xref: /minix3/lib/libm/src/e_j0.c (revision 2fe8fb192fe7e8720e3e7a77f928da545e872a6a)
1*2fe8fb19SBen Gras /* @(#)e_j0.c 5.1 93/09/24 */
2*2fe8fb19SBen Gras /*
3*2fe8fb19SBen Gras  * ====================================================
4*2fe8fb19SBen Gras  * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
5*2fe8fb19SBen Gras  *
6*2fe8fb19SBen Gras  * Developed at SunPro, a Sun Microsystems, Inc. business.
7*2fe8fb19SBen Gras  * Permission to use, copy, modify, and distribute this
8*2fe8fb19SBen Gras  * software is freely granted, provided that this notice
9*2fe8fb19SBen Gras  * is preserved.
10*2fe8fb19SBen Gras  * ====================================================
11*2fe8fb19SBen Gras  */
12*2fe8fb19SBen Gras 
13*2fe8fb19SBen Gras #include <sys/cdefs.h>
14*2fe8fb19SBen Gras #if defined(LIBM_SCCS) && !defined(lint)
15*2fe8fb19SBen Gras __RCSID("$NetBSD: e_j0.c,v 1.12 2007/08/20 16:01:38 drochner Exp $");
16*2fe8fb19SBen Gras #endif
17*2fe8fb19SBen Gras 
18*2fe8fb19SBen Gras /* __ieee754_j0(x), __ieee754_y0(x)
19*2fe8fb19SBen Gras  * Bessel function of the first and second kinds of order zero.
20*2fe8fb19SBen Gras  * Method -- j0(x):
21*2fe8fb19SBen Gras  *	1. For tiny x, we use j0(x) = 1 - x^2/4 + x^4/64 - ...
22*2fe8fb19SBen Gras  *	2. Reduce x to |x| since j0(x)=j0(-x),  and
23*2fe8fb19SBen Gras  *	   for x in (0,2)
24*2fe8fb19SBen Gras  *		j0(x) = 1-z/4+ z^2*R0/S0,  where z = x*x;
25*2fe8fb19SBen Gras  *	   (precision:  |j0-1+z/4-z^2R0/S0 |<2**-63.67 )
26*2fe8fb19SBen Gras  *	   for x in (2,inf)
27*2fe8fb19SBen Gras  * 		j0(x) = sqrt(2/(pi*x))*(p0(x)*cos(x0)-q0(x)*sin(x0))
28*2fe8fb19SBen Gras  * 	   where x0 = x-pi/4. It is better to compute sin(x0),cos(x0)
29*2fe8fb19SBen Gras  *	   as follow:
30*2fe8fb19SBen Gras  *		cos(x0) = cos(x)cos(pi/4)+sin(x)sin(pi/4)
31*2fe8fb19SBen Gras  *			= 1/sqrt(2) * (cos(x) + sin(x))
32*2fe8fb19SBen Gras  *		sin(x0) = sin(x)cos(pi/4)-cos(x)sin(pi/4)
33*2fe8fb19SBen Gras  *			= 1/sqrt(2) * (sin(x) - cos(x))
34*2fe8fb19SBen Gras  * 	   (To avoid cancellation, use
35*2fe8fb19SBen Gras  *		sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
36*2fe8fb19SBen Gras  * 	    to compute the worse one.)
37*2fe8fb19SBen Gras  *
38*2fe8fb19SBen Gras  *	3 Special cases
39*2fe8fb19SBen Gras  *		j0(nan)= nan
40*2fe8fb19SBen Gras  *		j0(0) = 1
41*2fe8fb19SBen Gras  *		j0(inf) = 0
42*2fe8fb19SBen Gras  *
43*2fe8fb19SBen Gras  * Method -- y0(x):
44*2fe8fb19SBen Gras  *	1. For x<2.
45*2fe8fb19SBen Gras  *	   Since
46*2fe8fb19SBen Gras  *		y0(x) = 2/pi*(j0(x)*(ln(x/2)+Euler) + x^2/4 - ...)
47*2fe8fb19SBen Gras  *	   therefore y0(x)-2/pi*j0(x)*ln(x) is an even function.
48*2fe8fb19SBen Gras  *	   We use the following function to approximate y0,
49*2fe8fb19SBen Gras  *		y0(x) = U(z)/V(z) + (2/pi)*(j0(x)*ln(x)), z= x^2
50*2fe8fb19SBen Gras  *	   where
51*2fe8fb19SBen Gras  *		U(z) = u00 + u01*z + ... + u06*z^6
52*2fe8fb19SBen Gras  *		V(z) = 1  + v01*z + ... + v04*z^4
53*2fe8fb19SBen Gras  *	   with absolute approximation error bounded by 2**-72.
54*2fe8fb19SBen Gras  *	   Note: For tiny x, U/V = u0 and j0(x)~1, hence
55*2fe8fb19SBen Gras  *		y0(tiny) = u0 + (2/pi)*ln(tiny), (choose tiny<2**-27)
56*2fe8fb19SBen Gras  *	2. For x>=2.
57*2fe8fb19SBen Gras  * 		y0(x) = sqrt(2/(pi*x))*(p0(x)*cos(x0)+q0(x)*sin(x0))
58*2fe8fb19SBen Gras  * 	   where x0 = x-pi/4. It is better to compute sin(x0),cos(x0)
59*2fe8fb19SBen Gras  *	   by the method mentioned above.
60*2fe8fb19SBen Gras  *	3. Special cases: y0(0)=-inf, y0(x<0)=NaN, y0(inf)=0.
61*2fe8fb19SBen Gras  */
62*2fe8fb19SBen Gras 
63*2fe8fb19SBen Gras #include "namespace.h"
64*2fe8fb19SBen Gras #include "math.h"
65*2fe8fb19SBen Gras #include "math_private.h"
66*2fe8fb19SBen Gras 
67*2fe8fb19SBen Gras static double pzero(double), qzero(double);
68*2fe8fb19SBen Gras 
69*2fe8fb19SBen Gras static const double
70*2fe8fb19SBen Gras huge 	= 1e300,
71*2fe8fb19SBen Gras one	= 1.0,
72*2fe8fb19SBen Gras invsqrtpi=  5.64189583547756279280e-01, /* 0x3FE20DD7, 0x50429B6D */
73*2fe8fb19SBen Gras tpi      =  6.36619772367581382433e-01, /* 0x3FE45F30, 0x6DC9C883 */
74*2fe8fb19SBen Gras  		/* R0/S0 on [0, 2.00] */
75*2fe8fb19SBen Gras R02  =  1.56249999999999947958e-02, /* 0x3F8FFFFF, 0xFFFFFFFD */
76*2fe8fb19SBen Gras R03  = -1.89979294238854721751e-04, /* 0xBF28E6A5, 0xB61AC6E9 */
77*2fe8fb19SBen Gras R04  =  1.82954049532700665670e-06, /* 0x3EBEB1D1, 0x0C503919 */
78*2fe8fb19SBen Gras R05  = -4.61832688532103189199e-09, /* 0xBE33D5E7, 0x73D63FCE */
79*2fe8fb19SBen Gras S01  =  1.56191029464890010492e-02, /* 0x3F8FFCE8, 0x82C8C2A4 */
80*2fe8fb19SBen Gras S02  =  1.16926784663337450260e-04, /* 0x3F1EA6D2, 0xDD57DBF4 */
81*2fe8fb19SBen Gras S03  =  5.13546550207318111446e-07, /* 0x3EA13B54, 0xCE84D5A9 */
82*2fe8fb19SBen Gras S04  =  1.16614003333790000205e-09; /* 0x3E1408BC, 0xF4745D8F */
83*2fe8fb19SBen Gras 
84*2fe8fb19SBen Gras static const double zero = 0.0;
85*2fe8fb19SBen Gras 
86*2fe8fb19SBen Gras double
__ieee754_j0(double x)87*2fe8fb19SBen Gras __ieee754_j0(double x)
88*2fe8fb19SBen Gras {
89*2fe8fb19SBen Gras 	double z, s,c,ss,cc,r,u,v;
90*2fe8fb19SBen Gras 	int32_t hx,ix;
91*2fe8fb19SBen Gras 
92*2fe8fb19SBen Gras 	GET_HIGH_WORD(hx,x);
93*2fe8fb19SBen Gras 	ix = hx&0x7fffffff;
94*2fe8fb19SBen Gras 	if(ix>=0x7ff00000) return one/(x*x);
95*2fe8fb19SBen Gras 	x = fabs(x);
96*2fe8fb19SBen Gras 	if(ix >= 0x40000000) {	/* |x| >= 2.0 */
97*2fe8fb19SBen Gras 		s = sin(x);
98*2fe8fb19SBen Gras 		c = cos(x);
99*2fe8fb19SBen Gras 		ss = s-c;
100*2fe8fb19SBen Gras 		cc = s+c;
101*2fe8fb19SBen Gras 		if(ix<0x7fe00000) {  /* make sure x+x not overflow */
102*2fe8fb19SBen Gras 		    z = -cos(x+x);
103*2fe8fb19SBen Gras 		    if ((s*c)<zero) cc = z/ss;
104*2fe8fb19SBen Gras 		    else 	    ss = z/cc;
105*2fe8fb19SBen Gras 		}
106*2fe8fb19SBen Gras 	/*
107*2fe8fb19SBen Gras 	 * j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x)
108*2fe8fb19SBen Gras 	 * y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x)
109*2fe8fb19SBen Gras 	 */
110*2fe8fb19SBen Gras 		if(ix>0x48000000) z = (invsqrtpi*cc)/sqrt(x);
111*2fe8fb19SBen Gras 		else {
112*2fe8fb19SBen Gras 		    u = pzero(x); v = qzero(x);
113*2fe8fb19SBen Gras 		    z = invsqrtpi*(u*cc-v*ss)/sqrt(x);
114*2fe8fb19SBen Gras 		}
115*2fe8fb19SBen Gras 		return z;
116*2fe8fb19SBen Gras 	}
117*2fe8fb19SBen Gras 	if(ix<0x3f200000) {	/* |x| < 2**-13 */
118*2fe8fb19SBen Gras 	    if(huge+x>one) {	/* raise inexact if x != 0 */
119*2fe8fb19SBen Gras 	        if(ix<0x3e400000) return one;	/* |x|<2**-27 */
120*2fe8fb19SBen Gras 	        else 	      return one - 0.25*x*x;
121*2fe8fb19SBen Gras 	    }
122*2fe8fb19SBen Gras 	}
123*2fe8fb19SBen Gras 	z = x*x;
124*2fe8fb19SBen Gras 	r =  z*(R02+z*(R03+z*(R04+z*R05)));
125*2fe8fb19SBen Gras 	s =  one+z*(S01+z*(S02+z*(S03+z*S04)));
126*2fe8fb19SBen Gras 	if(ix < 0x3FF00000) {	/* |x| < 1.00 */
127*2fe8fb19SBen Gras 	    return one + z*(-0.25+(r/s));
128*2fe8fb19SBen Gras 	} else {
129*2fe8fb19SBen Gras 	    u = 0.5*x;
130*2fe8fb19SBen Gras 	    return((one+u)*(one-u)+z*(r/s));
131*2fe8fb19SBen Gras 	}
132*2fe8fb19SBen Gras }
133*2fe8fb19SBen Gras 
134*2fe8fb19SBen Gras static const double
135*2fe8fb19SBen Gras u00  = -7.38042951086872317523e-02, /* 0xBFB2E4D6, 0x99CBD01F */
136*2fe8fb19SBen Gras u01  =  1.76666452509181115538e-01, /* 0x3FC69D01, 0x9DE9E3FC */
137*2fe8fb19SBen Gras u02  = -1.38185671945596898896e-02, /* 0xBF8C4CE8, 0xB16CFA97 */
138*2fe8fb19SBen Gras u03  =  3.47453432093683650238e-04, /* 0x3F36C54D, 0x20B29B6B */
139*2fe8fb19SBen Gras u04  = -3.81407053724364161125e-06, /* 0xBECFFEA7, 0x73D25CAD */
140*2fe8fb19SBen Gras u05  =  1.95590137035022920206e-08, /* 0x3E550057, 0x3B4EABD4 */
141*2fe8fb19SBen Gras u06  = -3.98205194132103398453e-11, /* 0xBDC5E43D, 0x693FB3C8 */
142*2fe8fb19SBen Gras v01  =  1.27304834834123699328e-02, /* 0x3F8A1270, 0x91C9C71A */
143*2fe8fb19SBen Gras v02  =  7.60068627350353253702e-05, /* 0x3F13ECBB, 0xF578C6C1 */
144*2fe8fb19SBen Gras v03  =  2.59150851840457805467e-07, /* 0x3E91642D, 0x7FF202FD */
145*2fe8fb19SBen Gras v04  =  4.41110311332675467403e-10; /* 0x3DFE5018, 0x3BD6D9EF */
146*2fe8fb19SBen Gras 
147*2fe8fb19SBen Gras double
__ieee754_y0(double x)148*2fe8fb19SBen Gras __ieee754_y0(double x)
149*2fe8fb19SBen Gras {
150*2fe8fb19SBen Gras 	double z, s,c,ss,cc,u,v;
151*2fe8fb19SBen Gras 	int32_t hx,ix,lx;
152*2fe8fb19SBen Gras 
153*2fe8fb19SBen Gras 	EXTRACT_WORDS(hx,lx,x);
154*2fe8fb19SBen Gras         ix = 0x7fffffff&hx;
155*2fe8fb19SBen Gras     /* Y0(NaN) is NaN, y0(-inf) is Nan, y0(inf) is 0  */
156*2fe8fb19SBen Gras 	if(ix>=0x7ff00000) return  one/(x+x*x);
157*2fe8fb19SBen Gras         if((ix|lx)==0) return -one/zero;
158*2fe8fb19SBen Gras         if(hx<0) return zero/zero;
159*2fe8fb19SBen Gras         if(ix >= 0x40000000) {  /* |x| >= 2.0 */
160*2fe8fb19SBen Gras         /* y0(x) = sqrt(2/(pi*x))*(p0(x)*sin(x0)+q0(x)*cos(x0))
161*2fe8fb19SBen Gras          * where x0 = x-pi/4
162*2fe8fb19SBen Gras          *      Better formula:
163*2fe8fb19SBen Gras          *              cos(x0) = cos(x)cos(pi/4)+sin(x)sin(pi/4)
164*2fe8fb19SBen Gras          *                      =  1/sqrt(2) * (sin(x) + cos(x))
165*2fe8fb19SBen Gras          *              sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4)
166*2fe8fb19SBen Gras          *                      =  1/sqrt(2) * (sin(x) - cos(x))
167*2fe8fb19SBen Gras          * To avoid cancellation, use
168*2fe8fb19SBen Gras          *              sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
169*2fe8fb19SBen Gras          * to compute the worse one.
170*2fe8fb19SBen Gras          */
171*2fe8fb19SBen Gras                 s = sin(x);
172*2fe8fb19SBen Gras                 c = cos(x);
173*2fe8fb19SBen Gras                 ss = s-c;
174*2fe8fb19SBen Gras                 cc = s+c;
175*2fe8fb19SBen Gras 	/*
176*2fe8fb19SBen Gras 	 * j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x)
177*2fe8fb19SBen Gras 	 * y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x)
178*2fe8fb19SBen Gras 	 */
179*2fe8fb19SBen Gras                 if(ix<0x7fe00000) {  /* make sure x+x not overflow */
180*2fe8fb19SBen Gras                     z = -cos(x+x);
181*2fe8fb19SBen Gras                     if ((s*c)<zero) cc = z/ss;
182*2fe8fb19SBen Gras                     else            ss = z/cc;
183*2fe8fb19SBen Gras                 }
184*2fe8fb19SBen Gras                 if(ix>0x48000000) z = (invsqrtpi*ss)/sqrt(x);
185*2fe8fb19SBen Gras                 else {
186*2fe8fb19SBen Gras                     u = pzero(x); v = qzero(x);
187*2fe8fb19SBen Gras                     z = invsqrtpi*(u*ss+v*cc)/sqrt(x);
188*2fe8fb19SBen Gras                 }
189*2fe8fb19SBen Gras                 return z;
190*2fe8fb19SBen Gras 	}
191*2fe8fb19SBen Gras 	if(ix<=0x3e400000) {	/* x < 2**-27 */
192*2fe8fb19SBen Gras 	    return(u00 + tpi*__ieee754_log(x));
193*2fe8fb19SBen Gras 	}
194*2fe8fb19SBen Gras 	z = x*x;
195*2fe8fb19SBen Gras 	u = u00+z*(u01+z*(u02+z*(u03+z*(u04+z*(u05+z*u06)))));
196*2fe8fb19SBen Gras 	v = one+z*(v01+z*(v02+z*(v03+z*v04)));
197*2fe8fb19SBen Gras 	return(u/v + tpi*(__ieee754_j0(x)*__ieee754_log(x)));
198*2fe8fb19SBen Gras }
199*2fe8fb19SBen Gras 
200*2fe8fb19SBen Gras /* The asymptotic expansions of pzero is
201*2fe8fb19SBen Gras  *	1 - 9/128 s^2 + 11025/98304 s^4 - ...,	where s = 1/x.
202*2fe8fb19SBen Gras  * For x >= 2, We approximate pzero by
203*2fe8fb19SBen Gras  * 	pzero(x) = 1 + (R/S)
204*2fe8fb19SBen Gras  * where  R = pR0 + pR1*s^2 + pR2*s^4 + ... + pR5*s^10
205*2fe8fb19SBen Gras  * 	  S = 1 + pS0*s^2 + ... + pS4*s^10
206*2fe8fb19SBen Gras  * and
207*2fe8fb19SBen Gras  *	| pzero(x)-1-R/S | <= 2  ** ( -60.26)
208*2fe8fb19SBen Gras  */
209*2fe8fb19SBen Gras static const double pR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
210*2fe8fb19SBen Gras   0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */
211*2fe8fb19SBen Gras  -7.03124999999900357484e-02, /* 0xBFB1FFFF, 0xFFFFFD32 */
212*2fe8fb19SBen Gras  -8.08167041275349795626e+00, /* 0xC02029D0, 0xB44FA779 */
213*2fe8fb19SBen Gras  -2.57063105679704847262e+02, /* 0xC0701102, 0x7B19E863 */
214*2fe8fb19SBen Gras  -2.48521641009428822144e+03, /* 0xC0A36A6E, 0xCD4DCAFC */
215*2fe8fb19SBen Gras  -5.25304380490729545272e+03, /* 0xC0B4850B, 0x36CC643D */
216*2fe8fb19SBen Gras };
217*2fe8fb19SBen Gras static const double pS8[5] = {
218*2fe8fb19SBen Gras   1.16534364619668181717e+02, /* 0x405D2233, 0x07A96751 */
219*2fe8fb19SBen Gras   3.83374475364121826715e+03, /* 0x40ADF37D, 0x50596938 */
220*2fe8fb19SBen Gras   4.05978572648472545552e+04, /* 0x40E3D2BB, 0x6EB6B05F */
221*2fe8fb19SBen Gras   1.16752972564375915681e+05, /* 0x40FC810F, 0x8F9FA9BD */
222*2fe8fb19SBen Gras   4.76277284146730962675e+04, /* 0x40E74177, 0x4F2C49DC */
223*2fe8fb19SBen Gras };
224*2fe8fb19SBen Gras 
225*2fe8fb19SBen Gras static const double pR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
226*2fe8fb19SBen Gras  -1.14125464691894502584e-11, /* 0xBDA918B1, 0x47E495CC */
227*2fe8fb19SBen Gras  -7.03124940873599280078e-02, /* 0xBFB1FFFF, 0xE69AFBC6 */
228*2fe8fb19SBen Gras  -4.15961064470587782438e+00, /* 0xC010A370, 0xF90C6BBF */
229*2fe8fb19SBen Gras  -6.76747652265167261021e+01, /* 0xC050EB2F, 0x5A7D1783 */
230*2fe8fb19SBen Gras  -3.31231299649172967747e+02, /* 0xC074B3B3, 0x6742CC63 */
231*2fe8fb19SBen Gras  -3.46433388365604912451e+02, /* 0xC075A6EF, 0x28A38BD7 */
232*2fe8fb19SBen Gras };
233*2fe8fb19SBen Gras static const double pS5[5] = {
234*2fe8fb19SBen Gras   6.07539382692300335975e+01, /* 0x404E6081, 0x0C98C5DE */
235*2fe8fb19SBen Gras   1.05125230595704579173e+03, /* 0x40906D02, 0x5C7E2864 */
236*2fe8fb19SBen Gras   5.97897094333855784498e+03, /* 0x40B75AF8, 0x8FBE1D60 */
237*2fe8fb19SBen Gras   9.62544514357774460223e+03, /* 0x40C2CCB8, 0xFA76FA38 */
238*2fe8fb19SBen Gras   2.40605815922939109441e+03, /* 0x40A2CC1D, 0xC70BE864 */
239*2fe8fb19SBen Gras };
240*2fe8fb19SBen Gras 
241*2fe8fb19SBen Gras static const double pR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
242*2fe8fb19SBen Gras  -2.54704601771951915620e-09, /* 0xBE25E103, 0x6FE1AA86 */
243*2fe8fb19SBen Gras  -7.03119616381481654654e-02, /* 0xBFB1FFF6, 0xF7C0E24B */
244*2fe8fb19SBen Gras  -2.40903221549529611423e+00, /* 0xC00345B2, 0xAEA48074 */
245*2fe8fb19SBen Gras  -2.19659774734883086467e+01, /* 0xC035F74A, 0x4CB94E14 */
246*2fe8fb19SBen Gras  -5.80791704701737572236e+01, /* 0xC04D0A22, 0x420A1A45 */
247*2fe8fb19SBen Gras  -3.14479470594888503854e+01, /* 0xC03F72AC, 0xA892D80F */
248*2fe8fb19SBen Gras };
249*2fe8fb19SBen Gras static const double pS3[5] = {
250*2fe8fb19SBen Gras   3.58560338055209726349e+01, /* 0x4041ED92, 0x84077DD3 */
251*2fe8fb19SBen Gras   3.61513983050303863820e+02, /* 0x40769839, 0x464A7C0E */
252*2fe8fb19SBen Gras   1.19360783792111533330e+03, /* 0x4092A66E, 0x6D1061D6 */
253*2fe8fb19SBen Gras   1.12799679856907414432e+03, /* 0x40919FFC, 0xB8C39B7E */
254*2fe8fb19SBen Gras   1.73580930813335754692e+02, /* 0x4065B296, 0xFC379081 */
255*2fe8fb19SBen Gras };
256*2fe8fb19SBen Gras 
257*2fe8fb19SBen Gras static const double pR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
258*2fe8fb19SBen Gras  -8.87534333032526411254e-08, /* 0xBE77D316, 0xE927026D */
259*2fe8fb19SBen Gras  -7.03030995483624743247e-02, /* 0xBFB1FF62, 0x495E1E42 */
260*2fe8fb19SBen Gras  -1.45073846780952986357e+00, /* 0xBFF73639, 0x8A24A843 */
261*2fe8fb19SBen Gras  -7.63569613823527770791e+00, /* 0xC01E8AF3, 0xEDAFA7F3 */
262*2fe8fb19SBen Gras  -1.11931668860356747786e+01, /* 0xC02662E6, 0xC5246303 */
263*2fe8fb19SBen Gras  -3.23364579351335335033e+00, /* 0xC009DE81, 0xAF8FE70F */
264*2fe8fb19SBen Gras };
265*2fe8fb19SBen Gras static const double pS2[5] = {
266*2fe8fb19SBen Gras   2.22202997532088808441e+01, /* 0x40363865, 0x908B5959 */
267*2fe8fb19SBen Gras   1.36206794218215208048e+02, /* 0x4061069E, 0x0EE8878F */
268*2fe8fb19SBen Gras   2.70470278658083486789e+02, /* 0x4070E786, 0x42EA079B */
269*2fe8fb19SBen Gras   1.53875394208320329881e+02, /* 0x40633C03, 0x3AB6FAFF */
270*2fe8fb19SBen Gras   1.46576176948256193810e+01, /* 0x402D50B3, 0x44391809 */
271*2fe8fb19SBen Gras };
272*2fe8fb19SBen Gras 
273*2fe8fb19SBen Gras static double
pzero(double x)274*2fe8fb19SBen Gras pzero(double x)
275*2fe8fb19SBen Gras {
276*2fe8fb19SBen Gras 	const double *p,*q;
277*2fe8fb19SBen Gras 	double z,r,s;
278*2fe8fb19SBen Gras 	int32_t ix;
279*2fe8fb19SBen Gras 
280*2fe8fb19SBen Gras 	p = q = 0;
281*2fe8fb19SBen Gras 	GET_HIGH_WORD(ix,x);
282*2fe8fb19SBen Gras 	ix &= 0x7fffffff;
283*2fe8fb19SBen Gras 	if(ix>=0x40200000)     {p = pR8; q= pS8;}
284*2fe8fb19SBen Gras 	else if(ix>=0x40122E8B){p = pR5; q= pS5;}
285*2fe8fb19SBen Gras 	else if(ix>=0x4006DB6D){p = pR3; q= pS3;}
286*2fe8fb19SBen Gras 	else if(ix>=0x40000000){p = pR2; q= pS2;}
287*2fe8fb19SBen Gras 	z = one/(x*x);
288*2fe8fb19SBen Gras 	r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
289*2fe8fb19SBen Gras 	s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*q[4]))));
290*2fe8fb19SBen Gras 	return one+ r/s;
291*2fe8fb19SBen Gras }
292*2fe8fb19SBen Gras 
293*2fe8fb19SBen Gras 
294*2fe8fb19SBen Gras /* For x >= 8, the asymptotic expansions of qzero is
295*2fe8fb19SBen Gras  *	-1/8 s + 75/1024 s^3 - ..., where s = 1/x.
296*2fe8fb19SBen Gras  * We approximate pzero by
297*2fe8fb19SBen Gras  * 	qzero(x) = s*(-1.25 + (R/S))
298*2fe8fb19SBen Gras  * where  R = qR0 + qR1*s^2 + qR2*s^4 + ... + qR5*s^10
299*2fe8fb19SBen Gras  * 	  S = 1 + qS0*s^2 + ... + qS5*s^12
300*2fe8fb19SBen Gras  * and
301*2fe8fb19SBen Gras  *	| qzero(x)/s +1.25-R/S | <= 2  ** ( -61.22)
302*2fe8fb19SBen Gras  */
303*2fe8fb19SBen Gras static const double qR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
304*2fe8fb19SBen Gras   0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */
305*2fe8fb19SBen Gras   7.32421874999935051953e-02, /* 0x3FB2BFFF, 0xFFFFFE2C */
306*2fe8fb19SBen Gras   1.17682064682252693899e+01, /* 0x40278952, 0x5BB334D6 */
307*2fe8fb19SBen Gras   5.57673380256401856059e+02, /* 0x40816D63, 0x15301825 */
308*2fe8fb19SBen Gras   8.85919720756468632317e+03, /* 0x40C14D99, 0x3E18F46D */
309*2fe8fb19SBen Gras   3.70146267776887834771e+04, /* 0x40E212D4, 0x0E901566 */
310*2fe8fb19SBen Gras };
311*2fe8fb19SBen Gras static const double qS8[6] = {
312*2fe8fb19SBen Gras   1.63776026895689824414e+02, /* 0x406478D5, 0x365B39BC */
313*2fe8fb19SBen Gras   8.09834494656449805916e+03, /* 0x40BFA258, 0x4E6B0563 */
314*2fe8fb19SBen Gras   1.42538291419120476348e+05, /* 0x41016652, 0x54D38C3F */
315*2fe8fb19SBen Gras   8.03309257119514397345e+05, /* 0x412883DA, 0x83A52B43 */
316*2fe8fb19SBen Gras   8.40501579819060512818e+05, /* 0x4129A66B, 0x28DE0B3D */
317*2fe8fb19SBen Gras  -3.43899293537866615225e+05, /* 0xC114FD6D, 0x2C9530C5 */
318*2fe8fb19SBen Gras };
319*2fe8fb19SBen Gras 
320*2fe8fb19SBen Gras static const double qR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
321*2fe8fb19SBen Gras   1.84085963594515531381e-11, /* 0x3DB43D8F, 0x29CC8CD9 */
322*2fe8fb19SBen Gras   7.32421766612684765896e-02, /* 0x3FB2BFFF, 0xD172B04C */
323*2fe8fb19SBen Gras   5.83563508962056953777e+00, /* 0x401757B0, 0xB9953DD3 */
324*2fe8fb19SBen Gras   1.35111577286449829671e+02, /* 0x4060E392, 0x0A8788E9 */
325*2fe8fb19SBen Gras   1.02724376596164097464e+03, /* 0x40900CF9, 0x9DC8C481 */
326*2fe8fb19SBen Gras   1.98997785864605384631e+03, /* 0x409F17E9, 0x53C6E3A6 */
327*2fe8fb19SBen Gras };
328*2fe8fb19SBen Gras static const double qS5[6] = {
329*2fe8fb19SBen Gras   8.27766102236537761883e+01, /* 0x4054B1B3, 0xFB5E1543 */
330*2fe8fb19SBen Gras   2.07781416421392987104e+03, /* 0x40A03BA0, 0xDA21C0CE */
331*2fe8fb19SBen Gras   1.88472887785718085070e+04, /* 0x40D267D2, 0x7B591E6D */
332*2fe8fb19SBen Gras   5.67511122894947329769e+04, /* 0x40EBB5E3, 0x97E02372 */
333*2fe8fb19SBen Gras   3.59767538425114471465e+04, /* 0x40E19118, 0x1F7A54A0 */
334*2fe8fb19SBen Gras  -5.35434275601944773371e+03, /* 0xC0B4EA57, 0xBEDBC609 */
335*2fe8fb19SBen Gras };
336*2fe8fb19SBen Gras 
337*2fe8fb19SBen Gras static const double qR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
338*2fe8fb19SBen Gras   4.37741014089738620906e-09, /* 0x3E32CD03, 0x6ADECB82 */
339*2fe8fb19SBen Gras   7.32411180042911447163e-02, /* 0x3FB2BFEE, 0x0E8D0842 */
340*2fe8fb19SBen Gras   3.34423137516170720929e+00, /* 0x400AC0FC, 0x61149CF5 */
341*2fe8fb19SBen Gras   4.26218440745412650017e+01, /* 0x40454F98, 0x962DAEDD */
342*2fe8fb19SBen Gras   1.70808091340565596283e+02, /* 0x406559DB, 0xE25EFD1F */
343*2fe8fb19SBen Gras   1.66733948696651168575e+02, /* 0x4064D77C, 0x81FA21E0 */
344*2fe8fb19SBen Gras };
345*2fe8fb19SBen Gras static const double qS3[6] = {
346*2fe8fb19SBen Gras   4.87588729724587182091e+01, /* 0x40486122, 0xBFE343A6 */
347*2fe8fb19SBen Gras   7.09689221056606015736e+02, /* 0x40862D83, 0x86544EB3 */
348*2fe8fb19SBen Gras   3.70414822620111362994e+03, /* 0x40ACF04B, 0xE44DFC63 */
349*2fe8fb19SBen Gras   6.46042516752568917582e+03, /* 0x40B93C6C, 0xD7C76A28 */
350*2fe8fb19SBen Gras   2.51633368920368957333e+03, /* 0x40A3A8AA, 0xD94FB1C0 */
351*2fe8fb19SBen Gras  -1.49247451836156386662e+02, /* 0xC062A7EB, 0x201CF40F */
352*2fe8fb19SBen Gras };
353*2fe8fb19SBen Gras 
354*2fe8fb19SBen Gras static const double qR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
355*2fe8fb19SBen Gras   1.50444444886983272379e-07, /* 0x3E84313B, 0x54F76BDB */
356*2fe8fb19SBen Gras   7.32234265963079278272e-02, /* 0x3FB2BEC5, 0x3E883E34 */
357*2fe8fb19SBen Gras   1.99819174093815998816e+00, /* 0x3FFFF897, 0xE727779C */
358*2fe8fb19SBen Gras   1.44956029347885735348e+01, /* 0x402CFDBF, 0xAAF96FE5 */
359*2fe8fb19SBen Gras   3.16662317504781540833e+01, /* 0x403FAA8E, 0x29FBDC4A */
360*2fe8fb19SBen Gras   1.62527075710929267416e+01, /* 0x403040B1, 0x71814BB4 */
361*2fe8fb19SBen Gras };
362*2fe8fb19SBen Gras static const double qS2[6] = {
363*2fe8fb19SBen Gras   3.03655848355219184498e+01, /* 0x403E5D96, 0xF7C07AED */
364*2fe8fb19SBen Gras   2.69348118608049844624e+02, /* 0x4070D591, 0xE4D14B40 */
365*2fe8fb19SBen Gras   8.44783757595320139444e+02, /* 0x408A6645, 0x22B3BF22 */
366*2fe8fb19SBen Gras   8.82935845112488550512e+02, /* 0x408B977C, 0x9C5CC214 */
367*2fe8fb19SBen Gras   2.12666388511798828631e+02, /* 0x406A9553, 0x0E001365 */
368*2fe8fb19SBen Gras  -5.31095493882666946917e+00, /* 0xC0153E6A, 0xF8B32931 */
369*2fe8fb19SBen Gras };
370*2fe8fb19SBen Gras 
371*2fe8fb19SBen Gras static double
qzero(double x)372*2fe8fb19SBen Gras qzero(double x)
373*2fe8fb19SBen Gras {
374*2fe8fb19SBen Gras 	const double *p,*q;
375*2fe8fb19SBen Gras 	double s,r,z;
376*2fe8fb19SBen Gras 	int32_t ix;
377*2fe8fb19SBen Gras 
378*2fe8fb19SBen Gras 	p = q = 0;
379*2fe8fb19SBen Gras 	GET_HIGH_WORD(ix,x);
380*2fe8fb19SBen Gras 	ix &= 0x7fffffff;
381*2fe8fb19SBen Gras 	if(ix>=0x40200000)     {p = qR8; q= qS8;}
382*2fe8fb19SBen Gras 	else if(ix>=0x40122E8B){p = qR5; q= qS5;}
383*2fe8fb19SBen Gras 	else if(ix>=0x4006DB6D){p = qR3; q= qS3;}
384*2fe8fb19SBen Gras 	else if(ix>=0x40000000){p = qR2; q= qS2;}
385*2fe8fb19SBen Gras 	z = one/(x*x);
386*2fe8fb19SBen Gras 	r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
387*2fe8fb19SBen Gras 	s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*(q[4]+z*q[5])))));
388*2fe8fb19SBen Gras 	return (-.125 + r/s)/x;
389*2fe8fb19SBen Gras }
390