erfl.c raw
1 /* origin: OpenBSD /usr/src/lib/libm/src/ld80/e_erfl.c */
2 /*
3 * ====================================================
4 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
5 *
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
9 * is preserved.
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 /* double erf(double x)
28 * double erfc(double x)
29 * x
30 * 2 |\
31 * erf(x) = --------- | exp(-t*t)dt
32 * sqrt(pi) \|
33 * 0
34 *
35 * erfc(x) = 1-erf(x)
36 * Note that
37 * erf(-x) = -erf(x)
38 * erfc(-x) = 2 - erfc(x)
39 *
40 * Method:
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 + ....)
47 * and that
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.
53 *
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]
63 *
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))
66 * z=1/x^2
67 * erf(x) = 1 - erfc(x)
68 *
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))
72 * if -6.666<x<0
73 * = 2.0 - tiny (if x <= -6.666)
74 * z=1/x^2
75 * erf(x) = sign(x)*(1.0 - erfc(x)) if x < 6.666, else
76 * erf(x) = sign(x)*(1.0 - tiny)
77 * Note1:
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);
83 * Note2:
84 * Here 4 and 5 make use of the asymptotic series
85 * exp(-x*x)
86 * erfc(x) ~ ---------- * ( 1 + Poly(1/x^2) )
87 * x*sqrt(pi)
88 *
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
92 * = 2 - tiny if x<0
93 *
94 * 7. Special case:
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
98 */
99
100
101 #include "libm.h"
102
103 #if LDBL_MANT_DIG == 53 && LDBL_MAX_EXP == 1024
104 long double erfl(long double x)
105 {
106 return erf(x);
107 }
108 long double erfcl(long double x)
109 {
110 return erfc(x);
111 }
112 #elif LDBL_MANT_DIG == 64 && LDBL_MAX_EXP == 16384
113 static const long double
114 erx = 0.845062911510467529296875L,
115
116 /*
117 * Coefficients for approximation to erf on [0,0.84375]
118 */
119 /* 8 * (2/sqrt(pi) - 1) */
120 efx8 = 1.0270333367641005911692712249723613735048E0L,
121 pp[6] = {
122 1.122751350964552113068262337278335028553E6L,
123 -2.808533301997696164408397079650699163276E6L,
124 -3.314325479115357458197119660818768924100E5L,
125 -6.848684465326256109712135497895525446398E4L,
126 -2.657817695110739185591505062971929859314E3L,
127 -1.655310302737837556654146291646499062882E2L,
128 },
129 qq[6] = {
130 8.745588372054466262548908189000448124232E6L,
131 3.746038264792471129367533128637019611485E6L,
132 7.066358783162407559861156173539693900031E5L,
133 7.448928604824620999413120955705448117056E4L,
134 4.511583986730994111992253980546131408924E3L,
135 1.368902937933296323345610240009071254014E2L,
136 /* 1.000000000000000000000000000000000000000E0 */
137 },
138
139 /*
140 * Coefficients for approximation to erf in [0.84375,1.25]
141 */
142 /* erf(x+1) = 0.845062911510467529296875 + pa(x)/qa(x)
143 -0.15625 <= x <= +.25
144 Peak relative error 8.5e-22 */
145 pa[8] = {
146 -1.076952146179812072156734957705102256059E0L,
147 1.884814957770385593365179835059971587220E2L,
148 -5.339153975012804282890066622962070115606E1L,
149 4.435910679869176625928504532109635632618E1L,
150 1.683219516032328828278557309642929135179E1L,
151 -2.360236618396952560064259585299045804293E0L,
152 1.852230047861891953244413872297940938041E0L,
153 9.394994446747752308256773044667843200719E-2L,
154 },
155 qa[7] = {
156 4.559263722294508998149925774781887811255E2L,
157 3.289248982200800575749795055149780689738E2L,
158 2.846070965875643009598627918383314457912E2L,
159 1.398715859064535039433275722017479994465E2L,
160 6.060190733759793706299079050985358190726E1L,
161 2.078695677795422351040502569964299664233E1L,
162 4.641271134150895940966798357442234498546E0L,
163 /* 1.000000000000000000000000000000000000000E0 */
164 },
165
166 /*
167 * Coefficients for approximation to erfc in [1.25,1/0.35]
168 */
169 /* erfc(1/x) = x exp (-1/x^2 - 0.5625 + ra(x^2)/sa(x^2))
170 1/2.85711669921875 < 1/x < 1/1.25
171 Peak relative error 3.1e-21 */
172 ra[] = {
173 1.363566591833846324191000679620738857234E-1L,
174 1.018203167219873573808450274314658434507E1L,
175 1.862359362334248675526472871224778045594E2L,
176 1.411622588180721285284945138667933330348E3L,
177 5.088538459741511988784440103218342840478E3L,
178 8.928251553922176506858267311750789273656E3L,
179 7.264436000148052545243018622742770549982E3L,
180 2.387492459664548651671894725748959751119E3L,
181 2.220916652813908085449221282808458466556E2L,
182 },
183 sa[] = {
184 -1.382234625202480685182526402169222331847E1L,
185 -3.315638835627950255832519203687435946482E2L,
186 -2.949124863912936259747237164260785326692E3L,
187 -1.246622099070875940506391433635999693661E4L,
188 -2.673079795851665428695842853070996219632E4L,
189 -2.880269786660559337358397106518918220991E4L,
190 -1.450600228493968044773354186390390823713E4L,
191 -2.874539731125893533960680525192064277816E3L,
192 -1.402241261419067750237395034116942296027E2L,
193 /* 1.000000000000000000000000000000000000000E0 */
194 },
195
196 /*
197 * Coefficients for approximation to erfc in [1/.35,107]
198 */
199 /* erfc(1/x) = x exp (-1/x^2 - 0.5625 + rb(x^2)/sb(x^2))
200 1/6.6666259765625 < 1/x < 1/2.85711669921875
201 Peak relative error 4.2e-22 */
202 rb[] = {
203 -4.869587348270494309550558460786501252369E-5L,
204 -4.030199390527997378549161722412466959403E-3L,
205 -9.434425866377037610206443566288917589122E-2L,
206 -9.319032754357658601200655161585539404155E-1L,
207 -4.273788174307459947350256581445442062291E0L,
208 -8.842289940696150508373541814064198259278E0L,
209 -7.069215249419887403187988144752613025255E0L,
210 -1.401228723639514787920274427443330704764E0L,
211 },
212 sb[] = {
213 4.936254964107175160157544545879293019085E-3L,
214 1.583457624037795744377163924895349412015E-1L,
215 1.850647991850328356622940552450636420484E0L,
216 9.927611557279019463768050710008450625415E0L,
217 2.531667257649436709617165336779212114570E1L,
218 2.869752886406743386458304052862814690045E1L,
219 1.182059497870819562441683560749192539345E1L,
220 /* 1.000000000000000000000000000000000000000E0 */
221 },
222 /* erfc(1/x) = x exp (-1/x^2 - 0.5625 + rc(x^2)/sc(x^2))
223 1/107 <= 1/x <= 1/6.6666259765625
224 Peak relative error 1.1e-21 */
225 rc[] = {
226 -8.299617545269701963973537248996670806850E-5L,
227 -6.243845685115818513578933902532056244108E-3L,
228 -1.141667210620380223113693474478394397230E-1L,
229 -7.521343797212024245375240432734425789409E-1L,
230 -1.765321928311155824664963633786967602934E0L,
231 -1.029403473103215800456761180695263439188E0L,
232 },
233 sc[] = {
234 8.413244363014929493035952542677768808601E-3L,
235 2.065114333816877479753334599639158060979E-1L,
236 1.639064941530797583766364412782135680148E0L,
237 4.936788463787115555582319302981666347450E0L,
238 5.005177727208955487404729933261347679090E0L,
239 /* 1.000000000000000000000000000000000000000E0 */
240 };
241
242 static long double erfc1(long double x)
243 {
244 long double s,P,Q;
245
246 s = fabsl(x) - 1;
247 P = pa[0] + s * (pa[1] + s * (pa[2] +
248 s * (pa[3] + s * (pa[4] + s * (pa[5] + s * (pa[6] + s * pa[7]))))));
249 Q = qa[0] + s * (qa[1] + s * (qa[2] +
250 s * (qa[3] + s * (qa[4] + s * (qa[5] + s * (qa[6] + s))))));
251 return 1 - erx - P / Q;
252 }
253
254 static long double erfc2(uint32_t ix, long double x)
255 {
256 union ldshape u;
257 long double s,z,R,S;
258
259 if (ix < 0x3fffa000) /* 0.84375 <= |x| < 1.25 */
260 return erfc1(x);
261
262 x = fabsl(x);
263 s = 1 / (x * x);
264 if (ix < 0x4000b6db) { /* 1.25 <= |x| < 2.857 ~ 1/.35 */
265 R = ra[0] + s * (ra[1] + s * (ra[2] + s * (ra[3] + s * (ra[4] +
266 s * (ra[5] + s * (ra[6] + s * (ra[7] + s * ra[8])))))));
267 S = sa[0] + s * (sa[1] + s * (sa[2] + s * (sa[3] + s * (sa[4] +
268 s * (sa[5] + s * (sa[6] + s * (sa[7] + s * (sa[8] + s))))))));
269 } else if (ix < 0x4001d555) { /* 2.857 <= |x| < 6.6666259765625 */
270 R = rb[0] + s * (rb[1] + s * (rb[2] + s * (rb[3] + s * (rb[4] +
271 s * (rb[5] + s * (rb[6] + s * rb[7]))))));
272 S = sb[0] + s * (sb[1] + s * (sb[2] + s * (sb[3] + s * (sb[4] +
273 s * (sb[5] + s * (sb[6] + s))))));
274 } else { /* 6.666 <= |x| < 107 (erfc only) */
275 R = rc[0] + s * (rc[1] + s * (rc[2] + s * (rc[3] +
276 s * (rc[4] + s * rc[5]))));
277 S = sc[0] + s * (sc[1] + s * (sc[2] + s * (sc[3] +
278 s * (sc[4] + s))));
279 }
280 u.f = x;
281 u.i.m &= -1ULL << 40;
282 z = u.f;
283 return expl(-z*z - 0.5625) * expl((z - x) * (z + x) + R / S) / x;
284 }
285
286 long double erfl(long double x)
287 {
288 long double r, s, z, y;
289 union ldshape u = {x};
290 uint32_t ix = (u.i.se & 0x7fffU)<<16 | u.i.m>>48;
291 int sign = u.i.se >> 15;
292
293 if (ix >= 0x7fff0000)
294 /* erf(nan)=nan, erf(+-inf)=+-1 */
295 return 1 - 2*sign + 1/x;
296 if (ix < 0x3ffed800) { /* |x| < 0.84375 */
297 if (ix < 0x3fde8000) { /* |x| < 2**-33 */
298 return 0.125 * (8 * x + efx8 * x); /* avoid underflow */
299 }
300 z = x * x;
301 r = pp[0] + z * (pp[1] +
302 z * (pp[2] + z * (pp[3] + z * (pp[4] + z * pp[5]))));
303 s = qq[0] + z * (qq[1] +
304 z * (qq[2] + z * (qq[3] + z * (qq[4] + z * (qq[5] + z)))));
305 y = r / s;
306 return x + x * y;
307 }
308 if (ix < 0x4001d555) /* |x| < 6.6666259765625 */
309 y = 1 - erfc2(ix,x);
310 else
311 y = 1 - 0x1p-16382L;
312 return sign ? -y : y;
313 }
314
315 long double erfcl(long double x)
316 {
317 long double r, s, z, y;
318 union ldshape u = {x};
319 uint32_t ix = (u.i.se & 0x7fffU)<<16 | u.i.m>>48;
320 int sign = u.i.se >> 15;
321
322 if (ix >= 0x7fff0000)
323 /* erfc(nan) = nan, erfc(+-inf) = 0,2 */
324 return 2*sign + 1/x;
325 if (ix < 0x3ffed800) { /* |x| < 0.84375 */
326 if (ix < 0x3fbe0000) /* |x| < 2**-65 */
327 return 1.0 - x;
328 z = x * x;
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)))));
333 y = r / s;
334 if (ix < 0x3ffd8000) /* x < 1/4 */
335 return 1.0 - (x + x * y);
336 return 0.5 - (x - 0.5 + x * y);
337 }
338 if (ix < 0x4005d600) /* |x| < 107 */
339 return sign ? 2 - erfc2(ix,x) : erfc2(ix,x);
340 y = 0x1p-16382L;
341 return sign ? 2 - y : y*y;
342 }
343 #elif LDBL_MANT_DIG == 113 && LDBL_MAX_EXP == 16384
344 // TODO: broken implementation to make things compile
345 long double erfl(long double x)
346 {
347 return erf(x);
348 }
349 long double erfcl(long double x)
350 {
351 return erfc(x);
352 }
353 #endif
354