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_clog_rl */
30 #include "complex_wrapper.h"
31 #include "longdouble.h"
32
33 /* INDENT OFF */
34 /*
35 * long double __k_clog_rl(long double x, long double y, long double *e);
36 *
37 * Compute real part of complex natural logarithm of x+iy in extra precision
38 *
39 * __k_clog_rl returns log(hypot(x, y)) with a correction term e.
40 *
41 * Accuracy: quad 140 bits, intel extended 91 bits.
42 *
43 * Method.
44 * Assume X > Y >= 0 . Let X = 2**nx * x, Y = 2**nx * y, where 1 <= x < 2.
45 * Let Z = X*X + Y*Y. Then Z = 2**(nx+nx) * z, where z = x*x + y*y.
46 * Note that z < 8.
47 * Let Z = x*x + y*y. Z can be normalized as Z = 2**N * z, 1 <= z < 2.
48 * We further break down z into 1 + zk + zh + zt, where
49 * zk = K*(2**-7) matches z to 7.5 significant bits, 0 <= K <= 2**(-7)-1
50 * zh = (z-zk) rounded to half of the current significant bits
51 * zt = (z-zk-zh) rounded.
52 *
53 * z - (1+zk) (zh+zt)
54 * Let s = ------------ = ---------------, then
55 * z + (1+zk) 2(1+zk)+zh+zt
56 * z
57 * log(Z) = N*log2 + log(z) = N*log2 + log(1+zk) + log(------)
58 * 1+zk
59 * 1+s
60 * = N * log2 + log(1 +zk) + log(---)
61 * 1-s
62 *
63 * 3 5
64 * = N*log2 + log(1+zk) + 2s + 1/12(2s) + 1/80(2s) + ...
65 *
66 *
67 * Note 1. For IEEE double precision, a fifteen degree odd polynomial
68 * 2s + P1*(2s)^3 + P2*(2s)^5 + P3*(2s)^7 + ... + P7*(2s)^15
69 * is generated by a special remez algorithm to
70 * approx log((1+s)/(1-s)) accurte to 145 bits.
71 * Note 2. 2s can be computed accurately as s2h+s2t by
72 * r = 2/((zh+zt)+2(1+zk))
73 * s2 = r*(zh+zt)
74 * s2h = s2 rounded to double; v = 0.5*s2h;
75 * s2t = r*((((zh-s2h*(1+zk))-v*zh)+zt)-v*zt)
76 */
77 /* INDENT ON */
78
79 static const long double
80 zero = 0.0L,
81 half = 0.5L,
82 two = 2.0L,
83 two240 = 1.7668470647783843295832975007429185158274839e+72L, /* 2^240 */
84
85 /* first 48 bits of ln2 */
86 ln2_h = 0.693147180559943620892227045260369777679443359375L,
87 ln2_t = 1.68852500507619780679039605677498525525412068e-15L,
88 P1 = .083333333333333333333333333333333333341023785768375L,
89 P2 = .01249999999999999999999999999999679085402075766159375L,
90 P3 = .002232142857142857142857143310092047621284490564671875L,
91 P4 = .00043402777777777777774746781319264872413156956512109375L,
92 P5 = .0000887784090909101756336594019277185263940665468935546875L,
93 P6 = .000018780048055589639895360927834628371268354778446533203125L,
94 P7 = .000004069227854328982921366736003458838031087153635406494140625L;
95
96 /*
97 * T[2k, 2k+1] = log(1+k*2**-7) for k = 0, ..., 2**7 - 1,
98 * with T[2k] * 2^48 is an int
99 */
100
101 static const long double TBL_log1k[] = {
102 0.0000000000000000000000000000000000000000e+00L,
103 0.0000000000000000000000000000000000000000e+00L,
104 7.7821404420532758194894995540380477905273e-03L,
105 1.6731279734005070987158875984584325351222e-15L,
106 1.5504186535963526694104075431823730468750e-02L,
107 1.7274567499706106231054091184928671990316e-15L,
108 2.3167059281533397552266251295804977416992e-02L,
109 9.8067653290966648493916241687661877474892e-16L,
110 3.0771658666751022792595904320478439331055e-02L,
111 2.6655784323032762937247606420524589813624e-15L,
112 3.8318864302134159061097307130694389343262e-02L,
113 2.4401326580179931029010027013316092332340e-15L,
114 4.5809536031292452662455616518855094909668e-02L,
115 1.7505042236510958082472042641283104263139e-15L,
116 5.3244514518809182845870964229106903076172e-02L,
117 3.1000199992295574218738634002122149891138e-15L,
118 6.0624621816433688081815489567816257476807e-02L,
119 1.1544987906424726040058093958345197512800e-15L,
120 6.7950661908504628172522643581032752990723e-02L,
121 3.1212220426341915966610439115772728417386e-15L,
122 7.5223421237584631171557703055441379547119e-02L,
123 2.8945270476369282210350897509258766743153e-15L,
124 8.2443669211073711267090402543544769287109e-02L,
125 8.8000106966612476303662698634483335676886e-16L,
126 8.9612158689686083334891009144484996795654e-02L,
127 1.0492850604602339995319895311151740799226e-15L,
128 9.6729626458550654888313147239387035369873e-02L,
129 4.5740725790924807640164516707244620870662e-16L,
130 1.0379679368164218544734467286616563796997e-01L,
131 1.3793787171308978090503366050174239822054e-15L,
132 1.1081436634028918319927470292896032333374e-01L,
133 9.3099553146639425160476473362380086036919e-16L,
134 1.1778303565638026384476688690483570098877e-01L,
135 3.1906940272225656860040797111813146690890e-15L,
136 1.2470347850095464536934741772711277008057e-01L,
137 2.5904940590976537504984110469214193890052e-15L,
138 1.3157635778871679121948545798659324645996e-01L,
139 2.4813692306707028899159917911012100567219e-15L,
140 1.3840232285911824305912887211889028549194e-01L,
141 8.9262619700148275890190121571708972000380e-16L,
142 1.4518200984449691759436973370611667633057e-01L,
143 9.7968756533003444764719201050911636480025e-16L,
144 1.5191604202583874894116888754069805145264e-01L,
145 3.2261306345373561864598749471119213018106e-15L,
146 1.5860503017663774016909883357584476470947e-01L,
147 8.4392427234104999681053621980394827998735e-16L,
148 1.6524957289530561865831259638071060180664e-01L,
149 1.5442172988528965297119225948270579746101e-15L,
150 1.7185025692665689689420105423778295516968e-01L,
151 2.3254458978918173643097657009894831132739e-15L,
152 1.7840765747281750464026117697358131408691e-01L,
153 7.9247913906453736065426776912520942036896e-16L,
154 1.8492233849401173984006163664162158966064e-01L,
155 2.5282384195601762803134514624610774126020e-16L,
156 1.9139485299962899489401024766266345977783e-01L,
157 4.5971528855989864541366920731297729269228e-16L,
158 1.9782574332991842425144568551331758499146e-01L,
159 1.4561111263856836438840838027526567191527e-15L,
160 2.0421554142868814096800633706152439117432e-01L,
161 2.7505358140491347148810394262840919337709e-15L,
162 2.1056476910734645002776233013719320297241e-01L,
163 3.1876417904825951583107481283088861928977e-15L,
164 2.1687393830061196808856038842350244522095e-01L,
165 2.3915305291373208450532580201045871599499e-15L,
166 2.2314355131420882116799475625157356262207e-01L,
167 9.3459830033405826094075253077304795996257e-16L,
168 2.2937410106484534821902343537658452987671e-01L,
169 4.8177245728966955534167425511952551974164e-16L,
170 2.3556607131276408040321257431060075759888e-01L,
171 2.8286743756446304426525380844720043381780e-15L,
172 2.4171993688714366044223424978554248809814e-01L,
173 1.5077020732661279714120052415509585052975e-15L,
174 2.4783616390458007572306087240576744079590e-01L,
175 1.1810575418933407573072030113600980623171e-15L,
176 2.5391520998096339667426946107298135757446e-01L,
177 4.7463053836833625309891834934881898560705e-17L,
178 2.5995752443692410338371701072901487350464e-01L,
179 1.9635883624838132961710716735786266795913e-15L,
180 2.6596354849713677026556979399174451828003e-01L,
181 1.1710735561325457988709887923652142233351e-15L,
182 2.7193371548364098089223261922597885131836e-01L,
183 7.7793943687530702031066421537496360004376e-16L,
184 2.7786845100345303194444568362087011337280e-01L,
185 3.2742419043493025311197092322146237692165e-15L,
186 2.8376817313064250924981024581938982009888e-01L,
187 2.0890970909765308649465619266075677112425e-15L,
188 2.8963329258304071345264674164354801177979e-01L,
189 1.9634262463138821209582240742801727823629e-15L,
190 2.9546421289383317798638017848134040832520e-01L,
191 2.6984003017275736237868564402005801750600e-15L,
192 3.0126133057816062432721082586795091629028e-01L,
193 1.1566856647123658045763670687640673680383e-15L,
194 3.0702503529490954292668902780860662460327e-01L,
195 2.3191484355127267712770857311812090801833e-15L,
196 3.1275571000389490450288576539605855941772e-01L,
197 1.9838833607942922604727420618882220398852e-15L,
198 3.1845373111853447767316538374871015548706e-01L,
199 1.3813708182984188944010814590398164268227e-16L,
200 3.2411946865421015218089451082050800323486e-01L,
201 1.8239097762496144793489474731253815376404e-15L,
202 3.2975328637246548169059678912162780761719e-01L,
203 2.5001238260227991620033344720809714552230e-15L,
204 3.3535554192113536942088103387504816055298e-01L,
205 2.4608362985459391180385214539620341910962e-15L,
206 3.4092658697059263772644044365733861923218e-01L,
207 5.7257864875612301758921090406373771458003e-16L,
208 3.4646676734620740489845047704875469207764e-01L,
209 1.1760200117113770182586341947822306069951e-15L,
210 3.5197642315717558858523261733353137969971e-01L,
211 2.5960702148389259075462896448369304790506e-15L,
212 3.5745588892180180096147523727267980575562e-01L,
213 1.9732645342528682246686790561260072184839e-15L,
214 3.6290549368936808605212718248367309570312e-01L,
215 3.6708569716349381675043725477739939978160e-16L,
216 3.6832556115870573876236448995769023895264e-01L,
217 1.9142858656640927085879445412821643247628e-15L,
218 3.7371640979358389245135185774415731430054e-01L,
219 1.8836966497497166619234389157276681281343e-16L,
220 3.7907835293496816575498087331652641296387e-01L,
221 1.2926358724723144934459175417385013725801e-15L,
222 3.8441169891033055705520382616668939590454e-01L,
223 1.4826795862363146014726140088145939341729e-15L,
224 3.8971675114002479745067830663174390792847e-01L,
225 4.1591978529737177695912258866565331189698e-16L,
226 3.9499380824086571806219581048935651779175e-01L,
227 3.2600441982258756252505182317625310732365e-15L,
228 4.0024316412701210765590076334774494171143e-01L,
229 5.9927342433864738622836851475469574662703e-16L,
230 4.0546510810816371872533636633306741714478e-01L,
231 6.6325267674913128171942721503283748008372e-16L,
232 4.1065992498526782128465129062533378601074e-01L,
233 5.6464965491255048900165082436455718077885e-16L,
234 4.1582789514371043537721561733633279800415e-01L,
235 5.3023611327561856950735176370587227509442e-16L,
236 4.2096929464412724541944044176489114761353e-01L,
237 2.3907094267197419048248363335257046791153e-15L,
238 4.2608439531089814522601955104619264602661e-01L,
239 1.9178985253285492839728700574592375309985e-15L,
240 4.3117346481836804628073878120630979537964e-01L,
241 3.2945784336977492852031005044499611665595e-15L,
242 4.3623676677491474151793227065354585647583e-01L,
243 3.3288311090524075754441878570852962903891e-15L,
244 4.4127456080487448275562201160937547683716e-01L,
245 7.4673387443005192574852544613692268411229e-16L,
246 4.4628710262841764233598951250314712524414e-01L,
247 1.8691966006681165218815050615460959199251e-15L,
248 4.5127464413945617138779198285192251205444e-01L,
249 2.4137569004002270899666314791611479063976e-15L,
250 4.5623743348158640742440184112638235092163e-01L,
251 1.1869564036970375473975162509216610120281e-15L,
252 4.6117571512216670726047595962882041931152e-01L,
253 3.4591075239659690349392915732654828400811e-15L,
254 4.6608972992459740680715185590088367462158e-01L,
255 1.8177514673916038857252366108673570603067e-15L,
256 4.7097971521878889689105562865734100341797e-01L,
257 2.1156558422273990182479555421331461933366e-15L,
258 4.7584590486996347635795245878398418426514e-01L,
259 4.3790725712752039722791012358345927696967e-16L,
260 4.8068852934575190261057286988943815231323e-01L,
261 5.0660455855585733988956280680891477171499e-18L,
262 4.8550781578169832641833636444061994552612e-01L,
263 2.4813834547127501689550526444948043590905e-15L,
264 4.9030398804519137456736643798649311065674e-01L,
265 2.4635829797216592537498738468934647345741e-15L,
266 4.9507726679784980206022737547755241394043e-01L,
267 1.7125377372093652812514167461480115600063e-15L,
268 4.9982786955644797899367404170334339141846e-01L,
269 1.3508276573735437007500942002018098437396e-15L,
270 5.0455601075239187025545106735080480575562e-01L,
271 3.4168028574643873701242268618467347998876e-15L,
272 5.0926190178980590417268103919923305511475e-01L,
273 2.0426313938800290907697638200502614622891e-15L,
274 5.1394575110223428282552049495279788970947e-01L,
275 3.3975485593321419703400672813719873194659e-17L,
276 5.1860776420804555186805373523384332656860e-01L,
277 8.0284923261130955371987633083003284697416e-17L,
278 5.2324814376454753528378205373883247375488e-01L,
279 3.0123302517119603836788558832352723470118e-16L,
280 5.2786708962084105678513878956437110900879e-01L,
281 1.3283287534282139298545497336570406582397e-15L,
282 5.3246479886946929127589100971817970275879e-01L,
283 2.5525980327137419625398485590148417041921e-15L,
284 5.3704146589688050994482182431966066360474e-01L,
285 3.1446219074198341716354190061340477078626e-15L,
286 5.4159728243274329884116014000028371810913e-01L,
287 1.0727353821639001503808606766770295812627e-15L,
288 5.4613243759813556721383065450936555862427e-01L,
289 8.3168566554721843605240702438699163825794e-17L,
290 5.5064711795266063631970610003918409347534e-01L,
291 1.6429402420791657293666192255419538448840e-15L,
292 5.5514150754050106684189813677221536636353e-01L,
293 5.2587358222274368868380660194332415847228e-16L,
294 5.5961578793542088305912329815328121185303e-01L,
295 1.8032117652023735453816330571171114110385e-15L,
296 5.6407013828480145889443519990891218185425e-01L,
297 1.5071769490901812785299634348367857600711e-15L,
298 5.6850473535266843327917740680277347564697e-01L,
299 2.7879956135806418878792935692629147550413e-16L,
300 5.7291975356178426181941176764667034149170e-01L,
301 1.2472733449589795907271346997596471822345e-15L,
302 5.7731536503482061561953742057085037231445e-01L,
303 2.9886985746409486460291929160223207644146e-15L,
304 5.8169173963462128540413687005639076232910e-01L,
305 1.1971164738836689815783808674399742176950e-15L,
306 5.8604904500357690722012193873524665832520e-01L,
307 1.3016839974975520776911897855504474452726e-15L,
308 5.9038744660217545856539800297468900680542e-01L,
309 9.1607651870514890975077236127894522134392e-16L,
310 5.9470710774668944509357970673590898513794e-01L,
311 3.3444207638397932963480545729233567201211e-15L,
312 5.9900818964608149030937056522816419601440e-01L,
313 1.9090722294592334873060460706130642200729e-15L,
314 6.0329085143808214297678205184638500213623e-01L,
315 2.1193638031348149256035110177854940281795e-15L,
316 6.0755525022453937822319858241826295852661e-01L,
317 2.4172778865703728624133665395876418941354e-15L,
318 6.1180154110599005434778518974781036376953e-01L,
319 2.8491821045766810044199163148675291775782e-15L,
320 6.1602987721551372146677749697118997573853e-01L,
321 2.9818078843122551067455400545109858745295e-16L,
322 6.2024040975185457114093878772109746932983e-01L,
323 2.9577105558448461493874424529516311623184e-15L,
324 6.2443328801189323939979658462107181549072e-01L,
325 2.6164274215943360130441858075903119505815e-16L,
326 6.2860865942237253989333112258464097976685e-01L,
327 1.5978509770831895426601797458058854400463e-15L,
328 6.3276666957103699928666173946112394332886e-01L,
329 8.3025912472904245581515990140161946934461e-16L,
330 6.3690746223706895534633076749742031097412e-01L,
331 2.7627416365968377888021629180796328536455e-16L,
332 6.4103117942092779912854894064366817474365e-01L,
333 3.4919270523937617243719652995048419893186e-15L,
334 6.4513796137358170312836591619998216629028e-01L,
335 2.9985368625799347497396478978681548584217e-15L,
336 6.4922794662510696639401430729776620864868e-01L,
337 2.8524968256626075449136225882322854909611e-15L,
338 6.5330127201274379444839723873883485794067e-01L,
339 1.8443102186424720390266302263929355424008e-15L,
340 6.5735807270835877602621621917933225631714e-01L,
341 1.2541156738040666039091970075936624723645e-15L,
342 6.6139848224536379461824253667145967483521e-01L,
343 1.2136419933020381912633127333149145382797e-15L,
344 6.6542263254508782210905337706208229064941e-01L,
345 2.6268410392329445778904988886114643307320e-15L,
346 6.6943065394262646350398426875472068786621e-01L,
347 2.8037949010021747828222575923191438798877e-15L,
348 6.7342267521216570003161905333399772644043e-01L,
349 1.0202663413354670195383104149875619397268e-15L,
350 6.7739882359180469961756898555904626846313e-01L,
351 1.4411921136244383020300914304078010801275e-15L,
352 6.8135922480790256372529256623238325119019e-01L,
353 5.0522277899333570619054540068138110661023e-16L,
354 6.8530400309891703614084690343588590621948e-01L,
355 2.3804032011755313470802014258958896193599e-15L,
356 6.8923328123880622797514661215245723724365e-01L,
357 2.7523497677256621466659891416404053623832e-15L,
358 };
359
360 /*
361 * Compute N*log2 + log(1+zk+zh+zt) in extra precision
362 */
363 static long double k_log_NKzl(int N, int K, long double zh, long double *zt)
364 {
365 long double y, r, w, s2, s2h, s2t, t, zk, v, P;
366 double dzk;
367
368 #if !defined(__x86)
369 unsigned lx, ly;
370 int j;
371 #endif
372
373 ((int *)&dzk)[HIWORD] = 0x3ff00000 + (K << 13);
374 ((int *)&dzk)[LOWORD] = 0;
375 t = zh + (*zt);
376 zk = (long double) dzk;
377 r = two / (t + two * zk);
378 s2h = s2 = r * t;
379 /* split s2 into correctly rounded half */
380
381 #if defined(__x86)
382 ((unsigned *)&s2h)[0] = 0; /* 32 bits chopped */
383 #else
384
385 lx = ((unsigned *)&s2h)[2]; /* 56 bits rounded */
386 j = ((lx >> 24) + 1) >> 1;
387 ((unsigned *)&s2h)[2] = (j << 25);
388 lx = ((unsigned *)&s2h)[1];
389 ly = lx + (j >> 7);
390 ((unsigned *)&s2h)[1] = ly;
391 ((unsigned *)&s2h)[0] += (ly == 0 && lx != 0);
392 ((unsigned *)&s2h)[3] = 0;
393 #endif
394
395 v = half * s2h;
396 w = s2 * s2;
397 s2t = r * ((((zh - s2h * zk) - v * zh) + (*zt)) - v * (*zt));
398 P = s2t + (w * s2) * ((P1 + w * P2) + (w * w) * ((P3 + w * P4)
399 + (w * w) * (P5 + w * P6 + (w * w) * P7)));
400 P += N * ln2_t + TBL_log1k[K + K + 1];
401 t = N*ln2_h + TBL_log1k[K+K];
402 y = t + (P + s2h);
403 P -= ((y - t) - s2h);
404 *zt = P;
405 return (y);
406 }
407
408 long double
409 __k_clog_rl(long double x, long double y, long double *er)
410 {
411 long double t1, t2, t3, t4, tk, z, wh, w, zh, zk;
412 int n, k, ix, iy, iz, nx, ny, nz, i;
413 double dk;
414
415 #if !defined(__x86)
416 int j;
417 unsigned lx, ly;
418 #endif
419
420 ix = HI_XWORD(x) & ~0x80000000;
421 iy = HI_XWORD(y) & ~0x80000000;
422 y = fabsl(y); x = fabsl(x);
423 if (ix < iy || (ix < 0x7fff0000 && ix == iy && x < y)) {
424 /* force x >= y */
425 tk = x; x = y; y = tk;
426 n = ix, ix = iy; iy = n;
427 }
428 *er = zero;
429 nx = ix >> 16; ny = iy >> 16;
430 if (nx >= 0x7fff) { /* x or y is Inf or NaN */
431 if (isinfl(x))
432 return (x);
433 else if (isinfl(y))
434 return (y);
435 else
436 return (x+y);
437 }
438 /*
439 * for tiny y:(double y < 2^-35, extended y < 2^-46, quad y < 2^-70)
440 *
441 * log(sqrt(1 + y**2)) = y**2 / 2 - y**4 / 8 + ... = y**2 / 2
442 */
443 #if defined(__x86)
444 if (x == 1.0L && ny < (0x3fff - 46)) {
445 #else
446 if (x == 1.0L && ny < (0x3fff - 70)) {
447 #endif
448
449 t2 = y * y;
450 if (ny >= 8305) { /* compute er = tail of t2 */
451 dk = (double) y;
452
453 #if defined(__x86)
454 ((unsigned *)&dk)[LOWORD] &= 0xfffe0000;
455 #endif
456
457 wh = (long double) dk;
458 *er = half * ((y - wh) * (y + wh) - (t2 - wh * wh));
459 }
460 return (half * t2);
461 }
462 /*
463 * x or y is subnormal or zero
464 */
465 if (nx == 0) {
466 if (x == 0.0L)
467 return (-1.0L / x);
468 else {
469 x *= two240;
470 y *= two240;
471 ix = HI_XWORD(x);
472 iy = HI_XWORD(y);
473 nx = (ix >> 16) - 240;
474 ny = (iy >> 16) - 240;
475 /* guard subnormal flush to 0 */
476 if (x == 0.0L)
477 return (-1.0L / x);
478 }
479 } else if (ny == 0) { /* y subnormal, scale it */
480 y *= two240;
481 iy = HI_XWORD(y);
482 ny = (iy >> 16) - 240;
483 }
484 n = nx - ny;
485 /*
486 * When y is zero or when x >> y, i.e., n > 62, 78, 122 for DBLE,
487 * EXTENDED, QUAD respectively,
488 * log(x) = log(sqrt(x * x + y * y)) to 27 extra bits.
489 */
490
491 #if defined(__x86)
492 if (n > 78 || y == 0.0L) {
493 #else
494 if (n > 122 || y == 0.0L) {
495 #endif
496
497 XFSCALE(x, (0x3fff - (ix >> 16)));
498 i = ((ix & 0xffff) + 0x100) >> 9; /* 7.5 bits of x */
499 zk = 1.0L + ((long double) i) * 0.0078125L;
500 z = x - zk;
501 dk = (double)z;
502
503 #if defined(__x86)
504 ((unsigned *)&dk)[LOWORD] &= 0xfffe0000;
505 #endif
506
507 zh = (long double)dk;
508 k = i & 0x7f; /* index of zk */
509 n = nx - 0x3fff;
510 *er = z - zh;
511 if (i == 0x80) { /* if zk = 2.0, adjust scaling */
512 n += 1;
513 zh *= 0.5L; *er *= 0.5L;
514 }
515 w = k_log_NKzl(n, k, zh, er);
516 } else {
517 /*
518 * compute z = x*x + y*y
519 */
520 XFSCALE(x, (0x3fff - (ix >> 16)));
521 XFSCALE(y, (0x3fff - n - (iy >> 16)));
522 ix = (ix & 0xffff) | 0x3fff0000;
523 iy = (iy & 0xffff) | (0x3fff0000 - (n << 16));
524 nx -= 0x3fff;
525 t1 = x * x; t2 = y * y;
526 wh = x;
527
528 /* split x into correctly rounded half */
529 #if defined(__x86)
530 ((unsigned *)&wh)[0] = 0; /* 32 bits chopped */
531 #else
532 lx = ((unsigned *)&wh)[2]; /* 56 rounded */
533 j = ((lx >> 24) + 1) >> 1;
534 ((unsigned *)&wh)[2] = (j << 25);
535 lx = ((unsigned *)&wh)[1];
536 ly = lx + (j >> 7);
537 ((unsigned *)&wh)[1] = ly;
538 ((unsigned *)&wh)[0] += (ly == 0 && lx != 0);
539 ((unsigned *)&wh)[3] = 0;
540 #endif
541
542 z = t1+t2;
543 /*
544 * higher precision simulation x*x = t1 + t3, y*y = t2 + t4
545 */
546 tk = wh - x;
547 t3 = tk * tk - (two * wh * tk - (wh * wh - t1));
548 wh = y;
549
550 /* split y into correctly rounded half */
551 #if defined(__x86)
552 ((unsigned *)&wh)[0] = 0; /* 32 bits chopped */
553 #else
554 ly = ((unsigned *)&wh)[2]; /* 56 bits rounded */
555 j = ((ly >> 24) + 1) >> 1;
556 ((unsigned *)&wh)[2] = (j << 25);
557 lx = ((unsigned *)&wh)[1];
558 ly = lx + (j >> 7);
559 ((unsigned *)&wh)[1] = ly;
560 ((unsigned *)&wh)[0] += (ly == 0 && lx != 0);
561 ((unsigned *)&wh)[3] = 0;
562 #endif
563
564 tk = wh - y;
565 t4 = tk * tk - (two * wh * tk - (wh * wh - t2));
566 /*
567 * find zk matches z to 7.5 bits
568 */
569 iz = HI_XWORD(z);
570 k = ((iz & 0xffff) + 0x100) >> 9; /* 7.5 bits of x */
571 nz = (iz >> 16) - 0x3fff + (k >> 7);
572 k &= 0x7f;
573 zk = 1.0L + ((long double) k) * 0.0078125L;
574 if (nz == 1) zk += zk;
575 else if (nz == 2) zk *= 4.0L;
576 else if (nz == 3) zk *= 8.0L;
577 /*
578 * order t1, t2, t3, t4 according to their size
579 */
580 if (t2 >= fabsl(t3)) {
581 if (fabsl(t3) < fabsl(t4)) {
582 wh = t3; t3 = t4; t4 = wh;
583 }
584 } else {
585 wh = t2; t2 = t3; t3 = wh;
586 }
587 /*
588 * higher precision simulation: x * x + y * y = t1 + t2 + t3 + t4
589 * = zk(7 bits) + zh(24 bits) + *er(tail) and call k_log_NKz
590 */
591 tk = t1 - zk;
592 zh = ((tk + t2) + t3) + t4;
593
594 /* split zh into correctly rounded half */
595 #if defined(__x86)
596 ((unsigned *)&zh)[0] = 0;
597 #else
598 ly = ((unsigned *)&zh)[2];
599 j = ((ly >> 24) + 1) >> 1;
600 ((unsigned *)&zh)[2] = (j << 25);
601 lx = ((unsigned *)&zh)[1];
602 ly = lx + (j >> 7);
603 ((unsigned *)&zh)[1] = ly;
604 ((unsigned *)&zh)[0] += (ly == 0 && lx != 0);
605 ((unsigned *)&zh)[3] = 0;
606 #endif
607
608 w = fabsl(zh);
609 if (w >= fabsl(t2))
610 {
611 *er = (((tk - zh) + t2) + t3) + t4;
612 }
613
614 else {
615
616 if (n == 0) {
617 wh = half * zk;
618 wh = (t1 - wh) - (wh - t2);
619 } else
620 wh = tk + t2;
621 if (w >= fabsl(t3))
622 *er = ((wh - zh) + t3) + t4;
623 else {
624 z = t3;
625 t3 += t4;
626 t4 -= t3 - z;
627 if (w >= fabsl(t3))
628 *er = ((wh - zh) + t3) + t4;
629 else
630 *er = ((wh + t3) - zh) + t4;
631 }
632 }
633 if (nz == 3) {
634 zh *= 0.125L; *er *= 0.125L;
635 } else if (nz == 2) {
636 zh *= 0.25L; *er *= 0.25L;
637 } else if (nz == 1) {
638 zh *= half; *er *= half;
639 }
640 nz += nx + nx;
641 w = half * k_log_NKzl(nz, k, zh, er);
642 *er *= half;
643 }
644 return (w);
645 }