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 /* Modified by Naohiko Shimizu/Tokai University, Japan 1997/08/26,
13 for performance improvement on pipelined processors.
14 */
15
16 /* __ieee754_j0(x), __ieee754_y0(x)
17 * Bessel function of the first and second kinds of order zero.
18 * Method -- j0(x):
19 * 1. For tiny x, we use j0(x) = 1 - x^2/4 + x^4/64 - ...
20 * 2. Reduce x to |x| since j0(x)=j0(-x), and
21 * for x in (0,2)
22 * j0(x) = 1-z/4+ z^2*R0/S0, where z = x*x;
23 * (precision: |j0-1+z/4-z^2R0/S0 |<2**-63.67 )
24 * for x in (2,inf)
25 * j0(x) = sqrt(2/(pi*x))*(p0(x)*cos(x0)-q0(x)*sin(x0))
26 * where x0 = x-pi/4. It is better to compute sin(x0),cos(x0)
27 * as follow:
28 * cos(x0) = cos(x)cos(pi/4)+sin(x)sin(pi/4)
29 * = 1/sqrt(2) * (cos(x) + sin(x))
30 * sin(x0) = sin(x)cos(pi/4)-cos(x)sin(pi/4)
31 * = 1/sqrt(2) * (sin(x) - cos(x))
32 * (To avoid cancellation, use
33 * sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
34 * to compute the worse one.)
35 *
36 * 3 Special cases
37 * j0(nan)= nan
38 * j0(0) = 1
39 * j0(inf) = 0
40 *
41 * Method -- y0(x):
42 * 1. For x<2.
43 * Since
44 * y0(x) = 2/pi*(j0(x)*(ln(x/2)+Euler) + x^2/4 - ...)
45 * therefore y0(x)-2/pi*j0(x)*ln(x) is an even function.
46 * We use the following function to approximate y0,
47 * y0(x) = U(z)/V(z) + (2/pi)*(j0(x)*ln(x)), z= x^2
48 * where
49 * U(z) = u00 + u01*z + ... + u06*z^6
50 * V(z) = 1 + v01*z + ... + v04*z^4
51 * with absolute approximation error bounded by 2**-72.
52 * Note: For tiny x, U/V = u0 and j0(x)~1, hence
53 * y0(tiny) = u0 + (2/pi)*ln(tiny), (choose tiny<2**-27)
54 * 2. For x>=2.
55 * y0(x) = sqrt(2/(pi*x))*(p0(x)*cos(x0)+q0(x)*sin(x0))
56 * where x0 = x-pi/4. It is better to compute sin(x0),cos(x0)
57 * by the method mentioned above.
58 * 3. Special cases: y0(0)=-inf, y0(x<0)=NaN, y0(inf)=0.
59 */
60
61 #include <math.h>
62 #include <math-barriers.h>
63 #include <math_private.h>
64 #include <libm-alias-finite.h>
65
66 static double pzero (double), qzero (double);
67
68 static const double
69 huge = 1e300,
70 one = 1.0,
71 invsqrtpi = 5.64189583547756279280e-01, /* 0x3FE20DD7, 0x50429B6D */
72 tpi = 6.36619772367581382433e-01, /* 0x3FE45F30, 0x6DC9C883 */
73 /* R0/S0 on [0, 2.00] */
74 R[] = { 0.0, 0.0, 1.56249999999999947958e-02, /* 0x3F8FFFFF, 0xFFFFFFFD */
75 -1.89979294238854721751e-04, /* 0xBF28E6A5, 0xB61AC6E9 */
76 1.82954049532700665670e-06, /* 0x3EBEB1D1, 0x0C503919 */
77 -4.61832688532103189199e-09 }, /* 0xBE33D5E7, 0x73D63FCE */
78 S[] = { 0.0, 1.56191029464890010492e-02, /* 0x3F8FFCE8, 0x82C8C2A4 */
79 1.16926784663337450260e-04, /* 0x3F1EA6D2, 0xDD57DBF4 */
80 5.13546550207318111446e-07, /* 0x3EA13B54, 0xCE84D5A9 */
81 1.16614003333790000205e-09 }; /* 0x3E1408BC, 0xF4745D8F */
82
83 static const double zero = 0.0;
84
85 double
86 __ieee754_j0 (double x)
87 {
88 double z, s, c, ss, cc, r, u, v, r1, r2, s1, s2, z2, z4;
89 int32_t hx, ix;
90
91 GET_HIGH_WORD (hx, x);
92 ix = hx & 0x7fffffff;
93 if (ix >= 0x7ff00000)
94 return one / (x * x);
95 x = fabs (x);
96 if (ix >= 0x40000000) /* |x| >= 2.0 */
97 {
98 __sincos (x, &s, &c);
99 ss = s - c;
100 cc = s + c;
101 if (ix < 0x7fe00000) /* make sure x+x not overflow */
102 {
103 z = -__cos (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) / sqrt (x);
115 else
116 {
117 u = pzero (x); v = qzero (x);
118 z = invsqrtpi * (u * cc - v * ss) / sqrt (x);
119 }
120 return z;
121 }
122 if (ix < 0x3f200000) /* |x| < 2**-13 */
123 {
124 math_force_eval (huge + x); /* raise inexact if x != 0 */
125 if (ix < 0x3e400000)
126 return one; /* |x|<2**-27 */
127 else
128 return one - 0.25 * x * x;
129 }
130 z = x * x;
131 r1 = z * R[2]; z2 = z * z;
132 r2 = R[3] + z * R[4]; z4 = z2 * z2;
133 r = r1 + z2 * r2 + z4 * R[5];
134 s1 = one + z * S[1];
135 s2 = S[2] + z * S[3];
136 s = s1 + z2 * s2 + z4 * S[4];
137 if (ix < 0x3FF00000) /* |x| < 1.00 */
138 {
139 return one + z * (-0.25 + (r / s));
140 }
141 else
142 {
143 u = 0.5 * x;
144 return ((one + u) * (one - u) + z * (r / s));
145 }
146 }
147 libm_alias_finite (__ieee754_j0, __j0)
148
149 static const double
150 U[] = { -7.38042951086872317523e-02, /* 0xBFB2E4D6, 0x99CBD01F */
151 1.76666452509181115538e-01, /* 0x3FC69D01, 0x9DE9E3FC */
152 -1.38185671945596898896e-02, /* 0xBF8C4CE8, 0xB16CFA97 */
153 3.47453432093683650238e-04, /* 0x3F36C54D, 0x20B29B6B */
154 -3.81407053724364161125e-06, /* 0xBECFFEA7, 0x73D25CAD */
155 1.95590137035022920206e-08, /* 0x3E550057, 0x3B4EABD4 */
156 -3.98205194132103398453e-11 }, /* 0xBDC5E43D, 0x693FB3C8 */
157 V[] = { 1.27304834834123699328e-02, /* 0x3F8A1270, 0x91C9C71A */
158 7.60068627350353253702e-05, /* 0x3F13ECBB, 0xF578C6C1 */
159 2.59150851840457805467e-07, /* 0x3E91642D, 0x7FF202FD */
160 4.41110311332675467403e-10 }; /* 0x3DFE5018, 0x3BD6D9EF */
161
162 double
163 __ieee754_y0 (double x)
164 {
165 double z, s, c, ss, cc, u, v, z2, z4, z6, u1, u2, u3, v1, v2;
166 int32_t hx, ix, lx;
167
168 EXTRACT_WORDS (hx, lx, x);
169 ix = 0x7fffffff & hx;
170 /* Y0(NaN) is NaN, y0(-inf) is Nan, y0(inf) is 0, y0(0) is -inf. */
171 if (ix >= 0x7ff00000)
172 return one / (x + x * x);
173 if ((ix | lx) == 0)
174 return -1 / zero; /* -inf and divide by zero exception. */
175 if (hx < 0)
176 return zero / (zero * x);
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 __sincos (x, &s, &c);
190 ss = s - c;
191 cc = s + c;
192 /*
193 * j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x)
194 * y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x)
195 */
196 if (ix < 0x7fe00000) /* make sure x+x not overflow */
197 {
198 z = -__cos (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) / sqrt (x);
206 else
207 {
208 u = pzero (x); v = qzero (x);
209 z = invsqrtpi * (u * ss + v * cc) / sqrt (x);
210 }
211 return z;
212 }
213 if (ix <= 0x3e400000) /* x < 2**-27 */
214 {
215 return (U[0] + tpi * __ieee754_log (x));
216 }
217 z = x * x;
218 u1 = U[0] + z * U[1]; z2 = z * z;
219 u2 = U[2] + z * U[3]; z4 = z2 * z2;
220 u3 = U[4] + z * U[5]; z6 = z4 * z2;
221 u = u1 + z2 * u2 + z4 * u3 + z6 * U[6];
222 v1 = one + z * V[0];
223 v2 = V[1] + z * V[2];
224 v = v1 + z2 * v2 + z4 * V[3];
225 return (u / v + tpi * (__ieee754_j0 (x) * __ieee754_log (x)));
226 }
227 libm_alias_finite (__ieee754_y0, __y0)
228
229 /* The asymptotic expansions of pzero is
230 * 1 - 9/128 s^2 + 11025/98304 s^4 - ..., where s = 1/x.
231 * For x >= 2, We approximate pzero by
232 * pzero(x) = 1 + (R/S)
233 * where R = pR0 + pR1*s^2 + pR2*s^4 + ... + pR5*s^10
234 * S = 1 + pS0*s^2 + ... + pS4*s^10
235 * and
236 * | pzero(x)-1-R/S | <= 2 ** ( -60.26)
237 */
238 static const double pR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
239 0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */
240 -7.03124999999900357484e-02, /* 0xBFB1FFFF, 0xFFFFFD32 */
241 -8.08167041275349795626e+00, /* 0xC02029D0, 0xB44FA779 */
242 -2.57063105679704847262e+02, /* 0xC0701102, 0x7B19E863 */
243 -2.48521641009428822144e+03, /* 0xC0A36A6E, 0xCD4DCAFC */
244 -5.25304380490729545272e+03, /* 0xC0B4850B, 0x36CC643D */
245 };
246 static const double pS8[5] = {
247 1.16534364619668181717e+02, /* 0x405D2233, 0x07A96751 */
248 3.83374475364121826715e+03, /* 0x40ADF37D, 0x50596938 */
249 4.05978572648472545552e+04, /* 0x40E3D2BB, 0x6EB6B05F */
250 1.16752972564375915681e+05, /* 0x40FC810F, 0x8F9FA9BD */
251 4.76277284146730962675e+04, /* 0x40E74177, 0x4F2C49DC */
252 };
253
254 static const double pR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
255 -1.14125464691894502584e-11, /* 0xBDA918B1, 0x47E495CC */
256 -7.03124940873599280078e-02, /* 0xBFB1FFFF, 0xE69AFBC6 */
257 -4.15961064470587782438e+00, /* 0xC010A370, 0xF90C6BBF */
258 -6.76747652265167261021e+01, /* 0xC050EB2F, 0x5A7D1783 */
259 -3.31231299649172967747e+02, /* 0xC074B3B3, 0x6742CC63 */
260 -3.46433388365604912451e+02, /* 0xC075A6EF, 0x28A38BD7 */
261 };
262 static const double pS5[5] = {
263 6.07539382692300335975e+01, /* 0x404E6081, 0x0C98C5DE */
264 1.05125230595704579173e+03, /* 0x40906D02, 0x5C7E2864 */
265 5.97897094333855784498e+03, /* 0x40B75AF8, 0x8FBE1D60 */
266 9.62544514357774460223e+03, /* 0x40C2CCB8, 0xFA76FA38 */
267 2.40605815922939109441e+03, /* 0x40A2CC1D, 0xC70BE864 */
268 };
269
270 static const double pR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
271 -2.54704601771951915620e-09, /* 0xBE25E103, 0x6FE1AA86 */
272 -7.03119616381481654654e-02, /* 0xBFB1FFF6, 0xF7C0E24B */
273 -2.40903221549529611423e+00, /* 0xC00345B2, 0xAEA48074 */
274 -2.19659774734883086467e+01, /* 0xC035F74A, 0x4CB94E14 */
275 -5.80791704701737572236e+01, /* 0xC04D0A22, 0x420A1A45 */
276 -3.14479470594888503854e+01, /* 0xC03F72AC, 0xA892D80F */
277 };
278 static const double pS3[5] = {
279 3.58560338055209726349e+01, /* 0x4041ED92, 0x84077DD3 */
280 3.61513983050303863820e+02, /* 0x40769839, 0x464A7C0E */
281 1.19360783792111533330e+03, /* 0x4092A66E, 0x6D1061D6 */
282 1.12799679856907414432e+03, /* 0x40919FFC, 0xB8C39B7E */
283 1.73580930813335754692e+02, /* 0x4065B296, 0xFC379081 */
284 };
285
286 static const double pR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
287 -8.87534333032526411254e-08, /* 0xBE77D316, 0xE927026D */
288 -7.03030995483624743247e-02, /* 0xBFB1FF62, 0x495E1E42 */
289 -1.45073846780952986357e+00, /* 0xBFF73639, 0x8A24A843 */
290 -7.63569613823527770791e+00, /* 0xC01E8AF3, 0xEDAFA7F3 */
291 -1.11931668860356747786e+01, /* 0xC02662E6, 0xC5246303 */
292 -3.23364579351335335033e+00, /* 0xC009DE81, 0xAF8FE70F */
293 };
294 static const double pS2[5] = {
295 2.22202997532088808441e+01, /* 0x40363865, 0x908B5959 */
296 1.36206794218215208048e+02, /* 0x4061069E, 0x0EE8878F */
297 2.70470278658083486789e+02, /* 0x4070E786, 0x42EA079B */
298 1.53875394208320329881e+02, /* 0x40633C03, 0x3AB6FAFF */
299 1.46576176948256193810e+01, /* 0x402D50B3, 0x44391809 */
300 };
301
302 static double
303 pzero (double x)
304 {
305 const double *p, *q;
306 double z, r, s, z2, z4, r1, r2, r3, s1, s2, s3;
307 int32_t ix;
308 GET_HIGH_WORD (ix, x);
309 ix &= 0x7fffffff;
310 /* ix >= 0x40000000 for all calls to this function. */
311 if (ix >= 0x41b00000)
312 {
313 return one;
314 }
315 else if (ix >= 0x40200000)
316 {
317 p = pR8; q = pS8;
318 }
319 else if (ix >= 0x40122E8B)
320 {
321 p = pR5; q = pS5;
322 }
323 else if (ix >= 0x4006DB6D)
324 {
325 p = pR3; q = pS3;
326 }
327 else
328 {
329 p = pR2; q = pS2;
330 }
331 z = one / (x * x);
332 r1 = p[0] + z * p[1]; z2 = z * z;
333 r2 = p[2] + z * p[3]; z4 = z2 * z2;
334 r3 = p[4] + z * p[5];
335 r = r1 + z2 * r2 + z4 * r3;
336 s1 = one + z * q[0];
337 s2 = q[1] + z * q[2];
338 s3 = q[3] + z * q[4];
339 s = s1 + z2 * s2 + z4 * s3;
340 return one + r / s;
341 }
342
343
344 /* For x >= 8, the asymptotic expansions of qzero is
345 * -1/8 s + 75/1024 s^3 - ..., where s = 1/x.
346 * We approximate pzero by
347 * qzero(x) = s*(-1.25 + (R/S))
348 * where R = qR0 + qR1*s^2 + qR2*s^4 + ... + qR5*s^10
349 * S = 1 + qS0*s^2 + ... + qS5*s^12
350 * and
351 * | qzero(x)/s +1.25-R/S | <= 2 ** ( -61.22)
352 */
353 static const double qR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
354 0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */
355 7.32421874999935051953e-02, /* 0x3FB2BFFF, 0xFFFFFE2C */
356 1.17682064682252693899e+01, /* 0x40278952, 0x5BB334D6 */
357 5.57673380256401856059e+02, /* 0x40816D63, 0x15301825 */
358 8.85919720756468632317e+03, /* 0x40C14D99, 0x3E18F46D */
359 3.70146267776887834771e+04, /* 0x40E212D4, 0x0E901566 */
360 };
361 static const double qS8[6] = {
362 1.63776026895689824414e+02, /* 0x406478D5, 0x365B39BC */
363 8.09834494656449805916e+03, /* 0x40BFA258, 0x4E6B0563 */
364 1.42538291419120476348e+05, /* 0x41016652, 0x54D38C3F */
365 8.03309257119514397345e+05, /* 0x412883DA, 0x83A52B43 */
366 8.40501579819060512818e+05, /* 0x4129A66B, 0x28DE0B3D */
367 -3.43899293537866615225e+05, /* 0xC114FD6D, 0x2C9530C5 */
368 };
369
370 static const double qR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
371 1.84085963594515531381e-11, /* 0x3DB43D8F, 0x29CC8CD9 */
372 7.32421766612684765896e-02, /* 0x3FB2BFFF, 0xD172B04C */
373 5.83563508962056953777e+00, /* 0x401757B0, 0xB9953DD3 */
374 1.35111577286449829671e+02, /* 0x4060E392, 0x0A8788E9 */
375 1.02724376596164097464e+03, /* 0x40900CF9, 0x9DC8C481 */
376 1.98997785864605384631e+03, /* 0x409F17E9, 0x53C6E3A6 */
377 };
378 static const double qS5[6] = {
379 8.27766102236537761883e+01, /* 0x4054B1B3, 0xFB5E1543 */
380 2.07781416421392987104e+03, /* 0x40A03BA0, 0xDA21C0CE */
381 1.88472887785718085070e+04, /* 0x40D267D2, 0x7B591E6D */
382 5.67511122894947329769e+04, /* 0x40EBB5E3, 0x97E02372 */
383 3.59767538425114471465e+04, /* 0x40E19118, 0x1F7A54A0 */
384 -5.35434275601944773371e+03, /* 0xC0B4EA57, 0xBEDBC609 */
385 };
386
387 static const double qR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
388 4.37741014089738620906e-09, /* 0x3E32CD03, 0x6ADECB82 */
389 7.32411180042911447163e-02, /* 0x3FB2BFEE, 0x0E8D0842 */
390 3.34423137516170720929e+00, /* 0x400AC0FC, 0x61149CF5 */
391 4.26218440745412650017e+01, /* 0x40454F98, 0x962DAEDD */
392 1.70808091340565596283e+02, /* 0x406559DB, 0xE25EFD1F */
393 1.66733948696651168575e+02, /* 0x4064D77C, 0x81FA21E0 */
394 };
395 static const double qS3[6] = {
396 4.87588729724587182091e+01, /* 0x40486122, 0xBFE343A6 */
397 7.09689221056606015736e+02, /* 0x40862D83, 0x86544EB3 */
398 3.70414822620111362994e+03, /* 0x40ACF04B, 0xE44DFC63 */
399 6.46042516752568917582e+03, /* 0x40B93C6C, 0xD7C76A28 */
400 2.51633368920368957333e+03, /* 0x40A3A8AA, 0xD94FB1C0 */
401 -1.49247451836156386662e+02, /* 0xC062A7EB, 0x201CF40F */
402 };
403
404 static const double qR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
405 1.50444444886983272379e-07, /* 0x3E84313B, 0x54F76BDB */
406 7.32234265963079278272e-02, /* 0x3FB2BEC5, 0x3E883E34 */
407 1.99819174093815998816e+00, /* 0x3FFFF897, 0xE727779C */
408 1.44956029347885735348e+01, /* 0x402CFDBF, 0xAAF96FE5 */
409 3.16662317504781540833e+01, /* 0x403FAA8E, 0x29FBDC4A */
410 1.62527075710929267416e+01, /* 0x403040B1, 0x71814BB4 */
411 };
412 static const double qS2[6] = {
413 3.03655848355219184498e+01, /* 0x403E5D96, 0xF7C07AED */
414 2.69348118608049844624e+02, /* 0x4070D591, 0xE4D14B40 */
415 8.44783757595320139444e+02, /* 0x408A6645, 0x22B3BF22 */
416 8.82935845112488550512e+02, /* 0x408B977C, 0x9C5CC214 */
417 2.12666388511798828631e+02, /* 0x406A9553, 0x0E001365 */
418 -5.31095493882666946917e+00, /* 0xC0153E6A, 0xF8B32931 */
419 };
420
421 static double
422 qzero (double x)
423 {
424 const double *p, *q;
425 double s, r, z, z2, z4, z6, r1, r2, r3, s1, s2, s3;
426 int32_t ix;
427 GET_HIGH_WORD (ix, x);
428 ix &= 0x7fffffff;
429 /* ix >= 0x40000000 for all calls to this function. */
430 if (ix >= 0x41b00000)
431 {
432 return -.125 / x;
433 }
434 else if (ix >= 0x40200000)
435 {
436 p = qR8; q = qS8;
437 }
438 else if (ix >= 0x40122E8B)
439 {
440 p = qR5; q = qS5;
441 }
442 else if (ix >= 0x4006DB6D)
443 {
444 p = qR3; q = qS3;
445 }
446 else
447 {
448 p = qR2; q = qS2;
449 }
450 z = one / (x * x);
451 r1 = p[0] + z * p[1]; z2 = z * z;
452 r2 = p[2] + z * p[3]; z4 = z2 * z2;
453 r3 = p[4] + z * p[5]; z6 = z4 * z2;
454 r = r1 + z2 * r2 + z4 * r3;
455 s1 = one + z * q[0];
456 s2 = q[1] + z * q[2];
457 s3 = q[3] + z * q[4];
458 s = s1 + z2 * s2 + z4 * s3 + z6 * q[5];
459 return (-.125 + r / s) / x;
460 }