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[bitrig.git] / lib / libm / src / ld80 / s_erfl.c
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1 /*
2 * ====================================================
3 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
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 * ====================================================
13 * Copyright (c) 2008 Stephen L. Moshier <steve@moshier.net>
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.
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.
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
36 * erfc(x) = 1-erf(x)
37 * Note that
38 * erf(-x) = -erf(x)
39 * erfc(-x) = 2 - erfc(x)
41 * Method:
42 * 1. For |x| in [0, 0.84375]
43 * erf(x) = x + x*R(x^2)
44 * erfc(x) = 1 - erf(x) if x in [-.84375,0.25]
45 * = 0.5 + ((0.5-x)-x*R) if x in [0.25,0.84375]
46 * Remark. The formula is derived by noting
47 * erf(x) = (2/sqrt(pi))*(x - x^3/3 + x^5/10 - x^7/42 + ....)
48 * and that
49 * 2/sqrt(pi) = 1.128379167095512573896158903121545171688
50 * is close to one. The interval is chosen because the fix
51 * point of erf(x) is near 0.6174 (i.e., erf(x)=x when x is
52 * near 0.6174), and by some experiment, 0.84375 is chosen to
53 * guarantee the error is less than one ulp for erf.
55 * 2. For |x| in [0.84375,1.25], let s = |x| - 1, and
56 * c = 0.84506291151 rounded to single (24 bits)
57 * erf(x) = sign(x) * (c + P1(s)/Q1(s))
58 * erfc(x) = (1-c) - P1(s)/Q1(s) if x > 0
59 * 1+(c+P1(s)/Q1(s)) if x < 0
60 * Remark: here we use the taylor series expansion at x=1.
61 * erf(1+s) = erf(1) + s*Poly(s)
62 * = 0.845.. + P1(s)/Q1(s)
63 * Note that |P1/Q1|< 0.078 for x in [0.84375,1.25]
65 * 3. For x in [1.25,1/0.35(~2.857143)],
66 * erfc(x) = (1/x)*exp(-x*x-0.5625+R1(z)/S1(z))
67 * z=1/x^2
68 * erf(x) = 1 - erfc(x)
70 * 4. For x in [1/0.35,107]
71 * erfc(x) = (1/x)*exp(-x*x-0.5625+R2/S2) if x > 0
72 * = 2.0 - (1/x)*exp(-x*x-0.5625+R2(z)/S2(z))
73 * if -6.666<x<0
74 * = 2.0 - tiny (if x <= -6.666)
75 * z=1/x^2
76 * erf(x) = sign(x)*(1.0 - erfc(x)) if x < 6.666, else
77 * erf(x) = sign(x)*(1.0 - tiny)
78 * Note1:
79 * To compute exp(-x*x-0.5625+R/S), let s be a single
80 * precision number and s := x; then
81 * -x*x = -s*s + (s-x)*(s+x)
82 * exp(-x*x-0.5626+R/S) =
83 * exp(-s*s-0.5625)*exp((s-x)*(s+x)+R/S);
84 * Note2:
85 * Here 4 and 5 make use of the asymptotic series
86 * exp(-x*x)
87 * erfc(x) ~ ---------- * ( 1 + Poly(1/x^2) )
88 * x*sqrt(pi)
90 * 5. For inf > x >= 107
91 * erf(x) = sign(x) *(1 - tiny) (raise inexact)
92 * erfc(x) = tiny*tiny (raise underflow) if x > 0
93 * = 2 - tiny if x<0
95 * 7. Special case:
96 * erf(0) = 0, erf(inf) = 1, erf(-inf) = -1,
97 * erfc(0) = 1, erfc(inf) = 0, erfc(-inf) = 2,
98 * erfc/erf(NaN) is NaN
102 #include <math.h>
104 #include "math_private.h"
106 static const long double
107 tiny = 1e-4931L,
108 half = 0.5L,
109 one = 1.0L,
110 two = 2.0L,
111 /* c = (float)0.84506291151 */
112 erx = 0.845062911510467529296875L,
114 * Coefficients for approximation to erf on [0,0.84375]
116 /* 2/sqrt(pi) - 1 */
117 efx = 1.2837916709551257389615890312154517168810E-1L,
118 /* 8 * (2/sqrt(pi) - 1) */
119 efx8 = 1.0270333367641005911692712249723613735048E0L,
121 pp[6] = {
122 1.122751350964552113068262337278335028553E6L,
123 -2.808533301997696164408397079650699163276E6L,
124 -3.314325479115357458197119660818768924100E5L,
125 -6.848684465326256109712135497895525446398E4L,
126 -2.657817695110739185591505062971929859314E3L,
127 -1.655310302737837556654146291646499062882E2L,
130 qq[6] = {
131 8.745588372054466262548908189000448124232E6L,
132 3.746038264792471129367533128637019611485E6L,
133 7.066358783162407559861156173539693900031E5L,
134 7.448928604824620999413120955705448117056E4L,
135 4.511583986730994111992253980546131408924E3L,
136 1.368902937933296323345610240009071254014E2L,
137 /* 1.000000000000000000000000000000000000000E0 */
141 * Coefficients for approximation to erf in [0.84375,1.25]
143 /* erf(x+1) = 0.845062911510467529296875 + pa(x)/qa(x)
144 -0.15625 <= x <= +.25
145 Peak relative error 8.5e-22 */
147 pa[8] = {
148 -1.076952146179812072156734957705102256059E0L,
149 1.884814957770385593365179835059971587220E2L,
150 -5.339153975012804282890066622962070115606E1L,
151 4.435910679869176625928504532109635632618E1L,
152 1.683219516032328828278557309642929135179E1L,
153 -2.360236618396952560064259585299045804293E0L,
154 1.852230047861891953244413872297940938041E0L,
155 9.394994446747752308256773044667843200719E-2L,
158 qa[7] = {
159 4.559263722294508998149925774781887811255E2L,
160 3.289248982200800575749795055149780689738E2L,
161 2.846070965875643009598627918383314457912E2L,
162 1.398715859064535039433275722017479994465E2L,
163 6.060190733759793706299079050985358190726E1L,
164 2.078695677795422351040502569964299664233E1L,
165 4.641271134150895940966798357442234498546E0L,
166 /* 1.000000000000000000000000000000000000000E0 */
170 * Coefficients for approximation to erfc in [1.25,1/0.35]
172 /* erfc(1/x) = x exp (-1/x^2 - 0.5625 + ra(x^2)/sa(x^2))
173 1/2.85711669921875 < 1/x < 1/1.25
174 Peak relative error 3.1e-21 */
176 ra[] = {
177 1.363566591833846324191000679620738857234E-1L,
178 1.018203167219873573808450274314658434507E1L,
179 1.862359362334248675526472871224778045594E2L,
180 1.411622588180721285284945138667933330348E3L,
181 5.088538459741511988784440103218342840478E3L,
182 8.928251553922176506858267311750789273656E3L,
183 7.264436000148052545243018622742770549982E3L,
184 2.387492459664548651671894725748959751119E3L,
185 2.220916652813908085449221282808458466556E2L,
188 sa[] = {
189 -1.382234625202480685182526402169222331847E1L,
190 -3.315638835627950255832519203687435946482E2L,
191 -2.949124863912936259747237164260785326692E3L,
192 -1.246622099070875940506391433635999693661E4L,
193 -2.673079795851665428695842853070996219632E4L,
194 -2.880269786660559337358397106518918220991E4L,
195 -1.450600228493968044773354186390390823713E4L,
196 -2.874539731125893533960680525192064277816E3L,
197 -1.402241261419067750237395034116942296027E2L,
198 /* 1.000000000000000000000000000000000000000E0 */
201 * Coefficients for approximation to erfc in [1/.35,107]
203 /* erfc(1/x) = x exp (-1/x^2 - 0.5625 + rb(x^2)/sb(x^2))
204 1/6.6666259765625 < 1/x < 1/2.85711669921875
205 Peak relative error 4.2e-22 */
206 rb[] = {
207 -4.869587348270494309550558460786501252369E-5L,
208 -4.030199390527997378549161722412466959403E-3L,
209 -9.434425866377037610206443566288917589122E-2L,
210 -9.319032754357658601200655161585539404155E-1L,
211 -4.273788174307459947350256581445442062291E0L,
212 -8.842289940696150508373541814064198259278E0L,
213 -7.069215249419887403187988144752613025255E0L,
214 -1.401228723639514787920274427443330704764E0L,
217 sb[] = {
218 4.936254964107175160157544545879293019085E-3L,
219 1.583457624037795744377163924895349412015E-1L,
220 1.850647991850328356622940552450636420484E0L,
221 9.927611557279019463768050710008450625415E0L,
222 2.531667257649436709617165336779212114570E1L,
223 2.869752886406743386458304052862814690045E1L,
224 1.182059497870819562441683560749192539345E1L,
225 /* 1.000000000000000000000000000000000000000E0 */
227 /* erfc(1/x) = x exp (-1/x^2 - 0.5625 + rc(x^2)/sc(x^2))
228 1/107 <= 1/x <= 1/6.6666259765625
229 Peak relative error 1.1e-21 */
230 rc[] = {
231 -8.299617545269701963973537248996670806850E-5L,
232 -6.243845685115818513578933902532056244108E-3L,
233 -1.141667210620380223113693474478394397230E-1L,
234 -7.521343797212024245375240432734425789409E-1L,
235 -1.765321928311155824664963633786967602934E0L,
236 -1.029403473103215800456761180695263439188E0L,
239 sc[] = {
240 8.413244363014929493035952542677768808601E-3L,
241 2.065114333816877479753334599639158060979E-1L,
242 1.639064941530797583766364412782135680148E0L,
243 4.936788463787115555582319302981666347450E0L,
244 5.005177727208955487404729933261347679090E0L,
245 /* 1.000000000000000000000000000000000000000E0 */
248 long double
249 erfl(long double x)
251 long double R, S, P, Q, s, y, z, r;
252 int32_t ix, i;
253 u_int32_t se, i0, i1;
255 GET_LDOUBLE_WORDS (se, i0, i1, x);
256 ix = se & 0x7fff;
258 if (ix >= 0x7fff)
259 { /* erf(nan)=nan */
260 i = ((se & 0xffff) >> 15) << 1;
261 return (long double) (1 - i) + one / x; /* erf(+-inf)=+-1 */
264 ix = (ix << 16) | (i0 >> 16);
265 if (ix < 0x3ffed800) /* |x|<0.84375 */
267 if (ix < 0x3fde8000) /* |x|<2**-33 */
269 if (ix < 0x00080000)
270 return 0.125 * (8.0 * x + efx8 * x); /*avoid underflow */
271 return x + efx * x;
273 z = x * x;
274 r = pp[0] + z * (pp[1]
275 + z * (pp[2] + z * (pp[3] + z * (pp[4] + z * pp[5]))));
276 s = qq[0] + z * (qq[1]
277 + z * (qq[2] + z * (qq[3] + z * (qq[4] + z * (qq[5] + z)))));
278 y = r / s;
279 return x + x * y;
281 if (ix < 0x3fffa000) /* 1.25 */
282 { /* 0.84375 <= |x| < 1.25 */
283 s = fabsl (x) - one;
284 P = pa[0] + s * (pa[1] + s * (pa[2]
285 + s * (pa[3] + s * (pa[4] + s * (pa[5] + s * (pa[6] + s * pa[7]))))));
286 Q = qa[0] + s * (qa[1] + s * (qa[2]
287 + s * (qa[3] + s * (qa[4] + s * (qa[5] + s * (qa[6] + s))))));
288 if ((se & 0x8000) == 0)
289 return erx + P / Q;
290 else
291 return -erx - P / Q;
293 if (ix >= 0x4001d555) /* 6.6666259765625 */
294 { /* inf>|x|>=6.666 */
295 if ((se & 0x8000) == 0)
296 return one - tiny;
297 else
298 return tiny - one;
300 x = fabsl (x);
301 s = one / (x * x);
302 if (ix < 0x4000b6db) /* 2.85711669921875 */
304 R = ra[0] + s * (ra[1] + s * (ra[2] + s * (ra[3] + s * (ra[4] +
305 s * (ra[5] + s * (ra[6] + s * (ra[7] + s * ra[8])))))));
306 S = sa[0] + s * (sa[1] + s * (sa[2] + s * (sa[3] + s * (sa[4] +
307 s * (sa[5] + s * (sa[6] + s * (sa[7] + s * (sa[8] + s))))))));
309 else
310 { /* |x| >= 1/0.35 */
311 R = rb[0] + s * (rb[1] + s * (rb[2] + s * (rb[3] + s * (rb[4] +
312 s * (rb[5] + s * (rb[6] + s * rb[7]))))));
313 S = sb[0] + s * (sb[1] + s * (sb[2] + s * (sb[3] + s * (sb[4] +
314 s * (sb[5] + s * (sb[6] + s))))));
316 z = x;
317 GET_LDOUBLE_WORDS (i, i0, i1, z);
318 i1 = 0;
319 SET_LDOUBLE_WORDS (z, i, i0, i1);
321 expl (-z * z - 0.5625) * expl ((z - x) * (z + x) + R / S);
322 if ((se & 0x8000) == 0)
323 return one - r / x;
324 else
325 return r / x - one;
328 long double
329 erfcl(long double x)
331 int32_t hx, ix;
332 long double R, S, P, Q, s, y, z, r;
333 u_int32_t se, i0, i1;
335 GET_LDOUBLE_WORDS (se, i0, i1, x);
336 ix = se & 0x7fff;
337 if (ix >= 0x7fff)
338 { /* erfc(nan)=nan */
339 /* erfc(+-inf)=0,2 */
340 return (long double) (((se & 0xffff) >> 15) << 1) + one / x;
343 ix = (ix << 16) | (i0 >> 16);
344 if (ix < 0x3ffed800) /* |x|<0.84375 */
346 if (ix < 0x3fbe0000) /* |x|<2**-65 */
347 return one - x;
348 z = x * x;
349 r = pp[0] + z * (pp[1]
350 + z * (pp[2] + z * (pp[3] + z * (pp[4] + z * pp[5]))));
351 s = qq[0] + z * (qq[1]
352 + z * (qq[2] + z * (qq[3] + z * (qq[4] + z * (qq[5] + z)))));
353 y = r / s;
354 if (ix < 0x3ffd8000) /* x<1/4 */
356 return one - (x + x * y);
358 else
360 r = x * y;
361 r += (x - half);
362 return half - r;
365 if (ix < 0x3fffa000) /* 1.25 */
366 { /* 0.84375 <= |x| < 1.25 */
367 s = fabsl (x) - one;
368 P = pa[0] + s * (pa[1] + s * (pa[2]
369 + s * (pa[3] + s * (pa[4] + s * (pa[5] + s * (pa[6] + s * pa[7]))))));
370 Q = qa[0] + s * (qa[1] + s * (qa[2]
371 + s * (qa[3] + s * (qa[4] + s * (qa[5] + s * (qa[6] + s))))));
372 if ((se & 0x8000) == 0)
374 z = one - erx;
375 return z - P / Q;
377 else
379 z = erx + P / Q;
380 return one + z;
383 if (ix < 0x4005d600) /* 107 */
384 { /* |x|<107 */
385 x = fabsl (x);
386 s = one / (x * x);
387 if (ix < 0x4000b6db) /* 2.85711669921875 */
388 { /* |x| < 1/.35 ~ 2.857143 */
389 R = ra[0] + s * (ra[1] + s * (ra[2] + s * (ra[3] + s * (ra[4] +
390 s * (ra[5] + s * (ra[6] + s * (ra[7] + s * ra[8])))))));
391 S = sa[0] + s * (sa[1] + s * (sa[2] + s * (sa[3] + s * (sa[4] +
392 s * (sa[5] + s * (sa[6] + s * (sa[7] + s * (sa[8] + s))))))));
394 else if (ix < 0x4001d555) /* 6.6666259765625 */
395 { /* 6.666 > |x| >= 1/.35 ~ 2.857143 */
396 R = rb[0] + s * (rb[1] + s * (rb[2] + s * (rb[3] + s * (rb[4] +
397 s * (rb[5] + s * (rb[6] + s * rb[7]))))));
398 S = sb[0] + s * (sb[1] + s * (sb[2] + s * (sb[3] + s * (sb[4] +
399 s * (sb[5] + s * (sb[6] + s))))));
401 else
402 { /* |x| >= 6.666 */
403 if (se & 0x8000)
404 return two - tiny; /* x < -6.666 */
406 R = rc[0] + s * (rc[1] + s * (rc[2] + s * (rc[3] +
407 s * (rc[4] + s * rc[5]))));
408 S = sc[0] + s * (sc[1] + s * (sc[2] + s * (sc[3] +
409 s * (sc[4] + s))));
411 z = x;
412 GET_LDOUBLE_WORDS (hx, i0, i1, z);
413 i1 = 0;
414 i0 &= 0xffffff00;
415 SET_LDOUBLE_WORDS (z, hx, i0, i1);
416 r = expl (-z * z - 0.5625) *
417 expl ((z - x) * (z + x) + R / S);
418 if ((se & 0x8000) == 0)
419 return r / x;
420 else
421 return two - r / x;
423 else
425 if ((se & 0x8000) == 0)
426 return tiny * tiny;
427 else
428 return two - tiny;