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#include <sys/cdefs.h>
14#if defined(LIBM_SCCS) && !defined(lint)
15__RCSID("$NetBSD: s_erf.c,v 1.10 1999/07/02 15:37:42 simonb Exp $");
16#endif
17
18/* double erf(double x)
19 * double erfc(double x)
20 *			     x
21 *		      2      |\
22 *     erf(x)  =  ---------  | exp(-t*t)dt
23 *	 	   sqrt(pi) \|
24 *			     0
25 *
26 *     erfc(x) =  1-erf(x)
27 *  Note that
28 *		erf(-x) = -erf(x)
29 *		erfc(-x) = 2 - erfc(x)
30 *
31 * Method:
32 *	1. For |x| in [0, 0.84375]
33 *	    erf(x)  = x + x*R(x^2)
34 *          erfc(x) = 1 - erf(x)           if x in [-.84375,0.25]
35 *                  = 0.5 + ((0.5-x)-x*R)  if x in [0.25,0.84375]
36 *	   where R = P/Q where P is an odd poly of degree 8 and
37 *	   Q is an odd poly of degree 10.
38 *						 -57.90
39 *			| R - (erf(x)-x)/x | <= 2
40 *
41 *
42 *	   Remark. The formula is derived by noting
43 *          erf(x) = (2/sqrt(pi))*(x - x^3/3 + x^5/10 - x^7/42 + ....)
44 *	   and that
45 *          2/sqrt(pi) = 1.128379167095512573896158903121545171688
46 *	   is close to one. The interval is chosen because the fix
47 *	   point of erf(x) is near 0.6174 (i.e., erf(x)=x when x is
48 *	   near 0.6174), and by some experiment, 0.84375 is chosen to
49 * 	   guarantee the error is less than one ulp for erf.
50 *
51 *      2. For |x| in [0.84375,1.25], let s = |x| - 1, and
52 *         c = 0.84506291151 rounded to single (24 bits)
53 *         	erf(x)  = sign(x) * (c  + P1(s)/Q1(s))
54 *         	erfc(x) = (1-c)  - P1(s)/Q1(s) if x > 0
55 *			  1+(c+P1(s)/Q1(s))    if x < 0
56 *         	|P1/Q1 - (erf(|x|)-c)| <= 2**-59.06
57 *	   Remark: here we use the taylor series expansion at x=1.
58 *		erf(1+s) = erf(1) + s*Poly(s)
59 *			 = 0.845.. + P1(s)/Q1(s)
60 *	   That is, we use rational approximation to approximate
61 *			erf(1+s) - (c = (single)0.84506291151)
62 *	   Note that |P1/Q1|< 0.078 for x in [0.84375,1.25]
63 *	   where
64 *		P1(s) = degree 6 poly in s
65 *		Q1(s) = degree 6 poly in s
66 *
67 *      3. For x in [1.25,1/0.35(~2.857143)],
68 *         	erfc(x) = (1/x)*exp(-x*x-0.5625+R1/S1)
69 *         	erf(x)  = 1 - erfc(x)
70 *	   where
71 *		R1(z) = degree 7 poly in z, (z=1/x^2)
72 *		S1(z) = degree 8 poly in z
73 *
74 *      4. For x in [1/0.35,28]
75 *         	erfc(x) = (1/x)*exp(-x*x-0.5625+R2/S2) if x > 0
76 *			= 2.0 - (1/x)*exp(-x*x-0.5625+R2/S2) if -6<x<0
77 *			= 2.0 - tiny		(if x <= -6)
78 *         	erf(x)  = sign(x)*(1.0 - erfc(x)) if x < 6, else
79 *         	erf(x)  = sign(x)*(1.0 - tiny)
80 *	   where
81 *		R2(z) = degree 6 poly in z, (z=1/x^2)
82 *		S2(z) = degree 7 poly in z
83 *
84 *      Note1:
85 *	   To compute exp(-x*x-0.5625+R/S), let s be a single
86 *	   precision number and s := x; then
87 *		-x*x = -s*s + (s-x)*(s+x)
88 *	        exp(-x*x-0.5626+R/S) =
89 *			exp(-s*s-0.5625)*exp((s-x)*(s+x)+R/S);
90 *      Note2:
91 *	   Here 4 and 5 make use of the asymptotic series
92 *			  exp(-x*x)
93 *		erfc(x) ~ ---------- * ( 1 + Poly(1/x^2) )
94 *			  x*sqrt(pi)
95 *	   We use rational approximation to approximate
96 *      	g(s)=f(1/x^2) = log(erfc(x)*x) - x*x + 0.5625
97 *	   Here is the error bound for R1/S1 and R2/S2
98 *      	|R1/S1 - f(x)|  < 2**(-62.57)
99 *      	|R2/S2 - f(x)|  < 2**(-61.52)
100 *
101 *      5. For inf > x >= 28
102 *         	erf(x)  = sign(x) *(1 - tiny)  (raise inexact)
103 *         	erfc(x) = tiny*tiny (raise underflow) if x > 0
104 *			= 2 - tiny if x<0
105 *
106 *      7. Special case:
107 *         	erf(0)  = 0, erf(inf)  = 1, erf(-inf) = -1,
108 *         	erfc(0) = 1, erfc(inf) = 0, erfc(-inf) = 2,
109 *	   	erfc/erf(NaN) is NaN
110 */
111
112
113#include "math.h"
114#include "math_private.h"
115
116static const double
117tiny	    = 1e-300,
118half=  5.00000000000000000000e-01, /* 0x3FE00000, 0x00000000 */
119one =  1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
120two =  2.00000000000000000000e+00, /* 0x40000000, 0x00000000 */
121	/* c = (float)0.84506291151 */
122erx =  8.45062911510467529297e-01, /* 0x3FEB0AC1, 0x60000000 */
123/*
124 * Coefficients for approximation to  erf on [0,0.84375]
125 */
126efx =  1.28379167095512586316e-01, /* 0x3FC06EBA, 0x8214DB69 */
127efx8=  1.02703333676410069053e+00, /* 0x3FF06EBA, 0x8214DB69 */
128pp0  =  1.28379167095512558561e-01, /* 0x3FC06EBA, 0x8214DB68 */
129pp1  = -3.25042107247001499370e-01, /* 0xBFD4CD7D, 0x691CB913 */
130pp2  = -2.84817495755985104766e-02, /* 0xBF9D2A51, 0xDBD7194F */
131pp3  = -5.77027029648944159157e-03, /* 0xBF77A291, 0x236668E4 */
132pp4  = -2.37630166566501626084e-05, /* 0xBEF8EAD6, 0x120016AC */
133qq1  =  3.97917223959155352819e-01, /* 0x3FD97779, 0xCDDADC09 */
134qq2  =  6.50222499887672944485e-02, /* 0x3FB0A54C, 0x5536CEBA */
135qq3  =  5.08130628187576562776e-03, /* 0x3F74D022, 0xC4D36B0F */
136qq4  =  1.32494738004321644526e-04, /* 0x3F215DC9, 0x221C1A10 */
137qq5  = -3.96022827877536812320e-06, /* 0xBED09C43, 0x42A26120 */
138/*
139 * Coefficients for approximation to  erf  in [0.84375,1.25]
140 */
141pa0  = -2.36211856075265944077e-03, /* 0xBF6359B8, 0xBEF77538 */
142pa1  =  4.14856118683748331666e-01, /* 0x3FDA8D00, 0xAD92B34D */
143pa2  = -3.72207876035701323847e-01, /* 0xBFD7D240, 0xFBB8C3F1 */
144pa3  =  3.18346619901161753674e-01, /* 0x3FD45FCA, 0x805120E4 */
145pa4  = -1.10894694282396677476e-01, /* 0xBFBC6398, 0x3D3E28EC */
146pa5  =  3.54783043256182359371e-02, /* 0x3FA22A36, 0x599795EB */
147pa6  = -2.16637559486879084300e-03, /* 0xBF61BF38, 0x0A96073F */
148qa1  =  1.06420880400844228286e-01, /* 0x3FBB3E66, 0x18EEE323 */
149qa2  =  5.40397917702171048937e-01, /* 0x3FE14AF0, 0x92EB6F33 */
150qa3  =  7.18286544141962662868e-02, /* 0x3FB2635C, 0xD99FE9A7 */
151qa4  =  1.26171219808761642112e-01, /* 0x3FC02660, 0xE763351F */
152qa5  =  1.36370839120290507362e-02, /* 0x3F8BEDC2, 0x6B51DD1C */
153qa6  =  1.19844998467991074170e-02, /* 0x3F888B54, 0x5735151D */
154/*
155 * Coefficients for approximation to  erfc in [1.25,1/0.35]
156 */
157ra0  = -9.86494403484714822705e-03, /* 0xBF843412, 0x600D6435 */
158ra1  = -6.93858572707181764372e-01, /* 0xBFE63416, 0xE4BA7360 */
159ra2  = -1.05586262253232909814e+01, /* 0xC0251E04, 0x41B0E726 */
160ra3  = -6.23753324503260060396e+01, /* 0xC04F300A, 0xE4CBA38D */
161ra4  = -1.62396669462573470355e+02, /* 0xC0644CB1, 0x84282266 */
162ra5  = -1.84605092906711035994e+02, /* 0xC067135C, 0xEBCCABB2 */
163ra6  = -8.12874355063065934246e+01, /* 0xC0545265, 0x57E4D2F2 */
164ra7  = -9.81432934416914548592e+00, /* 0xC023A0EF, 0xC69AC25C */
165sa1  =  1.96512716674392571292e+01, /* 0x4033A6B9, 0xBD707687 */
166sa2  =  1.37657754143519042600e+02, /* 0x4061350C, 0x526AE721 */
167sa3  =  4.34565877475229228821e+02, /* 0x407B290D, 0xD58A1A71 */
168sa4  =  6.45387271733267880336e+02, /* 0x40842B19, 0x21EC2868 */
169sa5  =  4.29008140027567833386e+02, /* 0x407AD021, 0x57700314 */
170sa6  =  1.08635005541779435134e+02, /* 0x405B28A3, 0xEE48AE2C */
171sa7  =  6.57024977031928170135e+00, /* 0x401A47EF, 0x8E484A93 */
172sa8  = -6.04244152148580987438e-02, /* 0xBFAEEFF2, 0xEE749A62 */
173/*
174 * Coefficients for approximation to  erfc in [1/.35,28]
175 */
176rb0  = -9.86494292470009928597e-03, /* 0xBF843412, 0x39E86F4A */
177rb1  = -7.99283237680523006574e-01, /* 0xBFE993BA, 0x70C285DE */
178rb2  = -1.77579549177547519889e+01, /* 0xC031C209, 0x555F995A */
179rb3  = -1.60636384855821916062e+02, /* 0xC064145D, 0x43C5ED98 */
180rb4  = -6.37566443368389627722e+02, /* 0xC083EC88, 0x1375F228 */
181rb5  = -1.02509513161107724954e+03, /* 0xC0900461, 0x6A2E5992 */
182rb6  = -4.83519191608651397019e+02, /* 0xC07E384E, 0x9BDC383F */
183sb1  =  3.03380607434824582924e+01, /* 0x403E568B, 0x261D5190 */
184sb2  =  3.25792512996573918826e+02, /* 0x40745CAE, 0x221B9F0A */
185sb3  =  1.53672958608443695994e+03, /* 0x409802EB, 0x189D5118 */
186sb4  =  3.19985821950859553908e+03, /* 0x40A8FFB7, 0x688C246A */
187sb5  =  2.55305040643316442583e+03, /* 0x40A3F219, 0xCEDF3BE6 */
188sb6  =  4.74528541206955367215e+02, /* 0x407DA874, 0xE79FE763 */
189sb7  = -2.24409524465858183362e+01; /* 0xC03670E2, 0x42712D62 */
190
191double
192erf(double x)
193{
194	int32_t hx,ix,i;
195	double R,S,P,Q,s,y,z,r;
196	GET_HIGH_WORD(hx,x);
197	ix = hx&0x7fffffff;
198	if(ix>=0x7ff00000) {		/* erf(nan)=nan */
199	    i = ((u_int32_t)hx>>31)<<1;
200	    return (double)(1-i)+one/x;	/* erf(+-inf)=+-1 */
201	}
202
203	if(ix < 0x3feb0000) {		/* |x|<0.84375 */
204	    if(ix < 0x3e300000) { 	/* |x|<2**-28 */
205	        if (ix < 0x00800000)
206		    return 0.125*(8.0*x+efx8*x);  /*avoid underflow */
207		return x + efx*x;
208	    }
209	    z = x*x;
210	    r = pp0+z*(pp1+z*(pp2+z*(pp3+z*pp4)));
211	    s = one+z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*qq5))));
212	    y = r/s;
213	    return x + x*y;
214	}
215	if(ix < 0x3ff40000) {		/* 0.84375 <= |x| < 1.25 */
216	    s = fabs(x)-one;
217	    P = pa0+s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6)))));
218	    Q = one+s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6)))));
219	    if(hx>=0) return erx + P/Q; else return -erx - P/Q;
220	}
221	if (ix >= 0x40180000) {		/* inf>|x|>=6 */
222	    if(hx>=0) return one-tiny; else return tiny-one;
223	}
224	x = fabs(x);
225 	s = one/(x*x);
226	if(ix< 0x4006DB6E) {	/* |x| < 1/0.35 */
227	    R=ra0+s*(ra1+s*(ra2+s*(ra3+s*(ra4+s*(
228				ra5+s*(ra6+s*ra7))))));
229	    S=one+s*(sa1+s*(sa2+s*(sa3+s*(sa4+s*(
230				sa5+s*(sa6+s*(sa7+s*sa8)))))));
231	} else {	/* |x| >= 1/0.35 */
232	    R=rb0+s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*(
233				rb5+s*rb6)))));
234	    S=one+s*(sb1+s*(sb2+s*(sb3+s*(sb4+s*(
235				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
243double
244erfc(double x)
245{
246	int32_t hx,ix;
247	double R,S,P,Q,s,y,z,r;
248	GET_HIGH_WORD(hx,x);
249	ix = hx&0x7fffffff;
250	if(ix>=0x7ff00000) {			/* erfc(nan)=nan */
251						/* erfc(+-inf)=0,2 */
252	    return (double)(((u_int32_t)hx>>31)<<1)+one/x;
253	}
254
255	if(ix < 0x3feb0000) {		/* |x|<0.84375 */
256	    if(ix < 0x3c700000)  	/* |x|<2**-56 */
257		return one-x;
258	    z = x*x;
259	    r = pp0+z*(pp1+z*(pp2+z*(pp3+z*pp4)));
260	    s = one+z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*qq5))));
261	    y = r/s;
262	    if(hx < 0x3fd00000) {  	/* x<1/4 */
263		return one-(x+x*y);
264	    } else {
265		r = x*y;
266		r += (x-half);
267	        return half - r ;
268	    }
269	}
270	if(ix < 0x3ff40000) {		/* 0.84375 <= |x| < 1.25 */
271	    s = fabs(x)-one;
272	    P = pa0+s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6)))));
273	    Q = one+s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6)))));
274	    if(hx>=0) {
275	        z  = one-erx; return z - P/Q;
276	    } else {
277		z = erx+P/Q; return one+z;
278	    }
279	}
280	if (ix < 0x403c0000) {		/* |x|<28 */
281	    x = fabs(x);
282 	    s = one/(x*x);
283	    if(ix< 0x4006DB6D) {	/* |x| < 1/.35 ~ 2.857143*/
284	        R=ra0+s*(ra1+s*(ra2+s*(ra3+s*(ra4+s*(
285				ra5+s*(ra6+s*ra7))))));
286	        S=one+s*(sa1+s*(sa2+s*(sa3+s*(sa4+s*(
287				sa5+s*(sa6+s*(sa7+s*sa8)))))));
288	    } else {			/* |x| >= 1/.35 ~ 2.857143 */
289		if(hx<0&&ix>=0x40180000) return two-tiny;/* x < -6 */
290	        R=rb0+s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*(
291				rb5+s*rb6)))));
292	        S=one+s*(sb1+s*(sb2+s*(sb3+s*(sb4+s*(
293				sb5+s*(sb6+s*sb7))))));
294	    }
295	    z  = x;
296	    SET_LOW_WORD(z,0);
297	    r  =  __ieee754_exp(-z*z-0.5625)*
298			__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