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Annotation of src/lib/libm/src/s_erf.c, Revision 1.10

1.1       jtc         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
1.10    ! simonb      8:  * software is freely granted, provided that this notice
1.1       jtc         9:  * is preserved.
                     10:  * ====================================================
                     11:  */
1.3       jtc        12:
1.9       lukem      13: #include <sys/cdefs.h>
1.7       jtc        14: #if defined(LIBM_SCCS) && !defined(lint)
1.10    ! simonb     15: __RCSID("$NetBSD: s_erf.c,v 1.9 1997/10/09 11:31:35 lukem Exp $");
1.3       jtc        16: #endif
1.1       jtc        17:
                     18: /* double erf(double x)
                     19:  * double erfc(double x)
                     20:  *                          x
                     21:  *                   2      |\
                     22:  *     erf(x)  =  ---------  | exp(-t*t)dt
1.10    ! simonb     23:  *                sqrt(pi) \|
1.1       jtc        24:  *                          0
                     25:  *
                     26:  *     erfc(x) =  1-erf(x)
1.10    ! simonb     27:  *  Note that
1.1       jtc        28:  *             erf(-x) = -erf(x)
                     29:  *             erfc(-x) = 2 - erfc(x)
                     30:  *
                     31:  * Method:
                     32:  *     1. For |x| in [0, 0.84375]
                     33:  *         erf(x)  = x + x*R(x^2)
                     34:  *          erfc(x) = 1 - erf(x)           if x in [-.84375,0.25]
                     35:  *                  = 0.5 + ((0.5-x)-x*R)  if x in [0.25,0.84375]
                     36:  *        where R = P/Q where P is an odd poly of degree 8 and
                     37:  *        Q is an odd poly of degree 10.
                     38:  *                                              -57.90
                     39:  *                     | R - (erf(x)-x)/x | <= 2
1.10    ! simonb     40:  *
1.1       jtc        41:  *
                     42:  *        Remark. The formula is derived by noting
                     43:  *          erf(x) = (2/sqrt(pi))*(x - x^3/3 + x^5/10 - x^7/42 + ....)
                     44:  *        and that
                     45:  *          2/sqrt(pi) = 1.128379167095512573896158903121545171688
                     46:  *        is close to one. The interval is chosen because the fix
                     47:  *        point of erf(x) is near 0.6174 (i.e., erf(x)=x when x is
                     48:  *        near 0.6174), and by some experiment, 0.84375 is chosen to
                     49:  *        guarantee the error is less than one ulp for erf.
                     50:  *
                     51:  *      2. For |x| in [0.84375,1.25], let s = |x| - 1, and
                     52:  *         c = 0.84506291151 rounded to single (24 bits)
                     53:  *             erf(x)  = sign(x) * (c  + P1(s)/Q1(s))
                     54:  *             erfc(x) = (1-c)  - P1(s)/Q1(s) if x > 0
                     55:  *                       1+(c+P1(s)/Q1(s))    if x < 0
                     56:  *             |P1/Q1 - (erf(|x|)-c)| <= 2**-59.06
                     57:  *        Remark: here we use the taylor series expansion at x=1.
                     58:  *             erf(1+s) = erf(1) + s*Poly(s)
                     59:  *                      = 0.845.. + P1(s)/Q1(s)
                     60:  *        That is, we use rational approximation to approximate
                     61:  *                     erf(1+s) - (c = (single)0.84506291151)
                     62:  *        Note that |P1/Q1|< 0.078 for x in [0.84375,1.25]
1.10    ! simonb     63:  *        where
1.1       jtc        64:  *             P1(s) = degree 6 poly in s
                     65:  *             Q1(s) = degree 6 poly in s
                     66:  *
1.10    ! simonb     67:  *      3. For x in [1.25,1/0.35(~2.857143)],
1.1       jtc        68:  *             erfc(x) = (1/x)*exp(-x*x-0.5625+R1/S1)
                     69:  *             erf(x)  = 1 - erfc(x)
1.10    ! simonb     70:  *        where
1.1       jtc        71:  *             R1(z) = degree 7 poly in z, (z=1/x^2)
                     72:  *             S1(z) = degree 8 poly in z
                     73:  *
                     74:  *      4. For x in [1/0.35,28]
                     75:  *             erfc(x) = (1/x)*exp(-x*x-0.5625+R2/S2) if x > 0
                     76:  *                     = 2.0 - (1/x)*exp(-x*x-0.5625+R2/S2) if -6<x<0
                     77:  *                     = 2.0 - tiny            (if x <= -6)
                     78:  *             erf(x)  = sign(x)*(1.0 - erfc(x)) if x < 6, else
                     79:  *             erf(x)  = sign(x)*(1.0 - tiny)
                     80:  *        where
                     81:  *             R2(z) = degree 6 poly in z, (z=1/x^2)
                     82:  *             S2(z) = degree 7 poly in z
                     83:  *
                     84:  *      Note1:
                     85:  *        To compute exp(-x*x-0.5625+R/S), let s be a single
                     86:  *        precision number and s := x; then
                     87:  *             -x*x = -s*s + (s-x)*(s+x)
1.10    ! simonb     88:  *             exp(-x*x-0.5626+R/S) =
1.1       jtc        89:  *                     exp(-s*s-0.5625)*exp((s-x)*(s+x)+R/S);
                     90:  *      Note2:
                     91:  *        Here 4 and 5 make use of the asymptotic series
                     92:  *                       exp(-x*x)
                     93:  *             erfc(x) ~ ---------- * ( 1 + Poly(1/x^2) )
                     94:  *                       x*sqrt(pi)
                     95:  *        We use rational approximation to approximate
                     96:  *             g(s)=f(1/x^2) = log(erfc(x)*x) - x*x + 0.5625
                     97:  *        Here is the error bound for R1/S1 and R2/S2
                     98:  *             |R1/S1 - f(x)|  < 2**(-62.57)
                     99:  *             |R2/S2 - f(x)|  < 2**(-61.52)
                    100:  *
                    101:  *      5. For inf > x >= 28
                    102:  *             erf(x)  = sign(x) *(1 - tiny)  (raise inexact)
                    103:  *             erfc(x) = tiny*tiny (raise underflow) if x > 0
                    104:  *                     = 2 - tiny if x<0
                    105:  *
                    106:  *      7. Special case:
                    107:  *             erf(0)  = 0, erf(inf)  = 1, erf(-inf) = -1,
1.10    ! simonb    108:  *             erfc(0) = 1, erfc(inf) = 0, erfc(-inf) = 2,
1.1       jtc       109:  *             erfc/erf(NaN) is NaN
                    110:  */
                    111:
                    112:
1.5       jtc       113: #include "math.h"
                    114: #include "math_private.h"
1.1       jtc       115:
                    116: #ifdef __STDC__
                    117: static const double
                    118: #else
                    119: static double
                    120: #endif
                    121: tiny       = 1e-300,
                    122: half=  5.00000000000000000000e-01, /* 0x3FE00000, 0x00000000 */
                    123: one =  1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
                    124: two =  2.00000000000000000000e+00, /* 0x40000000, 0x00000000 */
                    125:        /* c = (float)0.84506291151 */
                    126: erx =  8.45062911510467529297e-01, /* 0x3FEB0AC1, 0x60000000 */
                    127: /*
                    128:  * Coefficients for approximation to  erf on [0,0.84375]
                    129:  */
                    130: efx =  1.28379167095512586316e-01, /* 0x3FC06EBA, 0x8214DB69 */
                    131: efx8=  1.02703333676410069053e+00, /* 0x3FF06EBA, 0x8214DB69 */
                    132: pp0  =  1.28379167095512558561e-01, /* 0x3FC06EBA, 0x8214DB68 */
                    133: pp1  = -3.25042107247001499370e-01, /* 0xBFD4CD7D, 0x691CB913 */
                    134: pp2  = -2.84817495755985104766e-02, /* 0xBF9D2A51, 0xDBD7194F */
                    135: pp3  = -5.77027029648944159157e-03, /* 0xBF77A291, 0x236668E4 */
                    136: pp4  = -2.37630166566501626084e-05, /* 0xBEF8EAD6, 0x120016AC */
                    137: qq1  =  3.97917223959155352819e-01, /* 0x3FD97779, 0xCDDADC09 */
                    138: qq2  =  6.50222499887672944485e-02, /* 0x3FB0A54C, 0x5536CEBA */
                    139: qq3  =  5.08130628187576562776e-03, /* 0x3F74D022, 0xC4D36B0F */
                    140: qq4  =  1.32494738004321644526e-04, /* 0x3F215DC9, 0x221C1A10 */
                    141: qq5  = -3.96022827877536812320e-06, /* 0xBED09C43, 0x42A26120 */
                    142: /*
1.10    ! simonb    143:  * Coefficients for approximation to  erf  in [0.84375,1.25]
1.1       jtc       144:  */
                    145: pa0  = -2.36211856075265944077e-03, /* 0xBF6359B8, 0xBEF77538 */
                    146: pa1  =  4.14856118683748331666e-01, /* 0x3FDA8D00, 0xAD92B34D */
                    147: pa2  = -3.72207876035701323847e-01, /* 0xBFD7D240, 0xFBB8C3F1 */
                    148: pa3  =  3.18346619901161753674e-01, /* 0x3FD45FCA, 0x805120E4 */
                    149: pa4  = -1.10894694282396677476e-01, /* 0xBFBC6398, 0x3D3E28EC */
                    150: pa5  =  3.54783043256182359371e-02, /* 0x3FA22A36, 0x599795EB */
                    151: pa6  = -2.16637559486879084300e-03, /* 0xBF61BF38, 0x0A96073F */
                    152: qa1  =  1.06420880400844228286e-01, /* 0x3FBB3E66, 0x18EEE323 */
                    153: qa2  =  5.40397917702171048937e-01, /* 0x3FE14AF0, 0x92EB6F33 */
                    154: qa3  =  7.18286544141962662868e-02, /* 0x3FB2635C, 0xD99FE9A7 */
                    155: qa4  =  1.26171219808761642112e-01, /* 0x3FC02660, 0xE763351F */
                    156: qa5  =  1.36370839120290507362e-02, /* 0x3F8BEDC2, 0x6B51DD1C */
                    157: qa6  =  1.19844998467991074170e-02, /* 0x3F888B54, 0x5735151D */
                    158: /*
                    159:  * Coefficients for approximation to  erfc in [1.25,1/0.35]
                    160:  */
                    161: ra0  = -9.86494403484714822705e-03, /* 0xBF843412, 0x600D6435 */
                    162: ra1  = -6.93858572707181764372e-01, /* 0xBFE63416, 0xE4BA7360 */
                    163: ra2  = -1.05586262253232909814e+01, /* 0xC0251E04, 0x41B0E726 */
                    164: ra3  = -6.23753324503260060396e+01, /* 0xC04F300A, 0xE4CBA38D */
                    165: ra4  = -1.62396669462573470355e+02, /* 0xC0644CB1, 0x84282266 */
                    166: ra5  = -1.84605092906711035994e+02, /* 0xC067135C, 0xEBCCABB2 */
                    167: ra6  = -8.12874355063065934246e+01, /* 0xC0545265, 0x57E4D2F2 */
                    168: ra7  = -9.81432934416914548592e+00, /* 0xC023A0EF, 0xC69AC25C */
                    169: sa1  =  1.96512716674392571292e+01, /* 0x4033A6B9, 0xBD707687 */
                    170: sa2  =  1.37657754143519042600e+02, /* 0x4061350C, 0x526AE721 */
                    171: sa3  =  4.34565877475229228821e+02, /* 0x407B290D, 0xD58A1A71 */
                    172: sa4  =  6.45387271733267880336e+02, /* 0x40842B19, 0x21EC2868 */
                    173: sa5  =  4.29008140027567833386e+02, /* 0x407AD021, 0x57700314 */
                    174: sa6  =  1.08635005541779435134e+02, /* 0x405B28A3, 0xEE48AE2C */
                    175: sa7  =  6.57024977031928170135e+00, /* 0x401A47EF, 0x8E484A93 */
                    176: sa8  = -6.04244152148580987438e-02, /* 0xBFAEEFF2, 0xEE749A62 */
                    177: /*
                    178:  * Coefficients for approximation to  erfc in [1/.35,28]
                    179:  */
                    180: rb0  = -9.86494292470009928597e-03, /* 0xBF843412, 0x39E86F4A */
                    181: rb1  = -7.99283237680523006574e-01, /* 0xBFE993BA, 0x70C285DE */
                    182: rb2  = -1.77579549177547519889e+01, /* 0xC031C209, 0x555F995A */
                    183: rb3  = -1.60636384855821916062e+02, /* 0xC064145D, 0x43C5ED98 */
                    184: rb4  = -6.37566443368389627722e+02, /* 0xC083EC88, 0x1375F228 */
                    185: rb5  = -1.02509513161107724954e+03, /* 0xC0900461, 0x6A2E5992 */
                    186: rb6  = -4.83519191608651397019e+02, /* 0xC07E384E, 0x9BDC383F */
                    187: sb1  =  3.03380607434824582924e+01, /* 0x403E568B, 0x261D5190 */
                    188: sb2  =  3.25792512996573918826e+02, /* 0x40745CAE, 0x221B9F0A */
                    189: sb3  =  1.53672958608443695994e+03, /* 0x409802EB, 0x189D5118 */
                    190: sb4  =  3.19985821950859553908e+03, /* 0x40A8FFB7, 0x688C246A */
                    191: sb5  =  2.55305040643316442583e+03, /* 0x40A3F219, 0xCEDF3BE6 */
                    192: sb6  =  4.74528541206955367215e+02, /* 0x407DA874, 0xE79FE763 */
                    193: sb7  = -2.24409524465858183362e+01; /* 0xC03670E2, 0x42712D62 */
                    194:
                    195: #ifdef __STDC__
1.10    ! simonb    196:        double erf(double x)
1.1       jtc       197: #else
1.10    ! simonb    198:        double erf(x)
1.1       jtc       199:        double x;
                    200: #endif
                    201: {
1.6       jtc       202:        int32_t hx,ix,i;
1.1       jtc       203:        double R,S,P,Q,s,y,z,r;
1.5       jtc       204:        GET_HIGH_WORD(hx,x);
1.1       jtc       205:        ix = hx&0x7fffffff;
                    206:        if(ix>=0x7ff00000) {            /* erf(nan)=nan */
1.6       jtc       207:            i = ((u_int32_t)hx>>31)<<1;
1.1       jtc       208:            return (double)(1-i)+one/x; /* erf(+-inf)=+-1 */
                    209:        }
                    210:
                    211:        if(ix < 0x3feb0000) {           /* |x|<0.84375 */
                    212:            if(ix < 0x3e300000) {       /* |x|<2**-28 */
1.10    ! simonb    213:                if (ix < 0x00800000)
1.1       jtc       214:                    return 0.125*(8.0*x+efx8*x);  /*avoid underflow */
                    215:                return x + efx*x;
                    216:            }
                    217:            z = x*x;
                    218:            r = pp0+z*(pp1+z*(pp2+z*(pp3+z*pp4)));
                    219:            s = one+z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*qq5))));
                    220:            y = r/s;
                    221:            return x + x*y;
                    222:        }
                    223:        if(ix < 0x3ff40000) {           /* 0.84375 <= |x| < 1.25 */
                    224:            s = fabs(x)-one;
                    225:            P = pa0+s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6)))));
                    226:            Q = one+s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6)))));
                    227:            if(hx>=0) return erx + P/Q; else return -erx - P/Q;
                    228:        }
                    229:        if (ix >= 0x40180000) {         /* inf>|x|>=6 */
                    230:            if(hx>=0) return one-tiny; else return tiny-one;
                    231:        }
                    232:        x = fabs(x);
                    233:        s = one/(x*x);
                    234:        if(ix< 0x4006DB6E) {    /* |x| < 1/0.35 */
                    235:            R=ra0+s*(ra1+s*(ra2+s*(ra3+s*(ra4+s*(
                    236:                                ra5+s*(ra6+s*ra7))))));
                    237:            S=one+s*(sa1+s*(sa2+s*(sa3+s*(sa4+s*(
                    238:                                sa5+s*(sa6+s*(sa7+s*sa8)))))));
                    239:        } else {        /* |x| >= 1/0.35 */
                    240:            R=rb0+s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*(
                    241:                                rb5+s*rb6)))));
                    242:            S=one+s*(sb1+s*(sb2+s*(sb3+s*(sb4+s*(
                    243:                                sb5+s*(sb6+s*sb7))))));
                    244:        }
1.10    ! simonb    245:        z  = x;
1.5       jtc       246:        SET_LOW_WORD(z,0);
1.1       jtc       247:        r  =  __ieee754_exp(-z*z-0.5625)*__ieee754_exp((z-x)*(z+x)+R/S);
                    248:        if(hx>=0) return one-r/x; else return  r/x-one;
                    249: }
                    250:
                    251: #ifdef __STDC__
1.10    ! simonb    252:        double erfc(double x)
1.1       jtc       253: #else
1.10    ! simonb    254:        double erfc(x)
1.1       jtc       255:        double x;
                    256: #endif
                    257: {
1.6       jtc       258:        int32_t hx,ix;
1.1       jtc       259:        double R,S,P,Q,s,y,z,r;
1.5       jtc       260:        GET_HIGH_WORD(hx,x);
1.1       jtc       261:        ix = hx&0x7fffffff;
                    262:        if(ix>=0x7ff00000) {                    /* erfc(nan)=nan */
                    263:                                                /* erfc(+-inf)=0,2 */
1.6       jtc       264:            return (double)(((u_int32_t)hx>>31)<<1)+one/x;
1.1       jtc       265:        }
                    266:
                    267:        if(ix < 0x3feb0000) {           /* |x|<0.84375 */
                    268:            if(ix < 0x3c700000)         /* |x|<2**-56 */
                    269:                return one-x;
                    270:            z = x*x;
                    271:            r = pp0+z*(pp1+z*(pp2+z*(pp3+z*pp4)));
                    272:            s = one+z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*qq5))));
                    273:            y = r/s;
                    274:            if(hx < 0x3fd00000) {       /* x<1/4 */
                    275:                return one-(x+x*y);
                    276:            } else {
                    277:                r = x*y;
                    278:                r += (x-half);
                    279:                return half - r ;
                    280:            }
                    281:        }
                    282:        if(ix < 0x3ff40000) {           /* 0.84375 <= |x| < 1.25 */
                    283:            s = fabs(x)-one;
                    284:            P = pa0+s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6)))));
                    285:            Q = one+s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6)))));
                    286:            if(hx>=0) {
1.10    ! simonb    287:                z  = one-erx; return z - P/Q;
1.1       jtc       288:            } else {
                    289:                z = erx+P/Q; return one+z;
                    290:            }
                    291:        }
                    292:        if (ix < 0x403c0000) {          /* |x|<28 */
                    293:            x = fabs(x);
                    294:            s = one/(x*x);
                    295:            if(ix< 0x4006DB6D) {        /* |x| < 1/.35 ~ 2.857143*/
                    296:                R=ra0+s*(ra1+s*(ra2+s*(ra3+s*(ra4+s*(
                    297:                                ra5+s*(ra6+s*ra7))))));
                    298:                S=one+s*(sa1+s*(sa2+s*(sa3+s*(sa4+s*(
                    299:                                sa5+s*(sa6+s*(sa7+s*sa8)))))));
                    300:            } else {                    /* |x| >= 1/.35 ~ 2.857143 */
                    301:                if(hx<0&&ix>=0x40180000) return two-tiny;/* x < -6 */
                    302:                R=rb0+s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*(
                    303:                                rb5+s*rb6)))));
                    304:                S=one+s*(sb1+s*(sb2+s*(sb3+s*(sb4+s*(
                    305:                                sb5+s*(sb6+s*sb7))))));
                    306:            }
                    307:            z  = x;
1.5       jtc       308:            SET_LOW_WORD(z,0);
1.1       jtc       309:            r  =  __ieee754_exp(-z*z-0.5625)*
                    310:                        __ieee754_exp((z-x)*(z+x)+R/S);
                    311:            if(hx>0) return r/x; else return two-r/x;
                    312:        } else {
                    313:            if(hx>0) return tiny*tiny; else return two-tiny;
                    314:        }
                    315: }

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