1/* @(#)s_erf.c 5.1 93/09/24 */
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/* Modified by Naohiko Shimizu/Tokai University, Japan 1997/08/25,
13 for performance improvement on pipelined processors.
14*/
15
16#if defined(LIBM_SCCS) && !defined(lint)
17static char rcsid[] = "$NetBSD: s_erf.c,v 1.8 1995/05/10 20:47:05 jtc Exp $";
18#endif
19
20/* double erf(double x)
21 * double erfc(double x)
22 * x
23 * 2 |\
24 * erf(x) = --------- | exp(-t*t)dt
25 * sqrt(pi) \|
26 * 0
27 *
28 * erfc(x) = 1-erf(x)
29 * Note that
30 * erf(-x) = -erf(x)
31 * erfc(-x) = 2 - erfc(x)
32 *
33 * Method:
34 * 1. For |x| in [0, 0.84375]
35 * erf(x) = x + x*R(x^2)
36 * erfc(x) = 1 - erf(x) if x in [-.84375,0.25]
37 * = 0.5 + ((0.5-x)-x*R) if x in [0.25,0.84375]
38 * where R = P/Q where P is an odd poly of degree 8 and
39 * Q is an odd poly of degree 10.
40 * -57.90
41 * | R - (erf(x)-x)/x | <= 2
42 *
43 *
44 * Remark. The formula is derived by noting
45 * erf(x) = (2/sqrt(pi))*(x - x^3/3 + x^5/10 - x^7/42 + ....)
46 * and that
47 * 2/sqrt(pi) = 1.128379167095512573896158903121545171688
48 * is close to one. The interval is chosen because the fix
49 * point of erf(x) is near 0.6174 (i.e., erf(x)=x when x is
50 * near 0.6174), and by some experiment, 0.84375 is chosen to
51 * guarantee the error is less than one ulp for erf.
52 *
53 * 2. For |x| in [0.84375,1.25], let s = |x| - 1, and
54 * c = 0.84506291151 rounded to single (24 bits)
55 * erf(x) = sign(x) * (c + P1(s)/Q1(s))
56 * erfc(x) = (1-c) - P1(s)/Q1(s) if x > 0
57 * 1+(c+P1(s)/Q1(s)) if x < 0
58 * |P1/Q1 - (erf(|x|)-c)| <= 2**-59.06
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 * That is, we use rational approximation to approximate
63 * erf(1+s) - (c = (single)0.84506291151)
64 * Note that |P1/Q1|< 0.078 for x in [0.84375,1.25]
65 * where
66 * P1(s) = degree 6 poly in s
67 * Q1(s) = degree 6 poly in s
68 *
69 * 3. For x in [1.25,1/0.35(~2.857143)],
70 * erfc(x) = (1/x)*exp(-x*x-0.5625+R1/S1)
71 * erf(x) = 1 - erfc(x)
72 * where
73 * R1(z) = degree 7 poly in z, (z=1/x^2)
74 * S1(z) = degree 8 poly in z
75 *
76 * 4. For x in [1/0.35,28]
77 * erfc(x) = (1/x)*exp(-x*x-0.5625+R2/S2) if x > 0
78 * = 2.0 - (1/x)*exp(-x*x-0.5625+R2/S2) if -6<x<0
79 * = 2.0 - tiny (if x <= -6)
80 * erf(x) = sign(x)*(1.0 - erfc(x)) if x < 6, else
81 * erf(x) = sign(x)*(1.0 - tiny)
82 * where
83 * R2(z) = degree 6 poly in z, (z=1/x^2)
84 * S2(z) = degree 7 poly in z
85 *
86 * Note1:
87 * To compute exp(-x*x-0.5625+R/S), let s be a single
88 * precision number and s := x; then
89 * -x*x = -s*s + (s-x)*(s+x)
90 * exp(-x*x-0.5626+R/S) =
91 * exp(-s*s-0.5625)*exp((s-x)*(s+x)+R/S);
92 * Note2:
93 * Here 4 and 5 make use of the asymptotic series
94 * exp(-x*x)
95 * erfc(x) ~ ---------- * ( 1 + Poly(1/x^2) )
96 * x*sqrt(pi)
97 * We use rational approximation to approximate
98 * g(s)=f(1/x^2) = log(erfc(x)*x) - x*x + 0.5625
99 * Here is the error bound for R1/S1 and R2/S2
100 * |R1/S1 - f(x)| < 2**(-62.57)
101 * |R2/S2 - f(x)| < 2**(-61.52)
102 *
103 * 5. For inf > x >= 28
104 * erf(x) = sign(x) *(1 - tiny) (raise inexact)
105 * erfc(x) = tiny*tiny (raise underflow) if x > 0
106 * = 2 - tiny if x<0
107 *
108 * 7. Special case:
109 * erf(0) = 0, erf(inf) = 1, erf(-inf) = -1,
110 * erfc(0) = 1, erfc(inf) = 0, erfc(-inf) = 2,
111 * erfc/erf(NaN) is NaN
112 */
113
114
115#include <errno.h>
116#include <float.h>
117#include <math.h>
118#include <math_private.h>
119#include <fix-int-fp-convert-zero.h>
120
121static const double
122 tiny = 1e-300,
123 half = 5.00000000000000000000e-01, /* 0x3FE00000, 0x00000000 */
124 one = 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
125 two = 2.00000000000000000000e+00, /* 0x40000000, 0x00000000 */
126/* c = (float)0.84506291151 */
127 erx = 8.45062911510467529297e-01, /* 0x3FEB0AC1, 0x60000000 */
128/*
129 * Coefficients for approximation to erf on [0,0.84375]
130 */
131 efx = 1.28379167095512586316e-01, /* 0x3FC06EBA, 0x8214DB69 */
132 pp[] = { 1.28379167095512558561e-01, /* 0x3FC06EBA, 0x8214DB68 */
133 -3.25042107247001499370e-01, /* 0xBFD4CD7D, 0x691CB913 */
134 -2.84817495755985104766e-02, /* 0xBF9D2A51, 0xDBD7194F */
135 -5.77027029648944159157e-03, /* 0xBF77A291, 0x236668E4 */
136 -2.37630166566501626084e-05 }, /* 0xBEF8EAD6, 0x120016AC */
137 qq[] = { 0.0, 3.97917223959155352819e-01, /* 0x3FD97779, 0xCDDADC09 */
138 6.50222499887672944485e-02, /* 0x3FB0A54C, 0x5536CEBA */
139 5.08130628187576562776e-03, /* 0x3F74D022, 0xC4D36B0F */
140 1.32494738004321644526e-04, /* 0x3F215DC9, 0x221C1A10 */
141 -3.96022827877536812320e-06 }, /* 0xBED09C43, 0x42A26120 */
142/*
143 * Coefficients for approximation to erf in [0.84375,1.25]
144 */
145 pa[] = { -2.36211856075265944077e-03, /* 0xBF6359B8, 0xBEF77538 */
146 4.14856118683748331666e-01, /* 0x3FDA8D00, 0xAD92B34D */
147 -3.72207876035701323847e-01, /* 0xBFD7D240, 0xFBB8C3F1 */
148 3.18346619901161753674e-01, /* 0x3FD45FCA, 0x805120E4 */
149 -1.10894694282396677476e-01, /* 0xBFBC6398, 0x3D3E28EC */
150 3.54783043256182359371e-02, /* 0x3FA22A36, 0x599795EB */
151 -2.16637559486879084300e-03 }, /* 0xBF61BF38, 0x0A96073F */
152 qa[] = { 0.0, 1.06420880400844228286e-01, /* 0x3FBB3E66, 0x18EEE323 */
153 5.40397917702171048937e-01, /* 0x3FE14AF0, 0x92EB6F33 */
154 7.18286544141962662868e-02, /* 0x3FB2635C, 0xD99FE9A7 */
155 1.26171219808761642112e-01, /* 0x3FC02660, 0xE763351F */
156 1.36370839120290507362e-02, /* 0x3F8BEDC2, 0x6B51DD1C */
157 1.19844998467991074170e-02 }, /* 0x3F888B54, 0x5735151D */
158/*
159 * Coefficients for approximation to erfc in [1.25,1/0.35]
160 */
161 ra[] = { -9.86494403484714822705e-03, /* 0xBF843412, 0x600D6435 */
162 -6.93858572707181764372e-01, /* 0xBFE63416, 0xE4BA7360 */
163 -1.05586262253232909814e+01, /* 0xC0251E04, 0x41B0E726 */
164 -6.23753324503260060396e+01, /* 0xC04F300A, 0xE4CBA38D */
165 -1.62396669462573470355e+02, /* 0xC0644CB1, 0x84282266 */
166 -1.84605092906711035994e+02, /* 0xC067135C, 0xEBCCABB2 */
167 -8.12874355063065934246e+01, /* 0xC0545265, 0x57E4D2F2 */
168 -9.81432934416914548592e+00 }, /* 0xC023A0EF, 0xC69AC25C */
169 sa[] = { 0.0, 1.96512716674392571292e+01, /* 0x4033A6B9, 0xBD707687 */
170 1.37657754143519042600e+02, /* 0x4061350C, 0x526AE721 */
171 4.34565877475229228821e+02, /* 0x407B290D, 0xD58A1A71 */
172 6.45387271733267880336e+02, /* 0x40842B19, 0x21EC2868 */
173 4.29008140027567833386e+02, /* 0x407AD021, 0x57700314 */
174 1.08635005541779435134e+02, /* 0x405B28A3, 0xEE48AE2C */
175 6.57024977031928170135e+00, /* 0x401A47EF, 0x8E484A93 */
176 -6.04244152148580987438e-02 }, /* 0xBFAEEFF2, 0xEE749A62 */
177/*
178 * Coefficients for approximation to erfc in [1/.35,28]
179 */
180 rb[] = { -9.86494292470009928597e-03, /* 0xBF843412, 0x39E86F4A */
181 -7.99283237680523006574e-01, /* 0xBFE993BA, 0x70C285DE */
182 -1.77579549177547519889e+01, /* 0xC031C209, 0x555F995A */
183 -1.60636384855821916062e+02, /* 0xC064145D, 0x43C5ED98 */
184 -6.37566443368389627722e+02, /* 0xC083EC88, 0x1375F228 */
185 -1.02509513161107724954e+03, /* 0xC0900461, 0x6A2E5992 */
186 -4.83519191608651397019e+02 }, /* 0xC07E384E, 0x9BDC383F */
187 sb[] = { 0.0, 3.03380607434824582924e+01, /* 0x403E568B, 0x261D5190 */
188 3.25792512996573918826e+02, /* 0x40745CAE, 0x221B9F0A */
189 1.53672958608443695994e+03, /* 0x409802EB, 0x189D5118 */
190 3.19985821950859553908e+03, /* 0x40A8FFB7, 0x688C246A */
191 2.55305040643316442583e+03, /* 0x40A3F219, 0xCEDF3BE6 */
192 4.74528541206955367215e+02, /* 0x407DA874, 0xE79FE763 */
193 -2.24409524465858183362e+01 }; /* 0xC03670E2, 0x42712D62 */
194
195double
196__erf (double x)
197{
198 int32_t hx, ix, i;
199 double R, S, P, Q, s, y, z, r;
200 GET_HIGH_WORD (hx, x);
201 ix = hx & 0x7fffffff;
202 if (ix >= 0x7ff00000) /* erf(nan)=nan */
203 {
204 i = ((u_int32_t) hx >> 31) << 1;
205 return (double) (1 - i) + one / x; /* erf(+-inf)=+-1 */
206 }
207
208 if (ix < 0x3feb0000) /* |x|<0.84375 */
209 {
210 double r1, r2, s1, s2, s3, z2, z4;
211 if (ix < 0x3e300000) /* |x|<2**-28 */
212 {
213 if (ix < 0x00800000)
214 {
215 /* Avoid spurious underflow. */
216 double ret = 0.0625 * (16.0 * x + (16.0 * efx) * x);
217 math_check_force_underflow (ret);
218 return ret;
219 }
220 return x + efx * x;
221 }
222 z = x * x;
223 r1 = pp[0] + z * pp[1]; z2 = z * z;
224 r2 = pp[2] + z * pp[3]; z4 = z2 * z2;
225 s1 = one + z * qq[1];
226 s2 = qq[2] + z * qq[3];
227 s3 = qq[4] + z * qq[5];
228 r = r1 + z2 * r2 + z4 * pp[4];
229 s = s1 + z2 * s2 + z4 * s3;
230 y = r / s;
231 return x + x * y;
232 }
233 if (ix < 0x3ff40000) /* 0.84375 <= |x| < 1.25 */
234 {
235 double s2, s4, s6, P1, P2, P3, P4, Q1, Q2, Q3, Q4;
236 s = fabs (x) - one;
237 P1 = pa[0] + s * pa[1]; s2 = s * s;
238 Q1 = one + s * qa[1]; s4 = s2 * s2;
239 P2 = pa[2] + s * pa[3]; s6 = s4 * s2;
240 Q2 = qa[2] + s * qa[3];
241 P3 = pa[4] + s * pa[5];
242 Q3 = qa[4] + s * qa[5];
243 P4 = pa[6];
244 Q4 = qa[6];
245 P = P1 + s2 * P2 + s4 * P3 + s6 * P4;
246 Q = Q1 + s2 * Q2 + s4 * Q3 + s6 * Q4;
247 if (hx >= 0)
248 return erx + P / Q;
249 else
250 return -erx - P / Q;
251 }
252 if (ix >= 0x40180000) /* inf>|x|>=6 */
253 {
254 if (hx >= 0)
255 return one - tiny;
256 else
257 return tiny - one;
258 }
259 x = fabs (x);
260 s = one / (x * x);
261 if (ix < 0x4006DB6E) /* |x| < 1/0.35 */
262 {
263 double R1, R2, R3, R4, S1, S2, S3, S4, s2, s4, s6, s8;
264 R1 = ra[0] + s * ra[1]; s2 = s * s;
265 S1 = one + s * sa[1]; s4 = s2 * s2;
266 R2 = ra[2] + s * ra[3]; s6 = s4 * s2;
267 S2 = sa[2] + s * sa[3]; s8 = s4 * s4;
268 R3 = ra[4] + s * ra[5];
269 S3 = sa[4] + s * sa[5];
270 R4 = ra[6] + s * ra[7];
271 S4 = sa[6] + s * sa[7];
272 R = R1 + s2 * R2 + s4 * R3 + s6 * R4;
273 S = S1 + s2 * S2 + s4 * S3 + s6 * S4 + s8 * sa[8];
274 }
275 else /* |x| >= 1/0.35 */
276 {
277 double R1, R2, R3, S1, S2, S3, S4, s2, s4, s6;
278 R1 = rb[0] + s * rb[1]; s2 = s * s;
279 S1 = one + s * sb[1]; s4 = s2 * s2;
280 R2 = rb[2] + s * rb[3]; s6 = s4 * s2;
281 S2 = sb[2] + s * sb[3];
282 R3 = rb[4] + s * rb[5];
283 S3 = sb[4] + s * sb[5];
284 S4 = sb[6] + s * sb[7];
285 R = R1 + s2 * R2 + s4 * R3 + s6 * rb[6];
286 S = S1 + s2 * S2 + s4 * S3 + s6 * S4;
287 }
288 z = x;
289 SET_LOW_WORD (z, 0);
290 r = __ieee754_exp (-z * z - 0.5625) *
291 __ieee754_exp ((z - x) * (z + x) + R / S);
292 if (hx >= 0)
293 return one - r / x;
294 else
295 return r / x - one;
296}
297weak_alias (__erf, erf)
298#ifdef NO_LONG_DOUBLE
299strong_alias (__erf, __erfl)
300weak_alias (__erf, erfl)
301#endif
302
303double
304__erfc (double x)
305{
306 int32_t hx, ix;
307 double R, S, P, Q, s, y, z, r;
308 GET_HIGH_WORD (hx, x);
309 ix = hx & 0x7fffffff;
310 if (ix >= 0x7ff00000) /* erfc(nan)=nan */
311 { /* erfc(+-inf)=0,2 */
312 double ret = (double) (((u_int32_t) hx >> 31) << 1) + one / x;
313 if (FIX_INT_FP_CONVERT_ZERO && ret == 0.0)
314 return 0.0;
315 return ret;
316 }
317
318 if (ix < 0x3feb0000) /* |x|<0.84375 */
319 {
320 double r1, r2, s1, s2, s3, z2, z4;
321 if (ix < 0x3c700000) /* |x|<2**-56 */
322 return one - x;
323 z = x * x;
324 r1 = pp[0] + z * pp[1]; z2 = z * z;
325 r2 = pp[2] + z * pp[3]; z4 = z2 * z2;
326 s1 = one + z * qq[1];
327 s2 = qq[2] + z * qq[3];
328 s3 = qq[4] + z * qq[5];
329 r = r1 + z2 * r2 + z4 * pp[4];
330 s = s1 + z2 * s2 + z4 * s3;
331 y = r / s;
332 if (hx < 0x3fd00000) /* x<1/4 */
333 {
334 return one - (x + x * y);
335 }
336 else
337 {
338 r = x * y;
339 r += (x - half);
340 return half - r;
341 }
342 }
343 if (ix < 0x3ff40000) /* 0.84375 <= |x| < 1.25 */
344 {
345 double s2, s4, s6, P1, P2, P3, P4, Q1, Q2, Q3, Q4;
346 s = fabs (x) - one;
347 P1 = pa[0] + s * pa[1]; s2 = s * s;
348 Q1 = one + s * qa[1]; s4 = s2 * s2;
349 P2 = pa[2] + s * pa[3]; s6 = s4 * s2;
350 Q2 = qa[2] + s * qa[3];
351 P3 = pa[4] + s * pa[5];
352 Q3 = qa[4] + s * qa[5];
353 P4 = pa[6];
354 Q4 = qa[6];
355 P = P1 + s2 * P2 + s4 * P3 + s6 * P4;
356 Q = Q1 + s2 * Q2 + s4 * Q3 + s6 * Q4;
357 if (hx >= 0)
358 {
359 z = one - erx; return z - P / Q;
360 }
361 else
362 {
363 z = erx + P / Q; return one + z;
364 }
365 }
366 if (ix < 0x403c0000) /* |x|<28 */
367 {
368 x = fabs (x);
369 s = one / (x * x);
370 if (ix < 0x4006DB6D) /* |x| < 1/.35 ~ 2.857143*/
371 {
372 double R1, R2, R3, R4, S1, S2, S3, S4, s2, s4, s6, s8;
373 R1 = ra[0] + s * ra[1]; s2 = s * s;
374 S1 = one + s * sa[1]; s4 = s2 * s2;
375 R2 = ra[2] + s * ra[3]; s6 = s4 * s2;
376 S2 = sa[2] + s * sa[3]; s8 = s4 * s4;
377 R3 = ra[4] + s * ra[5];
378 S3 = sa[4] + s * sa[5];
379 R4 = ra[6] + s * ra[7];
380 S4 = sa[6] + s * sa[7];
381 R = R1 + s2 * R2 + s4 * R3 + s6 * R4;
382 S = S1 + s2 * S2 + s4 * S3 + s6 * S4 + s8 * sa[8];
383 }
384 else /* |x| >= 1/.35 ~ 2.857143 */
385 {
386 double R1, R2, R3, S1, S2, S3, S4, s2, s4, s6;
387 if (hx < 0 && ix >= 0x40180000)
388 return two - tiny; /* x < -6 */
389 R1 = rb[0] + s * rb[1]; s2 = s * s;
390 S1 = one + s * sb[1]; s4 = s2 * s2;
391 R2 = rb[2] + s * rb[3]; s6 = s4 * s2;
392 S2 = sb[2] + s * sb[3];
393 R3 = rb[4] + s * rb[5];
394 S3 = sb[4] + s * sb[5];
395 S4 = sb[6] + s * sb[7];
396 R = R1 + s2 * R2 + s4 * R3 + s6 * rb[6];
397 S = S1 + s2 * S2 + s4 * S3 + s6 * S4;
398 }
399 z = x;
400 SET_LOW_WORD (z, 0);
401 r = __ieee754_exp (-z * z - 0.5625) *
402 __ieee754_exp ((z - x) * (z + x) + R / S);
403 if (hx > 0)
404 {
405 double ret = math_narrow_eval (r / x);
406 if (ret == 0)
407 __set_errno (ERANGE);
408 return ret;
409 }
410 else
411 return two - r / x;
412 }
413 else
414 {
415 if (hx > 0)
416 {
417 __set_errno (ERANGE);
418 return tiny * tiny;
419 }
420 else
421 return two - tiny;
422 }
423}
424weak_alias (__erfc, erfc)
425#ifdef NO_LONG_DOUBLE
426strong_alias (__erfc, __erfcl)
427weak_alias (__erfc, erfcl)
428#endif
429