1 /* @(#)e_j1.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_j1(x), __ieee754_y1(x)
17 * Bessel function of the first and second kinds of order zero.
18 * Method -- j1(x):
19 * 1. For tiny x, we use j1(x) = x/2 - x^3/16 + x^5/384 - ...
20 * 2. Reduce x to |x| since j1(x)=-j1(-x), and
21 * for x in (0,2)
22 * j1(x) = x/2 + x*z*R0/S0, where z = x*x;
23 * (precision: |j1/x - 1/2 - R0/S0 |<2**-61.51 )
24 * for x in (2,inf)
25 * j1(x) = sqrt(2/(pi*x))*(p1(x)*cos(x1)-q1(x)*sin(x1))
26 * y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x1)+q1(x)*cos(x1))
27 * where x1 = x-3*pi/4. It is better to compute sin(x1),cos(x1)
28 * as follow:
29 * cos(x1) = cos(x)cos(3pi/4)+sin(x)sin(3pi/4)
30 * = 1/sqrt(2) * (sin(x) - cos(x))
31 * sin(x1) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4)
32 * = -1/sqrt(2) * (sin(x) + cos(x))
33 * (To avoid cancellation, use
34 * sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
35 * to compute the worse one.)
36 *
37 * 3 Special cases
38 * j1(nan)= nan
39 * j1(0) = 0
40 * j1(inf) = 0
41 *
42 * Method -- y1(x):
43 * 1. screen out x<=0 cases: y1(0)=-inf, y1(x<0)=NaN
44 * 2. For x<2.
45 * Since
46 * y1(x) = 2/pi*(j1(x)*(ln(x/2)+Euler)-1/x-x/2+5/64*x^3-...)
47 * therefore y1(x)-2/pi*j1(x)*ln(x)-1/x is an odd function.
48 * We use the following function to approximate y1,
49 * y1(x) = x*U(z)/V(z) + (2/pi)*(j1(x)*ln(x)-1/x), z= x^2
50 * where for x in [0,2] (abs err less than 2**-65.89)
51 * U(z) = U0[0] + U0[1]*z + ... + U0[4]*z^4
52 * V(z) = 1 + v0[0]*z + ... + v0[4]*z^5
53 * Note: For tiny x, 1/x dominate y1 and hence
54 * y1(tiny) = -2/pi/tiny, (choose tiny<2**-54)
55 * 3. For x>=2.
56 * y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x1)+q1(x)*cos(x1))
57 * where x1 = x-3*pi/4. It is better to compute sin(x1),cos(x1)
58 * by method mentioned above.
59 */
60
61 #include <errno.h>
62 #include <float.h>
63 #include <math.h>
64 #include <math-narrow-eval.h>
65 #include <math_private.h>
66 #include <math-underflow.h>
67 #include <libm-alias-finite.h>
68
69 static double pone (double), qone (double);
70
71 static const double
72 huge = 1e300,
73 one = 1.0,
74 invsqrtpi = 5.64189583547756279280e-01, /* 0x3FE20DD7, 0x50429B6D */
75 tpi = 6.36619772367581382433e-01, /* 0x3FE45F30, 0x6DC9C883 */
76 /* R0/S0 on [0,2] */
77 R[] = { -6.25000000000000000000e-02, /* 0xBFB00000, 0x00000000 */
78 1.40705666955189706048e-03, /* 0x3F570D9F, 0x98472C61 */
79 -1.59955631084035597520e-05, /* 0xBEF0C5C6, 0xBA169668 */
80 4.96727999609584448412e-08 }, /* 0x3E6AAAFA, 0x46CA0BD9 */
81 S[] = { 0.0, 1.91537599538363460805e-02, /* 0x3F939D0B, 0x12637E53 */
82 1.85946785588630915560e-04, /* 0x3F285F56, 0xB9CDF664 */
83 1.17718464042623683263e-06, /* 0x3EB3BFF8, 0x333F8498 */
84 5.04636257076217042715e-09, /* 0x3E35AC88, 0xC97DFF2C */
85 1.23542274426137913908e-11 }; /* 0x3DAB2ACF, 0xCFB97ED8 */
86
87 static const double zero = 0.0;
88
89 double
__ieee754_j1(double x)90 __ieee754_j1 (double x)
91 {
92 double z, s, c, ss, cc, r, u, v, y, r1, r2, s1, s2, s3, z2, z4;
93 int32_t hx, ix;
94
95 GET_HIGH_WORD (hx, x);
96 ix = hx & 0x7fffffff;
97 if (__glibc_unlikely (ix >= 0x7ff00000))
98 return one / x;
99 y = fabs (x);
100 if (ix >= 0x40000000) /* |x| >= 2.0 */
101 {
102 __sincos (y, &s, &c);
103 ss = -s - c;
104 cc = s - c;
105 if (ix < 0x7fe00000) /* make sure y+y not overflow */
106 {
107 z = __cos (y + y);
108 if ((s * c) > zero)
109 cc = z / ss;
110 else
111 ss = z / cc;
112 }
113 /*
114 * j1(x) = 1/sqrt(pi) * (P(1,x)*cc - Q(1,x)*ss) / sqrt(x)
115 * y1(x) = 1/sqrt(pi) * (P(1,x)*ss + Q(1,x)*cc) / sqrt(x)
116 */
117 if (ix > 0x48000000)
118 z = (invsqrtpi * cc) / sqrt (y);
119 else
120 {
121 u = pone (y); v = qone (y);
122 z = invsqrtpi * (u * cc - v * ss) / sqrt (y);
123 }
124 if (hx < 0)
125 return -z;
126 else
127 return z;
128 }
129 if (__glibc_unlikely (ix < 0x3e400000)) /* |x|<2**-27 */
130 {
131 if (huge + x > one) /* inexact if x!=0 necessary */
132 {
133 double ret = math_narrow_eval (0.5 * x);
134 math_check_force_underflow (ret);
135 if (ret == 0 && x != 0)
136 __set_errno (ERANGE);
137 return ret;
138 }
139 }
140 z = x * x;
141 r1 = z * R[0]; z2 = z * z;
142 r2 = R[1] + z * R[2]; z4 = z2 * z2;
143 r = r1 + z2 * r2 + z4 * R[3];
144 r *= x;
145 s1 = one + z * S[1];
146 s2 = S[2] + z * S[3];
147 s3 = S[4] + z * S[5];
148 s = s1 + z2 * s2 + z4 * s3;
149 return (x * 0.5 + r / s);
150 }
151 libm_alias_finite (__ieee754_j1, __j1)
152
153 static const double U0[5] = {
154 -1.96057090646238940668e-01, /* 0xBFC91866, 0x143CBC8A */
155 5.04438716639811282616e-02, /* 0x3FA9D3C7, 0x76292CD1 */
156 -1.91256895875763547298e-03, /* 0xBF5F55E5, 0x4844F50F */
157 2.35252600561610495928e-05, /* 0x3EF8AB03, 0x8FA6B88E */
158 -9.19099158039878874504e-08, /* 0xBE78AC00, 0x569105B8 */
159 };
160 static const double V0[5] = {
161 1.99167318236649903973e-02, /* 0x3F94650D, 0x3F4DA9F0 */
162 2.02552581025135171496e-04, /* 0x3F2A8C89, 0x6C257764 */
163 1.35608801097516229404e-06, /* 0x3EB6C05A, 0x894E8CA6 */
164 6.22741452364621501295e-09, /* 0x3E3ABF1D, 0x5BA69A86 */
165 1.66559246207992079114e-11, /* 0x3DB25039, 0xDACA772A */
166 };
167
168 double
__ieee754_y1(double x)169 __ieee754_y1 (double x)
170 {
171 double z, s, c, ss, cc, u, v, u1, u2, v1, v2, v3, z2, z4;
172 int32_t hx, ix, lx;
173
174 EXTRACT_WORDS (hx, lx, x);
175 ix = 0x7fffffff & hx;
176 /* if Y1(NaN) is NaN, Y1(-inf) is NaN, Y1(inf) is 0 */
177 if (__glibc_unlikely (ix >= 0x7ff00000))
178 return one / (x + x * x);
179 if (__glibc_unlikely ((ix | lx) == 0))
180 return -1 / zero; /* -inf and divide by zero exception. */
181 /* -inf and overflow exception. */;
182 if (__glibc_unlikely (hx < 0))
183 return zero / (zero * x);
184 if (ix >= 0x40000000) /* |x| >= 2.0 */
185 {
186 __sincos (x, &s, &c);
187 ss = -s - c;
188 cc = s - c;
189 if (ix < 0x7fe00000) /* make sure x+x not overflow */
190 {
191 z = __cos (x + x);
192 if ((s * c) > zero)
193 cc = z / ss;
194 else
195 ss = z / cc;
196 }
197 /* y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x0)+q1(x)*cos(x0))
198 * where x0 = x-3pi/4
199 * Better formula:
200 * cos(x0) = cos(x)cos(3pi/4)+sin(x)sin(3pi/4)
201 * = 1/sqrt(2) * (sin(x) - cos(x))
202 * sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4)
203 * = -1/sqrt(2) * (cos(x) + sin(x))
204 * To avoid cancellation, use
205 * sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
206 * to compute the worse one.
207 */
208 if (ix > 0x48000000)
209 z = (invsqrtpi * ss) / sqrt (x);
210 else
211 {
212 u = pone (x); v = qone (x);
213 z = invsqrtpi * (u * ss + v * cc) / sqrt (x);
214 }
215 return z;
216 }
217 if (__glibc_unlikely (ix <= 0x3c900000)) /* x < 2**-54 */
218 {
219 z = -tpi / x;
220 if (isinf (z))
221 __set_errno (ERANGE);
222 return z;
223 }
224 z = x * x;
225 u1 = U0[0] + z * U0[1]; z2 = z * z;
226 u2 = U0[2] + z * U0[3]; z4 = z2 * z2;
227 u = u1 + z2 * u2 + z4 * U0[4];
228 v1 = one + z * V0[0];
229 v2 = V0[1] + z * V0[2];
230 v3 = V0[3] + z * V0[4];
231 v = v1 + z2 * v2 + z4 * v3;
232 return (x * (u / v) + tpi * (__ieee754_j1 (x) * __ieee754_log (x) - one / x));
233 }
234 libm_alias_finite (__ieee754_y1, __y1)
235
236 /* For x >= 8, the asymptotic expansions of pone is
237 * 1 + 15/128 s^2 - 4725/2^15 s^4 - ..., where s = 1/x.
238 * We approximate pone by
239 * pone(x) = 1 + (R/S)
240 * where R = pr0 + pr1*s^2 + pr2*s^4 + ... + pr5*s^10
241 * S = 1 + ps0*s^2 + ... + ps4*s^10
242 * and
243 * | pone(x)-1-R/S | <= 2 ** ( -60.06)
244 */
245
246 static const double pr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
247 0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */
248 1.17187499999988647970e-01, /* 0x3FBDFFFF, 0xFFFFFCCE */
249 1.32394806593073575129e+01, /* 0x402A7A9D, 0x357F7FCE */
250 4.12051854307378562225e+02, /* 0x4079C0D4, 0x652EA590 */
251 3.87474538913960532227e+03, /* 0x40AE457D, 0xA3A532CC */
252 7.91447954031891731574e+03, /* 0x40BEEA7A, 0xC32782DD */
253 };
254 static const double ps8[5] = {
255 1.14207370375678408436e+02, /* 0x405C8D45, 0x8E656CAC */
256 3.65093083420853463394e+03, /* 0x40AC85DC, 0x964D274F */
257 3.69562060269033463555e+04, /* 0x40E20B86, 0x97C5BB7F */
258 9.76027935934950801311e+04, /* 0x40F7D42C, 0xB28F17BB */
259 3.08042720627888811578e+04, /* 0x40DE1511, 0x697A0B2D */
260 };
261
262 static const double pr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
263 1.31990519556243522749e-11, /* 0x3DAD0667, 0xDAE1CA7D */
264 1.17187493190614097638e-01, /* 0x3FBDFFFF, 0xE2C10043 */
265 6.80275127868432871736e+00, /* 0x401B3604, 0x6E6315E3 */
266 1.08308182990189109773e+02, /* 0x405B13B9, 0x452602ED */
267 5.17636139533199752805e+02, /* 0x40802D16, 0xD052D649 */
268 5.28715201363337541807e+02, /* 0x408085B8, 0xBB7E0CB7 */
269 };
270 static const double ps5[5] = {
271 5.92805987221131331921e+01, /* 0x404DA3EA, 0xA8AF633D */
272 9.91401418733614377743e+02, /* 0x408EFB36, 0x1B066701 */
273 5.35326695291487976647e+03, /* 0x40B4E944, 0x5706B6FB */
274 7.84469031749551231769e+03, /* 0x40BEA4B0, 0xB8A5BB15 */
275 1.50404688810361062679e+03, /* 0x40978030, 0x036F5E51 */
276 };
277
278 static const double pr3[6] = {
279 3.02503916137373618024e-09, /* 0x3E29FC21, 0xA7AD9EDD */
280 1.17186865567253592491e-01, /* 0x3FBDFFF5, 0x5B21D17B */
281 3.93297750033315640650e+00, /* 0x400F76BC, 0xE85EAD8A */
282 3.51194035591636932736e+01, /* 0x40418F48, 0x9DA6D129 */
283 9.10550110750781271918e+01, /* 0x4056C385, 0x4D2C1837 */
284 4.85590685197364919645e+01, /* 0x4048478F, 0x8EA83EE5 */
285 };
286 static const double ps3[5] = {
287 3.47913095001251519989e+01, /* 0x40416549, 0xA134069C */
288 3.36762458747825746741e+02, /* 0x40750C33, 0x07F1A75F */
289 1.04687139975775130551e+03, /* 0x40905B7C, 0x5037D523 */
290 8.90811346398256432622e+02, /* 0x408BD67D, 0xA32E31E9 */
291 1.03787932439639277504e+02, /* 0x4059F26D, 0x7C2EED53 */
292 };
293
294 static const double pr2[6] = { /* for x in [2.8570,2]=1/[0.3499,0.5] */
295 1.07710830106873743082e-07, /* 0x3E7CE9D4, 0xF65544F4 */
296 1.17176219462683348094e-01, /* 0x3FBDFF42, 0xBE760D83 */
297 2.36851496667608785174e+00, /* 0x4002F2B7, 0xF98FAEC0 */
298 1.22426109148261232917e+01, /* 0x40287C37, 0x7F71A964 */
299 1.76939711271687727390e+01, /* 0x4031B1A8, 0x177F8EE2 */
300 5.07352312588818499250e+00, /* 0x40144B49, 0xA574C1FE */
301 };
302 static const double ps2[5] = {
303 2.14364859363821409488e+01, /* 0x40356FBD, 0x8AD5ECDC */
304 1.25290227168402751090e+02, /* 0x405F5293, 0x14F92CD5 */
305 2.32276469057162813669e+02, /* 0x406D08D8, 0xD5A2DBD9 */
306 1.17679373287147100768e+02, /* 0x405D6B7A, 0xDA1884A9 */
307 8.36463893371618283368e+00, /* 0x4020BAB1, 0xF44E5192 */
308 };
309
310 static double
pone(double x)311 pone (double x)
312 {
313 const double *p, *q;
314 double z, r, s, r1, r2, r3, s1, s2, s3, z2, z4;
315 int32_t ix;
316 GET_HIGH_WORD (ix, x);
317 ix &= 0x7fffffff;
318 /* ix >= 0x40000000 for all calls to this function. */
319 if (ix >= 0x41b00000)
320 {
321 return one;
322 }
323 else if (ix >= 0x40200000)
324 {
325 p = pr8; q = ps8;
326 }
327 else if (ix >= 0x40122E8B)
328 {
329 p = pr5; q = ps5;
330 }
331 else if (ix >= 0x4006DB6D)
332 {
333 p = pr3; q = ps3;
334 }
335 else
336 {
337 p = pr2; q = ps2;
338 }
339 z = one / (x * x);
340 r1 = p[0] + z * p[1]; z2 = z * z;
341 r2 = p[2] + z * p[3]; z4 = z2 * z2;
342 r3 = p[4] + z * p[5];
343 r = r1 + z2 * r2 + z4 * r3;
344 s1 = one + z * q[0];
345 s2 = q[1] + z * q[2];
346 s3 = q[3] + z * q[4];
347 s = s1 + z2 * s2 + z4 * s3;
348 return one + r / s;
349 }
350
351
352 /* For x >= 8, the asymptotic expansions of qone is
353 * 3/8 s - 105/1024 s^3 - ..., where s = 1/x.
354 * We approximate pone by
355 * qone(x) = s*(0.375 + (R/S))
356 * where R = qr1*s^2 + qr2*s^4 + ... + qr5*s^10
357 * S = 1 + qs1*s^2 + ... + qs6*s^12
358 * and
359 * | qone(x)/s -0.375-R/S | <= 2 ** ( -61.13)
360 */
361
362 static const double qr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
363 0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */
364 -1.02539062499992714161e-01, /* 0xBFBA3FFF, 0xFFFFFDF3 */
365 -1.62717534544589987888e+01, /* 0xC0304591, 0xA26779F7 */
366 -7.59601722513950107896e+02, /* 0xC087BCD0, 0x53E4B576 */
367 -1.18498066702429587167e+04, /* 0xC0C724E7, 0x40F87415 */
368 -4.84385124285750353010e+04, /* 0xC0E7A6D0, 0x65D09C6A */
369 };
370 static const double qs8[6] = {
371 1.61395369700722909556e+02, /* 0x40642CA6, 0xDE5BCDE5 */
372 7.82538599923348465381e+03, /* 0x40BE9162, 0xD0D88419 */
373 1.33875336287249578163e+05, /* 0x4100579A, 0xB0B75E98 */
374 7.19657723683240939863e+05, /* 0x4125F653, 0x72869C19 */
375 6.66601232617776375264e+05, /* 0x412457D2, 0x7719AD5C */
376 -2.94490264303834643215e+05, /* 0xC111F969, 0x0EA5AA18 */
377 };
378
379 static const double qr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
380 -2.08979931141764104297e-11, /* 0xBDB6FA43, 0x1AA1A098 */
381 -1.02539050241375426231e-01, /* 0xBFBA3FFF, 0xCB597FEF */
382 -8.05644828123936029840e+00, /* 0xC0201CE6, 0xCA03AD4B */
383 -1.83669607474888380239e+02, /* 0xC066F56D, 0x6CA7B9B0 */
384 -1.37319376065508163265e+03, /* 0xC09574C6, 0x6931734F */
385 -2.61244440453215656817e+03, /* 0xC0A468E3, 0x88FDA79D */
386 };
387 static const double qs5[6] = {
388 8.12765501384335777857e+01, /* 0x405451B2, 0xFF5A11B2 */
389 1.99179873460485964642e+03, /* 0x409F1F31, 0xE77BF839 */
390 1.74684851924908907677e+04, /* 0x40D10F1F, 0x0D64CE29 */
391 4.98514270910352279316e+04, /* 0x40E8576D, 0xAABAD197 */
392 2.79480751638918118260e+04, /* 0x40DB4B04, 0xCF7C364B */
393 -4.71918354795128470869e+03, /* 0xC0B26F2E, 0xFCFFA004 */
394 };
395
396 static const double qr3[6] = {
397 -5.07831226461766561369e-09, /* 0xBE35CFA9, 0xD38FC84F */
398 -1.02537829820837089745e-01, /* 0xBFBA3FEB, 0x51AEED54 */
399 -4.61011581139473403113e+00, /* 0xC01270C2, 0x3302D9FF */
400 -5.78472216562783643212e+01, /* 0xC04CEC71, 0xC25D16DA */
401 -2.28244540737631695038e+02, /* 0xC06C87D3, 0x4718D55F */
402 -2.19210128478909325622e+02, /* 0xC06B66B9, 0x5F5C1BF6 */
403 };
404 static const double qs3[6] = {
405 4.76651550323729509273e+01, /* 0x4047D523, 0xCCD367E4 */
406 6.73865112676699709482e+02, /* 0x40850EEB, 0xC031EE3E */
407 3.38015286679526343505e+03, /* 0x40AA684E, 0x448E7C9A */
408 5.54772909720722782367e+03, /* 0x40B5ABBA, 0xA61D54A6 */
409 1.90311919338810798763e+03, /* 0x409DBC7A, 0x0DD4DF4B */
410 -1.35201191444307340817e+02, /* 0xC060E670, 0x290A311F */
411 };
412
413 static const double qr2[6] = { /* for x in [2.8570,2]=1/[0.3499,0.5] */
414 -1.78381727510958865572e-07, /* 0xBE87F126, 0x44C626D2 */
415 -1.02517042607985553460e-01, /* 0xBFBA3E8E, 0x9148B010 */
416 -2.75220568278187460720e+00, /* 0xC0060484, 0x69BB4EDA */
417 -1.96636162643703720221e+01, /* 0xC033A9E2, 0xC168907F */
418 -4.23253133372830490089e+01, /* 0xC04529A3, 0xDE104AAA */
419 -2.13719211703704061733e+01, /* 0xC0355F36, 0x39CF6E52 */
420 };
421 static const double qs2[6] = {
422 2.95333629060523854548e+01, /* 0x403D888A, 0x78AE64FF */
423 2.52981549982190529136e+02, /* 0x406F9F68, 0xDB821CBA */
424 7.57502834868645436472e+02, /* 0x4087AC05, 0xCE49A0F7 */
425 7.39393205320467245656e+02, /* 0x40871B25, 0x48D4C029 */
426 1.55949003336666123687e+02, /* 0x40637E5E, 0x3C3ED8D4 */
427 -4.95949898822628210127e+00, /* 0xC013D686, 0xE71BE86B */
428 };
429
430 static double
qone(double x)431 qone (double x)
432 {
433 const double *p, *q;
434 double s, r, z, r1, r2, r3, s1, s2, s3, z2, z4, z6;
435 int32_t ix;
436 GET_HIGH_WORD (ix, x);
437 ix &= 0x7fffffff;
438 /* ix >= 0x40000000 for all calls to this function. */
439 if (ix >= 0x41b00000)
440 {
441 return .375 / x;
442 }
443 else if (ix >= 0x40200000)
444 {
445 p = qr8; q = qs8;
446 }
447 else if (ix >= 0x40122E8B)
448 {
449 p = qr5; q = qs5;
450 }
451 else if (ix >= 0x4006DB6D)
452 {
453 p = qr3; q = qs3;
454 }
455 else
456 {
457 p = qr2; q = qs2;
458 }
459 z = one / (x * x);
460 r1 = p[0] + z * p[1]; z2 = z * z;
461 r2 = p[2] + z * p[3]; z4 = z2 * z2;
462 r3 = p[4] + z * p[5]; z6 = z4 * z2;
463 r = r1 + z2 * r2 + z4 * r3;
464 s1 = one + z * q[0];
465 s2 = q[1] + z * q[2];
466 s3 = q[3] + z * q[4];
467 s = s1 + z2 * s2 + z4 * s3 + z6 * q[5];
468 return (.375 + r / s) / x;
469 }
470