1 /* origin: FreeBSD /usr/src/lib/msun/src/s_erff.c */
2 /*
3 * Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com.
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 "libm.h"
17
18 static const float
19 erx = 8.4506291151e-01, /* 0x3f58560b */
20 /*
21 * Coefficients for approximation to erf on [0,0.84375]
22 */
23 efx8 = 1.0270333290e+00, /* 0x3f8375d4 */
24 pp0 = 1.2837916613e-01, /* 0x3e0375d4 */
25 pp1 = -3.2504209876e-01, /* 0xbea66beb */
26 pp2 = -2.8481749818e-02, /* 0xbce9528f */
27 pp3 = -5.7702702470e-03, /* 0xbbbd1489 */
28 pp4 = -2.3763017452e-05, /* 0xb7c756b1 */
29 qq1 = 3.9791721106e-01, /* 0x3ecbbbce */
30 qq2 = 6.5022252500e-02, /* 0x3d852a63 */
31 qq3 = 5.0813062117e-03, /* 0x3ba68116 */
32 qq4 = 1.3249473704e-04, /* 0x390aee49 */
33 qq5 = -3.9602282413e-06, /* 0xb684e21a */
34 /*
35 * Coefficients for approximation to erf in [0.84375,1.25]
36 */
37 pa0 = -2.3621185683e-03, /* 0xbb1acdc6 */
38 pa1 = 4.1485610604e-01, /* 0x3ed46805 */
39 pa2 = -3.7220788002e-01, /* 0xbebe9208 */
40 pa3 = 3.1834661961e-01, /* 0x3ea2fe54 */
41 pa4 = -1.1089469492e-01, /* 0xbde31cc2 */
42 pa5 = 3.5478305072e-02, /* 0x3d1151b3 */
43 pa6 = -2.1663755178e-03, /* 0xbb0df9c0 */
44 qa1 = 1.0642088205e-01, /* 0x3dd9f331 */
45 qa2 = 5.4039794207e-01, /* 0x3f0a5785 */
46 qa3 = 7.1828655899e-02, /* 0x3d931ae7 */
47 qa4 = 1.2617121637e-01, /* 0x3e013307 */
48 qa5 = 1.3637083583e-02, /* 0x3c5f6e13 */
49 qa6 = 1.1984500103e-02, /* 0x3c445aa3 */
50 /*
51 * Coefficients for approximation to erfc in [1.25,1/0.35]
52 */
53 ra0 = -9.8649440333e-03, /* 0xbc21a093 */
54 ra1 = -6.9385856390e-01, /* 0xbf31a0b7 */
55 ra2 = -1.0558626175e+01, /* 0xc128f022 */
56 ra3 = -6.2375331879e+01, /* 0xc2798057 */
57 ra4 = -1.6239666748e+02, /* 0xc322658c */
58 ra5 = -1.8460508728e+02, /* 0xc3389ae7 */
59 ra6 = -8.1287437439e+01, /* 0xc2a2932b */
60 ra7 = -9.8143291473e+00, /* 0xc11d077e */
61 sa1 = 1.9651271820e+01, /* 0x419d35ce */
62 sa2 = 1.3765776062e+02, /* 0x4309a863 */
63 sa3 = 4.3456588745e+02, /* 0x43d9486f */
64 sa4 = 6.4538726807e+02, /* 0x442158c9 */
65 sa5 = 4.2900814819e+02, /* 0x43d6810b */
66 sa6 = 1.0863500214e+02, /* 0x42d9451f */
67 sa7 = 6.5702495575e+00, /* 0x40d23f7c */
68 sa8 = -6.0424413532e-02, /* 0xbd777f97 */
69 /*
70 * Coefficients for approximation to erfc in [1/.35,28]
71 */
72 rb0 = -9.8649431020e-03, /* 0xbc21a092 */
73 rb1 = -7.9928326607e-01, /* 0xbf4c9dd4 */
74 rb2 = -1.7757955551e+01, /* 0xc18e104b */
75 rb3 = -1.6063638306e+02, /* 0xc320a2ea */
76 rb4 = -6.3756646729e+02, /* 0xc41f6441 */
77 rb5 = -1.0250950928e+03, /* 0xc480230b */
78 rb6 = -4.8351919556e+02, /* 0xc3f1c275 */
79 sb1 = 3.0338060379e+01, /* 0x41f2b459 */
80 sb2 = 3.2579251099e+02, /* 0x43a2e571 */
81 sb3 = 1.5367296143e+03, /* 0x44c01759 */
82 sb4 = 3.1998581543e+03, /* 0x4547fdbb */
83 sb5 = 2.5530502930e+03, /* 0x451f90ce */
84 sb6 = 4.7452853394e+02, /* 0x43ed43a7 */
85 sb7 = -2.2440952301e+01; /* 0xc1b38712 */
86
erfc1(float x)87 static float erfc1(float x)
88 {
89 float_t s,P,Q;
90
91 s = fabsf(x) - 1;
92 P = pa0+s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6)))));
93 Q = 1+s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6)))));
94 return 1 - erx - P/Q;
95 }
96
erfc2(uint32_t ix,float x)97 static float erfc2(uint32_t ix, float x)
98 {
99 float_t s,R,S;
100 float z;
101
102 if (ix < 0x3fa00000) /* |x| < 1.25 */
103 return erfc1(x);
104
105 x = fabsf(x);
106 s = 1/(x*x);
107 if (ix < 0x4036db6d) { /* |x| < 1/0.35 */
108 R = ra0+s*(ra1+s*(ra2+s*(ra3+s*(ra4+s*(
109 ra5+s*(ra6+s*ra7))))));
110 S = 1.0f+s*(sa1+s*(sa2+s*(sa3+s*(sa4+s*(
111 sa5+s*(sa6+s*(sa7+s*sa8)))))));
112 } else { /* |x| >= 1/0.35 */
113 R = rb0+s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*(
114 rb5+s*rb6)))));
115 S = 1.0f+s*(sb1+s*(sb2+s*(sb3+s*(sb4+s*(
116 sb5+s*(sb6+s*sb7))))));
117 }
118 GET_FLOAT_WORD(ix, x);
119 SET_FLOAT_WORD(z, ix&0xffffe000);
120 return expf(-z*z - 0.5625f) * expf((z-x)*(z+x) + R/S)/x;
121 }
122
erff(float x)123 float erff(float x)
124 {
125 float r,s,z,y;
126 uint32_t ix;
127 int sign;
128
129 GET_FLOAT_WORD(ix, x);
130 sign = ix>>31;
131 ix &= 0x7fffffff;
132 if (ix >= 0x7f800000) {
133 /* erf(nan)=nan, erf(+-inf)=+-1 */
134 return 1-2*sign + 1/x;
135 }
136 if (ix < 0x3f580000) { /* |x| < 0.84375 */
137 if (ix < 0x31800000) { /* |x| < 2**-28 */
138 /*avoid underflow */
139 return 0.125f*(8*x + efx8*x);
140 }
141 z = x*x;
142 r = pp0+z*(pp1+z*(pp2+z*(pp3+z*pp4)));
143 s = 1+z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*qq5))));
144 y = r/s;
145 return x + x*y;
146 }
147 if (ix < 0x40c00000) /* |x| < 6 */
148 y = 1 - erfc2(ix,x);
149 else
150 y = 1 - 0x1p-120f;
151 return sign ? -y : y;
152 }
153
erfcf(float x)154 float erfcf(float x)
155 {
156 float r,s,z,y;
157 uint32_t ix;
158 int sign;
159
160 GET_FLOAT_WORD(ix, x);
161 sign = ix>>31;
162 ix &= 0x7fffffff;
163 if (ix >= 0x7f800000) {
164 /* erfc(nan)=nan, erfc(+-inf)=0,2 */
165 return 2*sign + 1/x;
166 }
167
168 if (ix < 0x3f580000) { /* |x| < 0.84375 */
169 if (ix < 0x23800000) /* |x| < 2**-56 */
170 return 1.0f - x;
171 z = x*x;
172 r = pp0+z*(pp1+z*(pp2+z*(pp3+z*pp4)));
173 s = 1.0f+z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*qq5))));
174 y = r/s;
175 if (sign || ix < 0x3e800000) /* x < 1/4 */
176 return 1.0f - (x+x*y);
177 return 0.5f - (x - 0.5f + x*y);
178 }
179 if (ix < 0x41e00000) { /* |x| < 28 */
180 return sign ? 2 - erfc2(ix,x) : erfc2(ix,x);
181 }
182 return sign ? 2 - 0x1p-120f : 0x1p-120f*0x1p-120f;
183 }
184