1
2 /* @(#)e_j0.c 5.1 93/09/24 */
3 /*
4 * ====================================================
5 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
6 *
7 * Developed at SunPro, 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 /* j0(x), y0(x)
15 * Bessel function of the first and second kinds of order zero.
16 * Method -- j0(x):
17 * 1. For tiny x, we use j0(x) = 1 - x^2/4 + x^4/64 - ...
18 * 2. Reduce x to |x| since j0(x)=j0(-x), and
19 * for x in (0,2)
20 * j0(x) = 1-z/4+ z^2*R0/S0, where z = x*x;
21 * (precision: |j0-1+z/4-z^2R0/S0 |<2**-63.67 )
22 * for x in (2,inf)
23 * j0(x) = sqrt(2/(pi*x))*(p0(x)*cos(x0)-q0(x)*sin(x0))
24 * where x0 = x-pi/4. It is better to compute sin(x0),cos(x0)
25 * as follow:
26 * cos(x0) = cos(x)cos(pi/4)+sin(x)sin(pi/4)
27 * = 1/sqrt(2) * (cos(x) + sin(x))
28 * sin(x0) = sin(x)cos(pi/4)-cos(x)sin(pi/4)
29 * = 1/sqrt(2) * (sin(x) - cos(x))
30 * (To avoid cancellation, use
31 * sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
32 * to compute the worse one.)
33 *
34 * 3 Special cases
35 * j0(nan)= nan
36 * j0(0) = 1
37 * j0(inf) = 0
38 *
39 * Method -- y0(x):
40 * 1. For x<2.
41 * Since
42 * y0(x) = 2/pi*(j0(x)*(ln(x/2)+Euler) + x^2/4 - ...)
43 * therefore y0(x)-2/pi*j0(x)*ln(x) is an even function.
44 * We use the following function to approximate y0,
45 * y0(x) = U(z)/V(z) + (2/pi)*(j0(x)*ln(x)), z= x^2
46 * where
47 * U(z) = u00 + u01*z + ... + u06*z^6
48 * V(z) = 1 + v01*z + ... + v04*z^4
49 * with absolute approximation error bounded by 2**-72.
50 * Note: For tiny x, U/V = u0 and j0(x)~1, hence
51 * y0(tiny) = u0 + (2/pi)*ln(tiny), (choose tiny<2**-27)
52 * 2. For x>=2.
53 * y0(x) = sqrt(2/(pi*x))*(p0(x)*cos(x0)+q0(x)*sin(x0))
54 * where x0 = x-pi/4. It is better to compute sin(x0),cos(x0)
55 * by the method mentioned above.
56 * 3. Special cases: y0(0)=-inf, y0(x<0)=NaN, y0(inf)=0.
57 */
58
59 #include "fdlibm.h"
60
61 #ifdef _NEED_FLOAT64
62
63 static __float64 pzero(__float64);
64 static __float64 qzero(__float64);
65
66 static const __float64
67 huge = _F_64(1e300), one = _F_64(1.0),
68 invsqrtpi = _F_64(5.64189583547756279280e-01), /* 0x3FE20DD7, 0x50429B6D */
69 tpi = _F_64(6.36619772367581382433e-01), /* 0x3FE45F30, 0x6DC9C883 */
70 /* R0/S0 on [0, 2.00] */
71 R02 = _F_64(1.56249999999999947958e-02), /* 0x3F8FFFFF, 0xFFFFFFFD */
72 R03 = _F_64(-1.89979294238854721751e-04), /* 0xBF28E6A5, 0xB61AC6E9 */
73 R04 = _F_64(1.82954049532700665670e-06), /* 0x3EBEB1D1, 0x0C503919 */
74 R05 = _F_64(-4.61832688532103189199e-09), /* 0xBE33D5E7, 0x73D63FCE */
75 S01 = _F_64(1.56191029464890010492e-02), /* 0x3F8FFCE8, 0x82C8C2A4 */
76 S02 = _F_64(1.16926784663337450260e-04), /* 0x3F1EA6D2, 0xDD57DBF4 */
77 S03 = _F_64(5.13546550207318111446e-07), /* 0x3EA13B54, 0xCE84D5A9 */
78 S04 = _F_64(1.16614003333790000205e-09); /* 0x3E1408BC, 0xF4745D8F */
79
80 static const __float64 zero = _F_64(0.0);
81
82 __float64
j064(__float64 x)83 j064(__float64 x)
84 {
85 __float64 z, s, c, ss, cc, r, u, v;
86 __int32_t hx, ix;
87
88 if (isnan(x))
89 return x + x;
90
91 if (isinf(x))
92 return _F_64(0.0);
93
94 GET_HIGH_WORD(hx, x);
95 ix = hx & 0x7fffffff;
96 x = fabs64(x);
97 if (ix >= 0x40000000) { /* |x| >= 2.0 */
98 s = sin64(x);
99 c = cos64(x);
100 ss = s - c;
101 cc = s + c;
102 if (ix < 0x7fe00000) { /* make sure x+x not overflow */
103 z = -cos64(x + x);
104 if ((s * c) < zero)
105 cc = z / ss;
106 else
107 ss = z / cc;
108 }
109 /*
110 * j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x)
111 * y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x)
112 */
113 if (ix > 0x48000000)
114 z = (invsqrtpi * cc) / sqrt64(x);
115 else {
116 u = pzero(x);
117 v = qzero(x);
118 z = invsqrtpi * (u * cc - v * ss) / sqrt64(x);
119 }
120 return z;
121 }
122 if (ix < 0x3f200000) { /* |x| < 2**-13 */
123 if (huge + x > one) { /* raise inexact if x != 0 */
124 if (ix < 0x3e400000)
125 return one; /* |x|<2**-27 */
126 else
127 return one - _F_64(0.25) * x * x;
128 }
129 }
130 z = x * x;
131 r = z * (R02 + z * (R03 + z * (R04 + z * R05)));
132 s = one + z * (S01 + z * (S02 + z * (S03 + z * S04)));
133 if (ix < 0x3FF00000) { /* |x| < 1.00 */
134 return one + z * (_F_64(-0.25) + (r / s));
135 } else {
136 u = _F_64(0.5) * x;
137 return ((one + u) * (one - u) + z * (r / s));
138 }
139 }
140
141 _MATH_ALIAS_d_d(j0)
142
143 static const __float64 u00 =
144 _F_64(-7.38042951086872317523e-02), /* 0xBFB2E4D6, 0x99CBD01F */
145 u01 = _F_64(1.76666452509181115538e-01), /* 0x3FC69D01, 0x9DE9E3FC */
146 u02 = _F_64(-1.38185671945596898896e-02), /* 0xBF8C4CE8, 0xB16CFA97 */
147 u03 = _F_64(3.47453432093683650238e-04), /* 0x3F36C54D, 0x20B29B6B */
148 u04 = _F_64(-3.81407053724364161125e-06), /* 0xBECFFEA7, 0x73D25CAD */
149 u05 = _F_64(1.95590137035022920206e-08), /* 0x3E550057, 0x3B4EABD4 */
150 u06 = _F_64(-3.98205194132103398453e-11), /* 0xBDC5E43D, 0x693FB3C8 */
151 v01 = _F_64(1.27304834834123699328e-02), /* 0x3F8A1270, 0x91C9C71A */
152 v02 = _F_64(7.60068627350353253702e-05), /* 0x3F13ECBB, 0xF578C6C1 */
153 v03 = _F_64(2.59150851840457805467e-07), /* 0x3E91642D, 0x7FF202FD */
154 v04 = _F_64(4.41110311332675467403e-10); /* 0x3DFE5018, 0x3BD6D9EF */
155
156 __float64
y064(__float64 x)157 y064(__float64 x)
158 {
159 __float64 z, s, c, ss, cc, u, v;
160 __int32_t hx, ix, lx;
161
162 EXTRACT_WORDS(hx, lx, x);
163 ix = 0x7fffffff & hx;
164
165 if ((ix | lx) == 0)
166 return __math_divzero(1);
167
168 if (isnan(x))
169 return x + x;
170
171 if (hx < 0)
172 return __math_invalid(x);
173
174 if (ix >= 0x7ff00000)
175 return _F_64(0.0);
176
177 if (ix >= 0x40000000) { /* |x| >= 2.0 */
178 /* y0(x) = sqrt(2/(pi*x))*(p0(x)*sin(x0)+q0(x)*cos(x0))
179 * where x0 = x-pi/4
180 * Better formula:
181 * cos(x0) = cos(x)cos(pi/4)+sin(x)sin(pi/4)
182 * = 1/sqrt(2) * (sin(x) + cos(x))
183 * sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4)
184 * = 1/sqrt(2) * (sin(x) - cos(x))
185 * To avoid cancellation, use
186 * sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
187 * to compute the worse one.
188 */
189 s = sin64(x);
190 c = cos64(x);
191 ss = s - c;
192 cc = s + c;
193 /*
194 * j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x)
195 * y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x)
196 */
197 if (ix < 0x7fe00000) { /* make sure x+x not overflow */
198 z = -cos64(x + x);
199 if ((s * c) < zero)
200 cc = z / ss;
201 else
202 ss = z / cc;
203 }
204 if (ix > 0x48000000)
205 z = (invsqrtpi * ss) / sqrt64(x);
206 else {
207 u = pzero(x);
208 v = qzero(x);
209 z = invsqrtpi * (u * ss + v * cc) / sqrt64(x);
210 }
211 return z;
212 }
213 if (ix <= 0x3e400000) { /* x < 2**-27 */
214 return (u00 + tpi * log64(x));
215 }
216 z = x * x;
217 u = u00 +
218 z * (u01 + z * (u02 + z * (u03 + z * (u04 + z * (u05 + z * u06)))));
219 v = one + z * (v01 + z * (v02 + z * (v03 + z * v04)));
220 return (u / v + tpi * (j064(x) * log64(x)));
221 }
222
223 _MATH_ALIAS_d_d(y0)
224
225 /* The asymptotic expansions of pzero is
226 * 1 - 9/128 s^2 + 11025/98304 s^4 - ..., where s = 1/x.
227 * For x >= 2, We approximate pzero by
228 * pzero(x) = 1 + (R/S)
229 * where R = pR0 + pR1*s^2 + pR2*s^4 + ... + pR5*s^10
230 * S = 1 + pS0*s^2 + ... + pS4*s^10
231 * and
232 * | pzero(x)-1-R/S | <= 2 ** ( -60.26)
233 */
234 static const __float64 pR8[6] = {
235 /* for x in [inf, 8]=1/[0,0.125] */
236 _F_64(0.00000000000000000000e+00), /* 0x00000000, 0x00000000 */
237 _F_64(-7.03124999999900357484e-02), /* 0xBFB1FFFF, 0xFFFFFD32 */
238 _F_64(-8.08167041275349795626e+00), /* 0xC02029D0, 0xB44FA779 */
239 _F_64(-2.57063105679704847262e+02), /* 0xC0701102, 0x7B19E863 */
240 _F_64(-2.48521641009428822144e+03), /* 0xC0A36A6E, 0xCD4DCAFC */
241 _F_64(-5.25304380490729545272e+03), /* 0xC0B4850B, 0x36CC643D */
242 };
243 static const __float64 pS8[5] = {
244 _F_64(1.16534364619668181717e+02), /* 0x405D2233, 0x07A96751 */
245 _F_64(3.83374475364121826715e+03), /* 0x40ADF37D, 0x50596938 */
246 _F_64(4.05978572648472545552e+04), /* 0x40E3D2BB, 0x6EB6B05F */
247 _F_64(1.16752972564375915681e+05), /* 0x40FC810F, 0x8F9FA9BD */
248 _F_64(4.76277284146730962675e+04), /* 0x40E74177, 0x4F2C49DC */
249 };
250
251 static const __float64 pR5[6] = {
252 /* for x in [8,4.5454]=1/[0.125,0.22001] */
253 _F_64(-1.14125464691894502584e-11), /* 0xBDA918B1, 0x47E495CC */
254 _F_64(-7.03124940873599280078e-02), /* 0xBFB1FFFF, 0xE69AFBC6 */
255 _F_64(-4.15961064470587782438e+00), /* 0xC010A370, 0xF90C6BBF */
256 _F_64(-6.76747652265167261021e+01), /* 0xC050EB2F, 0x5A7D1783 */
257 _F_64(-3.31231299649172967747e+02), /* 0xC074B3B3, 0x6742CC63 */
258 _F_64(-3.46433388365604912451e+02), /* 0xC075A6EF, 0x28A38BD7 */
259 };
260 static const __float64 pS5[5] = {
261 _F_64(6.07539382692300335975e+01), /* 0x404E6081, 0x0C98C5DE */
262 _F_64(1.05125230595704579173e+03), /* 0x40906D02, 0x5C7E2864 */
263 _F_64(5.97897094333855784498e+03), /* 0x40B75AF8, 0x8FBE1D60 */
264 _F_64(9.62544514357774460223e+03), /* 0x40C2CCB8, 0xFA76FA38 */
265 _F_64(2.40605815922939109441e+03), /* 0x40A2CC1D, 0xC70BE864 */
266 };
267
268 static const __float64 pR3[6] = {
269 /* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
270 _F_64(-2.54704601771951915620e-09), /* 0xBE25E103, 0x6FE1AA86 */
271 _F_64(-7.03119616381481654654e-02), /* 0xBFB1FFF6, 0xF7C0E24B */
272 _F_64(-2.40903221549529611423e+00), /* 0xC00345B2, 0xAEA48074 */
273 _F_64(-2.19659774734883086467e+01), /* 0xC035F74A, 0x4CB94E14 */
274 _F_64(-5.80791704701737572236e+01), /* 0xC04D0A22, 0x420A1A45 */
275 _F_64(-3.14479470594888503854e+01), /* 0xC03F72AC, 0xA892D80F */
276 };
277 static const __float64 pS3[5] = {
278 _F_64(3.58560338055209726349e+01), /* 0x4041ED92, 0x84077DD3 */
279 _F_64(3.61513983050303863820e+02), /* 0x40769839, 0x464A7C0E */
280 _F_64(1.19360783792111533330e+03), /* 0x4092A66E, 0x6D1061D6 */
281 _F_64(1.12799679856907414432e+03), /* 0x40919FFC, 0xB8C39B7E */
282 _F_64(1.73580930813335754692e+02), /* 0x4065B296, 0xFC379081 */
283 };
284
285 static const __float64 pR2[6] = {
286 /* for x in [2.8570,2]=1/[0.3499,0.5] */
287 _F_64(-8.87534333032526411254e-08), /* 0xBE77D316, 0xE927026D */
288 _F_64(-7.03030995483624743247e-02), /* 0xBFB1FF62, 0x495E1E42 */
289 _F_64(-1.45073846780952986357e+00), /* 0xBFF73639, 0x8A24A843 */
290 _F_64(-7.63569613823527770791e+00), /* 0xC01E8AF3, 0xEDAFA7F3 */
291 _F_64(-1.11931668860356747786e+01), /* 0xC02662E6, 0xC5246303 */
292 _F_64(-3.23364579351335335033e+00), /* 0xC009DE81, 0xAF8FE70F */
293 };
294 static const __float64 pS2[5] = {
295 _F_64(2.22202997532088808441e+01), /* 0x40363865, 0x908B5959 */
296 _F_64(1.36206794218215208048e+02), /* 0x4061069E, 0x0EE8878F */
297 _F_64(2.70470278658083486789e+02), /* 0x4070E786, 0x42EA079B */
298 _F_64(1.53875394208320329881e+02), /* 0x40633C03, 0x3AB6FAFF */
299 _F_64(1.46576176948256193810e+01), /* 0x402D50B3, 0x44391809 */
300 };
301
302 static __float64
pzero(__float64 x)303 pzero(__float64 x)
304 {
305 const __float64 *p, *q;
306 __float64 z, r, s;
307 __int32_t ix;
308 GET_HIGH_WORD(ix, x);
309 ix &= 0x7fffffff;
310 if (ix >= 0x41b00000) {
311 return one;
312 } else if (ix >= 0x40200000) {
313 p = pR8;
314 q = pS8;
315 } else if (ix >= 0x40122E8B) {
316 p = pR5;
317 q = pS5;
318 } else if (ix >= 0x4006DB6D) {
319 p = pR3;
320 q = pS3;
321 } else {
322 p = pR2;
323 q = pS2;
324 }
325 z = one / (x * x);
326 r = p[0] + z * (p[1] + z * (p[2] + z * (p[3] + z * (p[4] + z * p[5]))));
327 s = one + z * (q[0] + z * (q[1] + z * (q[2] + z * (q[3] + z * q[4]))));
328 return one + r / s;
329 }
330
331 /* For x >= 8, the asymptotic expansions of qzero is
332 * -1/8 s + 75/1024 s^3 - ..., where s = 1/x.
333 * We approximate qzero by
334 * qzero(x) = s*(-1.25 + (R/S))
335 * where R = qR0 + qR1*s^2 + qR2*s^4 + ... + qR5*s^10
336 * S = 1 + qS0*s^2 + ... + qS5*s^12
337 * and
338 * | qzero(x)/s +1.25-R/S | <= 2 ** ( -61.22)
339 */
340 static const __float64 qR8[6] = {
341 /* for x in [inf, 8]=1/[0,0.125] */
342 _F_64(0.00000000000000000000e+00), /* 0x00000000, 0x00000000 */
343 _F_64(7.32421874999935051953e-02), /* 0x3FB2BFFF, 0xFFFFFE2C */
344 _F_64(1.17682064682252693899e+01), /* 0x40278952, 0x5BB334D6 */
345 _F_64(5.57673380256401856059e+02), /* 0x40816D63, 0x15301825 */
346 _F_64(8.85919720756468632317e+03), /* 0x40C14D99, 0x3E18F46D */
347 _F_64(3.70146267776887834771e+04), /* 0x40E212D4, 0x0E901566 */
348 };
349 static const __float64 qS8[6] = {
350 _F_64(1.63776026895689824414e+02), /* 0x406478D5, 0x365B39BC */
351 _F_64(8.09834494656449805916e+03), /* 0x40BFA258, 0x4E6B0563 */
352 _F_64(1.42538291419120476348e+05), /* 0x41016652, 0x54D38C3F */
353 _F_64(8.03309257119514397345e+05), /* 0x412883DA, 0x83A52B43 */
354 _F_64(8.40501579819060512818e+05), /* 0x4129A66B, 0x28DE0B3D */
355 _F_64(-3.43899293537866615225e+05), /* 0xC114FD6D, 0x2C9530C5 */
356 };
357
358 static const __float64 qR5[6] = {
359 /* for x in [8,4.5454]=1/[0.125,0.22001] */
360 _F_64(1.84085963594515531381e-11), /* 0x3DB43D8F, 0x29CC8CD9 */
361 _F_64(7.32421766612684765896e-02), /* 0x3FB2BFFF, 0xD172B04C */
362 _F_64(5.83563508962056953777e+00), /* 0x401757B0, 0xB9953DD3 */
363 _F_64(1.35111577286449829671e+02), /* 0x4060E392, 0x0A8788E9 */
364 _F_64(1.02724376596164097464e+03), /* 0x40900CF9, 0x9DC8C481 */
365 _F_64(1.98997785864605384631e+03), /* 0x409F17E9, 0x53C6E3A6 */
366 };
367 static const __float64 qS5[6] = {
368 _F_64(8.27766102236537761883e+01), /* 0x4054B1B3, 0xFB5E1543 */
369 _F_64(2.07781416421392987104e+03), /* 0x40A03BA0, 0xDA21C0CE */
370 _F_64(1.88472887785718085070e+04), /* 0x40D267D2, 0x7B591E6D */
371 _F_64(5.67511122894947329769e+04), /* 0x40EBB5E3, 0x97E02372 */
372 _F_64(3.59767538425114471465e+04), /* 0x40E19118, 0x1F7A54A0 */
373 _F_64(-5.35434275601944773371e+03), /* 0xC0B4EA57, 0xBEDBC609 */
374 };
375
376 static const __float64 qR3[6] = {
377 /* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
378 _F_64(4.37741014089738620906e-09), /* 0x3E32CD03, 0x6ADECB82 */
379 _F_64(7.32411180042911447163e-02), /* 0x3FB2BFEE, 0x0E8D0842 */
380 _F_64(3.34423137516170720929e+00), /* 0x400AC0FC, 0x61149CF5 */
381 _F_64(4.26218440745412650017e+01), /* 0x40454F98, 0x962DAEDD */
382 _F_64(1.70808091340565596283e+02), /* 0x406559DB, 0xE25EFD1F */
383 _F_64(1.66733948696651168575e+02), /* 0x4064D77C, 0x81FA21E0 */
384 };
385 static const __float64 qS3[6] = {
386 _F_64(4.87588729724587182091e+01), /* 0x40486122, 0xBFE343A6 */
387 _F_64(7.09689221056606015736e+02), /* 0x40862D83, 0x86544EB3 */
388 _F_64(3.70414822620111362994e+03), /* 0x40ACF04B, 0xE44DFC63 */
389 _F_64(6.46042516752568917582e+03), /* 0x40B93C6C, 0xD7C76A28 */
390 _F_64(2.51633368920368957333e+03), /* 0x40A3A8AA, 0xD94FB1C0 */
391 _F_64(-1.49247451836156386662e+02), /* 0xC062A7EB, 0x201CF40F */
392 };
393
394 static const __float64 qR2[6] = {
395 /* for x in [2.8570,2]=1/[0.3499,0.5] */
396 _F_64(1.50444444886983272379e-07), /* 0x3E84313B, 0x54F76BDB */
397 _F_64(7.32234265963079278272e-02), /* 0x3FB2BEC5, 0x3E883E34 */
398 _F_64(1.99819174093815998816e+00), /* 0x3FFFF897, 0xE727779C */
399 _F_64(1.44956029347885735348e+01), /* 0x402CFDBF, 0xAAF96FE5 */
400 _F_64(3.16662317504781540833e+01), /* 0x403FAA8E, 0x29FBDC4A */
401 _F_64(1.62527075710929267416e+01), /* 0x403040B1, 0x71814BB4 */
402 };
403 static const __float64 qS2[6] = {
404 _F_64(3.03655848355219184498e+01), /* 0x403E5D96, 0xF7C07AED */
405 _F_64(2.69348118608049844624e+02), /* 0x4070D591, 0xE4D14B40 */
406 _F_64(8.44783757595320139444e+02), /* 0x408A6645, 0x22B3BF22 */
407 _F_64(8.82935845112488550512e+02), /* 0x408B977C, 0x9C5CC214 */
408 _F_64(2.12666388511798828631e+02), /* 0x406A9553, 0x0E001365 */
409 _F_64(-5.31095493882666946917e+00), /* 0xC0153E6A, 0xF8B32931 */
410 };
411
412 static __float64
qzero(__float64 x)413 qzero(__float64 x)
414 {
415 const __float64 *p, *q;
416 __float64 s, r, z;
417 __int32_t ix;
418 GET_HIGH_WORD(ix, x);
419 ix &= 0x7fffffff;
420 if (ix >= 0x41b00000) {
421 return _F_64(-.125) / x;
422 } else if (ix >= 0x40200000) {
423 p = qR8;
424 q = qS8;
425 } else if (ix >= 0x40122E8B) {
426 p = qR5;
427 q = qS5;
428 } else if (ix >= 0x4006DB6D) {
429 p = qR3;
430 q = qS3;
431 } else {
432 p = qR2;
433 q = qS2;
434 }
435 z = one / (x * x);
436 r = p[0] + z * (p[1] + z * (p[2] + z * (p[3] + z * (p[4] + z * p[5]))));
437 s = one +
438 z * (q[0] +
439 z * (q[1] + z * (q[2] + z * (q[3] + z * (q[4] + z * q[5])))));
440 return (_F_64(-.125) + r / s) / x;
441 }
442
443 #endif /* _NEED_FLOAT64 */
444
