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