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 __FBSDID("$FreeBSD$");
18
19 #include "math.h"
20 #include "math_private.h"
21
22 static const float
23 tiny = 1e-30,
24 half= 5.0000000000e-01, /* 0x3F000000 */
25 one = 1.0000000000e+00, /* 0x3F800000 */
26 two = 2.0000000000e+00, /* 0x40000000 */
27 /*
28 * Coefficients for approximation to erf on [0,0.84375]
29 */
30 efx = 1.2837916613e-01, /* 0x3e0375d4 */
31 efx8= 1.0270333290e+00, /* 0x3f8375d4 */
32 /*
33 * Domain [0, 0.84375], range ~[-5.4446e-10,5.5197e-10]:
34 * |(erf(x) - x)/x - p(x)/q(x)| < 2**-31.
35 */
36 pp0 = 1.28379166e-01F, /* 0x1.06eba8p-3 */
37 pp1 = -3.36030394e-01F, /* -0x1.58185ap-2 */
38 pp2 = -1.86260219e-03F, /* -0x1.e8451ep-10 */
39 qq1 = 3.12324286e-01F, /* 0x1.3fd1f0p-2 */
40 qq2 = 2.16070302e-02F, /* 0x1.620274p-6 */
41 qq3 = -1.98859419e-03F, /* -0x1.04a626p-9 */
42 /*
43 * Domain [0.84375, 1.25], range ~[-1.953e-11,1.940e-11]:
44 * |(erf(x) - erx) - p(x)/q(x)| < 2**-36.
45 */
46 erx = 8.42697144e-01F, /* 0x1.af7600p-1. erf(1) rounded to 16 bits. */
47 pa0 = 3.64939137e-06F, /* 0x1.e9d022p-19 */
48 pa1 = 4.15109694e-01F, /* 0x1.a91284p-2 */
49 pa2 = -1.65179938e-01F, /* -0x1.5249dcp-3 */
50 pa3 = 1.10914491e-01F, /* 0x1.c64e46p-4 */
51 qa1 = 6.02074385e-01F, /* 0x1.344318p-1 */
52 qa2 = 5.35934687e-01F, /* 0x1.126608p-1 */
53 qa3 = 1.68576106e-01F, /* 0x1.593e6ep-3 */
54 qa4 = 5.62181212e-02F, /* 0x1.cc89f2p-5 */
55 /*
56 * Domain [1.25,1/0.35], range ~[-7.043e-10,7.457e-10]:
57 * |log(x*erfc(x)) + x**2 + 0.5625 - r(x)/s(x)| < 2**-30
58 */
59 ra0 = -9.87132732e-03F, /* -0x1.4376b2p-7 */
60 ra1 = -5.53605914e-01F, /* -0x1.1b723cp-1 */
61 ra2 = -2.17589188e+00F, /* -0x1.1683a0p+1 */
62 ra3 = -1.43268085e+00F, /* -0x1.6ec42cp+0 */
63 sa1 = 5.45995426e+00F, /* 0x1.5d6fe4p+2 */
64 sa2 = 6.69798088e+00F, /* 0x1.acabb8p+2 */
65 sa3 = 1.43113089e+00F, /* 0x1.6e5e98p+0 */
66 sa4 = -5.77397496e-02F, /* -0x1.d90108p-5 */
67 /*
68 * Domain [1/0.35, 11], range ~[-2.264e-13,2.336e-13]:
69 * |log(x*erfc(x)) + x**2 + 0.5625 - r(x)/s(x)| < 2**-42
70 */
71 rb0 = -9.86494310e-03F, /* -0x1.434124p-7 */
72 rb1 = -6.25171244e-01F, /* -0x1.401672p-1 */
73 rb2 = -6.16498327e+00F, /* -0x1.8a8f16p+2 */
74 rb3 = -1.66696873e+01F, /* -0x1.0ab70ap+4 */
75 rb4 = -9.53764343e+00F, /* -0x1.313460p+3 */
76 sb1 = 1.26884899e+01F, /* 0x1.96081cp+3 */
77 sb2 = 4.51839523e+01F, /* 0x1.6978bcp+5 */
78 sb3 = 4.72810211e+01F, /* 0x1.7a3f88p+5 */
79 sb4 = 8.93033314e+00F; /* 0x1.1dc54ap+3 */
80
81 float
erff(float x)82 erff(float x)
83 {
84 int32_t hx,ix,i;
85 float R,S,P,Q,s,y,z,r;
86 GET_FLOAT_WORD(hx,x);
87 ix = hx&0x7fffffff;
88 if(ix>=0x7f800000) { /* erf(nan)=nan */
89 i = ((u_int32_t)hx>>31)<<1;
90 return (float)(1-i)+one/x; /* erf(+-inf)=+-1 */
91 }
92
93 if(ix < 0x3f580000) { /* |x|<0.84375 */
94 if(ix < 0x38800000) { /* |x|<2**-14 */
95 if (ix < 0x04000000) /* |x|<0x1p-119 */
96 return (8*x+efx8*x)/8; /* avoid spurious underflow */
97 return x + efx*x;
98 }
99 z = x*x;
100 r = pp0+z*(pp1+z*pp2);
101 s = one+z*(qq1+z*(qq2+z*qq3));
102 y = r/s;
103 return x + x*y;
104 }
105 if(ix < 0x3fa00000) { /* 0.84375 <= |x| < 1.25 */
106 s = fabsf(x)-one;
107 P = pa0+s*(pa1+s*(pa2+s*pa3));
108 Q = one+s*(qa1+s*(qa2+s*(qa3+s*qa4)));
109 if(hx>=0) return erx + P/Q; else return -erx - P/Q;
110 }
111 if (ix >= 0x40800000) { /* inf>|x|>=4 */
112 if(hx>=0) return one-tiny; else return tiny-one;
113 }
114 x = fabsf(x);
115 s = one/(x*x);
116 if(ix< 0x4036DB6E) { /* |x| < 1/0.35 */
117 R=ra0+s*(ra1+s*(ra2+s*ra3));
118 S=one+s*(sa1+s*(sa2+s*(sa3+s*sa4)));
119 } else { /* |x| >= 1/0.35 */
120 R=rb0+s*(rb1+s*(rb2+s*(rb3+s*rb4)));
121 S=one+s*(sb1+s*(sb2+s*(sb3+s*sb4)));
122 }
123 SET_FLOAT_WORD(z,hx&0xffffe000);
124 r = expf(-z*z-0.5625F)*expf((z-x)*(z+x)+R/S);
125 if(hx>=0) return one-r/x; else return r/x-one;
126 }
127
128 float
erfcf(float x)129 erfcf(float x)
130 {
131 int32_t hx,ix;
132 float R,S,P,Q,s,y,z,r;
133 GET_FLOAT_WORD(hx,x);
134 ix = hx&0x7fffffff;
135 if(ix>=0x7f800000) { /* erfc(nan)=nan */
136 /* erfc(+-inf)=0,2 */
137 return (float)(((u_int32_t)hx>>31)<<1)+one/x;
138 }
139
140 if(ix < 0x3f580000) { /* |x|<0.84375 */
141 if(ix < 0x33800000) /* |x|<2**-24 */
142 return one-x;
143 z = x*x;
144 r = pp0+z*(pp1+z*pp2);
145 s = one+z*(qq1+z*(qq2+z*qq3));
146 y = r/s;
147 if(hx < 0x3e800000) { /* x<1/4 */
148 return one-(x+x*y);
149 } else {
150 r = x*y;
151 r += (x-half);
152 return half - r ;
153 }
154 }
155 if(ix < 0x3fa00000) { /* 0.84375 <= |x| < 1.25 */
156 s = fabsf(x)-one;
157 P = pa0+s*(pa1+s*(pa2+s*pa3));
158 Q = one+s*(qa1+s*(qa2+s*(qa3+s*qa4)));
159 if(hx>=0) {
160 z = one-erx; return z - P/Q;
161 } else {
162 z = erx+P/Q; return one+z;
163 }
164 }
165 if (ix < 0x41300000) { /* |x|<11 */
166 x = fabsf(x);
167 s = one/(x*x);
168 if(ix< 0x4036DB6D) { /* |x| < 1/.35 ~ 2.857143*/
169 R=ra0+s*(ra1+s*(ra2+s*ra3));
170 S=one+s*(sa1+s*(sa2+s*(sa3+s*sa4)));
171 } else { /* |x| >= 1/.35 ~ 2.857143 */
172 if(hx<0&&ix>=0x40a00000) return two-tiny;/* x < -5 */
173 R=rb0+s*(rb1+s*(rb2+s*(rb3+s*rb4)));
174 S=one+s*(sb1+s*(sb2+s*(sb3+s*sb4)));
175 }
176 SET_FLOAT_WORD(z,hx&0xffffe000);
177 r = expf(-z*z-0.5625F)*expf((z-x)*(z+x)+R/S);
178 if(hx>0) return r/x; else return two-r/x;
179 } else {
180 if(hx>0) return tiny*tiny; else return two-tiny;
181 }
182 }
183