1 /*
2 * ====================================================
3 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
4 *
5 * Developed at SunSoft, a Sun Microsystems, Inc. business.
6 * Permission to use, copy, modify, and distribute this
7 * software is freely granted, provided that this notice
8 * is preserved.
9 * ====================================================
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 #include "math.h"
58 #include "math_private.h"
59
60 static __inline double pone(double), qone(double);
61
62 static const volatile double vone = 1, vzero = 0;
63
64 static const double
65 huge = 1e300,
66 one = 1.0,
67 invsqrtpi= 5.64189583547756279280e-01, /* 0x3FE20DD7, 0x50429B6D */
68 tpi = 6.36619772367581382433e-01, /* 0x3FE45F30, 0x6DC9C883 */
69 /* R0/S0 on [0,2] */
70 r00 = -6.25000000000000000000e-02, /* 0xBFB00000, 0x00000000 */
71 r01 = 1.40705666955189706048e-03, /* 0x3F570D9F, 0x98472C61 */
72 r02 = -1.59955631084035597520e-05, /* 0xBEF0C5C6, 0xBA169668 */
73 r03 = 4.96727999609584448412e-08, /* 0x3E6AAAFA, 0x46CA0BD9 */
74 s01 = 1.91537599538363460805e-02, /* 0x3F939D0B, 0x12637E53 */
75 s02 = 1.85946785588630915560e-04, /* 0x3F285F56, 0xB9CDF664 */
76 s03 = 1.17718464042623683263e-06, /* 0x3EB3BFF8, 0x333F8498 */
77 s04 = 5.04636257076217042715e-09, /* 0x3E35AC88, 0xC97DFF2C */
78 s05 = 1.23542274426137913908e-11; /* 0x3DAB2ACF, 0xCFB97ED8 */
79
80 static const double zero = 0.0;
81
82 double
j1(double x)83 j1(double x)
84 {
85 double z, s,c,ss,cc,r,u,v,y;
86 int32_t hx,ix;
87
88 GET_HIGH_WORD(hx,x);
89 ix = hx&0x7fffffff;
90 if(ix>=0x7ff00000) return one/x;
91 y = fabs(x);
92 if(ix >= 0x40000000) { /* |x| >= 2.0 */
93 sincos(y, &s, &c);
94 ss = -s-c;
95 cc = s-c;
96 if(ix<0x7fe00000) { /* make sure y+y not overflow */
97 z = cos(y+y);
98 if ((s*c)>zero) cc = z/ss;
99 else ss = z/cc;
100 }
101 /*
102 * j1(x) = 1/sqrt(pi) * (P(1,x)*cc - Q(1,x)*ss) / sqrt(x)
103 * y1(x) = 1/sqrt(pi) * (P(1,x)*ss + Q(1,x)*cc) / sqrt(x)
104 */
105 if(ix>0x48000000) z = (invsqrtpi*cc)/sqrt(y);
106 else {
107 u = pone(y); v = qone(y);
108 z = invsqrtpi*(u*cc-v*ss)/sqrt(y);
109 }
110 if(hx<0) return -z;
111 else return z;
112 }
113 if(ix<0x3e400000) { /* |x|<2**-27 */
114 if(huge+x>one) return 0.5*x;/* inexact if x!=0 necessary */
115 }
116 z = x*x;
117 r = z*(r00+z*(r01+z*(r02+z*r03)));
118 s = one+z*(s01+z*(s02+z*(s03+z*(s04+z*s05))));
119 r *= x;
120 return(x*0.5+r/s);
121 }
122
123 static const double U0[5] = {
124 -1.96057090646238940668e-01, /* 0xBFC91866, 0x143CBC8A */
125 5.04438716639811282616e-02, /* 0x3FA9D3C7, 0x76292CD1 */
126 -1.91256895875763547298e-03, /* 0xBF5F55E5, 0x4844F50F */
127 2.35252600561610495928e-05, /* 0x3EF8AB03, 0x8FA6B88E */
128 -9.19099158039878874504e-08, /* 0xBE78AC00, 0x569105B8 */
129 };
130 static const double V0[5] = {
131 1.99167318236649903973e-02, /* 0x3F94650D, 0x3F4DA9F0 */
132 2.02552581025135171496e-04, /* 0x3F2A8C89, 0x6C257764 */
133 1.35608801097516229404e-06, /* 0x3EB6C05A, 0x894E8CA6 */
134 6.22741452364621501295e-09, /* 0x3E3ABF1D, 0x5BA69A86 */
135 1.66559246207992079114e-11, /* 0x3DB25039, 0xDACA772A */
136 };
137
138 double
y1(double x)139 y1(double x)
140 {
141 double z, s,c,ss,cc,u,v;
142 int32_t hx,ix,lx;
143
144 EXTRACT_WORDS(hx,lx,x);
145 ix = 0x7fffffff&hx;
146 /*
147 * y1(NaN) = NaN.
148 * y1(Inf) = 0.
149 * y1(-Inf) = NaN and raise invalid exception.
150 */
151 if(ix>=0x7ff00000) return vone/(x+x*x);
152 /* y1(+-0) = -inf and raise divide-by-zero exception. */
153 if((ix|lx)==0) return -one/vzero;
154 /* y1(x<0) = NaN and raise invalid exception. */
155 if(hx<0) return vzero/vzero;
156 if(ix >= 0x40000000) { /* |x| >= 2.0 */
157 sincos(x, &s, &c);
158 ss = -s-c;
159 cc = s-c;
160 if(ix<0x7fe00000) { /* make sure x+x not overflow */
161 z = cos(x+x);
162 if ((s*c)>zero) cc = z/ss;
163 else ss = z/cc;
164 }
165 /* y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x0)+q1(x)*cos(x0))
166 * where x0 = x-3pi/4
167 * Better formula:
168 * cos(x0) = cos(x)cos(3pi/4)+sin(x)sin(3pi/4)
169 * = 1/sqrt(2) * (sin(x) - cos(x))
170 * sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4)
171 * = -1/sqrt(2) * (cos(x) + sin(x))
172 * To avoid cancellation, use
173 * sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
174 * to compute the worse one.
175 */
176 if(ix>0x48000000) z = (invsqrtpi*ss)/sqrt(x);
177 else {
178 u = pone(x); v = qone(x);
179 z = invsqrtpi*(u*ss+v*cc)/sqrt(x);
180 }
181 return z;
182 }
183 if(ix<=0x3c900000) { /* x < 2**-54 */
184 return(-tpi/x);
185 }
186 z = x*x;
187 u = U0[0]+z*(U0[1]+z*(U0[2]+z*(U0[3]+z*U0[4])));
188 v = one+z*(V0[0]+z*(V0[1]+z*(V0[2]+z*(V0[3]+z*V0[4]))));
189 return(x*(u/v) + tpi*(j1(x)*log(x)-one/x));
190 }
191
192 /* For x >= 8, the asymptotic expansions of pone is
193 * 1 + 15/128 s^2 - 4725/2^15 s^4 - ..., where s = 1/x.
194 * We approximate pone by
195 * pone(x) = 1 + (R/S)
196 * where R = pr0 + pr1*s^2 + pr2*s^4 + ... + pr5*s^10
197 * S = 1 + ps0*s^2 + ... + ps4*s^10
198 * and
199 * | pone(x)-1-R/S | <= 2 ** ( -60.06)
200 */
201
202 static const double pr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
203 0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */
204 1.17187499999988647970e-01, /* 0x3FBDFFFF, 0xFFFFFCCE */
205 1.32394806593073575129e+01, /* 0x402A7A9D, 0x357F7FCE */
206 4.12051854307378562225e+02, /* 0x4079C0D4, 0x652EA590 */
207 3.87474538913960532227e+03, /* 0x40AE457D, 0xA3A532CC */
208 7.91447954031891731574e+03, /* 0x40BEEA7A, 0xC32782DD */
209 };
210 static const double ps8[5] = {
211 1.14207370375678408436e+02, /* 0x405C8D45, 0x8E656CAC */
212 3.65093083420853463394e+03, /* 0x40AC85DC, 0x964D274F */
213 3.69562060269033463555e+04, /* 0x40E20B86, 0x97C5BB7F */
214 9.76027935934950801311e+04, /* 0x40F7D42C, 0xB28F17BB */
215 3.08042720627888811578e+04, /* 0x40DE1511, 0x697A0B2D */
216 };
217
218 static const double pr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
219 1.31990519556243522749e-11, /* 0x3DAD0667, 0xDAE1CA7D */
220 1.17187493190614097638e-01, /* 0x3FBDFFFF, 0xE2C10043 */
221 6.80275127868432871736e+00, /* 0x401B3604, 0x6E6315E3 */
222 1.08308182990189109773e+02, /* 0x405B13B9, 0x452602ED */
223 5.17636139533199752805e+02, /* 0x40802D16, 0xD052D649 */
224 5.28715201363337541807e+02, /* 0x408085B8, 0xBB7E0CB7 */
225 };
226 static const double ps5[5] = {
227 5.92805987221131331921e+01, /* 0x404DA3EA, 0xA8AF633D */
228 9.91401418733614377743e+02, /* 0x408EFB36, 0x1B066701 */
229 5.35326695291487976647e+03, /* 0x40B4E944, 0x5706B6FB */
230 7.84469031749551231769e+03, /* 0x40BEA4B0, 0xB8A5BB15 */
231 1.50404688810361062679e+03, /* 0x40978030, 0x036F5E51 */
232 };
233
234 static const double pr3[6] = {
235 3.02503916137373618024e-09, /* 0x3E29FC21, 0xA7AD9EDD */
236 1.17186865567253592491e-01, /* 0x3FBDFFF5, 0x5B21D17B */
237 3.93297750033315640650e+00, /* 0x400F76BC, 0xE85EAD8A */
238 3.51194035591636932736e+01, /* 0x40418F48, 0x9DA6D129 */
239 9.10550110750781271918e+01, /* 0x4056C385, 0x4D2C1837 */
240 4.85590685197364919645e+01, /* 0x4048478F, 0x8EA83EE5 */
241 };
242 static const double ps3[5] = {
243 3.47913095001251519989e+01, /* 0x40416549, 0xA134069C */
244 3.36762458747825746741e+02, /* 0x40750C33, 0x07F1A75F */
245 1.04687139975775130551e+03, /* 0x40905B7C, 0x5037D523 */
246 8.90811346398256432622e+02, /* 0x408BD67D, 0xA32E31E9 */
247 1.03787932439639277504e+02, /* 0x4059F26D, 0x7C2EED53 */
248 };
249
250 static const double pr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
251 1.07710830106873743082e-07, /* 0x3E7CE9D4, 0xF65544F4 */
252 1.17176219462683348094e-01, /* 0x3FBDFF42, 0xBE760D83 */
253 2.36851496667608785174e+00, /* 0x4002F2B7, 0xF98FAEC0 */
254 1.22426109148261232917e+01, /* 0x40287C37, 0x7F71A964 */
255 1.76939711271687727390e+01, /* 0x4031B1A8, 0x177F8EE2 */
256 5.07352312588818499250e+00, /* 0x40144B49, 0xA574C1FE */
257 };
258 static const double ps2[5] = {
259 2.14364859363821409488e+01, /* 0x40356FBD, 0x8AD5ECDC */
260 1.25290227168402751090e+02, /* 0x405F5293, 0x14F92CD5 */
261 2.32276469057162813669e+02, /* 0x406D08D8, 0xD5A2DBD9 */
262 1.17679373287147100768e+02, /* 0x405D6B7A, 0xDA1884A9 */
263 8.36463893371618283368e+00, /* 0x4020BAB1, 0xF44E5192 */
264 };
265
266 static __inline double
pone(double x)267 pone(double x)
268 {
269 const double *p,*q;
270 double z,r,s;
271 int32_t ix;
272 GET_HIGH_WORD(ix,x);
273 ix &= 0x7fffffff;
274 if(ix>=0x40200000) {p = pr8; q= ps8;}
275 else if(ix>=0x40122E8B){p = pr5; q= ps5;}
276 else if(ix>=0x4006DB6D){p = pr3; q= ps3;}
277 else {p = pr2; q= ps2;} /* ix>=0x40000000 */
278 z = one/(x*x);
279 r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
280 s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*q[4]))));
281 return one+ r/s;
282 }
283
284
285 /* For x >= 8, the asymptotic expansions of qone is
286 * 3/8 s - 105/1024 s^3 - ..., where s = 1/x.
287 * We approximate pone by
288 * qone(x) = s*(0.375 + (R/S))
289 * where R = qr1*s^2 + qr2*s^4 + ... + qr5*s^10
290 * S = 1 + qs1*s^2 + ... + qs6*s^12
291 * and
292 * | qone(x)/s -0.375-R/S | <= 2 ** ( -61.13)
293 */
294
295 static const double qr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
296 0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */
297 -1.02539062499992714161e-01, /* 0xBFBA3FFF, 0xFFFFFDF3 */
298 -1.62717534544589987888e+01, /* 0xC0304591, 0xA26779F7 */
299 -7.59601722513950107896e+02, /* 0xC087BCD0, 0x53E4B576 */
300 -1.18498066702429587167e+04, /* 0xC0C724E7, 0x40F87415 */
301 -4.84385124285750353010e+04, /* 0xC0E7A6D0, 0x65D09C6A */
302 };
303 static const double qs8[6] = {
304 1.61395369700722909556e+02, /* 0x40642CA6, 0xDE5BCDE5 */
305 7.82538599923348465381e+03, /* 0x40BE9162, 0xD0D88419 */
306 1.33875336287249578163e+05, /* 0x4100579A, 0xB0B75E98 */
307 7.19657723683240939863e+05, /* 0x4125F653, 0x72869C19 */
308 6.66601232617776375264e+05, /* 0x412457D2, 0x7719AD5C */
309 -2.94490264303834643215e+05, /* 0xC111F969, 0x0EA5AA18 */
310 };
311
312 static const double qr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
313 -2.08979931141764104297e-11, /* 0xBDB6FA43, 0x1AA1A098 */
314 -1.02539050241375426231e-01, /* 0xBFBA3FFF, 0xCB597FEF */
315 -8.05644828123936029840e+00, /* 0xC0201CE6, 0xCA03AD4B */
316 -1.83669607474888380239e+02, /* 0xC066F56D, 0x6CA7B9B0 */
317 -1.37319376065508163265e+03, /* 0xC09574C6, 0x6931734F */
318 -2.61244440453215656817e+03, /* 0xC0A468E3, 0x88FDA79D */
319 };
320 static const double qs5[6] = {
321 8.12765501384335777857e+01, /* 0x405451B2, 0xFF5A11B2 */
322 1.99179873460485964642e+03, /* 0x409F1F31, 0xE77BF839 */
323 1.74684851924908907677e+04, /* 0x40D10F1F, 0x0D64CE29 */
324 4.98514270910352279316e+04, /* 0x40E8576D, 0xAABAD197 */
325 2.79480751638918118260e+04, /* 0x40DB4B04, 0xCF7C364B */
326 -4.71918354795128470869e+03, /* 0xC0B26F2E, 0xFCFFA004 */
327 };
328
329 static const double qr3[6] = {
330 -5.07831226461766561369e-09, /* 0xBE35CFA9, 0xD38FC84F */
331 -1.02537829820837089745e-01, /* 0xBFBA3FEB, 0x51AEED54 */
332 -4.61011581139473403113e+00, /* 0xC01270C2, 0x3302D9FF */
333 -5.78472216562783643212e+01, /* 0xC04CEC71, 0xC25D16DA */
334 -2.28244540737631695038e+02, /* 0xC06C87D3, 0x4718D55F */
335 -2.19210128478909325622e+02, /* 0xC06B66B9, 0x5F5C1BF6 */
336 };
337 static const double qs3[6] = {
338 4.76651550323729509273e+01, /* 0x4047D523, 0xCCD367E4 */
339 6.73865112676699709482e+02, /* 0x40850EEB, 0xC031EE3E */
340 3.38015286679526343505e+03, /* 0x40AA684E, 0x448E7C9A */
341 5.54772909720722782367e+03, /* 0x40B5ABBA, 0xA61D54A6 */
342 1.90311919338810798763e+03, /* 0x409DBC7A, 0x0DD4DF4B */
343 -1.35201191444307340817e+02, /* 0xC060E670, 0x290A311F */
344 };
345
346 static const double qr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
347 -1.78381727510958865572e-07, /* 0xBE87F126, 0x44C626D2 */
348 -1.02517042607985553460e-01, /* 0xBFBA3E8E, 0x9148B010 */
349 -2.75220568278187460720e+00, /* 0xC0060484, 0x69BB4EDA */
350 -1.96636162643703720221e+01, /* 0xC033A9E2, 0xC168907F */
351 -4.23253133372830490089e+01, /* 0xC04529A3, 0xDE104AAA */
352 -2.13719211703704061733e+01, /* 0xC0355F36, 0x39CF6E52 */
353 };
354 static const double qs2[6] = {
355 2.95333629060523854548e+01, /* 0x403D888A, 0x78AE64FF */
356 2.52981549982190529136e+02, /* 0x406F9F68, 0xDB821CBA */
357 7.57502834868645436472e+02, /* 0x4087AC05, 0xCE49A0F7 */
358 7.39393205320467245656e+02, /* 0x40871B25, 0x48D4C029 */
359 1.55949003336666123687e+02, /* 0x40637E5E, 0x3C3ED8D4 */
360 -4.95949898822628210127e+00, /* 0xC013D686, 0xE71BE86B */
361 };
362
363 static __inline double
qone(double x)364 qone(double x)
365 {
366 const double *p,*q;
367 double s,r,z;
368 int32_t ix;
369 GET_HIGH_WORD(ix,x);
370 ix &= 0x7fffffff;
371 if(ix>=0x40200000) {p = qr8; q= qs8;}
372 else if(ix>=0x40122E8B){p = qr5; q= qs5;}
373 else if(ix>=0x4006DB6D){p = qr3; q= qs3;}
374 else {p = qr2; q= qs2;} /* ix>=0x40000000 */
375 z = one/(x*x);
376 r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
377 s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*(q[4]+z*q[5])))));
378 return (.375 + r/s)/x;
379 }
380