1 /* origin: OpenBSD /usr/src/lib/libm/src/ld80/e_erfl.c */
3 * ====================================================
4 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
6 * Developed at SunPro, a Sun Microsystems, Inc. business.
7 * Permission to use, copy, modify, and distribute this
8 * software is freely granted, provided that this notice
10 * ====================================================
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.
27 /* double erf(double x)
28 * double erfc(double x)
31 * erf(x) = --------- | exp(-t*t)dt
38 * erfc(-x) = 2 - erfc(x)
41 * 1. For |x| in [0, 0.84375]
42 * erf(x) = x + x*R(x^2)
43 * erfc(x) = 1 - erf(x) if x in [-.84375,0.25]
44 * = 0.5 + ((0.5-x)-x*R) if x in [0.25,0.84375]
45 * Remark. The formula is derived by noting
46 * erf(x) = (2/sqrt(pi))*(x - x^3/3 + x^5/10 - x^7/42 + ....)
48 * 2/sqrt(pi) = 1.128379167095512573896158903121545171688
49 * is close to one. The interval is chosen because the fix
50 * point of erf(x) is near 0.6174 (i.e., erf(x)=x when x is
51 * near 0.6174), and by some experiment, 0.84375 is chosen to
52 * guarantee the error is less than one ulp for erf.
54 * 2. For |x| in [0.84375,1.25], let s = |x| - 1, and
55 * c = 0.84506291151 rounded to single (24 bits)
56 * erf(x) = sign(x) * (c + P1(s)/Q1(s))
57 * erfc(x) = (1-c) - P1(s)/Q1(s) if x > 0
58 * 1+(c+P1(s)/Q1(s)) if x < 0
59 * Remark: here we use the taylor series expansion at x=1.
60 * erf(1+s) = erf(1) + s*Poly(s)
61 * = 0.845.. + P1(s)/Q1(s)
62 * Note that |P1/Q1|< 0.078 for x in [0.84375,1.25]
64 * 3. For x in [1.25,1/0.35(~2.857143)],
65 * erfc(x) = (1/x)*exp(-x*x-0.5625+R1(z)/S1(z))
67 * erf(x) = 1 - erfc(x)
69 * 4. For x in [1/0.35,107]
70 * erfc(x) = (1/x)*exp(-x*x-0.5625+R2/S2) if x > 0
71 * = 2.0 - (1/x)*exp(-x*x-0.5625+R2(z)/S2(z))
73 * = 2.0 - tiny (if x <= -6.666)
75 * erf(x) = sign(x)*(1.0 - erfc(x)) if x < 6.666, else
76 * erf(x) = sign(x)*(1.0 - tiny)
78 * To compute exp(-x*x-0.5625+R/S), let s be a single
79 * precision number and s := x; then
80 * -x*x = -s*s + (s-x)*(s+x)
81 * exp(-x*x-0.5626+R/S) =
82 * exp(-s*s-0.5625)*exp((s-x)*(s+x)+R/S);
84 * Here 4 and 5 make use of the asymptotic series
86 * erfc(x) ~ ---------- * ( 1 + Poly(1/x^2) )
89 * 5. For inf > x >= 107
90 * erf(x) = sign(x) *(1 - tiny) (raise inexact)
91 * erfc(x) = tiny*tiny (raise underflow) if x > 0
95 * erf(0) = 0, erf(inf) = 1, erf(-inf) = -1,
96 * erfc(0) = 1, erfc(inf) = 0, erfc(-inf) = 2,
97 * erfc/erf(NaN) is NaN
103 #if LDBL_MANT_DIG == 53 && LDBL_MAX_EXP == 1024
104 long double erfl(long double x)
108 #elif LDBL_MANT_DIG == 64 && LDBL_MAX_EXP == 16384
109 static const long double
114 /* c = (float)0.84506291151 */
115 erx = 0.845062911510467529296875L,
118 * Coefficients for approximation to erf on [0,0.84375]
121 efx = 1.2837916709551257389615890312154517168810E-1L,
122 /* 8 * (2/sqrt(pi) - 1) */
123 efx8 = 1.0270333367641005911692712249723613735048E0L,
125 1.122751350964552113068262337278335028553E6L,
126 -2.808533301997696164408397079650699163276E6L,
127 -3.314325479115357458197119660818768924100E5L,
128 -6.848684465326256109712135497895525446398E4L,
129 -2.657817695110739185591505062971929859314E3L,
130 -1.655310302737837556654146291646499062882E2L,
133 8.745588372054466262548908189000448124232E6L,
134 3.746038264792471129367533128637019611485E6L,
135 7.066358783162407559861156173539693900031E5L,
136 7.448928604824620999413120955705448117056E4L,
137 4.511583986730994111992253980546131408924E3L,
138 1.368902937933296323345610240009071254014E2L,
139 /* 1.000000000000000000000000000000000000000E0 */
143 * Coefficients for approximation to erf in [0.84375,1.25]
145 /* erf(x+1) = 0.845062911510467529296875 + pa(x)/qa(x)
146 -0.15625 <= x <= +.25
147 Peak relative error 8.5e-22 */
149 -1.076952146179812072156734957705102256059E0L,
150 1.884814957770385593365179835059971587220E2L,
151 -5.339153975012804282890066622962070115606E1L,
152 4.435910679869176625928504532109635632618E1L,
153 1.683219516032328828278557309642929135179E1L,
154 -2.360236618396952560064259585299045804293E0L,
155 1.852230047861891953244413872297940938041E0L,
156 9.394994446747752308256773044667843200719E-2L,
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 1.363566591833846324191000679620738857234E-1L,
177 1.018203167219873573808450274314658434507E1L,
178 1.862359362334248675526472871224778045594E2L,
179 1.411622588180721285284945138667933330348E3L,
180 5.088538459741511988784440103218342840478E3L,
181 8.928251553922176506858267311750789273656E3L,
182 7.264436000148052545243018622742770549982E3L,
183 2.387492459664548651671894725748959751119E3L,
184 2.220916652813908085449221282808458466556E2L,
187 -1.382234625202480685182526402169222331847E1L,
188 -3.315638835627950255832519203687435946482E2L,
189 -2.949124863912936259747237164260785326692E3L,
190 -1.246622099070875940506391433635999693661E4L,
191 -2.673079795851665428695842853070996219632E4L,
192 -2.880269786660559337358397106518918220991E4L,
193 -1.450600228493968044773354186390390823713E4L,
194 -2.874539731125893533960680525192064277816E3L,
195 -1.402241261419067750237395034116942296027E2L,
196 /* 1.000000000000000000000000000000000000000E0 */
200 * Coefficients for approximation to erfc in [1/.35,107]
202 /* erfc(1/x) = x exp (-1/x^2 - 0.5625 + rb(x^2)/sb(x^2))
203 1/6.6666259765625 < 1/x < 1/2.85711669921875
204 Peak relative error 4.2e-22 */
206 -4.869587348270494309550558460786501252369E-5L,
207 -4.030199390527997378549161722412466959403E-3L,
208 -9.434425866377037610206443566288917589122E-2L,
209 -9.319032754357658601200655161585539404155E-1L,
210 -4.273788174307459947350256581445442062291E0L,
211 -8.842289940696150508373541814064198259278E0L,
212 -7.069215249419887403187988144752613025255E0L,
213 -1.401228723639514787920274427443330704764E0L,
216 4.936254964107175160157544545879293019085E-3L,
217 1.583457624037795744377163924895349412015E-1L,
218 1.850647991850328356622940552450636420484E0L,
219 9.927611557279019463768050710008450625415E0L,
220 2.531667257649436709617165336779212114570E1L,
221 2.869752886406743386458304052862814690045E1L,
222 1.182059497870819562441683560749192539345E1L,
223 /* 1.000000000000000000000000000000000000000E0 */
225 /* erfc(1/x) = x exp (-1/x^2 - 0.5625 + rc(x^2)/sc(x^2))
226 1/107 <= 1/x <= 1/6.6666259765625
227 Peak relative error 1.1e-21 */
229 -8.299617545269701963973537248996670806850E-5L,
230 -6.243845685115818513578933902532056244108E-3L,
231 -1.141667210620380223113693474478394397230E-1L,
232 -7.521343797212024245375240432734425789409E-1L,
233 -1.765321928311155824664963633786967602934E0L,
234 -1.029403473103215800456761180695263439188E0L,
237 8.413244363014929493035952542677768808601E-3L,
238 2.065114333816877479753334599639158060979E-1L,
239 1.639064941530797583766364412782135680148E0L,
240 4.936788463787115555582319302981666347450E0L,
241 5.005177727208955487404729933261347679090E0L,
242 /* 1.000000000000000000000000000000000000000E0 */
245 long double erfl(long double x)
247 long double R, S, P, Q, s, y, z, r;
251 GET_LDOUBLE_WORDS (se, i0, i1, x);
254 if (ix >= 0x7fff) { /* erf(nan)=nan */
255 i = ((se & 0xffff) >> 15) << 1;
256 return (long double)(1 - i) + one / x; /* erf(+-inf)=+-1 */
259 ix = (ix << 16) | (i0 >> 16);
260 if (ix < 0x3ffed800) { /* |x| < 0.84375 */
261 if (ix < 0x3fde8000) { /* |x| < 2**-33 */
263 return 0.125 * (8.0 * x + efx8 * x); /* avoid underflow */
267 r = pp[0] + z * (pp[1] +
268 z * (pp[2] + z * (pp[3] + z * (pp[4] + z * pp[5]))));
269 s = qq[0] + z * (qq[1] +
270 z * (qq[2] + z * (qq[3] + z * (qq[4] + z * (qq[5] + z)))));
274 if (ix < 0x3fffa000) { /* 0.84375 <= |x| < 1.25 */
276 P = pa[0] + s * (pa[1] + s * (pa[2] +
277 s * (pa[3] + s * (pa[4] + s * (pa[5] + s * (pa[6] + s * pa[7]))))));
278 Q = qa[0] + s * (qa[1] + s * (qa[2] +
279 s * (qa[3] + s * (qa[4] + s * (qa[5] + s * (qa[6] + s))))));
280 if ((se & 0x8000) == 0)
284 if (ix >= 0x4001d555) { /* inf > |x| >= 6.6666259765625 */
285 if ((se & 0x8000) == 0)
291 if (ix < 0x4000b6db) { /* 1.25 <= |x| < 2.85711669921875 ~ 1/.35 */
292 R = ra[0] + s * (ra[1] + s * (ra[2] + s * (ra[3] + s * (ra[4] +
293 s * (ra[5] + s * (ra[6] + s * (ra[7] + s * ra[8])))))));
294 S = sa[0] + s * (sa[1] + s * (sa[2] + s * (sa[3] + s * (sa[4] +
295 s * (sa[5] + s * (sa[6] + s * (sa[7] + s * (sa[8] + s))))))));
296 } else { /* 2.857 <= |x| < 6.667 */
297 R = rb[0] + s * (rb[1] + s * (rb[2] + s * (rb[3] + s * (rb[4] +
298 s * (rb[5] + s * (rb[6] + s * rb[7]))))));
299 S = sb[0] + s * (sb[1] + s * (sb[2] + s * (sb[3] + s * (sb[4] +
300 s * (sb[5] + s * (sb[6] + s))))));
303 GET_LDOUBLE_WORDS(i, i0, i1, z);
305 SET_LDOUBLE_WORDS(z, i, i0, i1);
306 r = expl(-z * z - 0.5625) * expl((z - x) * (z + x) + R / S);
307 if ((se & 0x8000) == 0)
312 long double erfcl(long double x)
315 long double R, S, P, Q, s, y, z, r;
318 GET_LDOUBLE_WORDS (se, i0, i1, x);
320 if (ix >= 0x7fff) { /* erfc(nan) = nan, erfc(+-inf) = 0,2 */
321 return (long double)(((se & 0xffff) >> 15) << 1) + one / x;
324 ix = (ix << 16) | (i0 >> 16);
325 if (ix < 0x3ffed800) { /* |x| < 0.84375 */
326 if (ix < 0x3fbe0000) /* |x| < 2**-65 */
329 r = pp[0] + z * (pp[1] +
330 z * (pp[2] + z * (pp[3] + z * (pp[4] + z * pp[5]))));
331 s = qq[0] + z * (qq[1] +
332 z * (qq[2] + z * (qq[3] + z * (qq[4] + z * (qq[5] + z)))));
334 if (ix < 0x3ffd8000) /* x < 1/4 */
335 return one - (x + x * y);
340 if (ix < 0x3fffa000) { /* 0.84375 <= |x| < 1.25 */
342 P = pa[0] + s * (pa[1] + s * (pa[2] +
343 s * (pa[3] + s * (pa[4] + s * (pa[5] + s * (pa[6] + s * pa[7]))))));
344 Q = qa[0] + s * (qa[1] + s * (qa[2] +
345 s * (qa[3] + s * (qa[4] + s * (qa[5] + s * (qa[6] + s))))));
346 if ((se & 0x8000) == 0) {
353 if (ix < 0x4005d600) { /* |x| < 107 */
356 if (ix < 0x4000b6db) { /* 1.25 <= |x| < 2.85711669921875 ~ 1/.35 */
357 R = ra[0] + s * (ra[1] + s * (ra[2] + s * (ra[3] + s * (ra[4] +
358 s * (ra[5] + s * (ra[6] + s * (ra[7] + s * ra[8])))))));
359 S = sa[0] + s * (sa[1] + s * (sa[2] + s * (sa[3] + s * (sa[4] +
360 s * (sa[5] + s * (sa[6] + s * (sa[7] + s * (sa[8] + s))))))));
361 } else if (ix < 0x4001d555) { /* 6.6666259765625 > |x| >= 1/.35 ~ 2.857143 */
362 R = rb[0] + s * (rb[1] + s * (rb[2] + s * (rb[3] + s * (rb[4] +
363 s * (rb[5] + s * (rb[6] + s * rb[7]))))));
364 S = sb[0] + s * (sb[1] + s * (sb[2] + s * (sb[3] + s * (sb[4] +
365 s * (sb[5] + s * (sb[6] + s))))));
366 } else { /* 107 > |x| >= 6.666 */
368 return two - tiny;/* x < -6.666 */
369 R = rc[0] + s * (rc[1] + s * (rc[2] + s * (rc[3] +
370 s * (rc[4] + s * rc[5]))));
371 S = sc[0] + s * (sc[1] + s * (sc[2] + s * (sc[3] +
375 GET_LDOUBLE_WORDS (hx, i0, i1, z);
378 SET_LDOUBLE_WORDS (z, hx, i0, i1);
379 r = expl (-z * z - 0.5625) *
380 expl ((z - x) * (z + x) + R / S);
381 if ((se & 0x8000) == 0)
386 if ((se & 0x8000) == 0)