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 use super::{cos, fabs, get_high_word, get_low_word, log, sin, sqrt};
58 const INVSQRTPI: f64 = 5.64189583547756279280e-01; /* 0x3FE20DD7, 0x50429B6D */
59 const TPI: f64 = 6.36619772367581382433e-01; /* 0x3FE45F30, 0x6DC9C883 */
60
61 /* common method when |x|>=2 */
common(ix: u32, x: f64, y0: bool) -> f6462 fn common(ix: u32, x: f64, y0: bool) -> f64 {
63 let s: f64;
64 let mut c: f64;
65 let mut ss: f64;
66 let mut cc: f64;
67 let z: f64;
68
69 /*
70 * j0(x) = sqrt(2/(pi*x))*(p0(x)*cos(x-pi/4)-q0(x)*sin(x-pi/4))
71 * y0(x) = sqrt(2/(pi*x))*(p0(x)*sin(x-pi/4)+q0(x)*cos(x-pi/4))
72 *
73 * sin(x-pi/4) = (sin(x) - cos(x))/sqrt(2)
74 * cos(x-pi/4) = (sin(x) + cos(x))/sqrt(2)
75 * sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
76 */
77 s = sin(x);
78 c = cos(x);
79 if y0 {
80 c = -c;
81 }
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(2.0 * x);
87 if s * c < 0.0 {
88 cc = z / ss;
89 } else {
90 ss = z / cc;
91 }
92 if ix < 0x48000000 {
93 if y0 {
94 ss = -ss;
95 }
96 cc = pzero(x) * cc - qzero(x) * ss;
97 }
98 }
99 return INVSQRTPI * cc / sqrt(x);
100 }
101
102 /* R0/S0 on [0, 2.00] */
103 const R02: f64 = 1.56249999999999947958e-02; /* 0x3F8FFFFF, 0xFFFFFFFD */
104 const R03: f64 = -1.89979294238854721751e-04; /* 0xBF28E6A5, 0xB61AC6E9 */
105 const R04: f64 = 1.82954049532700665670e-06; /* 0x3EBEB1D1, 0x0C503919 */
106 const R05: f64 = -4.61832688532103189199e-09; /* 0xBE33D5E7, 0x73D63FCE */
107 const S01: f64 = 1.56191029464890010492e-02; /* 0x3F8FFCE8, 0x82C8C2A4 */
108 const S02: f64 = 1.16926784663337450260e-04; /* 0x3F1EA6D2, 0xDD57DBF4 */
109 const S03: f64 = 5.13546550207318111446e-07; /* 0x3EA13B54, 0xCE84D5A9 */
110 const S04: f64 = 1.16614003333790000205e-09; /* 0x3E1408BC, 0xF4745D8F */
111
112 /// Zeroth order of the [Bessel function](https://en.wikipedia.org/wiki/Bessel_function) of the first kind (f64).
j0(mut x: f64) -> f64113 pub fn j0(mut x: f64) -> f64 {
114 let z: f64;
115 let r: f64;
116 let s: f64;
117 let mut ix: u32;
118
119 ix = get_high_word(x);
120 ix &= 0x7fffffff;
121
122 /* j0(+-inf)=0, j0(nan)=nan */
123 if ix >= 0x7ff00000 {
124 return 1.0 / (x * x);
125 }
126 x = fabs(x);
127
128 if ix >= 0x40000000 {
129 /* |x| >= 2 */
130 /* large ulp error near zeros: 2.4, 5.52, 8.6537,.. */
131 return common(ix, x, false);
132 }
133
134 /* 1 - x*x/4 + x*x*R(x^2)/S(x^2) */
135 if ix >= 0x3f200000 {
136 /* |x| >= 2**-13 */
137 /* up to 4ulp error close to 2 */
138 z = x * x;
139 r = z * (R02 + z * (R03 + z * (R04 + z * R05)));
140 s = 1.0 + z * (S01 + z * (S02 + z * (S03 + z * S04)));
141 return (1.0 + x / 2.0) * (1.0 - x / 2.0) + z * (r / s);
142 }
143
144 /* 1 - x*x/4 */
145 /* prevent underflow */
146 /* inexact should be raised when x!=0, this is not done correctly */
147 if ix >= 0x38000000 {
148 /* |x| >= 2**-127 */
149 x = 0.25 * x * x;
150 }
151 return 1.0 - x;
152 }
153
154 const U00: f64 = -7.38042951086872317523e-02; /* 0xBFB2E4D6, 0x99CBD01F */
155 const U01: f64 = 1.76666452509181115538e-01; /* 0x3FC69D01, 0x9DE9E3FC */
156 const U02: f64 = -1.38185671945596898896e-02; /* 0xBF8C4CE8, 0xB16CFA97 */
157 const U03: f64 = 3.47453432093683650238e-04; /* 0x3F36C54D, 0x20B29B6B */
158 const U04: f64 = -3.81407053724364161125e-06; /* 0xBECFFEA7, 0x73D25CAD */
159 const U05: f64 = 1.95590137035022920206e-08; /* 0x3E550057, 0x3B4EABD4 */
160 const U06: f64 = -3.98205194132103398453e-11; /* 0xBDC5E43D, 0x693FB3C8 */
161 const V01: f64 = 1.27304834834123699328e-02; /* 0x3F8A1270, 0x91C9C71A */
162 const V02: f64 = 7.60068627350353253702e-05; /* 0x3F13ECBB, 0xF578C6C1 */
163 const V03: f64 = 2.59150851840457805467e-07; /* 0x3E91642D, 0x7FF202FD */
164 const V04: f64 = 4.41110311332675467403e-10; /* 0x3DFE5018, 0x3BD6D9EF */
165
166 /// Zeroth order of the [Bessel function](https://en.wikipedia.org/wiki/Bessel_function) of the second kind (f64).
y0(x: f64) -> f64167 pub fn y0(x: f64) -> f64 {
168 let z: f64;
169 let u: f64;
170 let v: f64;
171 let ix: u32;
172 let lx: u32;
173
174 ix = get_high_word(x);
175 lx = get_low_word(x);
176
177 /* y0(nan)=nan, y0(<0)=nan, y0(0)=-inf, y0(inf)=0 */
178 if ((ix << 1) | lx) == 0 {
179 return -1.0 / 0.0;
180 }
181 if (ix >> 31) != 0 {
182 return 0.0 / 0.0;
183 }
184 if ix >= 0x7ff00000 {
185 return 1.0 / x;
186 }
187
188 if ix >= 0x40000000 {
189 /* x >= 2 */
190 /* large ulp errors near zeros: 3.958, 7.086,.. */
191 return common(ix, x, true);
192 }
193
194 /* U(x^2)/V(x^2) + (2/pi)*j0(x)*log(x) */
195 if ix >= 0x3e400000 {
196 /* x >= 2**-27 */
197 /* large ulp error near the first zero, x ~= 0.89 */
198 z = x * x;
199 u = U00 + z * (U01 + z * (U02 + z * (U03 + z * (U04 + z * (U05 + z * U06)))));
200 v = 1.0 + z * (V01 + z * (V02 + z * (V03 + z * V04)));
201 return u / v + TPI * (j0(x) * log(x));
202 }
203 return U00 + TPI * log(x);
204 }
205
206 /* The asymptotic expansions of pzero is
207 * 1 - 9/128 s^2 + 11025/98304 s^4 - ..., where s = 1/x.
208 * For x >= 2, We approximate pzero by
209 * pzero(x) = 1 + (R/S)
210 * where R = pR0 + pR1*s^2 + pR2*s^4 + ... + pR5*s^10
211 * S = 1 + pS0*s^2 + ... + pS4*s^10
212 * and
213 * | pzero(x)-1-R/S | <= 2 ** ( -60.26)
214 */
215 const PR8: [f64; 6] = [
216 /* for x in [inf, 8]=1/[0,0.125] */
217 0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */
218 -7.03124999999900357484e-02, /* 0xBFB1FFFF, 0xFFFFFD32 */
219 -8.08167041275349795626e+00, /* 0xC02029D0, 0xB44FA779 */
220 -2.57063105679704847262e+02, /* 0xC0701102, 0x7B19E863 */
221 -2.48521641009428822144e+03, /* 0xC0A36A6E, 0xCD4DCAFC */
222 -5.25304380490729545272e+03, /* 0xC0B4850B, 0x36CC643D */
223 ];
224 const PS8: [f64; 5] = [
225 1.16534364619668181717e+02, /* 0x405D2233, 0x07A96751 */
226 3.83374475364121826715e+03, /* 0x40ADF37D, 0x50596938 */
227 4.05978572648472545552e+04, /* 0x40E3D2BB, 0x6EB6B05F */
228 1.16752972564375915681e+05, /* 0x40FC810F, 0x8F9FA9BD */
229 4.76277284146730962675e+04, /* 0x40E74177, 0x4F2C49DC */
230 ];
231
232 const PR5: [f64; 6] = [
233 /* for x in [8,4.5454]=1/[0.125,0.22001] */
234 -1.14125464691894502584e-11, /* 0xBDA918B1, 0x47E495CC */
235 -7.03124940873599280078e-02, /* 0xBFB1FFFF, 0xE69AFBC6 */
236 -4.15961064470587782438e+00, /* 0xC010A370, 0xF90C6BBF */
237 -6.76747652265167261021e+01, /* 0xC050EB2F, 0x5A7D1783 */
238 -3.31231299649172967747e+02, /* 0xC074B3B3, 0x6742CC63 */
239 -3.46433388365604912451e+02, /* 0xC075A6EF, 0x28A38BD7 */
240 ];
241 const PS5: [f64; 5] = [
242 6.07539382692300335975e+01, /* 0x404E6081, 0x0C98C5DE */
243 1.05125230595704579173e+03, /* 0x40906D02, 0x5C7E2864 */
244 5.97897094333855784498e+03, /* 0x40B75AF8, 0x8FBE1D60 */
245 9.62544514357774460223e+03, /* 0x40C2CCB8, 0xFA76FA38 */
246 2.40605815922939109441e+03, /* 0x40A2CC1D, 0xC70BE864 */
247 ];
248
249 const PR3: [f64; 6] = [
250 /* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
251 -2.54704601771951915620e-09, /* 0xBE25E103, 0x6FE1AA86 */
252 -7.03119616381481654654e-02, /* 0xBFB1FFF6, 0xF7C0E24B */
253 -2.40903221549529611423e+00, /* 0xC00345B2, 0xAEA48074 */
254 -2.19659774734883086467e+01, /* 0xC035F74A, 0x4CB94E14 */
255 -5.80791704701737572236e+01, /* 0xC04D0A22, 0x420A1A45 */
256 -3.14479470594888503854e+01, /* 0xC03F72AC, 0xA892D80F */
257 ];
258 const PS3: [f64; 5] = [
259 3.58560338055209726349e+01, /* 0x4041ED92, 0x84077DD3 */
260 3.61513983050303863820e+02, /* 0x40769839, 0x464A7C0E */
261 1.19360783792111533330e+03, /* 0x4092A66E, 0x6D1061D6 */
262 1.12799679856907414432e+03, /* 0x40919FFC, 0xB8C39B7E */
263 1.73580930813335754692e+02, /* 0x4065B296, 0xFC379081 */
264 ];
265
266 const PR2: [f64; 6] = [
267 /* for x in [2.8570,2]=1/[0.3499,0.5] */
268 -8.87534333032526411254e-08, /* 0xBE77D316, 0xE927026D */
269 -7.03030995483624743247e-02, /* 0xBFB1FF62, 0x495E1E42 */
270 -1.45073846780952986357e+00, /* 0xBFF73639, 0x8A24A843 */
271 -7.63569613823527770791e+00, /* 0xC01E8AF3, 0xEDAFA7F3 */
272 -1.11931668860356747786e+01, /* 0xC02662E6, 0xC5246303 */
273 -3.23364579351335335033e+00, /* 0xC009DE81, 0xAF8FE70F */
274 ];
275 const PS2: [f64; 5] = [
276 2.22202997532088808441e+01, /* 0x40363865, 0x908B5959 */
277 1.36206794218215208048e+02, /* 0x4061069E, 0x0EE8878F */
278 2.70470278658083486789e+02, /* 0x4070E786, 0x42EA079B */
279 1.53875394208320329881e+02, /* 0x40633C03, 0x3AB6FAFF */
280 1.46576176948256193810e+01, /* 0x402D50B3, 0x44391809 */
281 ];
282
pzero(x: f64) -> f64283 fn pzero(x: f64) -> f64 {
284 let p: &[f64; 6];
285 let q: &[f64; 5];
286 let z: f64;
287 let r: f64;
288 let s: f64;
289 let mut ix: u32;
290
291 ix = get_high_word(x);
292 ix &= 0x7fffffff;
293 if ix >= 0x40200000 {
294 p = &PR8;
295 q = &PS8;
296 } else if ix >= 0x40122E8B {
297 p = &PR5;
298 q = &PS5;
299 } else if ix >= 0x4006DB6D {
300 p = &PR3;
301 q = &PS3;
302 } else
303 /*ix >= 0x40000000*/
304 {
305 p = &PR2;
306 q = &PS2;
307 }
308 z = 1.0 / (x * x);
309 r = p[0] + z * (p[1] + z * (p[2] + z * (p[3] + z * (p[4] + z * p[5]))));
310 s = 1.0 + z * (q[0] + z * (q[1] + z * (q[2] + z * (q[3] + z * q[4]))));
311 return 1.0 + r / s;
312 }
313
314 /* For x >= 8, the asymptotic expansions of qzero is
315 * -1/8 s + 75/1024 s^3 - ..., where s = 1/x.
316 * We approximate pzero by
317 * qzero(x) = s*(-1.25 + (R/S))
318 * where R = qR0 + qR1*s^2 + qR2*s^4 + ... + qR5*s^10
319 * S = 1 + qS0*s^2 + ... + qS5*s^12
320 * and
321 * | qzero(x)/s +1.25-R/S | <= 2 ** ( -61.22)
322 */
323 const QR8: [f64; 6] = [
324 /* for x in [inf, 8]=1/[0,0.125] */
325 0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */
326 7.32421874999935051953e-02, /* 0x3FB2BFFF, 0xFFFFFE2C */
327 1.17682064682252693899e+01, /* 0x40278952, 0x5BB334D6 */
328 5.57673380256401856059e+02, /* 0x40816D63, 0x15301825 */
329 8.85919720756468632317e+03, /* 0x40C14D99, 0x3E18F46D */
330 3.70146267776887834771e+04, /* 0x40E212D4, 0x0E901566 */
331 ];
332 const QS8: [f64; 6] = [
333 1.63776026895689824414e+02, /* 0x406478D5, 0x365B39BC */
334 8.09834494656449805916e+03, /* 0x40BFA258, 0x4E6B0563 */
335 1.42538291419120476348e+05, /* 0x41016652, 0x54D38C3F */
336 8.03309257119514397345e+05, /* 0x412883DA, 0x83A52B43 */
337 8.40501579819060512818e+05, /* 0x4129A66B, 0x28DE0B3D */
338 -3.43899293537866615225e+05, /* 0xC114FD6D, 0x2C9530C5 */
339 ];
340
341 const QR5: [f64; 6] = [
342 /* for x in [8,4.5454]=1/[0.125,0.22001] */
343 1.84085963594515531381e-11, /* 0x3DB43D8F, 0x29CC8CD9 */
344 7.32421766612684765896e-02, /* 0x3FB2BFFF, 0xD172B04C */
345 5.83563508962056953777e+00, /* 0x401757B0, 0xB9953DD3 */
346 1.35111577286449829671e+02, /* 0x4060E392, 0x0A8788E9 */
347 1.02724376596164097464e+03, /* 0x40900CF9, 0x9DC8C481 */
348 1.98997785864605384631e+03, /* 0x409F17E9, 0x53C6E3A6 */
349 ];
350 const QS5: [f64; 6] = [
351 8.27766102236537761883e+01, /* 0x4054B1B3, 0xFB5E1543 */
352 2.07781416421392987104e+03, /* 0x40A03BA0, 0xDA21C0CE */
353 1.88472887785718085070e+04, /* 0x40D267D2, 0x7B591E6D */
354 5.67511122894947329769e+04, /* 0x40EBB5E3, 0x97E02372 */
355 3.59767538425114471465e+04, /* 0x40E19118, 0x1F7A54A0 */
356 -5.35434275601944773371e+03, /* 0xC0B4EA57, 0xBEDBC609 */
357 ];
358
359 const QR3: [f64; 6] = [
360 /* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
361 4.37741014089738620906e-09, /* 0x3E32CD03, 0x6ADECB82 */
362 7.32411180042911447163e-02, /* 0x3FB2BFEE, 0x0E8D0842 */
363 3.34423137516170720929e+00, /* 0x400AC0FC, 0x61149CF5 */
364 4.26218440745412650017e+01, /* 0x40454F98, 0x962DAEDD */
365 1.70808091340565596283e+02, /* 0x406559DB, 0xE25EFD1F */
366 1.66733948696651168575e+02, /* 0x4064D77C, 0x81FA21E0 */
367 ];
368 const QS3: [f64; 6] = [
369 4.87588729724587182091e+01, /* 0x40486122, 0xBFE343A6 */
370 7.09689221056606015736e+02, /* 0x40862D83, 0x86544EB3 */
371 3.70414822620111362994e+03, /* 0x40ACF04B, 0xE44DFC63 */
372 6.46042516752568917582e+03, /* 0x40B93C6C, 0xD7C76A28 */
373 2.51633368920368957333e+03, /* 0x40A3A8AA, 0xD94FB1C0 */
374 -1.49247451836156386662e+02, /* 0xC062A7EB, 0x201CF40F */
375 ];
376
377 const QR2: [f64; 6] = [
378 /* for x in [2.8570,2]=1/[0.3499,0.5] */
379 1.50444444886983272379e-07, /* 0x3E84313B, 0x54F76BDB */
380 7.32234265963079278272e-02, /* 0x3FB2BEC5, 0x3E883E34 */
381 1.99819174093815998816e+00, /* 0x3FFFF897, 0xE727779C */
382 1.44956029347885735348e+01, /* 0x402CFDBF, 0xAAF96FE5 */
383 3.16662317504781540833e+01, /* 0x403FAA8E, 0x29FBDC4A */
384 1.62527075710929267416e+01, /* 0x403040B1, 0x71814BB4 */
385 ];
386 const QS2: [f64; 6] = [
387 3.03655848355219184498e+01, /* 0x403E5D96, 0xF7C07AED */
388 2.69348118608049844624e+02, /* 0x4070D591, 0xE4D14B40 */
389 8.44783757595320139444e+02, /* 0x408A6645, 0x22B3BF22 */
390 8.82935845112488550512e+02, /* 0x408B977C, 0x9C5CC214 */
391 2.12666388511798828631e+02, /* 0x406A9553, 0x0E001365 */
392 -5.31095493882666946917e+00, /* 0xC0153E6A, 0xF8B32931 */
393 ];
394
qzero(x: f64) -> f64395 fn qzero(x: f64) -> f64 {
396 let p: &[f64; 6];
397 let q: &[f64; 6];
398 let s: f64;
399 let r: f64;
400 let z: f64;
401 let mut ix: u32;
402
403 ix = get_high_word(x);
404 ix &= 0x7fffffff;
405 if ix >= 0x40200000 {
406 p = &QR8;
407 q = &QS8;
408 } else if ix >= 0x40122E8B {
409 p = &QR5;
410 q = &QS5;
411 } else if ix >= 0x4006DB6D {
412 p = &QR3;
413 q = &QS3;
414 } else
415 /*ix >= 0x40000000*/
416 {
417 p = &QR2;
418 q = &QS2;
419 }
420 z = 1.0 / (x * x);
421 r = p[0] + z * (p[1] + z * (p[2] + z * (p[3] + z * (p[4] + z * p[5]))));
422 s = 1.0 + z * (q[0] + z * (q[1] + z * (q[2] + z * (q[3] + z * (q[4] + z * q[5])))));
423 return (-0.125 + r / s) / x;
424 }
425