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)
17 static 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-narrow-eval.h>
119 #include <math_private.h>
120 #include <math-underflow.h>
121 #include <libm-alias-double.h>
122 #include <fix-int-fp-convert-zero.h>
123
124 static const double
125 tiny = 1e-300,
126 half = 5.00000000000000000000e-01, /* 0x3FE00000, 0x00000000 */
127 one = 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
128 two = 2.00000000000000000000e+00, /* 0x40000000, 0x00000000 */
129 /* c = (float)0.84506291151 */
130 erx = 8.45062911510467529297e-01, /* 0x3FEB0AC1, 0x60000000 */
131 /*
132 * Coefficients for approximation to erf on [0,0.84375]
133 */
134 efx = 1.28379167095512586316e-01, /* 0x3FC06EBA, 0x8214DB69 */
135 pp[] = { 1.28379167095512558561e-01, /* 0x3FC06EBA, 0x8214DB68 */
136 -3.25042107247001499370e-01, /* 0xBFD4CD7D, 0x691CB913 */
137 -2.84817495755985104766e-02, /* 0xBF9D2A51, 0xDBD7194F */
138 -5.77027029648944159157e-03, /* 0xBF77A291, 0x236668E4 */
139 -2.37630166566501626084e-05 }, /* 0xBEF8EAD6, 0x120016AC */
140 qq[] = { 0.0, 3.97917223959155352819e-01, /* 0x3FD97779, 0xCDDADC09 */
141 6.50222499887672944485e-02, /* 0x3FB0A54C, 0x5536CEBA */
142 5.08130628187576562776e-03, /* 0x3F74D022, 0xC4D36B0F */
143 1.32494738004321644526e-04, /* 0x3F215DC9, 0x221C1A10 */
144 -3.96022827877536812320e-06 }, /* 0xBED09C43, 0x42A26120 */
145 /*
146 * Coefficients for approximation to erf in [0.84375,1.25]
147 */
148 pa[] = { -2.36211856075265944077e-03, /* 0xBF6359B8, 0xBEF77538 */
149 4.14856118683748331666e-01, /* 0x3FDA8D00, 0xAD92B34D */
150 -3.72207876035701323847e-01, /* 0xBFD7D240, 0xFBB8C3F1 */
151 3.18346619901161753674e-01, /* 0x3FD45FCA, 0x805120E4 */
152 -1.10894694282396677476e-01, /* 0xBFBC6398, 0x3D3E28EC */
153 3.54783043256182359371e-02, /* 0x3FA22A36, 0x599795EB */
154 -2.16637559486879084300e-03 }, /* 0xBF61BF38, 0x0A96073F */
155 qa[] = { 0.0, 1.06420880400844228286e-01, /* 0x3FBB3E66, 0x18EEE323 */
156 5.40397917702171048937e-01, /* 0x3FE14AF0, 0x92EB6F33 */
157 7.18286544141962662868e-02, /* 0x3FB2635C, 0xD99FE9A7 */
158 1.26171219808761642112e-01, /* 0x3FC02660, 0xE763351F */
159 1.36370839120290507362e-02, /* 0x3F8BEDC2, 0x6B51DD1C */
160 1.19844998467991074170e-02 }, /* 0x3F888B54, 0x5735151D */
161 /*
162 * Coefficients for approximation to erfc in [1.25,1/0.35]
163 */
164 ra[] = { -9.86494403484714822705e-03, /* 0xBF843412, 0x600D6435 */
165 -6.93858572707181764372e-01, /* 0xBFE63416, 0xE4BA7360 */
166 -1.05586262253232909814e+01, /* 0xC0251E04, 0x41B0E726 */
167 -6.23753324503260060396e+01, /* 0xC04F300A, 0xE4CBA38D */
168 -1.62396669462573470355e+02, /* 0xC0644CB1, 0x84282266 */
169 -1.84605092906711035994e+02, /* 0xC067135C, 0xEBCCABB2 */
170 -8.12874355063065934246e+01, /* 0xC0545265, 0x57E4D2F2 */
171 -9.81432934416914548592e+00 }, /* 0xC023A0EF, 0xC69AC25C */
172 sa[] = { 0.0, 1.96512716674392571292e+01, /* 0x4033A6B9, 0xBD707687 */
173 1.37657754143519042600e+02, /* 0x4061350C, 0x526AE721 */
174 4.34565877475229228821e+02, /* 0x407B290D, 0xD58A1A71 */
175 6.45387271733267880336e+02, /* 0x40842B19, 0x21EC2868 */
176 4.29008140027567833386e+02, /* 0x407AD021, 0x57700314 */
177 1.08635005541779435134e+02, /* 0x405B28A3, 0xEE48AE2C */
178 6.57024977031928170135e+00, /* 0x401A47EF, 0x8E484A93 */
179 -6.04244152148580987438e-02 }, /* 0xBFAEEFF2, 0xEE749A62 */
180 /*
181 * Coefficients for approximation to erfc in [1/.35,28]
182 */
183 rb[] = { -9.86494292470009928597e-03, /* 0xBF843412, 0x39E86F4A */
184 -7.99283237680523006574e-01, /* 0xBFE993BA, 0x70C285DE */
185 -1.77579549177547519889e+01, /* 0xC031C209, 0x555F995A */
186 -1.60636384855821916062e+02, /* 0xC064145D, 0x43C5ED98 */
187 -6.37566443368389627722e+02, /* 0xC083EC88, 0x1375F228 */
188 -1.02509513161107724954e+03, /* 0xC0900461, 0x6A2E5992 */
189 -4.83519191608651397019e+02 }, /* 0xC07E384E, 0x9BDC383F */
190 sb[] = { 0.0, 3.03380607434824582924e+01, /* 0x403E568B, 0x261D5190 */
191 3.25792512996573918826e+02, /* 0x40745CAE, 0x221B9F0A */
192 1.53672958608443695994e+03, /* 0x409802EB, 0x189D5118 */
193 3.19985821950859553908e+03, /* 0x40A8FFB7, 0x688C246A */
194 2.55305040643316442583e+03, /* 0x40A3F219, 0xCEDF3BE6 */
195 4.74528541206955367215e+02, /* 0x407DA874, 0xE79FE763 */
196 -2.24409524465858183362e+01 }; /* 0xC03670E2, 0x42712D62 */
197
198 double
199 __erf (double x)
200 {
201 int32_t hx, ix, i;
202 double R, S, P, Q, s, y, z, r;
203 GET_HIGH_WORD (hx, x);
204 ix = hx & 0x7fffffff;
205 if (ix >= 0x7ff00000) /* erf(nan)=nan */
206 {
207 i = ((uint32_t) hx >> 31) << 1;
208 return (double) (1 - i) + one / x; /* erf(+-inf)=+-1 */
209 }
210
211 if (ix < 0x3feb0000) /* |x|<0.84375 */
212 {
213 double r1, r2, s1, s2, s3, z2, z4;
214 if (ix < 0x3e300000) /* |x|<2**-28 */
215 {
216 if (ix < 0x00800000)
217 {
218 /* Avoid spurious underflow. */
219 double ret = 0.0625 * (16.0 * x + (16.0 * efx) * x);
220 math_check_force_underflow (ret);
221 return ret;
222 }
223 return x + efx * x;
224 }
225 z = x * x;
226 r1 = pp[0] + z * pp[1]; z2 = z * z;
227 r2 = pp[2] + z * pp[3]; z4 = z2 * z2;
228 s1 = one + z * qq[1];
229 s2 = qq[2] + z * qq[3];
230 s3 = qq[4] + z * qq[5];
231 r = r1 + z2 * r2 + z4 * pp[4];
232 s = s1 + z2 * s2 + z4 * s3;
233 y = r / s;
234 return x + x * y;
235 }
236 if (ix < 0x3ff40000) /* 0.84375 <= |x| < 1.25 */
237 {
238 double s2, s4, s6, P1, P2, P3, P4, Q1, Q2, Q3, Q4;
239 s = fabs (x) - one;
240 P1 = pa[0] + s * pa[1]; s2 = s * s;
241 Q1 = one + s * qa[1]; s4 = s2 * s2;
242 P2 = pa[2] + s * pa[3]; s6 = s4 * s2;
243 Q2 = qa[2] + s * qa[3];
244 P3 = pa[4] + s * pa[5];
245 Q3 = qa[4] + s * qa[5];
246 P4 = pa[6];
247 Q4 = qa[6];
248 P = P1 + s2 * P2 + s4 * P3 + s6 * P4;
249 Q = Q1 + s2 * Q2 + s4 * Q3 + s6 * Q4;
250 if (hx >= 0)
251 return erx + P / Q;
252 else
253 return -erx - P / Q;
254 }
255 if (ix >= 0x40180000) /* inf>|x|>=6 */
256 {
257 if (hx >= 0)
258 return one - tiny;
259 else
260 return tiny - one;
261 }
262 x = fabs (x);
263 s = one / (x * x);
264 if (ix < 0x4006DB6E) /* |x| < 1/0.35 */
265 {
266 double R1, R2, R3, R4, S1, S2, S3, S4, s2, s4, s6, s8;
267 R1 = ra[0] + s * ra[1]; s2 = s * s;
268 S1 = one + s * sa[1]; s4 = s2 * s2;
269 R2 = ra[2] + s * ra[3]; s6 = s4 * s2;
270 S2 = sa[2] + s * sa[3]; s8 = s4 * s4;
271 R3 = ra[4] + s * ra[5];
272 S3 = sa[4] + s * sa[5];
273 R4 = ra[6] + s * ra[7];
274 S4 = sa[6] + s * sa[7];
275 R = R1 + s2 * R2 + s4 * R3 + s6 * R4;
276 S = S1 + s2 * S2 + s4 * S3 + s6 * S4 + s8 * sa[8];
277 }
278 else /* |x| >= 1/0.35 */
279 {
280 double R1, R2, R3, S1, S2, S3, S4, s2, s4, s6;
281 R1 = rb[0] + s * rb[1]; s2 = s * s;
282 S1 = one + s * sb[1]; s4 = s2 * s2;
283 R2 = rb[2] + s * rb[3]; s6 = s4 * s2;
284 S2 = sb[2] + s * sb[3];
285 R3 = rb[4] + s * rb[5];
286 S3 = sb[4] + s * sb[5];
287 S4 = sb[6] + s * sb[7];
288 R = R1 + s2 * R2 + s4 * R3 + s6 * rb[6];
289 S = S1 + s2 * S2 + s4 * S3 + s6 * S4;
290 }
291 z = x;
292 SET_LOW_WORD (z, 0);
293 r = __ieee754_exp (-z * z - 0.5625) *
294 __ieee754_exp ((z - x) * (z + x) + R / S);
295 if (hx >= 0)
296 return one - r / x;
297 else
298 return r / x - one;
299 }
300 libm_alias_double (__erf, erf)
301
302 double
303 __erfc (double x)
304 {
305 int32_t hx, ix;
306 double R, S, P, Q, s, y, z, r;
307 GET_HIGH_WORD (hx, x);
308 ix = hx & 0x7fffffff;
309 if (ix >= 0x7ff00000) /* erfc(nan)=nan */
310 { /* erfc(+-inf)=0,2 */
311 double ret = (double) (((uint32_t) hx >> 31) << 1) + one / x;
312 if (FIX_INT_FP_CONVERT_ZERO && ret == 0.0)
313 return 0.0;
314 return ret;
315 }
316
317 if (ix < 0x3feb0000) /* |x|<0.84375 */
318 {
319 double r1, r2, s1, s2, s3, z2, z4;
320 if (ix < 0x3c700000) /* |x|<2**-56 */
321 return one - x;
322 z = x * x;
323 r1 = pp[0] + z * pp[1]; z2 = z * z;
324 r2 = pp[2] + z * pp[3]; z4 = z2 * z2;
325 s1 = one + z * qq[1];
326 s2 = qq[2] + z * qq[3];
327 s3 = qq[4] + z * qq[5];
328 r = r1 + z2 * r2 + z4 * pp[4];
329 s = s1 + z2 * s2 + z4 * s3;
330 y = r / s;
331 if (hx < 0x3fd00000) /* x<1/4 */
332 {
333 return one - (x + x * y);
334 }
335 else
336 {
337 r = x * y;
338 r += (x - half);
339 return half - r;
340 }
341 }
342 if (ix < 0x3ff40000) /* 0.84375 <= |x| < 1.25 */
343 {
344 double s2, s4, s6, P1, P2, P3, P4, Q1, Q2, Q3, Q4;
345 s = fabs (x) - one;
346 P1 = pa[0] + s * pa[1]; s2 = s * s;
347 Q1 = one + s * qa[1]; s4 = s2 * s2;
348 P2 = pa[2] + s * pa[3]; s6 = s4 * s2;
349 Q2 = qa[2] + s * qa[3];
350 P3 = pa[4] + s * pa[5];
351 Q3 = qa[4] + s * qa[5];
352 P4 = pa[6];
353 Q4 = qa[6];
354 P = P1 + s2 * P2 + s4 * P3 + s6 * P4;
355 Q = Q1 + s2 * Q2 + s4 * Q3 + s6 * Q4;
356 if (hx >= 0)
357 {
358 z = one - erx; return z - P / Q;
359 }
360 else
361 {
362 z = erx + P / Q; return one + z;
363 }
364 }
365 if (ix < 0x403c0000) /* |x|<28 */
366 {
367 x = fabs (x);
368 s = one / (x * x);
369 if (ix < 0x4006DB6D) /* |x| < 1/.35 ~ 2.857143*/
370 {
371 double R1, R2, R3, R4, S1, S2, S3, S4, s2, s4, s6, s8;
372 R1 = ra[0] + s * ra[1]; s2 = s * s;
373 S1 = one + s * sa[1]; s4 = s2 * s2;
374 R2 = ra[2] + s * ra[3]; s6 = s4 * s2;
375 S2 = sa[2] + s * sa[3]; s8 = s4 * s4;
376 R3 = ra[4] + s * ra[5];
377 S3 = sa[4] + s * sa[5];
378 R4 = ra[6] + s * ra[7];
379 S4 = sa[6] + s * sa[7];
380 R = R1 + s2 * R2 + s4 * R3 + s6 * R4;
381 S = S1 + s2 * S2 + s4 * S3 + s6 * S4 + s8 * sa[8];
382 }
383 else /* |x| >= 1/.35 ~ 2.857143 */
384 {
385 double R1, R2, R3, S1, S2, S3, S4, s2, s4, s6;
386 if (hx < 0 && ix >= 0x40180000)
387 return two - tiny; /* x < -6 */
388 R1 = rb[0] + s * rb[1]; s2 = s * s;
389 S1 = one + s * sb[1]; s4 = s2 * s2;
390 R2 = rb[2] + s * rb[3]; s6 = s4 * s2;
391 S2 = sb[2] + s * sb[3];
392 R3 = rb[4] + s * rb[5];
393 S3 = sb[4] + s * sb[5];
394 S4 = sb[6] + s * sb[7];
395 R = R1 + s2 * R2 + s4 * R3 + s6 * rb[6];
396 S = S1 + s2 * S2 + s4 * S3 + s6 * S4;
397 }
398 z = x;
399 SET_LOW_WORD (z, 0);
400 r = __ieee754_exp (-z * z - 0.5625) *
401 __ieee754_exp ((z - x) * (z + x) + R / S);
402 if (hx > 0)
403 {
404 double ret = math_narrow_eval (r / x);
405 if (ret == 0)
406 __set_errno (ERANGE);
407 return ret;
408 }
409 else
410 return two - r / x;
411 }
412 else
413 {
414 if (hx > 0)
415 {
416 __set_errno (ERANGE);
417 return tiny * tiny;
418 }
419 else
420 return two - tiny;
421 }
422 }
423 libm_alias_double (__erfc, erfc)