1
2 /* @(#)e_j1.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 /* j1(x), y1(x)
15 * Bessel function of the first and second kinds of order zero.
16 * Method -- j1(x):
17 * 1. For tiny x, we use j1(x) = x/2 - x^3/16 + x^5/384 - ...
18 * 2. Reduce x to |x| since j1(x)=-j1(-x), and
19 * for x in (0,2)
20 * j1(x) = x/2 + x*z*R0/S0, where z = x*x;
21 * (precision: |j1/x - 1/2 - R0/S0 |<2**-61.51 )
22 * for x in (2,inf)
23 * j1(x) = sqrt(2/(pi*x))*(p1(x)*cos(x1)-q1(x)*sin(x1))
24 * y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x1)+q1(x)*cos(x1))
25 * where x1 = x-3*pi/4. It is better to compute sin(x1),cos(x1)
26 * as follow:
27 * cos(x1) = cos(x)cos(3pi/4)+sin(x)sin(3pi/4)
28 * = 1/sqrt(2) * (sin(x) - cos(x))
29 * sin(x1) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4)
30 * = -1/sqrt(2) * (sin(x) + cos(x))
31 * (To avoid cancellation, use
32 * sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
33 * to compute the worse one.)
34 *
35 * 3 Special cases
36 * j1(nan)= nan
37 * j1(0) = 0
38 * j1(inf) = 0
39 *
40 * Method -- y1(x):
41 * 1. screen out x<=0 cases: y1(0)=-inf, y1(x<0)=NaN
42 * 2. For x<2.
43 * Since
44 * y1(x) = 2/pi*(j1(x)*(ln(x/2)+Euler)-1/x-x/2+5/64*x^3-...)
45 * therefore y1(x)-2/pi*j1(x)*ln(x)-1/x is an odd function.
46 * We use the following function to approximate y1,
47 * y1(x) = x*U(z)/V(z) + (2/pi)*(j1(x)*ln(x)-1/x), z= x^2
48 * where for x in [0,2] (abs err less than 2**-65.89)
49 * U(z) = U0[0] + U0[1]*z + ... + U0[4]*z^4
50 * V(z) = 1 + v0[0]*z + ... + v0[4]*z^5
51 * Note: For tiny x, 1/x dominate y1 and hence
52 * y1(tiny) = -2/pi/tiny, (choose tiny<2**-54)
53 * 3. For x>=2.
54 * y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x1)+q1(x)*cos(x1))
55 * where x1 = x-3*pi/4. It is better to compute sin(x1),cos(x1)
56 * by method mentioned above.
57 */
58
59 #include "fdlibm.h"
60
61 #ifdef _NEED_FLOAT64
62
63 static __float64 pone(__float64);
64 static __float64 qone(__float64);
65
66 static const __float64
67 one = _F_64(1.0),
68 invsqrtpi =
69 _F_64(5.64189583547756279280e-01), /* 0x3FE20DD7, 0x50429B6D */
70 tpi = _F_64(6.36619772367581382433e-01), /* 0x3FE45F30, 0x6DC9C883 */
71 /* R0/S0 on [0,2] */
72 r00 = _F_64(-6.25000000000000000000e-02), /* 0xBFB00000, 0x00000000 */
73 r01 = _F_64(1.40705666955189706048e-03), /* 0x3F570D9F, 0x98472C61 */
74 r02 = _F_64(-1.59955631084035597520e-05), /* 0xBEF0C5C6, 0xBA169668 */
75 r03 = _F_64(4.96727999609584448412e-08), /* 0x3E6AAAFA, 0x46CA0BD9 */
76 s01 = _F_64(1.91537599538363460805e-02), /* 0x3F939D0B, 0x12637E53 */
77 s02 = _F_64(1.85946785588630915560e-04), /* 0x3F285F56, 0xB9CDF664 */
78 s03 = _F_64(1.17718464042623683263e-06), /* 0x3EB3BFF8, 0x333F8498 */
79 s04 = _F_64(5.04636257076217042715e-09), /* 0x3E35AC88, 0xC97DFF2C */
80 s05 = _F_64(1.23542274426137913908e-11); /* 0x3DAB2ACF, 0xCFB97ED8 */
81
82 static const __float64 zero = _F_64(0.0);
83
84 __float64
j164(__float64 x)85 j164(__float64 x)
86 {
87 __float64 z, s, c, ss, cc, r, u, v, y;
88 __int32_t hx, ix, lx;
89
90 if (isnan(x))
91 return x + x;
92
93 if (isinf(x))
94 return _F_64(0.0);
95
96 GET_HIGH_WORD(hx, x);
97 ix = hx & 0x7fffffff;
98 y = fabs64(x);
99 if (ix >= 0x40000000) { /* |x| >= 2.0 */
100 s = sin64(y);
101 c = cos64(y);
102 ss = -s - c;
103 cc = s - c;
104 if (ix < 0x7fe00000) { /* make sure y+y not overflow */
105 z = cos64(y + y);
106 if ((s * c) > zero)
107 cc = z / ss;
108 else
109 ss = z / cc;
110 }
111 /*
112 * j1(x) = 1/sqrt(pi) * (P(1,x)*cc - Q(1,x)*ss) / sqrt(x)
113 * y1(x) = 1/sqrt(pi) * (P(1,x)*ss + Q(1,x)*cc) / sqrt(x)
114 */
115 if (ix > 0x48000000)
116 z = (invsqrtpi * cc) / sqrt64(y);
117 else {
118 u = pone(y);
119 v = qone(y);
120 z = invsqrtpi * (u * cc - v * ss) / sqrt64(y);
121 }
122 if (hx < 0)
123 return -z;
124 else
125 return z;
126 }
127 if (ix < 0x3e400000) { /* |x|<2**-27 */
128 GET_LOW_WORD(lx, x);
129 if (ix == 0 && lx == 0)
130 return x;
131 return check_uflow(_F_64(0.5) * x); /* inexact if x!=0 necessary */
132 }
133 z = x * x;
134 r = z * (r00 + z * (r01 + z * (r02 + z * r03)));
135 s = one + z * (s01 + z * (s02 + z * (s03 + z * (s04 + z * s05))));
136 r *= x;
137 return (x * _F_64(0.5) + r / s);
138 }
139
140 _MATH_ALIAS_d_d(j1)
141
142 static const __float64 U0[5] = {
143 _F_64(-1.96057090646238940668e-01), /* 0xBFC91866, 0x143CBC8A */
144 _F_64(5.04438716639811282616e-02), /* 0x3FA9D3C7, 0x76292CD1 */
145 _F_64(-1.91256895875763547298e-03), /* 0xBF5F55E5, 0x4844F50F */
146 _F_64(2.35252600561610495928e-05), /* 0x3EF8AB03, 0x8FA6B88E */
147 _F_64(-9.19099158039878874504e-08), /* 0xBE78AC00, 0x569105B8 */
148 };
149 static const __float64 V0[5] = {
150 _F_64(1.99167318236649903973e-02), /* 0x3F94650D, 0x3F4DA9F0 */
151 _F_64(2.02552581025135171496e-04), /* 0x3F2A8C89, 0x6C257764 */
152 _F_64(1.35608801097516229404e-06), /* 0x3EB6C05A, 0x894E8CA6 */
153 _F_64(6.22741452364621501295e-09), /* 0x3E3ABF1D, 0x5BA69A86 */
154 _F_64(1.66559246207992079114e-11), /* 0x3DB25039, 0xDACA772A */
155 };
156
157 __float64
y164(__float64 x)158 y164(__float64 x)
159 {
160 __float64 z, s, c, ss, cc, u, v;
161 __int32_t hx, ix, lx;
162
163 EXTRACT_WORDS(hx, lx, x);
164 ix = 0x7fffffff & hx;
165 /* if Y1(NaN) is NaN, Y1(-inf) is NaN, Y1(inf) is 0 */
166 if ((ix | lx) == 0)
167 return __math_divzero(1);
168
169 if (isnan(x))
170 return x + x;
171
172 if (hx < 0)
173 return __math_invalid(x);
174
175 if (ix >= 0x7ff00000)
176 return _F_64(0.0);
177
178 if (ix >= 0x40000000) { /* |x| >= 2.0 */
179 s = sin64(x);
180 c = cos64(x);
181 ss = -s - c;
182 cc = s - c;
183 if (ix < 0x7fe00000) { /* make sure x+x not overflow */
184 z = cos64(x + x);
185 if ((s * c) > zero)
186 cc = z / ss;
187 else
188 ss = z / cc;
189 }
190 /* y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x0)+q1(x)*cos(x0))
191 * where x0 = x-3pi/4
192 * Better formula:
193 * cos(x0) = cos(x)cos(3pi/4)+sin(x)sin(3pi/4)
194 * = 1/sqrt(2) * (sin(x) - cos(x))
195 * sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4)
196 * = -1/sqrt(2) * (cos(x) + sin(x))
197 * To avoid cancellation, use
198 * sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
199 * to compute the worse one.
200 */
201 if (ix > 0x48000000)
202 z = (invsqrtpi * ss) / sqrt64(x);
203 else {
204 u = pone(x);
205 v = qone(x);
206 z = invsqrtpi * (u * ss + v * cc) / sqrt64(x);
207 }
208 return z;
209 }
210 if (ix <= 0x3c900000) { /* x < 2**-54 */
211 return check_oflow(-tpi / x);
212 }
213 z = x * x;
214 u = U0[0] + z * (U0[1] + z * (U0[2] + z * (U0[3] + z * U0[4])));
215 v = one + z * (V0[0] + z * (V0[1] + z * (V0[2] + z * (V0[3] + z * V0[4]))));
216 return (x * (u / v) + tpi * (j1(x) * log(x) - one / x));
217 }
218
219 _MATH_ALIAS_d_d(y1)
220
221 /* For x >= 8, the asymptotic expansions of pone is
222 * 1 + 15/128 s^2 - 4725/2^15 s^4 - ..., where s = 1/x.
223 * We approximate pone by
224 * pone(x) = 1 + (R/S)
225 * where R = pr0 + pr1*s^2 + pr2*s^4 + ... + pr5*s^10
226 * S = 1 + ps0*s^2 + ... + ps4*s^10
227 * and
228 * | pone(x)-1-R/S | <= 2 ** ( -60.06)
229 */
230
231 static const __float64 pr8[6] = {
232 /* for x in [inf, 8]=1/[0,0.125] */
233 _F_64(0.00000000000000000000e+00), /* 0x00000000, 0x00000000 */
234 _F_64(1.17187499999988647970e-01), /* 0x3FBDFFFF, 0xFFFFFCCE */
235 _F_64(1.32394806593073575129e+01), /* 0x402A7A9D, 0x357F7FCE */
236 _F_64(4.12051854307378562225e+02), /* 0x4079C0D4, 0x652EA590 */
237 _F_64(3.87474538913960532227e+03), /* 0x40AE457D, 0xA3A532CC */
238 _F_64(7.91447954031891731574e+03), /* 0x40BEEA7A, 0xC32782DD */
239 };
240 static const __float64 ps8[5] = {
241 _F_64(1.14207370375678408436e+02), /* 0x405C8D45, 0x8E656CAC */
242 _F_64(3.65093083420853463394e+03), /* 0x40AC85DC, 0x964D274F */
243 _F_64(3.69562060269033463555e+04), /* 0x40E20B86, 0x97C5BB7F */
244 _F_64(9.76027935934950801311e+04), /* 0x40F7D42C, 0xB28F17BB */
245 _F_64(3.08042720627888811578e+04), /* 0x40DE1511, 0x697A0B2D */
246 };
247
248 static const __float64 pr5[6] = {
249 /* for x in [8,4.5454]=1/[0.125,0.22001] */
250 _F_64(1.31990519556243522749e-11), /* 0x3DAD0667, 0xDAE1CA7D */
251 _F_64(1.17187493190614097638e-01), /* 0x3FBDFFFF, 0xE2C10043 */
252 _F_64(6.80275127868432871736e+00), /* 0x401B3604, 0x6E6315E3 */
253 _F_64(1.08308182990189109773e+02), /* 0x405B13B9, 0x452602ED */
254 _F_64(5.17636139533199752805e+02), /* 0x40802D16, 0xD052D649 */
255 _F_64(5.28715201363337541807e+02), /* 0x408085B8, 0xBB7E0CB7 */
256 };
257 static const __float64 ps5[5] = {
258 _F_64(5.92805987221131331921e+01), /* 0x404DA3EA, 0xA8AF633D */
259 _F_64(9.91401418733614377743e+02), /* 0x408EFB36, 0x1B066701 */
260 _F_64(5.35326695291487976647e+03), /* 0x40B4E944, 0x5706B6FB */
261 _F_64(7.84469031749551231769e+03), /* 0x40BEA4B0, 0xB8A5BB15 */
262 _F_64(1.50404688810361062679e+03), /* 0x40978030, 0x036F5E51 */
263 };
264
265 static const __float64 pr3[6] = {
266 _F_64(3.02503916137373618024e-09), /* 0x3E29FC21, 0xA7AD9EDD */
267 _F_64(1.17186865567253592491e-01), /* 0x3FBDFFF5, 0x5B21D17B */
268 _F_64(3.93297750033315640650e+00), /* 0x400F76BC, 0xE85EAD8A */
269 _F_64(3.51194035591636932736e+01), /* 0x40418F48, 0x9DA6D129 */
270 _F_64(9.10550110750781271918e+01), /* 0x4056C385, 0x4D2C1837 */
271 _F_64(4.85590685197364919645e+01), /* 0x4048478F, 0x8EA83EE5 */
272 };
273 static const __float64 ps3[5] = {
274 _F_64(3.47913095001251519989e+01), /* 0x40416549, 0xA134069C */
275 _F_64(3.36762458747825746741e+02), /* 0x40750C33, 0x07F1A75F */
276 _F_64(1.04687139975775130551e+03), /* 0x40905B7C, 0x5037D523 */
277 _F_64(8.90811346398256432622e+02), /* 0x408BD67D, 0xA32E31E9 */
278 _F_64(1.03787932439639277504e+02), /* 0x4059F26D, 0x7C2EED53 */
279 };
280
281 static const __float64 pr2[6] = {
282 /* for x in [2.8570,2]=1/[0.3499,0.5] */
283 _F_64(1.07710830106873743082e-07), /* 0x3E7CE9D4, 0xF65544F4 */
284 _F_64(1.17176219462683348094e-01), /* 0x3FBDFF42, 0xBE760D83 */
285 _F_64(2.36851496667608785174e+00), /* 0x4002F2B7, 0xF98FAEC0 */
286 _F_64(1.22426109148261232917e+01), /* 0x40287C37, 0x7F71A964 */
287 _F_64(1.76939711271687727390e+01), /* 0x4031B1A8, 0x177F8EE2 */
288 _F_64(5.07352312588818499250e+00), /* 0x40144B49, 0xA574C1FE */
289 };
290 static const __float64 ps2[5] = {
291 _F_64(2.14364859363821409488e+01), /* 0x40356FBD, 0x8AD5ECDC */
292 _F_64(1.25290227168402751090e+02), /* 0x405F5293, 0x14F92CD5 */
293 _F_64(2.32276469057162813669e+02), /* 0x406D08D8, 0xD5A2DBD9 */
294 _F_64(1.17679373287147100768e+02), /* 0x405D6B7A, 0xDA1884A9 */
295 _F_64(8.36463893371618283368e+00), /* 0x4020BAB1, 0xF44E5192 */
296 };
297
298 static __float64
pone(__float64 x)299 pone(__float64 x)
300 {
301 const __float64 *p, *q;
302 __float64 z, r, s;
303 __int32_t ix;
304 GET_HIGH_WORD(ix, x);
305 ix &= 0x7fffffff;
306 if (ix >= 0x41b00000) {
307 return one;
308 } else if (ix >= 0x40200000) {
309 p = pr8;
310 q = ps8;
311 } else if (ix >= 0x40122E8B) {
312 p = pr5;
313 q = ps5;
314 } else if (ix >= 0x4006DB6D) {
315 p = pr3;
316 q = ps3;
317 } else {
318 p = pr2;
319 q = ps2;
320 }
321 z = one / (x * x);
322 r = p[0] + z * (p[1] + z * (p[2] + z * (p[3] + z * (p[4] + z * p[5]))));
323 s = one + z * (q[0] + z * (q[1] + z * (q[2] + z * (q[3] + z * q[4]))));
324 return one + r / s;
325 }
326
327 /* For x >= 8, the asymptotic expansions of qone is
328 * 3/8 s - 105/1024 s^3 - ..., where s = 1/x.
329 * We approximate qone by
330 * qone(x) = s*(0.375 + (R/S))
331 * where R = qr1*s^2 + qr2*s^4 + ... + qr5*s^10
332 * S = 1 + qs1*s^2 + ... + qs6*s^12
333 * and
334 * | qone(x)/s -0.375-R/S | <= 2 ** ( -61.13)
335 */
336
337 static const __float64 qr8[6] = {
338 /* for x in [inf, 8]=1/[0,0.125] */
339 _F_64(0.00000000000000000000e+00), /* 0x00000000, 0x00000000 */
340 _F_64(-1.02539062499992714161e-01), /* 0xBFBA3FFF, 0xFFFFFDF3 */
341 _F_64(-1.62717534544589987888e+01), /* 0xC0304591, 0xA26779F7 */
342 _F_64(-7.59601722513950107896e+02), /* 0xC087BCD0, 0x53E4B576 */
343 _F_64(-1.18498066702429587167e+04), /* 0xC0C724E7, 0x40F87415 */
344 _F_64(-4.84385124285750353010e+04), /* 0xC0E7A6D0, 0x65D09C6A */
345 };
346 static const __float64 qs8[6] = {
347 _F_64(1.61395369700722909556e+02), /* 0x40642CA6, 0xDE5BCDE5 */
348 _F_64(7.82538599923348465381e+03), /* 0x40BE9162, 0xD0D88419 */
349 _F_64(1.33875336287249578163e+05), /* 0x4100579A, 0xB0B75E98 */
350 _F_64(7.19657723683240939863e+05), /* 0x4125F653, 0x72869C19 */
351 _F_64(6.66601232617776375264e+05), /* 0x412457D2, 0x7719AD5C */
352 _F_64(-2.94490264303834643215e+05), /* 0xC111F969, 0x0EA5AA18 */
353 };
354
355 static const __float64 qr5[6] = {
356 /* for x in [8,4.5454]=1/[0.125,0.22001] */
357 _F_64(-2.08979931141764104297e-11), /* 0xBDB6FA43, 0x1AA1A098 */
358 _F_64(-1.02539050241375426231e-01), /* 0xBFBA3FFF, 0xCB597FEF */
359 _F_64(-8.05644828123936029840e+00), /* 0xC0201CE6, 0xCA03AD4B */
360 _F_64(-1.83669607474888380239e+02), /* 0xC066F56D, 0x6CA7B9B0 */
361 _F_64(-1.37319376065508163265e+03), /* 0xC09574C6, 0x6931734F */
362 _F_64(-2.61244440453215656817e+03), /* 0xC0A468E3, 0x88FDA79D */
363 };
364 static const __float64 qs5[6] = {
365 _F_64(8.12765501384335777857e+01), /* 0x405451B2, 0xFF5A11B2 */
366 _F_64(1.99179873460485964642e+03), /* 0x409F1F31, 0xE77BF839 */
367 _F_64(1.74684851924908907677e+04), /* 0x40D10F1F, 0x0D64CE29 */
368 _F_64(4.98514270910352279316e+04), /* 0x40E8576D, 0xAABAD197 */
369 _F_64(2.79480751638918118260e+04), /* 0x40DB4B04, 0xCF7C364B */
370 _F_64(-4.71918354795128470869e+03), /* 0xC0B26F2E, 0xFCFFA004 */
371 };
372
373 static const __float64 qr3[6] = {
374 _F_64(-5.07831226461766561369e-09), /* 0xBE35CFA9, 0xD38FC84F */
375 _F_64(-1.02537829820837089745e-01), /* 0xBFBA3FEB, 0x51AEED54 */
376 _F_64(-4.61011581139473403113e+00), /* 0xC01270C2, 0x3302D9FF */
377 _F_64(-5.78472216562783643212e+01), /* 0xC04CEC71, 0xC25D16DA */
378 _F_64(-2.28244540737631695038e+02), /* 0xC06C87D3, 0x4718D55F */
379 _F_64(-2.19210128478909325622e+02), /* 0xC06B66B9, 0x5F5C1BF6 */
380 };
381 static const __float64 qs3[6] = {
382 _F_64(4.76651550323729509273e+01), /* 0x4047D523, 0xCCD367E4 */
383 _F_64(6.73865112676699709482e+02), /* 0x40850EEB, 0xC031EE3E */
384 _F_64(3.38015286679526343505e+03), /* 0x40AA684E, 0x448E7C9A */
385 _F_64(5.54772909720722782367e+03), /* 0x40B5ABBA, 0xA61D54A6 */
386 _F_64(1.90311919338810798763e+03), /* 0x409DBC7A, 0x0DD4DF4B */
387 _F_64(-1.35201191444307340817e+02), /* 0xC060E670, 0x290A311F */
388 };
389
390 static const __float64 qr2[6] = {
391 /* for x in [2.8570,2]=1/[0.3499,0.5] */
392 _F_64(-1.78381727510958865572e-07), /* 0xBE87F126, 0x44C626D2 */
393 _F_64(-1.02517042607985553460e-01), /* 0xBFBA3E8E, 0x9148B010 */
394 _F_64(-2.75220568278187460720e+00), /* 0xC0060484, 0x69BB4EDA */
395 _F_64(-1.96636162643703720221e+01), /* 0xC033A9E2, 0xC168907F */
396 _F_64(-4.23253133372830490089e+01), /* 0xC04529A3, 0xDE104AAA */
397 _F_64(-2.13719211703704061733e+01), /* 0xC0355F36, 0x39CF6E52 */
398 };
399 static const __float64 qs2[6] = {
400 _F_64(2.95333629060523854548e+01), /* 0x403D888A, 0x78AE64FF */
401 _F_64(2.52981549982190529136e+02), /* 0x406F9F68, 0xDB821CBA */
402 _F_64(7.57502834868645436472e+02), /* 0x4087AC05, 0xCE49A0F7 */
403 _F_64(7.39393205320467245656e+02), /* 0x40871B25, 0x48D4C029 */
404 _F_64(1.55949003336666123687e+02), /* 0x40637E5E, 0x3C3ED8D4 */
405 _F_64(-4.95949898822628210127e+00), /* 0xC013D686, 0xE71BE86B */
406 };
407
408 static __float64
qone(__float64 x)409 qone(__float64 x)
410 {
411 const __float64 *p, *q;
412 __float64 s, r, z;
413 __int32_t ix;
414 GET_HIGH_WORD(ix, x);
415 ix &= 0x7fffffff;
416 if (ix >= 0x41b00000) {
417 return .375 / x;
418 } else if (ix >= 0x40200000) {
419 p = qr8;
420 q = qs8;
421 } else if (ix >= 0x40122E8B) {
422 p = qr5;
423 q = qs5;
424 } else if (ix >= 0x4006DB6D) {
425 p = qr3;
426 q = qs3;
427 } else {
428 p = qr2;
429 q = qs2;
430 }
431 z = one / (x * x);
432 r = p[0] + z * (p[1] + z * (p[2] + z * (p[3] + z * (p[4] + z * p[5]))));
433 s = one +
434 z * (q[0] +
435 z * (q[1] + z * (q[2] + z * (q[3] + z * (q[4] + z * q[5])))));
436 return (.375 + r / s) / x;
437 }
438
439 #endif /* _NEED_FLOAT64 */
440
