s_erf.c revision 97409
18478Swollman/* @(#)s_erf.c 5.1 93/09/24 */ 28478Swollman/* 38478Swollman * ==================================================== 48478Swollman * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. 58478Swollman * 68478Swollman * Developed at SunPro, a Sun Microsystems, Inc. business. 78478Swollman * Permission to use, copy, modify, and distribute this 88478Swollman * software is freely granted, provided that this notice 98478Swollman * is preserved. 108478Swollman * ==================================================== 118478Swollman */ 128478Swollman 138478Swollman#ifndef lint 148478Swollmanstatic char rcsid[] = "$FreeBSD: head/lib/msun/src/s_erf.c 97409 2002-05-28 17:51:46Z alfred $"; 158478Swollman#endif 168478Swollman 178478Swollman/* double erf(double x) 188478Swollman * double erfc(double x) 198478Swollman * x 208478Swollman * 2 |\ 218478Swollman * erf(x) = --------- | exp(-t*t)dt 228478Swollman * sqrt(pi) \| 238478Swollman * 0 248478Swollman * 258478Swollman * erfc(x) = 1-erf(x) 268478Swollman * Note that 278478Swollman * erf(-x) = -erf(x) 288478Swollman * erfc(-x) = 2 - erfc(x) 298478Swollman * 30114589Sobrien * Method: 318478Swollman * 1. For |x| in [0, 0.84375] 3236999Scharnier * erf(x) = x + x*R(x^2) 338478Swollman * erfc(x) = 1 - erf(x) if x in [-.84375,0.25] 348478Swollman * = 0.5 + ((0.5-x)-x*R) if x in [0.25,0.84375] 358478Swollman * where R = P/Q where P is an odd poly of degree 8 and 368478Swollman * Q is an odd poly of degree 10. 378478Swollman * -57.90 3836999Scharnier * | R - (erf(x)-x)/x | <= 2 39114589Sobrien * 4036999Scharnier * 41114589Sobrien * Remark. The formula is derived by noting 42114589Sobrien * erf(x) = (2/sqrt(pi))*(x - x^3/3 + x^5/10 - x^7/42 + ....) 438478Swollman * and that 44136104Sdes * 2/sqrt(pi) = 1.128379167095512573896158903121545171688 45136104Sdes * is close to one. The interval is chosen because the fix 46136104Sdes * point of erf(x) is near 0.6174 (i.e., erf(x)=x when x is 47136104Sdes * near 0.6174), and by some experiment, 0.84375 is chosen to 4836999Scharnier * guarantee the error is less than one ulp for erf. 49242451Salfred * 50136104Sdes * 2. For |x| in [0.84375,1.25], let s = |x| - 1, and 51136104Sdes * c = 0.84506291151 rounded to single (24 bits) 52136104Sdes * erf(x) = sign(x) * (c + P1(s)/Q1(s)) 538478Swollman * erfc(x) = (1-c) - P1(s)/Q1(s) if x > 0 5478732Sdd * 1+(c+P1(s)/Q1(s)) if x < 0 5596381Salfred * |P1/Q1 - (erf(|x|)-c)| <= 2**-59.06 56136104Sdes * Remark: here we use the taylor series expansion at x=1. 578478Swollman * erf(1+s) = erf(1) + s*Poly(s) 588478Swollman * = 0.845.. + P1(s)/Q1(s) 59136104Sdes * That is, we use rational approximation to approximate 608478Swollman * erf(1+s) - (c = (single)0.84506291151) 61136104Sdes * Note that |P1/Q1|< 0.078 for x in [0.84375,1.25] 62136104Sdes * where 63136104Sdes * P1(s) = degree 6 poly in s 64242451Salfred * Q1(s) = degree 6 poly in s 65136104Sdes * 66242451Salfred * 3. For x in [1.25,1/0.35(~2.857143)], 67242486Salfred * erfc(x) = (1/x)*exp(-x*x-0.5625+R1/S1) 68136104Sdes * erf(x) = 1 - erfc(x) 69136104Sdes * where 70136104Sdes * R1(z) = degree 7 poly in z, (z=1/x^2) 71136104Sdes * S1(z) = degree 8 poly in z 72136104Sdes * 73136104Sdes * 4. For x in [1/0.35,28] 74136104Sdes * erfc(x) = (1/x)*exp(-x*x-0.5625+R2/S2) if x > 0 75163852Sjhb * = 2.0 - (1/x)*exp(-x*x-0.5625+R2/S2) if -6<x<0 76136104Sdes * = 2.0 - tiny (if x <= -6) 77158083Sps * erf(x) = sign(x)*(1.0 - erfc(x)) if x < 6, else 78136104Sdes * erf(x) = sign(x)*(1.0 - tiny) 79158083Sps * where 80136104Sdes * R2(z) = degree 6 poly in z, (z=1/x^2) 81163852Sjhb * S2(z) = degree 7 poly in z 82158083Sps * 83158083Sps * Note1: 84158083Sps * To compute exp(-x*x-0.5625+R/S), let s be a single 85163852Sjhb * precision number and s := x; then 86136104Sdes * -x*x = -s*s + (s-x)*(s+x) 87136104Sdes * exp(-x*x-0.5626+R/S) = 88136104Sdes * exp(-s*s-0.5625)*exp((s-x)*(s+x)+R/S); 89136104Sdes * Note2: 90136110Sdes * Here 4 and 5 make use of the asymptotic series 91136104Sdes * exp(-x*x) 92136104Sdes * erfc(x) ~ ---------- * ( 1 + Poly(1/x^2) ) 93136104Sdes * x*sqrt(pi) 94136104Sdes * We use rational approximation to approximate 95136104Sdes * g(s)=f(1/x^2) = log(erfc(x)*x) - x*x + 0.5625 96136104Sdes * Here is the error bound for R1/S1 and R2/S2 97242451Salfred * |R1/S1 - f(x)| < 2**(-62.57) 98242451Salfred * |R2/S2 - f(x)| < 2**(-61.52) 99242451Salfred * 100242451Salfred * 5. For inf > x >= 28 101242451Salfred * erf(x) = sign(x) *(1 - tiny) (raise inexact) 102242451Salfred * erfc(x) = tiny*tiny (raise underflow) if x > 0 103242451Salfred * = 2 - tiny if x<0 104242451Salfred * 105242451Salfred * 7. Special case: 106242451Salfred * erf(0) = 0, erf(inf) = 1, erf(-inf) = -1, 107242451Salfred * erfc(0) = 1, erfc(inf) = 0, erfc(-inf) = 2, 108242451Salfred * erfc/erf(NaN) is NaN 109242451Salfred */ 110242451Salfred 111242451Salfred 112242451Salfred#include "math.h" 113242451Salfred#include "math_private.h" 114242451Salfred 115242451Salfredstatic const double 116242451Salfredtiny = 1e-300, 117242451Salfredhalf= 5.00000000000000000000e-01, /* 0x3FE00000, 0x00000000 */ 118242451Salfredone = 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */ 119242451Salfredtwo = 2.00000000000000000000e+00, /* 0x40000000, 0x00000000 */ 120242451Salfred /* c = (float)0.84506291151 */ 121242451Salfrederx = 8.45062911510467529297e-01, /* 0x3FEB0AC1, 0x60000000 */ 122242451Salfred/* 1238478Swollman * Coefficients for approximation to erf on [0,0.84375] 12492542Simp */ 1258478Swollmanefx = 1.28379167095512586316e-01, /* 0x3FC06EBA, 0x8214DB69 */ 126136104Sdesefx8= 1.02703333676410069053e+00, /* 0x3FF06EBA, 0x8214DB69 */ 12793491Sphkpp0 = 1.28379167095512558561e-01, /* 0x3FC06EBA, 0x8214DB68 */ 12893491Sphkpp1 = -3.25042107247001499370e-01, /* 0xBFD4CD7D, 0x691CB913 */ 129242451Salfredpp2 = -2.84817495755985104766e-02, /* 0xBF9D2A51, 0xDBD7194F */ 1308478Swollmanpp3 = -5.77027029648944159157e-03, /* 0xBF77A291, 0x236668E4 */ 131242451Salfredpp4 = -2.37630166566501626084e-05, /* 0xBEF8EAD6, 0x120016AC */ 1328478Swollmanqq1 = 3.97917223959155352819e-01, /* 0x3FD97779, 0xCDDADC09 */ 133242451Salfredqq2 = 6.50222499887672944485e-02, /* 0x3FB0A54C, 0x5536CEBA */ 134242451Salfredqq3 = 5.08130628187576562776e-03, /* 0x3F74D022, 0xC4D36B0F */ 135242451Salfredqq4 = 1.32494738004321644526e-04, /* 0x3F215DC9, 0x221C1A10 */ 1368478Swollmanqq5 = -3.96022827877536812320e-06, /* 0xBED09C43, 0x42A26120 */ 1378478Swollman/* 1388478Swollman * Coefficients for approximation to erf in [0.84375,1.25] 1398478Swollman */ 1408478Swollmanpa0 = -2.36211856075265944077e-03, /* 0xBF6359B8, 0xBEF77538 */ 1418478Swollmanpa1 = 4.14856118683748331666e-01, /* 0x3FDA8D00, 0xAD92B34D */ 142136104Sdespa2 = -3.72207876035701323847e-01, /* 0xBFD7D240, 0xFBB8C3F1 */ 143136104Sdespa3 = 3.18346619901161753674e-01, /* 0x3FD45FCA, 0x805120E4 */ 1448478Swollmanpa4 = -1.10894694282396677476e-01, /* 0xBFBC6398, 0x3D3E28EC */ 1458478Swollmanpa5 = 3.54783043256182359371e-02, /* 0x3FA22A36, 0x599795EB */ 146242451Salfredpa6 = -2.16637559486879084300e-03, /* 0xBF61BF38, 0x0A96073F */ 147242451Salfredqa1 = 1.06420880400844228286e-01, /* 0x3FBB3E66, 0x18EEE323 */ 148242451Salfredqa2 = 5.40397917702171048937e-01, /* 0x3FE14AF0, 0x92EB6F33 */ 149242451Salfredqa3 = 7.18286544141962662868e-02, /* 0x3FB2635C, 0xD99FE9A7 */ 150242451Salfredqa4 = 1.26171219808761642112e-01, /* 0x3FC02660, 0xE763351F */ 151136104Sdesqa5 = 1.36370839120290507362e-02, /* 0x3F8BEDC2, 0x6B51DD1C */ 1528478Swollmanqa6 = 1.19844998467991074170e-02, /* 0x3F888B54, 0x5735151D */ 1538478Swollman/* 154136104Sdes * Coefficients for approximation to erfc in [1.25,1/0.35] 155291480Ssmh */ 156291480Ssmhra0 = -9.86494403484714822705e-03, /* 0xBF843412, 0x600D6435 */ 157291480Ssmhra1 = -6.93858572707181764372e-01, /* 0xBFE63416, 0xE4BA7360 */ 158291480Ssmhra2 = -1.05586262253232909814e+01, /* 0xC0251E04, 0x41B0E726 */ 159291480Ssmhra3 = -6.23753324503260060396e+01, /* 0xC04F300A, 0xE4CBA38D */ 160291480Ssmhra4 = -1.62396669462573470355e+02, /* 0xC0644CB1, 0x84282266 */ 161291480Ssmhra5 = -1.84605092906711035994e+02, /* 0xC067135C, 0xEBCCABB2 */ 162291480Ssmhra6 = -8.12874355063065934246e+01, /* 0xC0545265, 0x57E4D2F2 */ 163291480Ssmhra7 = -9.81432934416914548592e+00, /* 0xC023A0EF, 0xC69AC25C */ 164291480Ssmhsa1 = 1.96512716674392571292e+01, /* 0x4033A6B9, 0xBD707687 */ 165291480Ssmhsa2 = 1.37657754143519042600e+02, /* 0x4061350C, 0x526AE721 */ 166291480Ssmhsa3 = 4.34565877475229228821e+02, /* 0x407B290D, 0xD58A1A71 */ 167291480Ssmhsa4 = 6.45387271733267880336e+02, /* 0x40842B19, 0x21EC2868 */ 168291480Ssmhsa5 = 4.29008140027567833386e+02, /* 0x407AD021, 0x57700314 */ 169291480Ssmhsa6 = 1.08635005541779435134e+02, /* 0x405B28A3, 0xEE48AE2C */ 170291480Ssmhsa7 = 6.57024977031928170135e+00, /* 0x401A47EF, 0x8E484A93 */ 17193491Sphksa8 = -6.04244152148580987438e-02, /* 0xBFAEEFF2, 0xEE749A62 */ 172291480Ssmh/* 173291480Ssmh * Coefficients for approximation to erfc in [1/.35,28] 17493491Sphk */ 17594272Sphkrb0 = -9.86494292470009928597e-03, /* 0xBF843412, 0x39E86F4A */ 17693491Sphkrb1 = -7.99283237680523006574e-01, /* 0xBFE993BA, 0x70C285DE */ 17794272Sphkrb2 = -1.77579549177547519889e+01, /* 0xC031C209, 0x555F995A */ 17893491Sphkrb3 = -1.60636384855821916062e+02, /* 0xC064145D, 0x43C5ED98 */ 179291480Ssmhrb4 = -6.37566443368389627722e+02, /* 0xC083EC88, 0x1375F228 */ 1808478Swollmanrb5 = -1.02509513161107724954e+03, /* 0xC0900461, 0x6A2E5992 */ 18193491Sphkrb6 = -4.83519191608651397019e+02, /* 0xC07E384E, 0x9BDC383F */ 18293491Sphksb1 = 3.03380607434824582924e+01, /* 0x403E568B, 0x261D5190 */ 18393491Sphksb2 = 3.25792512996573918826e+02, /* 0x40745CAE, 0x221B9F0A */ 18493491Sphksb3 = 1.53672958608443695994e+03, /* 0x409802EB, 0x189D5118 */ 18594272Sphksb4 = 3.19985821950859553908e+03, /* 0x40A8FFB7, 0x688C246A */ 18693491Sphksb5 = 2.55305040643316442583e+03, /* 0x40A3F219, 0xCEDF3BE6 */ 18793491Sphksb6 = 4.74528541206955367215e+02, /* 0x407DA874, 0xE79FE763 */ 1888478Swollmansb7 = -2.24409524465858183362e+01; /* 0xC03670E2, 0x42712D62 */ 18993491Sphk 19094272Sphk double erf(double x) 1918478Swollman{ 19293491Sphk int32_t hx,ix,i; 1938478Swollman double R,S,P,Q,s,y,z,r; 194 GET_HIGH_WORD(hx,x); 195 ix = hx&0x7fffffff; 196 if(ix>=0x7ff00000) { /* erf(nan)=nan */ 197 i = ((u_int32_t)hx>>31)<<1; 198 return (double)(1-i)+one/x; /* erf(+-inf)=+-1 */ 199 } 200 201 if(ix < 0x3feb0000) { /* |x|<0.84375 */ 202 if(ix < 0x3e300000) { /* |x|<2**-28 */ 203 if (ix < 0x00800000) 204 return 0.125*(8.0*x+efx8*x); /*avoid underflow */ 205 return x + efx*x; 206 } 207 z = x*x; 208 r = pp0+z*(pp1+z*(pp2+z*(pp3+z*pp4))); 209 s = one+z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*qq5)))); 210 y = r/s; 211 return x + x*y; 212 } 213 if(ix < 0x3ff40000) { /* 0.84375 <= |x| < 1.25 */ 214 s = fabs(x)-one; 215 P = pa0+s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6))))); 216 Q = one+s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6))))); 217 if(hx>=0) return erx + P/Q; else return -erx - P/Q; 218 } 219 if (ix >= 0x40180000) { /* inf>|x|>=6 */ 220 if(hx>=0) return one-tiny; else return tiny-one; 221 } 222 x = fabs(x); 223 s = one/(x*x); 224 if(ix< 0x4006DB6E) { /* |x| < 1/0.35 */ 225 R=ra0+s*(ra1+s*(ra2+s*(ra3+s*(ra4+s*( 226 ra5+s*(ra6+s*ra7)))))); 227 S=one+s*(sa1+s*(sa2+s*(sa3+s*(sa4+s*( 228 sa5+s*(sa6+s*(sa7+s*sa8))))))); 229 } else { /* |x| >= 1/0.35 */ 230 R=rb0+s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*( 231 rb5+s*rb6))))); 232 S=one+s*(sb1+s*(sb2+s*(sb3+s*(sb4+s*( 233 sb5+s*(sb6+s*sb7)))))); 234 } 235 z = x; 236 SET_LOW_WORD(z,0); 237 r = __ieee754_exp(-z*z-0.5625)*__ieee754_exp((z-x)*(z+x)+R/S); 238 if(hx>=0) return one-r/x; else return r/x-one; 239} 240 241 double erfc(double x) 242{ 243 int32_t hx,ix; 244 double R,S,P,Q,s,y,z,r; 245 GET_HIGH_WORD(hx,x); 246 ix = hx&0x7fffffff; 247 if(ix>=0x7ff00000) { /* erfc(nan)=nan */ 248 /* erfc(+-inf)=0,2 */ 249 return (double)(((u_int32_t)hx>>31)<<1)+one/x; 250 } 251 252 if(ix < 0x3feb0000) { /* |x|<0.84375 */ 253 if(ix < 0x3c700000) /* |x|<2**-56 */ 254 return one-x; 255 z = x*x; 256 r = pp0+z*(pp1+z*(pp2+z*(pp3+z*pp4))); 257 s = one+z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*qq5)))); 258 y = r/s; 259 if(hx < 0x3fd00000) { /* x<1/4 */ 260 return one-(x+x*y); 261 } else { 262 r = x*y; 263 r += (x-half); 264 return half - r ; 265 } 266 } 267 if(ix < 0x3ff40000) { /* 0.84375 <= |x| < 1.25 */ 268 s = fabs(x)-one; 269 P = pa0+s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6))))); 270 Q = one+s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6))))); 271 if(hx>=0) { 272 z = one-erx; return z - P/Q; 273 } else { 274 z = erx+P/Q; return one+z; 275 } 276 } 277 if (ix < 0x403c0000) { /* |x|<28 */ 278 x = fabs(x); 279 s = one/(x*x); 280 if(ix< 0x4006DB6D) { /* |x| < 1/.35 ~ 2.857143*/ 281 R=ra0+s*(ra1+s*(ra2+s*(ra3+s*(ra4+s*( 282 ra5+s*(ra6+s*ra7)))))); 283 S=one+s*(sa1+s*(sa2+s*(sa3+s*(sa4+s*( 284 sa5+s*(sa6+s*(sa7+s*sa8))))))); 285 } else { /* |x| >= 1/.35 ~ 2.857143 */ 286 if(hx<0&&ix>=0x40180000) return two-tiny;/* x < -6 */ 287 R=rb0+s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*( 288 rb5+s*rb6))))); 289 S=one+s*(sb1+s*(sb2+s*(sb3+s*(sb4+s*( 290 sb5+s*(sb6+s*sb7)))))); 291 } 292 z = x; 293 SET_LOW_WORD(z,0); 294 r = __ieee754_exp(-z*z-0.5625)* 295 __ieee754_exp((z-x)*(z+x)+R/S); 296 if(hx>0) return r/x; else return two-r/x; 297 } else { 298 if(hx>0) return tiny*tiny; else return two-tiny; 299 } 300} 301