WebSVN - shark - Blame - Rev 3 - /shark/trunk/oslib/libm/msun/src/s

Rev	Author	Line No.	Line
2	pj	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	#ifndef lint
		14	static char rcsid[] = "$\Id: s_erf.c,v 1.2 1995/05/30 05:49:31 rgrimes Exp $";
		15	#endif
		16
		17	/* double erf(double x)
		18	* double erfc(double x)
		19	* x
		20	* 2 \|\
		21	* erf(x) = --------- \| exp(-t*t)dt
		22	* sqrt(pi) \\|
		23	* 0
		24	*
		25	* erfc(x) = 1-erf(x)
		26	* Note that
		27	* erf(-x) = -erf(x)
		28	* erfc(-x) = 2 - erfc(x)
		29	*
		30	* Method:
		31	* 1. For \|x\| in [0, 0.84375]
		32	* erf(x) = x + x*R(x^2)
		33	* erfc(x) = 1 - erf(x) if x in [-.84375,0.25]
		34	* = 0.5 + ((0.5-x)-x*R) if x in [0.25,0.84375]
		35	* where R = P/Q where P is an odd poly of degree 8 and
		36	* Q is an odd poly of degree 10.
		37	* -57.90
		38	* \| R - (erf(x)-x)/x \| <= 2
		39	*
		40	*
		41	* Remark. The formula is derived by noting
		42	* erf(x) = (2/sqrt(pi))*(x - x^3/3 + x^5/10 - x^7/42 + ....)
		43	* and that
		44	* 2/sqrt(pi) = 1.128379167095512573896158903121545171688
		45	* is close to one. The interval is chosen because the fix
		46	* point of erf(x) is near 0.6174 (i.e., erf(x)=x when x is
		47	* near 0.6174), and by some experiment, 0.84375 is chosen to
		48	* guarantee the error is less than one ulp for erf.
		49	*
		50	* 2. For \|x\| in [0.84375,1.25], let s = \|x\| - 1, and
		51	* c = 0.84506291151 rounded to single (24 bits)
		52	* erf(x) = sign(x) * (c + P1(s)/Q1(s))
		53	* erfc(x) = (1-c) - P1(s)/Q1(s) if x > 0
		54	* 1+(c+P1(s)/Q1(s)) if x < 0
		55	* \|P1/Q1 - (erf(\|x\|)-c)\| <= 2**-59.06
		56	* Remark: here we use the taylor series expansion at x=1.
		57	* erf(1+s) = erf(1) + s*Poly(s)
		58	* = 0.845.. + P1(s)/Q1(s)
		59	* That is, we use rational approximation to approximate
		60	* erf(1+s) - (c = (single)0.84506291151)
		61	* Note that \|P1/Q1\|< 0.078 for x in [0.84375,1.25]
		62	* where
		63	* P1(s) = degree 6 poly in s
		64	* Q1(s) = degree 6 poly in s
		65	*
		66	* 3. For x in [1.25,1/0.35(~2.857143)],
		67	* erfc(x) = (1/x)exp(-xx-0.5625+R1/S1)
		68	* erf(x) = 1 - erfc(x)
		69	* where
		70	* R1(z) = degree 7 poly in z, (z=1/x^2)
		71	* S1(z) = degree 8 poly in z
		72	*
		73	* 4. For x in [1/0.35,28]
		74	* erfc(x) = (1/x)exp(-xx-0.5625+R2/S2) if x > 0
		75	* = 2.0 - (1/x)exp(-xx-0.5625+R2/S2) if -6<x<0
		76	* = 2.0 - tiny (if x <= -6)
		77	* erf(x) = sign(x)*(1.0 - erfc(x)) if x < 6, else
		78	* erf(x) = sign(x)*(1.0 - tiny)
		79	* where
		80	* R2(z) = degree 6 poly in z, (z=1/x^2)
		81	* S2(z) = degree 7 poly in z
		82	*
		83	* Note1:
		84	* To compute exp(-x*x-0.5625+R/S), let s be a single
		85	* precision number and s := x; then
		86	* -xx = -ss + (s-x)*(s+x)
		87	* exp(-x*x-0.5626+R/S) =
		88	* exp(-ss-0.5625)exp((s-x)*(s+x)+R/S);
		89	* Note2:
		90	* Here 4 and 5 make use of the asymptotic series
		91	* exp(-x*x)
		92	* erfc(x) ~ ---------- * ( 1 + Poly(1/x^2) )
		93	* x*sqrt(pi)
		94	* We use rational approximation to approximate
		95	* g(s)=f(1/x^2) = log(erfc(x)x) - xx + 0.5625
		96	* Here is the error bound for R1/S1 and R2/S2
		97	* \|R1/S1 - f(x)\| < 2**(-62.57)
		98	* \|R2/S2 - f(x)\| < 2**(-61.52)
		99	*
		100	* 5. For inf > x >= 28
		101	* erf(x) = sign(x) *(1 - tiny) (raise inexact)
		102	* erfc(x) = tiny*tiny (raise underflow) if x > 0
		103	* = 2 - tiny if x<0
		104	*
		105	* 7. Special case:
		106	* erf(0) = 0, erf(inf) = 1, erf(-inf) = -1,
		107	* erfc(0) = 1, erfc(inf) = 0, erfc(-inf) = 2,
		108	* erfc/erf(NaN) is NaN
		109	*/
		110
		111
		112	#include "math.h"
		113	#include "math_private.h"
		114
		115	#ifdef __STDC__
		116	static const double
		117	#else
		118	static double
		119	#endif
		120	tiny = 1e-300,
		121	half= 5.00000000000000000000e-01, /* 0x3FE00000, 0x00000000 */
		122	one = 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
		123	two = 2.00000000000000000000e+00, /* 0x40000000, 0x00000000 */
		124	/* c = (float)0.84506291151 */
		125	erx = 8.45062911510467529297e-01, /* 0x3FEB0AC1, 0x60000000 */
		126	/*
		127	* Coefficients for approximation to erf on [0,0.84375]
		128	*/
		129	efx = 1.28379167095512586316e-01, /* 0x3FC06EBA, 0x8214DB69 */
		130	efx8= 1.02703333676410069053e+00, /* 0x3FF06EBA, 0x8214DB69 */
		131	pp0 = 1.28379167095512558561e-01, /* 0x3FC06EBA, 0x8214DB68 */
		132	pp1 = -3.25042107247001499370e-01, /* 0xBFD4CD7D, 0x691CB913 */
		133	pp2 = -2.84817495755985104766e-02, /* 0xBF9D2A51, 0xDBD7194F */
		134	pp3 = -5.77027029648944159157e-03, /* 0xBF77A291, 0x236668E4 */
		135	pp4 = -2.37630166566501626084e-05, /* 0xBEF8EAD6, 0x120016AC */
		136	qq1 = 3.97917223959155352819e-01, /* 0x3FD97779, 0xCDDADC09 */
		137	qq2 = 6.50222499887672944485e-02, /* 0x3FB0A54C, 0x5536CEBA */
		138	qq3 = 5.08130628187576562776e-03, /* 0x3F74D022, 0xC4D36B0F */
		139	qq4 = 1.32494738004321644526e-04, /* 0x3F215DC9, 0x221C1A10 */
		140	qq5 = -3.96022827877536812320e-06, /* 0xBED09C43, 0x42A26120 */
		141	/*
		142	* Coefficients for approximation to erf in [0.84375,1.25]
		143	*/
		144	pa0 = -2.36211856075265944077e-03, /* 0xBF6359B8, 0xBEF77538 */
		145	pa1 = 4.14856118683748331666e-01, /* 0x3FDA8D00, 0xAD92B34D */
		146	pa2 = -3.72207876035701323847e-01, /* 0xBFD7D240, 0xFBB8C3F1 */
		147	pa3 = 3.18346619901161753674e-01, /* 0x3FD45FCA, 0x805120E4 */
		148	pa4 = -1.10894694282396677476e-01, /* 0xBFBC6398, 0x3D3E28EC */
		149	pa5 = 3.54783043256182359371e-02, /* 0x3FA22A36, 0x599795EB */
		150	pa6 = -2.16637559486879084300e-03, /* 0xBF61BF38, 0x0A96073F */
		151	qa1 = 1.06420880400844228286e-01, /* 0x3FBB3E66, 0x18EEE323 */
		152	qa2 = 5.40397917702171048937e-01, /* 0x3FE14AF0, 0x92EB6F33 */
		153	qa3 = 7.18286544141962662868e-02, /* 0x3FB2635C, 0xD99FE9A7 */
		154	qa4 = 1.26171219808761642112e-01, /* 0x3FC02660, 0xE763351F */
		155	qa5 = 1.36370839120290507362e-02, /* 0x3F8BEDC2, 0x6B51DD1C */
		156	qa6 = 1.19844998467991074170e-02, /* 0x3F888B54, 0x5735151D */
		157	/*
		158	* Coefficients for approximation to erfc in [1.25,1/0.35]
		159	*/
		160	ra0 = -9.86494403484714822705e-03, /* 0xBF843412, 0x600D6435 */
		161	ra1 = -6.93858572707181764372e-01, /* 0xBFE63416, 0xE4BA7360 */
		162	ra2 = -1.05586262253232909814e+01, /* 0xC0251E04, 0x41B0E726 */
		163	ra3 = -6.23753324503260060396e+01, /* 0xC04F300A, 0xE4CBA38D */
		164	ra4 = -1.62396669462573470355e+02, /* 0xC0644CB1, 0x84282266 */
		165	ra5 = -1.84605092906711035994e+02, /* 0xC067135C, 0xEBCCABB2 */
		166	ra6 = -8.12874355063065934246e+01, /* 0xC0545265, 0x57E4D2F2 */
		167	ra7 = -9.81432934416914548592e+00, /* 0xC023A0EF, 0xC69AC25C */
		168	sa1 = 1.96512716674392571292e+01, /* 0x4033A6B9, 0xBD707687 */
		169	sa2 = 1.37657754143519042600e+02, /* 0x4061350C, 0x526AE721 */
		170	sa3 = 4.34565877475229228821e+02, /* 0x407B290D, 0xD58A1A71 */
		171	sa4 = 6.45387271733267880336e+02, /* 0x40842B19, 0x21EC2868 */
		172	sa5 = 4.29008140027567833386e+02, /* 0x407AD021, 0x57700314 */
		173	sa6 = 1.08635005541779435134e+02, /* 0x405B28A3, 0xEE48AE2C */
		174	sa7 = 6.57024977031928170135e+00, /* 0x401A47EF, 0x8E484A93 */
		175	sa8 = -6.04244152148580987438e-02, /* 0xBFAEEFF2, 0xEE749A62 */
		176	/*
		177	* Coefficients for approximation to erfc in [1/.35,28]
		178	*/
		179	rb0 = -9.86494292470009928597e-03, /* 0xBF843412, 0x39E86F4A */
		180	rb1 = -7.99283237680523006574e-01, /* 0xBFE993BA, 0x70C285DE */
		181	rb2 = -1.77579549177547519889e+01, /* 0xC031C209, 0x555F995A */
		182	rb3 = -1.60636384855821916062e+02, /* 0xC064145D, 0x43C5ED98 */
		183	rb4 = -6.37566443368389627722e+02, /* 0xC083EC88, 0x1375F228 */
		184	rb5 = -1.02509513161107724954e+03, /* 0xC0900461, 0x6A2E5992 */
		185	rb6 = -4.83519191608651397019e+02, /* 0xC07E384E, 0x9BDC383F */
		186	sb1 = 3.03380607434824582924e+01, /* 0x403E568B, 0x261D5190 */
		187	sb2 = 3.25792512996573918826e+02, /* 0x40745CAE, 0x221B9F0A */
		188	sb3 = 1.53672958608443695994e+03, /* 0x409802EB, 0x189D5118 */
		189	sb4 = 3.19985821950859553908e+03, /* 0x40A8FFB7, 0x688C246A */
		190	sb5 = 2.55305040643316442583e+03, /* 0x40A3F219, 0xCEDF3BE6 */
		191	sb6 = 4.74528541206955367215e+02, /* 0x407DA874, 0xE79FE763 */
		192	sb7 = -2.24409524465858183362e+01; /* 0xC03670E2, 0x42712D62 */
		193
		194	#ifdef __STDC__
		195	double erf(double x)
		196	#else
		197	double erf(x)
		198	double x;
		199	#endif
		200	{
		201	int32_t hx,ix,i;
		202	double R,S,P,Q,s,y,z,r;
		203	GET_HIGH_WORD(hx,x);
		204	ix = hx&0x7fffffff;
		205	if(ix>=0x7ff00000) { /* erf(nan)=nan */
		206	i = ((u_int32_t)hx>>31)<<1;
		207	return (double)(1-i)+one/x; /* erf(+-inf)=+-1 */
		208	}
		209
		210	if(ix < 0x3feb0000) { /* \|x\|<0.84375 */
		211	if(ix < 0x3e300000) { /* \|x\|<2*-28 /
		212	if (ix < 0x00800000)
		213	return 0.125(8.0x+efx8x); /avoid underflow */
		214	return x + efx*x;
		215	}
		216	z = x*x;
		217	r = pp0+z(pp1+z(pp2+z(pp3+zpp4)));
		218	s = one+z(qq1+z(qq2+z(qq3+z(qq4+z*qq5))));
		219	y = r/s;
		220	return x + x*y;
		221	}
		222	if(ix < 0x3ff40000) { /* 0.84375 <= \|x\| < 1.25 */
		223	s = fabs(x)-one;
		224	P = pa0+s(pa1+s(pa2+s(pa3+s(pa4+s(pa5+spa6)))));
		225	Q = one+s(qa1+s(qa2+s(qa3+s(qa4+s(qa5+sqa6)))));
		226	if(hx>=0) return erx + P/Q; else return -erx - P/Q;
		227	}
		228	if (ix >= 0x40180000) { /* inf>\|x\|>=6 */
		229	if(hx>=0) return one-tiny; else return tiny-one;
		230	}
		231	x = fabs(x);
		232	s = one/(x*x);
		233	if(ix< 0x4006DB6E) { /* \|x\| < 1/0.35 */
		234	R=ra0+s(ra1+s(ra2+s(ra3+s(ra4+s*(
		235	ra5+s(ra6+sra7))))));
		236	S=one+s(sa1+s(sa2+s(sa3+s(sa4+s*(
		237	sa5+s(sa6+s(sa7+s*sa8)))))));
		238	} else { /* \|x\| >= 1/0.35 */
		239	R=rb0+s(rb1+s(rb2+s(rb3+s(rb4+s*(
		240	rb5+s*rb6)))));
		241	S=one+s(sb1+s(sb2+s(sb3+s(sb4+s*(
		242	sb5+s(sb6+ssb7))))));
		243	}
		244	z = x;
		245	SET_LOW_WORD(z,0);
		246	r = __ieee754_exp(-zz-0.5625)__ieee754_exp((z-x)*(z+x)+R/S);
		247	if(hx>=0) return one-r/x; else return r/x-one;
		248	}
		249
		250	#ifdef __STDC__
		251	double erfc(double x)
		252	#else
		253	double erfc(x)
		254	double x;
		255	#endif
		256	{
		257	int32_t hx,ix;
		258	double R,S,P,Q,s,y,z,r;
		259	GET_HIGH_WORD(hx,x);
		260	ix = hx&0x7fffffff;
		261	if(ix>=0x7ff00000) { /* erfc(nan)=nan */
		262	/* erfc(+-inf)=0,2 */
		263	return (double)(((u_int32_t)hx>>31)<<1)+one/x;
		264	}
		265
		266	if(ix < 0x3feb0000) { /* \|x\|<0.84375 */
		267	if(ix < 0x3c700000) /* \|x\|<2*-56 /
		268	return one-x;
		269	z = x*x;
		270	r = pp0+z(pp1+z(pp2+z(pp3+zpp4)));
		271	s = one+z(qq1+z(qq2+z(qq3+z(qq4+z*qq5))));
		272	y = r/s;
		273	if(hx < 0x3fd00000) { /* x<1/4 */
		274	return one-(x+x*y);
		275	} else {
		276	r = x*y;
		277	r += (x-half);
		278	return half - r ;
		279	}
		280	}
		281	if(ix < 0x3ff40000) { /* 0.84375 <= \|x\| < 1.25 */
		282	s = fabs(x)-one;
		283	P = pa0+s(pa1+s(pa2+s(pa3+s(pa4+s(pa5+spa6)))));
		284	Q = one+s(qa1+s(qa2+s(qa3+s(qa4+s(qa5+sqa6)))));
		285	if(hx>=0) {
		286	z = one-erx; return z - P/Q;
		287	} else {
		288	z = erx+P/Q; return one+z;
		289	}
		290	}
		291	if (ix < 0x403c0000) { /* \|x\|<28 */
		292	x = fabs(x);
		293	s = one/(x*x);
		294	if(ix< 0x4006DB6D) { /* \|x\| < 1/.35 ~ 2.857143*/
		295	R=ra0+s(ra1+s(ra2+s(ra3+s(ra4+s*(
		296	ra5+s(ra6+sra7))))));
		297	S=one+s(sa1+s(sa2+s(sa3+s(sa4+s*(
		298	sa5+s(sa6+s(sa7+s*sa8)))))));
		299	} else { /* \|x\| >= 1/.35 ~ 2.857143 */
		300	if(hx<0&&ix>=0x40180000) return two-tiny;/* x < -6 */
		301	R=rb0+s(rb1+s(rb2+s(rb3+s(rb4+s*(
		302	rb5+s*rb6)))));
		303	S=one+s(sb1+s(sb2+s(sb3+s(sb4+s*(
		304	sb5+s(sb6+ssb7))))));
		305	}
		306	z = x;
		307	SET_LOW_WORD(z,0);
		308	r = __ieee754_exp(-zz-0.5625)
		309	__ieee754_exp((z-x)*(z+x)+R/S);
		310	if(hx>0) return r/x; else return two-r/x;
		311	} else {
		312	if(hx>0) return tiny*tiny; else return two-tiny;
		313	}
		314	}

Subversion Repositories shark

(root)/shark/trunk/oslib/libm/msun/src/s_erf.c - Rev 3