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| 1 /* origin: FreeBSD /usr/src/lib/msun/src/s_erf.c */ |
| 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 /* double erf(double x) |
| 13 * double erfc(double x) |
| 14 * x |
| 15 * 2 |\ |
| 16 * erf(x) = --------- | exp(-t*t)dt |
| 17 * sqrt(pi) \| |
| 18 * 0 |
| 19 * |
| 20 * erfc(x) = 1-erf(x) |
| 21 * Note that |
| 22 * erf(-x) = -erf(x) |
| 23 * erfc(-x) = 2 - erfc(x) |
| 24 * |
| 25 * Method: |
| 26 * 1. For |x| in [0, 0.84375] |
| 27 * erf(x) = x + x*R(x^2) |
| 28 * erfc(x) = 1 - erf(x) if x in [-.84375,0.25] |
| 29 * = 0.5 + ((0.5-x)-x*R) if x in [0.25,0.84375] |
| 30 * where R = P/Q where P is an odd poly of degree 8 and |
| 31 * Q is an odd poly of degree 10. |
| 32 * -57.90 |
| 33 * | R - (erf(x)-x)/x | <= 2 |
| 34 * |
| 35 * |
| 36 * Remark. The formula is derived by noting |
| 37 * erf(x) = (2/sqrt(pi))*(x - x^3/3 + x^5/10 - x^7/42 + ....) |
| 38 * and that |
| 39 * 2/sqrt(pi) = 1.128379167095512573896158903121545171688 |
| 40 * is close to one. The interval is chosen because the fix |
| 41 * point of erf(x) is near 0.6174 (i.e., erf(x)=x when x is |
| 42 * near 0.6174), and by some experiment, 0.84375 is chosen to |
| 43 * guarantee the error is less than one ulp for erf. |
| 44 * |
| 45 * 2. For |x| in [0.84375,1.25], let s = |x| - 1, and |
| 46 * c = 0.84506291151 rounded to single (24 bits) |
| 47 * erf(x) = sign(x) * (c + P1(s)/Q1(s)) |
| 48 * erfc(x) = (1-c) - P1(s)/Q1(s) if x > 0 |
| 49 * 1+(c+P1(s)/Q1(s)) if x < 0 |
| 50 * |P1/Q1 - (erf(|x|)-c)| <= 2**-59.06 |
| 51 * Remark: here we use the taylor series expansion at x=1. |
| 52 * erf(1+s) = erf(1) + s*Poly(s) |
| 53 * = 0.845.. + P1(s)/Q1(s) |
| 54 * That is, we use rational approximation to approximate |
| 55 * erf(1+s) - (c = (single)0.84506291151) |
| 56 * Note that |P1/Q1|< 0.078 for x in [0.84375,1.25] |
| 57 * where |
| 58 * P1(s) = degree 6 poly in s |
| 59 * Q1(s) = degree 6 poly in s |
| 60 * |
| 61 * 3. For x in [1.25,1/0.35(~2.857143)], |
| 62 * erfc(x) = (1/x)*exp(-x*x-0.5625+R1/S1) |
| 63 * erf(x) = 1 - erfc(x) |
| 64 * where |
| 65 * R1(z) = degree 7 poly in z, (z=1/x^2) |
| 66 * S1(z) = degree 8 poly in z |
| 67 * |
| 68 * 4. For x in [1/0.35,28] |
| 69 * erfc(x) = (1/x)*exp(-x*x-0.5625+R2/S2) if x > 0 |
| 70 * = 2.0 - (1/x)*exp(-x*x-0.5625+R2/S2) if -6<x<0 |
| 71 * = 2.0 - tiny (if x <= -6) |
| 72 * erf(x) = sign(x)*(1.0 - erfc(x)) if x < 6, else |
| 73 * erf(x) = sign(x)*(1.0 - tiny) |
| 74 * where |
| 75 * R2(z) = degree 6 poly in z, (z=1/x^2) |
| 76 * S2(z) = degree 7 poly in z |
| 77 * |
| 78 * Note1: |
| 79 * To compute exp(-x*x-0.5625+R/S), let s be a single |
| 80 * precision number and s := x; then |
| 81 * -x*x = -s*s + (s-x)*(s+x) |
| 82 * exp(-x*x-0.5626+R/S) = |
| 83 * exp(-s*s-0.5625)*exp((s-x)*(s+x)+R/S); |
| 84 * Note2: |
| 85 * Here 4 and 5 make use of the asymptotic series |
| 86 * exp(-x*x) |
| 87 * erfc(x) ~ ---------- * ( 1 + Poly(1/x^2) ) |
| 88 * x*sqrt(pi) |
| 89 * We use rational approximation to approximate |
| 90 * g(s)=f(1/x^2) = log(erfc(x)*x) - x*x + 0.5625 |
| 91 * Here is the error bound for R1/S1 and R2/S2 |
| 92 * |R1/S1 - f(x)| < 2**(-62.57) |
| 93 * |R2/S2 - f(x)| < 2**(-61.52) |
| 94 * |
| 95 * 5. For inf > x >= 28 |
| 96 * erf(x) = sign(x) *(1 - tiny) (raise inexact) |
| 97 * erfc(x) = tiny*tiny (raise underflow) if x > 0 |
| 98 * = 2 - tiny if x<0 |
| 99 * |
| 100 * 7. Special case: |
| 101 * erf(0) = 0, erf(inf) = 1, erf(-inf) = -1, |
| 102 * erfc(0) = 1, erfc(inf) = 0, erfc(-inf) = 2, |
| 103 * erfc/erf(NaN) is NaN |
| 104 */ |
| 105 |
| 106 #include "libm.h" |
| 107 |
| 108 static const double |
| 109 erx = 8.45062911510467529297e-01, /* 0x3FEB0AC1, 0x60000000 */ |
| 110 /* |
| 111 * Coefficients for approximation to erf on [0,0.84375] |
| 112 */ |
| 113 efx8 = 1.02703333676410069053e+00, /* 0x3FF06EBA, 0x8214DB69 */ |
| 114 pp0 = 1.28379167095512558561e-01, /* 0x3FC06EBA, 0x8214DB68 */ |
| 115 pp1 = -3.25042107247001499370e-01, /* 0xBFD4CD7D, 0x691CB913 */ |
| 116 pp2 = -2.84817495755985104766e-02, /* 0xBF9D2A51, 0xDBD7194F */ |
| 117 pp3 = -5.77027029648944159157e-03, /* 0xBF77A291, 0x236668E4 */ |
| 118 pp4 = -2.37630166566501626084e-05, /* 0xBEF8EAD6, 0x120016AC */ |
| 119 qq1 = 3.97917223959155352819e-01, /* 0x3FD97779, 0xCDDADC09 */ |
| 120 qq2 = 6.50222499887672944485e-02, /* 0x3FB0A54C, 0x5536CEBA */ |
| 121 qq3 = 5.08130628187576562776e-03, /* 0x3F74D022, 0xC4D36B0F */ |
| 122 qq4 = 1.32494738004321644526e-04, /* 0x3F215DC9, 0x221C1A10 */ |
| 123 qq5 = -3.96022827877536812320e-06, /* 0xBED09C43, 0x42A26120 */ |
| 124 /* |
| 125 * Coefficients for approximation to erf in [0.84375,1.25] |
| 126 */ |
| 127 pa0 = -2.36211856075265944077e-03, /* 0xBF6359B8, 0xBEF77538 */ |
| 128 pa1 = 4.14856118683748331666e-01, /* 0x3FDA8D00, 0xAD92B34D */ |
| 129 pa2 = -3.72207876035701323847e-01, /* 0xBFD7D240, 0xFBB8C3F1 */ |
| 130 pa3 = 3.18346619901161753674e-01, /* 0x3FD45FCA, 0x805120E4 */ |
| 131 pa4 = -1.10894694282396677476e-01, /* 0xBFBC6398, 0x3D3E28EC */ |
| 132 pa5 = 3.54783043256182359371e-02, /* 0x3FA22A36, 0x599795EB */ |
| 133 pa6 = -2.16637559486879084300e-03, /* 0xBF61BF38, 0x0A96073F */ |
| 134 qa1 = 1.06420880400844228286e-01, /* 0x3FBB3E66, 0x18EEE323 */ |
| 135 qa2 = 5.40397917702171048937e-01, /* 0x3FE14AF0, 0x92EB6F33 */ |
| 136 qa3 = 7.18286544141962662868e-02, /* 0x3FB2635C, 0xD99FE9A7 */ |
| 137 qa4 = 1.26171219808761642112e-01, /* 0x3FC02660, 0xE763351F */ |
| 138 qa5 = 1.36370839120290507362e-02, /* 0x3F8BEDC2, 0x6B51DD1C */ |
| 139 qa6 = 1.19844998467991074170e-02, /* 0x3F888B54, 0x5735151D */ |
| 140 /* |
| 141 * Coefficients for approximation to erfc in [1.25,1/0.35] |
| 142 */ |
| 143 ra0 = -9.86494403484714822705e-03, /* 0xBF843412, 0x600D6435 */ |
| 144 ra1 = -6.93858572707181764372e-01, /* 0xBFE63416, 0xE4BA7360 */ |
| 145 ra2 = -1.05586262253232909814e+01, /* 0xC0251E04, 0x41B0E726 */ |
| 146 ra3 = -6.23753324503260060396e+01, /* 0xC04F300A, 0xE4CBA38D */ |
| 147 ra4 = -1.62396669462573470355e+02, /* 0xC0644CB1, 0x84282266 */ |
| 148 ra5 = -1.84605092906711035994e+02, /* 0xC067135C, 0xEBCCABB2 */ |
| 149 ra6 = -8.12874355063065934246e+01, /* 0xC0545265, 0x57E4D2F2 */ |
| 150 ra7 = -9.81432934416914548592e+00, /* 0xC023A0EF, 0xC69AC25C */ |
| 151 sa1 = 1.96512716674392571292e+01, /* 0x4033A6B9, 0xBD707687 */ |
| 152 sa2 = 1.37657754143519042600e+02, /* 0x4061350C, 0x526AE721 */ |
| 153 sa3 = 4.34565877475229228821e+02, /* 0x407B290D, 0xD58A1A71 */ |
| 154 sa4 = 6.45387271733267880336e+02, /* 0x40842B19, 0x21EC2868 */ |
| 155 sa5 = 4.29008140027567833386e+02, /* 0x407AD021, 0x57700314 */ |
| 156 sa6 = 1.08635005541779435134e+02, /* 0x405B28A3, 0xEE48AE2C */ |
| 157 sa7 = 6.57024977031928170135e+00, /* 0x401A47EF, 0x8E484A93 */ |
| 158 sa8 = -6.04244152148580987438e-02, /* 0xBFAEEFF2, 0xEE749A62 */ |
| 159 /* |
| 160 * Coefficients for approximation to erfc in [1/.35,28] |
| 161 */ |
| 162 rb0 = -9.86494292470009928597e-03, /* 0xBF843412, 0x39E86F4A */ |
| 163 rb1 = -7.99283237680523006574e-01, /* 0xBFE993BA, 0x70C285DE */ |
| 164 rb2 = -1.77579549177547519889e+01, /* 0xC031C209, 0x555F995A */ |
| 165 rb3 = -1.60636384855821916062e+02, /* 0xC064145D, 0x43C5ED98 */ |
| 166 rb4 = -6.37566443368389627722e+02, /* 0xC083EC88, 0x1375F228 */ |
| 167 rb5 = -1.02509513161107724954e+03, /* 0xC0900461, 0x6A2E5992 */ |
| 168 rb6 = -4.83519191608651397019e+02, /* 0xC07E384E, 0x9BDC383F */ |
| 169 sb1 = 3.03380607434824582924e+01, /* 0x403E568B, 0x261D5190 */ |
| 170 sb2 = 3.25792512996573918826e+02, /* 0x40745CAE, 0x221B9F0A */ |
| 171 sb3 = 1.53672958608443695994e+03, /* 0x409802EB, 0x189D5118 */ |
| 172 sb4 = 3.19985821950859553908e+03, /* 0x40A8FFB7, 0x688C246A */ |
| 173 sb5 = 2.55305040643316442583e+03, /* 0x40A3F219, 0xCEDF3BE6 */ |
| 174 sb6 = 4.74528541206955367215e+02, /* 0x407DA874, 0xE79FE763 */ |
| 175 sb7 = -2.24409524465858183362e+01; /* 0xC03670E2, 0x42712D62 */ |
| 176 |
| 177 static double erfc1(double x) |
| 178 { |
| 179 double_t s,P,Q; |
| 180 |
| 181 s = fabs(x) - 1; |
| 182 P = pa0+s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6))))); |
| 183 Q = 1+s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6))))); |
| 184 return 1 - erx - P/Q; |
| 185 } |
| 186 |
| 187 static double erfc2(uint32_t ix, double x) |
| 188 { |
| 189 double_t s,R,S; |
| 190 double z; |
| 191 |
| 192 if (ix < 0x3ff40000) /* |x| < 1.25 */ |
| 193 return erfc1(x); |
| 194 |
| 195 x = fabs(x); |
| 196 s = 1/(x*x); |
| 197 if (ix < 0x4006db6d) { /* |x| < 1/.35 ~ 2.85714 */ |
| 198 R = ra0+s*(ra1+s*(ra2+s*(ra3+s*(ra4+s*( |
| 199 ra5+s*(ra6+s*ra7)))))); |
| 200 S = 1.0+s*(sa1+s*(sa2+s*(sa3+s*(sa4+s*( |
| 201 sa5+s*(sa6+s*(sa7+s*sa8))))))); |
| 202 } else { /* |x| > 1/.35 */ |
| 203 R = rb0+s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*( |
| 204 rb5+s*rb6))))); |
| 205 S = 1.0+s*(sb1+s*(sb2+s*(sb3+s*(sb4+s*( |
| 206 sb5+s*(sb6+s*sb7)))))); |
| 207 } |
| 208 z = x; |
| 209 SET_LOW_WORD(z,0); |
| 210 return exp(-z*z-0.5625)*exp((z-x)*(z+x)+R/S)/x; |
| 211 } |
| 212 |
| 213 double erf(double x) |
| 214 { |
| 215 double r,s,z,y; |
| 216 uint32_t ix; |
| 217 int sign; |
| 218 |
| 219 GET_HIGH_WORD(ix, x); |
| 220 sign = ix>>31; |
| 221 ix &= 0x7fffffff; |
| 222 if (ix >= 0x7ff00000) { |
| 223 /* erf(nan)=nan, erf(+-inf)=+-1 */ |
| 224 return 1-2*sign + 1/x; |
| 225 } |
| 226 if (ix < 0x3feb0000) { /* |x| < 0.84375 */ |
| 227 if (ix < 0x3e300000) { /* |x| < 2**-28 */ |
| 228 /* avoid underflow */ |
| 229 return 0.125*(8*x + efx8*x); |
| 230 } |
| 231 z = x*x; |
| 232 r = pp0+z*(pp1+z*(pp2+z*(pp3+z*pp4))); |
| 233 s = 1.0+z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*qq5)))); |
| 234 y = r/s; |
| 235 return x + x*y; |
| 236 } |
| 237 if (ix < 0x40180000) /* 0.84375 <= |x| < 6 */ |
| 238 y = 1 - erfc2(ix,x); |
| 239 else |
| 240 y = 1 - 0x1p-1022; |
| 241 return sign ? -y : y; |
| 242 } |
| 243 |
| 244 double erfc(double x) |
| 245 { |
| 246 double r,s,z,y; |
| 247 uint32_t ix; |
| 248 int sign; |
| 249 |
| 250 GET_HIGH_WORD(ix, x); |
| 251 sign = ix>>31; |
| 252 ix &= 0x7fffffff; |
| 253 if (ix >= 0x7ff00000) { |
| 254 /* erfc(nan)=nan, erfc(+-inf)=0,2 */ |
| 255 return 2*sign + 1/x; |
| 256 } |
| 257 if (ix < 0x3feb0000) { /* |x| < 0.84375 */ |
| 258 if (ix < 0x3c700000) /* |x| < 2**-56 */ |
| 259 return 1.0 - x; |
| 260 z = x*x; |
| 261 r = pp0+z*(pp1+z*(pp2+z*(pp3+z*pp4))); |
| 262 s = 1.0+z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*qq5)))); |
| 263 y = r/s; |
| 264 if (sign || ix < 0x3fd00000) { /* x < 1/4 */ |
| 265 return 1.0 - (x+x*y); |
| 266 } |
| 267 return 0.5 - (x - 0.5 + x*y); |
| 268 } |
| 269 if (ix < 0x403c0000) { /* 0.84375 <= |x| < 28 */ |
| 270 return sign ? 2 - erfc2(ix,x) : erfc2(ix,x); |
| 271 } |
| 272 return sign ? 2 - 0x1p-1022 : 0x1p-1022*0x1p-1022; |
| 273 } |
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Issue 1573973002:
Add a "fork" of musl as //fusl. (Closed)
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