xref: /freebsd/lib/msun/src/s_erff.c (revision 22d7dd834bc5cd189810e414701e3ad1e98102e4)
1 /* s_erff.c -- float version of s_erf.c.
2  * Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com.
3  */
4 
5 /*
6  * ====================================================
7  * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
8  *
9  * Developed at SunPro, a Sun Microsystems, Inc. business.
10  * Permission to use, copy, modify, and distribute this
11  * software is freely granted, provided that this notice
12  * is preserved.
13  * ====================================================
14  */
15 
16 #include <sys/cdefs.h>
17 #include "math.h"
18 #include "math_private.h"
19 
20 /* XXX Prevent compilers from erroneously constant folding: */
21 static const volatile float tiny = 1e-30;
22 
23 static const float
24 half= 0.5,
25 one = 1,
26 two = 2,
27 erx = 8.42697144e-01,			/* 0x3f57bb00 */
28 /*
29  * In the domain [0, 2**-14], only the first term in the power series
30  * expansion of erf(x) is used.  The magnitude of the first neglected
31  * terms is less than 2**-42.
32  */
33 efx = 1.28379166e-01, /* 0x3e0375d4 */
34 efx8= 1.02703333e+00, /* 0x3f8375d4 */
35 /*
36  * Domain [0, 0.84375], range ~[-5.4419e-10, 5.5179e-10]:
37  * |(erf(x) - x)/x - pp(x)/qq(x)| < 2**-31
38  */
39 pp0  =  1.28379166e-01, /* 0x3e0375d4 */
40 pp1  = -3.36030394e-01, /* 0xbeac0c2d */
41 pp2  = -1.86261395e-03, /* 0xbaf422f4 */
42 qq1  =  3.12324315e-01, /* 0x3e9fe8f9 */
43 qq2  =  2.16070414e-02, /* 0x3cb10140 */
44 qq3  = -1.98859372e-03, /* 0xbb025311 */
45 /*
46  * Domain [0.84375, 1.25], range ~[-1.023e-9, 1.023e-9]:
47  * |(erf(x) - erx) - pa(x)/qa(x)| < 2**-31
48  */
49 pa0  =  3.65041046e-06, /* 0x3674f993 */
50 pa1  =  4.15109307e-01, /* 0x3ed48935 */
51 pa2  = -2.09395722e-01, /* 0xbe566bd5 */
52 pa3  =  8.67677554e-02, /* 0x3db1b34b */
53 qa1  =  4.95560974e-01, /* 0x3efdba2b */
54 qa2  =  3.71248513e-01, /* 0x3ebe1449 */
55 qa3  =  3.92478965e-02, /* 0x3d20c267 */
56 /*
57  * Domain [1.25,1/0.35], range ~[-4.821e-9, 4.927e-9]:
58  * |log(x*erfc(x)) + x**2 + 0.5625 - ra(x)/sa(x)| < 2**-28
59  */
60 ra0  = -9.88156721e-03, /* 0xbc21e64c */
61 ra1  = -5.43658376e-01, /* 0xbf0b2d32 */
62 ra2  = -1.66828310e+00, /* 0xbfd58a4d */
63 ra3  = -6.91554189e-01, /* 0xbf3109b2 */
64 sa1  =  4.48581553e+00, /* 0x408f8bcd */
65 sa2  =  4.10799170e+00, /* 0x408374ab */
66 sa3  =  5.53855181e-01, /* 0x3f0dc974 */
67 /*
68  * Domain [2.85715, 11], range ~[-1.484e-9, 1.505e-9]:
69  * |log(x*erfc(x)) + x**2 + 0.5625 - rb(x)/sb(x)| < 2**-30
70  */
71 rb0  = -9.86496918e-03, /* 0xbc21a0ae */
72 rb1  = -5.48049808e-01, /* 0xbf0c4cfe */
73 rb2  = -1.84115684e+00, /* 0xbfebab07 */
74 sb1  =  4.87132740e+00, /* 0x409be1ea */
75 sb2  =  3.04982710e+00, /* 0x4043305e */
76 sb3  = -7.61900663e-01; /* 0xbf430bec */
77 
78 float
79 erff(float x)
80 {
81 	int32_t hx,ix,i;
82 	float R,S,P,Q,s,y,z,r;
83 	GET_FLOAT_WORD(hx,x);
84 	ix = hx&0x7fffffff;
85 	if(ix>=0x7f800000) {		/* erff(nan)=nan */
86 	    i = ((u_int32_t)hx>>31)<<1;
87 	    return (float)(1-i)+one/x;	/* erff(+-inf)=+-1 */
88 	}
89 
90 	if(ix < 0x3f580000) {		/* |x|<0.84375 */
91 	    if(ix < 0x38800000) { 	/* |x|<2**-14 */
92 	        if (ix < 0x04000000)	/* |x|<0x1p-119 */
93 		    return (8*x+efx8*x)/8;	/* avoid spurious underflow */
94 		return x + efx*x;
95 	    }
96 	    z = x*x;
97 	    r = pp0+z*(pp1+z*pp2);
98 	    s = one+z*(qq1+z*(qq2+z*qq3));
99 	    y = r/s;
100 	    return x + x*y;
101 	}
102 	if(ix < 0x3fa00000) {		/* 0.84375 <= |x| < 1.25 */
103 	    s = fabsf(x)-one;
104 	    P = pa0+s*(pa1+s*(pa2+s*pa3));
105 	    Q = one+s*(qa1+s*(qa2+s*qa3));
106 	    if(hx>=0) return erx + P/Q; else return -erx - P/Q;
107 	}
108 	if (ix >= 0x40800000) {		/* inf>|x|>=4 */
109 	    if(hx>=0) return one-tiny; else return tiny-one;
110 	}
111 	x = fabsf(x);
112  	s = one/(x*x);
113 	if(ix< 0x4036db8c) {	/* |x| < 2.85715 ~ 1/0.35 */
114 	    R=ra0+s*(ra1+s*(ra2+s*ra3));
115 	    S=one+s*(sa1+s*(sa2+s*sa3));
116 	} else {	/* |x| >= 2.85715 ~ 1/0.35 */
117 	    R=rb0+s*(rb1+s*rb2);
118 	    S=one+s*(sb1+s*(sb2+s*sb3));
119 	}
120 	SET_FLOAT_WORD(z,hx&0xffffe000);
121 	r  = expf(-z*z-0.5625F)*expf((z-x)*(z+x)+R/S);
122 	if(hx>=0) return one-r/x; else return  r/x-one;
123 }
124 
125 float
126 erfcf(float x)
127 {
128 	int32_t hx,ix;
129 	float R,S,P,Q,s,y,z,r;
130 	GET_FLOAT_WORD(hx,x);
131 	ix = hx&0x7fffffff;
132 	if(ix>=0x7f800000) {			/* erfcf(nan)=nan */
133 						/* erfcf(+-inf)=0,2 */
134 	    return (float)(((u_int32_t)hx>>31)<<1)+one/x;
135 	}
136 
137 	if(ix < 0x3f580000) {		/* |x|<0.84375 */
138 	    if(ix < 0x33800000)  	/* |x|<2**-24 */
139 		return one-x;
140 	    z = x*x;
141 	    r = pp0+z*(pp1+z*pp2);
142 	    s = one+z*(qq1+z*(qq2+z*qq3));
143 	    y = r/s;
144 	    if(hx < 0x3e800000) {  	/* x<1/4 */
145 		return one-(x+x*y);
146 	    } else {
147 		r = x*y;
148 		r += (x-half);
149 	        return half - r ;
150 	    }
151 	}
152 	if(ix < 0x3fa00000) {		/* 0.84375 <= |x| < 1.25 */
153 	    s = fabsf(x)-one;
154 	    P = pa0+s*(pa1+s*(pa2+s*pa3));
155 	    Q = one+s*(qa1+s*(qa2+s*qa3));
156 	    if(hx>=0) {
157 	        z  = one-erx; return z - P/Q;
158 	    } else {
159 		z = erx+P/Q; return one+z;
160 	    }
161 	}
162 	if (ix < 0x41300000) {		/* |x|<11 */
163 	    x = fabsf(x);
164  	    s = one/(x*x);
165 	    if(ix< 0x4036db8c) {	/* |x| < 2.85715 ~ 1/.35 */
166 		R=ra0+s*(ra1+s*(ra2+s*ra3));
167 		S=one+s*(sa1+s*(sa2+s*sa3));
168 	    } else {			/* |x| >= 2.85715 ~ 1/.35 */
169 		if(hx<0&&ix>=0x40a00000) return two-tiny;/* x < -5 */
170 		R=rb0+s*(rb1+s*rb2);
171 		S=one+s*(sb1+s*(sb2+s*sb3));
172 	    }
173 	    SET_FLOAT_WORD(z,hx&0xffffe000);
174 	    r  = expf(-z*z-0.5625F)*expf((z-x)*(z+x)+R/S);
175 	    if(hx>0) return r/x; else return two-r/x;
176 	} else {
177 	    if(hx>0) return tiny*tiny; else return two-tiny;
178 	}
179 }
180