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1 
2 /* @(#)e_j1.c 1.3 95/01/18 */
3 /*
4  * ====================================================
5  * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
6  *
7  * Developed at SunSoft, a Sun Microsystems, Inc. business.
8  * Permission to use, copy, modify, and distribute this
9  * software is freely granted, provided that this notice
10  * is preserved.
11  * ====================================================
12  */
13 
14 /* __ieee754_j1(x), __ieee754_y1(x)
15  * Bessel function of the first and second kinds of order zero.
16  * Method -- ieee_j1(x):
17  *	1. For tiny x, we use ieee_j1(x) = x/2 - x^3/16 + x^5/384 - ...
18  *	2. Reduce x to |x| since ieee_j1(x)=-ieee_j1(-x),  and
19  *	   for x in (0,2)
20  *		j1(x) = x/2 + x*z*R0/S0,  where z = x*x;
21  *	   (precision:  |j1/x - 1/2 - R0/S0 |<2**-61.51 )
22  *	   for x in (2,inf)
23  * 		j1(x) = ieee_sqrt(2/(pi*x))*(p1(x)*ieee_cos(x1)-q1(x)*ieee_sin(x1))
24  * 		y1(x) = ieee_sqrt(2/(pi*x))*(p1(x)*ieee_sin(x1)+q1(x)*ieee_cos(x1))
25  * 	   where x1 = x-3*pi/4. It is better to compute ieee_sin(x1),cos(x1)
26  *	   as follow:
27  *		cos(x1) =  ieee_cos(x)cos(3pi/4)+ieee_sin(x)sin(3pi/4)
28  *			=  1/ieee_sqrt(2) * (ieee_sin(x) - ieee_cos(x))
29  *		sin(x1) =  ieee_sin(x)cos(3pi/4)-ieee_cos(x)sin(3pi/4)
30  *			= -1/ieee_sqrt(2) * (ieee_sin(x) + ieee_cos(x))
31  * 	   (To avoid cancellation, use
32  *		sin(x) +- ieee_cos(x) = -ieee_cos(2x)/(ieee_sin(x) -+ ieee_cos(x))
33  * 	    to compute the worse one.)
34  *
35  *	3 Special cases
36  *		j1(nan)= nan
37  *		j1(0) = 0
38  *		j1(inf) = 0
39  *
40  * Method -- ieee_y1(x):
41  *	1. screen out x<=0 cases: ieee_y1(0)=-inf, ieee_y1(x<0)=NaN
42  *	2. For x<2.
43  *	   Since
44  *		y1(x) = 2/pi*(ieee_j1(x)*(ln(x/2)+Euler)-1/x-x/2+5/64*x^3-...)
45  *	   therefore ieee_y1(x)-2/pi*ieee_j1(x)*ln(x)-1/x is an odd function.
46  *	   We use the following function to approximate y1,
47  *		y1(x) = x*U(z)/V(z) + (2/pi)*(ieee_j1(x)*ln(x)-1/x), z= x^2
48  *	   where for x in [0,2] (abs err less than 2**-65.89)
49  *		U(z) = U0[0] + U0[1]*z + ... + U0[4]*z^4
50  *		V(z) = 1  + v0[0]*z + ... + v0[4]*z^5
51  *	   Note: For tiny x, 1/x dominate y1 and hence
52  *		y1(tiny) = -2/pi/tiny, (choose tiny<2**-54)
53  *	3. For x>=2.
54  * 		y1(x) = ieee_sqrt(2/(pi*x))*(p1(x)*ieee_sin(x1)+q1(x)*ieee_cos(x1))
55  * 	   where x1 = x-3*pi/4. It is better to compute ieee_sin(x1),cos(x1)
56  *	   by method mentioned above.
57  */
58 
59 #include "fdlibm.h"
60 
61 #ifdef __STDC__
62 static double pone(double), qone(double);
63 #else
64 static double pone(), qone();
65 #endif
66 
67 #ifdef __STDC__
68 static const double
69 #else
70 static double
71 #endif
72 huge    = 1e300,
73 one	= 1.0,
74 invsqrtpi=  5.64189583547756279280e-01, /* 0x3FE20DD7, 0x50429B6D */
75 tpi      =  6.36619772367581382433e-01, /* 0x3FE45F30, 0x6DC9C883 */
76 	/* R0/S0 on [0,2] */
77 r00  = -6.25000000000000000000e-02, /* 0xBFB00000, 0x00000000 */
78 r01  =  1.40705666955189706048e-03, /* 0x3F570D9F, 0x98472C61 */
79 r02  = -1.59955631084035597520e-05, /* 0xBEF0C5C6, 0xBA169668 */
80 r03  =  4.96727999609584448412e-08, /* 0x3E6AAAFA, 0x46CA0BD9 */
81 s01  =  1.91537599538363460805e-02, /* 0x3F939D0B, 0x12637E53 */
82 s02  =  1.85946785588630915560e-04, /* 0x3F285F56, 0xB9CDF664 */
83 s03  =  1.17718464042623683263e-06, /* 0x3EB3BFF8, 0x333F8498 */
84 s04  =  5.04636257076217042715e-09, /* 0x3E35AC88, 0xC97DFF2C */
85 s05  =  1.23542274426137913908e-11; /* 0x3DAB2ACF, 0xCFB97ED8 */
86 
87 static double zero    = 0.0;
88 
89 #ifdef __STDC__
__ieee754_j1(double x)90 	double __ieee754_j1(double x)
91 #else
92 	double __ieee754_j1(x)
93 	double x;
94 #endif
95 {
96 	double z, s,c,ss,cc,r,u,v,y;
97 	int hx,ix;
98 
99 	hx = __HI(x);
100 	ix = hx&0x7fffffff;
101 	if(ix>=0x7ff00000) return one/x;
102 	y = ieee_fabs(x);
103 	if(ix >= 0x40000000) {	/* |x| >= 2.0 */
104 		s = ieee_sin(y);
105 		c = ieee_cos(y);
106 		ss = -s-c;
107 		cc = s-c;
108 		if(ix<0x7fe00000) {  /* make sure y+y not overflow */
109 		    z = ieee_cos(y+y);
110 		    if ((s*c)>zero) cc = z/ss;
111 		    else 	    ss = z/cc;
112 		}
113 	/*
114 	 * ieee_j1(x) = 1/ieee_sqrt(pi) * (P(1,x)*cc - Q(1,x)*ss) / ieee_sqrt(x)
115 	 * ieee_y1(x) = 1/ieee_sqrt(pi) * (P(1,x)*ss + Q(1,x)*cc) / ieee_sqrt(x)
116 	 */
117 		if(ix>0x48000000) z = (invsqrtpi*cc)/ieee_sqrt(y);
118 		else {
119 		    u = pone(y); v = qone(y);
120 		    z = invsqrtpi*(u*cc-v*ss)/ieee_sqrt(y);
121 		}
122 		if(hx<0) return -z;
123 		else  	 return  z;
124 	}
125 	if(ix<0x3e400000) {	/* |x|<2**-27 */
126 	    if(huge+x>one) return 0.5*x;/* inexact if x!=0 necessary */
127 	}
128 	z = x*x;
129 	r =  z*(r00+z*(r01+z*(r02+z*r03)));
130 	s =  one+z*(s01+z*(s02+z*(s03+z*(s04+z*s05))));
131 	r *= x;
132 	return(x*0.5+r/s);
133 }
134 
135 #ifdef __STDC__
136 static const double U0[5] = {
137 #else
138 static double U0[5] = {
139 #endif
140  -1.96057090646238940668e-01, /* 0xBFC91866, 0x143CBC8A */
141   5.04438716639811282616e-02, /* 0x3FA9D3C7, 0x76292CD1 */
142  -1.91256895875763547298e-03, /* 0xBF5F55E5, 0x4844F50F */
143   2.35252600561610495928e-05, /* 0x3EF8AB03, 0x8FA6B88E */
144  -9.19099158039878874504e-08, /* 0xBE78AC00, 0x569105B8 */
145 };
146 #ifdef __STDC__
147 static const double V0[5] = {
148 #else
149 static double V0[5] = {
150 #endif
151   1.99167318236649903973e-02, /* 0x3F94650D, 0x3F4DA9F0 */
152   2.02552581025135171496e-04, /* 0x3F2A8C89, 0x6C257764 */
153   1.35608801097516229404e-06, /* 0x3EB6C05A, 0x894E8CA6 */
154   6.22741452364621501295e-09, /* 0x3E3ABF1D, 0x5BA69A86 */
155   1.66559246207992079114e-11, /* 0x3DB25039, 0xDACA772A */
156 };
157 
158 #ifdef __STDC__
__ieee754_y1(double x)159 	double __ieee754_y1(double x)
160 #else
161 	double __ieee754_y1(x)
162 	double x;
163 #endif
164 {
165 	double z, s,c,ss,cc,u,v;
166 	int hx,ix,lx;
167 
168         hx = __HI(x);
169         ix = 0x7fffffff&hx;
170         lx = __LO(x);
171     /* if Y1(NaN) is NaN, Y1(-inf) is NaN, Y1(inf) is 0 */
172 	if(ix>=0x7ff00000) return  one/(x+x*x);
173         if((ix|lx)==0) return -one/zero;
174         if(hx<0) return zero/zero;
175         if(ix >= 0x40000000) {  /* |x| >= 2.0 */
176                 s = ieee_sin(x);
177                 c = ieee_cos(x);
178                 ss = -s-c;
179                 cc = s-c;
180                 if(ix<0x7fe00000) {  /* make sure x+x not overflow */
181                     z = ieee_cos(x+x);
182                     if ((s*c)>zero) cc = z/ss;
183                     else            ss = z/cc;
184                 }
185         /* ieee_y1(x) = ieee_sqrt(2/(pi*x))*(p1(x)*ieee_sin(x0)+q1(x)*ieee_cos(x0))
186          * where x0 = x-3pi/4
187          *      Better formula:
188          *              ieee_cos(x0) = ieee_cos(x)cos(3pi/4)+ieee_sin(x)sin(3pi/4)
189          *                      =  1/ieee_sqrt(2) * (ieee_sin(x) - ieee_cos(x))
190          *              ieee_sin(x0) = ieee_sin(x)cos(3pi/4)-ieee_cos(x)sin(3pi/4)
191          *                      = -1/ieee_sqrt(2) * (ieee_cos(x) + ieee_sin(x))
192          * To avoid cancellation, use
193          *              ieee_sin(x) +- ieee_cos(x) = -ieee_cos(2x)/(ieee_sin(x) -+ ieee_cos(x))
194          * to compute the worse one.
195          */
196                 if(ix>0x48000000) z = (invsqrtpi*ss)/ieee_sqrt(x);
197                 else {
198                     u = pone(x); v = qone(x);
199                     z = invsqrtpi*(u*ss+v*cc)/ieee_sqrt(x);
200                 }
201                 return z;
202         }
203         if(ix<=0x3c900000) {    /* x < 2**-54 */
204             return(-tpi/x);
205         }
206         z = x*x;
207         u = U0[0]+z*(U0[1]+z*(U0[2]+z*(U0[3]+z*U0[4])));
208         v = one+z*(V0[0]+z*(V0[1]+z*(V0[2]+z*(V0[3]+z*V0[4]))));
209         return(x*(u/v) + tpi*(__ieee754_j1(x)*__ieee754_log(x)-one/x));
210 }
211 
212 /* For x >= 8, the asymptotic expansions of pone is
213  *	1 + 15/128 s^2 - 4725/2^15 s^4 - ...,	where s = 1/x.
214  * We approximate pone by
215  * 	pone(x) = 1 + (R/S)
216  * where  R = pr0 + pr1*s^2 + pr2*s^4 + ... + pr5*s^10
217  * 	  S = 1 + ps0*s^2 + ... + ps4*s^10
218  * and
219  *	| pone(x)-1-R/S | <= 2  ** ( -60.06)
220  */
221 
222 #ifdef __STDC__
223 static const double pr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
224 #else
225 static double pr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
226 #endif
227   0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */
228   1.17187499999988647970e-01, /* 0x3FBDFFFF, 0xFFFFFCCE */
229   1.32394806593073575129e+01, /* 0x402A7A9D, 0x357F7FCE */
230   4.12051854307378562225e+02, /* 0x4079C0D4, 0x652EA590 */
231   3.87474538913960532227e+03, /* 0x40AE457D, 0xA3A532CC */
232   7.91447954031891731574e+03, /* 0x40BEEA7A, 0xC32782DD */
233 };
234 #ifdef __STDC__
235 static const double ps8[5] = {
236 #else
237 static double ps8[5] = {
238 #endif
239   1.14207370375678408436e+02, /* 0x405C8D45, 0x8E656CAC */
240   3.65093083420853463394e+03, /* 0x40AC85DC, 0x964D274F */
241   3.69562060269033463555e+04, /* 0x40E20B86, 0x97C5BB7F */
242   9.76027935934950801311e+04, /* 0x40F7D42C, 0xB28F17BB */
243   3.08042720627888811578e+04, /* 0x40DE1511, 0x697A0B2D */
244 };
245 
246 #ifdef __STDC__
247 static const double pr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
248 #else
249 static double pr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
250 #endif
251   1.31990519556243522749e-11, /* 0x3DAD0667, 0xDAE1CA7D */
252   1.17187493190614097638e-01, /* 0x3FBDFFFF, 0xE2C10043 */
253   6.80275127868432871736e+00, /* 0x401B3604, 0x6E6315E3 */
254   1.08308182990189109773e+02, /* 0x405B13B9, 0x452602ED */
255   5.17636139533199752805e+02, /* 0x40802D16, 0xD052D649 */
256   5.28715201363337541807e+02, /* 0x408085B8, 0xBB7E0CB7 */
257 };
258 #ifdef __STDC__
259 static const double ps5[5] = {
260 #else
261 static double ps5[5] = {
262 #endif
263   5.92805987221131331921e+01, /* 0x404DA3EA, 0xA8AF633D */
264   9.91401418733614377743e+02, /* 0x408EFB36, 0x1B066701 */
265   5.35326695291487976647e+03, /* 0x40B4E944, 0x5706B6FB */
266   7.84469031749551231769e+03, /* 0x40BEA4B0, 0xB8A5BB15 */
267   1.50404688810361062679e+03, /* 0x40978030, 0x036F5E51 */
268 };
269 
270 #ifdef __STDC__
271 static const double pr3[6] = {
272 #else
273 static double pr3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
274 #endif
275   3.02503916137373618024e-09, /* 0x3E29FC21, 0xA7AD9EDD */
276   1.17186865567253592491e-01, /* 0x3FBDFFF5, 0x5B21D17B */
277   3.93297750033315640650e+00, /* 0x400F76BC, 0xE85EAD8A */
278   3.51194035591636932736e+01, /* 0x40418F48, 0x9DA6D129 */
279   9.10550110750781271918e+01, /* 0x4056C385, 0x4D2C1837 */
280   4.85590685197364919645e+01, /* 0x4048478F, 0x8EA83EE5 */
281 };
282 #ifdef __STDC__
283 static const double ps3[5] = {
284 #else
285 static double ps3[5] = {
286 #endif
287   3.47913095001251519989e+01, /* 0x40416549, 0xA134069C */
288   3.36762458747825746741e+02, /* 0x40750C33, 0x07F1A75F */
289   1.04687139975775130551e+03, /* 0x40905B7C, 0x5037D523 */
290   8.90811346398256432622e+02, /* 0x408BD67D, 0xA32E31E9 */
291   1.03787932439639277504e+02, /* 0x4059F26D, 0x7C2EED53 */
292 };
293 
294 #ifdef __STDC__
295 static const double pr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
296 #else
297 static double pr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
298 #endif
299   1.07710830106873743082e-07, /* 0x3E7CE9D4, 0xF65544F4 */
300   1.17176219462683348094e-01, /* 0x3FBDFF42, 0xBE760D83 */
301   2.36851496667608785174e+00, /* 0x4002F2B7, 0xF98FAEC0 */
302   1.22426109148261232917e+01, /* 0x40287C37, 0x7F71A964 */
303   1.76939711271687727390e+01, /* 0x4031B1A8, 0x177F8EE2 */
304   5.07352312588818499250e+00, /* 0x40144B49, 0xA574C1FE */
305 };
306 #ifdef __STDC__
307 static const double ps2[5] = {
308 #else
309 static double ps2[5] = {
310 #endif
311   2.14364859363821409488e+01, /* 0x40356FBD, 0x8AD5ECDC */
312   1.25290227168402751090e+02, /* 0x405F5293, 0x14F92CD5 */
313   2.32276469057162813669e+02, /* 0x406D08D8, 0xD5A2DBD9 */
314   1.17679373287147100768e+02, /* 0x405D6B7A, 0xDA1884A9 */
315   8.36463893371618283368e+00, /* 0x4020BAB1, 0xF44E5192 */
316 };
317 
318 #ifdef __STDC__
pone(double x)319 	static double pone(double x)
320 #else
321 	static double pone(x)
322 	double x;
323 #endif
324 {
325 #ifdef __STDC__
326 	const double *p,*q;
327 #else
328 	double *p,*q;
329 #endif
330 	double z,r,s;
331         int ix;
332         ix = 0x7fffffff&__HI(x);
333         if(ix>=0x40200000)     {p = pr8; q= ps8;}
334         else if(ix>=0x40122E8B){p = pr5; q= ps5;}
335         else if(ix>=0x4006DB6D){p = pr3; q= ps3;}
336         else if(ix>=0x40000000){p = pr2; q= ps2;}
337         z = one/(x*x);
338         r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
339         s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*q[4]))));
340         return one+ r/s;
341 }
342 
343 
344 /* For x >= 8, the asymptotic expansions of qone is
345  *	3/8 s - 105/1024 s^3 - ..., where s = 1/x.
346  * We approximate pone by
347  * 	qone(x) = s*(0.375 + (R/S))
348  * where  R = qr1*s^2 + qr2*s^4 + ... + qr5*s^10
349  * 	  S = 1 + qs1*s^2 + ... + qs6*s^12
350  * and
351  *	| qone(x)/s -0.375-R/S | <= 2  ** ( -61.13)
352  */
353 
354 #ifdef __STDC__
355 static const double qr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
356 #else
357 static double qr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
358 #endif
359   0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */
360  -1.02539062499992714161e-01, /* 0xBFBA3FFF, 0xFFFFFDF3 */
361  -1.62717534544589987888e+01, /* 0xC0304591, 0xA26779F7 */
362  -7.59601722513950107896e+02, /* 0xC087BCD0, 0x53E4B576 */
363  -1.18498066702429587167e+04, /* 0xC0C724E7, 0x40F87415 */
364  -4.84385124285750353010e+04, /* 0xC0E7A6D0, 0x65D09C6A */
365 };
366 #ifdef __STDC__
367 static const double qs8[6] = {
368 #else
369 static double qs8[6] = {
370 #endif
371   1.61395369700722909556e+02, /* 0x40642CA6, 0xDE5BCDE5 */
372   7.82538599923348465381e+03, /* 0x40BE9162, 0xD0D88419 */
373   1.33875336287249578163e+05, /* 0x4100579A, 0xB0B75E98 */
374   7.19657723683240939863e+05, /* 0x4125F653, 0x72869C19 */
375   6.66601232617776375264e+05, /* 0x412457D2, 0x7719AD5C */
376  -2.94490264303834643215e+05, /* 0xC111F969, 0x0EA5AA18 */
377 };
378 
379 #ifdef __STDC__
380 static const double qr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
381 #else
382 static double qr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
383 #endif
384  -2.08979931141764104297e-11, /* 0xBDB6FA43, 0x1AA1A098 */
385  -1.02539050241375426231e-01, /* 0xBFBA3FFF, 0xCB597FEF */
386  -8.05644828123936029840e+00, /* 0xC0201CE6, 0xCA03AD4B */
387  -1.83669607474888380239e+02, /* 0xC066F56D, 0x6CA7B9B0 */
388  -1.37319376065508163265e+03, /* 0xC09574C6, 0x6931734F */
389  -2.61244440453215656817e+03, /* 0xC0A468E3, 0x88FDA79D */
390 };
391 #ifdef __STDC__
392 static const double qs5[6] = {
393 #else
394 static double qs5[6] = {
395 #endif
396   8.12765501384335777857e+01, /* 0x405451B2, 0xFF5A11B2 */
397   1.99179873460485964642e+03, /* 0x409F1F31, 0xE77BF839 */
398   1.74684851924908907677e+04, /* 0x40D10F1F, 0x0D64CE29 */
399   4.98514270910352279316e+04, /* 0x40E8576D, 0xAABAD197 */
400   2.79480751638918118260e+04, /* 0x40DB4B04, 0xCF7C364B */
401  -4.71918354795128470869e+03, /* 0xC0B26F2E, 0xFCFFA004 */
402 };
403 
404 #ifdef __STDC__
405 static const double qr3[6] = {
406 #else
407 static double qr3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
408 #endif
409  -5.07831226461766561369e-09, /* 0xBE35CFA9, 0xD38FC84F */
410  -1.02537829820837089745e-01, /* 0xBFBA3FEB, 0x51AEED54 */
411  -4.61011581139473403113e+00, /* 0xC01270C2, 0x3302D9FF */
412  -5.78472216562783643212e+01, /* 0xC04CEC71, 0xC25D16DA */
413  -2.28244540737631695038e+02, /* 0xC06C87D3, 0x4718D55F */
414  -2.19210128478909325622e+02, /* 0xC06B66B9, 0x5F5C1BF6 */
415 };
416 #ifdef __STDC__
417 static const double qs3[6] = {
418 #else
419 static double qs3[6] = {
420 #endif
421   4.76651550323729509273e+01, /* 0x4047D523, 0xCCD367E4 */
422   6.73865112676699709482e+02, /* 0x40850EEB, 0xC031EE3E */
423   3.38015286679526343505e+03, /* 0x40AA684E, 0x448E7C9A */
424   5.54772909720722782367e+03, /* 0x40B5ABBA, 0xA61D54A6 */
425   1.90311919338810798763e+03, /* 0x409DBC7A, 0x0DD4DF4B */
426  -1.35201191444307340817e+02, /* 0xC060E670, 0x290A311F */
427 };
428 
429 #ifdef __STDC__
430 static const double qr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
431 #else
432 static double qr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
433 #endif
434  -1.78381727510958865572e-07, /* 0xBE87F126, 0x44C626D2 */
435  -1.02517042607985553460e-01, /* 0xBFBA3E8E, 0x9148B010 */
436  -2.75220568278187460720e+00, /* 0xC0060484, 0x69BB4EDA */
437  -1.96636162643703720221e+01, /* 0xC033A9E2, 0xC168907F */
438  -4.23253133372830490089e+01, /* 0xC04529A3, 0xDE104AAA */
439  -2.13719211703704061733e+01, /* 0xC0355F36, 0x39CF6E52 */
440 };
441 #ifdef __STDC__
442 static const double qs2[6] = {
443 #else
444 static double qs2[6] = {
445 #endif
446   2.95333629060523854548e+01, /* 0x403D888A, 0x78AE64FF */
447   2.52981549982190529136e+02, /* 0x406F9F68, 0xDB821CBA */
448   7.57502834868645436472e+02, /* 0x4087AC05, 0xCE49A0F7 */
449   7.39393205320467245656e+02, /* 0x40871B25, 0x48D4C029 */
450   1.55949003336666123687e+02, /* 0x40637E5E, 0x3C3ED8D4 */
451  -4.95949898822628210127e+00, /* 0xC013D686, 0xE71BE86B */
452 };
453 
454 #ifdef __STDC__
qone(double x)455 	static double qone(double x)
456 #else
457 	static double qone(x)
458 	double x;
459 #endif
460 {
461 #ifdef __STDC__
462 	const double *p,*q;
463 #else
464 	double *p,*q;
465 #endif
466 	double  s,r,z;
467 	int ix;
468 	ix = 0x7fffffff&__HI(x);
469 	if(ix>=0x40200000)     {p = qr8; q= qs8;}
470 	else if(ix>=0x40122E8B){p = qr5; q= qs5;}
471 	else if(ix>=0x4006DB6D){p = qr3; q= qs3;}
472 	else if(ix>=0x40000000){p = qr2; q= qs2;}
473 	z = one/(x*x);
474 	r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
475 	s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*(q[4]+z*q[5])))));
476 	return (.375 + r/s)/x;
477 }
478