e_j0.c revision 8870 
1245539Sandrew/* @(#)e_j0.c 5.1 93/09/24 */
2253396Sandrew/*
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
14static char rcsid[] = "$Id: e_j0.c,v 1.1.1.1 1994/08/19 09:39:43 jkh Exp $";
15#endif
16
17/* __ieee754_j0(x), __ieee754_y0(x)
18 * Bessel function of the first and second kinds of order zero.
19 * Method -- j0(x):
20 *	1. For tiny x, we use j0(x) = 1 - x^2/4 + x^4/64 - ...
21 *	2. Reduce x to |x| since j0(x)=j0(-x),  and
22 *	   for x in (0,2)
23 *		j0(x) = 1-z/4+ z^2*R0/S0,  where z = x*x;
24 *	   (precision:  |j0-1+z/4-z^2R0/S0 |<2**-63.67 )
25 *	   for x in (2,inf)
26 * 		j0(x) = sqrt(2/(pi*x))*(p0(x)*cos(x0)-q0(x)*sin(x0))
27 * 	   where x0 = x-pi/4. It is better to compute sin(x0),cos(x0)
28 *	   as follow:
29 *		cos(x0) = cos(x)cos(pi/4)+sin(x)sin(pi/4)
30 *			= 1/sqrt(2) * (cos(x) + sin(x))
31 *		sin(x0) = sin(x)cos(pi/4)-cos(x)sin(pi/4)
32 *			= 1/sqrt(2) * (sin(x) - cos(x))
33 * 	   (To avoid cancellation, use
34 *		sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
35 * 	    to compute the worse one.)
36 *
37 *	3 Special cases
38 *		j0(nan)= nan
39 *		j0(0) = 1
40 *		j0(inf) = 0
41 *
42 * Method -- y0(x):
43 *	1. For x<2.
44 *	   Since
45 *		y0(x) = 2/pi*(j0(x)*(ln(x/2)+Euler) + x^2/4 - ...)
46 *	   therefore y0(x)-2/pi*j0(x)*ln(x) is an even function.
47 *	   We use the following function to approximate y0,
48 *		y0(x) = U(z)/V(z) + (2/pi)*(j0(x)*ln(x)), z= x^2
49 *	   where
50 *		U(z) = u00 + u01*z + ... + u06*z^6
51 *		V(z) = 1  + v01*z + ... + v04*z^4
52 *	   with absolute approximation error bounded by 2**-72.
53 *	   Note: For tiny x, U/V = u0 and j0(x)~1, hence
54 *		y0(tiny) = u0 + (2/pi)*ln(tiny), (choose tiny<2**-27)
55 *	2. For x>=2.
56 * 		y0(x) = sqrt(2/(pi*x))*(p0(x)*cos(x0)+q0(x)*sin(x0))
57 * 	   where x0 = x-pi/4. It is better to compute sin(x0),cos(x0)
58 *	   by the method mentioned above.
59 *	3. Special cases: y0(0)=-inf, y0(x<0)=NaN, y0(inf)=0.
60 */
61
62#include "math.h"
63#include "math_private.h"
64
65#ifdef __STDC__
66static double pzero(double), qzero(double);
67#else
68static double pzero(), qzero();
69#endif
70
71#ifdef __STDC__
72static const double
73#else
74static double
75#endif
76huge 	= 1e300,
77one	= 1.0,
78invsqrtpi=  5.64189583547756279280e-01, /* 0x3FE20DD7, 0x50429B6D */
79tpi      =  6.36619772367581382433e-01, /* 0x3FE45F30, 0x6DC9C883 */
80 		/* R0/S0 on [0, 2.00] */
81R02  =  1.56249999999999947958e-02, /* 0x3F8FFFFF, 0xFFFFFFFD */
82R03  = -1.89979294238854721751e-04, /* 0xBF28E6A5, 0xB61AC6E9 */
83R04  =  1.82954049532700665670e-06, /* 0x3EBEB1D1, 0x0C503919 */
84R05  = -4.61832688532103189199e-09, /* 0xBE33D5E7, 0x73D63FCE */
85S01  =  1.56191029464890010492e-02, /* 0x3F8FFCE8, 0x82C8C2A4 */
86S02  =  1.16926784663337450260e-04, /* 0x3F1EA6D2, 0xDD57DBF4 */
87S03  =  5.13546550207318111446e-07, /* 0x3EA13B54, 0xCE84D5A9 */
88S04  =  1.16614003333790000205e-09; /* 0x3E1408BC, 0xF4745D8F */
89
90#ifdef __STDC__
91static const double zero = 0.0;
92#else
93static double zero = 0.0;
94#endif
95
96#ifdef __STDC__
97	double __ieee754_j0(double x)
98#else
99	double __ieee754_j0(x)
100	double x;
101#endif
102{
103	double z, s,c,ss,cc,r,u,v;
104	int32_t hx,ix;
105
106	GET_HIGH_WORD(hx,x);
107	ix = hx&0x7fffffff;
108	if(ix>=0x7ff00000) return one/(x*x);
109	x = fabs(x);
110	if(ix >= 0x40000000) {	/* |x| >= 2.0 */
111		s = sin(x);
112		c = cos(x);
113		ss = s-c;
114		cc = s+c;
115		if(ix<0x7fe00000) {  /* make sure x+x not overflow */
116		    z = -cos(x+x);
117		    if ((s*c)<zero) cc = z/ss;
118		    else 	    ss = z/cc;
119		}
120	/*
121	 * j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x)
122	 * y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x)
123	 */
124		if(ix>0x48000000) z = (invsqrtpi*cc)/sqrt(x);
125		else {
126		    u = pzero(x); v = qzero(x);
127		    z = invsqrtpi*(u*cc-v*ss)/sqrt(x);
128		}
129		return z;
130	}
131	if(ix<0x3f200000) {	/* |x| < 2**-13 */
132	    if(huge+x>one) {	/* raise inexact if x != 0 */
133	        if(ix<0x3e400000) return one;	/* |x|<2**-27 */
134	        else 	      return one - 0.25*x*x;
135	    }
136	}
137	z = x*x;
138	r =  z*(R02+z*(R03+z*(R04+z*R05)));
139	s =  one+z*(S01+z*(S02+z*(S03+z*S04)));
140	if(ix < 0x3FF00000) {	/* |x| < 1.00 */
141	    return one + z*(-0.25+(r/s));
142	} else {
143	    u = 0.5*x;
144	    return((one+u)*(one-u)+z*(r/s));
145	}
146}
147
148#ifdef __STDC__
149static const double
150#else
151static double
152#endif
153u00  = -7.38042951086872317523e-02, /* 0xBFB2E4D6, 0x99CBD01F */
154u01  =  1.76666452509181115538e-01, /* 0x3FC69D01, 0x9DE9E3FC */
155u02  = -1.38185671945596898896e-02, /* 0xBF8C4CE8, 0xB16CFA97 */
156u03  =  3.47453432093683650238e-04, /* 0x3F36C54D, 0x20B29B6B */
157u04  = -3.81407053724364161125e-06, /* 0xBECFFEA7, 0x73D25CAD */
158u05  =  1.95590137035022920206e-08, /* 0x3E550057, 0x3B4EABD4 */
159u06  = -3.98205194132103398453e-11, /* 0xBDC5E43D, 0x693FB3C8 */
160v01  =  1.27304834834123699328e-02, /* 0x3F8A1270, 0x91C9C71A */
161v02  =  7.60068627350353253702e-05, /* 0x3F13ECBB, 0xF578C6C1 */
162v03  =  2.59150851840457805467e-07, /* 0x3E91642D, 0x7FF202FD */
163v04  =  4.41110311332675467403e-10; /* 0x3DFE5018, 0x3BD6D9EF */
164
165#ifdef __STDC__
166	double __ieee754_y0(double x)
167#else
168	double __ieee754_y0(x)
169	double x;
170#endif
171{
172	double z, s,c,ss,cc,u,v;
173	int32_t hx,ix,lx;
174
175	EXTRACT_WORDS(hx,lx,x);
176        ix = 0x7fffffff&hx;
177    /* Y0(NaN) is NaN, y0(-inf) is Nan, y0(inf) is 0  */
178	if(ix>=0x7ff00000) return  one/(x+x*x);
179        if((ix|lx)==0) return -one/zero;
180        if(hx<0) return zero/zero;
181        if(ix >= 0x40000000) {  /* |x| >= 2.0 */
182        /* y0(x) = sqrt(2/(pi*x))*(p0(x)*sin(x0)+q0(x)*cos(x0))
183         * where x0 = x-pi/4
184         *      Better formula:
185         *              cos(x0) = cos(x)cos(pi/4)+sin(x)sin(pi/4)
186         *                      =  1/sqrt(2) * (sin(x) + cos(x))
187         *              sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4)
188         *                      =  1/sqrt(2) * (sin(x) - cos(x))
189         * To avoid cancellation, use
190         *              sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
191         * to compute the worse one.
192         */
193                s = sin(x);
194                c = cos(x);
195                ss = s-c;
196                cc = s+c;
197	/*
198	 * j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x)
199	 * y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x)
200	 */
201                if(ix<0x7fe00000) {  /* make sure x+x not overflow */
202                    z = -cos(x+x);
203                    if ((s*c)<zero) cc = z/ss;
204                    else            ss = z/cc;
205                }
206                if(ix>0x48000000) z = (invsqrtpi*ss)/sqrt(x);
207                else {
208                    u = pzero(x); v = qzero(x);
209                    z = invsqrtpi*(u*ss+v*cc)/sqrt(x);
210                }
211                return z;
212	}
213	if(ix<=0x3e400000) {	/* x < 2**-27 */
214	    return(u00 + tpi*__ieee754_log(x));
215	}
216	z = x*x;
217	u = u00+z*(u01+z*(u02+z*(u03+z*(u04+z*(u05+z*u06)))));
218	v = one+z*(v01+z*(v02+z*(v03+z*v04)));
219	return(u/v + tpi*(__ieee754_j0(x)*__ieee754_log(x)));
220}
221
222/* The asymptotic expansions of pzero is
223 *	1 - 9/128 s^2 + 11025/98304 s^4 - ...,	where s = 1/x.
224 * For x >= 2, We approximate pzero by
225 * 	pzero(x) = 1 + (R/S)
226 * where  R = pR0 + pR1*s^2 + pR2*s^4 + ... + pR5*s^10
227 * 	  S = 1 + pS0*s^2 + ... + pS4*s^10
228 * and
229 *	| pzero(x)-1-R/S | <= 2  ** ( -60.26)
230 */
231#ifdef __STDC__
232static const double pR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
233#else
234static double pR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
235#endif
236  0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */
237 -7.03124999999900357484e-02, /* 0xBFB1FFFF, 0xFFFFFD32 */
238 -8.08167041275349795626e+00, /* 0xC02029D0, 0xB44FA779 */
239 -2.57063105679704847262e+02, /* 0xC0701102, 0x7B19E863 */
240 -2.48521641009428822144e+03, /* 0xC0A36A6E, 0xCD4DCAFC */
241 -5.25304380490729545272e+03, /* 0xC0B4850B, 0x36CC643D */
242};
243#ifdef __STDC__
244static const double pS8[5] = {
245#else
246static double pS8[5] = {
247#endif
248  1.16534364619668181717e+02, /* 0x405D2233, 0x07A96751 */
249  3.83374475364121826715e+03, /* 0x40ADF37D, 0x50596938 */
250  4.05978572648472545552e+04, /* 0x40E3D2BB, 0x6EB6B05F */
251  1.16752972564375915681e+05, /* 0x40FC810F, 0x8F9FA9BD */
252  4.76277284146730962675e+04, /* 0x40E74177, 0x4F2C49DC */
253};
254
255#ifdef __STDC__
256static const double pR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
257#else
258static double pR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
259#endif
260 -1.14125464691894502584e-11, /* 0xBDA918B1, 0x47E495CC */
261 -7.03124940873599280078e-02, /* 0xBFB1FFFF, 0xE69AFBC6 */
262 -4.15961064470587782438e+00, /* 0xC010A370, 0xF90C6BBF */
263 -6.76747652265167261021e+01, /* 0xC050EB2F, 0x5A7D1783 */
264 -3.31231299649172967747e+02, /* 0xC074B3B3, 0x6742CC63 */
265 -3.46433388365604912451e+02, /* 0xC075A6EF, 0x28A38BD7 */
266};
267#ifdef __STDC__
268static const double pS5[5] = {
269#else
270static double pS5[5] = {
271#endif
272  6.07539382692300335975e+01, /* 0x404E6081, 0x0C98C5DE */
273  1.05125230595704579173e+03, /* 0x40906D02, 0x5C7E2864 */
274  5.97897094333855784498e+03, /* 0x40B75AF8, 0x8FBE1D60 */
275  9.62544514357774460223e+03, /* 0x40C2CCB8, 0xFA76FA38 */
276  2.40605815922939109441e+03, /* 0x40A2CC1D, 0xC70BE864 */
277};
278
279#ifdef __STDC__
280static const double pR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
281#else
282static double pR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
283#endif
284 -2.54704601771951915620e-09, /* 0xBE25E103, 0x6FE1AA86 */
285 -7.03119616381481654654e-02, /* 0xBFB1FFF6, 0xF7C0E24B */
286 -2.40903221549529611423e+00, /* 0xC00345B2, 0xAEA48074 */
287 -2.19659774734883086467e+01, /* 0xC035F74A, 0x4CB94E14 */
288 -5.80791704701737572236e+01, /* 0xC04D0A22, 0x420A1A45 */
289 -3.14479470594888503854e+01, /* 0xC03F72AC, 0xA892D80F */
290};
291#ifdef __STDC__
292static const double pS3[5] = {
293#else
294static double pS3[5] = {
295#endif
296  3.58560338055209726349e+01, /* 0x4041ED92, 0x84077DD3 */
297  3.61513983050303863820e+02, /* 0x40769839, 0x464A7C0E */
298  1.19360783792111533330e+03, /* 0x4092A66E, 0x6D1061D6 */
299  1.12799679856907414432e+03, /* 0x40919FFC, 0xB8C39B7E */
300  1.73580930813335754692e+02, /* 0x4065B296, 0xFC379081 */
301};
302
303#ifdef __STDC__
304static const double pR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
305#else
306static double pR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
307#endif
308 -8.87534333032526411254e-08, /* 0xBE77D316, 0xE927026D */
309 -7.03030995483624743247e-02, /* 0xBFB1FF62, 0x495E1E42 */
310 -1.45073846780952986357e+00, /* 0xBFF73639, 0x8A24A843 */
311 -7.63569613823527770791e+00, /* 0xC01E8AF3, 0xEDAFA7F3 */
312 -1.11931668860356747786e+01, /* 0xC02662E6, 0xC5246303 */
313 -3.23364579351335335033e+00, /* 0xC009DE81, 0xAF8FE70F */
314};
315#ifdef __STDC__
316static const double pS2[5] = {
317#else
318static double pS2[5] = {
319#endif
320  2.22202997532088808441e+01, /* 0x40363865, 0x908B5959 */
321  1.36206794218215208048e+02, /* 0x4061069E, 0x0EE8878F */
322  2.70470278658083486789e+02, /* 0x4070E786, 0x42EA079B */
323  1.53875394208320329881e+02, /* 0x40633C03, 0x3AB6FAFF */
324  1.46576176948256193810e+01, /* 0x402D50B3, 0x44391809 */
325};
326
327#ifdef __STDC__
328	static double pzero(double x)
329#else
330	static double pzero(x)
331	double x;
332#endif
333{
334#ifdef __STDC__
335	const double *p,*q;
336#else
337	double *p,*q;
338#endif
339	double z,r,s;
340	int32_t ix;
341	GET_HIGH_WORD(ix,x);
342	ix &= 0x7fffffff;
343	if(ix>=0x40200000)     {p = pR8; q= pS8;}
344	else if(ix>=0x40122E8B){p = pR5; q= pS5;}
345	else if(ix>=0x4006DB6D){p = pR3; q= pS3;}
346	else if(ix>=0x40000000){p = pR2; q= pS2;}
347	z = one/(x*x);
348	r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
349	s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*q[4]))));
350	return one+ r/s;
351}
352
353
354/* For x >= 8, the asymptotic expansions of qzero is
355 *	-1/8 s + 75/1024 s^3 - ..., where s = 1/x.
356 * We approximate pzero by
357 * 	qzero(x) = s*(-1.25 + (R/S))
358 * where  R = qR0 + qR1*s^2 + qR2*s^4 + ... + qR5*s^10
359 * 	  S = 1 + qS0*s^2 + ... + qS5*s^12
360 * and
361 *	| qzero(x)/s +1.25-R/S | <= 2  ** ( -61.22)
362 */
363#ifdef __STDC__
364static const double qR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
365#else
366static double qR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
367#endif
368  0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */
369  7.32421874999935051953e-02, /* 0x3FB2BFFF, 0xFFFFFE2C */
370  1.17682064682252693899e+01, /* 0x40278952, 0x5BB334D6 */
371  5.57673380256401856059e+02, /* 0x40816D63, 0x15301825 */
372  8.85919720756468632317e+03, /* 0x40C14D99, 0x3E18F46D */
373  3.70146267776887834771e+04, /* 0x40E212D4, 0x0E901566 */
374};
375#ifdef __STDC__
376static const double qS8[6] = {
377#else
378static double qS8[6] = {
379#endif
380  1.63776026895689824414e+02, /* 0x406478D5, 0x365B39BC */
381  8.09834494656449805916e+03, /* 0x40BFA258, 0x4E6B0563 */
382  1.42538291419120476348e+05, /* 0x41016652, 0x54D38C3F */
383  8.03309257119514397345e+05, /* 0x412883DA, 0x83A52B43 */
384  8.40501579819060512818e+05, /* 0x4129A66B, 0x28DE0B3D */
385 -3.43899293537866615225e+05, /* 0xC114FD6D, 0x2C9530C5 */
386};
387
388#ifdef __STDC__
389static const double qR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
390#else
391static double qR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
392#endif
393  1.84085963594515531381e-11, /* 0x3DB43D8F, 0x29CC8CD9 */
394  7.32421766612684765896e-02, /* 0x3FB2BFFF, 0xD172B04C */
395  5.83563508962056953777e+00, /* 0x401757B0, 0xB9953DD3 */
396  1.35111577286449829671e+02, /* 0x4060E392, 0x0A8788E9 */
397  1.02724376596164097464e+03, /* 0x40900CF9, 0x9DC8C481 */
398  1.98997785864605384631e+03, /* 0x409F17E9, 0x53C6E3A6 */
399};
400#ifdef __STDC__
401static const double qS5[6] = {
402#else
403static double qS5[6] = {
404#endif
405  8.27766102236537761883e+01, /* 0x4054B1B3, 0xFB5E1543 */
406  2.07781416421392987104e+03, /* 0x40A03BA0, 0xDA21C0CE */
407  1.88472887785718085070e+04, /* 0x40D267D2, 0x7B591E6D */
408  5.67511122894947329769e+04, /* 0x40EBB5E3, 0x97E02372 */
409  3.59767538425114471465e+04, /* 0x40E19118, 0x1F7A54A0 */
410 -5.35434275601944773371e+03, /* 0xC0B4EA57, 0xBEDBC609 */
411};
412
413#ifdef __STDC__
414static const double qR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
415#else
416static double qR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
417#endif
418  4.37741014089738620906e-09, /* 0x3E32CD03, 0x6ADECB82 */
419  7.32411180042911447163e-02, /* 0x3FB2BFEE, 0x0E8D0842 */
420  3.34423137516170720929e+00, /* 0x400AC0FC, 0x61149CF5 */
421  4.26218440745412650017e+01, /* 0x40454F98, 0x962DAEDD */
422  1.70808091340565596283e+02, /* 0x406559DB, 0xE25EFD1F */
423  1.66733948696651168575e+02, /* 0x4064D77C, 0x81FA21E0 */
424};
425#ifdef __STDC__
426static const double qS3[6] = {
427#else
428static double qS3[6] = {
429#endif
430  4.87588729724587182091e+01, /* 0x40486122, 0xBFE343A6 */
431  7.09689221056606015736e+02, /* 0x40862D83, 0x86544EB3 */
432  3.70414822620111362994e+03, /* 0x40ACF04B, 0xE44DFC63 */
433  6.46042516752568917582e+03, /* 0x40B93C6C, 0xD7C76A28 */
434  2.51633368920368957333e+03, /* 0x40A3A8AA, 0xD94FB1C0 */
435 -1.49247451836156386662e+02, /* 0xC062A7EB, 0x201CF40F */
436};
437
438#ifdef __STDC__
439static const double qR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
440#else
441static double qR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
442#endif
443  1.50444444886983272379e-07, /* 0x3E84313B, 0x54F76BDB */
444  7.32234265963079278272e-02, /* 0x3FB2BEC5, 0x3E883E34 */
445  1.99819174093815998816e+00, /* 0x3FFFF897, 0xE727779C */
446  1.44956029347885735348e+01, /* 0x402CFDBF, 0xAAF96FE5 */
447  3.16662317504781540833e+01, /* 0x403FAA8E, 0x29FBDC4A */
448  1.62527075710929267416e+01, /* 0x403040B1, 0x71814BB4 */
449};
450#ifdef __STDC__
451static const double qS2[6] = {
452#else
453static double qS2[6] = {
454#endif
455  3.03655848355219184498e+01, /* 0x403E5D96, 0xF7C07AED */
456  2.69348118608049844624e+02, /* 0x4070D591, 0xE4D14B40 */
457  8.44783757595320139444e+02, /* 0x408A6645, 0x22B3BF22 */
458  8.82935845112488550512e+02, /* 0x408B977C, 0x9C5CC214 */
459  2.12666388511798828631e+02, /* 0x406A9553, 0x0E001365 */
460 -5.31095493882666946917e+00, /* 0xC0153E6A, 0xF8B32931 */
461};
462
463#ifdef __STDC__
464	static double qzero(double x)
465#else
466	static double qzero(x)
467	double x;
468#endif
469{
470#ifdef __STDC__
471	const double *p,*q;
472#else
473	double *p,*q;
474#endif
475	double s,r,z;
476	int32_t ix;
477	GET_HIGH_WORD(ix,x);
478	ix &= 0x7fffffff;
479	if(ix>=0x40200000)     {p = qR8; q= qS8;}
480	else if(ix>=0x40122E8B){p = qR5; q= qS5;}
481	else if(ix>=0x4006DB6D){p = qR3; q= qS3;}
482	else if(ix>=0x40000000){p = qR2; q= qS2;}
483	z = one/(x*x);
484	r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
485	s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*(q[4]+z*q[5])))));
486	return (-.125 + r/s)/x;
487}
488