1 /* origin: FreeBSD /usr/src/lib/msun/src/e_j1.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 /* j1(x), y1(x)
13 * Bessel function of the first and second kinds of order zero.
14 * Method -- j1(x):
15 * 1. For tiny x, we use j1(x) = x/2 - x^3/16 + x^5/384 - ...
16 * 2. Reduce x to |x| since j1(x)=-j1(-x), and
17 * for x in (0,2)
18 * j1(x) = x/2 + x*z*R0/S0, where z = x*x;
19 * (precision: |j1/x - 1/2 - R0/S0 |<2**-61.51 )
20 * for x in (2,inf)
21 * j1(x) = sqrt(2/(pi*x))*(p1(x)*cos(x1)-q1(x)*sin(x1))
22 * y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x1)+q1(x)*cos(x1))
23 * where x1 = x-3*pi/4. It is better to compute sin(x1),cos(x1)
24 * as follow:
25 * cos(x1) = cos(x)cos(3pi/4)+sin(x)sin(3pi/4)
26 * = 1/sqrt(2) * (sin(x) - cos(x))
27 * sin(x1) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4)
28 * = -1/sqrt(2) * (sin(x) + cos(x))
29 * (To avoid cancellation, use
30 * sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
31 * to compute the worse one.)
32 *
33 * 3 Special cases
34 * j1(nan)= nan
35 * j1(0) = 0
36 * j1(inf) = 0
37 *
38 * Method -- y1(x):
39 * 1. screen out x<=0 cases: y1(0)=-inf, y1(x<0)=NaN
40 * 2. For x<2.
41 * Since
42 * y1(x) = 2/pi*(j1(x)*(ln(x/2)+Euler)-1/x-x/2+5/64*x^3-...)
43 * therefore y1(x)-2/pi*j1(x)*ln(x)-1/x is an odd function.
44 * We use the following function to approximate y1,
45 * y1(x) = x*U(z)/V(z) + (2/pi)*(j1(x)*ln(x)-1/x), z= x^2
46 * where for x in [0,2] (abs err less than 2**-65.89)
47 * U(z) = U0[0] + U0[1]*z + ... + U0[4]*z^4
48 * V(z) = 1 + v0[0]*z + ... + v0[4]*z^5
49 * Note: For tiny x, 1/x dominate y1 and hence
50 * y1(tiny) = -2/pi/tiny, (choose tiny<2**-54)
51 * 3. For x>=2.
52 * y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x1)+q1(x)*cos(x1))
53 * where x1 = x-3*pi/4. It is better to compute sin(x1),cos(x1)
54 * by method mentioned above.
55 */
56
57 use super::{cos, fabs, get_high_word, get_low_word, log, sin, sqrt};
58
59 const INVSQRTPI: f64 = 5.64189583547756279280e-01; /* 0x3FE20DD7, 0x50429B6D */
60 const TPI: f64 = 6.36619772367581382433e-01; /* 0x3FE45F30, 0x6DC9C883 */
61
common(ix: u32, x: f64, y1: bool, sign: bool) -> f6462 fn common(ix: u32, x: f64, y1: bool, sign: bool) -> f64 {
63 let z: f64;
64 let mut s: f64;
65 let c: f64;
66 let mut ss: f64;
67 let mut cc: f64;
68
69 /*
70 * j1(x) = sqrt(2/(pi*x))*(p1(x)*cos(x-3pi/4)-q1(x)*sin(x-3pi/4))
71 * y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x-3pi/4)+q1(x)*cos(x-3pi/4))
72 *
73 * sin(x-3pi/4) = -(sin(x) + cos(x))/sqrt(2)
74 * cos(x-3pi/4) = (sin(x) - cos(x))/sqrt(2)
75 * sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
76 */
77 s = sin(x);
78 if y1 {
79 s = -s;
80 }
81 c = cos(x);
82 cc = s - c;
83 if ix < 0x7fe00000 {
84 /* avoid overflow in 2*x */
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 y1 {
94 ss = -ss;
95 }
96 cc = pone(x) * cc - qone(x) * ss;
97 }
98 }
99 if sign {
100 cc = -cc;
101 }
102 return INVSQRTPI * cc / sqrt(x);
103 }
104
105 /* R0/S0 on [0,2] */
106 const R00: f64 = -6.25000000000000000000e-02; /* 0xBFB00000, 0x00000000 */
107 const R01: f64 = 1.40705666955189706048e-03; /* 0x3F570D9F, 0x98472C61 */
108 const R02: f64 = -1.59955631084035597520e-05; /* 0xBEF0C5C6, 0xBA169668 */
109 const R03: f64 = 4.96727999609584448412e-08; /* 0x3E6AAAFA, 0x46CA0BD9 */
110 const S01: f64 = 1.91537599538363460805e-02; /* 0x3F939D0B, 0x12637E53 */
111 const S02: f64 = 1.85946785588630915560e-04; /* 0x3F285F56, 0xB9CDF664 */
112 const S03: f64 = 1.17718464042623683263e-06; /* 0x3EB3BFF8, 0x333F8498 */
113 const S04: f64 = 5.04636257076217042715e-09; /* 0x3E35AC88, 0xC97DFF2C */
114 const S05: f64 = 1.23542274426137913908e-11; /* 0x3DAB2ACF, 0xCFB97ED8 */
115
116 /// First order of the [Bessel function](https://en.wikipedia.org/wiki/Bessel_function) of the first kind (f64).
j1(x: f64) -> f64117 pub fn j1(x: f64) -> f64 {
118 let mut z: f64;
119 let r: f64;
120 let s: f64;
121 let mut ix: u32;
122 let sign: bool;
123
124 ix = get_high_word(x);
125 sign = (ix >> 31) != 0;
126 ix &= 0x7fffffff;
127 if ix >= 0x7ff00000 {
128 return 1.0 / (x * x);
129 }
130 if ix >= 0x40000000 {
131 /* |x| >= 2 */
132 return common(ix, fabs(x), false, sign);
133 }
134 if ix >= 0x38000000 {
135 /* |x| >= 2**-127 */
136 z = x * x;
137 r = z * (R00 + z * (R01 + z * (R02 + z * R03)));
138 s = 1.0 + z * (S01 + z * (S02 + z * (S03 + z * (S04 + z * S05))));
139 z = r / s;
140 } else {
141 /* avoid underflow, raise inexact if x!=0 */
142 z = x;
143 }
144 return (0.5 + z) * x;
145 }
146
147 const U0: [f64; 5] = [
148 -1.96057090646238940668e-01, /* 0xBFC91866, 0x143CBC8A */
149 5.04438716639811282616e-02, /* 0x3FA9D3C7, 0x76292CD1 */
150 -1.91256895875763547298e-03, /* 0xBF5F55E5, 0x4844F50F */
151 2.35252600561610495928e-05, /* 0x3EF8AB03, 0x8FA6B88E */
152 -9.19099158039878874504e-08, /* 0xBE78AC00, 0x569105B8 */
153 ];
154 const V0: [f64; 5] = [
155 1.99167318236649903973e-02, /* 0x3F94650D, 0x3F4DA9F0 */
156 2.02552581025135171496e-04, /* 0x3F2A8C89, 0x6C257764 */
157 1.35608801097516229404e-06, /* 0x3EB6C05A, 0x894E8CA6 */
158 6.22741452364621501295e-09, /* 0x3E3ABF1D, 0x5BA69A86 */
159 1.66559246207992079114e-11, /* 0x3DB25039, 0xDACA772A */
160 ];
161
162 /// First order of the [Bessel function](https://en.wikipedia.org/wiki/Bessel_function) of the second kind (f64).
y1(x: f64) -> f64163 pub fn y1(x: f64) -> f64 {
164 let z: f64;
165 let u: f64;
166 let v: f64;
167 let ix: u32;
168 let lx: u32;
169
170 ix = get_high_word(x);
171 lx = get_low_word(x);
172
173 /* y1(nan)=nan, y1(<0)=nan, y1(0)=-inf, y1(inf)=0 */
174 if (ix << 1 | lx) == 0 {
175 return -1.0 / 0.0;
176 }
177 if (ix >> 31) != 0 {
178 return 0.0 / 0.0;
179 }
180 if ix >= 0x7ff00000 {
181 return 1.0 / x;
182 }
183
184 if ix >= 0x40000000 {
185 /* x >= 2 */
186 return common(ix, x, true, false);
187 }
188 if ix < 0x3c900000 {
189 /* x < 2**-54 */
190 return -TPI / x;
191 }
192 z = x * x;
193 u = U0[0] + z * (U0[1] + z * (U0[2] + z * (U0[3] + z * U0[4])));
194 v = 1.0 + z * (V0[0] + z * (V0[1] + z * (V0[2] + z * (V0[3] + z * V0[4]))));
195 return x * (u / v) + TPI * (j1(x) * log(x) - 1.0 / x);
196 }
197
198 /* For x >= 8, the asymptotic expansions of pone is
199 * 1 + 15/128 s^2 - 4725/2^15 s^4 - ..., where s = 1/x.
200 * We approximate pone by
201 * pone(x) = 1 + (R/S)
202 * where R = pr0 + pr1*s^2 + pr2*s^4 + ... + pr5*s^10
203 * S = 1 + ps0*s^2 + ... + ps4*s^10
204 * and
205 * | pone(x)-1-R/S | <= 2 ** ( -60.06)
206 */
207
208 const PR8: [f64; 6] = [
209 /* for x in [inf, 8]=1/[0,0.125] */
210 0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */
211 1.17187499999988647970e-01, /* 0x3FBDFFFF, 0xFFFFFCCE */
212 1.32394806593073575129e+01, /* 0x402A7A9D, 0x357F7FCE */
213 4.12051854307378562225e+02, /* 0x4079C0D4, 0x652EA590 */
214 3.87474538913960532227e+03, /* 0x40AE457D, 0xA3A532CC */
215 7.91447954031891731574e+03, /* 0x40BEEA7A, 0xC32782DD */
216 ];
217 const PS8: [f64; 5] = [
218 1.14207370375678408436e+02, /* 0x405C8D45, 0x8E656CAC */
219 3.65093083420853463394e+03, /* 0x40AC85DC, 0x964D274F */
220 3.69562060269033463555e+04, /* 0x40E20B86, 0x97C5BB7F */
221 9.76027935934950801311e+04, /* 0x40F7D42C, 0xB28F17BB */
222 3.08042720627888811578e+04, /* 0x40DE1511, 0x697A0B2D */
223 ];
224
225 const PR5: [f64; 6] = [
226 /* for x in [8,4.5454]=1/[0.125,0.22001] */
227 1.31990519556243522749e-11, /* 0x3DAD0667, 0xDAE1CA7D */
228 1.17187493190614097638e-01, /* 0x3FBDFFFF, 0xE2C10043 */
229 6.80275127868432871736e+00, /* 0x401B3604, 0x6E6315E3 */
230 1.08308182990189109773e+02, /* 0x405B13B9, 0x452602ED */
231 5.17636139533199752805e+02, /* 0x40802D16, 0xD052D649 */
232 5.28715201363337541807e+02, /* 0x408085B8, 0xBB7E0CB7 */
233 ];
234 const PS5: [f64; 5] = [
235 5.92805987221131331921e+01, /* 0x404DA3EA, 0xA8AF633D */
236 9.91401418733614377743e+02, /* 0x408EFB36, 0x1B066701 */
237 5.35326695291487976647e+03, /* 0x40B4E944, 0x5706B6FB */
238 7.84469031749551231769e+03, /* 0x40BEA4B0, 0xB8A5BB15 */
239 1.50404688810361062679e+03, /* 0x40978030, 0x036F5E51 */
240 ];
241
242 const PR3: [f64; 6] = [
243 3.02503916137373618024e-09, /* 0x3E29FC21, 0xA7AD9EDD */
244 1.17186865567253592491e-01, /* 0x3FBDFFF5, 0x5B21D17B */
245 3.93297750033315640650e+00, /* 0x400F76BC, 0xE85EAD8A */
246 3.51194035591636932736e+01, /* 0x40418F48, 0x9DA6D129 */
247 9.10550110750781271918e+01, /* 0x4056C385, 0x4D2C1837 */
248 4.85590685197364919645e+01, /* 0x4048478F, 0x8EA83EE5 */
249 ];
250 const PS3: [f64; 5] = [
251 3.47913095001251519989e+01, /* 0x40416549, 0xA134069C */
252 3.36762458747825746741e+02, /* 0x40750C33, 0x07F1A75F */
253 1.04687139975775130551e+03, /* 0x40905B7C, 0x5037D523 */
254 8.90811346398256432622e+02, /* 0x408BD67D, 0xA32E31E9 */
255 1.03787932439639277504e+02, /* 0x4059F26D, 0x7C2EED53 */
256 ];
257
258 const PR2: [f64; 6] = [
259 /* for x in [2.8570,2]=1/[0.3499,0.5] */
260 1.07710830106873743082e-07, /* 0x3E7CE9D4, 0xF65544F4 */
261 1.17176219462683348094e-01, /* 0x3FBDFF42, 0xBE760D83 */
262 2.36851496667608785174e+00, /* 0x4002F2B7, 0xF98FAEC0 */
263 1.22426109148261232917e+01, /* 0x40287C37, 0x7F71A964 */
264 1.76939711271687727390e+01, /* 0x4031B1A8, 0x177F8EE2 */
265 5.07352312588818499250e+00, /* 0x40144B49, 0xA574C1FE */
266 ];
267 const PS2: [f64; 5] = [
268 2.14364859363821409488e+01, /* 0x40356FBD, 0x8AD5ECDC */
269 1.25290227168402751090e+02, /* 0x405F5293, 0x14F92CD5 */
270 2.32276469057162813669e+02, /* 0x406D08D8, 0xD5A2DBD9 */
271 1.17679373287147100768e+02, /* 0x405D6B7A, 0xDA1884A9 */
272 8.36463893371618283368e+00, /* 0x4020BAB1, 0xF44E5192 */
273 ];
274
pone(x: f64) -> f64275 fn pone(x: f64) -> f64 {
276 let p: &[f64; 6];
277 let q: &[f64; 5];
278 let z: f64;
279 let r: f64;
280 let s: f64;
281 let mut ix: u32;
282
283 ix = get_high_word(x);
284 ix &= 0x7fffffff;
285 if ix >= 0x40200000 {
286 p = &PR8;
287 q = &PS8;
288 } else if ix >= 0x40122E8B {
289 p = &PR5;
290 q = &PS5;
291 } else if ix >= 0x4006DB6D {
292 p = &PR3;
293 q = &PS3;
294 } else
295 /*ix >= 0x40000000*/
296 {
297 p = &PR2;
298 q = &PS2;
299 }
300 z = 1.0 / (x * x);
301 r = p[0] + z * (p[1] + z * (p[2] + z * (p[3] + z * (p[4] + z * p[5]))));
302 s = 1.0 + z * (q[0] + z * (q[1] + z * (q[2] + z * (q[3] + z * q[4]))));
303 return 1.0 + r / s;
304 }
305
306 /* For x >= 8, the asymptotic expansions of qone is
307 * 3/8 s - 105/1024 s^3 - ..., where s = 1/x.
308 * We approximate pone by
309 * qone(x) = s*(0.375 + (R/S))
310 * where R = qr1*s^2 + qr2*s^4 + ... + qr5*s^10
311 * S = 1 + qs1*s^2 + ... + qs6*s^12
312 * and
313 * | qone(x)/s -0.375-R/S | <= 2 ** ( -61.13)
314 */
315
316 const QR8: [f64; 6] = [
317 /* for x in [inf, 8]=1/[0,0.125] */
318 0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */
319 -1.02539062499992714161e-01, /* 0xBFBA3FFF, 0xFFFFFDF3 */
320 -1.62717534544589987888e+01, /* 0xC0304591, 0xA26779F7 */
321 -7.59601722513950107896e+02, /* 0xC087BCD0, 0x53E4B576 */
322 -1.18498066702429587167e+04, /* 0xC0C724E7, 0x40F87415 */
323 -4.84385124285750353010e+04, /* 0xC0E7A6D0, 0x65D09C6A */
324 ];
325 const QS8: [f64; 6] = [
326 1.61395369700722909556e+02, /* 0x40642CA6, 0xDE5BCDE5 */
327 7.82538599923348465381e+03, /* 0x40BE9162, 0xD0D88419 */
328 1.33875336287249578163e+05, /* 0x4100579A, 0xB0B75E98 */
329 7.19657723683240939863e+05, /* 0x4125F653, 0x72869C19 */
330 6.66601232617776375264e+05, /* 0x412457D2, 0x7719AD5C */
331 -2.94490264303834643215e+05, /* 0xC111F969, 0x0EA5AA18 */
332 ];
333
334 const QR5: [f64; 6] = [
335 /* for x in [8,4.5454]=1/[0.125,0.22001] */
336 -2.08979931141764104297e-11, /* 0xBDB6FA43, 0x1AA1A098 */
337 -1.02539050241375426231e-01, /* 0xBFBA3FFF, 0xCB597FEF */
338 -8.05644828123936029840e+00, /* 0xC0201CE6, 0xCA03AD4B */
339 -1.83669607474888380239e+02, /* 0xC066F56D, 0x6CA7B9B0 */
340 -1.37319376065508163265e+03, /* 0xC09574C6, 0x6931734F */
341 -2.61244440453215656817e+03, /* 0xC0A468E3, 0x88FDA79D */
342 ];
343 const QS5: [f64; 6] = [
344 8.12765501384335777857e+01, /* 0x405451B2, 0xFF5A11B2 */
345 1.99179873460485964642e+03, /* 0x409F1F31, 0xE77BF839 */
346 1.74684851924908907677e+04, /* 0x40D10F1F, 0x0D64CE29 */
347 4.98514270910352279316e+04, /* 0x40E8576D, 0xAABAD197 */
348 2.79480751638918118260e+04, /* 0x40DB4B04, 0xCF7C364B */
349 -4.71918354795128470869e+03, /* 0xC0B26F2E, 0xFCFFA004 */
350 ];
351
352 const QR3: [f64; 6] = [
353 -5.07831226461766561369e-09, /* 0xBE35CFA9, 0xD38FC84F */
354 -1.02537829820837089745e-01, /* 0xBFBA3FEB, 0x51AEED54 */
355 -4.61011581139473403113e+00, /* 0xC01270C2, 0x3302D9FF */
356 -5.78472216562783643212e+01, /* 0xC04CEC71, 0xC25D16DA */
357 -2.28244540737631695038e+02, /* 0xC06C87D3, 0x4718D55F */
358 -2.19210128478909325622e+02, /* 0xC06B66B9, 0x5F5C1BF6 */
359 ];
360 const QS3: [f64; 6] = [
361 4.76651550323729509273e+01, /* 0x4047D523, 0xCCD367E4 */
362 6.73865112676699709482e+02, /* 0x40850EEB, 0xC031EE3E */
363 3.38015286679526343505e+03, /* 0x40AA684E, 0x448E7C9A */
364 5.54772909720722782367e+03, /* 0x40B5ABBA, 0xA61D54A6 */
365 1.90311919338810798763e+03, /* 0x409DBC7A, 0x0DD4DF4B */
366 -1.35201191444307340817e+02, /* 0xC060E670, 0x290A311F */
367 ];
368
369 const QR2: [f64; 6] = [
370 /* for x in [2.8570,2]=1/[0.3499,0.5] */
371 -1.78381727510958865572e-07, /* 0xBE87F126, 0x44C626D2 */
372 -1.02517042607985553460e-01, /* 0xBFBA3E8E, 0x9148B010 */
373 -2.75220568278187460720e+00, /* 0xC0060484, 0x69BB4EDA */
374 -1.96636162643703720221e+01, /* 0xC033A9E2, 0xC168907F */
375 -4.23253133372830490089e+01, /* 0xC04529A3, 0xDE104AAA */
376 -2.13719211703704061733e+01, /* 0xC0355F36, 0x39CF6E52 */
377 ];
378 const QS2: [f64; 6] = [
379 2.95333629060523854548e+01, /* 0x403D888A, 0x78AE64FF */
380 2.52981549982190529136e+02, /* 0x406F9F68, 0xDB821CBA */
381 7.57502834868645436472e+02, /* 0x4087AC05, 0xCE49A0F7 */
382 7.39393205320467245656e+02, /* 0x40871B25, 0x48D4C029 */
383 1.55949003336666123687e+02, /* 0x40637E5E, 0x3C3ED8D4 */
384 -4.95949898822628210127e+00, /* 0xC013D686, 0xE71BE86B */
385 ];
386
qone(x: f64) -> f64387 fn qone(x: f64) -> f64 {
388 let p: &[f64; 6];
389 let q: &[f64; 6];
390 let s: f64;
391 let r: f64;
392 let z: f64;
393 let mut ix: u32;
394
395 ix = get_high_word(x);
396 ix &= 0x7fffffff;
397 if ix >= 0x40200000 {
398 p = &QR8;
399 q = &QS8;
400 } else if ix >= 0x40122E8B {
401 p = &QR5;
402 q = &QS5;
403 } else if ix >= 0x4006DB6D {
404 p = &QR3;
405 q = &QS3;
406 } else
407 /*ix >= 0x40000000*/
408 {
409 p = &QR2;
410 q = &QS2;
411 }
412 z = 1.0 / (x * x);
413 r = p[0] + z * (p[1] + z * (p[2] + z * (p[3] + z * (p[4] + z * p[5]))));
414 s = 1.0 + z * (q[0] + z * (q[1] + z * (q[2] + z * (q[3] + z * (q[4] + z * q[5])))));
415 return (0.375 + r / s) / x;
416 }
417