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1 /* origin: FreeBSD /usr/src/lib/msun/src/s_erff.c */ | 1 /* origin: FreeBSD /usr/src/lib/msun/src/s_erff.c */ |
2 /* | 2 /* |
3 * Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com. | 3 * Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com. |
4 */ | 4 */ |
5 /* | 5 /* |
6 * ==================================================== | 6 * ==================================================== |
7 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. | 7 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. |
8 * | 8 * |
9 * Developed at SunPro, a Sun Microsystems, Inc. business. | 9 * Developed at SunPro, a Sun Microsystems, Inc. business. |
10 * Permission to use, copy, modify, and distribute this | 10 * Permission to use, copy, modify, and distribute this |
11 * software is freely granted, provided that this notice | 11 * software is freely granted, provided that this notice |
12 * is preserved. | 12 * is preserved. |
13 * ==================================================== | 13 * ==================================================== |
14 */ | 14 */ |
15 | 15 |
16 #include "libm.h" | 16 #include "libm.h" |
17 | 17 |
18 static const float | 18 static const float erx = 8.4506291151e-01, /* 0x3f58560b */ |
19 erx = 8.4506291151e-01, /* 0x3f58560b */ | 19 /* |
20 /* | 20 * Coefficients for approximation to
erf on [0,0.84375] |
21 * Coefficients for approximation to erf on [0,0.84375] | 21 */ |
22 */ | 22 efx8 = 1.0270333290e+00, /* 0x3f8375d4 */ |
23 efx8 = 1.0270333290e+00, /* 0x3f8375d4 */ | 23 pp0 = 1.2837916613e-01, /* 0x3e0375d4 */ |
24 pp0 = 1.2837916613e-01, /* 0x3e0375d4 */ | 24 pp1 = -3.2504209876e-01, /* 0xbea66beb */ |
25 pp1 = -3.2504209876e-01, /* 0xbea66beb */ | 25 pp2 = -2.8481749818e-02, /* 0xbce9528f */ |
26 pp2 = -2.8481749818e-02, /* 0xbce9528f */ | 26 pp3 = -5.7702702470e-03, /* 0xbbbd1489 */ |
27 pp3 = -5.7702702470e-03, /* 0xbbbd1489 */ | 27 pp4 = -2.3763017452e-05, /* 0xb7c756b1 */ |
28 pp4 = -2.3763017452e-05, /* 0xb7c756b1 */ | 28 qq1 = 3.9791721106e-01, /* 0x3ecbbbce */ |
29 qq1 = 3.9791721106e-01, /* 0x3ecbbbce */ | 29 qq2 = 6.5022252500e-02, /* 0x3d852a63 */ |
30 qq2 = 6.5022252500e-02, /* 0x3d852a63 */ | 30 qq3 = 5.0813062117e-03, /* 0x3ba68116 */ |
31 qq3 = 5.0813062117e-03, /* 0x3ba68116 */ | 31 qq4 = 1.3249473704e-04, /* 0x390aee49 */ |
32 qq4 = 1.3249473704e-04, /* 0x390aee49 */ | 32 qq5 = -3.9602282413e-06, /* 0xb684e21a */ |
33 qq5 = -3.9602282413e-06, /* 0xb684e21a */ | 33 /* |
34 /* | 34 * Coefficients for approximation to
erf in [0.84375,1.25] |
35 * Coefficients for approximation to erf in [0.84375,1.25] | 35 */ |
36 */ | 36 pa0 = -2.3621185683e-03, /* 0xbb1acdc6 */ |
37 pa0 = -2.3621185683e-03, /* 0xbb1acdc6 */ | 37 pa1 = 4.1485610604e-01, /* 0x3ed46805 */ |
38 pa1 = 4.1485610604e-01, /* 0x3ed46805 */ | 38 pa2 = -3.7220788002e-01, /* 0xbebe9208 */ |
39 pa2 = -3.7220788002e-01, /* 0xbebe9208 */ | 39 pa3 = 3.1834661961e-01, /* 0x3ea2fe54 */ |
40 pa3 = 3.1834661961e-01, /* 0x3ea2fe54 */ | 40 pa4 = -1.1089469492e-01, /* 0xbde31cc2 */ |
41 pa4 = -1.1089469492e-01, /* 0xbde31cc2 */ | 41 pa5 = 3.5478305072e-02, /* 0x3d1151b3 */ |
42 pa5 = 3.5478305072e-02, /* 0x3d1151b3 */ | 42 pa6 = -2.1663755178e-03, /* 0xbb0df9c0 */ |
43 pa6 = -2.1663755178e-03, /* 0xbb0df9c0 */ | 43 qa1 = 1.0642088205e-01, /* 0x3dd9f331 */ |
44 qa1 = 1.0642088205e-01, /* 0x3dd9f331 */ | 44 qa2 = 5.4039794207e-01, /* 0x3f0a5785 */ |
45 qa2 = 5.4039794207e-01, /* 0x3f0a5785 */ | 45 qa3 = 7.1828655899e-02, /* 0x3d931ae7 */ |
46 qa3 = 7.1828655899e-02, /* 0x3d931ae7 */ | 46 qa4 = 1.2617121637e-01, /* 0x3e013307 */ |
47 qa4 = 1.2617121637e-01, /* 0x3e013307 */ | 47 qa5 = 1.3637083583e-02, /* 0x3c5f6e13 */ |
48 qa5 = 1.3637083583e-02, /* 0x3c5f6e13 */ | 48 qa6 = 1.1984500103e-02, /* 0x3c445aa3 */ |
49 qa6 = 1.1984500103e-02, /* 0x3c445aa3 */ | 49 /* |
50 /* | 50 * Coefficients for approximation to
erfc in [1.25,1/0.35] |
51 * Coefficients for approximation to erfc in [1.25,1/0.35] | 51 */ |
52 */ | 52 ra0 = -9.8649440333e-03, /* 0xbc21a093 */ |
53 ra0 = -9.8649440333e-03, /* 0xbc21a093 */ | 53 ra1 = -6.9385856390e-01, /* 0xbf31a0b7 */ |
54 ra1 = -6.9385856390e-01, /* 0xbf31a0b7 */ | 54 ra2 = -1.0558626175e+01, /* 0xc128f022 */ |
55 ra2 = -1.0558626175e+01, /* 0xc128f022 */ | 55 ra3 = -6.2375331879e+01, /* 0xc2798057 */ |
56 ra3 = -6.2375331879e+01, /* 0xc2798057 */ | 56 ra4 = -1.6239666748e+02, /* 0xc322658c */ |
57 ra4 = -1.6239666748e+02, /* 0xc322658c */ | 57 ra5 = -1.8460508728e+02, /* 0xc3389ae7 */ |
58 ra5 = -1.8460508728e+02, /* 0xc3389ae7 */ | 58 ra6 = -8.1287437439e+01, /* 0xc2a2932b */ |
59 ra6 = -8.1287437439e+01, /* 0xc2a2932b */ | 59 ra7 = -9.8143291473e+00, /* 0xc11d077e */ |
60 ra7 = -9.8143291473e+00, /* 0xc11d077e */ | 60 sa1 = 1.9651271820e+01, /* 0x419d35ce */ |
61 sa1 = 1.9651271820e+01, /* 0x419d35ce */ | 61 sa2 = 1.3765776062e+02, /* 0x4309a863 */ |
62 sa2 = 1.3765776062e+02, /* 0x4309a863 */ | 62 sa3 = 4.3456588745e+02, /* 0x43d9486f */ |
63 sa3 = 4.3456588745e+02, /* 0x43d9486f */ | 63 sa4 = 6.4538726807e+02, /* 0x442158c9 */ |
64 sa4 = 6.4538726807e+02, /* 0x442158c9 */ | 64 sa5 = 4.2900814819e+02, /* 0x43d6810b */ |
65 sa5 = 4.2900814819e+02, /* 0x43d6810b */ | 65 sa6 = 1.0863500214e+02, /* 0x42d9451f */ |
66 sa6 = 1.0863500214e+02, /* 0x42d9451f */ | 66 sa7 = 6.5702495575e+00, /* 0x40d23f7c */ |
67 sa7 = 6.5702495575e+00, /* 0x40d23f7c */ | 67 sa8 = -6.0424413532e-02, /* 0xbd777f97 */ |
68 sa8 = -6.0424413532e-02, /* 0xbd777f97 */ | 68 /* |
69 /* | 69 * Coefficients for approximation to
erfc in [1/.35,28] |
70 * Coefficients for approximation to erfc in [1/.35,28] | 70 */ |
71 */ | 71 rb0 = -9.8649431020e-03, /* 0xbc21a092 */ |
72 rb0 = -9.8649431020e-03, /* 0xbc21a092 */ | 72 rb1 = -7.9928326607e-01, /* 0xbf4c9dd4 */ |
73 rb1 = -7.9928326607e-01, /* 0xbf4c9dd4 */ | 73 rb2 = -1.7757955551e+01, /* 0xc18e104b */ |
74 rb2 = -1.7757955551e+01, /* 0xc18e104b */ | 74 rb3 = -1.6063638306e+02, /* 0xc320a2ea */ |
75 rb3 = -1.6063638306e+02, /* 0xc320a2ea */ | 75 rb4 = -6.3756646729e+02, /* 0xc41f6441 */ |
76 rb4 = -6.3756646729e+02, /* 0xc41f6441 */ | 76 rb5 = -1.0250950928e+03, /* 0xc480230b */ |
77 rb5 = -1.0250950928e+03, /* 0xc480230b */ | 77 rb6 = -4.8351919556e+02, /* 0xc3f1c275 */ |
78 rb6 = -4.8351919556e+02, /* 0xc3f1c275 */ | 78 sb1 = 3.0338060379e+01, /* 0x41f2b459 */ |
79 sb1 = 3.0338060379e+01, /* 0x41f2b459 */ | 79 sb2 = 3.2579251099e+02, /* 0x43a2e571 */ |
80 sb2 = 3.2579251099e+02, /* 0x43a2e571 */ | 80 sb3 = 1.5367296143e+03, /* 0x44c01759 */ |
81 sb3 = 1.5367296143e+03, /* 0x44c01759 */ | 81 sb4 = 3.1998581543e+03, /* 0x4547fdbb */ |
82 sb4 = 3.1998581543e+03, /* 0x4547fdbb */ | 82 sb5 = 2.5530502930e+03, /* 0x451f90ce */ |
83 sb5 = 2.5530502930e+03, /* 0x451f90ce */ | 83 sb6 = 4.7452853394e+02, /* 0x43ed43a7 */ |
84 sb6 = 4.7452853394e+02, /* 0x43ed43a7 */ | 84 sb7 = -2.2440952301e+01; /* 0xc1b38712 */ |
85 sb7 = -2.2440952301e+01; /* 0xc1b38712 */ | |
86 | 85 |
87 static float erfc1(float x) | 86 static float erfc1(float x) { |
88 { | 87 float_t s, P, Q; |
89 » float_t s,P,Q; | |
90 | 88 |
91 » s = fabsf(x) - 1; | 89 s = fabsf(x) - 1; |
92 » P = pa0+s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6))))); | 90 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))))); | 91 Q = 1 + s * (qa1 + s * (qa2 + s * (qa3 + s * (qa4 + s * (qa5 + s * qa6))))); |
94 » return 1 - erx - P/Q; | 92 return 1 - erx - P / Q; |
95 } | 93 } |
96 | 94 |
97 static float erfc2(uint32_t ix, float x) | 95 static float erfc2(uint32_t ix, float x) { |
98 { | 96 float_t s, R, S; |
99 » float_t s,R,S; | 97 float z; |
100 » float z; | |
101 | 98 |
102 » if (ix < 0x3fa00000) /* |x| < 1.25 */ | 99 if (ix < 0x3fa00000) /* |x| < 1.25 */ |
103 » » return erfc1(x); | 100 return erfc1(x); |
104 | 101 |
105 » x = fabsf(x); | 102 x = fabsf(x); |
106 » s = 1/(x*x); | 103 s = 1 / (x * x); |
107 » if (ix < 0x4036db6d) { /* |x| < 1/0.35 */ | 104 if (ix < 0x4036db6d) { /* |x| < 1/0.35 */ |
108 » » R = ra0+s*(ra1+s*(ra2+s*(ra3+s*(ra4+s*( | 105 R = ra0 + |
109 » » ra5+s*(ra6+s*ra7)))))); | 106 s * (ra1 + |
110 » » S = 1.0f+s*(sa1+s*(sa2+s*(sa3+s*(sa4+s*( | 107 s * (ra2 + |
111 » » sa5+s*(sa6+s*(sa7+s*sa8))))))); | 108 s * (ra3 + s * (ra4 + s * (ra5 + s * (ra6 + s * ra7)))))); |
112 » } else { /* |x| >= 1/0.35 */ | 109 S = 1.0f + |
113 » » R = rb0+s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*( | 110 s * (sa1 + |
114 » » rb5+s*rb6))))); | 111 s * (sa2 + |
115 » » S = 1.0f+s*(sb1+s*(sb2+s*(sb3+s*(sb4+s*( | 112 s * (sa3 + |
116 » » sb5+s*(sb6+s*sb7)))))); | 113 s * (sa4 + |
117 » } | 114 s * (sa5 + s * (sa6 + s * (sa7 + s * sa8))))))); |
118 » GET_FLOAT_WORD(ix, x); | 115 } else { /* |x| >= 1/0.35 */ |
119 » SET_FLOAT_WORD(z, ix&0xffffe000); | 116 R = rb0 + |
120 » return expf(-z*z - 0.5625f) * expf((z-x)*(z+x) + R/S)/x; | 117 s * (rb1 + s * (rb2 + s * (rb3 + s * (rb4 + s * (rb5 + s * rb6))))); |
| 118 S = 1.0f + |
| 119 s * (sb1 + |
| 120 s * (sb2 + |
| 121 s * (sb3 + s * (sb4 + s * (sb5 + s * (sb6 + s * sb7)))))); |
| 122 } |
| 123 GET_FLOAT_WORD(ix, x); |
| 124 SET_FLOAT_WORD(z, ix & 0xffffe000); |
| 125 return expf(-z * z - 0.5625f) * expf((z - x) * (z + x) + R / S) / x; |
121 } | 126 } |
122 | 127 |
123 float erff(float x) | 128 float erff(float x) { |
124 { | 129 float r, s, z, y; |
125 » float r,s,z,y; | 130 uint32_t ix; |
126 » uint32_t ix; | 131 int sign; |
127 » int sign; | |
128 | 132 |
129 » GET_FLOAT_WORD(ix, x); | 133 GET_FLOAT_WORD(ix, x); |
130 » sign = ix>>31; | 134 sign = ix >> 31; |
131 » ix &= 0x7fffffff; | 135 ix &= 0x7fffffff; |
132 » if (ix >= 0x7f800000) { | 136 if (ix >= 0x7f800000) { |
133 » » /* erf(nan)=nan, erf(+-inf)=+-1 */ | 137 /* erf(nan)=nan, erf(+-inf)=+-1 */ |
134 » » return 1-2*sign + 1/x; | 138 return 1 - 2 * sign + 1 / x; |
135 » } | 139 } |
136 » if (ix < 0x3f580000) { /* |x| < 0.84375 */ | 140 if (ix < 0x3f580000) { /* |x| < 0.84375 */ |
137 » » if (ix < 0x31800000) { /* |x| < 2**-28 */ | 141 if (ix < 0x31800000) { /* |x| < 2**-28 */ |
138 » » » /*avoid underflow */ | 142 /*avoid underflow */ |
139 » » » return 0.125f*(8*x + efx8*x); | 143 return 0.125f * (8 * x + efx8 * x); |
140 » » } | 144 } |
141 » » z = x*x; | 145 z = x * x; |
142 » » r = pp0+z*(pp1+z*(pp2+z*(pp3+z*pp4))); | 146 r = pp0 + z * (pp1 + z * (pp2 + z * (pp3 + z * pp4))); |
143 » » s = 1+z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*qq5)))); | 147 s = 1 + z * (qq1 + z * (qq2 + z * (qq3 + z * (qq4 + z * qq5)))); |
144 » » y = r/s; | 148 y = r / s; |
145 » » return x + x*y; | 149 return x + x * y; |
146 » } | 150 } |
147 » if (ix < 0x40c00000) /* |x| < 6 */ | 151 if (ix < 0x40c00000) /* |x| < 6 */ |
148 » » y = 1 - erfc2(ix,x); | 152 y = 1 - erfc2(ix, x); |
149 » else | 153 else |
150 » » y = 1 - 0x1p-120f; | 154 y = 1 - 0x1p-120f; |
151 » return sign ? -y : y; | 155 return sign ? -y : y; |
152 } | 156 } |
153 | 157 |
154 float erfcf(float x) | 158 float erfcf(float x) { |
155 { | 159 float r, s, z, y; |
156 » float r,s,z,y; | 160 uint32_t ix; |
157 » uint32_t ix; | 161 int sign; |
158 » int sign; | |
159 | 162 |
160 » GET_FLOAT_WORD(ix, x); | 163 GET_FLOAT_WORD(ix, x); |
161 » sign = ix>>31; | 164 sign = ix >> 31; |
162 » ix &= 0x7fffffff; | 165 ix &= 0x7fffffff; |
163 » if (ix >= 0x7f800000) { | 166 if (ix >= 0x7f800000) { |
164 » » /* erfc(nan)=nan, erfc(+-inf)=0,2 */ | 167 /* erfc(nan)=nan, erfc(+-inf)=0,2 */ |
165 » » return 2*sign + 1/x; | 168 return 2 * sign + 1 / x; |
166 » } | 169 } |
167 | 170 |
168 » if (ix < 0x3f580000) { /* |x| < 0.84375 */ | 171 if (ix < 0x3f580000) { /* |x| < 0.84375 */ |
169 » » if (ix < 0x23800000) /* |x| < 2**-56 */ | 172 if (ix < 0x23800000) /* |x| < 2**-56 */ |
170 » » » return 1.0f - x; | 173 return 1.0f - x; |
171 » » z = x*x; | 174 z = x * x; |
172 » » r = pp0+z*(pp1+z*(pp2+z*(pp3+z*pp4))); | 175 r = pp0 + z * (pp1 + z * (pp2 + z * (pp3 + z * pp4))); |
173 » » s = 1.0f+z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*qq5)))); | 176 s = 1.0f + z * (qq1 + z * (qq2 + z * (qq3 + z * (qq4 + z * qq5)))); |
174 » » y = r/s; | 177 y = r / s; |
175 » » if (sign || ix < 0x3e800000) /* x < 1/4 */ | 178 if (sign || ix < 0x3e800000) /* x < 1/4 */ |
176 » » » return 1.0f - (x+x*y); | 179 return 1.0f - (x + x * y); |
177 » » return 0.5f - (x - 0.5f + x*y); | 180 return 0.5f - (x - 0.5f + x * y); |
178 » } | 181 } |
179 » if (ix < 0x41e00000) { /* |x| < 28 */ | 182 if (ix < 0x41e00000) { /* |x| < 28 */ |
180 » » return sign ? 2 - erfc2(ix,x) : erfc2(ix,x); | 183 return sign ? 2 - erfc2(ix, x) : erfc2(ix, x); |
181 » } | 184 } |
182 » return sign ? 2 - 0x1p-120f : 0x1p-120f*0x1p-120f; | 185 return sign ? 2 - 0x1p-120f : 0x1p-120f * 0x1p-120f; |
183 } | 186 } |
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