1 /*-
2 * Copyright (c) 1992, 1993
3 * The Regents of the University of California. All rights reserved.
4 *
5 * %sccs.include.redist.c%
6 */
7
8 #ifndef lint
9 static char sccsid[] = "@(#)erf.c 8.1 (Berkeley) 06/04/93";
10 #endif /* not lint */
11
12 /* Modified Nov 30, 1992 P. McILROY:
13 * Replaced expansions for x >= 1.25 (error 1.7ulp vs ~6ulp)
14 * Replaced even+odd with direct calculation for x < .84375,
15 * to avoid destructive cancellation.
16 *
17 * Performance of erfc(x):
18 * In 300000 trials in the range [.83, .84375] the
19 * maximum observed error was 3.6ulp.
20 *
21 * In [.84735,1.25] the maximum observed error was <2.5ulp in
22 * 100000 runs in the range [1.2, 1.25].
23 *
24 * In [1.25,26] (Not including subnormal results)
25 * the error is < 1.7ulp.
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 *
38 * Method:
39 * 1. Reduce x to |x| by erf(-x) = -erf(x)
40 * 2. For x in [0, 0.84375]
41 * erf(x) = x + x*P(x^2)
42 * erfc(x) = 1 - erf(x) if x<=0.25
43 * = 0.5 + ((0.5-x)-x*P) if x in [0.25,0.84375]
44 * where
45 * 2 2 4 20
46 * P = P(x ) = (p0 + p1 * x + p2 * x + ... + p10 * x )
47 * is an approximation to (erf(x)-x)/x with precision
48 *
49 * -56.45
50 * | P - (erf(x)-x)/x | <= 2
51 *
52 *
53 * Remark. The formula is derived by noting
54 * erf(x) = (2/sqrt(pi))*(x - x^3/3 + x^5/10 - x^7/42 + ....)
55 * and that
56 * 2/sqrt(pi) = 1.128379167095512573896158903121545171688
57 * is close to one. The interval is chosen because the fixed
58 * point of erf(x) is near 0.6174 (i.e., erf(x)=x when x is
59 * near 0.6174), and by some experiment, 0.84375 is chosen to
60 * guarantee the error is less than one ulp for erf.
61 *
62 * 3. For x in [0.84375,1.25], let s = x - 1, and
63 * c = 0.84506291151 rounded to single (24 bits)
64 * erf(x) = c + P1(s)/Q1(s)
65 * erfc(x) = (1-c) - P1(s)/Q1(s)
66 * |P1/Q1 - (erf(x)-c)| <= 2**-59.06
67 * Remark: here we use the taylor series expansion at x=1.
68 * erf(1+s) = erf(1) + s*Poly(s)
69 * = 0.845.. + P1(s)/Q1(s)
70 * That is, we use rational approximation to approximate
71 * erf(1+s) - (c = (single)0.84506291151)
72 * Note that |P1/Q1|< 0.078 for x in [0.84375,1.25]
73 * where
74 * P1(s) = degree 6 poly in s
75 * Q1(s) = degree 6 poly in s
76 *
77 * 4. For x in [1.25, 2]; [2, 4]
78 * erf(x) = 1.0 - tiny
79 * erfc(x) = (1/x)exp(-x*x-(.5*log(pi) -.5z + R(z)/S(z))
80 *
81 * Where z = 1/(x*x), R is degree 9, and S is degree 3;
82 *
83 * 5. For x in [4,28]
84 * erf(x) = 1.0 - tiny
85 * erfc(x) = (1/x)exp(-x*x-(.5*log(pi)+eps + zP(z))
86 *
87 * Where P is degree 14 polynomial in 1/(x*x).
88 *
89 * Notes:
90 * Here 4 and 5 make use of the asymptotic series
91 * exp(-x*x)
92 * erfc(x) ~ ---------- * ( 1 + Poly(1/x^2) );
93 * x*sqrt(pi)
94 *
95 * where for z = 1/(x*x)
96 * P(z) ~ z/2*(-1 + z*3/2*(1 + z*5/2*(-1 + z*7/2*(1 +...))))
97 *
98 * Thus we use rational approximation to approximate
99 * erfc*x*exp(x*x) ~ 1/sqrt(pi);
100 *
101 * The error bound for the target function, G(z) for
102 * the interval
103 * [4, 28]:
104 * |eps + 1/(z)P(z) - G(z)| < 2**(-56.61)
105 * for [2, 4]:
106 * |R(z)/S(z) - G(z)| < 2**(-58.24)
107 * for [1.25, 2]:
108 * |R(z)/S(z) - G(z)| < 2**(-58.12)
109 *
110 * 6. For inf > x >= 28
111 * erf(x) = 1 - tiny (raise inexact)
112 * erfc(x) = tiny*tiny (raise underflow)
113 *
114 * 7. Special cases:
115 * erf(0) = 0, erf(inf) = 1, erf(-inf) = -1,
116 * erfc(0) = 1, erfc(inf) = 0, erfc(-inf) = 2,
117 * erfc/erf(NaN) is NaN
118 */
119
120 #if defined(vax) || defined(tahoe)
121 #define _IEEE 0
122 #define TRUNC(x) (double) (float) (x)
123 #else
124 #define _IEEE 1
125 #define TRUNC(x) *(((int *) &x) + 1) &= 0xf8000000
126 #define infnan(x) 0.0
127 #endif
128
129 #ifdef _IEEE_LIBM
130 /*
131 * redefining "___function" to "function" in _IEEE_LIBM mode
132 */
133 #include "ieee_libm.h"
134 #endif
135
136 static double
137 tiny = 1e-300,
138 half = 0.5,
139 one = 1.0,
140 two = 2.0,
141 c = 8.45062911510467529297e-01, /* (float)0.84506291151 */
142 /*
143 * Coefficients for approximation to erf in [0,0.84375]
144 */
145 p0t8 = 1.02703333676410051049867154944018394163280,
146 p0 = 1.283791670955125638123339436800229927041e-0001,
147 p1 = -3.761263890318340796574473028946097022260e-0001,
148 p2 = 1.128379167093567004871858633779992337238e-0001,
149 p3 = -2.686617064084433642889526516177508374437e-0002,
150 p4 = 5.223977576966219409445780927846432273191e-0003,
151 p5 = -8.548323822001639515038738961618255438422e-0004,
152 p6 = 1.205520092530505090384383082516403772317e-0004,
153 p7 = -1.492214100762529635365672665955239554276e-0005,
154 p8 = 1.640186161764254363152286358441771740838e-0006,
155 p9 = -1.571599331700515057841960987689515895479e-0007,
156 p10= 1.073087585213621540635426191486561494058e-0008;
157 /*
158 * Coefficients for approximation to erf in [0.84375,1.25]
159 */
160 static double
161 pa0 = -2.362118560752659485957248365514511540287e-0003,
162 pa1 = 4.148561186837483359654781492060070469522e-0001,
163 pa2 = -3.722078760357013107593507594535478633044e-0001,
164 pa3 = 3.183466199011617316853636418691420262160e-0001,
165 pa4 = -1.108946942823966771253985510891237782544e-0001,
166 pa5 = 3.547830432561823343969797140537411825179e-0002,
167 pa6 = -2.166375594868790886906539848893221184820e-0003,
168 qa1 = 1.064208804008442270765369280952419863524e-0001,
169 qa2 = 5.403979177021710663441167681878575087235e-0001,
170 qa3 = 7.182865441419627066207655332170665812023e-0002,
171 qa4 = 1.261712198087616469108438860983447773726e-0001,
172 qa5 = 1.363708391202905087876983523620537833157e-0002,
173 qa6 = 1.198449984679910764099772682882189711364e-0002;
174 /*
175 * log(sqrt(pi)) for large x expansions.
176 * The tail (lsqrtPI_lo) is included in the rational
177 * approximations.
178 */
179 static double
180 lsqrtPI_hi = .5723649429247000819387380943226;
181 /*
182 * lsqrtPI_lo = .000000000000000005132975581353913;
183 *
184 * Coefficients for approximation to erfc in [2, 4]
185 */
186 static double
187 rb0 = -1.5306508387410807582e-010, /* includes lsqrtPI_lo */
188 rb1 = 2.15592846101742183841910806188e-008,
189 rb2 = 6.24998557732436510470108714799e-001,
190 rb3 = 8.24849222231141787631258921465e+000,
191 rb4 = 2.63974967372233173534823436057e+001,
192 rb5 = 9.86383092541570505318304640241e+000,
193 rb6 = -7.28024154841991322228977878694e+000,
194 rb7 = 5.96303287280680116566600190708e+000,
195 rb8 = -4.40070358507372993983608466806e+000,
196 rb9 = 2.39923700182518073731330332521e+000,
197 rb10 = -6.89257464785841156285073338950e-001,
198 sb1 = 1.56641558965626774835300238919e+001,
199 sb2 = 7.20522741000949622502957936376e+001,
200 sb3 = 9.60121069770492994166488642804e+001;
201 /*
202 * Coefficients for approximation to erfc in [1.25, 2]
203 */
204 static double
205 rc0 = -2.47925334685189288817e-007, /* includes lsqrtPI_lo */
206 rc1 = 1.28735722546372485255126993930e-005,
207 rc2 = 6.24664954087883916855616917019e-001,
208 rc3 = 4.69798884785807402408863708843e+000,
209 rc4 = 7.61618295853929705430118701770e+000,
210 rc5 = 9.15640208659364240872946538730e-001,
211 rc6 = -3.59753040425048631334448145935e-001,
212 rc7 = 1.42862267989304403403849619281e-001,
213 rc8 = -4.74392758811439801958087514322e-002,
214 rc9 = 1.09964787987580810135757047874e-002,
215 rc10 = -1.28856240494889325194638463046e-003,
216 sc1 = 9.97395106984001955652274773456e+000,
217 sc2 = 2.80952153365721279953959310660e+001,
218 sc3 = 2.19826478142545234106819407316e+001;
219 /*
220 * Coefficients for approximation to erfc in [4,28]
221 */
222 static double
223 rd0 = -2.1491361969012978677e-016, /* includes lsqrtPI_lo */
224 rd1 = -4.99999999999640086151350330820e-001,
225 rd2 = 6.24999999772906433825880867516e-001,
226 rd3 = -1.54166659428052432723177389562e+000,
227 rd4 = 5.51561147405411844601985649206e+000,
228 rd5 = -2.55046307982949826964613748714e+001,
229 rd6 = 1.43631424382843846387913799845e+002,
230 rd7 = -9.45789244999420134263345971704e+002,
231 rd8 = 6.94834146607051206956384703517e+003,
232 rd9 = -5.27176414235983393155038356781e+004,
233 rd10 = 3.68530281128672766499221324921e+005,
234 rd11 = -2.06466642800404317677021026611e+006,
235 rd12 = 7.78293889471135381609201431274e+006,
236 rd13 = -1.42821001129434127360582351685e+007;
237
erf(x)238 double erf(x)
239 double x;
240 {
241 double R,S,P,Q,ax,s,y,z,r,fabs(),exp();
242 if(!finite(x)) { /* erf(nan)=nan */
243 if (isnan(x))
244 return(x);
245 return (x > 0 ? one : -one); /* erf(+/-inf)= +/-1 */
246 }
247 if ((ax = x) < 0)
248 ax = - ax;
249 if (ax < .84375) {
250 if (ax < 3.7e-09) {
251 if (ax < 1.0e-308)
252 return 0.125*(8.0*x+p0t8*x); /*avoid underflow */
253 return x + p0*x;
254 }
255 y = x*x;
256 r = y*(p1+y*(p2+y*(p3+y*(p4+y*(p5+
257 y*(p6+y*(p7+y*(p8+y*(p9+y*p10)))))))));
258 return x + x*(p0+r);
259 }
260 if (ax < 1.25) { /* 0.84375 <= |x| < 1.25 */
261 s = fabs(x)-one;
262 P = pa0+s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6)))));
263 Q = one+s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6)))));
264 if (x>=0)
265 return (c + P/Q);
266 else
267 return (-c - P/Q);
268 }
269 if (ax >= 6.0) { /* inf>|x|>=6 */
270 if (x >= 0.0)
271 return (one-tiny);
272 else
273 return (tiny-one);
274 }
275 /* 1.25 <= |x| < 6 */
276 z = -ax*ax;
277 s = -one/z;
278 if (ax < 2.0) {
279 R = rc0+s*(rc1+s*(rc2+s*(rc3+s*(rc4+s*(rc5+
280 s*(rc6+s*(rc7+s*(rc8+s*(rc9+s*rc10)))))))));
281 S = one+s*(sc1+s*(sc2+s*sc3));
282 } else {
283 R = rb0+s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*(rb5+
284 s*(rb6+s*(rb7+s*(rb8+s*(rb9+s*rb10)))))))));
285 S = one+s*(sb1+s*(sb2+s*sb3));
286 }
287 y = (R/S -.5*s) - lsqrtPI_hi;
288 z += y;
289 z = exp(z)/ax;
290 if (x >= 0)
291 return (one-z);
292 else
293 return (z-one);
294 }
295
erfc(x)296 double erfc(x)
297 double x;
298 {
299 double R,S,P,Q,s,ax,y,z,r,fabs(),__exp__D();
300 if (!finite(x)) {
301 if (isnan(x)) /* erfc(NaN) = NaN */
302 return(x);
303 else if (x > 0) /* erfc(+-inf)=0,2 */
304 return 0.0;
305 else
306 return 2.0;
307 }
308 if ((ax = x) < 0)
309 ax = -ax;
310 if (ax < .84375) { /* |x|<0.84375 */
311 if (ax < 1.38777878078144568e-17) /* |x|<2**-56 */
312 return one-x;
313 y = x*x;
314 r = y*(p1+y*(p2+y*(p3+y*(p4+y*(p5+
315 y*(p6+y*(p7+y*(p8+y*(p9+y*p10)))))))));
316 if (ax < .0625) { /* |x|<2**-4 */
317 return (one-(x+x*(p0+r)));
318 } else {
319 r = x*(p0+r);
320 r += (x-half);
321 return (half - r);
322 }
323 }
324 if (ax < 1.25) { /* 0.84375 <= |x| < 1.25 */
325 s = ax-one;
326 P = pa0+s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6)))));
327 Q = one+s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6)))));
328 if (x>=0) {
329 z = one-c; return z - P/Q;
330 } else {
331 z = c+P/Q; return one+z;
332 }
333 }
334 if (ax >= 28) /* Out of range */
335 if (x>0)
336 return (tiny*tiny);
337 else
338 return (two-tiny);
339 z = ax;
340 TRUNC(z);
341 y = z - ax; y *= (ax+z);
342 z *= -z; /* Here z + y = -x^2 */
343 s = one/(-z-y); /* 1/(x*x) */
344 if (ax >= 4) { /* 6 <= ax */
345 R = s*(rd1+s*(rd2+s*(rd3+s*(rd4+s*(rd5+
346 s*(rd6+s*(rd7+s*(rd8+s*(rd9+s*(rd10
347 +s*(rd11+s*(rd12+s*rd13))))))))))));
348 y += rd0;
349 } else if (ax >= 2) {
350 R = rb0+s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*(rb5+
351 s*(rb6+s*(rb7+s*(rb8+s*(rb9+s*rb10)))))))));
352 S = one+s*(sb1+s*(sb2+s*sb3));
353 y += R/S;
354 R = -.5*s;
355 } else {
356 R = rc0+s*(rc1+s*(rc2+s*(rc3+s*(rc4+s*(rc5+
357 s*(rc6+s*(rc7+s*(rc8+s*(rc9+s*rc10)))))))));
358 S = one+s*(sc1+s*(sc2+s*sc3));
359 y += R/S;
360 R = -.5*s;
361 }
362 /* return exp(-x^2 - lsqrtPI_hi + R + y)/x; */
363 s = ((R + y) - lsqrtPI_hi) + z;
364 y = (((z-s) - lsqrtPI_hi) + R) + y;
365 r = __exp__D(s, y)/x;
366 if (x>0)
367 return r;
368 else
369 return two-r;
370 }
371