1 /*
2  * ====================================================
3  * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
4  *
5  * Developed at SunPro, a Sun Microsystems, Inc. business.
6  * Permission to use, copy, modify, and distribute this
7  * software is freely granted, provided that this notice
8  * is preserved.
9  * ====================================================
10  */
11 
12 /* __ieee754_j0(x), __ieee754_y0(x)
13  * Bessel function of the first and second kinds of order zero.
14  * Method -- j0(x):
15  *	1. For tiny x, we use j0(x) = 1 - x^2/4 + x^4/64 - ...
16  *	2. Reduce x to |x| since j0(x)=j0(-x),  and
17  *	   for x in (0,2)
18  *		j0(x) = 1-z/4+ z^2*R0/S0,  where z = x*x;
19  *	   (precision:  |j0-1+z/4-z^2R0/S0 |<2**-63.67 )
20  *	   for x in (2,inf)
21  * 		j0(x) = sqrt(2/(pi*x))*(p0(x)*cos(x0)-q0(x)*sin(x0))
22  * 	   where x0 = x-pi/4. It is better to compute sin(x0),cos(x0)
23  *	   as follow:
24  *		cos(x0) = cos(x)cos(pi/4)+sin(x)sin(pi/4)
25  *			= 1/sqrt(2) * (cos(x) + sin(x))
26  *		sin(x0) = sin(x)cos(pi/4)-cos(x)sin(pi/4)
27  *			= 1/sqrt(2) * (sin(x) - cos(x))
28  * 	   (To avoid cancellation, use
29  *		sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
30  * 	    to compute the worse one.)
31  *
32  *	3 Special cases
33  *		j0(nan)= nan
34  *		j0(0) = 1
35  *		j0(inf) = 0
36  *
37  * Method -- y0(x):
38  *	1. For x<2.
39  *	   Since
40  *		y0(x) = 2/pi*(j0(x)*(ln(x/2)+Euler) + x^2/4 - ...)
41  *	   therefore y0(x)-2/pi*j0(x)*ln(x) is an even function.
42  *	   We use the following function to approximate y0,
43  *		y0(x) = U(z)/V(z) + (2/pi)*(j0(x)*ln(x)), z= x^2
44  *	   where
45  *		U(z) = u00 + u01*z + ... + u06*z^6
46  *		V(z) = 1  + v01*z + ... + v04*z^4
47  *	   with absolute approximation error bounded by 2**-72.
48  *	   Note: For tiny x, U/V = u0 and j0(x)~1, hence
49  *		y0(tiny) = u0 + (2/pi)*ln(tiny), (choose tiny<2**-27)
50  *	2. For x>=2.
51  * 		y0(x) = sqrt(2/(pi*x))*(p0(x)*cos(x0)+q0(x)*sin(x0))
52  * 	   where x0 = x-pi/4. It is better to compute sin(x0),cos(x0)
53  *	   by the method mentioned above.
54  *	3. Special cases: y0(0)=-inf, y0(x<0)=NaN, y0(inf)=0.
55  */
56 
57 #include "math.h"
58 #include "math_private.h"
59 
60 static double pzero(double), qzero(double);
61 
62 static const double
63 huge 	= 1e300,
64 one	= 1.0,
65 invsqrtpi=  5.64189583547756279280e-01, /* 0x3FE20DD7, 0x50429B6D */
66 tpi      =  6.36619772367581382433e-01, /* 0x3FE45F30, 0x6DC9C883 */
67  		/* R0/S0 on [0, 2.00] */
68 R02  =  1.56249999999999947958e-02, /* 0x3F8FFFFF, 0xFFFFFFFD */
69 R03  = -1.89979294238854721751e-04, /* 0xBF28E6A5, 0xB61AC6E9 */
70 R04  =  1.82954049532700665670e-06, /* 0x3EBEB1D1, 0x0C503919 */
71 R05  = -4.61832688532103189199e-09, /* 0xBE33D5E7, 0x73D63FCE */
72 S01  =  1.56191029464890010492e-02, /* 0x3F8FFCE8, 0x82C8C2A4 */
73 S02  =  1.16926784663337450260e-04, /* 0x3F1EA6D2, 0xDD57DBF4 */
74 S03  =  5.13546550207318111446e-07, /* 0x3EA13B54, 0xCE84D5A9 */
75 S04  =  1.16614003333790000205e-09; /* 0x3E1408BC, 0xF4745D8F */
76 
77 static const double zero = 0.0;
78 
__ieee754_j0(double x)79 double __ieee754_j0(double x)
80 {
81 	double z, s,c,ss,cc,r,u,v;
82 	int32_t hx,ix;
83 
84 	GET_HIGH_WORD(hx,x);
85 	ix = hx&0x7fffffff;
86 	if(ix>=0x7ff00000) return one/(x*x);
87 	x = fabs(x);
88 	if(ix >= 0x40000000) {	/* |x| >= 2.0 */
89 		s = sin(x);
90 		c = cos(x);
91 		ss = s-c;
92 		cc = s+c;
93 		if(ix<0x7fe00000) {  /* make sure x+x not overflow */
94 		    z = -cos(x+x);
95 		    if ((s*c)<zero) cc = z/ss;
96 		    else 	    ss = z/cc;
97 		}
98 	/*
99 	 * j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x)
100 	 * y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x)
101 	 */
102 		if(ix>0x48000000) z = (invsqrtpi*cc)/sqrt(x);
103 		else {
104 		    u = pzero(x); v = qzero(x);
105 		    z = invsqrtpi*(u*cc-v*ss)/sqrt(x);
106 		}
107 		return z;
108 	}
109 	if(ix<0x3f200000) {	/* |x| < 2**-13 */
110 	    if(huge+x>one) {	/* raise inexact if x != 0 */
111 	        if(ix<0x3e400000) return one;	/* |x|<2**-27 */
112 	        else 	      return one - 0.25*x*x;
113 	    }
114 	}
115 	z = x*x;
116 	r =  z*(R02+z*(R03+z*(R04+z*R05)));
117 	s =  one+z*(S01+z*(S02+z*(S03+z*S04)));
118 	if(ix < 0x3FF00000) {	/* |x| < 1.00 */
119 	    return one + z*(-0.25+(r/s));
120 	} else {
121 	    u = 0.5*x;
122 	    return((one+u)*(one-u)+z*(r/s));
123 	}
124 }
125 
126 /*
127  * wrapper j0(double x)
128  */
129 #ifndef _IEEE_LIBM
j0(double x)130 double j0(double x)
131 {
132 	double z = __ieee754_j0(x);
133 	if (_LIB_VERSION == _IEEE_ || isnan(x))
134 		return z;
135 	if (fabs(x) > X_TLOSS)
136 		return __kernel_standard(x, x, 34); /* j0(|x|>X_TLOSS) */
137 	return z;
138 }
139 #else
140 strong_alias(__ieee754_j0, j0)
141 #endif
142 
143 static const double
144 u00  = -7.38042951086872317523e-02, /* 0xBFB2E4D6, 0x99CBD01F */
145 u01  =  1.76666452509181115538e-01, /* 0x3FC69D01, 0x9DE9E3FC */
146 u02  = -1.38185671945596898896e-02, /* 0xBF8C4CE8, 0xB16CFA97 */
147 u03  =  3.47453432093683650238e-04, /* 0x3F36C54D, 0x20B29B6B */
148 u04  = -3.81407053724364161125e-06, /* 0xBECFFEA7, 0x73D25CAD */
149 u05  =  1.95590137035022920206e-08, /* 0x3E550057, 0x3B4EABD4 */
150 u06  = -3.98205194132103398453e-11, /* 0xBDC5E43D, 0x693FB3C8 */
151 v01  =  1.27304834834123699328e-02, /* 0x3F8A1270, 0x91C9C71A */
152 v02  =  7.60068627350353253702e-05, /* 0x3F13ECBB, 0xF578C6C1 */
153 v03  =  2.59150851840457805467e-07, /* 0x3E91642D, 0x7FF202FD */
154 v04  =  4.41110311332675467403e-10; /* 0x3DFE5018, 0x3BD6D9EF */
155 
__ieee754_y0(double x)156 double __ieee754_y0(double x)
157 {
158 	double z, s,c,ss,cc,u,v;
159 	int32_t hx,ix,lx;
160 
161 	EXTRACT_WORDS(hx,lx,x);
162         ix = 0x7fffffff&hx;
163     /* Y0(NaN) is NaN, y0(-inf) is Nan, y0(inf) is 0  */
164 	if(ix>=0x7ff00000) return  one/(x+x*x);
165         if((ix|lx)==0) return -one/zero;
166         if(hx<0) return zero/zero;
167         if(ix >= 0x40000000) {  /* |x| >= 2.0 */
168         /* y0(x) = sqrt(2/(pi*x))*(p0(x)*sin(x0)+q0(x)*cos(x0))
169          * where x0 = x-pi/4
170          *      Better formula:
171          *              cos(x0) = cos(x)cos(pi/4)+sin(x)sin(pi/4)
172          *                      =  1/sqrt(2) * (sin(x) + cos(x))
173          *              sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4)
174          *                      =  1/sqrt(2) * (sin(x) - cos(x))
175          * To avoid cancellation, use
176          *              sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
177          * to compute the worse one.
178          */
179                 s = sin(x);
180                 c = cos(x);
181                 ss = s-c;
182                 cc = s+c;
183 	/*
184 	 * j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x)
185 	 * y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x)
186 	 */
187                 if(ix<0x7fe00000) {  /* make sure x+x not overflow */
188                     z = -cos(x+x);
189                     if ((s*c)<zero) cc = z/ss;
190                     else            ss = z/cc;
191                 }
192                 if(ix>0x48000000) z = (invsqrtpi*ss)/sqrt(x);
193                 else {
194                     u = pzero(x); v = qzero(x);
195                     z = invsqrtpi*(u*ss+v*cc)/sqrt(x);
196                 }
197                 return z;
198 	}
199 	if(ix<=0x3e400000) {	/* x < 2**-27 */
200 	    return(u00 + tpi*__ieee754_log(x));
201 	}
202 	z = x*x;
203 	u = u00+z*(u01+z*(u02+z*(u03+z*(u04+z*(u05+z*u06)))));
204 	v = one+z*(v01+z*(v02+z*(v03+z*v04)));
205 	return(u/v + tpi*(__ieee754_j0(x)*__ieee754_log(x)));
206 }
207 
208 /*
209  * wrapper y0(double x)
210  */
211 #ifndef _IEEE_LIBM
y0(double x)212 double y0(double x)
213 {
214 	double z = __ieee754_y0(x);
215 	if (_LIB_VERSION == _IEEE_ || isnan(x))
216 		return z;
217 	if (x <= 0.0) {
218 		if (x == 0.0) /* d= -one/(x-x); */
219 			return __kernel_standard(x, x, 8);
220 		/* d = zero/(x-x); */
221 		return __kernel_standard(x, x, 9);
222 	}
223 	if (x > X_TLOSS)
224 		return __kernel_standard(x, x, 35); /* y0(x>X_TLOSS) */
225 	return z;
226 }
227 #else
228 strong_alias(__ieee754_y0, y0)
229 #endif
230 
231 
232 /* The asymptotic expansions of pzero is
233  *	1 - 9/128 s^2 + 11025/98304 s^4 - ...,	where s = 1/x.
234  * For x >= 2, We approximate pzero by
235  * 	pzero(x) = 1 + (R/S)
236  * where  R = pR0 + pR1*s^2 + pR2*s^4 + ... + pR5*s^10
237  * 	  S = 1 + pS0*s^2 + ... + pS4*s^10
238  * and
239  *	| pzero(x)-1-R/S | <= 2  ** ( -60.26)
240  */
241 static const double pR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
242   0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */
243  -7.03124999999900357484e-02, /* 0xBFB1FFFF, 0xFFFFFD32 */
244  -8.08167041275349795626e+00, /* 0xC02029D0, 0xB44FA779 */
245  -2.57063105679704847262e+02, /* 0xC0701102, 0x7B19E863 */
246  -2.48521641009428822144e+03, /* 0xC0A36A6E, 0xCD4DCAFC */
247  -5.25304380490729545272e+03, /* 0xC0B4850B, 0x36CC643D */
248 };
249 static const double pS8[5] = {
250   1.16534364619668181717e+02, /* 0x405D2233, 0x07A96751 */
251   3.83374475364121826715e+03, /* 0x40ADF37D, 0x50596938 */
252   4.05978572648472545552e+04, /* 0x40E3D2BB, 0x6EB6B05F */
253   1.16752972564375915681e+05, /* 0x40FC810F, 0x8F9FA9BD */
254   4.76277284146730962675e+04, /* 0x40E74177, 0x4F2C49DC */
255 };
256 
257 static const double pR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
258  -1.14125464691894502584e-11, /* 0xBDA918B1, 0x47E495CC */
259  -7.03124940873599280078e-02, /* 0xBFB1FFFF, 0xE69AFBC6 */
260  -4.15961064470587782438e+00, /* 0xC010A370, 0xF90C6BBF */
261  -6.76747652265167261021e+01, /* 0xC050EB2F, 0x5A7D1783 */
262  -3.31231299649172967747e+02, /* 0xC074B3B3, 0x6742CC63 */
263  -3.46433388365604912451e+02, /* 0xC075A6EF, 0x28A38BD7 */
264 };
265 static const double pS5[5] = {
266   6.07539382692300335975e+01, /* 0x404E6081, 0x0C98C5DE */
267   1.05125230595704579173e+03, /* 0x40906D02, 0x5C7E2864 */
268   5.97897094333855784498e+03, /* 0x40B75AF8, 0x8FBE1D60 */
269   9.62544514357774460223e+03, /* 0x40C2CCB8, 0xFA76FA38 */
270   2.40605815922939109441e+03, /* 0x40A2CC1D, 0xC70BE864 */
271 };
272 
273 static const double pR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
274  -2.54704601771951915620e-09, /* 0xBE25E103, 0x6FE1AA86 */
275  -7.03119616381481654654e-02, /* 0xBFB1FFF6, 0xF7C0E24B */
276  -2.40903221549529611423e+00, /* 0xC00345B2, 0xAEA48074 */
277  -2.19659774734883086467e+01, /* 0xC035F74A, 0x4CB94E14 */
278  -5.80791704701737572236e+01, /* 0xC04D0A22, 0x420A1A45 */
279  -3.14479470594888503854e+01, /* 0xC03F72AC, 0xA892D80F */
280 };
281 static const double pS3[5] = {
282   3.58560338055209726349e+01, /* 0x4041ED92, 0x84077DD3 */
283   3.61513983050303863820e+02, /* 0x40769839, 0x464A7C0E */
284   1.19360783792111533330e+03, /* 0x4092A66E, 0x6D1061D6 */
285   1.12799679856907414432e+03, /* 0x40919FFC, 0xB8C39B7E */
286   1.73580930813335754692e+02, /* 0x4065B296, 0xFC379081 */
287 };
288 
289 static const double pR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
290  -8.87534333032526411254e-08, /* 0xBE77D316, 0xE927026D */
291  -7.03030995483624743247e-02, /* 0xBFB1FF62, 0x495E1E42 */
292  -1.45073846780952986357e+00, /* 0xBFF73639, 0x8A24A843 */
293  -7.63569613823527770791e+00, /* 0xC01E8AF3, 0xEDAFA7F3 */
294  -1.11931668860356747786e+01, /* 0xC02662E6, 0xC5246303 */
295  -3.23364579351335335033e+00, /* 0xC009DE81, 0xAF8FE70F */
296 };
297 static const double pS2[5] = {
298   2.22202997532088808441e+01, /* 0x40363865, 0x908B5959 */
299   1.36206794218215208048e+02, /* 0x4061069E, 0x0EE8878F */
300   2.70470278658083486789e+02, /* 0x4070E786, 0x42EA079B */
301   1.53875394208320329881e+02, /* 0x40633C03, 0x3AB6FAFF */
302   1.46576176948256193810e+01, /* 0x402D50B3, 0x44391809 */
303 };
304 
pzero(double x)305 static double pzero(double x)
306 {
307 	const double *p = 0,*q = 0;
308 	double z,r,s;
309 	int32_t ix;
310 	GET_HIGH_WORD(ix,x);
311 	ix &= 0x7fffffff;
312 	if(ix>=0x40200000)     {p = pR8; q= pS8;}
313 	else if(ix>=0x40122E8B){p = pR5; q= pS5;}
314 	else if(ix>=0x4006DB6D){p = pR3; q= pS3;}
315 	else if(ix>=0x40000000){p = pR2; q= pS2;}
316 	z = one/(x*x);
317 	r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
318 	s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*q[4]))));
319 	return one+ r/s;
320 }
321 
322 
323 /* For x >= 8, the asymptotic expansions of qzero is
324  *	-1/8 s + 75/1024 s^3 - ..., where s = 1/x.
325  * We approximate pzero by
326  * 	qzero(x) = s*(-1.25 + (R/S))
327  * where  R = qR0 + qR1*s^2 + qR2*s^4 + ... + qR5*s^10
328  * 	  S = 1 + qS0*s^2 + ... + qS5*s^12
329  * and
330  *	| qzero(x)/s +1.25-R/S | <= 2  ** ( -61.22)
331  */
332 static const double qR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
333   0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */
334   7.32421874999935051953e-02, /* 0x3FB2BFFF, 0xFFFFFE2C */
335   1.17682064682252693899e+01, /* 0x40278952, 0x5BB334D6 */
336   5.57673380256401856059e+02, /* 0x40816D63, 0x15301825 */
337   8.85919720756468632317e+03, /* 0x40C14D99, 0x3E18F46D */
338   3.70146267776887834771e+04, /* 0x40E212D4, 0x0E901566 */
339 };
340 static const double qS8[6] = {
341   1.63776026895689824414e+02, /* 0x406478D5, 0x365B39BC */
342   8.09834494656449805916e+03, /* 0x40BFA258, 0x4E6B0563 */
343   1.42538291419120476348e+05, /* 0x41016652, 0x54D38C3F */
344   8.03309257119514397345e+05, /* 0x412883DA, 0x83A52B43 */
345   8.40501579819060512818e+05, /* 0x4129A66B, 0x28DE0B3D */
346  -3.43899293537866615225e+05, /* 0xC114FD6D, 0x2C9530C5 */
347 };
348 
349 static const double qR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
350   1.84085963594515531381e-11, /* 0x3DB43D8F, 0x29CC8CD9 */
351   7.32421766612684765896e-02, /* 0x3FB2BFFF, 0xD172B04C */
352   5.83563508962056953777e+00, /* 0x401757B0, 0xB9953DD3 */
353   1.35111577286449829671e+02, /* 0x4060E392, 0x0A8788E9 */
354   1.02724376596164097464e+03, /* 0x40900CF9, 0x9DC8C481 */
355   1.98997785864605384631e+03, /* 0x409F17E9, 0x53C6E3A6 */
356 };
357 static const double qS5[6] = {
358   8.27766102236537761883e+01, /* 0x4054B1B3, 0xFB5E1543 */
359   2.07781416421392987104e+03, /* 0x40A03BA0, 0xDA21C0CE */
360   1.88472887785718085070e+04, /* 0x40D267D2, 0x7B591E6D */
361   5.67511122894947329769e+04, /* 0x40EBB5E3, 0x97E02372 */
362   3.59767538425114471465e+04, /* 0x40E19118, 0x1F7A54A0 */
363  -5.35434275601944773371e+03, /* 0xC0B4EA57, 0xBEDBC609 */
364 };
365 
366 static const double qR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
367   4.37741014089738620906e-09, /* 0x3E32CD03, 0x6ADECB82 */
368   7.32411180042911447163e-02, /* 0x3FB2BFEE, 0x0E8D0842 */
369   3.34423137516170720929e+00, /* 0x400AC0FC, 0x61149CF5 */
370   4.26218440745412650017e+01, /* 0x40454F98, 0x962DAEDD */
371   1.70808091340565596283e+02, /* 0x406559DB, 0xE25EFD1F */
372   1.66733948696651168575e+02, /* 0x4064D77C, 0x81FA21E0 */
373 };
374 static const double qS3[6] = {
375   4.87588729724587182091e+01, /* 0x40486122, 0xBFE343A6 */
376   7.09689221056606015736e+02, /* 0x40862D83, 0x86544EB3 */
377   3.70414822620111362994e+03, /* 0x40ACF04B, 0xE44DFC63 */
378   6.46042516752568917582e+03, /* 0x40B93C6C, 0xD7C76A28 */
379   2.51633368920368957333e+03, /* 0x40A3A8AA, 0xD94FB1C0 */
380  -1.49247451836156386662e+02, /* 0xC062A7EB, 0x201CF40F */
381 };
382 
383 static const double qR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
384   1.50444444886983272379e-07, /* 0x3E84313B, 0x54F76BDB */
385   7.32234265963079278272e-02, /* 0x3FB2BEC5, 0x3E883E34 */
386   1.99819174093815998816e+00, /* 0x3FFFF897, 0xE727779C */
387   1.44956029347885735348e+01, /* 0x402CFDBF, 0xAAF96FE5 */
388   3.16662317504781540833e+01, /* 0x403FAA8E, 0x29FBDC4A */
389   1.62527075710929267416e+01, /* 0x403040B1, 0x71814BB4 */
390 };
391 static const double qS2[6] = {
392   3.03655848355219184498e+01, /* 0x403E5D96, 0xF7C07AED */
393   2.69348118608049844624e+02, /* 0x4070D591, 0xE4D14B40 */
394   8.44783757595320139444e+02, /* 0x408A6645, 0x22B3BF22 */
395   8.82935845112488550512e+02, /* 0x408B977C, 0x9C5CC214 */
396   2.12666388511798828631e+02, /* 0x406A9553, 0x0E001365 */
397  -5.31095493882666946917e+00, /* 0xC0153E6A, 0xF8B32931 */
398 };
399 
qzero(double x)400 static double qzero(double x)
401 {
402 	const double *p=0,*q=0;
403 	double s,r,z;
404 	int32_t ix;
405 	GET_HIGH_WORD(ix,x);
406 	ix &= 0x7fffffff;
407 	if(ix>=0x40200000)     {p = qR8; q= qS8;}
408 	else if(ix>=0x40122E8B){p = qR5; q= qS5;}
409 	else if(ix>=0x4006DB6D){p = qR3; q= qS3;}
410 	else if(ix>=0x40000000){p = qR2; q= qS2;}
411 	z = one/(x*x);
412 	r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
413 	s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*(q[4]+z*q[5])))));
414 	return (-.125 + r/s)/x;
415 }
416