1 
2 /* @(#)e_j0.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 #include <sys/cdefs.h>
15 __FBSDID("$FreeBSD: head/lib/msun/src/e_j0.c 283032 2015-05-17 16:27:06Z kargl $");
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 static __inline double pzero(double), qzero(double);
66 
67 static const volatile double vone = 1, vzero = 0;
68 
69 static const double
70 huge 	= 1e300,
71 one	= 1.0,
72 invsqrtpi=  5.64189583547756279280e-01, /* 0x3FE20DD7, 0x50429B6D */
73 tpi      =  6.36619772367581382433e-01, /* 0x3FE45F30, 0x6DC9C883 */
74  		/* R0/S0 on [0, 2.00] */
75 R02  =  1.56249999999999947958e-02, /* 0x3F8FFFFF, 0xFFFFFFFD */
76 R03  = -1.89979294238854721751e-04, /* 0xBF28E6A5, 0xB61AC6E9 */
77 R04  =  1.82954049532700665670e-06, /* 0x3EBEB1D1, 0x0C503919 */
78 R05  = -4.61832688532103189199e-09, /* 0xBE33D5E7, 0x73D63FCE */
79 S01  =  1.56191029464890010492e-02, /* 0x3F8FFCE8, 0x82C8C2A4 */
80 S02  =  1.16926784663337450260e-04, /* 0x3F1EA6D2, 0xDD57DBF4 */
81 S03  =  5.13546550207318111446e-07, /* 0x3EA13B54, 0xCE84D5A9 */
82 S04  =  1.16614003333790000205e-09; /* 0x3E1408BC, 0xF4745D8F */
83 
84 static const double zero = 0.0;
85 
86 double
__ieee754_j0(double x)87 __ieee754_j0(double x)
88 {
89 	double z, s,c,ss,cc,r,u,v;
90 	int32_t hx,ix;
91 
92 	GET_HIGH_WORD(hx,x);
93 	ix = hx&0x7fffffff;
94 	if(ix>=0x7ff00000) return one/(x*x);
95 	x = fabs(x);
96 	if(ix >= 0x40000000) {	/* |x| >= 2.0 */
97 		s = sin(x);
98 		c = cos(x);
99 		ss = s-c;
100 		cc = s+c;
101 		if(ix<0x7fe00000) {  /* make sure x+x not overflow */
102 		    z = -cos(x+x);
103 		    if ((s*c)<zero) cc = z/ss;
104 		    else 	    ss = z/cc;
105 		}
106 	/*
107 	 * j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x)
108 	 * y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x)
109 	 */
110 		if(ix>0x48000000) z = (invsqrtpi*cc)/sqrt(x);
111 		else {
112 		    u = pzero(x); v = qzero(x);
113 		    z = invsqrtpi*(u*cc-v*ss)/sqrt(x);
114 		}
115 		return z;
116 	}
117 	if(ix<0x3f200000) {	/* |x| < 2**-13 */
118 	    if(huge+x>one) {	/* raise inexact if x != 0 */
119 	        if(ix<0x3e400000) return one;	/* |x|<2**-27 */
120 	        else 	      return one - x*x/4;
121 	    }
122 	}
123 	z = x*x;
124 	r =  z*(R02+z*(R03+z*(R04+z*R05)));
125 	s =  one+z*(S01+z*(S02+z*(S03+z*S04)));
126 	if(ix < 0x3FF00000) {	/* |x| < 1.00 */
127 	    return one + z*(-0.25+(r/s));
128 	} else {
129 	    u = 0.5*x;
130 	    return((one+u)*(one-u)+z*(r/s));
131 	}
132 }
133 
134 static const double
135 u00  = -7.38042951086872317523e-02, /* 0xBFB2E4D6, 0x99CBD01F */
136 u01  =  1.76666452509181115538e-01, /* 0x3FC69D01, 0x9DE9E3FC */
137 u02  = -1.38185671945596898896e-02, /* 0xBF8C4CE8, 0xB16CFA97 */
138 u03  =  3.47453432093683650238e-04, /* 0x3F36C54D, 0x20B29B6B */
139 u04  = -3.81407053724364161125e-06, /* 0xBECFFEA7, 0x73D25CAD */
140 u05  =  1.95590137035022920206e-08, /* 0x3E550057, 0x3B4EABD4 */
141 u06  = -3.98205194132103398453e-11, /* 0xBDC5E43D, 0x693FB3C8 */
142 v01  =  1.27304834834123699328e-02, /* 0x3F8A1270, 0x91C9C71A */
143 v02  =  7.60068627350353253702e-05, /* 0x3F13ECBB, 0xF578C6C1 */
144 v03  =  2.59150851840457805467e-07, /* 0x3E91642D, 0x7FF202FD */
145 v04  =  4.41110311332675467403e-10; /* 0x3DFE5018, 0x3BD6D9EF */
146 
147 double
__ieee754_y0(double x)148 __ieee754_y0(double x)
149 {
150 	double z, s,c,ss,cc,u,v;
151 	int32_t hx,ix,lx;
152 
153 	EXTRACT_WORDS(hx,lx,x);
154         ix = 0x7fffffff&hx;
155 	/*
156 	 * y0(NaN) = NaN.
157 	 * y0(Inf) = 0.
158 	 * y0(-Inf) = NaN and raise invalid exception.
159 	 */
160 	if(ix>=0x7ff00000) return vone/(x+x*x);
161 	/* y0(+-0) = -inf and raise divide-by-zero exception. */
162 	if((ix|lx)==0) return -one/vzero;
163 	/* y0(x<0) = NaN and raise invalid exception. */
164 	if(hx<0) return vzero/vzero;
165         if(ix >= 0x40000000) {  /* |x| >= 2.0 */
166         /* y0(x) = sqrt(2/(pi*x))*(p0(x)*sin(x0)+q0(x)*cos(x0))
167          * where x0 = x-pi/4
168          *      Better formula:
169          *              cos(x0) = cos(x)cos(pi/4)+sin(x)sin(pi/4)
170          *                      =  1/sqrt(2) * (sin(x) + cos(x))
171          *              sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4)
172          *                      =  1/sqrt(2) * (sin(x) - cos(x))
173          * To avoid cancellation, use
174          *              sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
175          * to compute the worse one.
176          */
177                 s = sin(x);
178                 c = cos(x);
179                 ss = s-c;
180                 cc = s+c;
181 	/*
182 	 * j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x)
183 	 * y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x)
184 	 */
185                 if(ix<0x7fe00000) {  /* make sure x+x not overflow */
186                     z = -cos(x+x);
187                     if ((s*c)<zero) cc = z/ss;
188                     else            ss = z/cc;
189                 }
190                 if(ix>0x48000000) z = (invsqrtpi*ss)/sqrt(x);
191                 else {
192                     u = pzero(x); v = qzero(x);
193                     z = invsqrtpi*(u*ss+v*cc)/sqrt(x);
194                 }
195                 return z;
196 	}
197 	if(ix<=0x3e400000) {	/* x < 2**-27 */
198 	    return(u00 + tpi*__ieee754_log(x));
199 	}
200 	z = x*x;
201 	u = u00+z*(u01+z*(u02+z*(u03+z*(u04+z*(u05+z*u06)))));
202 	v = one+z*(v01+z*(v02+z*(v03+z*v04)));
203 	return(u/v + tpi*(__ieee754_j0(x)*__ieee754_log(x)));
204 }
205 
206 /* The asymptotic expansions of pzero is
207  *	1 - 9/128 s^2 + 11025/98304 s^4 - ...,	where s = 1/x.
208  * For x >= 2, We approximate pzero by
209  * 	pzero(x) = 1 + (R/S)
210  * where  R = pR0 + pR1*s^2 + pR2*s^4 + ... + pR5*s^10
211  * 	  S = 1 + pS0*s^2 + ... + pS4*s^10
212  * and
213  *	| pzero(x)-1-R/S | <= 2  ** ( -60.26)
214  */
215 static const double pR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
216   0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */
217  -7.03124999999900357484e-02, /* 0xBFB1FFFF, 0xFFFFFD32 */
218  -8.08167041275349795626e+00, /* 0xC02029D0, 0xB44FA779 */
219  -2.57063105679704847262e+02, /* 0xC0701102, 0x7B19E863 */
220  -2.48521641009428822144e+03, /* 0xC0A36A6E, 0xCD4DCAFC */
221  -5.25304380490729545272e+03, /* 0xC0B4850B, 0x36CC643D */
222 };
223 static const double pS8[5] = {
224   1.16534364619668181717e+02, /* 0x405D2233, 0x07A96751 */
225   3.83374475364121826715e+03, /* 0x40ADF37D, 0x50596938 */
226   4.05978572648472545552e+04, /* 0x40E3D2BB, 0x6EB6B05F */
227   1.16752972564375915681e+05, /* 0x40FC810F, 0x8F9FA9BD */
228   4.76277284146730962675e+04, /* 0x40E74177, 0x4F2C49DC */
229 };
230 
231 static const double pR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
232  -1.14125464691894502584e-11, /* 0xBDA918B1, 0x47E495CC */
233  -7.03124940873599280078e-02, /* 0xBFB1FFFF, 0xE69AFBC6 */
234  -4.15961064470587782438e+00, /* 0xC010A370, 0xF90C6BBF */
235  -6.76747652265167261021e+01, /* 0xC050EB2F, 0x5A7D1783 */
236  -3.31231299649172967747e+02, /* 0xC074B3B3, 0x6742CC63 */
237  -3.46433388365604912451e+02, /* 0xC075A6EF, 0x28A38BD7 */
238 };
239 static const double pS5[5] = {
240   6.07539382692300335975e+01, /* 0x404E6081, 0x0C98C5DE */
241   1.05125230595704579173e+03, /* 0x40906D02, 0x5C7E2864 */
242   5.97897094333855784498e+03, /* 0x40B75AF8, 0x8FBE1D60 */
243   9.62544514357774460223e+03, /* 0x40C2CCB8, 0xFA76FA38 */
244   2.40605815922939109441e+03, /* 0x40A2CC1D, 0xC70BE864 */
245 };
246 
247 static const double pR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
248  -2.54704601771951915620e-09, /* 0xBE25E103, 0x6FE1AA86 */
249  -7.03119616381481654654e-02, /* 0xBFB1FFF6, 0xF7C0E24B */
250  -2.40903221549529611423e+00, /* 0xC00345B2, 0xAEA48074 */
251  -2.19659774734883086467e+01, /* 0xC035F74A, 0x4CB94E14 */
252  -5.80791704701737572236e+01, /* 0xC04D0A22, 0x420A1A45 */
253  -3.14479470594888503854e+01, /* 0xC03F72AC, 0xA892D80F */
254 };
255 static const double pS3[5] = {
256   3.58560338055209726349e+01, /* 0x4041ED92, 0x84077DD3 */
257   3.61513983050303863820e+02, /* 0x40769839, 0x464A7C0E */
258   1.19360783792111533330e+03, /* 0x4092A66E, 0x6D1061D6 */
259   1.12799679856907414432e+03, /* 0x40919FFC, 0xB8C39B7E */
260   1.73580930813335754692e+02, /* 0x4065B296, 0xFC379081 */
261 };
262 
263 static const double pR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
264  -8.87534333032526411254e-08, /* 0xBE77D316, 0xE927026D */
265  -7.03030995483624743247e-02, /* 0xBFB1FF62, 0x495E1E42 */
266  -1.45073846780952986357e+00, /* 0xBFF73639, 0x8A24A843 */
267  -7.63569613823527770791e+00, /* 0xC01E8AF3, 0xEDAFA7F3 */
268  -1.11931668860356747786e+01, /* 0xC02662E6, 0xC5246303 */
269  -3.23364579351335335033e+00, /* 0xC009DE81, 0xAF8FE70F */
270 };
271 static const double pS2[5] = {
272   2.22202997532088808441e+01, /* 0x40363865, 0x908B5959 */
273   1.36206794218215208048e+02, /* 0x4061069E, 0x0EE8878F */
274   2.70470278658083486789e+02, /* 0x4070E786, 0x42EA079B */
275   1.53875394208320329881e+02, /* 0x40633C03, 0x3AB6FAFF */
276   1.46576176948256193810e+01, /* 0x402D50B3, 0x44391809 */
277 };
278 
279 static __inline double
pzero(double x)280 pzero(double x)
281 {
282 	const double *p,*q;
283 	double z,r,s;
284 	int32_t ix;
285 	GET_HIGH_WORD(ix,x);
286 	ix &= 0x7fffffff;
287 	if(ix>=0x40200000)     {p = pR8; q= pS8;}
288 	else if(ix>=0x40122E8B){p = pR5; q= pS5;}
289 	else if(ix>=0x4006DB6D){p = pR3; q= pS3;}
290 	else                   {p = pR2; q= pS2;}	/* ix>=0x40000000 */
291 	z = one/(x*x);
292 	r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
293 	s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*q[4]))));
294 	return one+ r/s;
295 }
296 
297 
298 /* For x >= 8, the asymptotic expansions of qzero is
299  *	-1/8 s + 75/1024 s^3 - ..., where s = 1/x.
300  * We approximate pzero by
301  * 	qzero(x) = s*(-1.25 + (R/S))
302  * where  R = qR0 + qR1*s^2 + qR2*s^4 + ... + qR5*s^10
303  * 	  S = 1 + qS0*s^2 + ... + qS5*s^12
304  * and
305  *	| qzero(x)/s +1.25-R/S | <= 2  ** ( -61.22)
306  */
307 static const double qR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
308   0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */
309   7.32421874999935051953e-02, /* 0x3FB2BFFF, 0xFFFFFE2C */
310   1.17682064682252693899e+01, /* 0x40278952, 0x5BB334D6 */
311   5.57673380256401856059e+02, /* 0x40816D63, 0x15301825 */
312   8.85919720756468632317e+03, /* 0x40C14D99, 0x3E18F46D */
313   3.70146267776887834771e+04, /* 0x40E212D4, 0x0E901566 */
314 };
315 static const double qS8[6] = {
316   1.63776026895689824414e+02, /* 0x406478D5, 0x365B39BC */
317   8.09834494656449805916e+03, /* 0x40BFA258, 0x4E6B0563 */
318   1.42538291419120476348e+05, /* 0x41016652, 0x54D38C3F */
319   8.03309257119514397345e+05, /* 0x412883DA, 0x83A52B43 */
320   8.40501579819060512818e+05, /* 0x4129A66B, 0x28DE0B3D */
321  -3.43899293537866615225e+05, /* 0xC114FD6D, 0x2C9530C5 */
322 };
323 
324 static const double qR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
325   1.84085963594515531381e-11, /* 0x3DB43D8F, 0x29CC8CD9 */
326   7.32421766612684765896e-02, /* 0x3FB2BFFF, 0xD172B04C */
327   5.83563508962056953777e+00, /* 0x401757B0, 0xB9953DD3 */
328   1.35111577286449829671e+02, /* 0x4060E392, 0x0A8788E9 */
329   1.02724376596164097464e+03, /* 0x40900CF9, 0x9DC8C481 */
330   1.98997785864605384631e+03, /* 0x409F17E9, 0x53C6E3A6 */
331 };
332 static const double qS5[6] = {
333   8.27766102236537761883e+01, /* 0x4054B1B3, 0xFB5E1543 */
334   2.07781416421392987104e+03, /* 0x40A03BA0, 0xDA21C0CE */
335   1.88472887785718085070e+04, /* 0x40D267D2, 0x7B591E6D */
336   5.67511122894947329769e+04, /* 0x40EBB5E3, 0x97E02372 */
337   3.59767538425114471465e+04, /* 0x40E19118, 0x1F7A54A0 */
338  -5.35434275601944773371e+03, /* 0xC0B4EA57, 0xBEDBC609 */
339 };
340 
341 static const double qR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
342   4.37741014089738620906e-09, /* 0x3E32CD03, 0x6ADECB82 */
343   7.32411180042911447163e-02, /* 0x3FB2BFEE, 0x0E8D0842 */
344   3.34423137516170720929e+00, /* 0x400AC0FC, 0x61149CF5 */
345   4.26218440745412650017e+01, /* 0x40454F98, 0x962DAEDD */
346   1.70808091340565596283e+02, /* 0x406559DB, 0xE25EFD1F */
347   1.66733948696651168575e+02, /* 0x4064D77C, 0x81FA21E0 */
348 };
349 static const double qS3[6] = {
350   4.87588729724587182091e+01, /* 0x40486122, 0xBFE343A6 */
351   7.09689221056606015736e+02, /* 0x40862D83, 0x86544EB3 */
352   3.70414822620111362994e+03, /* 0x40ACF04B, 0xE44DFC63 */
353   6.46042516752568917582e+03, /* 0x40B93C6C, 0xD7C76A28 */
354   2.51633368920368957333e+03, /* 0x40A3A8AA, 0xD94FB1C0 */
355  -1.49247451836156386662e+02, /* 0xC062A7EB, 0x201CF40F */
356 };
357 
358 static const double qR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
359   1.50444444886983272379e-07, /* 0x3E84313B, 0x54F76BDB */
360   7.32234265963079278272e-02, /* 0x3FB2BEC5, 0x3E883E34 */
361   1.99819174093815998816e+00, /* 0x3FFFF897, 0xE727779C */
362   1.44956029347885735348e+01, /* 0x402CFDBF, 0xAAF96FE5 */
363   3.16662317504781540833e+01, /* 0x403FAA8E, 0x29FBDC4A */
364   1.62527075710929267416e+01, /* 0x403040B1, 0x71814BB4 */
365 };
366 static const double qS2[6] = {
367   3.03655848355219184498e+01, /* 0x403E5D96, 0xF7C07AED */
368   2.69348118608049844624e+02, /* 0x4070D591, 0xE4D14B40 */
369   8.44783757595320139444e+02, /* 0x408A6645, 0x22B3BF22 */
370   8.82935845112488550512e+02, /* 0x408B977C, 0x9C5CC214 */
371   2.12666388511798828631e+02, /* 0x406A9553, 0x0E001365 */
372  -5.31095493882666946917e+00, /* 0xC0153E6A, 0xF8B32931 */
373 };
374 
375 static __inline double
qzero(double x)376 qzero(double x)
377 {
378 	const double *p,*q;
379 	double s,r,z;
380 	int32_t ix;
381 	GET_HIGH_WORD(ix,x);
382 	ix &= 0x7fffffff;
383 	if(ix>=0x40200000)     {p = qR8; q= qS8;}
384 	else if(ix>=0x40122E8B){p = qR5; q= qS5;}
385 	else if(ix>=0x4006DB6D){p = qR3; q= qS3;}
386 	else                   {p = qR2; q= qS2;}	/* ix>=0x40000000 */
387 	z = one/(x*x);
388 	r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
389 	s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*(q[4]+z*q[5])))));
390 	return (-.125 + r/s)/x;
391 }
392