1 /*
2 * ====================================================
3 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
4 *
5 * Developed at SunPro, a Sun Microsystems, Inc. business.
6 * Permission to use, copy, modify, and distribute this
7 * software is freely granted, provided that this notice
8 * is preserved.
9 * ====================================================
10 */
11
12 /*
13 * Copyright (c) 2008 Stephen L. Moshier <steve@moshier.net>
14 *
15 * Permission to use, copy, modify, and distribute this software for any
16 * purpose with or without fee is hereby granted, provided that the above
17 * copyright notice and this permission notice appear in all copies.
18 *
19 * THE SOFTWARE IS PROVIDED "AS IS" AND THE AUTHOR DISCLAIMS ALL WARRANTIES
20 * WITH REGARD TO THIS SOFTWARE INCLUDING ALL IMPLIED WARRANTIES OF
21 * MERCHANTABILITY AND FITNESS. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR
22 * ANY SPECIAL, DIRECT, INDIRECT, OR CONSEQUENTIAL DAMAGES OR ANY DAMAGES
23 * WHATSOEVER RESULTING FROM LOSS OF USE, DATA OR PROFITS, WHETHER IN AN
24 * ACTION OF CONTRACT, NEGLIGENCE OR OTHER TORTIOUS ACTION, ARISING OUT OF
25 * OR IN CONNECTION WITH THE USE OR PERFORMANCE OF THIS SOFTWARE.
26 */
27
28 /* double erf(double x)
29 * double erfc(double x)
30 * x
31 * 2 |\
32 * erf(x) = --------- | exp(-t*t)dt
33 * sqrt(pi) \|
34 * 0
35 *
36 * erfc(x) = 1-erf(x)
37 * Note that
38 * erf(-x) = -erf(x)
39 * erfc(-x) = 2 - erfc(x)
40 *
41 * Method:
42 * 1. erf(x) = x + x*R(x^2) for |x| in [0, 7/8]
43 * Remark. The formula is derived by noting
44 * erf(x) = (2/sqrt(pi))*(x - x^3/3 + x^5/10 - x^7/42 + ....)
45 * and that
46 * 2/sqrt(pi) = 1.128379167095512573896158903121545171688
47 * is close to one.
48 *
49 * 1a. erf(x) = 1 - erfc(x), for |x| > 1.0
50 * erfc(x) = 1 - erf(x) if |x| < 1/4
51 *
52 * 2. For |x| in [7/8, 1], let s = |x| - 1, and
53 * c = 0.84506291151 rounded to single (24 bits)
54 * erf(s + c) = sign(x) * (c + P1(s)/Q1(s))
55 * Remark: here we use the taylor series expansion at x=1.
56 * erf(1+s) = erf(1) + s*Poly(s)
57 * = 0.845.. + P1(s)/Q1(s)
58 * Note that |P1/Q1|< 0.078 for x in [0.84375,1.25]
59 *
60 * 3. For x in [1/4, 5/4],
61 * erfc(s + const) = erfc(const) + s P1(s)/Q1(s)
62 * for const = 1/4, 3/8, ..., 9/8
63 * and 0 <= s <= 1/8 .
64 *
65 * 4. For x in [5/4, 107],
66 * erfc(x) = (1/x)*exp(-x*x-0.5625 + R(z))
67 * z=1/x^2
68 * The interval is partitioned into several segments
69 * of width 1/8 in 1/x.
70 *
71 * Note1:
72 * To compute exp(-x*x-0.5625+R/S), let s be a single
73 * precision number and s := x; then
74 * -x*x = -s*s + (s-x)*(s+x)
75 * exp(-x*x-0.5626+R/S) =
76 * exp(-s*s-0.5625)*exp((s-x)*(s+x)+R/S);
77 * Note2:
78 * Here 4 and 5 make use of the asymptotic series
79 * exp(-x*x)
80 * erfc(x) ~ ---------- * ( 1 + Poly(1/x^2) )
81 * x*sqrt(pi)
82 *
83 * 5. For inf > x >= 107
84 * erf(x) = sign(x) *(1 - tiny) (raise inexact)
85 * erfc(x) = tiny*tiny (raise underflow) if x > 0
86 * = 2 - tiny if x<0
87 *
88 * 7. Special case:
89 * erf(0) = 0, erf(inf) = 1, erf(-inf) = -1,
90 * erfc(0) = 1, erfc(inf) = 0, erfc(-inf) = 2,
91 * erfc/erf(NaN) is NaN
92 */
93
94
95
96 /* Evaluate P[n] x^n + P[n-1] x^(n-1) + ... + P[0] */
97
98 static long double
neval(long double x,const long double * p,int n)99 neval (long double x, const long double *p, int n)
100 {
101 long double y;
102
103 p += n;
104 y = *p--;
105 do
106 {
107 y = y * x + *p--;
108 }
109 while (--n > 0);
110 return y;
111 }
112
113
114 /* Evaluate x^n+1 + P[n] x^(n) + P[n-1] x^(n-1) + ... + P[0] */
115
116 static long double
deval(long double x,const long double * p,int n)117 deval (long double x, const long double *p, int n)
118 {
119 long double y;
120
121 p += n;
122 y = x + *p--;
123 do
124 {
125 y = y * x + *p--;
126 }
127 while (--n > 0);
128 return y;
129 }
130
131
132
133 static const long double
134 tiny = 1e-4931L,
135 one = 1.0L,
136 two = 2.0L,
137 /* 2/sqrt(pi) - 1 */
138 efx = 1.2837916709551257389615890312154517168810E-1L,
139 /* 8 * (2/sqrt(pi) - 1) */
140 efx8 = 1.0270333367641005911692712249723613735048E0L;
141
142
143 /* erf(x) = x + x R(x^2)
144 0 <= x <= 7/8
145 Peak relative error 1.8e-35 */
146 #define NTN1 8
147 static const long double TN1[NTN1 + 1] =
148 {
149 -3.858252324254637124543172907442106422373E10L,
150 9.580319248590464682316366876952214879858E10L,
151 1.302170519734879977595901236693040544854E10L,
152 2.922956950426397417800321486727032845006E9L,
153 1.764317520783319397868923218385468729799E8L,
154 1.573436014601118630105796794840834145120E7L,
155 4.028077380105721388745632295157816229289E5L,
156 1.644056806467289066852135096352853491530E4L,
157 3.390868480059991640235675479463287886081E1L
158 };
159 #define NTD1 8
160 static const long double TD1[NTD1 + 1] =
161 {
162 -3.005357030696532927149885530689529032152E11L,
163 -1.342602283126282827411658673839982164042E11L,
164 -2.777153893355340961288511024443668743399E10L,
165 -3.483826391033531996955620074072768276974E9L,
166 -2.906321047071299585682722511260895227921E8L,
167 -1.653347985722154162439387878512427542691E7L,
168 -6.245520581562848778466500301865173123136E5L,
169 -1.402124304177498828590239373389110545142E4L,
170 -1.209368072473510674493129989468348633579E2L
171 /* 1.0E0 */
172 };
173
174
175 /* erf(z+1) = erf_const + P(z)/Q(z)
176 -.125 <= z <= 0
177 Peak relative error 7.3e-36 */
178 static const long double erf_const = 0.845062911510467529296875L;
179 #define NTN2 8
180 static const long double TN2[NTN2 + 1] =
181 {
182 -4.088889697077485301010486931817357000235E1L,
183 7.157046430681808553842307502826960051036E3L,
184 -2.191561912574409865550015485451373731780E3L,
185 2.180174916555316874988981177654057337219E3L,
186 2.848578658049670668231333682379720943455E2L,
187 1.630362490952512836762810462174798925274E2L,
188 6.317712353961866974143739396865293596895E0L,
189 2.450441034183492434655586496522857578066E1L,
190 5.127662277706787664956025545897050896203E-1L
191 };
192 #define NTD2 8
193 static const long double TD2[NTD2 + 1] =
194 {
195 1.731026445926834008273768924015161048885E4L,
196 1.209682239007990370796112604286048173750E4L,
197 1.160950290217993641320602282462976163857E4L,
198 5.394294645127126577825507169061355698157E3L,
199 2.791239340533632669442158497532521776093E3L,
200 8.989365571337319032943005387378993827684E2L,
201 2.974016493766349409725385710897298069677E2L,
202 6.148192754590376378740261072533527271947E1L,
203 1.178502892490738445655468927408440847480E1L
204 /* 1.0E0 */
205 };
206
207
208 /* erfc(x + 0.25) = erfc(0.25) + x R(x)
209 0 <= x < 0.125
210 Peak relative error 1.4e-35 */
211 #define NRNr13 8
212 static const long double RNr13[NRNr13 + 1] =
213 {
214 -2.353707097641280550282633036456457014829E3L,
215 3.871159656228743599994116143079870279866E2L,
216 -3.888105134258266192210485617504098426679E2L,
217 -2.129998539120061668038806696199343094971E1L,
218 -8.125462263594034672468446317145384108734E1L,
219 8.151549093983505810118308635926270319660E0L,
220 -5.033362032729207310462422357772568553670E0L,
221 -4.253956621135136090295893547735851168471E-2L,
222 -8.098602878463854789780108161581050357814E-2L
223 };
224 #define NRDr13 7
225 static const long double RDr13[NRDr13 + 1] =
226 {
227 2.220448796306693503549505450626652881752E3L,
228 1.899133258779578688791041599040951431383E2L,
229 1.061906712284961110196427571557149268454E3L,
230 7.497086072306967965180978101974566760042E1L,
231 2.146796115662672795876463568170441327274E2L,
232 1.120156008362573736664338015952284925592E1L,
233 2.211014952075052616409845051695042741074E1L,
234 6.469655675326150785692908453094054988938E-1L
235 /* 1.0E0 */
236 };
237 /* erfc(0.25) = C13a + C13b to extra precision. */
238 static const long double C13a = 0.723663330078125L;
239 static const long double C13b = 1.0279753638067014931732235184287934646022E-5L;
240
241
242 /* erfc(x + 0.375) = erfc(0.375) + x R(x)
243 0 <= x < 0.125
244 Peak relative error 1.2e-35 */
245 #define NRNr14 8
246 static const long double RNr14[NRNr14 + 1] =
247 {
248 -2.446164016404426277577283038988918202456E3L,
249 6.718753324496563913392217011618096698140E2L,
250 -4.581631138049836157425391886957389240794E2L,
251 -2.382844088987092233033215402335026078208E1L,
252 -7.119237852400600507927038680970936336458E1L,
253 1.313609646108420136332418282286454287146E1L,
254 -6.188608702082264389155862490056401365834E0L,
255 -2.787116601106678287277373011101132659279E-2L,
256 -2.230395570574153963203348263549700967918E-2L
257 };
258 #define NRDr14 7
259 static const long double RDr14[NRDr14 + 1] =
260 {
261 2.495187439241869732696223349840963702875E3L,
262 2.503549449872925580011284635695738412162E2L,
263 1.159033560988895481698051531263861842461E3L,
264 9.493751466542304491261487998684383688622E1L,
265 2.276214929562354328261422263078480321204E2L,
266 1.367697521219069280358984081407807931847E1L,
267 2.276988395995528495055594829206582732682E1L,
268 7.647745753648996559837591812375456641163E-1L
269 /* 1.0E0 */
270 };
271 /* erfc(0.375) = C14a + C14b to extra precision. */
272 static const long double C14a = 0.5958709716796875L;
273 static const long double C14b = 1.2118885490201676174914080878232469565953E-5L;
274
275 /* erfc(x + 0.5) = erfc(0.5) + x R(x)
276 0 <= x < 0.125
277 Peak relative error 4.7e-36 */
278 #define NRNr15 8
279 static const long double RNr15[NRNr15 + 1] =
280 {
281 -2.624212418011181487924855581955853461925E3L,
282 8.473828904647825181073831556439301342756E2L,
283 -5.286207458628380765099405359607331669027E2L,
284 -3.895781234155315729088407259045269652318E1L,
285 -6.200857908065163618041240848728398496256E1L,
286 1.469324610346924001393137895116129204737E1L,
287 -6.961356525370658572800674953305625578903E0L,
288 5.145724386641163809595512876629030548495E-3L,
289 1.990253655948179713415957791776180406812E-2L
290 };
291 #define NRDr15 7
292 static const long double RDr15[NRDr15 + 1] =
293 {
294 2.986190760847974943034021764693341524962E3L,
295 5.288262758961073066335410218650047725985E2L,
296 1.363649178071006978355113026427856008978E3L,
297 1.921707975649915894241864988942255320833E2L,
298 2.588651100651029023069013885900085533226E2L,
299 2.628752920321455606558942309396855629459E1L,
300 2.455649035885114308978333741080991380610E1L,
301 1.378826653595128464383127836412100939126E0L
302 /* 1.0E0 */
303 };
304 /* erfc(0.5) = C15a + C15b to extra precision. */
305 static const long double C15a = 0.4794921875L;
306 static const long double C15b = 7.9346869534623172533461080354712635484242E-6L;
307
308 /* erfc(x + 0.625) = erfc(0.625) + x R(x)
309 0 <= x < 0.125
310 Peak relative error 5.1e-36 */
311 #define NRNr16 8
312 static const long double RNr16[NRNr16 + 1] =
313 {
314 -2.347887943200680563784690094002722906820E3L,
315 8.008590660692105004780722726421020136482E2L,
316 -5.257363310384119728760181252132311447963E2L,
317 -4.471737717857801230450290232600243795637E1L,
318 -4.849540386452573306708795324759300320304E1L,
319 1.140885264677134679275986782978655952843E1L,
320 -6.731591085460269447926746876983786152300E0L,
321 1.370831653033047440345050025876085121231E-1L,
322 2.022958279982138755020825717073966576670E-2L,
323 };
324 #define NRDr16 7
325 static const long double RDr16[NRDr16 + 1] =
326 {
327 3.075166170024837215399323264868308087281E3L,
328 8.730468942160798031608053127270430036627E2L,
329 1.458472799166340479742581949088453244767E3L,
330 3.230423687568019709453130785873540386217E2L,
331 2.804009872719893612081109617983169474655E2L,
332 4.465334221323222943418085830026979293091E1L,
333 2.612723259683205928103787842214809134746E1L,
334 2.341526751185244109722204018543276124997E0L,
335 /* 1.0E0 */
336 };
337 /* erfc(0.625) = C16a + C16b to extra precision. */
338 static const long double C16a = 0.3767547607421875L;
339 static const long double C16b = 4.3570693945275513594941232097252997287766E-6L;
340
341 /* erfc(x + 0.75) = erfc(0.75) + x R(x)
342 0 <= x < 0.125
343 Peak relative error 1.7e-35 */
344 #define NRNr17 8
345 static const long double RNr17[NRNr17 + 1] =
346 {
347 -1.767068734220277728233364375724380366826E3L,
348 6.693746645665242832426891888805363898707E2L,
349 -4.746224241837275958126060307406616817753E2L,
350 -2.274160637728782675145666064841883803196E1L,
351 -3.541232266140939050094370552538987982637E1L,
352 6.988950514747052676394491563585179503865E0L,
353 -5.807687216836540830881352383529281215100E0L,
354 3.631915988567346438830283503729569443642E-1L,
355 -1.488945487149634820537348176770282391202E-2L
356 };
357 #define NRDr17 7
358 static const long double RDr17[NRDr17 + 1] =
359 {
360 2.748457523498150741964464942246913394647E3L,
361 1.020213390713477686776037331757871252652E3L,
362 1.388857635935432621972601695296561952738E3L,
363 3.903363681143817750895999579637315491087E2L,
364 2.784568344378139499217928969529219886578E2L,
365 5.555800830216764702779238020065345401144E1L,
366 2.646215470959050279430447295801291168941E1L,
367 2.984905282103517497081766758550112011265E0L,
368 /* 1.0E0 */
369 };
370 /* erfc(0.75) = C17a + C17b to extra precision. */
371 static const long double C17a = 0.2888336181640625L;
372 static const long double C17b = 1.0748182422368401062165408589222625794046E-5L;
373
374
375 /* erfc(x + 0.875) = erfc(0.875) + x R(x)
376 0 <= x < 0.125
377 Peak relative error 2.2e-35 */
378 #define NRNr18 8
379 static const long double RNr18[NRNr18 + 1] =
380 {
381 -1.342044899087593397419622771847219619588E3L,
382 6.127221294229172997509252330961641850598E2L,
383 -4.519821356522291185621206350470820610727E2L,
384 1.223275177825128732497510264197915160235E1L,
385 -2.730789571382971355625020710543532867692E1L,
386 4.045181204921538886880171727755445395862E0L,
387 -4.925146477876592723401384464691452700539E0L,
388 5.933878036611279244654299924101068088582E-1L,
389 -5.557645435858916025452563379795159124753E-2L
390 };
391 #define NRDr18 7
392 static const long double RDr18[NRDr18 + 1] =
393 {
394 2.557518000661700588758505116291983092951E3L,
395 1.070171433382888994954602511991940418588E3L,
396 1.344842834423493081054489613250688918709E3L,
397 4.161144478449381901208660598266288188426E2L,
398 2.763670252219855198052378138756906980422E2L,
399 5.998153487868943708236273854747564557632E1L,
400 2.657695108438628847733050476209037025318E1L,
401 3.252140524394421868923289114410336976512E0L,
402 /* 1.0E0 */
403 };
404 /* erfc(0.875) = C18a + C18b to extra precision. */
405 static const long double C18a = 0.215911865234375L;
406 static const long double C18b = 1.3073705765341685464282101150637224028267E-5L;
407
408 /* erfc(x + 1.0) = erfc(1.0) + x R(x)
409 0 <= x < 0.125
410 Peak relative error 1.6e-35 */
411 #define NRNr19 8
412 static const long double RNr19[NRNr19 + 1] =
413 {
414 -1.139180936454157193495882956565663294826E3L,
415 6.134903129086899737514712477207945973616E2L,
416 -4.628909024715329562325555164720732868263E2L,
417 4.165702387210732352564932347500364010833E1L,
418 -2.286979913515229747204101330405771801610E1L,
419 1.870695256449872743066783202326943667722E0L,
420 -4.177486601273105752879868187237000032364E0L,
421 7.533980372789646140112424811291782526263E-1L,
422 -8.629945436917752003058064731308767664446E-2L
423 };
424 #define NRDr19 7
425 static const long double RDr19[NRDr19 + 1] =
426 {
427 2.744303447981132701432716278363418643778E3L,
428 1.266396359526187065222528050591302171471E3L,
429 1.466739461422073351497972255511919814273E3L,
430 4.868710570759693955597496520298058147162E2L,
431 2.993694301559756046478189634131722579643E2L,
432 6.868976819510254139741559102693828237440E1L,
433 2.801505816247677193480190483913753613630E1L,
434 3.604439909194350263552750347742663954481E0L,
435 /* 1.0E0 */
436 };
437 /* erfc(1.0) = C19a + C19b to extra precision. */
438 static const long double C19a = 0.15728759765625L;
439 static const long double C19b = 1.1609394035130658779364917390740703933002E-5L;
440
441 /* erfc(x + 1.125) = erfc(1.125) + x R(x)
442 0 <= x < 0.125
443 Peak relative error 3.6e-36 */
444 #define NRNr20 8
445 static const long double RNr20[NRNr20 + 1] =
446 {
447 -9.652706916457973956366721379612508047640E2L,
448 5.577066396050932776683469951773643880634E2L,
449 -4.406335508848496713572223098693575485978E2L,
450 5.202893466490242733570232680736966655434E1L,
451 -1.931311847665757913322495948705563937159E1L,
452 -9.364318268748287664267341457164918090611E-2L,
453 -3.306390351286352764891355375882586201069E0L,
454 7.573806045289044647727613003096916516475E-1L,
455 -9.611744011489092894027478899545635991213E-2L
456 };
457 #define NRDr20 7
458 static const long double RDr20[NRDr20 + 1] =
459 {
460 3.032829629520142564106649167182428189014E3L,
461 1.659648470721967719961167083684972196891E3L,
462 1.703545128657284619402511356932569292535E3L,
463 6.393465677731598872500200253155257708763E2L,
464 3.489131397281030947405287112726059221934E2L,
465 8.848641738570783406484348434387611713070E1L,
466 3.132269062552392974833215844236160958502E1L,
467 4.430131663290563523933419966185230513168E0L
468 /* 1.0E0 */
469 };
470 /* erfc(1.125) = C20a + C20b to extra precision. */
471 static const long double C20a = 0.111602783203125L;
472 static const long double C20b = 8.9850951672359304215530728365232161564636E-6L;
473
474 /* erfc(1/x) = 1/x exp (-1/x^2 - 0.5625 + R(1/x^2))
475 7/8 <= 1/x < 1
476 Peak relative error 1.4e-35 */
477 #define NRNr8 9
478 static const long double RNr8[NRNr8 + 1] =
479 {
480 3.587451489255356250759834295199296936784E1L,
481 5.406249749087340431871378009874875889602E2L,
482 2.931301290625250886238822286506381194157E3L,
483 7.359254185241795584113047248898753470923E3L,
484 9.201031849810636104112101947312492532314E3L,
485 5.749697096193191467751650366613289284777E3L,
486 1.710415234419860825710780802678697889231E3L,
487 2.150753982543378580859546706243022719599E2L,
488 8.740953582272147335100537849981160931197E0L,
489 4.876422978828717219629814794707963640913E-2L
490 };
491 #define NRDr8 8
492 static const long double RDr8[NRDr8 + 1] =
493 {
494 6.358593134096908350929496535931630140282E1L,
495 9.900253816552450073757174323424051765523E2L,
496 5.642928777856801020545245437089490805186E3L,
497 1.524195375199570868195152698617273739609E4L,
498 2.113829644500006749947332935305800887345E4L,
499 1.526438562626465706267943737310282977138E4L,
500 5.561370922149241457131421914140039411782E3L,
501 9.394035530179705051609070428036834496942E2L,
502 6.147019596150394577984175188032707343615E1L
503 /* 1.0E0 */
504 };
505
506 /* erfc(1/x) = 1/x exp (-1/x^2 - 0.5625 + R(1/x^2))
507 0.75 <= 1/x <= 0.875
508 Peak relative error 2.0e-36 */
509 #define NRNr7 9
510 static const long double RNr7[NRNr7 + 1] =
511 {
512 1.686222193385987690785945787708644476545E1L,
513 1.178224543567604215602418571310612066594E3L,
514 1.764550584290149466653899886088166091093E4L,
515 1.073758321890334822002849369898232811561E5L,
516 3.132840749205943137619839114451290324371E5L,
517 4.607864939974100224615527007793867585915E5L,
518 3.389781820105852303125270837910972384510E5L,
519 1.174042187110565202875011358512564753399E5L,
520 1.660013606011167144046604892622504338313E4L,
521 6.700393957480661937695573729183733234400E2L
522 };
523 #define NRDr7 9
524 static const long double RDr7[NRDr7 + 1] =
525 {
526 -1.709305024718358874701575813642933561169E3L,
527 -3.280033887481333199580464617020514788369E4L,
528 -2.345284228022521885093072363418750835214E5L,
529 -8.086758123097763971926711729242327554917E5L,
530 -1.456900414510108718402423999575992450138E6L,
531 -1.391654264881255068392389037292702041855E6L,
532 -6.842360801869939983674527468509852583855E5L,
533 -1.597430214446573566179675395199807533371E5L,
534 -1.488876130609876681421645314851760773480E4L,
535 -3.511762950935060301403599443436465645703E2L
536 /* 1.0E0 */
537 };
538
539 /* erfc(1/x) = 1/x exp(-1/x^2 - 0.5625 + R(1/x^2))
540 5/8 <= 1/x < 3/4
541 Peak relative error 1.9e-35 */
542 #define NRNr6 9
543 static const long double RNr6[NRNr6 + 1] =
544 {
545 1.642076876176834390623842732352935761108E0L,
546 1.207150003611117689000664385596211076662E2L,
547 2.119260779316389904742873816462800103939E3L,
548 1.562942227734663441801452930916044224174E4L,
549 5.656779189549710079988084081145693580479E4L,
550 1.052166241021481691922831746350942786299E5L,
551 9.949798524786000595621602790068349165758E4L,
552 4.491790734080265043407035220188849562856E4L,
553 8.377074098301530326270432059434791287601E3L,
554 4.506934806567986810091824791963991057083E2L
555 };
556 #define NRDr6 9
557 static const long double RDr6[NRDr6 + 1] =
558 {
559 -1.664557643928263091879301304019826629067E2L,
560 -3.800035902507656624590531122291160668452E3L,
561 -3.277028191591734928360050685359277076056E4L,
562 -1.381359471502885446400589109566587443987E5L,
563 -3.082204287382581873532528989283748656546E5L,
564 -3.691071488256738343008271448234631037095E5L,
565 -2.300482443038349815750714219117566715043E5L,
566 -6.873955300927636236692803579555752171530E4L,
567 -8.262158817978334142081581542749986845399E3L,
568 -2.517122254384430859629423488157361983661E2L
569 /* 1.00 */
570 };
571
572 /* erfc(1/x) = 1/x exp(-1/x^2 - 0.5625 + R(1/x^2))
573 1/2 <= 1/x < 5/8
574 Peak relative error 4.6e-36 */
575 #define NRNr5 10
576 static const long double RNr5[NRNr5 + 1] =
577 {
578 -3.332258927455285458355550878136506961608E-3L,
579 -2.697100758900280402659586595884478660721E-1L,
580 -6.083328551139621521416618424949137195536E0L,
581 -6.119863528983308012970821226810162441263E1L,
582 -3.176535282475593173248810678636522589861E2L,
583 -8.933395175080560925809992467187963260693E2L,
584 -1.360019508488475978060917477620199499560E3L,
585 -1.075075579828188621541398761300910213280E3L,
586 -4.017346561586014822824459436695197089916E2L,
587 -5.857581368145266249509589726077645791341E1L,
588 -2.077715925587834606379119585995758954399E0L
589 };
590 #define NRDr5 9
591 static const long double RDr5[NRDr5 + 1] =
592 {
593 3.377879570417399341550710467744693125385E-1L,
594 1.021963322742390735430008860602594456187E1L,
595 1.200847646592942095192766255154827011939E2L,
596 7.118915528142927104078182863387116942836E2L,
597 2.318159380062066469386544552429625026238E3L,
598 4.238729853534009221025582008928765281620E3L,
599 4.279114907284825886266493994833515580782E3L,
600 2.257277186663261531053293222591851737504E3L,
601 5.570475501285054293371908382916063822957E2L,
602 5.142189243856288981145786492585432443560E1L
603 /* 1.0E0 */
604 };
605
606 /* erfc(1/x) = 1/x exp(-1/x^2 - 0.5625 + R(1/x^2))
607 3/8 <= 1/x < 1/2
608 Peak relative error 2.0e-36 */
609 #define NRNr4 10
610 static const long double RNr4[NRNr4 + 1] =
611 {
612 3.258530712024527835089319075288494524465E-3L,
613 2.987056016877277929720231688689431056567E-1L,
614 8.738729089340199750734409156830371528862E0L,
615 1.207211160148647782396337792426311125923E2L,
616 8.997558632489032902250523945248208224445E2L,
617 3.798025197699757225978410230530640879762E3L,
618 9.113203668683080975637043118209210146846E3L,
619 1.203285891339933238608683715194034900149E4L,
620 8.100647057919140328536743641735339740855E3L,
621 2.383888249907144945837976899822927411769E3L,
622 2.127493573166454249221983582495245662319E2L
623 };
624 #define NRDr4 10
625 static const long double RDr4[NRDr4 + 1] =
626 {
627 -3.303141981514540274165450687270180479586E-1L,
628 -1.353768629363605300707949368917687066724E1L,
629 -2.206127630303621521950193783894598987033E2L,
630 -1.861800338758066696514480386180875607204E3L,
631 -8.889048775872605708249140016201753255599E3L,
632 -2.465888106627948210478692168261494857089E4L,
633 -3.934642211710774494879042116768390014289E4L,
634 -3.455077258242252974937480623730228841003E4L,
635 -1.524083977439690284820586063729912653196E4L,
636 -2.810541887397984804237552337349093953857E3L,
637 -1.343929553541159933824901621702567066156E2L
638 /* 1.0E0 */
639 };
640
641 /* erfc(1/x) = 1/x exp(-1/x^2 - 0.5625 + R(1/x^2))
642 1/4 <= 1/x < 3/8
643 Peak relative error 8.4e-37 */
644 #define NRNr3 11
645 static const long double RNr3[NRNr3 + 1] =
646 {
647 -1.952401126551202208698629992497306292987E-6L,
648 -2.130881743066372952515162564941682716125E-4L,
649 -8.376493958090190943737529486107282224387E-3L,
650 -1.650592646560987700661598877522831234791E-1L,
651 -1.839290818933317338111364667708678163199E0L,
652 -1.216278715570882422410442318517814388470E1L,
653 -4.818759344462360427612133632533779091386E1L,
654 -1.120994661297476876804405329172164436784E2L,
655 -1.452850765662319264191141091859300126931E2L,
656 -9.485207851128957108648038238656777241333E1L,
657 -2.563663855025796641216191848818620020073E1L,
658 -1.787995944187565676837847610706317833247E0L
659 };
660 #define NRDr3 10
661 static const long double RDr3[NRDr3 + 1] =
662 {
663 1.979130686770349481460559711878399476903E-4L,
664 1.156941716128488266238105813374635099057E-2L,
665 2.752657634309886336431266395637285974292E-1L,
666 3.482245457248318787349778336603569327521E0L,
667 2.569347069372696358578399521203959253162E1L,
668 1.142279000180457419740314694631879921561E2L,
669 3.056503977190564294341422623108332700840E2L,
670 4.780844020923794821656358157128719184422E2L,
671 4.105972727212554277496256802312730410518E2L,
672 1.724072188063746970865027817017067646246E2L,
673 2.815939183464818198705278118326590370435E1L
674 /* 1.0E0 */
675 };
676
677 /* erfc(1/x) = 1/x exp(-1/x^2 - 0.5625 + R(1/x^2))
678 1/8 <= 1/x < 1/4
679 Peak relative error 1.5e-36 */
680 #define NRNr2 11
681 static const long double RNr2[NRNr2 + 1] =
682 {
683 -2.638914383420287212401687401284326363787E-8L,
684 -3.479198370260633977258201271399116766619E-6L,
685 -1.783985295335697686382487087502222519983E-4L,
686 -4.777876933122576014266349277217559356276E-3L,
687 -7.450634738987325004070761301045014986520E-2L,
688 -7.068318854874733315971973707247467326619E-1L,
689 -4.113919921935944795764071670806867038732E0L,
690 -1.440447573226906222417767283691888875082E1L,
691 -2.883484031530718428417168042141288943905E1L,
692 -2.990886974328476387277797361464279931446E1L,
693 -1.325283914915104866248279787536128997331E1L,
694 -1.572436106228070195510230310658206154374E0L
695 };
696 #define NRDr2 10
697 static const long double RDr2[NRDr2 + 1] =
698 {
699 2.675042728136731923554119302571867799673E-6L,
700 2.170997868451812708585443282998329996268E-4L,
701 7.249969752687540289422684951196241427445E-3L,
702 1.302040375859768674620410563307838448508E-1L,
703 1.380202483082910888897654537144485285549E0L,
704 8.926594113174165352623847870299170069350E0L,
705 3.521089584782616472372909095331572607185E1L,
706 8.233547427533181375185259050330809105570E1L,
707 1.072971579885803033079469639073292840135E2L,
708 6.943803113337964469736022094105143158033E1L,
709 1.775695341031607738233608307835017282662E1L
710 /* 1.0E0 */
711 };
712
713 /* erfc(1/x) = 1/x exp(-1/x^2 - 0.5625 + R(1/x^2))
714 1/128 <= 1/x < 1/8
715 Peak relative error 2.2e-36 */
716 #define NRNr1 9
717 static const long double RNr1[NRNr1 + 1] =
718 {
719 -4.250780883202361946697751475473042685782E-8L,
720 -5.375777053288612282487696975623206383019E-6L,
721 -2.573645949220896816208565944117382460452E-4L,
722 -6.199032928113542080263152610799113086319E-3L,
723 -8.262721198693404060380104048479916247786E-2L,
724 -6.242615227257324746371284637695778043982E-1L,
725 -2.609874739199595400225113299437099626386E0L,
726 -5.581967563336676737146358534602770006970E0L,
727 -5.124398923356022609707490956634280573882E0L,
728 -1.290865243944292370661544030414667556649E0L
729 };
730 #define NRDr1 8
731 static const long double RDr1[NRDr1 + 1] =
732 {
733 4.308976661749509034845251315983612976224E-6L,
734 3.265390126432780184125233455960049294580E-4L,
735 9.811328839187040701901866531796570418691E-3L,
736 1.511222515036021033410078631914783519649E-1L,
737 1.289264341917429958858379585970225092274E0L,
738 6.147640356182230769548007536914983522270E0L,
739 1.573966871337739784518246317003956180750E1L,
740 1.955534123435095067199574045529218238263E1L,
741 9.472613121363135472247929109615785855865E0L
742 /* 1.0E0 */
743 };
744
745
746 long double
erfl(long double x)747 erfl(long double x)
748 {
749 long double a, y, z;
750 int32_t i, ix, sign;
751 ieee_quad_shape_type u;
752
753 u.value = x;
754 sign = u.parts32.mswhi;
755 ix = sign & 0x7fffffff;
756
757 if (ix >= 0x7fff0000)
758 { /* erf(nan)=nan */
759 i = ((sign & 0xffff0000) >> 31) << 1;
760 return (long double) (1 - i) + one / x; /* erf(+-inf)=+-1 */
761 }
762
763 if (ix >= 0x3fff0000) /* |x| >= 1.0 */
764 {
765 /* Don't call if erfcl would underflow */
766 if (ix < 0x4005ac00) {
767 u.parts32.mswhi = ix;
768 y = one - erfcl (u.value);
769 } else
770 y = one;
771 if (sign < 0)
772 y = -y;
773 return y;
774 /* return (one - erfcl (x)); */
775 }
776 u.parts32.mswhi = ix;
777 a = u.value;
778 z = x * x;
779 if (ix < 0x3ffec000) /* a < 0.875 */
780 {
781 if (ix < 0x3fc60000) /* |x|<2**-57 */
782 {
783 if (ix < 0x00080000)
784 return 0.125L * (8.0L * x + efx8 * x); /*avoid underflow */
785 return x + efx * x;
786 }
787 y = a + a * neval (z, TN1, NTN1) / deval (z, TD1, NTD1);
788 }
789 else
790 {
791 a = a - one;
792 y = erf_const + neval (a, TN2, NTN2) / deval (a, TD2, NTD2);
793 }
794
795 if (sign & 0x80000000) /* x < 0 */
796 y = -y;
797 return( y );
798 }
799
800 long double
erfcl(long double x)801 erfcl(long double x)
802 {
803 long double y, z, p, r;
804 int32_t i, ix, sign;
805 ieee_quad_shape_type u;
806
807 u.value = x;
808 sign = u.parts32.mswhi;
809 ix = sign & 0x7fffffff;
810 u.parts32.mswhi = ix;
811
812 if (ix >= 0x7fff0000)
813 { /* erfc(nan)=nan */
814 /* erfc(+-inf)=0,2 */
815 return (long double) (((u_int32_t) sign >> 31) << 1) + one / x;
816 }
817
818 if (ix < 0x3ffd0000) /* |x| <1/4 */
819 {
820 if (ix < 0x3f8d0000) /* |x|<2**-114 */
821 return one - x;
822 return one - erfl (x);
823 }
824 if (ix < 0x3fff4000) /* 1.25 */
825 {
826 x = u.value;
827 i = 8.0L * x;
828 switch (i)
829 {
830 default:
831 case 2:
832 z = x - 0.25L;
833 y = C13b + z * neval (z, RNr13, NRNr13) / deval (z, RDr13, NRDr13);
834 y += C13a;
835 break;
836 case 3:
837 z = x - 0.375L;
838 y = C14b + z * neval (z, RNr14, NRNr14) / deval (z, RDr14, NRDr14);
839 y += C14a;
840 break;
841 case 4:
842 z = x - 0.5L;
843 y = C15b + z * neval (z, RNr15, NRNr15) / deval (z, RDr15, NRDr15);
844 y += C15a;
845 break;
846 case 5:
847 z = x - 0.625L;
848 y = C16b + z * neval (z, RNr16, NRNr16) / deval (z, RDr16, NRDr16);
849 y += C16a;
850 break;
851 case 6:
852 z = x - 0.75L;
853 y = C17b + z * neval (z, RNr17, NRNr17) / deval (z, RDr17, NRDr17);
854 y += C17a;
855 break;
856 case 7:
857 z = x - 0.875L;
858 y = C18b + z * neval (z, RNr18, NRNr18) / deval (z, RDr18, NRDr18);
859 y += C18a;
860 break;
861 case 8:
862 z = x - 1.0L;
863 y = C19b + z * neval (z, RNr19, NRNr19) / deval (z, RDr19, NRDr19);
864 y += C19a;
865 break;
866 case 9:
867 z = x - 1.125L;
868 y = C20b + z * neval (z, RNr20, NRNr20) / deval (z, RDr20, NRDr20);
869 y += C20a;
870 break;
871 }
872 if (sign & 0x80000000)
873 y = 2.0L - y;
874 return y;
875 }
876 /* 1.25 < |x| < 107 */
877 if (ix < 0x4005ac00)
878 {
879 /* x < -9 */
880 if ((ix >= 0x40022000) && (sign & 0x80000000))
881 return two - tiny;
882
883 x = fabsl (x);
884 z = one / (x * x);
885 i = 8.0L / x;
886 switch (i)
887 {
888 default:
889 case 0:
890 p = neval (z, RNr1, NRNr1) / deval (z, RDr1, NRDr1);
891 break;
892 case 1:
893 p = neval (z, RNr2, NRNr2) / deval (z, RDr2, NRDr2);
894 break;
895 case 2:
896 p = neval (z, RNr3, NRNr3) / deval (z, RDr3, NRDr3);
897 break;
898 case 3:
899 p = neval (z, RNr4, NRNr4) / deval (z, RDr4, NRDr4);
900 break;
901 case 4:
902 p = neval (z, RNr5, NRNr5) / deval (z, RDr5, NRDr5);
903 break;
904 case 5:
905 p = neval (z, RNr6, NRNr6) / deval (z, RDr6, NRDr6);
906 break;
907 case 6:
908 p = neval (z, RNr7, NRNr7) / deval (z, RDr7, NRDr7);
909 break;
910 case 7:
911 p = neval (z, RNr8, NRNr8) / deval (z, RDr8, NRDr8);
912 break;
913 }
914 u.value = x;
915 u.parts32.lswlo = 0;
916 u.parts32.lswhi &= 0xfe000000;
917 z = u.value;
918 r = expl (-z * z - 0.5625L) *
919 expl ((z - x) * (z + x) + p);
920 if ((sign & 0x80000000) == 0)
921 return r / x;
922 else
923 return two - r / x;
924 }
925 else
926 {
927 if ((sign & 0x80000000) == 0)
928 return __math_uflowl(0);
929 else
930 return two - tiny;
931 }
932 }
933
