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[minix.git] / lib / libm / src / e_j0.c
blob96ade957cc162859bdafab4071a39dbfb3de4173
1 /* @(#)e_j0.c 5.1 93/09/24 */
2 /*
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 #include <sys/cdefs.h>
14 #if defined(LIBM_SCCS) && !defined(lint)
15 __RCSID("$NetBSD: e_j0.c,v 1.12 2007/08/20 16:01:38 drochner Exp $");
16 #endif
17
18 /* __ieee754_j0(x), __ieee754_y0(x)
19 * Bessel function of the first and second kinds of order zero.
20 * Method -- j0(x):
21 * 1. For tiny x, we use j0(x) = 1 - x^2/4 + x^4/64 - ...
22 * 2. Reduce x to |x| since j0(x)=j0(-x), and
23 * for x in (0,2)
24 * j0(x) = 1-z/4+ z^2*R0/S0, where z = x*x;
25 * (precision: |j0-1+z/4-z^2R0/S0 |<2**-63.67 )
26 * for x in (2,inf)
27 * j0(x) = sqrt(2/(pi*x))*(p0(x)*cos(x0)-q0(x)*sin(x0))
28 * where x0 = x-pi/4. It is better to compute sin(x0),cos(x0)
29 * as follow:
30 * cos(x0) = cos(x)cos(pi/4)+sin(x)sin(pi/4)
31 * = 1/sqrt(2) * (cos(x) + sin(x))
32 * sin(x0) = sin(x)cos(pi/4)-cos(x)sin(pi/4)
33 * = 1/sqrt(2) * (sin(x) - cos(x))
34 * (To avoid cancellation, use
35 * sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
36 * to compute the worse one.)
37 *
38 * 3 Special cases
39 * j0(nan)= nan
40 * j0(0) = 1
41 * j0(inf) = 0
42 *
43 * Method -- y0(x):
44 * 1. For x<2.
45 * Since
46 * y0(x) = 2/pi*(j0(x)*(ln(x/2)+Euler) + x^2/4 - ...)
47 * therefore y0(x)-2/pi*j0(x)*ln(x) is an even function.
48 * We use the following function to approximate y0,
49 * y0(x) = U(z)/V(z) + (2/pi)*(j0(x)*ln(x)), z= x^2
50 * where
51 * U(z) = u00 + u01*z + ... + u06*z^6
52 * V(z) = 1 + v01*z + ... + v04*z^4
53 * with absolute approximation error bounded by 2**-72.
54 * Note: For tiny x, U/V = u0 and j0(x)~1, hence
55 * y0(tiny) = u0 + (2/pi)*ln(tiny), (choose tiny<2**-27)
56 * 2. For x>=2.
57 * y0(x) = sqrt(2/(pi*x))*(p0(x)*cos(x0)+q0(x)*sin(x0))
58 * where x0 = x-pi/4. It is better to compute sin(x0),cos(x0)
59 * by the method mentioned above.
60 * 3. Special cases: y0(0)=-inf, y0(x<0)=NaN, y0(inf)=0.
61 */
62
63 #include "namespace.h"
64 #include "math.h"
65 #include "math_private.h"
66
67 static double pzero(double), qzero(double);
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
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 - 0.25*x*x;
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
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 /* Y0(NaN) is NaN, y0(-inf) is Nan, y0(inf) is 0 */
156 if(ix>=0x7ff00000) return one/(x+x*x);
157 if((ix|lx)==0) return -one/zero;
158 if(hx<0) return zero/zero;
159 if(ix >= 0x40000000) { /* |x| >= 2.0 */
160 /* y0(x) = sqrt(2/(pi*x))*(p0(x)*sin(x0)+q0(x)*cos(x0))
161 * where x0 = x-pi/4
162 * Better formula:
163 * cos(x0) = cos(x)cos(pi/4)+sin(x)sin(pi/4)
164 * = 1/sqrt(2) * (sin(x) + cos(x))
165 * sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4)
166 * = 1/sqrt(2) * (sin(x) - cos(x))
167 * To avoid cancellation, use
168 * sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
169 * to compute the worse one.
170 */
171 s = sin(x);
172 c = cos(x);
173 ss = s-c;
174 cc = s+c;
175 /*
176 * j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x)
177 * y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x)
178 */
179 if(ix<0x7fe00000) { /* make sure x+x not overflow */
180 z = -cos(x+x);
181 if ((s*c)<zero) cc = z/ss;
182 else ss = z/cc;
183 }
184 if(ix>0x48000000) z = (invsqrtpi*ss)/sqrt(x);
185 else {
186 u = pzero(x); v = qzero(x);
187 z = invsqrtpi*(u*ss+v*cc)/sqrt(x);
188 }
189 return z;
190 }
191 if(ix<=0x3e400000) { /* x < 2**-27 */
192 return(u00 + tpi*__ieee754_log(x));
193 }
194 z = x*x;
195 u = u00+z*(u01+z*(u02+z*(u03+z*(u04+z*(u05+z*u06)))));
196 v = one+z*(v01+z*(v02+z*(v03+z*v04)));
197 return(u/v + tpi*(__ieee754_j0(x)*__ieee754_log(x)));
198 }
199
200 /* The asymptotic expansions of pzero is
201 * 1 - 9/128 s^2 + 11025/98304 s^4 - ..., where s = 1/x.
202 * For x >= 2, We approximate pzero by
203 * pzero(x) = 1 + (R/S)
204 * where R = pR0 + pR1*s^2 + pR2*s^4 + ... + pR5*s^10
205 * S = 1 + pS0*s^2 + ... + pS4*s^10
206 * and
207 * | pzero(x)-1-R/S | <= 2 ** ( -60.26)
208 */
209 static const double pR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
210 0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */
211 -7.03124999999900357484e-02, /* 0xBFB1FFFF, 0xFFFFFD32 */
212 -8.08167041275349795626e+00, /* 0xC02029D0, 0xB44FA779 */
213 -2.57063105679704847262e+02, /* 0xC0701102, 0x7B19E863 */
214 -2.48521641009428822144e+03, /* 0xC0A36A6E, 0xCD4DCAFC */
215 -5.25304380490729545272e+03, /* 0xC0B4850B, 0x36CC643D */
216 };
217 static const double pS8[5] = {
218 1.16534364619668181717e+02, /* 0x405D2233, 0x07A96751 */
219 3.83374475364121826715e+03, /* 0x40ADF37D, 0x50596938 */
220 4.05978572648472545552e+04, /* 0x40E3D2BB, 0x6EB6B05F */
221 1.16752972564375915681e+05, /* 0x40FC810F, 0x8F9FA9BD */
222 4.76277284146730962675e+04, /* 0x40E74177, 0x4F2C49DC */
223 };
224
225 static const double pR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
226 -1.14125464691894502584e-11, /* 0xBDA918B1, 0x47E495CC */
227 -7.03124940873599280078e-02, /* 0xBFB1FFFF, 0xE69AFBC6 */
228 -4.15961064470587782438e+00, /* 0xC010A370, 0xF90C6BBF */
229 -6.76747652265167261021e+01, /* 0xC050EB2F, 0x5A7D1783 */
230 -3.31231299649172967747e+02, /* 0xC074B3B3, 0x6742CC63 */
231 -3.46433388365604912451e+02, /* 0xC075A6EF, 0x28A38BD7 */
232 };
233 static const double pS5[5] = {
234 6.07539382692300335975e+01, /* 0x404E6081, 0x0C98C5DE */
235 1.05125230595704579173e+03, /* 0x40906D02, 0x5C7E2864 */
236 5.97897094333855784498e+03, /* 0x40B75AF8, 0x8FBE1D60 */
237 9.62544514357774460223e+03, /* 0x40C2CCB8, 0xFA76FA38 */
238 2.40605815922939109441e+03, /* 0x40A2CC1D, 0xC70BE864 */
239 };
240
241 static const double pR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
242 -2.54704601771951915620e-09, /* 0xBE25E103, 0x6FE1AA86 */
243 -7.03119616381481654654e-02, /* 0xBFB1FFF6, 0xF7C0E24B */
244 -2.40903221549529611423e+00, /* 0xC00345B2, 0xAEA48074 */
245 -2.19659774734883086467e+01, /* 0xC035F74A, 0x4CB94E14 */
246 -5.80791704701737572236e+01, /* 0xC04D0A22, 0x420A1A45 */
247 -3.14479470594888503854e+01, /* 0xC03F72AC, 0xA892D80F */
248 };
249 static const double pS3[5] = {
250 3.58560338055209726349e+01, /* 0x4041ED92, 0x84077DD3 */
251 3.61513983050303863820e+02, /* 0x40769839, 0x464A7C0E */
252 1.19360783792111533330e+03, /* 0x4092A66E, 0x6D1061D6 */
253 1.12799679856907414432e+03, /* 0x40919FFC, 0xB8C39B7E */
254 1.73580930813335754692e+02, /* 0x4065B296, 0xFC379081 */
255 };
256
257 static const double pR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
258 -8.87534333032526411254e-08, /* 0xBE77D316, 0xE927026D */
259 -7.03030995483624743247e-02, /* 0xBFB1FF62, 0x495E1E42 */
260 -1.45073846780952986357e+00, /* 0xBFF73639, 0x8A24A843 */
261 -7.63569613823527770791e+00, /* 0xC01E8AF3, 0xEDAFA7F3 */
262 -1.11931668860356747786e+01, /* 0xC02662E6, 0xC5246303 */
263 -3.23364579351335335033e+00, /* 0xC009DE81, 0xAF8FE70F */
264 };
265 static const double pS2[5] = {
266 2.22202997532088808441e+01, /* 0x40363865, 0x908B5959 */
267 1.36206794218215208048e+02, /* 0x4061069E, 0x0EE8878F */
268 2.70470278658083486789e+02, /* 0x4070E786, 0x42EA079B */
269 1.53875394208320329881e+02, /* 0x40633C03, 0x3AB6FAFF */
270 1.46576176948256193810e+01, /* 0x402D50B3, 0x44391809 */
271 };
272
273 static double
274 pzero(double x)
275 {
276 const double *p,*q;
277 double z,r,s;
278 int32_t ix;
279
280 p = q = 0;
281 GET_HIGH_WORD(ix,x);
282 ix &= 0x7fffffff;
283 if(ix>=0x40200000) {p = pR8; q= pS8;}
284 else if(ix>=0x40122E8B){p = pR5; q= pS5;}
285 else if(ix>=0x4006DB6D){p = pR3; q= pS3;}
286 else if(ix>=0x40000000){p = pR2; q= pS2;}
287 z = one/(x*x);
288 r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
289 s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*q[4]))));
290 return one+ r/s;
291 }
292
293
294 /* For x >= 8, the asymptotic expansions of qzero is
295 * -1/8 s + 75/1024 s^3 - ..., where s = 1/x.
296 * We approximate pzero by
297 * qzero(x) = s*(-1.25 + (R/S))
298 * where R = qR0 + qR1*s^2 + qR2*s^4 + ... + qR5*s^10
299 * S = 1 + qS0*s^2 + ... + qS5*s^12
300 * and
301 * | qzero(x)/s +1.25-R/S | <= 2 ** ( -61.22)
302 */
303 static const double qR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
304 0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */
305 7.32421874999935051953e-02, /* 0x3FB2BFFF, 0xFFFFFE2C */
306 1.17682064682252693899e+01, /* 0x40278952, 0x5BB334D6 */
307 5.57673380256401856059e+02, /* 0x40816D63, 0x15301825 */
308 8.85919720756468632317e+03, /* 0x40C14D99, 0x3E18F46D */
309 3.70146267776887834771e+04, /* 0x40E212D4, 0x0E901566 */
310 };
311 static const double qS8[6] = {
312 1.63776026895689824414e+02, /* 0x406478D5, 0x365B39BC */
313 8.09834494656449805916e+03, /* 0x40BFA258, 0x4E6B0563 */
314 1.42538291419120476348e+05, /* 0x41016652, 0x54D38C3F */
315 8.03309257119514397345e+05, /* 0x412883DA, 0x83A52B43 */
316 8.40501579819060512818e+05, /* 0x4129A66B, 0x28DE0B3D */
317 -3.43899293537866615225e+05, /* 0xC114FD6D, 0x2C9530C5 */
318 };
319
320 static const double qR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
321 1.84085963594515531381e-11, /* 0x3DB43D8F, 0x29CC8CD9 */
322 7.32421766612684765896e-02, /* 0x3FB2BFFF, 0xD172B04C */
323 5.83563508962056953777e+00, /* 0x401757B0, 0xB9953DD3 */
324 1.35111577286449829671e+02, /* 0x4060E392, 0x0A8788E9 */
325 1.02724376596164097464e+03, /* 0x40900CF9, 0x9DC8C481 */
326 1.98997785864605384631e+03, /* 0x409F17E9, 0x53C6E3A6 */
327 };
328 static const double qS5[6] = {
329 8.27766102236537761883e+01, /* 0x4054B1B3, 0xFB5E1543 */
330 2.07781416421392987104e+03, /* 0x40A03BA0, 0xDA21C0CE */
331 1.88472887785718085070e+04, /* 0x40D267D2, 0x7B591E6D */
332 5.67511122894947329769e+04, /* 0x40EBB5E3, 0x97E02372 */
333 3.59767538425114471465e+04, /* 0x40E19118, 0x1F7A54A0 */
334 -5.35434275601944773371e+03, /* 0xC0B4EA57, 0xBEDBC609 */
335 };
336
337 static const double qR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
338 4.37741014089738620906e-09, /* 0x3E32CD03, 0x6ADECB82 */
339 7.32411180042911447163e-02, /* 0x3FB2BFEE, 0x0E8D0842 */
340 3.34423137516170720929e+00, /* 0x400AC0FC, 0x61149CF5 */
341 4.26218440745412650017e+01, /* 0x40454F98, 0x962DAEDD */
342 1.70808091340565596283e+02, /* 0x406559DB, 0xE25EFD1F */
343 1.66733948696651168575e+02, /* 0x4064D77C, 0x81FA21E0 */
344 };
345 static const double qS3[6] = {
346 4.87588729724587182091e+01, /* 0x40486122, 0xBFE343A6 */
347 7.09689221056606015736e+02, /* 0x40862D83, 0x86544EB3 */
348 3.70414822620111362994e+03, /* 0x40ACF04B, 0xE44DFC63 */
349 6.46042516752568917582e+03, /* 0x40B93C6C, 0xD7C76A28 */
350 2.51633368920368957333e+03, /* 0x40A3A8AA, 0xD94FB1C0 */
351 -1.49247451836156386662e+02, /* 0xC062A7EB, 0x201CF40F */
352 };
353
354 static const double qR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
355 1.50444444886983272379e-07, /* 0x3E84313B, 0x54F76BDB */
356 7.32234265963079278272e-02, /* 0x3FB2BEC5, 0x3E883E34 */
357 1.99819174093815998816e+00, /* 0x3FFFF897, 0xE727779C */
358 1.44956029347885735348e+01, /* 0x402CFDBF, 0xAAF96FE5 */
359 3.16662317504781540833e+01, /* 0x403FAA8E, 0x29FBDC4A */
360 1.62527075710929267416e+01, /* 0x403040B1, 0x71814BB4 */
361 };
362 static const double qS2[6] = {
363 3.03655848355219184498e+01, /* 0x403E5D96, 0xF7C07AED */
364 2.69348118608049844624e+02, /* 0x4070D591, 0xE4D14B40 */
365 8.44783757595320139444e+02, /* 0x408A6645, 0x22B3BF22 */
366 8.82935845112488550512e+02, /* 0x408B977C, 0x9C5CC214 */
367 2.12666388511798828631e+02, /* 0x406A9553, 0x0E001365 */
368 -5.31095493882666946917e+00, /* 0xC0153E6A, 0xF8B32931 */
369 };
370
371 static double
372 qzero(double x)
373 {
374 const double *p,*q;
375 double s,r,z;
376 int32_t ix;
377
378 p = q = 0;
379 GET_HIGH_WORD(ix,x);
380 ix &= 0x7fffffff;
381 if(ix>=0x40200000) {p = qR8; q= qS8;}
382 else if(ix>=0x40122E8B){p = qR5; q= qS5;}
383 else if(ix>=0x4006DB6D){p = qR3; q= qS3;}
384 else if(ix>=0x40000000){p = qR2; q= qS2;}
385 z = one/(x*x);
386 r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
387 s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*(q[4]+z*q[5])))));
388 return (-.125 + r/s)/x;
389 }
MINIX 3 (repo.or.cz mirror)