994141e-01L,
250 2.376123138237744569778442382812e-01L,
251 2.449786631041206419467926025391e-01L,
252 2.596296293195337057113647460938e-01L,
253 2.741674510762095451354980468750e-01L,
254 2.885873618070036172866821289062e-01L,
255 3.028848683461546897888183593750e-01L,
256 3.170557531993836164474487304688e-01L,
257 3.310960766393691301345825195312e-01L,
258 3.450021771714091300964355468750e-01L,
259 3.587706702528521418571472167969e-01L,
260 3.723984466632828116416931152344e-01L,
261 3.858826693613082170486450195312e-01L,
262 3.992207695264369249343872070312e-01L,
263 4.124104415532201528549194335938e-01L,
264 4.254496373469009995460510253906e-01L,
265 4.383365598041564226150512695312e-01L,
266 4.510696559445932507514953613281e-01L,
267 4.636476089945062994956970214844e-01L,
268 4.883339509833604097366333007812e-01L,
269 5.123894601128995418548583984375e-01L,
270 5.358112377580255270004272460938e-01L,
271 5.585993151180446147918701171875e-01L,
272 5.807563534472137689590454101562e-01L,
273 6.022873460315167903900146484375e-01L,
274 6.231993297114968299865722656250e-01L,
275 6.435011087451130151748657226562e-01L,
276 6.632029926404356956481933593750e-01L,
277 6.823165547102689743041992187500e-01L,
278 7.008544078562408685684204101562e-01L,
279 7.188299994450062513351440429688e-01L,
280 7.362574287690222263336181640625e-01L,
281 7.531512808054685592651367187500e-01L,
282 7.695264802314341068267822265625e-01L,
283 7.853981633670628070831298828125e-01L,
284 8.156919232569634914398193359375e-01L,
285 8.441539860796183347702026367188e-01L,
286 8.709034570492804050445556640625e-01L,
287 8.960553845390677452087402343750e-01L,
288 9.197196052409708499908447265625e-01L,
289 9.420000403188169002532958984375e-01L,
290 9.629943305626511573791503906250e-01L,
291 9.827937232330441474914550781250e-01L,
292 1.001483135391026735305786132812e+00L,
293 1.019141343887895345687866210938e+00L,
294 1.035841252654790878295898437500e+00L,
295 1.051650212146341800689697265625e+00L,
296 1.066630364861339330673217773438e+00L,
297 1.080839000176638364791870117188e+00L,
298 1.094328907318413257598876953125e+00L,
299 1.107148717623203992843627929688e+00L,
300 1.130953743588179349899291992188e+00L,
301 1.152571997139602899551391601562e+00L,
302 1.172273880802094936370849609375e+00L,
303 1.190289949532598257064819335938e+00L,
304 1.206817369908094406127929687500e+00L,
305 1.222025323193520307540893554688e+00L,
306 1.236059489194303750991821289062e+00L,
307 1.249045772012323141098022460938e+00L,
308 1.261093381792306900024414062500e+00L,
309 1.272297394927591085433959960938e+00L,
310 1.282740879338234663009643554688e+00L,
311 1.292496667709201574325561523438e+00L,
312 1.301628833636641502380371093750e+00L,
313 1.310193934943526983261108398438e+00L,
314 1.318242050707340240478515625000e+00L,
315 1.325817663222551345825195312500e+00L,
316 1.339705659542232751846313476562e+00L,
317 1.352127380669116973876953125000e+00L,
318 1.363300099968910217285156250000e+00L,
319 1.373400766868144273757934570312e+00L,
320 1.382574821356683969497680664062e+00L,
321 1.390942826867103576660156250000e+00L,
322 1.398605511989444494247436523438e+00L,
323 1.405647648964077234268188476562e+00L,
324 1.412141064181923866271972656250e+00L,
325 1.418146998155862092971801757812e+00L,
326 1.423717970959842205047607421875e+00L,
327 1.428899271879345178604125976562e+00L,
328 1.433730152435600757598876953125e+00L,
329 1.438244794495403766632080078125e+00L,
330 1.442473099101334810256958007812e+00L,
331 1.446441331878304481506347656250e+00L,
332 #endif
333 };
334
335 static const long double TBL_atan_lol[] = {
336 #if defined(__sparc)
337 1.4074869197628063802317202820414310039556e-36L,
338 -4.9596961594739925555730439437999675295505e-36L,
339 8.9527745625194648873931213446361849472788e-36L,
340 1.1880437423207895718180765843544965589427e-35L,
341 -2.7810278112045145378425375128234365381448e-37L,
342 1.4797220377023800327295536234315147262387e-36L,
343 -4.2169561400548198732870384801849639863829e-36L,
344 7.2431229666913484649930323656316023494680e-36L,
345 -2.1573430089839170299895679353790663182462e-36L,
346 -9.9515745405126723554452367298128605186305e-36L,
347 -3.9065558992324838181617569730397882363067e-36L,
348 5.5260292271793726813211980664661124518807e-36L,
349 8.8415722215914321807682254318036452043689e-36L,
350 -8.1767728791586179254193323628285599800711e-36L,
351 -1.3344123034656142243797113823028330070762e-36L,
352 -4.4927331207813382908930733924681325892188e-36L,
353 4.4945511471812490393201824336762495687730e-36L,
354 -1.6688081504279223555776724459648440567274e-35L,
355 1.5629757586107955769461086568937329684113e-35L,
356 -2.2389835563308078552507970385331510848109e-35L,
357 -4.8312321745547311551870450671182151367050e-36L,
358 -1.4336172352905832876958926610980698844309e-35L,
359 -8.7440181998899932802989174170960593316080e-36L,
360 5.9284636008529837445780360785464550143016e-36L,
361 -2.2376651248436241276061055295043514993630e-35L,
362 6.0745837599336105414280310756677442136480e-36L,
363 1.5372187110451949677792344762029967023093e-35L,
364 2.0976068056751156241657121582478790247159e-35L,
365 -5.5623956405495438060726862202622807523700e-36L,
366 1.9697366707832471841858411934897351901523e-35L,
367 2.1070311964479488509034733639424887543697e-35L,
368 -2.3027356362982001602256518510854229844561e-35L,
369 4.8950964225733349266861843522029764772843e-36L,
370 -7.2380143477794458213872723050820253166391e-36L,
371 1.6365648865703614031637443396049568858105e-35L,
372 -3.9885811958234530793729129919803234197399e-35L,
373 4.1587722120912613510417783923227421336929e-35L,
374 3.8347421454556472153684687377337135027394e-35L,
375 -9.2251178933638721723515896465489002497864e-36L,
376 1.4094619690455989526175736741854656192178e-36L,
377 3.3568857805472235270612851425810803679451e-35L,
378 3.9090991055522552395018106803232118803401e-35L,
379 5.2956416979654208140521862707297033857956e-36L,
380 -5.0960846819945514367847063923662507136721e-36L,
381 -4.4959014425277615858329680393918315204998e-35L,
382 3.8039226544551634266566857615962609653834e-35L,
383 -4.4056522872895512108308642196611689657618e-36L,
384 1.6025024192482161076223807753425619076948e-36L,
385 2.1679525325309452561992610065108380635264e-35L,
386 1.9844038013515422125715362925736754104066e-35L,
387 3.9139619471799746834505227353568432457241e-35L,
388 2.1113443807975453505518453436799561854730e-35L,
389 3.1558557277444692755039816944392770185432e-35L,
390 1.6295044520355461408265585619500238335614e-35L,
391 -3.5087245209270305856151230356171213582305e-35L,
392 2.9041041864282855679591055270946117300088e-35L,
393 -2.3128843453818356590931995209806627233282e-35L,
394 -7.7124923181471578439967973820714857839953e-35L,
395 2.7539027829886922429092063590445808781462e-35L,
396 -9.4500899453181308951084545990839335972452e-35L,
397 -7.3061755302032092337594946001641651543473e-35L,
398 -4.1736144813953752193952770157406952602798e-35L,
399 3.4369948356256407045344855262863733571105e-35L,
400 -6.3790243492298090907302084924276831116460e-35L,
401 -9.6842943816353261291004127866079538980649e-36L,
402 4.8746757539138870909275958326700072821615e-35L,
403 -8.7533886477084190884511601368582548254655e-35L,
404 1.4284743992327918892692551138086727754845e-35L,
405 5.7262776211073389542565625693479173445042e-35L,
406 -3.2254883148780411245594822270747948565684e-35L,
407 7.8853548190609877325965525252380833808405e-35L,
408 8.4081736739037194097515038365370730251333e-35L,
409 7.4722870357563683815078242981933587273670e-35L,
410 7.9977202825793435289434813600890494256112e-36L,
411 -8.0577840773362139054848492346292673645405e-35L,
412 1.4217746753670583065490040209048757624336e-35L,
413 1.2232486914221205004109743560319090913328e-35L,
414 8.9696055070830036447361957217943988339065e-35L,
415 -3.1480394435081884410686066739846269858951e-35L,
416 -5.0927146040715345013240642517608928352977e-35L,
417 -5.7431997715924136568133859432702789493569e-35L,
418 -4.3920451405083770279099766080476485439987e-35L,
419 9.1106753984907715563018666776308759323326e-35L,
420 -3.7032569014272841009512400773061537538358e-35L,
421 8.8167419429746714276909825405131416764489e-35L,
422 -3.8389341696028352503752312861740895209678e-36L,
423 -3.3462959341960891546340895508017603408404e-35L,
424 -3.9212626776786074383916188498955828634947e-35L,
425 -7.8340397396377867255864494568594088378648e-35L,
426 7.4681018632456986520600640340627309824469e-35L,
427 8.9110918618956918451135594876165314884113e-35L,
428 3.9418160632271890530431797145664308529115e-35L,
429 -4.1048114088580104820193435638327617443913e-35L,
430 -2.3165419451582153326383944756220900454330e-35L,
431 -1.8428312581525319409399330203703211113843e-35L,
432 7.1477316546709482345411712017906842769961e-35L,
433 2.9914501578435874662153637707016094237004e-35L,
434 #elif defined(__x86)
435 1.108243739551347953496477557317e-11L,
436 3.644022694535396219063202730280e-11L,
437 7.667835628314065801595065768845e-12L,
438 5.026377078169301918590803009109e-11L,
439 1.161327548990211907411719105561e-11L,
440 4.785569941615255008968280209991e-11L,
441 5.595107356360146549819920947848e-11L,
442 1.673930035747684999707469623769e-11L,
443 2.611250523102718193166964451527e-11L,
444 1.384250305661681615897729354721e-11L,
445 2.278105796029649304219088055497e-11L,
446 3.586371256902077123693302823191e-13L,
447 3.342842716722085763523965049902e-11L,
448 3.670968534386232233574504707347e-11L,
449 6.196832945990602657404893210974e-13L,
450 4.169679549603939604438777470618e-11L,
451 2.274351222528987867221331091414e-11L,
452 8.872382531858169709022188891298e-11L,
453 4.344925246387385146717580155420e-11L,
454 8.707377833692929105196832265348e-11L,
455 2.881671577173773513055821329154e-11L,
456 9.763393361566846205717315422347e-12L,
457 6.476296480975626822569454546857e-11L,
458 3.569597877124574002505169001136e-11L,
459 1.772007853877284712958549977698e-11L,
460 1.347141028196192304932683248872e-11L,
461 3.676555884905046507598141175404e-11L,
462 4.881564068032948912761478588710e-11L,
463 4.416715404487185607337693704681e-11L,
464 2.314128999621257979016734983553e-11L,
465 5.380138283056477968352133002913e-11L,
466 4.393022562414389595406841771063e-11L,
467 6.299816718559209976839402028537e-12L,
468 7.304511413053165996581483735843e-11L,
469 1.978381648117426221467592544212e-10L,
470 2.024381732686578226139414070989e-10L,
471 2.255178211796380992141612703464e-10L,
472 1.204566302442290648452508620986e-10L,
473 1.034473912921080457667329099995e-10L,
474 2.225691010059030834353745950874e-10L,
475 4.817137162794350606107263804151e-11L,
476 6.565755971506095086327587326326e-11L,
477 1.644791039522307629611529931429e-10L,
478 2.820930388953087163050126809014e-11L,
479 1.766182540818701085571546539514e-10L,
480 2.124059054092171070266466628320e-10L,
481 1.567258302596026515190288816001e-10L,
482 1.742241535800378094231540188685e-10L,
483 3.038550253253096300737572104929e-11L,
484 5.925991958164150280814584656688e-11L,
485 3.355266774764151155289750652594e-11L,
486 2.637254809561744853531409402995e-11L,
487 3.227621096606048365493782702458e-11L,
488 1.094459672377587282585894259882e-10L,
489 6.064676448464127209709358607166e-11L,
490 1.182850444360454453720999258140e-10L,
491 1.428492049425553288966601449688e-11L,
492 3.032079976125434624889374125094e-10L,
493 3.784543889504767060855636487744e-10L,
494 3.540092982887960328254439790467e-10L,
495 4.020318667701700464612998296302e-10L,
496 4.544042324059585739827798668654e-10L,
497 3.645299460952866120296998202703e-10L,
498 2.776662293911361485235212513020e-12L,
499 1.708865101734375304910370400700e-10L,
500 3.909810965716415233488278047493e-10L,
501 7.606461848875826105025137974947e-11L,
502 3.263814502297453347587046149712e-10L,
503 1.499334758629144388918183376012e-10L,
504 3.771581242675818925565576303133e-10L,
505 1.746932950084818923507049088298e-11L,
506 2.837781909176306820465786987027e-10L,
507 3.859312847318946163435901230778e-10L,
508 4.601335192895268187473357720101e-10L,
509 2.811262558622337888849804940684e-10L,
510 4.060360843532416964489955306249e-10L,
511 8.058369357752989796958168458531e-11L,
512 3.725546414244147566166855921414e-10L,
513 1.040286509953292907344053122733e-10L,
514 3.094968093808145773271362531155e-10L,
515 4.454811192340438979284756311844e-10L,
516 5.676678748199027602705574110388e-11L,
517 2.518376833121948163898128509842e-10L,
518 3.907837370041422778250991189943e-10L,
519 7.687158710333735613246114865100e-11L,
520 1.334418885622867537060685125566e-10L,
521 1.353147719826124443836432060856e-10L,
522 2.825131007652335581739282335732e-10L,
523 4.161925466840049254333079881002e-10L,
524 4.265713490956410156084891599630e-10L,
525 2.437693664320585461575989523716e-10L,
526 4.466519138542116247357297503086e-10L,
527 3.113875178143440979746983590908e-10L,
528 4.910822904159495654488736486097e-11L,
529 2.818831329324169810481585538618e-12L,
530 7.767009768334052125229252512543e-12L,
531 3.698307026936191862258804165254e-10L,
532 #endif
533 };
534
535 /*
536 * mx_atanl(x, err)
537 * Table look-up algorithm
538 * By K.C. Ng, March 9, 1989
539 *
540 * Algorithm.
541 *
542 * The algorithm is based on atan(x)=atan(y)+atan((x-y)/(1+x*y)).
543 * We use poly1(x) to approximate atan(x) for x in [0,1/8] with
544 * error (relative)
545 * |(atan(x)-poly1(x))/x|<= 2^-140
546 *
547 * and use poly2(x) to approximate atan(x) for x in [0,1/65] with
548 * error
549 * |atan(x)-poly2(x)|<= 2^-143.7
550 *
551 * Here poly1 and poly2 are odd polynomial with the following form:
552 * x + x^3*(a1+x^2*(a2+...))
553 *
554 * (0). Purge off Inf and NaN and 0
555 * (1). Reduce x to positive by atan(x) = -atan(-x).
556 * (2). For x <= 1/8, use
557 * (2.1) if x < 2^(-prec/2), atan(x) = x with inexact flag raised
558 * (2.2) Otherwise
559 * atan(x) = poly1(x)
560 * (3). For x >= 8 then (prec = 78)
561 * (3.1) if x >= 2^prec, atan(x) = atan(inf) - pio2_lo
562 * (3.2) if x >= 2^(prec/3), atan(x) = atan(inf) - 1/x
563 * (3.3) if x > 65, atan(x) = atan(inf) - poly2(1/x)
564 * (3.4) Otherwise, atan(x) = atan(inf) - poly1(1/x)
565 *
566 * (4). Now x is in (0.125, 8)
567 * Find y that match x to 4.5 bit after binary (easy).
568 * If iy is the high word of y, then
569 * single : j = (iy - 0x3e000000) >> 19
570 * double : j = (iy - 0x3fc00000) >> 16
571 * quad : j = (iy - 0x3ffc0000) >> 12
572 *
573 * Let s = (x-y)/(1+x*y). Then
574 * atan(x) = atan(y) + poly1(s)
575 * = _TBL_atan_hi[j] + (_TBL_atan_lo[j] + poly2(s) )
576 *
577 * Note. |s| <= 1.5384615385e-02 = 1/65. Maxium occurs at x = 1.03125
578 *
579 */
580
581 /* BEGIN CSTYLED */
582 /*
583 * p[0] - p[16] for atan(x) =
584 * x + x^3*(p1+x^2*(p2+...))
585 */
586 static const long double pe[] = {
587 1.0L,
588 0.0L,
589 #if defined(__sparc)
590 -0.33333333333333332870740406406184774823L,
591 -4.62592926927148558508441072595508240609e-18L,
592 0.19999999999999999722444243843710864894L,
593 2.77555756156289124602047010782090464486e-18L,
594 -0.14285714285714285615158658515611023176L,
595 -9.91270557700756738621231719241800559409e-19L,
596 #elif defined(__x86)
597 -0.33333333325572311878204345703125L,
598 -7.76102145512898763020833333192787755766644373e-11L,
599 0.19999999995343387126922607421875L,
600 4.65661287307739257812498949613909375938538636e-11L,
601 -0.142857142840512096881866455078125L,
602 -1.66307602609906877787419703858463013035681375e-11L,
603 #endif
604 };
605
606 static const long double p[] = { /* p[0] - p[16] */
607 1.0L,
608 -3.33333333333333333333333333333333333319278775586e-0001L,
609 1.99999999999999999999999999999999894961390937601e-0001L,
610 -1.42857142857142857142857142856866970385846301312e-0001L,
611 1.11111111111111111111111110742899094415954427738e-0001L,
612 -9.09090909090909090909087972707015549231951421806e-0002L,
613 7.69230769230769230767699003016385628597359717046e-0002L,
614 -6.66666666666666666113842763495291228025226575259e-0002L,
615 5.88235294117646915706902204947653640091126695962e-0002L,
616 -5.26315789473657016886225044679594035524579379810e-0002L,
617 4.76190476186633969331771169790375592681525481267e-0002L,
618 -4.34782608290146274616081389793141896576997370161e-0002L,
619 3.99999968161267722260103962788865225205057218988e-0002L,
620 -3.70368536844778256320786172745225703228683638328e-0002L,
621 3.44752320396524479494062858284036892703898522150e-0002L,
622 -3.20491216046653214683721787776813360591233428081e-0002L,
623 2.67632651033434456758550618122802167256870856514e-0002L,
624 };
625
626 /* q[0] - q[9] */
627 static const long double qe[] = {
628 1.0L,
629 0.0L,
630 #if defined(__sparc)
631 -0.33333333333333332870740406406184774823486804962158203125L,
632 -4.625929269271485585069345465471207312531868714634217630e-18L,
633 0.19999999999999999722444243843710864894092082977294921875L,
634 2.7755575615628864268260553912956813621977220359134667560e-18L,
635 #elif defined(__x86)
636 -0.33333333325572311878204345703125L,
637 -7.76102145512898763020833333042135150927893e-11L,
638 0.19999999995343387126922607421875L,
639 4.656612873077392578124507576697622106863058e-11L,
640 #endif
641 };
642
643 static const long double q[] = { /* q[0] - q[9] */
644 -3.33333333333333333333333333333333333304213515094e-0001L,
645 1.99999999999999999999999999999995075766976221077e-0001L,
646 -1.42857142857142857142857142570379604317921113079e-0001L,
647 1.11111111111111111111102923861900979127978214077e-0001L,
648 -9.09090909090909089586854075816999506863320031460e-0002L,
649 7.69230769230756334929213246003824644696974730368e-0002L,
650 -6.66666666589192433974402013508912138168133579856e-0002L,
651 5.88235013696778007696800252045588307023299350858e-0002L,
652 -5.25754959898164576495303840687699583228444695685e-0002L,
653 };
654
655 static const long double two8700 = 9.140338438955067659002088492701e+2618L, /* 2^8700 */
656 twom8700 = 1.094051392821643668051436593760e-2619L, /* 2^-8700 */
657 one = 1.0L,
658 zero = 0.0L,
659 pi = 3.1415926535897932384626433832795028841971693993751L,
660 pio2 = 1.57079632679489661923132169163975144209858469968755L,
661 pio4 = 0.785398163397448309615660845819875721049292349843776L,
662 pi3o4 = 2.356194490192344928846982537459627163147877049531329L,
663 #if defined(__sparc)
664 pi_lo = 8.67181013012378102479704402604335196876232e-35L,
665 pio2_lo = 4.33590506506189051239852201302167598438116e-35L,
666 pio4_lo = 2.16795253253094525619926100651083799219058e-35L,
667 pi3o4_lo = 6.50385759759283576859778301953251397657174e-35L;
668 #elif defined(__x86)
669 pi_lo = -5.01655761266833202355732708e-20L,
670 pio2_lo = -2.50827880633416601177866354e-20L,
671 pio4_lo = -1.25413940316708300588933177e-20L,
672 pi3o4_lo = -9.18342907192877118770525931e-20L;
673 #endif
674 /* END CSTYLED */
675
676 static long double
677 mx_atanl(long double x, long double *err)
678 {
679 long double y, z, r, s, t, w, s_h, s_l, x_h, x_l, zz[3], ee[2], z_h,
680 z_l, r_h, r_l, u, v;
681 int ix, iy, hx, i, j;
682 float fx;
683
684 hx = HI_XWORD(x);
685 ix = hx & (~0x80000000);
686
687 /* for |x| < 1/8 */
688 if (ix < 0x3ffc0000) {
689 if (ix < 0x3ff30000) { /* when |x| < 2**-12 */
690 if (ix < 0x3fc60000) { /* if |x| < 2**-prec/2 */
691 *err = (long double)((int)x);
692 return (x);
693 }
694
695 z = x * x;
696 t = q[8];
697
698 for (i = 7; i >= 0; i--)
699 t = q[i] + z * t;
700
701 t *= x * z;
702 r = x + t;
703 *err = t - (r - x);
704 return (r);
705 }
706
707 z = x * x;
708
709 /* use long double precision at p4 and on */
710 t = p[16];
711
712 for (i = 15; i >= 4; i--)
713 t = p[i] + z * t;
714
715 ee[0] = z * t;
716
717 x_h = x;
718 HALF(x_h);
719 z_h = z;
720 HALF(z_h);
721 x_l = x - x_h;
722 z_l = (x_h * x_h - z_h);
723 zz[0] = z;
724 zz[1] = z_h;
725 zz[2] = z_l + x_l * (x + x_h);
726
727 /* compute (1+z*(p1+z*(p2+z*(p3+e)))) */
728
729 mx_polyl(zz, pe, ee, 3);
730
731 /* finally x*(1+z*(p1+...)) */
732 r = x_h * ee[0];
733 t = x * ee[1] + x_l * ee[0];
734 s = t + r;
735 *err = t - (s - r);
736 return (s);
737 }
738
739 /* for |x| >= 8.0 */
740 if (ix >= 0x40020000) { /* x >= 8 */
741 x = fabsl(x);
742
743 if (ix >= 0x402e0000) { /* x >= 2**47 */
744 if (ix >= 0x408b0000) /* x >= 2**140 */
745 y = -pio2_lo;
746 else
747 y = one / x - pio2_lo;
748
749 if (hx >= 0) {
750 t = pio2 - y;
751 *err = -(y - (pio2 - t));
752 } else {
753 t = y - pio2;
754 *err = y - (pio2 + t);
755 }
756
757 return (t);
758 } else {
759 /* compute r = 1/x */
760 r = one / x;
761 z = r * r;
762 x_h = x;
763 HALF(x_h);
764 r_h = r;
765 HALF(r_h);
766 z_h = z;
767 HALF(z_h);
768 r_l = r * ((x_h - x) * r_h - (x_h * r_h - one));
769 z_l = (r_h * r_h - z_h);
770 zz[0] = z;
771 zz[1] = z_h;
772 zz[2] = z_l + r_l * (r + r_h);
773
774 if (ix < 0x40050400) { /* 8 < x < 65 */
775 /* use double precision at p4 and on */
776 t = p[16];
777
778 for (i = 15; i >= 4; i--)
779 t = p[i] + z * t;
780
781 ee[0] = z * t;
782 /* compute (1+z*(p1+z*(p2+z*(p3+e)))) */
783 mx_polyl(zz, pe, ee, 3);
784 } else { /* x < 65 < 2**47 */
785 /* use long double at q3 and on */
786 t = q[8];
787
788 for (i = 7; i >= 2; i--)
789 t = q[i] + z * t;
790
791 ee[0] = z * t;
792 /* compute (1+z*(q1+z*(q2+e))) */
793 mx_polyl(zz, qe, ee, 2);
794 }
795
796 /* pio2 - r*(1+...) */
797 v = r_h * ee[0];
798 t = pio2_lo - (r * ee[1] + r_l * ee[0]);
799
800 if (hx >= 0) {
801 s = pio2 - v;
802 t -= (v - (pio2 - s));
803 } else {
804 s = v - pio2;
805 t = -(t - (v - (s + pio2)));
806 }
807
808 w = s + t;
809 *err = t - (w - s);
810 return (w);
811 }
812 }
813
814 /* now x is between 1/8 and 8 */
815 iy = (ix + 0x00000800) & 0x7ffff000;
816 j = (iy - 0x3ffc0000) >> 12;
817 ((int *)&fx)[0] = 0x3e000000 + (j << 19);
818 y = (long double)fx;
819 x = fabsl(x);
820
821 w = (x - y);
822 v = 1.0L / (one + x * y);
823 s = w * v;
824 z = s * s;
825 /* use long double precision at q3 and on */
826 t = q[8];
827
828 for (i = 7; i >= 2; i--)
829 t = q[i] + z * t;
830
831 ee[0] = z * t;
832 s_h = s;
833 HALF(s_h);
834 z_h = z;
835 HALF(z_h);
836 x_h = x;
837 HALF(x_h);
838 t = one + x * y;
839 HALF(t);
840 r = -((x_h - x) * y - (x_h * y - (t - one)));
841 s_l = -v * (s_h * r - (w - s_h * t));
842 z_l = (s_h * s_h - z_h);
843 zz[0] = z;
844 zz[1] = z_h;
845 zz[2] = z_l + s_l * (s + s_h);
846 /* compute (1+z*(q1+z*(q2+e))) by call mx_poly */
847 mx_polyl(zz, qe, ee, 2);
848 v = s_h * ee[0];
849 t = TBL_atan_lol[j] + (s * ee[1] + s_l * ee[0]);
850 u = TBL_atan_hil[j];
851 s = u + v;
852 t += (v - (s - u));
853 w = s + t;
854 *err = t - (w - s);
855
856 if (hx < 0) {
857 w = -w;
858 *err = -*err;
859 }
860
861 return (w);
862 }
863
864 long double
865 __k_atan2l(long double y, long double x, long double *w)
866 {
867 long double t, xh, th, t1, t2, w1, w2;
868 int ix, iy, hx, hy;
869
870 hy = HI_XWORD(y);
871 hx = HI_XWORD(x);
872 iy = hy & ~0x80000000;
873 ix = hx & ~0x80000000;
874
875 *w = 0.0;
876
877 if (ix >= 0x7fff0000 || iy >= 0x7fff0000) { /* ignore inexact */
878 if (isnanl(x) || isnanl(y)) {
879 return (x * y);
880 } else if (iy < 0x7fff0000) {
881 if (hx >= 0) { /* ATAN2(+-finite, +inf) is +-0 */
882 *w *= y;
883 return (*w);
884 } else { /* ATAN2(+-finite, -inf) is +-pi */
885 *w = copysignl(pi_lo, y);
886 return (copysignl(pi, y));
887 }
888 } else if (ix < 0x7fff0000) {
889 /* ATAN2(+-inf, finite) is +-pi/2 */
890 *w = (hy >= 0) ? pio2_lo : -pio2_lo;
891 return ((hy >= 0) ? pio2 : -pio2);
892 } else if (hx > 0) { /* ATAN2(+-INF,+INF) = +-pi/4 */
893 *w = (hy >= 0) ? pio4_lo : -pio4_lo;
894 return ((hy >= 0) ? pio4 : -pio4);
895 } else { /* ATAN2(+-INF,-INF) = +-3pi/4 */
896 *w = (hy >= 0) ? pi3o4_lo : -pi3o4_lo;
897 return ((hy >= 0) ? pi3o4 : -pi3o4);
898 }
899 } else if (x == zero || y == zero) {
900 if (y == zero) {
901 if (hx >= 0) { /* ATAN2(+-0, +(0 <= x <= inf)) is +-0 */
902 return (y);
903 } else { /* ATAN2(+-0, -(0 <= x <= inf)) is +-pi */
904 *w = (hy >= 0) ? pi_lo : -pi_lo;
905 return ((hy >= 0) ? pi : -pi);
906 }
907 } else { /* ATAN2(+-(anything but 0 and NaN), 0) is +-pi/2 */
908 *w = (hy >= 0) ? pio2_lo : -pio2_lo;
909 return ((hy >= 0) ? pio2 : -pio2);
910 }
911 } else if (iy - ix > 0x00640000) { /* |x/y| < 2 ** -100 */
912 *w = (hy >= 0) ? pio2_lo : -pio2_lo;
913 return ((hy >= 0) ? pio2 : -pio2);
914 } else if (ix - iy > 0x00640000) { /* |y/x| < 2 ** -100 */
915 if (hx < 0) {
916 *w = (hy >= 0) ? pi_lo : -pi_lo;
917 return ((hy >= 0) ? pi : -pi);
918 } else {
919 t = y / x;
920 th = t;
921 HALF(th);
922 xh = x;
923 HALF(xh);
924 t1 = (x - xh) * t + xh * (t - th);
925 t2 = y - xh * th;
926 *w = (t2 - t1) / x;
927 return (t);
928 }
929 } else {
930 if (ix >= 0x5fff3000) {
931 x *= twom8700;
932 y *= twom8700;
933 } else if (ix < 0x203d0000) {
934 x *= two8700;
935 y *= two8700;
936 }
937
938 y = fabsl(y);
939 x = fabsl(x);
940 t = y / x;
941 th = t;
942 HALF(th);
943 xh = x;
944 HALF(xh);
945 t1 = (x - xh) * t + xh * (t - th);
946 t2 = y - xh * th;
947 w1 = mx_atanl(t, &w2);
948 w2 += (t2 - t1) / (x + y * t);
949
950 if (hx < 0) {
951 t1 = pi - w1;
952 t2 = pi - t1;
953 w2 = (pi_lo - w2) - (w1 - t2);
954 w1 = t1;
955 }
956
957 *w = (hy >= 0) ? w2 : -w2;
958 return ((hy >= 0) ? w1 : -w1);
959 }
960 }