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 #include <sys/cdefs.h> 14 __FBSDID("$FreeBSD$"); 15 16 /* double erf(double x) 17 * double erfc(double x) 18 * x 19 * 2 |\ 20 * erf(x) = --------- | exp(-t*t)dt 21 * sqrt(pi) \| 22 * 0 23 * 24 * erfc(x) = 1-erf(x) 25 * Note that 26 * erf(-x) = -erf(x) 27 * erfc(-x) = 2 - erfc(x) 28 * 29 * Method: 30 * 1. For |x| in [0, 0.84375] 31 * erf(x) = x + x*R(x^2) 32 * erfc(x) = 1 - erf(x) if x in [-.84375,0.25] 33 * = 0.5 + ((0.5-x)-x*R) if x in [0.25,0.84375] 34 * where R = P/Q where P is an odd poly of degree 8 and 35 * Q is an odd poly of degree 10. 36 * -57.90 37 * | R - (erf(x)-x)/x | <= 2 38 * 39 * 40 * Remark. The formula is derived by noting 41 * erf(x) = (2/sqrt(pi))*(x - x^3/3 + x^5/10 - x^7/42 + ....) 42 * and that 43 * 2/sqrt(pi) = 1.128379167095512573896158903121545171688 44 * is close to one. The interval is chosen because the fix 45 * point of erf(x) is near 0.6174 (i.e., erf(x)=x when x is 46 * near 0.6174), and by some experiment, 0.84375 is chosen to 47 * guarantee the error is less than one ulp for erf. 48 * 49 * 2. For |x| in [0.84375,1.25], let s = |x| - 1, and 50 * c = 0.84506291151 rounded to single (24 bits) 51 * erf(x) = sign(x) * (c + P1(s)/Q1(s)) 52 * erfc(x) = (1-c) - P1(s)/Q1(s) if x > 0 53 * 1+(c+P1(s)/Q1(s)) if x < 0 54 * |P1/Q1 - (erf(|x|)-c)| <= 2**-59.06 55 * Remark: here we use the taylor series expansion at x=1. 56 * erf(1+s) = erf(1) + s*Poly(s) 57 * = 0.845.. + P1(s)/Q1(s) 58 * That is, we use rational approximation to approximate 59 * erf(1+s) - (c = (single)0.84506291151) 60 * Note that |P1/Q1|< 0.078 for x in [0.84375,1.25] 61 * where 62 * P1(s) = degree 6 poly in s 63 * Q1(s) = degree 6 poly in s 64 * 65 * 3. For x in [1.25,1/0.35(~2.857143)], 66 * erfc(x) = (1/x)*exp(-x*x-0.5625+R1/S1) 67 * erf(x) = 1 - erfc(x) 68 * where 69 * R1(z) = degree 7 poly in z, (z=1/x^2) 70 * S1(z) = degree 8 poly in z 71 * 72 * 4. For x in [1/0.35,28] 73 * erfc(x) = (1/x)*exp(-x*x-0.5625+R2/S2) if x > 0 74 * = 2.0 - (1/x)*exp(-x*x-0.5625+R2/S2) if -6<x<0 75 * = 2.0 - tiny (if x <= -6) 76 * erf(x) = sign(x)*(1.0 - erfc(x)) if x < 6, else 77 * erf(x) = sign(x)*(1.0 - tiny) 78 * where 79 * R2(z) = degree 6 poly in z, (z=1/x^2) 80 * S2(z) = degree 7 poly in z 81 * 82 * Note1: 83 * To compute exp(-x*x-0.5625+R/S), let s be a single 84 * precision number and s := x; then 85 * -x*x = -s*s + (s-x)*(s+x) 86 * exp(-x*x-0.5626+R/S) = 87 * exp(-s*s-0.5625)*exp((s-x)*(s+x)+R/S); 88 * Note2: 89 * Here 4 and 5 make use of the asymptotic series 90 * exp(-x*x) 91 * erfc(x) ~ ---------- * ( 1 + Poly(1/x^2) ) 92 * x*sqrt(pi) 93 * We use rational approximation to approximate 94 * g(s)=f(1/x^2) = log(erfc(x)*x) - x*x + 0.5625 95 * Here is the error bound for R1/S1 and R2/S2 96 * |R1/S1 - f(x)| < 2**(-62.57) 97 * |R2/S2 - f(x)| < 2**(-61.52) 98 * 99 * 5. For inf > x >= 28 100 * erf(x) = sign(x) *(1 - tiny) (raise inexact) 101 * erfc(x) = tiny*tiny (raise underflow) if x > 0 102 * = 2 - tiny if x<0 103 * 104 * 7. Special case: 105 * erf(0) = 0, erf(inf) = 1, erf(-inf) = -1, 106 * erfc(0) = 1, erfc(inf) = 0, erfc(-inf) = 2, 107 * erfc/erf(NaN) is NaN 108 */ 109 110 111 #include "math.h" 112 #include "math_private.h" 113 114 /* XXX Prevent compilers from erroneously constant folding: */ 115 static const volatile double tiny= 1e-300; 116 117 static const double 118 half= 0.5, 119 one = 1, 120 two = 2, 121 /* c = (float)0.84506291151 */ 122 erx = 8.45062911510467529297e-01, /* 0x3FEB0AC1, 0x60000000 */ 123 /* 124 * In the domain [0, 2**-28], only the first term in the power series 125 * expansion of erf(x) is used. The magnitude of the first neglected 126 * terms is less than 2**-84. 127 */ 128 efx = 1.28379167095512586316e-01, /* 0x3FC06EBA, 0x8214DB69 */ 129 efx8= 1.02703333676410069053e+00, /* 0x3FF06EBA, 0x8214DB69 */ 130 /* 131 * Coefficients for approximation to erf on [0,0.84375] 132 */ 133 pp0 = 1.28379167095512558561e-01, /* 0x3FC06EBA, 0x8214DB68 */ 134 pp1 = -3.25042107247001499370e-01, /* 0xBFD4CD7D, 0x691CB913 */ 135 pp2 = -2.84817495755985104766e-02, /* 0xBF9D2A51, 0xDBD7194F */ 136 pp3 = -5.77027029648944159157e-03, /* 0xBF77A291, 0x236668E4 */ 137 pp4 = -2.37630166566501626084e-05, /* 0xBEF8EAD6, 0x120016AC */ 138 qq1 = 3.97917223959155352819e-01, /* 0x3FD97779, 0xCDDADC09 */ 139 qq2 = 6.50222499887672944485e-02, /* 0x3FB0A54C, 0x5536CEBA */ 140 qq3 = 5.08130628187576562776e-03, /* 0x3F74D022, 0xC4D36B0F */ 141 qq4 = 1.32494738004321644526e-04, /* 0x3F215DC9, 0x221C1A10 */ 142 qq5 = -3.96022827877536812320e-06, /* 0xBED09C43, 0x42A26120 */ 143 /* 144 * Coefficients for approximation to erf in [0.84375,1.25] 145 */ 146 pa0 = -2.36211856075265944077e-03, /* 0xBF6359B8, 0xBEF77538 */ 147 pa1 = 4.14856118683748331666e-01, /* 0x3FDA8D00, 0xAD92B34D */ 148 pa2 = -3.72207876035701323847e-01, /* 0xBFD7D240, 0xFBB8C3F1 */ 149 pa3 = 3.18346619901161753674e-01, /* 0x3FD45FCA, 0x805120E4 */ 150 pa4 = -1.10894694282396677476e-01, /* 0xBFBC6398, 0x3D3E28EC */ 151 pa5 = 3.54783043256182359371e-02, /* 0x3FA22A36, 0x599795EB */ 152 pa6 = -2.16637559486879084300e-03, /* 0xBF61BF38, 0x0A96073F */ 153 qa1 = 1.06420880400844228286e-01, /* 0x3FBB3E66, 0x18EEE323 */ 154 qa2 = 5.40397917702171048937e-01, /* 0x3FE14AF0, 0x92EB6F33 */ 155 qa3 = 7.18286544141962662868e-02, /* 0x3FB2635C, 0xD99FE9A7 */ 156 qa4 = 1.26171219808761642112e-01, /* 0x3FC02660, 0xE763351F */ 157 qa5 = 1.36370839120290507362e-02, /* 0x3F8BEDC2, 0x6B51DD1C */ 158 qa6 = 1.19844998467991074170e-02, /* 0x3F888B54, 0x5735151D */ 159 /* 160 * Coefficients for approximation to erfc in [1.25,1/0.35] 161 */ 162 ra0 = -9.86494403484714822705e-03, /* 0xBF843412, 0x600D6435 */ 163 ra1 = -6.93858572707181764372e-01, /* 0xBFE63416, 0xE4BA7360 */ 164 ra2 = -1.05586262253232909814e+01, /* 0xC0251E04, 0x41B0E726 */ 165 ra3 = -6.23753324503260060396e+01, /* 0xC04F300A, 0xE4CBA38D */ 166 ra4 = -1.62396669462573470355e+02, /* 0xC0644CB1, 0x84282266 */ 167 ra5 = -1.84605092906711035994e+02, /* 0xC067135C, 0xEBCCABB2 */ 168 ra6 = -8.12874355063065934246e+01, /* 0xC0545265, 0x57E4D2F2 */ 169 ra7 = -9.81432934416914548592e+00, /* 0xC023A0EF, 0xC69AC25C */ 170 sa1 = 1.96512716674392571292e+01, /* 0x4033A6B9, 0xBD707687 */ 171 sa2 = 1.37657754143519042600e+02, /* 0x4061350C, 0x526AE721 */ 172 sa3 = 4.34565877475229228821e+02, /* 0x407B290D, 0xD58A1A71 */ 173 sa4 = 6.45387271733267880336e+02, /* 0x40842B19, 0x21EC2868 */ 174 sa5 = 4.29008140027567833386e+02, /* 0x407AD021, 0x57700314 */ 175 sa6 = 1.08635005541779435134e+02, /* 0x405B28A3, 0xEE48AE2C */ 176 sa7 = 6.57024977031928170135e+00, /* 0x401A47EF, 0x8E484A93 */ 177 sa8 = -6.04244152148580987438e-02, /* 0xBFAEEFF2, 0xEE749A62 */ 178 /* 179 * Coefficients for approximation to erfc in [1/.35,28] 180 */ 181 rb0 = -9.86494292470009928597e-03, /* 0xBF843412, 0x39E86F4A */ 182 rb1 = -7.99283237680523006574e-01, /* 0xBFE993BA, 0x70C285DE */ 183 rb2 = -1.77579549177547519889e+01, /* 0xC031C209, 0x555F995A */ 184 rb3 = -1.60636384855821916062e+02, /* 0xC064145D, 0x43C5ED98 */ 185 rb4 = -6.37566443368389627722e+02, /* 0xC083EC88, 0x1375F228 */ 186 rb5 = -1.02509513161107724954e+03, /* 0xC0900461, 0x6A2E5992 */ 187 rb6 = -4.83519191608651397019e+02, /* 0xC07E384E, 0x9BDC383F */ 188 sb1 = 3.03380607434824582924e+01, /* 0x403E568B, 0x261D5190 */ 189 sb2 = 3.25792512996573918826e+02, /* 0x40745CAE, 0x221B9F0A */ 190 sb3 = 1.53672958608443695994e+03, /* 0x409802EB, 0x189D5118 */ 191 sb4 = 3.19985821950859553908e+03, /* 0x40A8FFB7, 0x688C246A */ 192 sb5 = 2.55305040643316442583e+03, /* 0x40A3F219, 0xCEDF3BE6 */ 193 sb6 = 4.74528541206955367215e+02, /* 0x407DA874, 0xE79FE763 */ 194 sb7 = -2.24409524465858183362e+01; /* 0xC03670E2, 0x42712D62 */ 195 196 double 197 erf(double x) 198 { 199 int32_t hx,ix,i; 200 double R,S,P,Q,s,y,z,r; 201 GET_HIGH_WORD(hx,x); 202 ix = hx&0x7fffffff; 203 if(ix>=0x7ff00000) { /* erf(nan)=nan */ 204 i = ((u_int32_t)hx>>31)<<1; 205 return (double)(1-i)+one/x; /* erf(+-inf)=+-1 */ 206 } 207 208 if(ix < 0x3feb0000) { /* |x|<0.84375 */ 209 if(ix < 0x3e300000) { /* |x|<2**-28 */ 210 if (ix < 0x00800000) 211 return (8*x+efx8*x)/8; /* avoid spurious underflow */ 212 return x + efx*x; 213 } 214 z = x*x; 215 r = pp0+z*(pp1+z*(pp2+z*(pp3+z*pp4))); 216 s = one+z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*qq5)))); 217 y = r/s; 218 return x + x*y; 219 } 220 if(ix < 0x3ff40000) { /* 0.84375 <= |x| < 1.25 */ 221 s = fabs(x)-one; 222 P = pa0+s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6))))); 223 Q = one+s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6))))); 224 if(hx>=0) return erx + P/Q; else return -erx - P/Q; 225 } 226 if (ix >= 0x40180000) { /* inf>|x|>=6 */ 227 if(hx>=0) return one-tiny; else return tiny-one; 228 } 229 x = fabs(x); 230 s = one/(x*x); 231 if(ix< 0x4006DB6E) { /* |x| < 1/0.35 */ 232 R=ra0+s*(ra1+s*(ra2+s*(ra3+s*(ra4+s*(ra5+s*(ra6+s*ra7)))))); 233 S=one+s*(sa1+s*(sa2+s*(sa3+s*(sa4+s*(sa5+s*(sa6+s*(sa7+ 234 s*sa8))))))); 235 } else { /* |x| >= 1/0.35 */ 236 R=rb0+s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*(rb5+s*rb6))))); 237 S=one+s*(sb1+s*(sb2+s*(sb3+s*(sb4+s*(sb5+s*(sb6+s*sb7)))))); 238 } 239 z = x; 240 SET_LOW_WORD(z,0); 241 r = __ieee754_exp(-z*z-0.5625)*__ieee754_exp((z-x)*(z+x)+R/S); 242 if(hx>=0) return one-r/x; else return r/x-one; 243 } 244 245 #if (LDBL_MANT_DIG == 53) 246 __weak_reference(erf, erfl); 247 #endif 248 249 double 250 erfc(double x) 251 { 252 int32_t hx,ix; 253 double R,S,P,Q,s,y,z,r; 254 GET_HIGH_WORD(hx,x); 255 ix = hx&0x7fffffff; 256 if(ix>=0x7ff00000) { /* erfc(nan)=nan */ 257 /* erfc(+-inf)=0,2 */ 258 return (double)(((u_int32_t)hx>>31)<<1)+one/x; 259 } 260 261 if(ix < 0x3feb0000) { /* |x|<0.84375 */ 262 if(ix < 0x3c700000) /* |x|<2**-56 */ 263 return one-x; 264 z = x*x; 265 r = pp0+z*(pp1+z*(pp2+z*(pp3+z*pp4))); 266 s = one+z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*qq5)))); 267 y = r/s; 268 if(hx < 0x3fd00000) { /* x<1/4 */ 269 return one-(x+x*y); 270 } else { 271 r = x*y; 272 r += (x-half); 273 return half - r ; 274 } 275 } 276 if(ix < 0x3ff40000) { /* 0.84375 <= |x| < 1.25 */ 277 s = fabs(x)-one; 278 P = pa0+s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6))))); 279 Q = one+s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6))))); 280 if(hx>=0) { 281 z = one-erx; return z - P/Q; 282 } else { 283 z = erx+P/Q; return one+z; 284 } 285 } 286 if (ix < 0x403c0000) { /* |x|<28 */ 287 x = fabs(x); 288 s = one/(x*x); 289 if(ix< 0x4006DB6D) { /* |x| < 1/.35 ~ 2.857143*/ 290 R=ra0+s*(ra1+s*(ra2+s*(ra3+s*(ra4+s*(ra5+s*(ra6+s*ra7)))))); 291 S=one+s*(sa1+s*(sa2+s*(sa3+s*(sa4+s*(sa5+s*(sa6+s*(sa7+ 292 s*sa8))))))); 293 } else { /* |x| >= 1/.35 ~ 2.857143 */ 294 if(hx<0&&ix>=0x40180000) return two-tiny;/* x < -6 */ 295 R=rb0+s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*(rb5+s*rb6))))); 296 S=one+s*(sb1+s*(sb2+s*(sb3+s*(sb4+s*(sb5+s*(sb6+s*sb7)))))); 297 } 298 z = x; 299 SET_LOW_WORD(z,0); 300 r = __ieee754_exp(-z*z-0.5625)*__ieee754_exp((z-x)*(z+x)+R/S); 301 if(hx>0) return r/x; else return two-r/x; 302 } else { 303 if(hx>0) return tiny*tiny; else return two-tiny; 304 } 305 } 306 307 #if (LDBL_MANT_DIG == 53) 308 __weak_reference(erfc, erfcl); 309 #endif 310