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/* Modified by Naohiko Shimizu/Tokai University, Japan 1997/08/25, 13 for performance improvement on pipelined processors. 14*/ 15 16#if defined(LIBM_SCCS) && !defined(lint) 17static char rcsid[] = "$NetBSD: s_erf.c,v 1.8 1995/05/10 20:47:05 jtc Exp $"; 18#endif 19 20/* double erf(double x) 21 * double erfc(double x) 22 * x 23 * 2 |\ 24 * erf(x) = --------- | exp(-t*t)dt 25 * sqrt(pi) \| 26 * 0 27 * 28 * erfc(x) = 1-erf(x) 29 * Note that 30 * erf(-x) = -erf(x) 31 * erfc(-x) = 2 - erfc(x) 32 * 33 * Method: 34 * 1. For |x| in [0, 0.84375] 35 * erf(x) = x + x*R(x^2) 36 * erfc(x) = 1 - erf(x) if x in [-.84375,0.25] 37 * = 0.5 + ((0.5-x)-x*R) if x in [0.25,0.84375] 38 * where R = P/Q where P is an odd poly of degree 8 and 39 * Q is an odd poly of degree 10. 40 * -57.90 41 * | R - (erf(x)-x)/x | <= 2 42 * 43 * 44 * Remark. The formula is derived by noting 45 * erf(x) = (2/sqrt(pi))*(x - x^3/3 + x^5/10 - x^7/42 + ....) 46 * and that 47 * 2/sqrt(pi) = 1.128379167095512573896158903121545171688 48 * is close to one. The interval is chosen because the fix 49 * point of erf(x) is near 0.6174 (i.e., erf(x)=x when x is 50 * near 0.6174), and by some experiment, 0.84375 is chosen to 51 * guarantee the error is less than one ulp for erf. 52 * 53 * 2. For |x| in [0.84375,1.25], let s = |x| - 1, and 54 * c = 0.84506291151 rounded to single (24 bits) 55 * erf(x) = sign(x) * (c + P1(s)/Q1(s)) 56 * erfc(x) = (1-c) - P1(s)/Q1(s) if x > 0 57 * 1+(c+P1(s)/Q1(s)) if x < 0 58 * |P1/Q1 - (erf(|x|)-c)| <= 2**-59.06 59 * Remark: here we use the taylor series expansion at x=1. 60 * erf(1+s) = erf(1) + s*Poly(s) 61 * = 0.845.. + P1(s)/Q1(s) 62 * That is, we use rational approximation to approximate 63 * erf(1+s) - (c = (single)0.84506291151) 64 * Note that |P1/Q1|< 0.078 for x in [0.84375,1.25] 65 * where 66 * P1(s) = degree 6 poly in s 67 * Q1(s) = degree 6 poly in s 68 * 69 * 3. For x in [1.25,1/0.35(~2.857143)], 70 * erfc(x) = (1/x)*exp(-x*x-0.5625+R1/S1) 71 * erf(x) = 1 - erfc(x) 72 * where 73 * R1(z) = degree 7 poly in z, (z=1/x^2) 74 * S1(z) = degree 8 poly in z 75 * 76 * 4. For x in [1/0.35,28] 77 * erfc(x) = (1/x)*exp(-x*x-0.5625+R2/S2) if x > 0 78 * = 2.0 - (1/x)*exp(-x*x-0.5625+R2/S2) if -6<x<0 79 * = 2.0 - tiny (if x <= -6) 80 * erf(x) = sign(x)*(1.0 - erfc(x)) if x < 6, else 81 * erf(x) = sign(x)*(1.0 - tiny) 82 * where 83 * R2(z) = degree 6 poly in z, (z=1/x^2) 84 * S2(z) = degree 7 poly in z 85 * 86 * Note1: 87 * To compute exp(-x*x-0.5625+R/S), let s be a single 88 * precision number and s := x; then 89 * -x*x = -s*s + (s-x)*(s+x) 90 * exp(-x*x-0.5626+R/S) = 91 * exp(-s*s-0.5625)*exp((s-x)*(s+x)+R/S); 92 * Note2: 93 * Here 4 and 5 make use of the asymptotic series 94 * exp(-x*x) 95 * erfc(x) ~ ---------- * ( 1 + Poly(1/x^2) ) 96 * x*sqrt(pi) 97 * We use rational approximation to approximate 98 * g(s)=f(1/x^2) = log(erfc(x)*x) - x*x + 0.5625 99 * Here is the error bound for R1/S1 and R2/S2 100 * |R1/S1 - f(x)| < 2**(-62.57) 101 * |R2/S2 - f(x)| < 2**(-61.52) 102 * 103 * 5. For inf > x >= 28 104 * erf(x) = sign(x) *(1 - tiny) (raise inexact) 105 * erfc(x) = tiny*tiny (raise underflow) if x > 0 106 * = 2 - tiny if x<0 107 * 108 * 7. Special case: 109 * erf(0) = 0, erf(inf) = 1, erf(-inf) = -1, 110 * erfc(0) = 1, erfc(inf) = 0, erfc(-inf) = 2, 111 * erfc/erf(NaN) is NaN 112 */ 113 114 115#include "math.h" 116#include "math_private.h" 117 118#ifdef __STDC__ 119static const double 120#else 121static double 122#endif 123tiny = 1e-300, 124half= 5.00000000000000000000e-01, /* 0x3FE00000, 0x00000000 */ 125one = 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */ 126two = 2.00000000000000000000e+00, /* 0x40000000, 0x00000000 */ 127 /* c = (float)0.84506291151 */ 128erx = 8.45062911510467529297e-01, /* 0x3FEB0AC1, 0x60000000 */ 129/* 130 * Coefficients for approximation to erf on [0,0.84375] 131 */ 132efx = 1.28379167095512586316e-01, /* 0x3FC06EBA, 0x8214DB69 */ 133efx8= 1.02703333676410069053e+00, /* 0x3FF06EBA, 0x8214DB69 */ 134pp[] = {1.28379167095512558561e-01, /* 0x3FC06EBA, 0x8214DB68 */ 135 -3.25042107247001499370e-01, /* 0xBFD4CD7D, 0x691CB913 */ 136 -2.84817495755985104766e-02, /* 0xBF9D2A51, 0xDBD7194F */ 137 -5.77027029648944159157e-03, /* 0xBF77A291, 0x236668E4 */ 138 -2.37630166566501626084e-05}, /* 0xBEF8EAD6, 0x120016AC */ 139qq[] = {0.0, 3.97917223959155352819e-01, /* 0x3FD97779, 0xCDDADC09 */ 140 6.50222499887672944485e-02, /* 0x3FB0A54C, 0x5536CEBA */ 141 5.08130628187576562776e-03, /* 0x3F74D022, 0xC4D36B0F */ 142 1.32494738004321644526e-04, /* 0x3F215DC9, 0x221C1A10 */ 143 -3.96022827877536812320e-06}, /* 0xBED09C43, 0x42A26120 */ 144/* 145 * Coefficients for approximation to erf in [0.84375,1.25] 146 */ 147pa[] = {-2.36211856075265944077e-03, /* 0xBF6359B8, 0xBEF77538 */ 148 4.14856118683748331666e-01, /* 0x3FDA8D00, 0xAD92B34D */ 149 -3.72207876035701323847e-01, /* 0xBFD7D240, 0xFBB8C3F1 */ 150 3.18346619901161753674e-01, /* 0x3FD45FCA, 0x805120E4 */ 151 -1.10894694282396677476e-01, /* 0xBFBC6398, 0x3D3E28EC */ 152 3.54783043256182359371e-02, /* 0x3FA22A36, 0x599795EB */ 153 -2.16637559486879084300e-03}, /* 0xBF61BF38, 0x0A96073F */ 154qa[] = {0.0, 1.06420880400844228286e-01, /* 0x3FBB3E66, 0x18EEE323 */ 155 5.40397917702171048937e-01, /* 0x3FE14AF0, 0x92EB6F33 */ 156 7.18286544141962662868e-02, /* 0x3FB2635C, 0xD99FE9A7 */ 157 1.26171219808761642112e-01, /* 0x3FC02660, 0xE763351F */ 158 1.36370839120290507362e-02, /* 0x3F8BEDC2, 0x6B51DD1C */ 159 1.19844998467991074170e-02}, /* 0x3F888B54, 0x5735151D */ 160/* 161 * Coefficients for approximation to erfc in [1.25,1/0.35] 162 */ 163ra[] = {-9.86494403484714822705e-03, /* 0xBF843412, 0x600D6435 */ 164 -6.93858572707181764372e-01, /* 0xBFE63416, 0xE4BA7360 */ 165 -1.05586262253232909814e+01, /* 0xC0251E04, 0x41B0E726 */ 166 -6.23753324503260060396e+01, /* 0xC04F300A, 0xE4CBA38D */ 167 -1.62396669462573470355e+02, /* 0xC0644CB1, 0x84282266 */ 168 -1.84605092906711035994e+02, /* 0xC067135C, 0xEBCCABB2 */ 169 -8.12874355063065934246e+01, /* 0xC0545265, 0x57E4D2F2 */ 170 -9.81432934416914548592e+00}, /* 0xC023A0EF, 0xC69AC25C */ 171sa[] = {0.0,1.96512716674392571292e+01, /* 0x4033A6B9, 0xBD707687 */ 172 1.37657754143519042600e+02, /* 0x4061350C, 0x526AE721 */ 173 4.34565877475229228821e+02, /* 0x407B290D, 0xD58A1A71 */ 174 6.45387271733267880336e+02, /* 0x40842B19, 0x21EC2868 */ 175 4.29008140027567833386e+02, /* 0x407AD021, 0x57700314 */ 176 1.08635005541779435134e+02, /* 0x405B28A3, 0xEE48AE2C */ 177 6.57024977031928170135e+00, /* 0x401A47EF, 0x8E484A93 */ 178 -6.04244152148580987438e-02}, /* 0xBFAEEFF2, 0xEE749A62 */ 179/* 180 * Coefficients for approximation to erfc in [1/.35,28] 181 */ 182rb[] = {-9.86494292470009928597e-03, /* 0xBF843412, 0x39E86F4A */ 183 -7.99283237680523006574e-01, /* 0xBFE993BA, 0x70C285DE */ 184 -1.77579549177547519889e+01, /* 0xC031C209, 0x555F995A */ 185 -1.60636384855821916062e+02, /* 0xC064145D, 0x43C5ED98 */ 186 -6.37566443368389627722e+02, /* 0xC083EC88, 0x1375F228 */ 187 -1.02509513161107724954e+03, /* 0xC0900461, 0x6A2E5992 */ 188 -4.83519191608651397019e+02}, /* 0xC07E384E, 0x9BDC383F */ 189sb[] = {0.0,3.03380607434824582924e+01, /* 0x403E568B, 0x261D5190 */ 190 3.25792512996573918826e+02, /* 0x40745CAE, 0x221B9F0A */ 191 1.53672958608443695994e+03, /* 0x409802EB, 0x189D5118 */ 192 3.19985821950859553908e+03, /* 0x40A8FFB7, 0x688C246A */ 193 2.55305040643316442583e+03, /* 0x40A3F219, 0xCEDF3BE6 */ 194 4.74528541206955367215e+02, /* 0x407DA874, 0xE79FE763 */ 195 -2.24409524465858183362e+01}; /* 0xC03670E2, 0x42712D62 */ 196 197#ifdef __STDC__ 198 double __erf(double x) 199#else 200 double __erf(x) 201 double x; 202#endif 203{ 204 int32_t hx,ix,i; 205 double R,S,P,Q,s,y,z,r; 206 GET_HIGH_WORD(hx,x); 207 ix = hx&0x7fffffff; 208 if(ix>=0x7ff00000) { /* erf(nan)=nan */ 209 i = ((u_int32_t)hx>>31)<<1; 210 return (double)(1-i)+one/x; /* erf(+-inf)=+-1 */ 211 } 212 213 if(ix < 0x3feb0000) { /* |x|<0.84375 */ 214 double r1,r2,s1,s2,s3,z2,z4; 215 if(ix < 0x3e300000) { /* |x|<2**-28 */ 216 if (ix < 0x00800000) 217 return 0.125*(8.0*x+efx8*x); /*avoid underflow */ 218 return x + efx*x; 219 } 220 z = x*x; 221#ifdef DO_NOT_USE_THIS 222 r = pp0+z*(pp1+z*(pp2+z*(pp3+z*pp4))); 223 s = one+z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*qq5)))); 224#else 225 r1 = pp[0]+z*pp[1]; z2=z*z; 226 r2 = pp[2]+z*pp[3]; z4=z2*z2; 227 s1 = one+z*qq[1]; 228 s2 = qq[2]+z*qq[3]; 229 s3 = qq[4]+z*qq[5]; 230 r = r1 + z2*r2 + z4*pp[4]; 231 s = s1 + z2*s2 + z4*s3; 232#endif 233 y = r/s; 234 return x + x*y; 235 } 236 if(ix < 0x3ff40000) { /* 0.84375 <= |x| < 1.25 */ 237 double s2,s4,s6,P1,P2,P3,P4,Q1,Q2,Q3,Q4; 238 s = fabs(x)-one; 239#ifdef DO_NOT_USE_THIS 240 P = pa0+s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6))))); 241 Q = one+s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6))))); 242#else 243 P1 = pa[0]+s*pa[1]; s2=s*s; 244 Q1 = one+s*qa[1]; s4=s2*s2; 245 P2 = pa[2]+s*pa[3]; s6=s4*s2; 246 Q2 = qa[2]+s*qa[3]; 247 P3 = pa[4]+s*pa[5]; 248 Q3 = qa[4]+s*qa[5]; 249 P4 = pa[6]; 250 Q4 = qa[6]; 251 P = P1 + s2*P2 + s4*P3 + s6*P4; 252 Q = Q1 + s2*Q2 + s4*Q3 + s6*Q4; 253#endif 254 if(hx>=0) return erx + P/Q; else return -erx - P/Q; 255 } 256 if (ix >= 0x40180000) { /* inf>|x|>=6 */ 257 if(hx>=0) return one-tiny; else return tiny-one; 258 } 259 x = fabs(x); 260 s = one/(x*x); 261 if(ix< 0x4006DB6E) { /* |x| < 1/0.35 */ 262#ifdef DO_NOT_USE_THIS 263 R=ra0+s*(ra1+s*(ra2+s*(ra3+s*(ra4+s*( 264 ra5+s*(ra6+s*ra7)))))); 265 S=one+s*(sa1+s*(sa2+s*(sa3+s*(sa4+s*( 266 sa5+s*(sa6+s*(sa7+s*sa8))))))); 267#else 268 double R1,R2,R3,R4,S1,S2,S3,S4,s2,s4,s6,s8; 269 R1 = ra[0]+s*ra[1];s2 = s*s; 270 S1 = one+s*sa[1]; s4 = s2*s2; 271 R2 = ra[2]+s*ra[3];s6 = s4*s2; 272 S2 = sa[2]+s*sa[3];s8 = s4*s4; 273 R3 = ra[4]+s*ra[5]; 274 S3 = sa[4]+s*sa[5]; 275 R4 = ra[6]+s*ra[7]; 276 S4 = sa[6]+s*sa[7]; 277 R = R1 + s2*R2 + s4*R3 + s6*R4; 278 S = S1 + s2*S2 + s4*S3 + s6*S4 + s8*sa[8]; 279#endif 280 } else { /* |x| >= 1/0.35 */ 281#ifdef DO_NOT_USE_THIS 282 R=rb0+s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*( 283 rb5+s*rb6))))); 284 S=one+s*(sb1+s*(sb2+s*(sb3+s*(sb4+s*( 285 sb5+s*(sb6+s*sb7)))))); 286#else 287 double R1,R2,R3,S1,S2,S3,S4,s2,s4,s6; 288 R1 = rb[0]+s*rb[1];s2 = s*s; 289 S1 = one+s*sb[1]; s4 = s2*s2; 290 R2 = rb[2]+s*rb[3];s6 = s4*s2; 291 S2 = sb[2]+s*sb[3]; 292 R3 = rb[4]+s*rb[5]; 293 S3 = sb[4]+s*sb[5]; 294 S4 = sb[6]+s*sb[7]; 295 R = R1 + s2*R2 + s4*R3 + s6*rb[6]; 296 S = S1 + s2*S2 + s4*S3 + s6*S4; 297#endif 298 } 299 z = x; 300 SET_LOW_WORD(z,0); 301 r = __ieee754_exp(-z*z-0.5625)*__ieee754_exp((z-x)*(z+x)+R/S); 302 if(hx>=0) return one-r/x; else return r/x-one; 303} 304weak_alias (__erf, erf) 305#ifdef NO_LONG_DOUBLE 306strong_alias (__erf, __erfl) 307weak_alias (__erf, erfl) 308#endif 309 310#ifdef __STDC__ 311 double __erfc(double x) 312#else 313 double __erfc(x) 314 double x; 315#endif 316{ 317 int32_t hx,ix; 318 double R,S,P,Q,s,y,z,r; 319 GET_HIGH_WORD(hx,x); 320 ix = hx&0x7fffffff; 321 if(ix>=0x7ff00000) { /* erfc(nan)=nan */ 322 /* erfc(+-inf)=0,2 */ 323 return (double)(((u_int32_t)hx>>31)<<1)+one/x; 324 } 325 326 if(ix < 0x3feb0000) { /* |x|<0.84375 */ 327 double r1,r2,s1,s2,s3,z2,z4; 328 if(ix < 0x3c700000) /* |x|<2**-56 */ 329 return one-x; 330 z = x*x; 331#ifdef DO_NOT_USE_THIS 332 r = pp0+z*(pp1+z*(pp2+z*(pp3+z*pp4))); 333 s = one+z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*qq5)))); 334#else 335 r1 = pp[0]+z*pp[1]; z2=z*z; 336 r2 = pp[2]+z*pp[3]; z4=z2*z2; 337 s1 = one+z*qq[1]; 338 s2 = qq[2]+z*qq[3]; 339 s3 = qq[4]+z*qq[5]; 340 r = r1 + z2*r2 + z4*pp[4]; 341 s = s1 + z2*s2 + z4*s3; 342#endif 343 y = r/s; 344 if(hx < 0x3fd00000) { /* x<1/4 */ 345 return one-(x+x*y); 346 } else { 347 r = x*y; 348 r += (x-half); 349 return half - r ; 350 } 351 } 352 if(ix < 0x3ff40000) { /* 0.84375 <= |x| < 1.25 */ 353 double s2,s4,s6,P1,P2,P3,P4,Q1,Q2,Q3,Q4; 354 s = fabs(x)-one; 355#ifdef DO_NOT_USE_THIS 356 P = pa0+s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6))))); 357 Q = one+s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6))))); 358#else 359 P1 = pa[0]+s*pa[1]; s2=s*s; 360 Q1 = one+s*qa[1]; s4=s2*s2; 361 P2 = pa[2]+s*pa[3]; s6=s4*s2; 362 Q2 = qa[2]+s*qa[3]; 363 P3 = pa[4]+s*pa[5]; 364 Q3 = qa[4]+s*qa[5]; 365 P4 = pa[6]; 366 Q4 = qa[6]; 367 P = P1 + s2*P2 + s4*P3 + s6*P4; 368 Q = Q1 + s2*Q2 + s4*Q3 + s6*Q4; 369#endif 370 if(hx>=0) { 371 z = one-erx; return z - P/Q; 372 } else { 373 z = erx+P/Q; return one+z; 374 } 375 } 376 if (ix < 0x403c0000) { /* |x|<28 */ 377 x = fabs(x); 378 s = one/(x*x); 379 if(ix< 0x4006DB6D) { /* |x| < 1/.35 ~ 2.857143*/ 380#ifdef DO_NOT_USE_THIS 381 R=ra0+s*(ra1+s*(ra2+s*(ra3+s*(ra4+s*( 382 ra5+s*(ra6+s*ra7)))))); 383 S=one+s*(sa1+s*(sa2+s*(sa3+s*(sa4+s*( 384 sa5+s*(sa6+s*(sa7+s*sa8))))))); 385#else 386 double R1,R2,R3,R4,S1,S2,S3,S4,s2,s4,s6,s8; 387 R1 = ra[0]+s*ra[1];s2 = s*s; 388 S1 = one+s*sa[1]; s4 = s2*s2; 389 R2 = ra[2]+s*ra[3];s6 = s4*s2; 390 S2 = sa[2]+s*sa[3];s8 = s4*s4; 391 R3 = ra[4]+s*ra[5]; 392 S3 = sa[4]+s*sa[5]; 393 R4 = ra[6]+s*ra[7]; 394 S4 = sa[6]+s*sa[7]; 395 R = R1 + s2*R2 + s4*R3 + s6*R4; 396 S = S1 + s2*S2 + s4*S3 + s6*S4 + s8*sa[8]; 397#endif 398 } else { /* |x| >= 1/.35 ~ 2.857143 */ 399 double R1,R2,R3,S1,S2,S3,S4,s2,s4,s6; 400 if(hx<0&&ix>=0x40180000) return two-tiny;/* x < -6 */ 401#ifdef DO_NOT_USE_THIS 402 R=rb0+s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*( 403 rb5+s*rb6))))); 404 S=one+s*(sb1+s*(sb2+s*(sb3+s*(sb4+s*( 405 sb5+s*(sb6+s*sb7)))))); 406#else 407 R1 = rb[0]+s*rb[1];s2 = s*s; 408 S1 = one+s*sb[1]; s4 = s2*s2; 409 R2 = rb[2]+s*rb[3];s6 = s4*s2; 410 S2 = sb[2]+s*sb[3]; 411 R3 = rb[4]+s*rb[5]; 412 S3 = sb[4]+s*sb[5]; 413 S4 = sb[6]+s*sb[7]; 414 R = R1 + s2*R2 + s4*R3 + s6*rb[6]; 415 S = S1 + s2*S2 + s4*S3 + s6*S4; 416#endif 417 } 418 z = x; 419 SET_LOW_WORD(z,0); 420 r = __ieee754_exp(-z*z-0.5625)* 421 __ieee754_exp((z-x)*(z+x)+R/S); 422 if(hx>0) return r/x; else return two-r/x; 423 } else { 424 if(hx>0) return tiny*tiny; else return two-tiny; 425 } 426} 427weak_alias (__erfc, erfc) 428#ifdef NO_LONG_DOUBLE 429strong_alias (__erfc, __erfcl) 430weak_alias (__erfc, erfcl) 431#endif 432