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. 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.
54 *
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]
64 *
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)
69 *
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)
89 *
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
94 *
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
99 */
100
101
102
103
104 static const long double
105 tiny = 1e-4931L,
106 half = 0.5L,
107 one = 1.0L,
108 two = 2.0L,
109 /* c = (float)0.84506291151 */
110 erx = 0.845062911510467529296875L,
111 /*
112 * Coefficients for approximation to erf on [0,0.84375]
113 */
114 /* 2/sqrt(pi) - 1 */
115 efx = 1.2837916709551257389615890312154517168810E-1L,
116 /* 8 * (2/sqrt(pi) - 1) */
117 efx8 = 1.0270333367641005911692712249723613735048E0L,
118
119 pp[6] = {
120 1.122751350964552113068262337278335028553E6L,
121 -2.808533301997696164408397079650699163276E6L,
122 -3.314325479115357458197119660818768924100E5L,
123 -6.848684465326256109712135497895525446398E4L,
124 -2.657817695110739185591505062971929859314E3L,
125 -1.655310302737837556654146291646499062882E2L,
126 },
127
128 qq[6] = {
129 8.745588372054466262548908189000448124232E6L,
130 3.746038264792471129367533128637019611485E6L,
131 7.066358783162407559861156173539693900031E5L,
132 7.448928604824620999413120955705448117056E4L,
133 4.511583986730994111992253980546131408924E3L,
134 1.368902937933296323345610240009071254014E2L,
135 /* 1.000000000000000000000000000000000000000E0 */
136 },
137
138 /*
139 * Coefficients for approximation to erf in [0.84375,1.25]
140 */
141 /* erf(x+1) = 0.845062911510467529296875 + pa(x)/qa(x)
142 -0.15625 <= x <= +.25
143 Peak relative error 8.5e-22 */
144
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
156 qa[7] = {
157 4.559263722294508998149925774781887811255E2L,
158 3.289248982200800575749795055149780689738E2L,
159 2.846070965875643009598627918383314457912E2L,
160 1.398715859064535039433275722017479994465E2L,
161 6.060190733759793706299079050985358190726E1L,
162 2.078695677795422351040502569964299664233E1L,
163 4.641271134150895940966798357442234498546E0L,
164 /* 1.000000000000000000000000000000000000000E0 */
165 },
166
167 /*
168 * Coefficients for approximation to erfc in [1.25,1/0.35]
169 */
170 /* erfc(1/x) = x exp (-1/x^2 - 0.5625 + ra(x^2)/sa(x^2))
171 1/2.85711669921875 < 1/x < 1/1.25
172 Peak relative error 3.1e-21 */
173
174 ra[] = {
175 1.363566591833846324191000679620738857234E-1L,
176 1.018203167219873573808450274314658434507E1L,
177 1.862359362334248675526472871224778045594E2L,
178 1.411622588180721285284945138667933330348E3L,
179 5.088538459741511988784440103218342840478E3L,
180 8.928251553922176506858267311750789273656E3L,
181 7.264436000148052545243018622742770549982E3L,
182 2.387492459664548651671894725748959751119E3L,
183 2.220916652813908085449221282808458466556E2L,
184 },
185
186 sa[] = {
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 */
197 },
198 /*
199 * Coefficients for approximation to erfc in [1/.35,107]
200 */
201 /* erfc(1/x) = x exp (-1/x^2 - 0.5625 + rb(x^2)/sb(x^2))
202 1/6.6666259765625 < 1/x < 1/2.85711669921875
203 Peak relative error 4.2e-22 */
204 rb[] = {
205 -4.869587348270494309550558460786501252369E-5L,
206 -4.030199390527997378549161722412466959403E-3L,
207 -9.434425866377037610206443566288917589122E-2L,
208 -9.319032754357658601200655161585539404155E-1L,
209 -4.273788174307459947350256581445442062291E0L,
210 -8.842289940696150508373541814064198259278E0L,
211 -7.069215249419887403187988144752613025255E0L,
212 -1.401228723639514787920274427443330704764E0L,
213 },
214
215 sb[] = {
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 */
224 },
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 */
228 rc[] = {
229 -8.299617545269701963973537248996670806850E-5L,
230 -6.243845685115818513578933902532056244108E-3L,
231 -1.141667210620380223113693474478394397230E-1L,
232 -7.521343797212024245375240432734425789409E-1L,
233 -1.765321928311155824664963633786967602934E0L,
234 -1.029403473103215800456761180695263439188E0L,
235 },
236
237 sc[] = {
238 8.413244363014929493035952542677768808601E-3L,
239 2.065114333816877479753334599639158060979E-1L,
240 1.639064941530797583766364412782135680148E0L,
241 4.936788463787115555582319302981666347450E0L,
242 5.005177727208955487404729933261347679090E0L,
243 /* 1.000000000000000000000000000000000000000E0 */
244 };
245
246 long double
erfl(long double x)247 erfl(long double x)
248 {
249 long double R, S, P, Q, s, y, z, r;
250 int32_t ix, i;
251 u_int32_t se, i0, i1;
252
253 GET_LDOUBLE_WORDS (se, i0, i1, x);
254 ix = se & 0x7fff;
255
256 if (ix >= 0x7fff)
257 { /* erf(nan)=nan */
258 i = ((se & 0xffff) >> 15) << 1;
259 return (long double) (1 - i) + one / x; /* erf(+-inf)=+-1 */
260 }
261
262 ix = (ix << 16) | (i0 >> 16);
263 if (ix < 0x3ffed800) /* |x|<0.84375 */
264 {
265 if (ix < 0x3fde8000) /* |x|<2**-33 */
266 {
267 if (ix < 0x00080000)
268 return 0.125L * (8.0L * x + efx8 * x); /*avoid underflow */
269 return x + efx * x;
270 }
271 z = x * x;
272 r = pp[0] + z * (pp[1]
273 + z * (pp[2] + z * (pp[3] + z * (pp[4] + z * pp[5]))));
274 s = qq[0] + z * (qq[1]
275 + z * (qq[2] + z * (qq[3] + z * (qq[4] + z * (qq[5] + z)))));
276 y = r / s;
277 return x + x * y;
278 }
279 if (ix < 0x3fffa000) /* 1.25 */
280 { /* 0.84375 <= |x| < 1.25 */
281 s = fabsl (x) - one;
282 P = pa[0] + s * (pa[1] + s * (pa[2]
283 + s * (pa[3] + s * (pa[4] + s * (pa[5] + s * (pa[6] + s * pa[7]))))));
284 Q = qa[0] + s * (qa[1] + s * (qa[2]
285 + s * (qa[3] + s * (qa[4] + s * (qa[5] + s * (qa[6] + s))))));
286 if ((se & 0x8000) == 0)
287 return erx + P / Q;
288 else
289 return -erx - P / Q;
290 }
291 if (ix >= 0x4001d555) /* 6.6666259765625 */
292 { /* inf>|x|>=6.666 */
293 if ((se & 0x8000) == 0)
294 return one - tiny;
295 else
296 return tiny - one;
297 }
298 x = fabsl (x);
299 s = one / (x * x);
300 if (ix < 0x4000b6db) /* 2.85711669921875 */
301 {
302 R = ra[0] + s * (ra[1] + s * (ra[2] + s * (ra[3] + s * (ra[4] +
303 s * (ra[5] + s * (ra[6] + s * (ra[7] + s * ra[8])))))));
304 S = sa[0] + s * (sa[1] + s * (sa[2] + s * (sa[3] + s * (sa[4] +
305 s * (sa[5] + s * (sa[6] + s * (sa[7] + s * (sa[8] + s))))))));
306 }
307 else
308 { /* |x| >= 1/0.35 */
309 R = rb[0] + s * (rb[1] + s * (rb[2] + s * (rb[3] + s * (rb[4] +
310 s * (rb[5] + s * (rb[6] + s * rb[7]))))));
311 S = sb[0] + s * (sb[1] + s * (sb[2] + s * (sb[3] + s * (sb[4] +
312 s * (sb[5] + s * (sb[6] + s))))));
313 }
314 z = x;
315 GET_LDOUBLE_WORDS (i, i0, i1, z);
316 i1 = 0;
317 SET_LDOUBLE_WORDS (z, i, i0, i1);
318 r =
319 expl (-z * z - 0.5625L) * expl ((z - x) * (z + x) + R / S);
320 if ((se & 0x8000) == 0)
321 return one - r / x;
322 else
323 return r / x - one;
324 }
325
326 long double
erfcl(long double x)327 erfcl(long double x)
328 {
329 int32_t hx, ix;
330 long double R, S, P, Q, s, y, z, r;
331 u_int32_t se, i0, i1;
332
333 GET_LDOUBLE_WORDS (se, i0, i1, x);
334 ix = se & 0x7fff;
335 if (ix >= 0x7fff)
336 { /* erfc(nan)=nan */
337 /* erfc(+-inf)=0,2 */
338 return (long double) (((se & 0xffff) >> 15) << 1) + one / x;
339 }
340
341 ix = (ix << 16) | (i0 >> 16);
342 if (ix < 0x3ffed800) /* |x|<0.84375 */
343 {
344 if (ix < 0x3fbe0000) /* |x|<2**-65 */
345 return one - x;
346 z = x * x;
347 r = pp[0] + z * (pp[1]
348 + z * (pp[2] + z * (pp[3] + z * (pp[4] + z * pp[5]))));
349 s = qq[0] + z * (qq[1]
350 + z * (qq[2] + z * (qq[3] + z * (qq[4] + z * (qq[5] + z)))));
351 y = r / s;
352 if (ix < 0x3ffd8000) /* x<1/4 */
353 {
354 return one - (x + x * y);
355 }
356 else
357 {
358 r = x * y;
359 r += (x - half);
360 return half - r;
361 }
362 }
363 if (ix < 0x3fffa000) /* 1.25 */
364 { /* 0.84375 <= |x| < 1.25 */
365 s = fabsl (x) - one;
366 P = pa[0] + s * (pa[1] + s * (pa[2]
367 + s * (pa[3] + s * (pa[4] + s * (pa[5] + s * (pa[6] + s * pa[7]))))));
368 Q = qa[0] + s * (qa[1] + s * (qa[2]
369 + s * (qa[3] + s * (qa[4] + s * (qa[5] + s * (qa[6] + s))))));
370 if ((se & 0x8000) == 0)
371 {
372 z = one - erx;
373 return z - P / Q;
374 }
375 else
376 {
377 z = erx + P / Q;
378 return one + z;
379 }
380 }
381 if (ix < 0x4005d600) /* 107 */
382 { /* |x|<107 */
383 x = fabsl (x);
384 s = one / (x * x);
385 if (ix < 0x4000b6db) /* 2.85711669921875 */
386 { /* |x| < 1/.35 ~ 2.857143 */
387 R = ra[0] + s * (ra[1] + s * (ra[2] + s * (ra[3] + s * (ra[4] +
388 s * (ra[5] + s * (ra[6] + s * (ra[7] + s * ra[8])))))));
389 S = sa[0] + s * (sa[1] + s * (sa[2] + s * (sa[3] + s * (sa[4] +
390 s * (sa[5] + s * (sa[6] + s * (sa[7] + s * (sa[8] + s))))))));
391 }
392 else if (ix < 0x4001d555) /* 6.6666259765625 */
393 { /* 6.666 > |x| >= 1/.35 ~ 2.857143 */
394 R = rb[0] + s * (rb[1] + s * (rb[2] + s * (rb[3] + s * (rb[4] +
395 s * (rb[5] + s * (rb[6] + s * rb[7]))))));
396 S = sb[0] + s * (sb[1] + s * (sb[2] + s * (sb[3] + s * (sb[4] +
397 s * (sb[5] + s * (sb[6] + s))))));
398 }
399 else
400 { /* |x| >= 6.666 */
401 if (se & 0x8000)
402 return two - tiny; /* x < -6.666 */
403
404 R = rc[0] + s * (rc[1] + s * (rc[2] + s * (rc[3] +
405 s * (rc[4] + s * rc[5]))));
406 S = sc[0] + s * (sc[1] + s * (sc[2] + s * (sc[3] +
407 s * (sc[4] + s))));
408 }
409 z = x;
410 GET_LDOUBLE_WORDS (hx, i0, i1, z);
411 i1 = 0;
412 i0 &= 0xffffff00;
413 SET_LDOUBLE_WORDS (z, hx, i0, i1);
414 r = expl (-z * z - 0.5625L) *
415 expl ((z - x) * (z + x) + R / S);
416 if ((se & 0x8000) == 0)
417 return r / x;
418 else
419 return two - r / x;
420 }
421 else
422 {
423 if ((se & 0x8000) == 0)
424 return __math_uflowl(0);
425 else
426 return two - tiny;
427 }
428 }
429
