```   1 /*
3  *
4  * The contents of this file are subject to the terms of the
6  * You may not use this file except in compliance with the License.
7  *
8  * You can obtain a copy of the license at usr/src/OPENSOLARIS.LICENSE
9  * or http://www.opensolaris.org/os/licensing.
10  * See the License for the specific language governing permissions
11  * and limitations under the License.
12  *
13  * When distributing Covered Code, include this CDDL HEADER in each
15  * If applicable, add the following below this CDDL HEADER, with the
16  * fields enclosed by brackets "[]" replaced with your own identifying
18  *
20  */
21
22 /*
24  */
25
26 /*
28  * Use is subject to license terms.
29  */
30
31 #pragma weak __erf = erf
32 #pragma weak __erfc = erfc
33
34
35 /*
36  * double erf(double x)
37  * double erfc(double x)
38  *                           x
39  *                    2      |\
40  *     erf(x)  =  ---------  | exp(-t*t)dt
41  *                 sqrt(pi) \|
42  *                           0
43  *
44  *     erfc(x) =  1-erf(x)
45  *  Note that
46  *              erf(-x) = -erf(x)
47  *              erfc(-x) = 2 - erfc(x)
48  *
49  * Method:
50  *      1. For |x| in [0, 0.84375]
51  *          erf(x)  = x + x*R(x^2)
52  *          erfc(x) = 1 - erf(x)           if x in [-.84375,0.25]
53  *                  = 0.5 + ((0.5-x)-x*R)  if x in [0.25,0.84375]
54  *         where R = P/Q where P is an odd poly of degree 8 and
55  *         Q is an odd poly of degree 10.
56  *                                               -57.90
57  *                      | R - (erf(x)-x)/x | <= 2
58  *
59  *
60  *         Remark. The formula is derived by noting
61  *          erf(x) = (2/sqrt(pi))*(x - x^3/3 + x^5/10 - x^7/42 + ....)
62  *         and that
63  *          2/sqrt(pi) = 1.128379167095512573896158903121545171688
64  *         is close to one. The interval is chosen because the fix
65  *         point of erf(x) is near 0.6174 (i.e., erf(x)=x when x is
66  *         near 0.6174), and by some experiment, 0.84375 is chosen to
67  *         guarantee the error is less than one ulp for erf.
68  *
69  *      2. For |x| in [0.84375,1.25], let s = |x| - 1, and
70  *         c = 0.84506291151 rounded to single (24 bits)
71  *              erf(x)  = sign(x) * (c  + P1(s)/Q1(s))
72  *              erfc(x) = (1-c)  - P1(s)/Q1(s) if x > 0
73  *                        1+(c+P1(s)/Q1(s))    if x < 0
74  *              |P1/Q1 - (erf(|x|)-c)| <= 2**-59.06
75  *         Remark: here we use the taylor series expansion at x=1.
76  *              erf(1+s) = erf(1) + s*Poly(s)
77  *                       = 0.845.. + P1(s)/Q1(s)
78  *         That is, we use rational approximation to approximate
79  *                      erf(1+s) - (c = (single)0.84506291151)
80  *         Note that |P1/Q1|< 0.078 for x in [0.84375,1.25]
81  *         where
82  *              P1(s) = degree 6 poly in s
83  *              Q1(s) = degree 6 poly in s
84  *
85  *      3. For x in [1.25,1/0.35(~2.857143)],
86  *              erfc(x) = (1/x)*exp(-x*x-0.5625+R1/S1)
87  *              erf(x)  = 1 - erfc(x)
88  *         where
89  *              R1(z) = degree 7 poly in z, (z=1/x^2)
90  *              S1(z) = degree 8 poly in z
91  *
92  *      4. For x in [1/0.35,28]
93  *              erfc(x) = (1/x)*exp(-x*x-0.5625+R2/S2) if x > 0
94  *                      = 2.0 - (1/x)*exp(-x*x-0.5625+R2/S2) if -6<x<0
95  *                      = 2.0 - tiny            (if x <= -6)
96  *              erf(x)  = sign(x)*(1.0 - erfc(x)) if x < 6, else
97  *              erf(x)  = sign(x)*(1.0 - tiny)
98  *         where
99  *              R2(z) = degree 6 poly in z, (z=1/x^2)
100  *              S2(z) = degree 7 poly in z
101  *
102  *      Note1:
103  *         To compute exp(-x*x-0.5625+R/S), let s be a single
104  *         precision number and s := x; then
105  *              -x*x = -s*s + (s-x)*(s+x)
106  *              exp(-x*x-0.5626+R/S) =
107  *                      exp(-s*s-0.5625)*exp((s-x)*(s+x)+R/S);
108  *      Note2:
109  *         Here 4 and 5 make use of the asymptotic series
110  *                        exp(-x*x)
111  *              erfc(x) ~ ---------- * ( 1 + Poly(1/x^2) )
112  *                        x*sqrt(pi)
113  *         We use rational approximation to approximate
114  *              g(s)=f(1/x^2) = log(erfc(x)*x) - x*x + 0.5625
115  *         Here is the error bound for R1/S1 and R2/S2
116  *              |R1/S1 - f(x)|  < 2**(-62.57)
117  *              |R2/S2 - f(x)|  < 2**(-61.52)
118  *
119  *      5. For inf > x >= 28
120  *              erf(x)  = sign(x) *(1 - tiny)  (raise inexact)
121  *              erfc(x) = tiny*tiny (raise underflow) if x > 0
122  *                      = 2 - tiny if x<0
123  *
124  *      7. Special case:
125  *              erf(0)  = 0, erf(inf)  = 1, erf(-inf) = -1,
126  *              erfc(0) = 1, erfc(inf) = 0, erfc(-inf) = 2,
127  *      erfc/erf(NaN) is NaN
128  */
129
130 #include "libm_macros.h"
131 #include <math.h>
132
133 static const double xxx[] = {
134 /* tiny */
135         1e-300,
136 /* half */ 5.00000000000000000000e-01,  /* 3FE00000, 00000000 */
137 /* one */ 1.00000000000000000000e+00,   /* 3FF00000, 00000000 */
138 /* two */ 2.00000000000000000000e+00,   /* 40000000, 00000000 */
139 /* erx */ 8.45062911510467529297e-01,   /* 3FEB0AC1, 60000000 */
140
141 /*
142  * Coefficients for approximation to  erf on [0,0.84375]
143  */
144 /* efx */ 1.28379167095512586316e-01,   /* 3FC06EBA, 8214DB69 */
145 /* efx8 */ 1.02703333676410069053e+00,  /* 3FF06EBA, 8214DB69 */
146 /* pp0 */ 1.28379167095512558561e-01,   /* 3FC06EBA, 8214DB68 */
147 /* pp1 */ -3.25042107247001499370e-01,  /* BFD4CD7D, 691CB913 */
148 /* pp2 */ -2.84817495755985104766e-02,  /* BF9D2A51, DBD7194F */
149 /* pp3 */ -5.77027029648944159157e-03,  /* BF77A291, 236668E4 */
150 /* pp4 */ -2.37630166566501626084e-05,  /* BEF8EAD6, 120016AC */
151 /* qq1 */ 3.97917223959155352819e-01,   /* 3FD97779, CDDADC09 */
152 /* qq2 */ 6.50222499887672944485e-02,   /* 3FB0A54C, 5536CEBA */
153 /* qq3 */ 5.08130628187576562776e-03,   /* 3F74D022, C4D36B0F */
154 /* qq4 */ 1.32494738004321644526e-04,   /* 3F215DC9, 221C1A10 */
155 /* qq5 */ -3.96022827877536812320e-06,  /* BED09C43, 42A26120 */
156
157 /*
158  * Coefficients for approximation to  erf  in [0.84375,1.25]
159  */
160 /* pa0 */ -2.36211856075265944077e-03,  /* BF6359B8, BEF77538 */
161 /* pa1 */ 4.14856118683748331666e-01,   /* 3FDA8D00, AD92B34D */
162 /* pa2 */ -3.72207876035701323847e-01,  /* BFD7D240, FBB8C3F1 */
163 /* pa3 */ 3.18346619901161753674e-01,   /* 3FD45FCA, 805120E4 */
164 /* pa4 */ -1.10894694282396677476e-01,  /* BFBC6398, 3D3E28EC */
165 /* pa5 */ 3.54783043256182359371e-02,   /* 3FA22A36, 599795EB */
166 /* pa6 */ -2.16637559486879084300e-03,  /* BF61BF38, 0A96073F */
167 /* qa1 */ 1.06420880400844228286e-01,   /* 3FBB3E66, 18EEE323 */
168 /* qa2 */ 5.40397917702171048937e-01,   /* 3FE14AF0, 92EB6F33 */
169 /* qa3 */ 7.18286544141962662868e-02,   /* 3FB2635C, D99FE9A7 */
170 /* qa4 */ 1.26171219808761642112e-01,   /* 3FC02660, E763351F */
171 /* qa5 */ 1.36370839120290507362e-02,   /* 3F8BEDC2, 6B51DD1C */
172 /* qa6 */ 1.19844998467991074170e-02,   /* 3F888B54, 5735151D */
173
174 /*
175  * Coefficients for approximation to  erfc in [1.25,1/0.35]
176  */
177 /* ra0 */ -9.86494403484714822705e-03,  /* BF843412, 600D6435 */
178 /* ra1 */ -6.93858572707181764372e-01,  /* BFE63416, E4BA7360 */
179 /* ra2 */ -1.05586262253232909814e+01,  /* C0251E04, 41B0E726 */
180 /* ra3 */ -6.23753324503260060396e+01,  /* C04F300A, E4CBA38D */
181 /* ra4 */ -1.62396669462573470355e+02,  /* C0644CB1, 84282266 */
182 /* ra5 */ -1.84605092906711035994e+02,  /* C067135C, EBCCABB2 */
183 /* ra6 */ -8.12874355063065934246e+01,  /* C0545265, 57E4D2F2 */
184 /* ra7 */ -9.81432934416914548592e+00,  /* C023A0EF, C69AC25C */
185 /* sa1 */ 1.96512716674392571292e+01,   /* 4033A6B9, BD707687 */
186 /* sa2 */ 1.37657754143519042600e+02,   /* 4061350C, 526AE721 */
187 /* sa3 */ 4.34565877475229228821e+02,   /* 407B290D, D58A1A71 */
188 /* sa4 */ 6.45387271733267880336e+02,   /* 40842B19, 21EC2868 */
189 /* sa5 */ 4.29008140027567833386e+02,   /* 407AD021, 57700314 */
190 /* sa6 */ 1.08635005541779435134e+02,   /* 405B28A3, EE48AE2C */
191 /* sa7 */ 6.57024977031928170135e+00,   /* 401A47EF, 8E484A93 */
192 /* sa8 */ -6.04244152148580987438e-02,  /* BFAEEFF2, EE749A62 */
193
194 /*
195  * Coefficients for approximation to  erfc in [1/.35,28]
196  */
197 /* rb0 */ -9.86494292470009928597e-03,  /* BF843412, 39E86F4A */
198 /* rb1 */ -7.99283237680523006574e-01,  /* BFE993BA, 70C285DE */
199 /* rb2 */ -1.77579549177547519889e+01,  /* C031C209, 555F995A */
200 /* rb3 */ -1.60636384855821916062e+02,  /* C064145D, 43C5ED98 */
201 /* rb4 */ -6.37566443368389627722e+02,  /* C083EC88, 1375F228 */
202 /* rb5 */ -1.02509513161107724954e+03,  /* C0900461, 6A2E5992 */
203 /* rb6 */ -4.83519191608651397019e+02,  /* C07E384E, 9BDC383F */
204 /* sb1 */ 3.03380607434824582924e+01,   /* 403E568B, 261D5190 */
205 /* sb2 */ 3.25792512996573918826e+02,   /* 40745CAE, 221B9F0A */
206 /* sb3 */ 1.53672958608443695994e+03,   /* 409802EB, 189D5118 */
207 /* sb4 */ 3.19985821950859553908e+03,   /* 40A8FFB7, 688C246A */
208 /* sb5 */ 2.55305040643316442583e+03,   /* 40A3F219, CEDF3BE6 */
209 /* sb6 */ 4.74528541206955367215e+02,   /* 407DA874, E79FE763 */
210 /* sb7 */ -2.24409524465858183362e+01   /* C03670E2, 42712D62 */
211 };
212
213 #define tiny            xxx[0]
214 #define half            xxx[1]
215 #define one             xxx[2]
216 #define two             xxx[3]
217 #define erx             xxx[4]
218
219 /*
220  * Coefficients for approximation to  erf on [0,0.84375]
221  */
222 #define efx             xxx[5]
223 #define efx8            xxx[6]
224 #define pp0             xxx[7]
225 #define pp1             xxx[8]
226 #define pp2             xxx[9]
227 #define pp3             xxx[10]
228 #define pp4             xxx[11]
229 #define qq1             xxx[12]
230 #define qq2             xxx[13]
231 #define qq3             xxx[14]
232 #define qq4             xxx[15]
233 #define qq5             xxx[16]
234
235 /*
236  * Coefficients for approximation to  erf  in [0.84375,1.25]
237  */
238 #define pa0             xxx[17]
239 #define pa1             xxx[18]
240 #define pa2             xxx[19]
241 #define pa3             xxx[20]
242 #define pa4             xxx[21]
243 #define pa5             xxx[22]
244 #define pa6             xxx[23]
245 #define qa1             xxx[24]
246 #define qa2             xxx[25]
247 #define qa3             xxx[26]
248 #define qa4             xxx[27]
249 #define qa5             xxx[28]
250 #define qa6             xxx[29]
251
252 /*
253  * Coefficients for approximation to  erfc in [1.25,1/0.35]
254  */
255 #define ra0             xxx[30]
256 #define ra1             xxx[31]
257 #define ra2             xxx[32]
258 #define ra3             xxx[33]
259 #define ra4             xxx[34]
260 #define ra5             xxx[35]
261 #define ra6             xxx[36]
262 #define ra7             xxx[37]
263 #define sa1             xxx[38]
264 #define sa2             xxx[39]
265 #define sa3             xxx[40]
266 #define sa4             xxx[41]
267 #define sa5             xxx[42]
268 #define sa6             xxx[43]
269 #define sa7             xxx[44]
270 #define sa8             xxx[45]
271
272 /*
273  * Coefficients for approximation to  erfc in [1/.35,28]
274  */
275 #define rb0             xxx[46]
276 #define rb1             xxx[47]
277 #define rb2             xxx[48]
278 #define rb3             xxx[49]
279 #define rb4             xxx[50]
280 #define rb5             xxx[51]
281 #define rb6             xxx[52]
282 #define sb1             xxx[53]
283 #define sb2             xxx[54]
284 #define sb3             xxx[55]
285 #define sb4             xxx[56]
286 #define sb5             xxx[57]
287 #define sb6             xxx[58]
288 #define sb7             xxx[59]
289
290 double
291 erf(double x)
292 {
293         int hx, ix, i;
294         double R, S, P, Q, s, y, z, r;
295
296         hx = ((int *)&x)[HIWORD];
297         ix = hx & 0x7fffffff;
298
299         if (ix >= 0x7ff00000) {              /* erf(nan)=nan */
301                 if (ix >= 0x7ff80000)        /* assumes sparc-like QNaN */
302                         return (x);
303 #endif
304                 i = ((unsigned)hx >> 31) << 1;
305                 return ((double)(1 - i) + one / x);     /* erf(+-inf)=+-1 */
306         }
307
308         if (ix < 0x3feb0000) {                               /* |x|<0.84375 */
309                 if (ix < 0x3e300000) {                       /* |x|<2**-28 */
310                         if (ix < 0x00800000)         /* avoid underflow */
311                                 return (0.125 * (8.0 * x + efx8 * x));
312
313                         return (x + efx * x);
314                 }
315
316                 z = x * x;
317                 r = pp0 + z * (pp1 + z * (pp2 + z * (pp3 + z * pp4)));
318                 s = one + z * (qq1 + z * (qq2 + z * (qq3 + z * (qq4 + z *
319                     qq5))));
320                 y = r / s;
321                 return (x + x * y);
322         }
323
324         if (ix < 0x3ff40000) {               /* 0.84375 <= |x| < 1.25 */
325                 s = fabs(x) - one;
326                 P = pa0 + s * (pa1 + s * (pa2 + s * (pa3 + s * (pa4 + s * (pa5 +
327                     s * pa6)))));
328                 Q = one + s * (qa1 + s * (qa2 + s * (qa3 + s * (qa4 + s * (qa5 +
329                     s * qa6)))));
330
331                 if (hx >= 0)
332                         return (erx + P / Q);
333                 else
334                         return (-erx - P / Q);
335         }
336
337         if (ix >= 0x40180000) {              /* inf > |x| >= 6 */
338                 if (hx >= 0)
339                         return (one - tiny);
340                 else
341                         return (tiny - one);
342         }
343
344         x = fabs(x);
345         s = one / (x * x);
346
347         if (ix < 0x4006DB6E) {               /* |x| < 1/0.35 */
348                 R = ra0 + s * (ra1 + s * (ra2 + s * (ra3 + s * (ra4 + s * (ra5 +
349                     s * (ra6 + s * ra7))))));
350                 S = one + s * (sa1 + s * (sa2 + s * (sa3 + s * (sa4 + s * (sa5 +
351                     s * (sa6 + s * (sa7 + s * sa8)))))));
352         } else {                        /* |x| >= 1/0.35 */
353                 R = rb0 + s * (rb1 + s * (rb2 + s * (rb3 + s * (rb4 + s * (rb5 +
354                     s * rb6)))));
355                 S = one + s * (sb1 + s * (sb2 + s * (sb3 + s * (sb4 + s * (sb5 +
356                     s * (sb6 + s * sb7))))));
357         }
358
359         z = x;
360         ((int *)&z)[LOWORD] = 0;
361         r = exp(-z * z - 0.5625) * exp((z - x) * (z + x) + R / S);
362
363         if (hx >= 0)
364                 return (one - r / x);
365         else
366                 return (r / x - one);
367 }
368
369 double
370 erfc(double x)
371 {
372         int hx, ix;
373         double R, S, P, Q, s, y, z, r;
374
375         hx = ((int *)&x)[HIWORD];
376         ix = hx & 0x7fffffff;
377
378         if (ix >= 0x7ff00000) {              /* erfc(nan)=nan */
380                 if (ix >= 0x7ff80000)        /* assumes sparc-like QNaN */
381                         return (x);
382 #endif
383                 /* erfc(+-inf)=0,2 */
384                 return ((double)(((unsigned)hx >> 31) << 1) + one / x);
385         }
386
387         if (ix < 0x3feb0000) {               /* |x| < 0.84375 */
388                 if (ix < 0x3c700000) /* |x| < 2**-56 */
389                         return (one - x);
390
391                 z = x * x;
392                 r = pp0 + z * (pp1 + z * (pp2 + z * (pp3 + z * pp4)));
393                 s = one + z * (qq1 + z * (qq2 + z * (qq3 + z * (qq4 + z *
394                     qq5))));
395                 y = r / s;
396
397                 if (hx < 0x3fd00000) {       /* x < 1/4 */
398                         return (one - (x + x * y));
399                 } else {
400                         r = x * y;
401                         r += (x - half);
402                         return (half - r);
403                 }
404         }
405
406         if (ix < 0x3ff40000) {               /* 0.84375 <= |x| < 1.25 */
407                 s = fabs(x) - one;
408                 P = pa0 + s * (pa1 + s * (pa2 + s * (pa3 + s * (pa4 + s * (pa5 +
409                     s * pa6)))));
410                 Q = one + s * (qa1 + s * (qa2 + s * (qa3 + s * (qa4 + s * (qa5 +
411                     s * qa6)))));
412
413                 if (hx >= 0) {
414                         z = one - erx;
415                         return (z - P / Q);
416                 } else {
417                         z = erx + P / Q;
418                         return (one + z);
419                 }
420         }
421
422         if (ix < 0x403c0000) {               /* |x|<28 */
423                 x = fabs(x);
424                 s = one / (x * x);
425
426                 if (ix < 0x4006DB6D) {       /* |x| < 1/.35 ~ 2.857143 */
427                         R = ra0 + s * (ra1 + s * (ra2 + s * (ra3 + s * (ra4 +
428                             s * (ra5 + s * (ra6 + s * ra7))))));
429                         S = one + s * (sa1 + s * (sa2 + s * (sa3 + s * (sa4 +
430                             s * (sa5 + s * (sa6 + s * (sa7 + s * sa8)))))));
431                 } else {
432                         /* |x| >= 1/.35 ~ 2.857143 */
433                         if (hx < 0 && ix >= 0x40180000)
434                                 return (two - tiny);    /* x < -6 */
435
436                         R = rb0 + s * (rb1 + s * (rb2 + s * (rb3 + s * (rb4 +
437                             s * (rb5 + s * rb6)))));
438                         S = one + s * (sb1 + s * (sb2 + s * (sb3 + s * (sb4 +
439                             s * (sb5 + s * (sb6 + s * sb7))))));
440                 }
441
442                 z = x;
443                 ((int *)&z)[LOWORD] = 0;
444                 r = exp(-z * z - 0.5625) * exp((z - x) * (z + x) + R / S);
445
446                 if (hx > 0)
447                         return (r / x);
448                 else
449                         return (two - r / x);
450         } else {
451                 if (hx > 0)
452                         return (tiny * tiny);
453                 else
454                         return (two - tiny);
455         }
456 }
```