1 /* 2 * ==================================================== 3 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. 4 * 5 * Developed at SunPro, a Sun Microsystems, Inc. business. 6 * Permission to use, copy, modify, and distribute this 7 * software is freely granted, provided that this notice 8 * is preserved. 9 * ==================================================== 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 <float.h> 107 #include "math.h" 108 #include "math_private.h" 109 110 /* XXX Prevent compilers from erroneously constant folding: */ 111 static const volatile double tiny= 1e-300; 112 113 static const double 114 half= 0.5, 115 one = 1, 116 two = 2, 117 /* c = (float)0.84506291151 */ 118 erx = 8.45062911510467529297e-01, /* 0x3FEB0AC1, 0x60000000 */ 119 /* 120 * In the domain [0, 2**-28], only the first term in the power series 121 * expansion of erf(x) is used. The magnitude of the first neglected 122 * terms is less than 2**-84. 123 */ 124 efx = 1.28379167095512586316e-01, /* 0x3FC06EBA, 0x8214DB69 */ 125 efx8= 1.02703333676410069053e+00, /* 0x3FF06EBA, 0x8214DB69 */ 126 /* 127 * Coefficients for approximation to erf on [0,0.84375] 128 */ 129 pp0 = 1.28379167095512558561e-01, /* 0x3FC06EBA, 0x8214DB68 */ 130 pp1 = -3.25042107247001499370e-01, /* 0xBFD4CD7D, 0x691CB913 */ 131 pp2 = -2.84817495755985104766e-02, /* 0xBF9D2A51, 0xDBD7194F */ 132 pp3 = -5.77027029648944159157e-03, /* 0xBF77A291, 0x236668E4 */ 133 pp4 = -2.37630166566501626084e-05, /* 0xBEF8EAD6, 0x120016AC */ 134 qq1 = 3.97917223959155352819e-01, /* 0x3FD97779, 0xCDDADC09 */ 135 qq2 = 6.50222499887672944485e-02, /* 0x3FB0A54C, 0x5536CEBA */ 136 qq3 = 5.08130628187576562776e-03, /* 0x3F74D022, 0xC4D36B0F */ 137 qq4 = 1.32494738004321644526e-04, /* 0x3F215DC9, 0x221C1A10 */ 138 qq5 = -3.96022827877536812320e-06, /* 0xBED09C43, 0x42A26120 */ 139 /* 140 * Coefficients for approximation to erf in [0.84375,1.25] 141 */ 142 pa0 = -2.36211856075265944077e-03, /* 0xBF6359B8, 0xBEF77538 */ 143 pa1 = 4.14856118683748331666e-01, /* 0x3FDA8D00, 0xAD92B34D */ 144 pa2 = -3.72207876035701323847e-01, /* 0xBFD7D240, 0xFBB8C3F1 */ 145 pa3 = 3.18346619901161753674e-01, /* 0x3FD45FCA, 0x805120E4 */ 146 pa4 = -1.10894694282396677476e-01, /* 0xBFBC6398, 0x3D3E28EC */ 147 pa5 = 3.54783043256182359371e-02, /* 0x3FA22A36, 0x599795EB */ 148 pa6 = -2.16637559486879084300e-03, /* 0xBF61BF38, 0x0A96073F */ 149 qa1 = 1.06420880400844228286e-01, /* 0x3FBB3E66, 0x18EEE323 */ 150 qa2 = 5.40397917702171048937e-01, /* 0x3FE14AF0, 0x92EB6F33 */ 151 qa3 = 7.18286544141962662868e-02, /* 0x3FB2635C, 0xD99FE9A7 */ 152 qa4 = 1.26171219808761642112e-01, /* 0x3FC02660, 0xE763351F */ 153 qa5 = 1.36370839120290507362e-02, /* 0x3F8BEDC2, 0x6B51DD1C */ 154 qa6 = 1.19844998467991074170e-02, /* 0x3F888B54, 0x5735151D */ 155 /* 156 * Coefficients for approximation to erfc in [1.25,1/0.35] 157 */ 158 ra0 = -9.86494403484714822705e-03, /* 0xBF843412, 0x600D6435 */ 159 ra1 = -6.93858572707181764372e-01, /* 0xBFE63416, 0xE4BA7360 */ 160 ra2 = -1.05586262253232909814e+01, /* 0xC0251E04, 0x41B0E726 */ 161 ra3 = -6.23753324503260060396e+01, /* 0xC04F300A, 0xE4CBA38D */ 162 ra4 = -1.62396669462573470355e+02, /* 0xC0644CB1, 0x84282266 */ 163 ra5 = -1.84605092906711035994e+02, /* 0xC067135C, 0xEBCCABB2 */ 164 ra6 = -8.12874355063065934246e+01, /* 0xC0545265, 0x57E4D2F2 */ 165 ra7 = -9.81432934416914548592e+00, /* 0xC023A0EF, 0xC69AC25C */ 166 sa1 = 1.96512716674392571292e+01, /* 0x4033A6B9, 0xBD707687 */ 167 sa2 = 1.37657754143519042600e+02, /* 0x4061350C, 0x526AE721 */ 168 sa3 = 4.34565877475229228821e+02, /* 0x407B290D, 0xD58A1A71 */ 169 sa4 = 6.45387271733267880336e+02, /* 0x40842B19, 0x21EC2868 */ 170 sa5 = 4.29008140027567833386e+02, /* 0x407AD021, 0x57700314 */ 171 sa6 = 1.08635005541779435134e+02, /* 0x405B28A3, 0xEE48AE2C */ 172 sa7 = 6.57024977031928170135e+00, /* 0x401A47EF, 0x8E484A93 */ 173 sa8 = -6.04244152148580987438e-02, /* 0xBFAEEFF2, 0xEE749A62 */ 174 /* 175 * Coefficients for approximation to erfc in [1/.35,28] 176 */ 177 rb0 = -9.86494292470009928597e-03, /* 0xBF843412, 0x39E86F4A */ 178 rb1 = -7.99283237680523006574e-01, /* 0xBFE993BA, 0x70C285DE */ 179 rb2 = -1.77579549177547519889e+01, /* 0xC031C209, 0x555F995A */ 180 rb3 = -1.60636384855821916062e+02, /* 0xC064145D, 0x43C5ED98 */ 181 rb4 = -6.37566443368389627722e+02, /* 0xC083EC88, 0x1375F228 */ 182 rb5 = -1.02509513161107724954e+03, /* 0xC0900461, 0x6A2E5992 */ 183 rb6 = -4.83519191608651397019e+02, /* 0xC07E384E, 0x9BDC383F */ 184 sb1 = 3.03380607434824582924e+01, /* 0x403E568B, 0x261D5190 */ 185 sb2 = 3.25792512996573918826e+02, /* 0x40745CAE, 0x221B9F0A */ 186 sb3 = 1.53672958608443695994e+03, /* 0x409802EB, 0x189D5118 */ 187 sb4 = 3.19985821950859553908e+03, /* 0x40A8FFB7, 0x688C246A */ 188 sb5 = 2.55305040643316442583e+03, /* 0x40A3F219, 0xCEDF3BE6 */ 189 sb6 = 4.74528541206955367215e+02, /* 0x407DA874, 0xE79FE763 */ 190 sb7 = -2.24409524465858183362e+01; /* 0xC03670E2, 0x42712D62 */ 191 192 double 193 erf(double x) 194 { 195 int32_t hx,ix,i; 196 double R,S,P,Q,s,y,z,r; 197 GET_HIGH_WORD(hx,x); 198 ix = hx&0x7fffffff; 199 if(ix>=0x7ff00000) { /* erf(nan)=nan */ 200 i = ((u_int32_t)hx>>31)<<1; 201 return (double)(1-i)+one/x; /* erf(+-inf)=+-1 */ 202 } 203 204 if(ix < 0x3feb0000) { /* |x|<0.84375 */ 205 if(ix < 0x3e300000) { /* |x|<2**-28 */ 206 if (ix < 0x00800000) 207 return (8*x+efx8*x)/8; /* avoid spurious underflow */ 208 return x + efx*x; 209 } 210 z = x*x; 211 r = pp0+z*(pp1+z*(pp2+z*(pp3+z*pp4))); 212 s = one+z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*qq5)))); 213 y = r/s; 214 return x + x*y; 215 } 216 if(ix < 0x3ff40000) { /* 0.84375 <= |x| < 1.25 */ 217 s = fabs(x)-one; 218 P = pa0+s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6))))); 219 Q = one+s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6))))); 220 if(hx>=0) return erx + P/Q; else return -erx - P/Q; 221 } 222 if (ix >= 0x40180000) { /* inf>|x|>=6 */ 223 if(hx>=0) return one-tiny; else return tiny-one; 224 } 225 x = fabs(x); 226 s = one/(x*x); 227 if(ix< 0x4006DB6E) { /* |x| < 1/0.35 */ 228 R=ra0+s*(ra1+s*(ra2+s*(ra3+s*(ra4+s*(ra5+s*(ra6+s*ra7)))))); 229 S=one+s*(sa1+s*(sa2+s*(sa3+s*(sa4+s*(sa5+s*(sa6+s*(sa7+ 230 s*sa8))))))); 231 } else { /* |x| >= 1/0.35 */ 232 R=rb0+s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*(rb5+s*rb6))))); 233 S=one+s*(sb1+s*(sb2+s*(sb3+s*(sb4+s*(sb5+s*(sb6+s*sb7)))))); 234 } 235 z = x; 236 SET_LOW_WORD(z,0); 237 r = exp(-z*z-0.5625)*exp((z-x)*(z+x)+R/S); 238 if(hx>=0) return one-r/x; else return r/x-one; 239 } 240 241 #if (LDBL_MANT_DIG == 53) 242 __weak_reference(erf, erfl); 243 #endif 244 245 double 246 erfc(double x) 247 { 248 int32_t hx,ix; 249 double R,S,P,Q,s,y,z,r; 250 GET_HIGH_WORD(hx,x); 251 ix = hx&0x7fffffff; 252 if(ix>=0x7ff00000) { /* erfc(nan)=nan */ 253 /* erfc(+-inf)=0,2 */ 254 return (double)(((u_int32_t)hx>>31)<<1)+one/x; 255 } 256 257 if(ix < 0x3feb0000) { /* |x|<0.84375 */ 258 if(ix < 0x3c700000) /* |x|<2**-56 */ 259 return one-x; 260 z = x*x; 261 r = pp0+z*(pp1+z*(pp2+z*(pp3+z*pp4))); 262 s = one+z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*qq5)))); 263 y = r/s; 264 if(hx < 0x3fd00000) { /* x<1/4 */ 265 return one-(x+x*y); 266 } else { 267 r = x*y; 268 r += (x-half); 269 return half - r ; 270 } 271 } 272 if(ix < 0x3ff40000) { /* 0.84375 <= |x| < 1.25 */ 273 s = fabs(x)-one; 274 P = pa0+s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6))))); 275 Q = one+s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6))))); 276 if(hx>=0) { 277 z = one-erx; return z - P/Q; 278 } else { 279 z = erx+P/Q; return one+z; 280 } 281 } 282 if (ix < 0x403c0000) { /* |x|<28 */ 283 x = fabs(x); 284 s = one/(x*x); 285 if(ix< 0x4006DB6D) { /* |x| < 1/.35 ~ 2.857143*/ 286 R=ra0+s*(ra1+s*(ra2+s*(ra3+s*(ra4+s*(ra5+s*(ra6+s*ra7)))))); 287 S=one+s*(sa1+s*(sa2+s*(sa3+s*(sa4+s*(sa5+s*(sa6+s*(sa7+ 288 s*sa8))))))); 289 } else { /* |x| >= 1/.35 ~ 2.857143 */ 290 if(hx<0&&ix>=0x40180000) return two-tiny;/* x < -6 */ 291 R=rb0+s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*(rb5+s*rb6))))); 292 S=one+s*(sb1+s*(sb2+s*(sb3+s*(sb4+s*(sb5+s*(sb6+s*sb7)))))); 293 } 294 z = x; 295 SET_LOW_WORD(z,0); 296 r = exp(-z*z-0.5625)*exp((z-x)*(z+x)+R/S); 297 if(hx>0) return r/x; else return two-r/x; 298 } else { 299 if(hx>0) return tiny*tiny; else return two-tiny; 300 } 301 } 302 303 #if (LDBL_MANT_DIG == 53) 304 __weak_reference(erfc, erfcl); 305 #endif 306