gonzui

    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: /* LINTLIBRARY */
   14: 
   15: /* double erf(double x)
   16:  * double erfc(double x)
   17:  *                           x
   18:  *                    2      |\
   19:  *     erf(x)  =  ---------  | exp(-t*t)dt
   20:  *              sqrt(pi) \| 
   21:  *                           0
   22:  *
   23:  *     erfc(x) =  1-erf(x)
   24:  *  Note that 
   25:  *              erf(-x) = -erf(x)
   26:  *              erfc(-x) = 2 - erfc(x)
   27:  *
   28:  * Method:
   29:  *      1. For |x| in [0, 0.84375]
   30:  *          erf(x)  = x + x*R(x^2)
   31:  *          erfc(x) = 1 - erf(x)           if x in [-.84375,0.25]
   32:  *                  = 0.5 + ((0.5-x)-x*R)  if x in [0.25,0.84375]
   33:  *         where R = P/Q where P is an odd poly of degree 8 and
   34:  *         Q is an odd poly of degree 10.
   35:  *                                               -57.90
   36:  *                      | R - (erf(x)-x)/x | <= 2
   37:  *      
   38:  *
   39:  *         Remark. The formula is derived by noting
   40:  *          erf(x) = (2/sqrt(pi))*(x - x^3/3 + x^5/10 - x^7/42 + ....)
   41:  *         and that
   42:  *          2/sqrt(pi) = 1.128379167095512573896158903121545171688
   43:  *         is close to one. The interval is chosen because the fix
   44:  *         point of erf(x) is near 0.6174 (i.e., erf(x)=x when x is
   45:  *         near 0.6174), and by some experiment, 0.84375 is chosen to
   46:  *         guarantee the error is less than one ulp for erf.
   47:  *
   48:  *      2. For |x| in [0.84375,1.25], let s = |x| - 1, and
   49:  *         c = 0.84506291151 rounded to single (24 bits)
   50:  *              erf(x)  = sign(x) * (c  + P1(s)/Q1(s))
   51:  *              erfc(x) = (1-c)  - P1(s)/Q1(s) if x > 0
   52:  *                        1+(c+P1(s)/Q1(s))    if x < 0
   53:  *              |P1/Q1 - (erf(|x|)-c)| <= 2**-59.06
   54:  *         Remark: here we use the taylor series expansion at x=1.
   55:  *              erf(1+s) = erf(1) + s*Poly(s)
   56:  *                       = 0.845.. + P1(s)/Q1(s)
   57:  *         That is, we use rational approximation to approximate
   58:  *                      erf(1+s) - (c = (single)0.84506291151)
   59:  *         Note that |P1/Q1|< 0.078 for x in [0.84375,1.25]
   60:  *         where 
   61:  *              P1(s) = degree 6 poly in s
   62:  *              Q1(s) = degree 6 poly in s
   63:  *
   64:  *      3. For x in [1.25,1/0.35(~2.857143)], 
   65:  *              erfc(x) = (1/x)*exp(-x*x-0.5625+R1/S1)
   66:  *              erf(x)  = 1 - erfc(x)
   67:  *         where 
   68:  *              R1(z) = degree 7 poly in z, (z=1/x^2)
   69:  *              S1(z) = degree 8 poly in z
   70:  *
   71:  *      4. For x in [1/0.35,28]
   72:  *              erfc(x) = (1/x)*exp(-x*x-0.5625+R2/S2) if x > 0
   73:  *                      = 2.0 - (1/x)*exp(-x*x-0.5625+R2/S2) if -6<x<0
   74:  *                      = 2.0 - tiny               (if x <= -6)
   75:  *              erf(x)  = sign(x)*(1.0 - erfc(x)) if x < 6, else
   76:  *              erf(x)  = sign(x)*(1.0 - tiny)
   77:  *         where
   78:  *              R2(z) = degree 6 poly in z, (z=1/x^2)
   79:  *              S2(z) = degree 7 poly in z
   80:  *
   81:  *      Note1:
   82:  *         To compute exp(-x*x-0.5625+R/S), let s be a single
   83:  *         precision number and s := x; then
   84:  *              -x*x = -s*s + (s-x)*(s+x)
   85:  *              exp(-x*x-0.5626+R/S) = 
   86:  *                      exp(-s*s-0.5625)*exp((s-x)*(s+x)+R/S);
   87:  *      Note2:
   88:  *         Here 4 and 5 make use of the asymptotic series
   89:  *                        exp(-x*x)
   90:  *              erfc(x) ~ ---------- * ( 1 + Poly(1/x^2) )
   91:  *                        x*sqrt(pi)
   92:  *         We use rational approximation to approximate
   93:  *              g(s)=f(1/x^2) = log(erfc(x)*x) - x*x + 0.5625
   94:  *         Here is the error bound for R1/S1 and R2/S2
   95:  *              |R1/S1 - f(x)|  < 2**(-62.57)
   96:  *              |R2/S2 - f(x)|  < 2**(-61.52)
   97:  *
   98:  *      5. For inf > x >= 28
   99:  *              erf(x)  = sign(x) *(1 - tiny)  (raise inexact)
  100:  *              erfc(x) = tiny*tiny (raise underflow) if x > 0
  101:  *                      = 2 - tiny if x<0
  102:  *
  103:  *      7. Special case:
  104:  *              erf(0)  = 0, erf(inf)  = 1, erf(-inf) = -1,
  105:  *              erfc(0) = 1, erfc(inf) = 0, erfc(-inf) = 2, 
  106:  *           erfc/erf(NaN) is NaN
  107:  */
  108: 
  109: #include <sys/cdefs.h>
  110: #include <float.h>
  111: #include <math.h>
  112: 
  113: #include "math_private.h"
  114: 
  115: static const double
  116: tiny        = 1e-300,
  117: half=  5.00000000000000000000e-01, /* 0x3FE00000, 0x00000000 */
  118: one =  1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
  119: two =  2.00000000000000000000e+00, /* 0x40000000, 0x00000000 */
  120:         /* c = (float)0.84506291151 */
  121: erx =  8.45062911510467529297e-01, /* 0x3FEB0AC1, 0x60000000 */
  122: /*
  123:  * Coefficients for approximation to  erf on [0,0.84375]
  124:  */
  125: efx =  1.28379167095512586316e-01, /* 0x3FC06EBA, 0x8214DB69 */
  126: efx8=  1.02703333676410069053e+00, /* 0x3FF06EBA, 0x8214DB69 */
  127: pp0  =  1.28379167095512558561e-01, /* 0x3FC06EBA, 0x8214DB68 */
  128: pp1  = -3.25042107247001499370e-01, /* 0xBFD4CD7D, 0x691CB913 */
  129: pp2  = -2.84817495755985104766e-02, /* 0xBF9D2A51, 0xDBD7194F */
  130: pp3  = -5.77027029648944159157e-03, /* 0xBF77A291, 0x236668E4 */
  131: pp4  = -2.37630166566501626084e-05, /* 0xBEF8EAD6, 0x120016AC */
  132: qq1  =  3.97917223959155352819e-01, /* 0x3FD97779, 0xCDDADC09 */
  133: qq2  =  6.50222499887672944485e-02, /* 0x3FB0A54C, 0x5536CEBA */
  134: qq3  =  5.08130628187576562776e-03, /* 0x3F74D022, 0xC4D36B0F */
  135: qq4  =  1.32494738004321644526e-04, /* 0x3F215DC9, 0x221C1A10 */
  136: qq5  = -3.96022827877536812320e-06, /* 0xBED09C43, 0x42A26120 */
  137: /*
  138:  * Coefficients for approximation to  erf  in [0.84375,1.25] 
  139:  */
  140: pa0  = -2.36211856075265944077e-03, /* 0xBF6359B8, 0xBEF77538 */
  141: pa1  =  4.14856118683748331666e-01, /* 0x3FDA8D00, 0xAD92B34D */
  142: pa2  = -3.72207876035701323847e-01, /* 0xBFD7D240, 0xFBB8C3F1 */
  143: pa3  =  3.18346619901161753674e-01, /* 0x3FD45FCA, 0x805120E4 */
  144: pa4  = -1.10894694282396677476e-01, /* 0xBFBC6398, 0x3D3E28EC */
  145: pa5  =  3.54783043256182359371e-02, /* 0x3FA22A36, 0x599795EB */
  146: pa6  = -2.16637559486879084300e-03, /* 0xBF61BF38, 0x0A96073F */
  147: qa1  =  1.06420880400844228286e-01, /* 0x3FBB3E66, 0x18EEE323 */
  148: qa2  =  5.40397917702171048937e-01, /* 0x3FE14AF0, 0x92EB6F33 */
  149: qa3  =  7.18286544141962662868e-02, /* 0x3FB2635C, 0xD99FE9A7 */
  150: qa4  =  1.26171219808761642112e-01, /* 0x3FC02660, 0xE763351F */
  151: qa5  =  1.36370839120290507362e-02, /* 0x3F8BEDC2, 0x6B51DD1C */
  152: qa6  =  1.19844998467991074170e-02, /* 0x3F888B54, 0x5735151D */
  153: /*
  154:  * Coefficients for approximation to  erfc in [1.25,1/0.35]
  155:  */
  156: ra0  = -9.86494403484714822705e-03, /* 0xBF843412, 0x600D6435 */
  157: ra1  = -6.93858572707181764372e-01, /* 0xBFE63416, 0xE4BA7360 */
  158: ra2  = -1.05586262253232909814e+01, /* 0xC0251E04, 0x41B0E726 */
  159: ra3  = -6.23753324503260060396e+01, /* 0xC04F300A, 0xE4CBA38D */
  160: ra4  = -1.62396669462573470355e+02, /* 0xC0644CB1, 0x84282266 */
  161: ra5  = -1.84605092906711035994e+02, /* 0xC067135C, 0xEBCCABB2 */
  162: ra6  = -8.12874355063065934246e+01, /* 0xC0545265, 0x57E4D2F2 */
  163: ra7  = -9.81432934416914548592e+00, /* 0xC023A0EF, 0xC69AC25C */
  164: sa1  =  1.96512716674392571292e+01, /* 0x4033A6B9, 0xBD707687 */
  165: sa2  =  1.37657754143519042600e+02, /* 0x4061350C, 0x526AE721 */
  166: sa3  =  4.34565877475229228821e+02, /* 0x407B290D, 0xD58A1A71 */
  167: sa4  =  6.45387271733267880336e+02, /* 0x40842B19, 0x21EC2868 */
  168: sa5  =  4.29008140027567833386e+02, /* 0x407AD021, 0x57700314 */
  169: sa6  =  1.08635005541779435134e+02, /* 0x405B28A3, 0xEE48AE2C */
  170: sa7  =  6.57024977031928170135e+00, /* 0x401A47EF, 0x8E484A93 */
  171: sa8  = -6.04244152148580987438e-02, /* 0xBFAEEFF2, 0xEE749A62 */
  172: /*
  173:  * Coefficients for approximation to  erfc in [1/.35,28]
  174:  */
  175: rb0  = -9.86494292470009928597e-03, /* 0xBF843412, 0x39E86F4A */
  176: rb1  = -7.99283237680523006574e-01, /* 0xBFE993BA, 0x70C285DE */
  177: rb2  = -1.77579549177547519889e+01, /* 0xC031C209, 0x555F995A */
  178: rb3  = -1.60636384855821916062e+02, /* 0xC064145D, 0x43C5ED98 */
  179: rb4  = -6.37566443368389627722e+02, /* 0xC083EC88, 0x1375F228 */
  180: rb5  = -1.02509513161107724954e+03, /* 0xC0900461, 0x6A2E5992 */
  181: rb6  = -4.83519191608651397019e+02, /* 0xC07E384E, 0x9BDC383F */
  182: sb1  =  3.03380607434824582924e+01, /* 0x403E568B, 0x261D5190 */
  183: sb2  =  3.25792512996573918826e+02, /* 0x40745CAE, 0x221B9F0A */
  184: sb3  =  1.53672958608443695994e+03, /* 0x409802EB, 0x189D5118 */
  185: sb4  =  3.19985821950859553908e+03, /* 0x40A8FFB7, 0x688C246A */
  186: sb5  =  2.55305040643316442583e+03, /* 0x40A3F219, 0xCEDF3BE6 */
  187: sb6  =  4.74528541206955367215e+02, /* 0x407DA874, 0xE79FE763 */
  188: sb7  = -2.24409524465858183362e+01; /* 0xC03670E2, 0x42712D62 */
  189: 
  190: double
  191: erf(double x) 
  192: {
  193:         int32_t hx,ix,i;
  194:         double R,S,P,Q,s,y,z,r;
  195:         GET_HIGH_WORD(hx,x);
  196:         ix = hx&0x7fffffff;
  197:         if(ix>=0x7ff00000) {           /* erf(nan)=nan */
  198:             i = ((u_int32_t)hx>>31)<<1;
  199:             return (double)(1-i)+one/x;        /* erf(+-inf)=+-1 */
  200:         }
  201: 
  202:         if(ix < 0x3feb0000) {          /* |x|<0.84375 */
  203:             if(ix < 0x3e300000) {      /* |x|<2**-28 */
  204:                 if (ix < 0x00800000) 
  205:                     return 0.125*(8.0*x+efx8*x);  /*avoid underflow */
  206:                 return x + efx*x;
  207:             }
  208:             z = x*x;
  209:             r = pp0+z*(pp1+z*(pp2+z*(pp3+z*pp4)));
  210:             s = one+z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*qq5))));
  211:             y = r/s;
  212:             return x + x*y;
  213:         }
  214:         if(ix < 0x3ff40000) {          /* 0.84375 <= |x| < 1.25 */
  215:             s = fabs(x)-one;
  216:             P = pa0+s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6)))));
  217:             Q = one+s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6)))));
  218:             if(hx>=0) return erx + P/Q; else return -erx - P/Q;
  219:         }
  220:         if (ix >= 0x40180000) {                /* inf>|x|>=6 */
  221:             if(hx>=0) return one-tiny; else return tiny-one;
  222:         }
  223:         x = fabs(x);
  224:         s = one/(x*x);
  225:         if(ix< 0x4006DB6E) {   /* |x| < 1/0.35 */
  226:             R=ra0+s*(ra1+s*(ra2+s*(ra3+s*(ra4+s*(
  227:                                 ra5+s*(ra6+s*ra7))))));
  228:             S=one+s*(sa1+s*(sa2+s*(sa3+s*(sa4+s*(
  229:                                 sa5+s*(sa6+s*(sa7+s*sa8)))))));
  230:         } else {       /* |x| >= 1/0.35 */
  231:             R=rb0+s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*(
  232:                                 rb5+s*rb6)))));
  233:             S=one+s*(sb1+s*(sb2+s*(sb3+s*(sb4+s*(
  234:                                 sb5+s*(sb6+s*sb7))))));
  235:         }
  236:         z  = x;  
  237:         SET_LOW_WORD(z,0);
  238:         r  =  exp(-z*z-0.5625)*exp((z-x)*(z+x)+R/S);
  239:         if(hx>=0) return one-r/x; else return  r/x-one;
  240: }
  241: 
  242: double
  243: erfc(double x) 
  244: {
  245:         int32_t hx,ix;
  246:         double R,S,P,Q,s,y,z,r;
  247:         GET_HIGH_WORD(hx,x);
  248:         ix = hx&0x7fffffff;
  249:         if(ix>=0x7ff00000) {                   /* erfc(nan)=nan */
  250:                                                 /* erfc(+-inf)=0,2 */
  251:             return (double)(((u_int32_t)hx>>31)<<1)+one/x;
  252:         }
  253: 
  254:         if(ix < 0x3feb0000) {          /* |x|<0.84375 */
  255:             if(ix < 0x3c700000)        /* |x|<2**-56 */
  256:                 return one-x;
  257:             z = x*x;
  258:             r = pp0+z*(pp1+z*(pp2+z*(pp3+z*pp4)));
  259:             s = one+z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*qq5))));
  260:             y = r/s;
  261:             if(hx < 0x3fd00000) {      /* x<1/4 */
  262:                 return one-(x+x*y);
  263:             } else {
  264:                 r = x*y;
  265:                 r += (x-half);
  266:                 return half - r ;
  267:             }
  268:         }
  269:         if(ix < 0x3ff40000) {          /* 0.84375 <= |x| < 1.25 */
  270:             s = fabs(x)-one;
  271:             P = pa0+s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6)))));
  272:             Q = one+s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6)))));
  273:             if(hx>=0) {
  274:                 z  = one-erx; return z - P/Q; 
  275:             } else {
  276:                 z = erx+P/Q; return one+z;
  277:             }
  278:         }
  279:         if (ix < 0x403c0000) {         /* |x|<28 */
  280:             x = fabs(x);
  281:             s = one/(x*x);
  282:             if(ix< 0x4006DB6D) {       /* |x| < 1/.35 ~ 2.857143*/
  283:                 R=ra0+s*(ra1+s*(ra2+s*(ra3+s*(ra4+s*(
  284:                                 ra5+s*(ra6+s*ra7))))));
  285:                 S=one+s*(sa1+s*(sa2+s*(sa3+s*(sa4+s*(
  286:                                 sa5+s*(sa6+s*(sa7+s*sa8)))))));
  287:             } else {                   /* |x| >= 1/.35 ~ 2.857143 */
  288:                 if(hx<0&&ix>=0x40180000) return two-tiny;/* x < -6 */
  289:                 R=rb0+s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*(
  290:                                 rb5+s*rb6)))));
  291:                 S=one+s*(sb1+s*(sb2+s*(sb3+s*(sb4+s*(
  292:                                 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: #ifdef  lint
  305: /* PROTOLIB1 */
  306: long double erfl(long double);
  307: long double erfcl(long double);
  308: #else   /* lint */
  309: __weak_alias(erfl, erf);
  310: __weak_alias(erfcl, erfc);
  311: #endif  /* lint */
  312: #endif  /* LDBL_MANT_DIG == 53 */