s_erf.c revision 268587 
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__FBSDID("$FreeBSD: head/lib/msun/src/s_erf.c 268587 2014-07-13 15:15:51Z kargl $");
15
16/* double erf(double x)
17 * double erfc(double x)
18 *			     x
19 *		      2      |\
20 *     erf(x)  =  ---------  | exp(-t*t)dt
21 *	 	   sqrt(pi) \|
22 *			     0
23 *
24 *     erfc(x) =  1-erf(x)
25 *  Note that
26 *		erf(-x) = -erf(x)
27 *		erfc(-x) = 2 - erfc(x)
28 *
29 * Method:
30 *	1. For |x| in [0, 0.84375]
31 *	    erf(x)  = x + x*R(x^2)
32 *          erfc(x) = 1 - erf(x)           if x in [-.84375,0.25]
33 *                  = 0.5 + ((0.5-x)-x*R)  if x in [0.25,0.84375]
34 *	   where R = P/Q where P is an odd poly of degree 8 and
35 *	   Q is an odd poly of degree 10.
36 *						 -57.90
37 *			| R - (erf(x)-x)/x | <= 2
38 *
39 *
40 *	   Remark. The formula is derived by noting
41 *          erf(x) = (2/sqrt(pi))*(x - x^3/3 + x^5/10 - x^7/42 + ....)
42 *	   and that
43 *          2/sqrt(pi) = 1.128379167095512573896158903121545171688
44 *	   is close to one. The interval is chosen because the fix
45 *	   point of erf(x) is near 0.6174 (i.e., erf(x)=x when x is
46 *	   near 0.6174), and by some experiment, 0.84375 is chosen to
47 * 	   guarantee the error is less than one ulp for erf.
48 *
49 *      2. For |x| in [0.84375,1.25], let s = |x| - 1, and
50 *         c = 0.84506291151 rounded to single (24 bits)
51 *         	erf(x)  = sign(x) * (c  + P1(s)/Q1(s))
52 *         	erfc(x) = (1-c)  - P1(s)/Q1(s) if x > 0
53 *			  1+(c+P1(s)/Q1(s))    if x < 0
54 *         	|P1/Q1 - (erf(|x|)-c)| <= 2**-59.06
55 *	   Remark: here we use the taylor series expansion at x=1.
56 *		erf(1+s) = erf(1) + s*Poly(s)
57 *			 = 0.845.. + P1(s)/Q1(s)
58 *	   That is, we use rational approximation to approximate
59 *			erf(1+s) - (c = (single)0.84506291151)
60 *	   Note that |P1/Q1|< 0.078 for x in [0.84375,1.25]
61 *	   where
62 *		P1(s) = degree 6 poly in s
63 *		Q1(s) = degree 6 poly in s
64 *
65 *      3. For x in [1.25,1/0.35(~2.857143)],
66 *         	erfc(x) = (1/x)*exp(-x*x-0.5625+R1/S1)
67 *         	erf(x)  = 1 - erfc(x)
68 *	   where
69 *		R1(z) = degree 7 poly in z, (z=1/x^2)
70 *		S1(z) = degree 8 poly in z
71 *
72 *      4. For x in [1/0.35,28]
73 *         	erfc(x) = (1/x)*exp(-x*x-0.5625+R2/S2) if x > 0
74 *			= 2.0 - (1/x)*exp(-x*x-0.5625+R2/S2) if -6<x<0
75 *			= 2.0 - tiny		(if x <= -6)
76 *         	erf(x)  = sign(x)*(1.0 - erfc(x)) if x < 6, else
77 *         	erf(x)  = sign(x)*(1.0 - tiny)
78 *	   where
79 *		R2(z) = degree 6 poly in z, (z=1/x^2)
80 *		S2(z) = degree 7 poly in z
81 *
82 *      Note1:
83 *	   To compute exp(-x*x-0.5625+R/S), let s be a single
84 *	   precision number and s := x; then
85 *		-x*x = -s*s + (s-x)*(s+x)
86 *	        exp(-x*x-0.5626+R/S) =
87 *			exp(-s*s-0.5625)*exp((s-x)*(s+x)+R/S);
88 *      Note2:
89 *	   Here 4 and 5 make use of the asymptotic series
90 *			  exp(-x*x)
91 *		erfc(x) ~ ---------- * ( 1 + Poly(1/x^2) )
92 *			  x*sqrt(pi)
93 *	   We use rational approximation to approximate
94 *      	g(s)=f(1/x^2) = log(erfc(x)*x) - x*x + 0.5625
95 *	   Here is the error bound for R1/S1 and R2/S2
96 *      	|R1/S1 - f(x)|  < 2**(-62.57)
97 *      	|R2/S2 - f(x)|  < 2**(-61.52)
98 *
99 *      5. For inf > x >= 28
100 *         	erf(x)  = sign(x) *(1 - tiny)  (raise inexact)
101 *         	erfc(x) = tiny*tiny (raise underflow) if x > 0
102 *			= 2 - tiny if x<0
103 *
104 *      7. Special case:
105 *         	erf(0)  = 0, erf(inf)  = 1, erf(-inf) = -1,
106 *         	erfc(0) = 1, erfc(inf) = 0, erfc(-inf) = 2,
107 *	   	erfc/erf(NaN) is NaN
108 */
109
110
111#include "math.h"
112#include "math_private.h"
113
114static const double
115tiny	    = 1e-300,
116half=  5.00000000000000000000e-01, /* 0x3FE00000, 0x00000000 */
117one =  1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
118two =  2.00000000000000000000e+00, /* 0x40000000, 0x00000000 */
119/* c = (float)0.84506291151 */
120erx =  8.45062911510467529297e-01, /* 0x3FEB0AC1, 0x60000000 */
121efx =  1.28379167095512586316e-01, /* 0x3FC06EBA, 0x8214DB69 */
122efx8=  1.02703333676410069053e+00, /* 0x3FF06EBA, 0x8214DB69 */
123/*
124 * Coefficients for approximation to erf on [0,0.84375]
125 */
126pp0  =  1.28379167095512558561e-01, /* 0x3FC06EBA, 0x8214DB68 */
127pp1  = -3.25042107247001499370e-01, /* 0xBFD4CD7D, 0x691CB913 */
128pp2  = -2.84817495755985104766e-02, /* 0xBF9D2A51, 0xDBD7194F */
129pp3  = -5.77027029648944159157e-03, /* 0xBF77A291, 0x236668E4 */
130pp4  = -2.37630166566501626084e-05, /* 0xBEF8EAD6, 0x120016AC */
131qq1  =  3.97917223959155352819e-01, /* 0x3FD97779, 0xCDDADC09 */
132qq2  =  6.50222499887672944485e-02, /* 0x3FB0A54C, 0x5536CEBA */
133qq3  =  5.08130628187576562776e-03, /* 0x3F74D022, 0xC4D36B0F */
134qq4  =  1.32494738004321644526e-04, /* 0x3F215DC9, 0x221C1A10 */
135qq5  = -3.96022827877536812320e-06, /* 0xBED09C43, 0x42A26120 */
136/*
137 * Coefficients for approximation to erf in [0.84375,1.25]
138 */
139pa0  = -2.36211856075265944077e-03, /* 0xBF6359B8, 0xBEF77538 */
140pa1  =  4.14856118683748331666e-01, /* 0x3FDA8D00, 0xAD92B34D */
141pa2  = -3.72207876035701323847e-01, /* 0xBFD7D240, 0xFBB8C3F1 */
142pa3  =  3.18346619901161753674e-01, /* 0x3FD45FCA, 0x805120E4 */
143pa4  = -1.10894694282396677476e-01, /* 0xBFBC6398, 0x3D3E28EC */
144pa5  =  3.54783043256182359371e-02, /* 0x3FA22A36, 0x599795EB */
145pa6  = -2.16637559486879084300e-03, /* 0xBF61BF38, 0x0A96073F */
146qa1  =  1.06420880400844228286e-01, /* 0x3FBB3E66, 0x18EEE323 */
147qa2  =  5.40397917702171048937e-01, /* 0x3FE14AF0, 0x92EB6F33 */
148qa3  =  7.18286544141962662868e-02, /* 0x3FB2635C, 0xD99FE9A7 */
149qa4  =  1.26171219808761642112e-01, /* 0x3FC02660, 0xE763351F */
150qa5  =  1.36370839120290507362e-02, /* 0x3F8BEDC2, 0x6B51DD1C */
151qa6  =  1.19844998467991074170e-02, /* 0x3F888B54, 0x5735151D */
152/*
153 * Coefficients for approximation to erfc in [1.25,1/0.35]
154 */
155ra0  = -9.86494403484714822705e-03, /* 0xBF843412, 0x600D6435 */
156ra1  = -6.93858572707181764372e-01, /* 0xBFE63416, 0xE4BA7360 */
157ra2  = -1.05586262253232909814e+01, /* 0xC0251E04, 0x41B0E726 */
158ra3  = -6.23753324503260060396e+01, /* 0xC04F300A, 0xE4CBA38D */
159ra4  = -1.62396669462573470355e+02, /* 0xC0644CB1, 0x84282266 */
160ra5  = -1.84605092906711035994e+02, /* 0xC067135C, 0xEBCCABB2 */
161ra6  = -8.12874355063065934246e+01, /* 0xC0545265, 0x57E4D2F2 */
162ra7  = -9.81432934416914548592e+00, /* 0xC023A0EF, 0xC69AC25C */
163sa1  =  1.96512716674392571292e+01, /* 0x4033A6B9, 0xBD707687 */
164sa2  =  1.37657754143519042600e+02, /* 0x4061350C, 0x526AE721 */
165sa3  =  4.34565877475229228821e+02, /* 0x407B290D, 0xD58A1A71 */
166sa4  =  6.45387271733267880336e+02, /* 0x40842B19, 0x21EC2868 */
167sa5  =  4.29008140027567833386e+02, /* 0x407AD021, 0x57700314 */
168sa6  =  1.08635005541779435134e+02, /* 0x405B28A3, 0xEE48AE2C */
169sa7  =  6.57024977031928170135e+00, /* 0x401A47EF, 0x8E484A93 */
170sa8  = -6.04244152148580987438e-02, /* 0xBFAEEFF2, 0xEE749A62 */
171/*
172 * Coefficients for approximation to erfc in [1/.35,28]
173 */
174rb0  = -9.86494292470009928597e-03, /* 0xBF843412, 0x39E86F4A */
175rb1  = -7.99283237680523006574e-01, /* 0xBFE993BA, 0x70C285DE */
176rb2  = -1.77579549177547519889e+01, /* 0xC031C209, 0x555F995A */
177rb3  = -1.60636384855821916062e+02, /* 0xC064145D, 0x43C5ED98 */
178rb4  = -6.37566443368389627722e+02, /* 0xC083EC88, 0x1375F228 */
179rb5  = -1.02509513161107724954e+03, /* 0xC0900461, 0x6A2E5992 */
180rb6  = -4.83519191608651397019e+02, /* 0xC07E384E, 0x9BDC383F */
181sb1  =  3.03380607434824582924e+01, /* 0x403E568B, 0x261D5190 */
182sb2  =  3.25792512996573918826e+02, /* 0x40745CAE, 0x221B9F0A */
183sb3  =  1.53672958608443695994e+03, /* 0x409802EB, 0x189D5118 */
184sb4  =  3.19985821950859553908e+03, /* 0x40A8FFB7, 0x688C246A */
185sb5  =  2.55305040643316442583e+03, /* 0x40A3F219, 0xCEDF3BE6 */
186sb6  =  4.74528541206955367215e+02, /* 0x407DA874, 0xE79FE763 */
187sb7  = -2.24409524465858183362e+01; /* 0xC03670E2, 0x42712D62 */
188
189double
190erf(double x)
191{
192	int32_t hx,ix,i;
193	double R,S,P,Q,s,y,z,r;
194	GET_HIGH_WORD(hx,x);
195	ix = hx&0x7fffffff;
196	if(ix>=0x7ff00000) {		/* erf(nan)=nan */
197	    i = ((u_int32_t)hx>>31)<<1;
198	    return (double)(1-i)+one/x;	/* erf(+-inf)=+-1 */
199	}
200
201	if(ix < 0x3feb0000) {		/* |x|<0.84375 */
202	    if(ix < 0x3e300000) { 	/* |x|<2**-28 */
203	        if (ix < 0x00800000)
204		    return (8*x+efx8*x)/8;	/* avoid spurious underflow */
205		return x + efx*x;
206	    }
207	    z = x*x;
208	    r = pp0+z*(pp1+z*(pp2+z*(pp3+z*pp4)));
209	    s = one+z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*qq5))));
210	    y = r/s;
211	    return x + x*y;
212	}
213	if(ix < 0x3ff40000) {		/* 0.84375 <= |x| < 1.25 */
214	    s = fabs(x)-one;
215	    P = pa0+s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6)))));
216	    Q = one+s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6)))));
217	    if(hx>=0) return erx + P/Q; else return -erx - P/Q;
218	}
219	if (ix >= 0x40180000) {		/* inf>|x|>=6 */
220	    if(hx>=0) return one-tiny; else return tiny-one;
221	}
222	x = fabs(x);
223 	s = one/(x*x);
224	if(ix< 0x4006DB6E) {	/* |x| < 1/0.35 */
225	    R=ra0+s*(ra1+s*(ra2+s*(ra3+s*(ra4+s*(ra5+s*(ra6+s*ra7))))));
226	    S=one+s*(sa1+s*(sa2+s*(sa3+s*(sa4+s*(sa5+s*(sa6+s*(sa7+
227		s*sa8)))))));
228	} else {	/* |x| >= 1/0.35 */
229	    R=rb0+s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*(rb5+s*rb6)))));
230	    S=one+s*(sb1+s*(sb2+s*(sb3+s*(sb4+s*(sb5+s*(sb6+s*sb7))))));
231	}
232	z  = x;
233	SET_LOW_WORD(z,0);
234	r  =  __ieee754_exp(-z*z-0.5625)*__ieee754_exp((z-x)*(z+x)+R/S);
235	if(hx>=0) return one-r/x; else return  r/x-one;
236}
237
238double
239erfc(double x)
240{
241	int32_t hx,ix;
242	double R,S,P,Q,s,y,z,r;
243	GET_HIGH_WORD(hx,x);
244	ix = hx&0x7fffffff;
245	if(ix>=0x7ff00000) {			/* erfc(nan)=nan */
246						/* erfc(+-inf)=0,2 */
247	    return (double)(((u_int32_t)hx>>31)<<1)+one/x;
248	}
249
250	if(ix < 0x3feb0000) {		/* |x|<0.84375 */
251	    if(ix < 0x3c700000)  	/* |x|<2**-56 */
252		return one-x;
253	    z = x*x;
254	    r = pp0+z*(pp1+z*(pp2+z*(pp3+z*pp4)));
255	    s = one+z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*qq5))));
256	    y = r/s;
257	    if(hx < 0x3fd00000) {  	/* x<1/4 */
258		return one-(x+x*y);
259	    } else {
260		r = x*y;
261		r += (x-half);
262	        return half - r ;
263	    }
264	}
265	if(ix < 0x3ff40000) {		/* 0.84375 <= |x| < 1.25 */
266	    s = fabs(x)-one;
267	    P = pa0+s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6)))));
268	    Q = one+s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6)))));
269	    if(hx>=0) {
270	        z  = one-erx; return z - P/Q;
271	    } else {
272		z = erx+P/Q; return one+z;
273	    }
274	}
275	if (ix < 0x403c0000) {		/* |x|<28 */
276	    x = fabs(x);
277 	    s = one/(x*x);
278	    if(ix< 0x4006DB6D) {	/* |x| < 1/.35 ~ 2.857143*/
279		R=ra0+s*(ra1+s*(ra2+s*(ra3+s*(ra4+s*(ra5+s*(ra6+s*ra7))))));
280		S=one+s*(sa1+s*(sa2+s*(sa3+s*(sa4+s*(sa5+s*(sa6+s*(sa7+
281		    s*sa8)))))));
282	    } else {			/* |x| >= 1/.35 ~ 2.857143 */
283		if(hx<0&&ix>=0x40180000) return two-tiny;/* x < -6 */
284		R=rb0+s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*(rb5+s*rb6)))));
285		S=one+s*(sb1+s*(sb2+s*(sb3+s*(sb4+s*(sb5+s*(sb6+s*sb7))))));
286	    }
287	    z  = x;
288	    SET_LOW_WORD(z,0);
289	    r  =  __ieee754_exp(-z*z-0.5625)*__ieee754_exp((z-x)*(z+x)+R/S);
290	    if(hx>0) return r/x; else return two-r/x;
291	} else {
292	    if(hx>0) return tiny*tiny; else return two-tiny;
293	}
294}
295