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
110 erx = 0.845062911510467529296875L,
113 * Coefficients for approximation to erf on [0,0.84375]
115 /* 8 * (2/sqrt(pi) - 1) */
116 efx8 = 1.0270333367641005911692712249723613735048E0L,
118 1.122751350964552113068262337278335028553E6L,
119 -2.808533301997696164408397079650699163276E6L,
120 -3.314325479115357458197119660818768924100E5L,
121 -6.848684465326256109712135497895525446398E4L,
122 -2.657817695110739185591505062971929859314E3L,
123 -1.655310302737837556654146291646499062882E2L,
126 8.745588372054466262548908189000448124232E6L,
127 3.746038264792471129367533128637019611485E6L,
128 7.066358783162407559861156173539693900031E5L,
129 7.448928604824620999413120955705448117056E4L,
130 4.511583986730994111992253980546131408924E3L,
131 1.368902937933296323345610240009071254014E2L,
132 /* 1.000000000000000000000000000000000000000E0 */
136 * Coefficients for approximation to erf in [0.84375,1.25]
138 /* erf(x+1) = 0.845062911510467529296875 + pa(x)/qa(x)
139 -0.15625 <= x <= +.25
140 Peak relative error 8.5e-22 */
142 -1.076952146179812072156734957705102256059E0L,
143 1.884814957770385593365179835059971587220E2L,
144 -5.339153975012804282890066622962070115606E1L,
145 4.435910679869176625928504532109635632618E1L,
146 1.683219516032328828278557309642929135179E1L,
147 -2.360236618396952560064259585299045804293E0L,
148 1.852230047861891953244413872297940938041E0L,
149 9.394994446747752308256773044667843200719E-2L,
152 4.559263722294508998149925774781887811255E2L,
153 3.289248982200800575749795055149780689738E2L,
154 2.846070965875643009598627918383314457912E2L,
155 1.398715859064535039433275722017479994465E2L,
156 6.060190733759793706299079050985358190726E1L,
157 2.078695677795422351040502569964299664233E1L,
158 4.641271134150895940966798357442234498546E0L,
159 /* 1.000000000000000000000000000000000000000E0 */
163 * Coefficients for approximation to erfc in [1.25,1/0.35]
165 /* erfc(1/x) = x exp (-1/x^2 - 0.5625 + ra(x^2)/sa(x^2))
166 1/2.85711669921875 < 1/x < 1/1.25
167 Peak relative error 3.1e-21 */
169 1.363566591833846324191000679620738857234E-1L,
170 1.018203167219873573808450274314658434507E1L,
171 1.862359362334248675526472871224778045594E2L,
172 1.411622588180721285284945138667933330348E3L,
173 5.088538459741511988784440103218342840478E3L,
174 8.928251553922176506858267311750789273656E3L,
175 7.264436000148052545243018622742770549982E3L,
176 2.387492459664548651671894725748959751119E3L,
177 2.220916652813908085449221282808458466556E2L,
180 -1.382234625202480685182526402169222331847E1L,
181 -3.315638835627950255832519203687435946482E2L,
182 -2.949124863912936259747237164260785326692E3L,
183 -1.246622099070875940506391433635999693661E4L,
184 -2.673079795851665428695842853070996219632E4L,
185 -2.880269786660559337358397106518918220991E4L,
186 -1.450600228493968044773354186390390823713E4L,
187 -2.874539731125893533960680525192064277816E3L,
188 -1.402241261419067750237395034116942296027E2L,
189 /* 1.000000000000000000000000000000000000000E0 */
193 * Coefficients for approximation to erfc in [1/.35,107]
195 /* erfc(1/x) = x exp (-1/x^2 - 0.5625 + rb(x^2)/sb(x^2))
196 1/6.6666259765625 < 1/x < 1/2.85711669921875
197 Peak relative error 4.2e-22 */
199 -4.869587348270494309550558460786501252369E-5L,
200 -4.030199390527997378549161722412466959403E-3L,
201 -9.434425866377037610206443566288917589122E-2L,
202 -9.319032754357658601200655161585539404155E-1L,
203 -4.273788174307459947350256581445442062291E0L,
204 -8.842289940696150508373541814064198259278E0L,
205 -7.069215249419887403187988144752613025255E0L,
206 -1.401228723639514787920274427443330704764E0L,
209 4.936254964107175160157544545879293019085E-3L,
210 1.583457624037795744377163924895349412015E-1L,
211 1.850647991850328356622940552450636420484E0L,
212 9.927611557279019463768050710008450625415E0L,
213 2.531667257649436709617165336779212114570E1L,
214 2.869752886406743386458304052862814690045E1L,
215 1.182059497870819562441683560749192539345E1L,
216 /* 1.000000000000000000000000000000000000000E0 */
218 /* erfc(1/x) = x exp (-1/x^2 - 0.5625 + rc(x^2)/sc(x^2))
219 1/107 <= 1/x <= 1/6.6666259765625
220 Peak relative error 1.1e-21 */
222 -8.299617545269701963973537248996670806850E-5L,
223 -6.243845685115818513578933902532056244108E-3L,
224 -1.141667210620380223113693474478394397230E-1L,
225 -7.521343797212024245375240432734425789409E-1L,
226 -1.765321928311155824664963633786967602934E0L,
227 -1.029403473103215800456761180695263439188E0L,
230 8.413244363014929493035952542677768808601E-3L,
231 2.065114333816877479753334599639158060979E-1L,
232 1.639064941530797583766364412782135680148E0L,
233 4.936788463787115555582319302981666347450E0L,
234 5.005177727208955487404729933261347679090E0L,
235 /* 1.000000000000000000000000000000000000000E0 */
238 static long double erfc1(long double x)
243 P = pa[0] + s * (pa[1] + s * (pa[2] +
244 s * (pa[3] + s * (pa[4] + s * (pa[5] + s * (pa[6] + s * pa[7]))))));
245 Q = qa[0] + s * (qa[1] + s * (qa[2] +
246 s * (qa[3] + s * (qa[4] + s * (qa[5] + s * (qa[6] + s))))));
247 return 1 - erx - P / Q;
250 static long double erfc2(uint32_t ix, long double x)
255 if (ix < 0x3fffa000) /* 0.84375 <= |x| < 1.25 */
260 if (ix < 0x4000b6db) { /* 1.25 <= |x| < 2.857 ~ 1/.35 */
261 R = ra[0] + s * (ra[1] + s * (ra[2] + s * (ra[3] + s * (ra[4] +
262 s * (ra[5] + s * (ra[6] + s * (ra[7] + s * ra[8])))))));
263 S = sa[0] + s * (sa[1] + s * (sa[2] + s * (sa[3] + s * (sa[4] +
264 s * (sa[5] + s * (sa[6] + s * (sa[7] + s * (sa[8] + s))))))));
265 } else { /* 2.857 <= |x| */
266 R = rb[0] + s * (rb[1] + s * (rb[2] + s * (rb[3] + s * (rb[4] +
267 s * (rb[5] + s * (rb[6] + s * rb[7]))))));
268 S = sb[0] + s * (sb[1] + s * (sb[2] + s * (sb[3] + s * (sb[4] +
269 s * (sb[5] + s * (sb[6] + s))))));
271 if (ix < 0x4000b6db) { /* 1.25 <= |x| < 2.85711669921875 ~ 1/.35 */
272 R = ra[0] + s * (ra[1] + s * (ra[2] + s * (ra[3] + s * (ra[4] +
273 s * (ra[5] + s * (ra[6] + s * (ra[7] + s * ra[8])))))));
274 S = sa[0] + s * (sa[1] + s * (sa[2] + s * (sa[3] + s * (sa[4] +
275 s * (sa[5] + s * (sa[6] + s * (sa[7] + s * (sa[8] + s))))))));
276 } else if (ix < 0x4001d555) { /* 6.6666259765625 > |x| >= 1/.35 ~ 2.857143 */
277 R = rb[0] + s * (rb[1] + s * (rb[2] + s * (rb[3] + s * (rb[4] +
278 s * (rb[5] + s * (rb[6] + s * rb[7]))))));
279 S = sb[0] + s * (sb[1] + s * (sb[2] + s * (sb[3] + s * (sb[4] +
280 s * (sb[5] + s * (sb[6] + s))))));
281 } else { /* 107 > |x| >= 6.666 */
282 R = rc[0] + s * (rc[1] + s * (rc[2] + s * (rc[3] +
283 s * (rc[4] + s * rc[5]))));
284 S = sc[0] + s * (sc[1] + s * (sc[2] + s * (sc[3] +
288 GET_LDOUBLE_WORDS(ix, i0, i1, z);
291 SET_LDOUBLE_WORDS(z, ix, i0, i1);
292 return expl(-z*z - 0.5625) * expl((z - x) * (z + x) + R / S) / x;
295 long double erfl(long double x)
297 long double r, s, z, y;
301 GET_LDOUBLE_WORDS(ix, i0, i1, x);
305 /* erf(nan)=nan, erf(+-inf)=+-1 */
306 return 1 - 2*sign + 1/x;
308 ix = (ix << 16) | (i0 >> 16);
309 if (ix < 0x3ffed800) { /* |x| < 0.84375 */
310 if (ix < 0x3fde8000) { /* |x| < 2**-33 */
311 return 0.125 * (8 * x + efx8 * x); /* avoid underflow */
314 r = pp[0] + z * (pp[1] +
315 z * (pp[2] + z * (pp[3] + z * (pp[4] + z * pp[5]))));
316 s = qq[0] + z * (qq[1] +
317 z * (qq[2] + z * (qq[3] + z * (qq[4] + z * (qq[5] + z)))));
321 if (ix < 0x4001d555) /* |x| < 6.6666259765625 */
325 return sign ? -y : y;
328 long double erfcl(long double x)
330 long double r, s, z, y;
334 GET_LDOUBLE_WORDS(ix, i0, i1, x);
338 /* erfc(nan) = nan, erfc(+-inf) = 0,2 */
341 ix = (ix << 16) | (i0 >> 16);
342 if (ix < 0x3ffed800) { /* |x| < 0.84375 */
343 if (ix < 0x3fbe0000) /* |x| < 2**-65 */
346 r = pp[0] + z * (pp[1] +
347 z * (pp[2] + z * (pp[3] + z * (pp[4] + z * pp[5]))));
348 s = qq[0] + z * (qq[1] +
349 z * (qq[2] + z * (qq[3] + z * (qq[4] + z * (qq[5] + z)))));
351 if (ix < 0x3ffd8000) /* x < 1/4 */
352 return 1.0 - (x + x * y);
353 return 0.5 - (x - 0.5 + x * y);
355 if (ix < 0x4005d600) /* |x| < 107 */
356 return sign ? 2 - erfc2(ix,x) : erfc2(ix,x);
357 return sign ? 2 - 0x1p-16382L : 0x1p-16382L*0x1p-16382L;