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 108static const double erx = 8.45062911510467529297e-01, /* 0x3FEB0AC1, 0x60000000 */ 109 /* 110 * Coefficients for approximation to erf on [0,0.84375] 111 */ 112 efx8 = 1.02703333676410069053e+00, /* 0x3FF06EBA, 0x8214DB69 */ 113 pp0 = 1.28379167095512558561e-01, /* 0x3FC06EBA, 0x8214DB68 */ 114 pp1 = -3.25042107247001499370e-01, /* 0xBFD4CD7D, 0x691CB913 */ 115 pp2 = -2.84817495755985104766e-02, /* 0xBF9D2A51, 0xDBD7194F */ 116 pp3 = -5.77027029648944159157e-03, /* 0xBF77A291, 0x236668E4 */ 117 pp4 = -2.37630166566501626084e-05, /* 0xBEF8EAD6, 0x120016AC */ 118 qq1 = 3.97917223959155352819e-01, /* 0x3FD97779, 0xCDDADC09 */ 119 qq2 = 6.50222499887672944485e-02, /* 0x3FB0A54C, 0x5536CEBA */ 120 qq3 = 5.08130628187576562776e-03, /* 0x3F74D022, 0xC4D36B0F */ 121 qq4 = 1.32494738004321644526e-04, /* 0x3F215DC9, 0x221C1A10 */ 122 qq5 = -3.96022827877536812320e-06, /* 0xBED09C43, 0x42A26120 */ 123 /* 124 * Coefficients for approximation to erf in [0.84375,1.25] 125 */ 126 pa0 = -2.36211856075265944077e-03, /* 0xBF6359B8, 0xBEF77538 */ 127 pa1 = 4.14856118683748331666e-01, /* 0x3FDA8D00, 0xAD92B34D */ 128 pa2 = -3.72207876035701323847e-01, /* 0xBFD7D240, 0xFBB8C3F1 */ 129 pa3 = 3.18346619901161753674e-01, /* 0x3FD45FCA, 0x805120E4 */ 130 pa4 = -1.10894694282396677476e-01, /* 0xBFBC6398, 0x3D3E28EC */ 131 pa5 = 3.54783043256182359371e-02, /* 0x3FA22A36, 0x599795EB */ 132 pa6 = -2.16637559486879084300e-03, /* 0xBF61BF38, 0x0A96073F */ 133 qa1 = 1.06420880400844228286e-01, /* 0x3FBB3E66, 0x18EEE323 */ 134 qa2 = 5.40397917702171048937e-01, /* 0x3FE14AF0, 0x92EB6F33 */ 135 qa3 = 7.18286544141962662868e-02, /* 0x3FB2635C, 0xD99FE9A7 */ 136 qa4 = 1.26171219808761642112e-01, /* 0x3FC02660, 0xE763351F */ 137 qa5 = 1.36370839120290507362e-02, /* 0x3F8BEDC2, 0x6B51DD1C */ 138 qa6 = 1.19844998467991074170e-02, /* 0x3F888B54, 0x5735151D */ 139 /* 140 * Coefficients for approximation to erfc in [1.25,1/0.35] 141 */ 142 ra0 = -9.86494403484714822705e-03, /* 0xBF843412, 0x600D6435 */ 143 ra1 = -6.93858572707181764372e-01, /* 0xBFE63416, 0xE4BA7360 */ 144 ra2 = -1.05586262253232909814e+01, /* 0xC0251E04, 0x41B0E726 */ 145 ra3 = -6.23753324503260060396e+01, /* 0xC04F300A, 0xE4CBA38D */ 146 ra4 = -1.62396669462573470355e+02, /* 0xC0644CB1, 0x84282266 */ 147 ra5 = -1.84605092906711035994e+02, /* 0xC067135C, 0xEBCCABB2 */ 148 ra6 = -8.12874355063065934246e+01, /* 0xC0545265, 0x57E4D2F2 */ 149 ra7 = -9.81432934416914548592e+00, /* 0xC023A0EF, 0xC69AC25C */ 150 sa1 = 1.96512716674392571292e+01, /* 0x4033A6B9, 0xBD707687 */ 151 sa2 = 1.37657754143519042600e+02, /* 0x4061350C, 0x526AE721 */ 152 sa3 = 4.34565877475229228821e+02, /* 0x407B290D, 0xD58A1A71 */ 153 sa4 = 6.45387271733267880336e+02, /* 0x40842B19, 0x21EC2868 */ 154 sa5 = 4.29008140027567833386e+02, /* 0x407AD021, 0x57700314 */ 155 sa6 = 1.08635005541779435134e+02, /* 0x405B28A3, 0xEE48AE2C */ 156 sa7 = 6.57024977031928170135e+00, /* 0x401A47EF, 0x8E484A93 */ 157 sa8 = -6.04244152148580987438e-02, /* 0xBFAEEFF2, 0xEE749A62 */ 158 /* 159 * Coefficients for approximation to erfc in [1/.35,28] 160 */ 161 rb0 = -9.86494292470009928597e-03, /* 0xBF843412, 0x39E86F4A */ 162 rb1 = -7.99283237680523006574e-01, /* 0xBFE993BA, 0x70C285DE */ 163 rb2 = -1.77579549177547519889e+01, /* 0xC031C209, 0x555F995A */ 164 rb3 = -1.60636384855821916062e+02, /* 0xC064145D, 0x43C5ED98 */ 165 rb4 = -6.37566443368389627722e+02, /* 0xC083EC88, 0x1375F228 */ 166 rb5 = -1.02509513161107724954e+03, /* 0xC0900461, 0x6A2E5992 */ 167 rb6 = -4.83519191608651397019e+02, /* 0xC07E384E, 0x9BDC383F */ 168 sb1 = 3.03380607434824582924e+01, /* 0x403E568B, 0x261D5190 */ 169 sb2 = 3.25792512996573918826e+02, /* 0x40745CAE, 0x221B9F0A */ 170 sb3 = 1.53672958608443695994e+03, /* 0x409802EB, 0x189D5118 */ 171 sb4 = 3.19985821950859553908e+03, /* 0x40A8FFB7, 0x688C246A */ 172 sb5 = 2.55305040643316442583e+03, /* 0x40A3F219, 0xCEDF3BE6 */ 173 sb6 = 4.74528541206955367215e+02, /* 0x407DA874, 0xE79FE763 */ 174 sb7 = -2.24409524465858183362e+01; /* 0xC03670E2, 0x42712D62 */ 175 176static double erfc1(double x) { 177 double_t s, P, Q; 178 179 s = fabs(x) - 1; 180 P = pa0 + s * (pa1 + s * (pa2 + s * (pa3 + s * (pa4 + s * (pa5 + s * pa6))))); 181 Q = 1 + s * (qa1 + s * (qa2 + s * (qa3 + s * (qa4 + s * (qa5 + s * qa6))))); 182 return 1 - erx - P / Q; 183} 184 185static double erfc2(uint32_t ix, double x) { 186 double_t s, R, S; 187 double z; 188 189 if (ix < 0x3ff40000) /* |x| < 1.25 */ 190 return erfc1(x); 191 192 x = fabs(x); 193 s = 1 / (x * x); 194 if (ix < 0x4006db6d) { /* |x| < 1/.35 ~ 2.85714 */ 195 R = ra0 + s * (ra1 + s * (ra2 + s * (ra3 + s * (ra4 + s * (ra5 + s * (ra6 + s * ra7)))))); 196 S = 1.0 + 197 s * (sa1 + 198 s * (sa2 + s * (sa3 + s * (sa4 + s * (sa5 + s * (sa6 + s * (sa7 + s * sa8))))))); 199 } else { /* |x| > 1/.35 */ 200 R = rb0 + s * (rb1 + s * (rb2 + s * (rb3 + s * (rb4 + s * (rb5 + s * rb6))))); 201 S = 1.0 + s * (sb1 + s * (sb2 + s * (sb3 + s * (sb4 + s * (sb5 + s * (sb6 + s * sb7)))))); 202 } 203 z = x; 204 SET_LOW_WORD(z, 0); 205 return exp(-z * z - 0.5625) * exp((z - x) * (z + x) + R / S) / x; 206} 207 208double erf(double x) { 209 double r, s, z, y; 210 uint32_t ix; 211 int sign; 212 213 GET_HIGH_WORD(ix, x); 214 sign = ix >> 31; 215 ix &= 0x7fffffff; 216 if (ix >= 0x7ff00000) { 217 /* erf(nan)=nan, erf(+-inf)=+-1 */ 218 return 1 - 2 * sign + 1 / x; 219 } 220 if (ix < 0x3feb0000) { /* |x| < 0.84375 */ 221 if (ix < 0x3e300000) { /* |x| < 2**-28 */ 222 /* avoid underflow */ 223 return 0.125 * (8 * x + efx8 * x); 224 } 225 z = x * x; 226 r = pp0 + z * (pp1 + z * (pp2 + z * (pp3 + z * pp4))); 227 s = 1.0 + z * (qq1 + z * (qq2 + z * (qq3 + z * (qq4 + z * qq5)))); 228 y = r / s; 229 return x + x * y; 230 } 231 if (ix < 0x40180000) /* 0.84375 <= |x| < 6 */ 232 y = 1 - erfc2(ix, x); 233 else 234 y = 1 - 0x1p-1022; 235 return sign ? -y : y; 236} 237 238double erfc(double x) { 239 double r, s, z, y; 240 uint32_t ix; 241 int sign; 242 243 GET_HIGH_WORD(ix, x); 244 sign = ix >> 31; 245 ix &= 0x7fffffff; 246 if (ix >= 0x7ff00000) { 247 /* erfc(nan)=nan, erfc(+-inf)=0,2 */ 248 return 2 * sign + 1 / x; 249 } 250 if (ix < 0x3feb0000) { /* |x| < 0.84375 */ 251 if (ix < 0x3c700000) /* |x| < 2**-56 */ 252 return 1.0 - x; 253 z = x * x; 254 r = pp0 + z * (pp1 + z * (pp2 + z * (pp3 + z * pp4))); 255 s = 1.0 + z * (qq1 + z * (qq2 + z * (qq3 + z * (qq4 + z * qq5)))); 256 y = r / s; 257 if (sign || ix < 0x3fd00000) { /* x < 1/4 */ 258 return 1.0 - (x + x * y); 259 } 260 return 0.5 - (x - 0.5 + x * y); 261 } 262 if (ix < 0x403c0000) { /* 0.84375 <= |x| < 28 */ 263 return sign ? 2 - erfc2(ix, x) : erfc2(ix, x); 264 } 265 return sign ? 2 - 0x1p-1022 : 0x1p-1022 * 0x1p-1022; 266} 267