source: asp3_tinet_ecnl_rx/trunk/musl-1.1.18/src/math/j0.c@ 337

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1/* origin: FreeBSD /usr/src/lib/msun/src/e_j0.c */
2/*
3 * ====================================================
4 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
5 *
6 * Developed at SunSoft, 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/* j0(x), 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 "libm.h"
58
59static double pzero(double), qzero(double);
60
61static const double
62invsqrtpi = 5.64189583547756279280e-01, /* 0x3FE20DD7, 0x50429B6D */
63tpi = 6.36619772367581382433e-01; /* 0x3FE45F30, 0x6DC9C883 */
64
65/* common method when |x|>=2 */
66static double common(uint32_t ix, double x, int y0)
67{
68 double s,c,ss,cc,z;
69
70 /*
71 * j0(x) = sqrt(2/(pi*x))*(p0(x)*cos(x-pi/4)-q0(x)*sin(x-pi/4))
72 * y0(x) = sqrt(2/(pi*x))*(p0(x)*sin(x-pi/4)+q0(x)*cos(x-pi/4))
73 *
74 * sin(x-pi/4) = (sin(x) - cos(x))/sqrt(2)
75 * cos(x-pi/4) = (sin(x) + cos(x))/sqrt(2)
76 * sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
77 */
78 s = sin(x);
79 c = cos(x);
80 if (y0)
81 c = -c;
82 cc = s+c;
83 /* avoid overflow in 2*x, big ulp error when x>=0x1p1023 */
84 if (ix < 0x7fe00000) {
85 ss = s-c;
86 z = -cos(2*x);
87 if (s*c < 0)
88 cc = z/ss;
89 else
90 ss = z/cc;
91 if (ix < 0x48000000) {
92 if (y0)
93 ss = -ss;
94 cc = pzero(x)*cc-qzero(x)*ss;
95 }
96 }
97 return invsqrtpi*cc/sqrt(x);
98}
99
100/* R0/S0 on [0, 2.00] */
101static const double
102R02 = 1.56249999999999947958e-02, /* 0x3F8FFFFF, 0xFFFFFFFD */
103R03 = -1.89979294238854721751e-04, /* 0xBF28E6A5, 0xB61AC6E9 */
104R04 = 1.82954049532700665670e-06, /* 0x3EBEB1D1, 0x0C503919 */
105R05 = -4.61832688532103189199e-09, /* 0xBE33D5E7, 0x73D63FCE */
106S01 = 1.56191029464890010492e-02, /* 0x3F8FFCE8, 0x82C8C2A4 */
107S02 = 1.16926784663337450260e-04, /* 0x3F1EA6D2, 0xDD57DBF4 */
108S03 = 5.13546550207318111446e-07, /* 0x3EA13B54, 0xCE84D5A9 */
109S04 = 1.16614003333790000205e-09; /* 0x3E1408BC, 0xF4745D8F */
110
111double j0(double x)
112{
113 double z,r,s;
114 uint32_t ix;
115
116 GET_HIGH_WORD(ix, x);
117 ix &= 0x7fffffff;
118
119 /* j0(+-inf)=0, j0(nan)=nan */
120 if (ix >= 0x7ff00000)
121 return 1/(x*x);
122 x = fabs(x);
123
124 if (ix >= 0x40000000) { /* |x| >= 2 */
125 /* large ulp error near zeros: 2.4, 5.52, 8.6537,.. */
126 return common(ix,x,0);
127 }
128
129 /* 1 - x*x/4 + x*x*R(x^2)/S(x^2) */
130 if (ix >= 0x3f200000) { /* |x| >= 2**-13 */
131 /* up to 4ulp error close to 2 */
132 z = x*x;
133 r = z*(R02+z*(R03+z*(R04+z*R05)));
134 s = 1+z*(S01+z*(S02+z*(S03+z*S04)));
135 return (1+x/2)*(1-x/2) + z*(r/s);
136 }
137
138 /* 1 - x*x/4 */
139 /* prevent underflow */
140 /* inexact should be raised when x!=0, this is not done correctly */
141 if (ix >= 0x38000000) /* |x| >= 2**-127 */
142 x = 0.25*x*x;
143 return 1 - x;
144}
145
146static const double
147u00 = -7.38042951086872317523e-02, /* 0xBFB2E4D6, 0x99CBD01F */
148u01 = 1.76666452509181115538e-01, /* 0x3FC69D01, 0x9DE9E3FC */
149u02 = -1.38185671945596898896e-02, /* 0xBF8C4CE8, 0xB16CFA97 */
150u03 = 3.47453432093683650238e-04, /* 0x3F36C54D, 0x20B29B6B */
151u04 = -3.81407053724364161125e-06, /* 0xBECFFEA7, 0x73D25CAD */
152u05 = 1.95590137035022920206e-08, /* 0x3E550057, 0x3B4EABD4 */
153u06 = -3.98205194132103398453e-11, /* 0xBDC5E43D, 0x693FB3C8 */
154v01 = 1.27304834834123699328e-02, /* 0x3F8A1270, 0x91C9C71A */
155v02 = 7.60068627350353253702e-05, /* 0x3F13ECBB, 0xF578C6C1 */
156v03 = 2.59150851840457805467e-07, /* 0x3E91642D, 0x7FF202FD */
157v04 = 4.41110311332675467403e-10; /* 0x3DFE5018, 0x3BD6D9EF */
158
159double y0(double x)
160{
161 double z,u,v;
162 uint32_t ix,lx;
163
164 EXTRACT_WORDS(ix, lx, x);
165
166 /* y0(nan)=nan, y0(<0)=nan, y0(0)=-inf, y0(inf)=0 */
167 if ((ix<<1 | lx) == 0)
168 return -1/0.0;
169 if (ix>>31)
170 return 0/0.0;
171 if (ix >= 0x7ff00000)
172 return 1/x;
173
174 if (ix >= 0x40000000) { /* x >= 2 */
175 /* large ulp errors near zeros: 3.958, 7.086,.. */
176 return common(ix,x,1);
177 }
178
179 /* U(x^2)/V(x^2) + (2/pi)*j0(x)*log(x) */
180 if (ix >= 0x3e400000) { /* x >= 2**-27 */
181 /* large ulp error near the first zero, x ~= 0.89 */
182 z = x*x;
183 u = u00+z*(u01+z*(u02+z*(u03+z*(u04+z*(u05+z*u06)))));
184 v = 1.0+z*(v01+z*(v02+z*(v03+z*v04)));
185 return u/v + tpi*(j0(x)*log(x));
186 }
187 return u00 + tpi*log(x);
188}
189
190/* The asymptotic expansions of pzero is
191 * 1 - 9/128 s^2 + 11025/98304 s^4 - ..., where s = 1/x.
192 * For x >= 2, We approximate pzero by
193 * pzero(x) = 1 + (R/S)
194 * where R = pR0 + pR1*s^2 + pR2*s^4 + ... + pR5*s^10
195 * S = 1 + pS0*s^2 + ... + pS4*s^10
196 * and
197 * | pzero(x)-1-R/S | <= 2 ** ( -60.26)
198 */
199static const double pR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
200 0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */
201 -7.03124999999900357484e-02, /* 0xBFB1FFFF, 0xFFFFFD32 */
202 -8.08167041275349795626e+00, /* 0xC02029D0, 0xB44FA779 */
203 -2.57063105679704847262e+02, /* 0xC0701102, 0x7B19E863 */
204 -2.48521641009428822144e+03, /* 0xC0A36A6E, 0xCD4DCAFC */
205 -5.25304380490729545272e+03, /* 0xC0B4850B, 0x36CC643D */
206};
207static const double pS8[5] = {
208 1.16534364619668181717e+02, /* 0x405D2233, 0x07A96751 */
209 3.83374475364121826715e+03, /* 0x40ADF37D, 0x50596938 */
210 4.05978572648472545552e+04, /* 0x40E3D2BB, 0x6EB6B05F */
211 1.16752972564375915681e+05, /* 0x40FC810F, 0x8F9FA9BD */
212 4.76277284146730962675e+04, /* 0x40E74177, 0x4F2C49DC */
213};
214
215static const double pR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
216 -1.14125464691894502584e-11, /* 0xBDA918B1, 0x47E495CC */
217 -7.03124940873599280078e-02, /* 0xBFB1FFFF, 0xE69AFBC6 */
218 -4.15961064470587782438e+00, /* 0xC010A370, 0xF90C6BBF */
219 -6.76747652265167261021e+01, /* 0xC050EB2F, 0x5A7D1783 */
220 -3.31231299649172967747e+02, /* 0xC074B3B3, 0x6742CC63 */
221 -3.46433388365604912451e+02, /* 0xC075A6EF, 0x28A38BD7 */
222};
223static const double pS5[5] = {
224 6.07539382692300335975e+01, /* 0x404E6081, 0x0C98C5DE */
225 1.05125230595704579173e+03, /* 0x40906D02, 0x5C7E2864 */
226 5.97897094333855784498e+03, /* 0x40B75AF8, 0x8FBE1D60 */
227 9.62544514357774460223e+03, /* 0x40C2CCB8, 0xFA76FA38 */
228 2.40605815922939109441e+03, /* 0x40A2CC1D, 0xC70BE864 */
229};
230
231static const double pR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
232 -2.54704601771951915620e-09, /* 0xBE25E103, 0x6FE1AA86 */
233 -7.03119616381481654654e-02, /* 0xBFB1FFF6, 0xF7C0E24B */
234 -2.40903221549529611423e+00, /* 0xC00345B2, 0xAEA48074 */
235 -2.19659774734883086467e+01, /* 0xC035F74A, 0x4CB94E14 */
236 -5.80791704701737572236e+01, /* 0xC04D0A22, 0x420A1A45 */
237 -3.14479470594888503854e+01, /* 0xC03F72AC, 0xA892D80F */
238};
239static const double pS3[5] = {
240 3.58560338055209726349e+01, /* 0x4041ED92, 0x84077DD3 */
241 3.61513983050303863820e+02, /* 0x40769839, 0x464A7C0E */
242 1.19360783792111533330e+03, /* 0x4092A66E, 0x6D1061D6 */
243 1.12799679856907414432e+03, /* 0x40919FFC, 0xB8C39B7E */
244 1.73580930813335754692e+02, /* 0x4065B296, 0xFC379081 */
245};
246
247static const double pR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
248 -8.87534333032526411254e-08, /* 0xBE77D316, 0xE927026D */
249 -7.03030995483624743247e-02, /* 0xBFB1FF62, 0x495E1E42 */
250 -1.45073846780952986357e+00, /* 0xBFF73639, 0x8A24A843 */
251 -7.63569613823527770791e+00, /* 0xC01E8AF3, 0xEDAFA7F3 */
252 -1.11931668860356747786e+01, /* 0xC02662E6, 0xC5246303 */
253 -3.23364579351335335033e+00, /* 0xC009DE81, 0xAF8FE70F */
254};
255static const double pS2[5] = {
256 2.22202997532088808441e+01, /* 0x40363865, 0x908B5959 */
257 1.36206794218215208048e+02, /* 0x4061069E, 0x0EE8878F */
258 2.70470278658083486789e+02, /* 0x4070E786, 0x42EA079B */
259 1.53875394208320329881e+02, /* 0x40633C03, 0x3AB6FAFF */
260 1.46576176948256193810e+01, /* 0x402D50B3, 0x44391809 */
261};
262
263static double pzero(double x)
264{
265 const double *p,*q;
266 double_t z,r,s;
267 uint32_t ix;
268
269 GET_HIGH_WORD(ix, x);
270 ix &= 0x7fffffff;
271 if (ix >= 0x40200000){p = pR8; q = pS8;}
272 else if (ix >= 0x40122E8B){p = pR5; q = pS5;}
273 else if (ix >= 0x4006DB6D){p = pR3; q = pS3;}
274 else /*ix >= 0x40000000*/ {p = pR2; q = pS2;}
275 z = 1.0/(x*x);
276 r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
277 s = 1.0+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*q[4]))));
278 return 1.0 + r/s;
279}
280
281
282/* For x >= 8, the asymptotic expansions of qzero is
283 * -1/8 s + 75/1024 s^3 - ..., where s = 1/x.
284 * We approximate pzero by
285 * qzero(x) = s*(-1.25 + (R/S))
286 * where R = qR0 + qR1*s^2 + qR2*s^4 + ... + qR5*s^10
287 * S = 1 + qS0*s^2 + ... + qS5*s^12
288 * and
289 * | qzero(x)/s +1.25-R/S | <= 2 ** ( -61.22)
290 */
291static const double qR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
292 0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */
293 7.32421874999935051953e-02, /* 0x3FB2BFFF, 0xFFFFFE2C */
294 1.17682064682252693899e+01, /* 0x40278952, 0x5BB334D6 */
295 5.57673380256401856059e+02, /* 0x40816D63, 0x15301825 */
296 8.85919720756468632317e+03, /* 0x40C14D99, 0x3E18F46D */
297 3.70146267776887834771e+04, /* 0x40E212D4, 0x0E901566 */
298};
299static const double qS8[6] = {
300 1.63776026895689824414e+02, /* 0x406478D5, 0x365B39BC */
301 8.09834494656449805916e+03, /* 0x40BFA258, 0x4E6B0563 */
302 1.42538291419120476348e+05, /* 0x41016652, 0x54D38C3F */
303 8.03309257119514397345e+05, /* 0x412883DA, 0x83A52B43 */
304 8.40501579819060512818e+05, /* 0x4129A66B, 0x28DE0B3D */
305 -3.43899293537866615225e+05, /* 0xC114FD6D, 0x2C9530C5 */
306};
307
308static const double qR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
309 1.84085963594515531381e-11, /* 0x3DB43D8F, 0x29CC8CD9 */
310 7.32421766612684765896e-02, /* 0x3FB2BFFF, 0xD172B04C */
311 5.83563508962056953777e+00, /* 0x401757B0, 0xB9953DD3 */
312 1.35111577286449829671e+02, /* 0x4060E392, 0x0A8788E9 */
313 1.02724376596164097464e+03, /* 0x40900CF9, 0x9DC8C481 */
314 1.98997785864605384631e+03, /* 0x409F17E9, 0x53C6E3A6 */
315};
316static const double qS5[6] = {
317 8.27766102236537761883e+01, /* 0x4054B1B3, 0xFB5E1543 */
318 2.07781416421392987104e+03, /* 0x40A03BA0, 0xDA21C0CE */
319 1.88472887785718085070e+04, /* 0x40D267D2, 0x7B591E6D */
320 5.67511122894947329769e+04, /* 0x40EBB5E3, 0x97E02372 */
321 3.59767538425114471465e+04, /* 0x40E19118, 0x1F7A54A0 */
322 -5.35434275601944773371e+03, /* 0xC0B4EA57, 0xBEDBC609 */
323};
324
325static const double qR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
326 4.37741014089738620906e-09, /* 0x3E32CD03, 0x6ADECB82 */
327 7.32411180042911447163e-02, /* 0x3FB2BFEE, 0x0E8D0842 */
328 3.34423137516170720929e+00, /* 0x400AC0FC, 0x61149CF5 */
329 4.26218440745412650017e+01, /* 0x40454F98, 0x962DAEDD */
330 1.70808091340565596283e+02, /* 0x406559DB, 0xE25EFD1F */
331 1.66733948696651168575e+02, /* 0x4064D77C, 0x81FA21E0 */
332};
333static const double qS3[6] = {
334 4.87588729724587182091e+01, /* 0x40486122, 0xBFE343A6 */
335 7.09689221056606015736e+02, /* 0x40862D83, 0x86544EB3 */
336 3.70414822620111362994e+03, /* 0x40ACF04B, 0xE44DFC63 */
337 6.46042516752568917582e+03, /* 0x40B93C6C, 0xD7C76A28 */
338 2.51633368920368957333e+03, /* 0x40A3A8AA, 0xD94FB1C0 */
339 -1.49247451836156386662e+02, /* 0xC062A7EB, 0x201CF40F */
340};
341
342static const double qR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
343 1.50444444886983272379e-07, /* 0x3E84313B, 0x54F76BDB */
344 7.32234265963079278272e-02, /* 0x3FB2BEC5, 0x3E883E34 */
345 1.99819174093815998816e+00, /* 0x3FFFF897, 0xE727779C */
346 1.44956029347885735348e+01, /* 0x402CFDBF, 0xAAF96FE5 */
347 3.16662317504781540833e+01, /* 0x403FAA8E, 0x29FBDC4A */
348 1.62527075710929267416e+01, /* 0x403040B1, 0x71814BB4 */
349};
350static const double qS2[6] = {
351 3.03655848355219184498e+01, /* 0x403E5D96, 0xF7C07AED */
352 2.69348118608049844624e+02, /* 0x4070D591, 0xE4D14B40 */
353 8.44783757595320139444e+02, /* 0x408A6645, 0x22B3BF22 */
354 8.82935845112488550512e+02, /* 0x408B977C, 0x9C5CC214 */
355 2.12666388511798828631e+02, /* 0x406A9553, 0x0E001365 */
356 -5.31095493882666946917e+00, /* 0xC0153E6A, 0xF8B32931 */
357};
358
359static double qzero(double x)
360{
361 const double *p,*q;
362 double_t s,r,z;
363 uint32_t ix;
364
365 GET_HIGH_WORD(ix, x);
366 ix &= 0x7fffffff;
367 if (ix >= 0x40200000){p = qR8; q = qS8;}
368 else if (ix >= 0x40122E8B){p = qR5; q = qS5;}
369 else if (ix >= 0x4006DB6D){p = qR3; q = qS3;}
370 else /*ix >= 0x40000000*/ {p = qR2; q = qS2;}
371 z = 1.0/(x*x);
372 r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
373 s = 1.0+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*(q[4]+z*q[5])))));
374 return (-.125 + r/s)/x;
375}
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