xref: /freebsd/lib/msun/src/e_j1.c (revision 7fdf597e96a02165cfe22ff357b857d5fa15ed8a)
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
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
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
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
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