1 /* @(#)s_erf.c 5.1 93/09/24 */ 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 13 /* double erf(double x) 14 * double erfc(double x) 15 * x 16 * 2 |\ 17 * erf(x) = --------- | exp(-t*t)dt 18 * sqrt(pi) \| 19 * 0 20 * 21 * erfc(x) = 1-erf(x) 22 * Note that 23 * erf(-x) = -erf(x) 24 * erfc(-x) = 2 - erfc(x) 25 * 26 * Method: 27 * 1. For |x| in [0, 0.84375] 28 * erf(x) = x + x*R(x^2) 29 * erfc(x) = 1 - erf(x) if x in [-.84375,0.25] 30 * = 0.5 + ((0.5-x)-x*R) if x in [0.25,0.84375] 31 * where R = P/Q where P is an odd poly of degree 8 and 32 * Q is an odd poly of degree 10. 33 * -57.90 34 * | R - (erf(x)-x)/x | <= 2 35 * 36 * 37 * Remark. The formula is derived by noting 38 * erf(x) = (2/sqrt(pi))*(x - x^3/3 + x^5/10 - x^7/42 + ....) 39 * and that 40 * 2/sqrt(pi) = 1.128379167095512573896158903121545171688 41 * is close to one. The interval is chosen because the fix 42 * point of erf(x) is near 0.6174 (i.e., erf(x)=x when x is 43 * near 0.6174), and by some experiment, 0.84375 is chosen to 44 * guarantee the error is less than one ulp for erf. 45 * 46 * 2. For |x| in [0.84375,1.25], let s = |x| - 1, and 47 * c = 0.84506291151 rounded to single (24 bits) 48 * erf(x) = sign(x) * (c + P1(s)/Q1(s)) 49 * erfc(x) = (1-c) - P1(s)/Q1(s) if x > 0 50 * 1+(c+P1(s)/Q1(s)) if x < 0 51 * |P1/Q1 - (erf(|x|)-c)| <= 2**-59.06 52 * Remark: here we use the taylor series expansion at x=1. 53 * erf(1+s) = erf(1) + s*Poly(s) 54 * = 0.845.. + P1(s)/Q1(s) 55 * That is, we use rational approximation to approximate 56 * erf(1+s) - (c = (single)0.84506291151) 57 * Note that |P1/Q1|< 0.078 for x in [0.84375,1.25] 58 * where 59 * P1(s) = degree 6 poly in s 60 * Q1(s) = degree 6 poly in s 61 * 62 * 3. For x in [1.25,1/0.35(~2.857143)], 63 * erfc(x) = (1/x)*exp(-x*x-0.5625+R1/S1) 64 * erf(x) = 1 - erfc(x) 65 * where 66 * R1(z) = degree 7 poly in z, (z=1/x^2) 67 * S1(z) = degree 8 poly in z 68 * 69 * 4. For x in [1/0.35,28] 70 * erfc(x) = (1/x)*exp(-x*x-0.5625+R2/S2) if x > 0 71 * = 2.0 - (1/x)*exp(-x*x-0.5625+R2/S2) if -6<x<0 72 * = 2.0 - tiny (if x <= -6) 73 * erf(x) = sign(x)*(1.0 - erfc(x)) if x < 6, else 74 * erf(x) = sign(x)*(1.0 - tiny) 75 * where 76 * R2(z) = degree 6 poly in z, (z=1/x^2) 77 * S2(z) = degree 7 poly in z 78 * 79 * Note1: 80 * To compute exp(-x*x-0.5625+R/S), let s be a single 81 * precision number and s := x; then 82 * -x*x = -s*s + (s-x)*(s+x) 83 * exp(-x*x-0.5626+R/S) = 84 * exp(-s*s-0.5625)*exp((s-x)*(s+x)+R/S); 85 * Note2: 86 * Here 4 and 5 make use of the asymptotic series 87 * exp(-x*x) 88 * erfc(x) ~ ---------- * ( 1 + Poly(1/x^2) ) 89 * x*sqrt(pi) 90 * We use rational approximation to approximate 91 * g(s)=f(1/x^2) = log(erfc(x)*x) - x*x + 0.5625 92 * Here is the error bound for R1/S1 and R2/S2 93 * |R1/S1 - f(x)| < 2**(-62.57) 94 * |R2/S2 - f(x)| < 2**(-61.52) 95 * 96 * 5. For inf > x >= 28 97 * erf(x) = sign(x) *(1 - tiny) (raise inexact) 98 * erfc(x) = tiny*tiny (raise underflow) if x > 0 99 * = 2 - tiny if x<0 100 * 101 * 7. Special case: 102 * erf(0) = 0, erf(inf) = 1, erf(-inf) = -1, 103 * erfc(0) = 1, erfc(inf) = 0, erfc(-inf) = 2, 104 * erfc/erf(NaN) is NaN 105 */ 106 107 #include <float.h> 108 #include <math.h> 109 110 #include "math_private.h" 111 112 static const double 113 tiny = 1e-300, 114 half= 5.00000000000000000000e-01, /* 0x3FE00000, 0x00000000 */ 115 one = 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */ 116 two = 2.00000000000000000000e+00, /* 0x40000000, 0x00000000 */ 117 /* c = (float)0.84506291151 */ 118 erx = 8.45062911510467529297e-01, /* 0x3FEB0AC1, 0x60000000 */ 119 /* 120 * Coefficients for approximation to erf on [0,0.84375] 121 */ 122 efx = 1.28379167095512586316e-01, /* 0x3FC06EBA, 0x8214DB69 */ 123 efx8= 1.02703333676410069053e+00, /* 0x3FF06EBA, 0x8214DB69 */ 124 pp0 = 1.28379167095512558561e-01, /* 0x3FC06EBA, 0x8214DB68 */ 125 pp1 = -3.25042107247001499370e-01, /* 0xBFD4CD7D, 0x691CB913 */ 126 pp2 = -2.84817495755985104766e-02, /* 0xBF9D2A51, 0xDBD7194F */ 127 pp3 = -5.77027029648944159157e-03, /* 0xBF77A291, 0x236668E4 */ 128 pp4 = -2.37630166566501626084e-05, /* 0xBEF8EAD6, 0x120016AC */ 129 qq1 = 3.97917223959155352819e-01, /* 0x3FD97779, 0xCDDADC09 */ 130 qq2 = 6.50222499887672944485e-02, /* 0x3FB0A54C, 0x5536CEBA */ 131 qq3 = 5.08130628187576562776e-03, /* 0x3F74D022, 0xC4D36B0F */ 132 qq4 = 1.32494738004321644526e-04, /* 0x3F215DC9, 0x221C1A10 */ 133 qq5 = -3.96022827877536812320e-06, /* 0xBED09C43, 0x42A26120 */ 134 /* 135 * Coefficients for approximation to erf in [0.84375,1.25] 136 */ 137 pa0 = -2.36211856075265944077e-03, /* 0xBF6359B8, 0xBEF77538 */ 138 pa1 = 4.14856118683748331666e-01, /* 0x3FDA8D00, 0xAD92B34D */ 139 pa2 = -3.72207876035701323847e-01, /* 0xBFD7D240, 0xFBB8C3F1 */ 140 pa3 = 3.18346619901161753674e-01, /* 0x3FD45FCA, 0x805120E4 */ 141 pa4 = -1.10894694282396677476e-01, /* 0xBFBC6398, 0x3D3E28EC */ 142 pa5 = 3.54783043256182359371e-02, /* 0x3FA22A36, 0x599795EB */ 143 pa6 = -2.16637559486879084300e-03, /* 0xBF61BF38, 0x0A96073F */ 144 qa1 = 1.06420880400844228286e-01, /* 0x3FBB3E66, 0x18EEE323 */ 145 qa2 = 5.40397917702171048937e-01, /* 0x3FE14AF0, 0x92EB6F33 */ 146 qa3 = 7.18286544141962662868e-02, /* 0x3FB2635C, 0xD99FE9A7 */ 147 qa4 = 1.26171219808761642112e-01, /* 0x3FC02660, 0xE763351F */ 148 qa5 = 1.36370839120290507362e-02, /* 0x3F8BEDC2, 0x6B51DD1C */ 149 qa6 = 1.19844998467991074170e-02, /* 0x3F888B54, 0x5735151D */ 150 /* 151 * Coefficients for approximation to erfc in [1.25,1/0.35] 152 */ 153 ra0 = -9.86494403484714822705e-03, /* 0xBF843412, 0x600D6435 */ 154 ra1 = -6.93858572707181764372e-01, /* 0xBFE63416, 0xE4BA7360 */ 155 ra2 = -1.05586262253232909814e+01, /* 0xC0251E04, 0x41B0E726 */ 156 ra3 = -6.23753324503260060396e+01, /* 0xC04F300A, 0xE4CBA38D */ 157 ra4 = -1.62396669462573470355e+02, /* 0xC0644CB1, 0x84282266 */ 158 ra5 = -1.84605092906711035994e+02, /* 0xC067135C, 0xEBCCABB2 */ 159 ra6 = -8.12874355063065934246e+01, /* 0xC0545265, 0x57E4D2F2 */ 160 ra7 = -9.81432934416914548592e+00, /* 0xC023A0EF, 0xC69AC25C */ 161 sa1 = 1.96512716674392571292e+01, /* 0x4033A6B9, 0xBD707687 */ 162 sa2 = 1.37657754143519042600e+02, /* 0x4061350C, 0x526AE721 */ 163 sa3 = 4.34565877475229228821e+02, /* 0x407B290D, 0xD58A1A71 */ 164 sa4 = 6.45387271733267880336e+02, /* 0x40842B19, 0x21EC2868 */ 165 sa5 = 4.29008140027567833386e+02, /* 0x407AD021, 0x57700314 */ 166 sa6 = 1.08635005541779435134e+02, /* 0x405B28A3, 0xEE48AE2C */ 167 sa7 = 6.57024977031928170135e+00, /* 0x401A47EF, 0x8E484A93 */ 168 sa8 = -6.04244152148580987438e-02, /* 0xBFAEEFF2, 0xEE749A62 */ 169 /* 170 * Coefficients for approximation to erfc in [1/.35,28] 171 */ 172 rb0 = -9.86494292470009928597e-03, /* 0xBF843412, 0x39E86F4A */ 173 rb1 = -7.99283237680523006574e-01, /* 0xBFE993BA, 0x70C285DE */ 174 rb2 = -1.77579549177547519889e+01, /* 0xC031C209, 0x555F995A */ 175 rb3 = -1.60636384855821916062e+02, /* 0xC064145D, 0x43C5ED98 */ 176 rb4 = -6.37566443368389627722e+02, /* 0xC083EC88, 0x1375F228 */ 177 rb5 = -1.02509513161107724954e+03, /* 0xC0900461, 0x6A2E5992 */ 178 rb6 = -4.83519191608651397019e+02, /* 0xC07E384E, 0x9BDC383F */ 179 sb1 = 3.03380607434824582924e+01, /* 0x403E568B, 0x261D5190 */ 180 sb2 = 3.25792512996573918826e+02, /* 0x40745CAE, 0x221B9F0A */ 181 sb3 = 1.53672958608443695994e+03, /* 0x409802EB, 0x189D5118 */ 182 sb4 = 3.19985821950859553908e+03, /* 0x40A8FFB7, 0x688C246A */ 183 sb5 = 2.55305040643316442583e+03, /* 0x40A3F219, 0xCEDF3BE6 */ 184 sb6 = 4.74528541206955367215e+02, /* 0x407DA874, 0xE79FE763 */ 185 sb7 = -2.24409524465858183362e+01; /* 0xC03670E2, 0x42712D62 */ 186 187 double 188 erf(double x) 189 { 190 int32_t hx,ix,i; 191 double R,S,P,Q,s,y,z,r; 192 GET_HIGH_WORD(hx,x); 193 ix = hx&0x7fffffff; 194 if(ix>=0x7ff00000) { /* erf(nan)=nan */ 195 i = ((u_int32_t)hx>>31)<<1; 196 return (double)(1-i)+one/x; /* erf(+-inf)=+-1 */ 197 } 198 199 if(ix < 0x3feb0000) { /* |x|<0.84375 */ 200 if(ix < 0x3e300000) { /* |x|<2**-28 */ 201 if (ix < 0x00800000) 202 return 0.125*(8.0*x+efx8*x); /*avoid underflow */ 203 return x + efx*x; 204 } 205 z = x*x; 206 r = pp0+z*(pp1+z*(pp2+z*(pp3+z*pp4))); 207 s = one+z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*qq5)))); 208 y = r/s; 209 return x + x*y; 210 } 211 if(ix < 0x3ff40000) { /* 0.84375 <= |x| < 1.25 */ 212 s = fabs(x)-one; 213 P = pa0+s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6))))); 214 Q = one+s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6))))); 215 if(hx>=0) return erx + P/Q; else return -erx - P/Q; 216 } 217 if (ix >= 0x40180000) { /* inf>|x|>=6 */ 218 if(hx>=0) return one-tiny; else return tiny-one; 219 } 220 x = fabs(x); 221 s = one/(x*x); 222 if(ix< 0x4006DB6E) { /* |x| < 1/0.35 */ 223 R=ra0+s*(ra1+s*(ra2+s*(ra3+s*(ra4+s*( 224 ra5+s*(ra6+s*ra7)))))); 225 S=one+s*(sa1+s*(sa2+s*(sa3+s*(sa4+s*( 226 sa5+s*(sa6+s*(sa7+s*sa8))))))); 227 } else { /* |x| >= 1/0.35 */ 228 R=rb0+s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*( 229 rb5+s*rb6))))); 230 S=one+s*(sb1+s*(sb2+s*(sb3+s*(sb4+s*( 231 sb5+s*(sb6+s*sb7)))))); 232 } 233 z = x; 234 SET_LOW_WORD(z,0); 235 r = exp(-z*z-0.5625)*exp((z-x)*(z+x)+R/S); 236 if(hx>=0) return one-r/x; else return r/x-one; 237 } 238 239 double 240 erfc(double x) 241 { 242 int32_t hx,ix; 243 double R,S,P,Q,s,y,z,r; 244 GET_HIGH_WORD(hx,x); 245 ix = hx&0x7fffffff; 246 if(ix>=0x7ff00000) { /* erfc(nan)=nan */ 247 /* erfc(+-inf)=0,2 */ 248 return (double)(((u_int32_t)hx>>31)<<1)+one/x; 249 } 250 251 if(ix < 0x3feb0000) { /* |x|<0.84375 */ 252 if(ix < 0x3c700000) /* |x|<2**-56 */ 253 return one-x; 254 z = x*x; 255 r = pp0+z*(pp1+z*(pp2+z*(pp3+z*pp4))); 256 s = one+z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*qq5)))); 257 y = r/s; 258 if(hx < 0x3fd00000) { /* x<1/4 */ 259 return one-(x+x*y); 260 } else { 261 r = x*y; 262 r += (x-half); 263 return half - r ; 264 } 265 } 266 if(ix < 0x3ff40000) { /* 0.84375 <= |x| < 1.25 */ 267 s = fabs(x)-one; 268 P = pa0+s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6))))); 269 Q = one+s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6))))); 270 if(hx>=0) { 271 z = one-erx; return z - P/Q; 272 } else { 273 z = erx+P/Q; return one+z; 274 } 275 } 276 if (ix < 0x403c0000) { /* |x|<28 */ 277 x = fabs(x); 278 s = one/(x*x); 279 if(ix< 0x4006DB6D) { /* |x| < 1/.35 ~ 2.857143*/ 280 R=ra0+s*(ra1+s*(ra2+s*(ra3+s*(ra4+s*( 281 ra5+s*(ra6+s*ra7)))))); 282 S=one+s*(sa1+s*(sa2+s*(sa3+s*(sa4+s*( 283 sa5+s*(sa6+s*(sa7+s*sa8))))))); 284 } else { /* |x| >= 1/.35 ~ 2.857143 */ 285 if(hx<0&&ix>=0x40180000) return two-tiny;/* x < -6 */ 286 R=rb0+s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*( 287 rb5+s*rb6))))); 288 S=one+s*(sb1+s*(sb2+s*(sb3+s*(sb4+s*( 289 sb5+s*(sb6+s*sb7)))))); 290 } 291 z = x; 292 SET_LOW_WORD(z,0); 293 r = exp(-z*z-0.5625) * exp((z-x)*(z+x)+R/S); 294 if(hx>0) return r/x; else return two-r/x; 295 } else { 296 if(hx>0) return tiny*tiny; else return two-tiny; 297 } 298 } 299 300 #if LDBL_MANT_DIG == DBL_MANT_DIG 301 __strong_alias(erfl, erf); 302 __strong_alias(erfcl, erf); 303 #endif /* LDBL_MANT_DIG == DBL_MANT_DIG */ 304