libm: Sync with FreeBSD (gains 6 long double functions)
[dragonfly.git] / lib / libm / src / s_erf.c
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
13
14 /* double erf(double x)
15  * double erfc(double x)
16  *                           x
17  *                    2      |\
18  *     erf(x)  =  ---------  | exp(-t*t)dt
19  *                 sqrt(pi) \|
20  *                           0
21  *
22  *     erfc(x) =  1-erf(x)
23  *  Note that
24  *              erf(-x) = -erf(x)
25  *              erfc(-x) = 2 - erfc(x)
26  *
27  * Method:
28  *      1. For |x| in [0, 0.84375]
29  *          erf(x)  = x + x*R(x^2)
30  *          erfc(x) = 1 - erf(x)           if x in [-.84375,0.25]
31  *                  = 0.5 + ((0.5-x)-x*R)  if x in [0.25,0.84375]
32  *         where R = P/Q where P is an odd poly of degree 8 and
33  *         Q is an odd poly of degree 10.
34  *                                               -57.90
35  *                      | R - (erf(x)-x)/x | <= 2
36  *
37  *
38  *         Remark. The formula is derived by noting
39  *          erf(x) = (2/sqrt(pi))*(x - x^3/3 + x^5/10 - x^7/42 + ....)
40  *         and that
41  *          2/sqrt(pi) = 1.128379167095512573896158903121545171688
42  *         is close to one. The interval is chosen because the fix
43  *         point of erf(x) is near 0.6174 (i.e., erf(x)=x when x is
44  *         near 0.6174), and by some experiment, 0.84375 is chosen to
45  *         guarantee the error is less than one ulp for erf.
46  *
47  *      2. For |x| in [0.84375,1.25], let s = |x| - 1, and
48  *         c = 0.84506291151 rounded to single (24 bits)
49  *              erf(x)  = sign(x) * (c  + P1(s)/Q1(s))
50  *              erfc(x) = (1-c)  - P1(s)/Q1(s) if x > 0
51  *                        1+(c+P1(s)/Q1(s))    if x < 0
52  *              |P1/Q1 - (erf(|x|)-c)| <= 2**-59.06
53  *         Remark: here we use the taylor series expansion at x=1.
54  *              erf(1+s) = erf(1) + s*Poly(s)
55  *                       = 0.845.. + P1(s)/Q1(s)
56  *         That is, we use rational approximation to approximate
57  *                      erf(1+s) - (c = (single)0.84506291151)
58  *         Note that |P1/Q1|< 0.078 for x in [0.84375,1.25]
59  *         where
60  *              P1(s) = degree 6 poly in s
61  *              Q1(s) = degree 6 poly in s
62  *
63  *      3. For x in [1.25,1/0.35(~2.857143)],
64  *              erfc(x) = (1/x)*exp(-x*x-0.5625+R1/S1)
65  *              erf(x)  = 1 - erfc(x)
66  *         where
67  *              R1(z) = degree 7 poly in z, (z=1/x^2)
68  *              S1(z) = degree 8 poly in z
69  *
70  *      4. For x in [1/0.35,28]
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/S2) if -6<x<0
73  *                      = 2.0 - tiny            (if x <= -6)
74  *              erf(x)  = sign(x)*(1.0 - erfc(x)) if x < 6, else
75  *              erf(x)  = sign(x)*(1.0 - tiny)
76  *         where
77  *              R2(z) = degree 6 poly in z, (z=1/x^2)
78  *              S2(z) = degree 7 poly in z
79  *
80  *      Note1:
81  *         To compute exp(-x*x-0.5625+R/S), let s be a single
82  *         precision number and s := x; then
83  *              -x*x = -s*s + (s-x)*(s+x)
84  *              exp(-x*x-0.5626+R/S) =
85  *                      exp(-s*s-0.5625)*exp((s-x)*(s+x)+R/S);
86  *      Note2:
87  *         Here 4 and 5 make use of the asymptotic series
88  *                        exp(-x*x)
89  *              erfc(x) ~ ---------- * ( 1 + Poly(1/x^2) )
90  *                        x*sqrt(pi)
91  *         We use rational approximation to approximate
92  *              g(s)=f(1/x^2) = log(erfc(x)*x) - x*x + 0.5625
93  *         Here is the error bound for R1/S1 and R2/S2
94  *              |R1/S1 - f(x)|  < 2**(-62.57)
95  *              |R2/S2 - f(x)|  < 2**(-61.52)
96  *
97  *      5. For inf > x >= 28
98  *              erf(x)  = sign(x) *(1 - tiny)  (raise inexact)
99  *              erfc(x) = tiny*tiny (raise underflow) if x > 0
100  *                      = 2 - tiny if x<0
101  *
102  *      7. Special case:
103  *              erf(0)  = 0, erf(inf)  = 1, erf(-inf) = -1,
104  *              erfc(0) = 1, erfc(inf) = 0, erfc(-inf) = 2,
105  *              erfc/erf(NaN) is NaN
106  */
107
108
109 #include "math.h"
110 #include "math_private.h"
111
112 /* XXX Prevent compilers from erroneously constant folding: */
113 static const volatile double tiny= 1e-300;
114
115 static const double
116 half= 0.5,
117 one = 1,
118 two = 2,
119 /* c = (float)0.84506291151 */
120 erx =  8.45062911510467529297e-01, /* 0x3FEB0AC1, 0x60000000 */
121 /*
122  * In the domain [0, 2**-28], only the first term in the power series
123  * expansion of erf(x) is used.  The magnitude of the first neglected
124  * terms is less than 2**-84.
125  */
126 efx =  1.28379167095512586316e-01, /* 0x3FC06EBA, 0x8214DB69 */
127 efx8=  1.02703333676410069053e+00, /* 0x3FF06EBA, 0x8214DB69 */
128 /*
129  * Coefficients for approximation to erf on [0,0.84375]
130  */
131 pp0  =  1.28379167095512558561e-01, /* 0x3FC06EBA, 0x8214DB68 */
132 pp1  = -3.25042107247001499370e-01, /* 0xBFD4CD7D, 0x691CB913 */
133 pp2  = -2.84817495755985104766e-02, /* 0xBF9D2A51, 0xDBD7194F */
134 pp3  = -5.77027029648944159157e-03, /* 0xBF77A291, 0x236668E4 */
135 pp4  = -2.37630166566501626084e-05, /* 0xBEF8EAD6, 0x120016AC */
136 qq1  =  3.97917223959155352819e-01, /* 0x3FD97779, 0xCDDADC09 */
137 qq2  =  6.50222499887672944485e-02, /* 0x3FB0A54C, 0x5536CEBA */
138 qq3  =  5.08130628187576562776e-03, /* 0x3F74D022, 0xC4D36B0F */
139 qq4  =  1.32494738004321644526e-04, /* 0x3F215DC9, 0x221C1A10 */
140 qq5  = -3.96022827877536812320e-06, /* 0xBED09C43, 0x42A26120 */
141 /*
142  * Coefficients for approximation to erf in [0.84375,1.25]
143  */
144 pa0  = -2.36211856075265944077e-03, /* 0xBF6359B8, 0xBEF77538 */
145 pa1  =  4.14856118683748331666e-01, /* 0x3FDA8D00, 0xAD92B34D */
146 pa2  = -3.72207876035701323847e-01, /* 0xBFD7D240, 0xFBB8C3F1 */
147 pa3  =  3.18346619901161753674e-01, /* 0x3FD45FCA, 0x805120E4 */
148 pa4  = -1.10894694282396677476e-01, /* 0xBFBC6398, 0x3D3E28EC */
149 pa5  =  3.54783043256182359371e-02, /* 0x3FA22A36, 0x599795EB */
150 pa6  = -2.16637559486879084300e-03, /* 0xBF61BF38, 0x0A96073F */
151 qa1  =  1.06420880400844228286e-01, /* 0x3FBB3E66, 0x18EEE323 */
152 qa2  =  5.40397917702171048937e-01, /* 0x3FE14AF0, 0x92EB6F33 */
153 qa3  =  7.18286544141962662868e-02, /* 0x3FB2635C, 0xD99FE9A7 */
154 qa4  =  1.26171219808761642112e-01, /* 0x3FC02660, 0xE763351F */
155 qa5  =  1.36370839120290507362e-02, /* 0x3F8BEDC2, 0x6B51DD1C */
156 qa6  =  1.19844998467991074170e-02, /* 0x3F888B54, 0x5735151D */
157 /*
158  * Coefficients for approximation to erfc in [1.25,1/0.35]
159  */
160 ra0  = -9.86494403484714822705e-03, /* 0xBF843412, 0x600D6435 */
161 ra1  = -6.93858572707181764372e-01, /* 0xBFE63416, 0xE4BA7360 */
162 ra2  = -1.05586262253232909814e+01, /* 0xC0251E04, 0x41B0E726 */
163 ra3  = -6.23753324503260060396e+01, /* 0xC04F300A, 0xE4CBA38D */
164 ra4  = -1.62396669462573470355e+02, /* 0xC0644CB1, 0x84282266 */
165 ra5  = -1.84605092906711035994e+02, /* 0xC067135C, 0xEBCCABB2 */
166 ra6  = -8.12874355063065934246e+01, /* 0xC0545265, 0x57E4D2F2 */
167 ra7  = -9.81432934416914548592e+00, /* 0xC023A0EF, 0xC69AC25C */
168 sa1  =  1.96512716674392571292e+01, /* 0x4033A6B9, 0xBD707687 */
169 sa2  =  1.37657754143519042600e+02, /* 0x4061350C, 0x526AE721 */
170 sa3  =  4.34565877475229228821e+02, /* 0x407B290D, 0xD58A1A71 */
171 sa4  =  6.45387271733267880336e+02, /* 0x40842B19, 0x21EC2868 */
172 sa5  =  4.29008140027567833386e+02, /* 0x407AD021, 0x57700314 */
173 sa6  =  1.08635005541779435134e+02, /* 0x405B28A3, 0xEE48AE2C */
174 sa7  =  6.57024977031928170135e+00, /* 0x401A47EF, 0x8E484A93 */
175 sa8  = -6.04244152148580987438e-02, /* 0xBFAEEFF2, 0xEE749A62 */
176 /*
177  * Coefficients for approximation to erfc in [1/.35,28]
178  */
179 rb0  = -9.86494292470009928597e-03, /* 0xBF843412, 0x39E86F4A */
180 rb1  = -7.99283237680523006574e-01, /* 0xBFE993BA, 0x70C285DE */
181 rb2  = -1.77579549177547519889e+01, /* 0xC031C209, 0x555F995A */
182 rb3  = -1.60636384855821916062e+02, /* 0xC064145D, 0x43C5ED98 */
183 rb4  = -6.37566443368389627722e+02, /* 0xC083EC88, 0x1375F228 */
184 rb5  = -1.02509513161107724954e+03, /* 0xC0900461, 0x6A2E5992 */
185 rb6  = -4.83519191608651397019e+02, /* 0xC07E384E, 0x9BDC383F */
186 sb1  =  3.03380607434824582924e+01, /* 0x403E568B, 0x261D5190 */
187 sb2  =  3.25792512996573918826e+02, /* 0x40745CAE, 0x221B9F0A */
188 sb3  =  1.53672958608443695994e+03, /* 0x409802EB, 0x189D5118 */
189 sb4  =  3.19985821950859553908e+03, /* 0x40A8FFB7, 0x688C246A */
190 sb5  =  2.55305040643316442583e+03, /* 0x40A3F219, 0xCEDF3BE6 */
191 sb6  =  4.74528541206955367215e+02, /* 0x407DA874, 0xE79FE763 */
192 sb7  = -2.24409524465858183362e+01; /* 0xC03670E2, 0x42712D62 */
193
194 double
195 erf(double x)
196 {
197         int32_t hx,ix,i;
198         double R,S,P,Q,s,y,z,r;
199         GET_HIGH_WORD(hx,x);
200         ix = hx&0x7fffffff;
201         if(ix>=0x7ff00000) {            /* erf(nan)=nan */
202             i = ((u_int32_t)hx>>31)<<1;
203             return (double)(1-i)+one/x; /* erf(+-inf)=+-1 */
204         }
205
206         if(ix < 0x3feb0000) {           /* |x|<0.84375 */
207             if(ix < 0x3e300000) {       /* |x|<2**-28 */
208                 if (ix < 0x00800000)
209                     return (8*x+efx8*x)/8;      /* avoid spurious underflow */
210                 return x + efx*x;
211             }
212             z = x*x;
213             r = pp0+z*(pp1+z*(pp2+z*(pp3+z*pp4)));
214             s = one+z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*qq5))));
215             y = r/s;
216             return x + x*y;
217         }
218         if(ix < 0x3ff40000) {           /* 0.84375 <= |x| < 1.25 */
219             s = fabs(x)-one;
220             P = pa0+s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6)))));
221             Q = one+s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6)))));
222             if(hx>=0) return erx + P/Q; else return -erx - P/Q;
223         }
224         if (ix >= 0x40180000) {         /* inf>|x|>=6 */
225             if(hx>=0) return one-tiny; else return tiny-one;
226         }
227         x = fabs(x);
228         s = one/(x*x);
229         if(ix< 0x4006DB6E) {    /* |x| < 1/0.35 */
230             R=ra0+s*(ra1+s*(ra2+s*(ra3+s*(ra4+s*(ra5+s*(ra6+s*ra7))))));
231             S=one+s*(sa1+s*(sa2+s*(sa3+s*(sa4+s*(sa5+s*(sa6+s*(sa7+
232                 s*sa8)))))));
233         } else {        /* |x| >= 1/0.35 */
234             R=rb0+s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*(rb5+s*rb6)))));
235             S=one+s*(sb1+s*(sb2+s*(sb3+s*(sb4+s*(sb5+s*(sb6+s*sb7))))));
236         }
237         z  = x;
238         SET_LOW_WORD(z,0);
239         r  =  __ieee754_exp(-z*z-0.5625)*__ieee754_exp((z-x)*(z+x)+R/S);
240         if(hx>=0) return one-r/x; else return  r/x-one;
241 }
242
243 #if (LDBL_MANT_DIG == 53)
244 __weak_reference(erf, erfl);
245 #endif
246
247 double
248 erfc(double x)
249 {
250         int32_t hx,ix;
251         double R,S,P,Q,s,y,z,r;
252         GET_HIGH_WORD(hx,x);
253         ix = hx&0x7fffffff;
254         if(ix>=0x7ff00000) {                    /* erfc(nan)=nan */
255                                                 /* erfc(+-inf)=0,2 */
256             return (double)(((u_int32_t)hx>>31)<<1)+one/x;
257         }
258
259         if(ix < 0x3feb0000) {           /* |x|<0.84375 */
260             if(ix < 0x3c700000)         /* |x|<2**-56 */
261                 return one-x;
262             z = x*x;
263             r = pp0+z*(pp1+z*(pp2+z*(pp3+z*pp4)));
264             s = one+z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*qq5))));
265             y = r/s;
266             if(hx < 0x3fd00000) {       /* x<1/4 */
267                 return one-(x+x*y);
268             } else {
269                 r = x*y;
270                 r += (x-half);
271                 return half - r ;
272             }
273         }
274         if(ix < 0x3ff40000) {           /* 0.84375 <= |x| < 1.25 */
275             s = fabs(x)-one;
276             P = pa0+s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6)))));
277             Q = one+s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6)))));
278             if(hx>=0) {
279                 z  = one-erx; return z - P/Q;
280             } else {
281                 z = erx+P/Q; return one+z;
282             }
283         }
284         if (ix < 0x403c0000) {          /* |x|<28 */
285             x = fabs(x);
286             s = one/(x*x);
287             if(ix< 0x4006DB6D) {        /* |x| < 1/.35 ~ 2.857143*/
288                 R=ra0+s*(ra1+s*(ra2+s*(ra3+s*(ra4+s*(ra5+s*(ra6+s*ra7))))));
289                 S=one+s*(sa1+s*(sa2+s*(sa3+s*(sa4+s*(sa5+s*(sa6+s*(sa7+
290                     s*sa8)))))));
291             } else {                    /* |x| >= 1/.35 ~ 2.857143 */
292                 if(hx<0&&ix>=0x40180000) return two-tiny;/* x < -6 */
293                 R=rb0+s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*(rb5+s*rb6)))));
294                 S=one+s*(sb1+s*(sb2+s*(sb3+s*(sb4+s*(sb5+s*(sb6+s*sb7))))));
295             }
296             z  = x;
297             SET_LOW_WORD(z,0);
298             r  =  __ieee754_exp(-z*z-0.5625)*__ieee754_exp((z-x)*(z+x)+R/S);
299             if(hx>0) return r/x; else return two-r/x;
300         } else {
301             if(hx>0) return tiny*tiny; else return two-tiny;
302         }
303 }
304
305 #if (LDBL_MANT_DIG == 53)
306 __weak_reference(erfc, erfcl);
307 #endif