GCC Code Coverage Report
Directory: ./ Exec Total Coverage
File: lib/libm/src/e_j1.c Lines: 71 71 100.0 %
Date: 2017-11-07 Branches: 38 48 79.2 %

Line Branch Exec Source
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
13
/* j1(x), y1(x)
14
 * Bessel function of the first and second kinds of order zero.
15
 * Method -- j1(x):
16
 *	1. For tiny x, we use j1(x) = x/2 - x^3/16 + x^5/384 - ...
17
 *	2. Reduce x to |x| since j1(x)=-j1(-x),  and
18
 *	   for x in (0,2)
19
 *		j1(x) = x/2 + x*z*R0/S0,  where z = x*x;
20
 *	   (precision:  |j1/x - 1/2 - R0/S0 |<2**-61.51 )
21
 *	   for x in (2,inf)
22
 * 		j1(x) = sqrt(2/(pi*x))*(p1(x)*cos(x1)-q1(x)*sin(x1))
23
 * 		y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x1)+q1(x)*cos(x1))
24
 * 	   where x1 = x-3*pi/4. It is better to compute sin(x1),cos(x1)
25
 *	   as follow:
26
 *		cos(x1) =  cos(x)cos(3pi/4)+sin(x)sin(3pi/4)
27
 *			=  1/sqrt(2) * (sin(x) - cos(x))
28
 *		sin(x1) =  sin(x)cos(3pi/4)-cos(x)sin(3pi/4)
29
 *			= -1/sqrt(2) * (sin(x) + cos(x))
30
 * 	   (To avoid cancellation, use
31
 *		sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
32
 * 	    to compute the worse one.)
33
 *
34
 *	3 Special cases
35
 *		j1(nan)= nan
36
 *		j1(0) = 0
37
 *		j1(inf) = 0
38
 *
39
 * Method -- y1(x):
40
 *	1. screen out x<=0 cases: y1(0)=-inf, y1(x<0)=NaN
41
 *	2. For x<2.
42
 *	   Since
43
 *		y1(x) = 2/pi*(j1(x)*(ln(x/2)+Euler)-1/x-x/2+5/64*x^3-...)
44
 *	   therefore y1(x)-2/pi*j1(x)*ln(x)-1/x is an odd function.
45
 *	   We use the following function to approximate y1,
46
 *		y1(x) = x*U(z)/V(z) + (2/pi)*(j1(x)*ln(x)-1/x), z= x^2
47
 *	   where for x in [0,2] (abs err less than 2**-65.89)
48
 *		U(z) = U0[0] + U0[1]*z + ... + U0[4]*z^4
49
 *		V(z) = 1  + v0[0]*z + ... + v0[4]*z^5
50
 *	   Note: For tiny x, 1/x dominate y1 and hence
51
 *		y1(tiny) = -2/pi/tiny, (choose tiny<2**-54)
52
 *	3. For x>=2.
53
 * 		y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x1)+q1(x)*cos(x1))
54
 * 	   where x1 = x-3*pi/4. It is better to compute sin(x1),cos(x1)
55
 *	   by method mentioned above.
56
 */
57
58
#include "math.h"
59
#include "math_private.h"
60
61
static double pone(double), qone(double);
62
63
static const double
64
huge    = 1e300,
65
one	= 1.0,
66
invsqrtpi=  5.64189583547756279280e-01, /* 0x3FE20DD7, 0x50429B6D */
67
tpi      =  6.36619772367581382433e-01, /* 0x3FE45F30, 0x6DC9C883 */
68
	/* R0/S0 on [0,2] */
69
r00  = -6.25000000000000000000e-02, /* 0xBFB00000, 0x00000000 */
70
r01  =  1.40705666955189706048e-03, /* 0x3F570D9F, 0x98472C61 */
71
r02  = -1.59955631084035597520e-05, /* 0xBEF0C5C6, 0xBA169668 */
72
r03  =  4.96727999609584448412e-08, /* 0x3E6AAAFA, 0x46CA0BD9 */
73
s01  =  1.91537599538363460805e-02, /* 0x3F939D0B, 0x12637E53 */
74
s02  =  1.85946785588630915560e-04, /* 0x3F285F56, 0xB9CDF664 */
75
s03  =  1.17718464042623683263e-06, /* 0x3EB3BFF8, 0x333F8498 */
76
s04  =  5.04636257076217042715e-09, /* 0x3E35AC88, 0xC97DFF2C */
77
s05  =  1.23542274426137913908e-11; /* 0x3DAB2ACF, 0xCFB97ED8 */
78
79
static const double zero    = 0.0;
80
81
double
82
j1(double x)
83
{
84
	double z, s,c,ss,cc,r,u,v,y;
85
	int32_t hx,ix;
86
87
26620
	GET_HIGH_WORD(hx,x);
88
13310
	ix = hx&0x7fffffff;
89
13310
	if(ix>=0x7ff00000) return one/x;
90
13310
	y = fabs(x);
91
13310
	if(ix >= 0x40000000) {	/* |x| >= 2.0 */
92
7060
		s = sin(y);
93
7060
		c = cos(y);
94
7060
		ss = -s-c;
95
7060
		cc = s-c;
96
7060
		if(ix<0x7fe00000) {  /* make sure y+y not overflow */
97
7060
		    z = cos(y+y);
98
11930
		    if ((s*c)>zero) cc = z/ss;
99
2190
		    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
7060
		if(ix>0x48000000) z = (invsqrtpi*cc)/sqrt(y);
106
		else {
107
7060
		    u = pone(y); v = qone(y);
108
7060
		    z = invsqrtpi*(u*cc-v*ss)/sqrt(y);
109
		}
110
7770
		if(hx<0) return -z;
111
6350
		else  	 return  z;
112
	}
113
6250
	if(ix<0x3e400000) {	/* |x|<2**-27 */
114
20
	    if(huge+x>one) return 0.5*x;/* inexact if x!=0 necessary */
115
	}
116
6240
	z = x*x;
117
6240
	r =  z*(r00+z*(r01+z*(r02+z*r03)));
118
6240
	s =  one+z*(s01+z*(s02+z*(s03+z*(s04+z*s05))));
119
6240
	r *= x;
120
6240
	return(x*0.5+r/s);
121
13310
}
122
DEF_NONSTD(j1);
123
124
static const double U0[5] = {
125
 -1.96057090646238940668e-01, /* 0xBFC91866, 0x143CBC8A */
126
  5.04438716639811282616e-02, /* 0x3FA9D3C7, 0x76292CD1 */
127
 -1.91256895875763547298e-03, /* 0xBF5F55E5, 0x4844F50F */
128
  2.35252600561610495928e-05, /* 0x3EF8AB03, 0x8FA6B88E */
129
 -9.19099158039878874504e-08, /* 0xBE78AC00, 0x569105B8 */
130
};
131
static const double V0[5] = {
132
  1.99167318236649903973e-02, /* 0x3F94650D, 0x3F4DA9F0 */
133
  2.02552581025135171496e-04, /* 0x3F2A8C89, 0x6C257764 */
134
  1.35608801097516229404e-06, /* 0x3EB6C05A, 0x894E8CA6 */
135
  6.22741452364621501295e-09, /* 0x3E3ABF1D, 0x5BA69A86 */
136
  1.66559246207992079114e-11, /* 0x3DB25039, 0xDACA772A */
137
};
138
139
double
140
y1(double x)
141
{
142
	double z, s,c,ss,cc,u,v;
143
	int32_t hx,ix,lx;
144
145
19740
	EXTRACT_WORDS(hx,lx,x);
146
9870
        ix = 0x7fffffff&hx;
147
    /* if Y1(NaN) is NaN, Y1(-inf) is NaN, Y1(inf) is 0 */
148
9870
	if(ix>=0x7ff00000) return  one/(x+x*x);
149
9870
        if((ix|lx)==0) return -one/zero;
150
9890
        if(hx<0) return zero/zero;
151
9850
        if(ix >= 0x40000000) {  /* |x| >= 2.0 */
152
6760
                s = sin(x);
153
6760
                c = cos(x);
154
6760
                ss = -s-c;
155
6760
                cc = s-c;
156
6760
                if(ix<0x7fe00000) {  /* make sure x+x not overflow */
157
6760
                    z = cos(x+x);
158
9440
                    if ((s*c)>zero) cc = z/ss;
159
4080
                    else            ss = z/cc;
160
                }
161
        /* y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x0)+q1(x)*cos(x0))
162
         * where x0 = x-3pi/4
163
         *      Better formula:
164
         *              cos(x0) = cos(x)cos(3pi/4)+sin(x)sin(3pi/4)
165
         *                      =  1/sqrt(2) * (sin(x) - cos(x))
166
         *              sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4)
167
         *                      = -1/sqrt(2) * (cos(x) + sin(x))
168
         * To avoid cancellation, use
169
         *              sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
170
         * to compute the worse one.
171
         */
172
6760
                if(ix>0x48000000) z = (invsqrtpi*ss)/sqrt(x);
173
                else {
174
6760
                    u = pone(x); v = qone(x);
175
6760
                    z = invsqrtpi*(u*ss+v*cc)/sqrt(x);
176
                }
177
6760
                return z;
178
        }
179
3090
        if(ix<=0x3c900000) {    /* x < 2**-54 */
180
20
            return(-tpi/x);
181
        }
182
3070
        z = x*x;
183
3070
        u = U0[0]+z*(U0[1]+z*(U0[2]+z*(U0[3]+z*U0[4])));
184
3070
        v = one+z*(V0[0]+z*(V0[1]+z*(V0[2]+z*(V0[3]+z*V0[4]))));
185
3070
        return(x*(u/v) + tpi*(j1(x)*log(x)-one/x));
186
9870
}
187
DEF_NONSTD(y1);
188
189
/* For x >= 8, the asymptotic expansions of pone is
190
 *	1 + 15/128 s^2 - 4725/2^15 s^4 - ...,	where s = 1/x.
191
 * We approximate pone by
192
 * 	pone(x) = 1 + (R/S)
193
 * where  R = pr0 + pr1*s^2 + pr2*s^4 + ... + pr5*s^10
194
 * 	  S = 1 + ps0*s^2 + ... + ps4*s^10
195
 * and
196
 *	| pone(x)-1-R/S | <= 2  ** ( -60.06)
197
 */
198
199
static const double pr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
200
  0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */
201
  1.17187499999988647970e-01, /* 0x3FBDFFFF, 0xFFFFFCCE */
202
  1.32394806593073575129e+01, /* 0x402A7A9D, 0x357F7FCE */
203
  4.12051854307378562225e+02, /* 0x4079C0D4, 0x652EA590 */
204
  3.87474538913960532227e+03, /* 0x40AE457D, 0xA3A532CC */
205
  7.91447954031891731574e+03, /* 0x40BEEA7A, 0xC32782DD */
206
};
207
static const double ps8[5] = {
208
  1.14207370375678408436e+02, /* 0x405C8D45, 0x8E656CAC */
209
  3.65093083420853463394e+03, /* 0x40AC85DC, 0x964D274F */
210
  3.69562060269033463555e+04, /* 0x40E20B86, 0x97C5BB7F */
211
  9.76027935934950801311e+04, /* 0x40F7D42C, 0xB28F17BB */
212
  3.08042720627888811578e+04, /* 0x40DE1511, 0x697A0B2D */
213
};
214
215
static const double pr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
216
  1.31990519556243522749e-11, /* 0x3DAD0667, 0xDAE1CA7D */
217
  1.17187493190614097638e-01, /* 0x3FBDFFFF, 0xE2C10043 */
218
  6.80275127868432871736e+00, /* 0x401B3604, 0x6E6315E3 */
219
  1.08308182990189109773e+02, /* 0x405B13B9, 0x452602ED */
220
  5.17636139533199752805e+02, /* 0x40802D16, 0xD052D649 */
221
  5.28715201363337541807e+02, /* 0x408085B8, 0xBB7E0CB7 */
222
};
223
static const double ps5[5] = {
224
  5.92805987221131331921e+01, /* 0x404DA3EA, 0xA8AF633D */
225
  9.91401418733614377743e+02, /* 0x408EFB36, 0x1B066701 */
226
  5.35326695291487976647e+03, /* 0x40B4E944, 0x5706B6FB */
227
  7.84469031749551231769e+03, /* 0x40BEA4B0, 0xB8A5BB15 */
228
  1.50404688810361062679e+03, /* 0x40978030, 0x036F5E51 */
229
};
230
231
static const double pr3[6] = {
232
  3.02503916137373618024e-09, /* 0x3E29FC21, 0xA7AD9EDD */
233
  1.17186865567253592491e-01, /* 0x3FBDFFF5, 0x5B21D17B */
234
  3.93297750033315640650e+00, /* 0x400F76BC, 0xE85EAD8A */
235
  3.51194035591636932736e+01, /* 0x40418F48, 0x9DA6D129 */
236
  9.10550110750781271918e+01, /* 0x4056C385, 0x4D2C1837 */
237
  4.85590685197364919645e+01, /* 0x4048478F, 0x8EA83EE5 */
238
};
239
static const double ps3[5] = {
240
  3.47913095001251519989e+01, /* 0x40416549, 0xA134069C */
241
  3.36762458747825746741e+02, /* 0x40750C33, 0x07F1A75F */
242
  1.04687139975775130551e+03, /* 0x40905B7C, 0x5037D523 */
243
  8.90811346398256432622e+02, /* 0x408BD67D, 0xA32E31E9 */
244
  1.03787932439639277504e+02, /* 0x4059F26D, 0x7C2EED53 */
245
};
246
247
static const double pr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
248
  1.07710830106873743082e-07, /* 0x3E7CE9D4, 0xF65544F4 */
249
  1.17176219462683348094e-01, /* 0x3FBDFF42, 0xBE760D83 */
250
  2.36851496667608785174e+00, /* 0x4002F2B7, 0xF98FAEC0 */
251
  1.22426109148261232917e+01, /* 0x40287C37, 0x7F71A964 */
252
  1.76939711271687727390e+01, /* 0x4031B1A8, 0x177F8EE2 */
253
  5.07352312588818499250e+00, /* 0x40144B49, 0xA574C1FE */
254
};
255
static const double ps2[5] = {
256
  2.14364859363821409488e+01, /* 0x40356FBD, 0x8AD5ECDC */
257
  1.25290227168402751090e+02, /* 0x405F5293, 0x14F92CD5 */
258
  2.32276469057162813669e+02, /* 0x406D08D8, 0xD5A2DBD9 */
259
  1.17679373287147100768e+02, /* 0x405D6B7A, 0xDA1884A9 */
260
  8.36463893371618283368e+00, /* 0x4020BAB1, 0xF44E5192 */
261
};
262
263
static double
264
pone(double x)
265
{
266
	const double *p,*q;
267
	double z,r,s;
268
        int32_t ix;
269
27640
	GET_HIGH_WORD(ix,x);
270
13820
	ix &= 0x7fffffff;
271
23420
        if(ix>=0x40200000)     {p = pr8; q= ps8;}
272
6040
        else if(ix>=0x40122E8B){p = pr5; q= ps5;}
273
3820
        else if(ix>=0x4006DB6D){p = pr3; q= ps3;}
274
1960
        else if(ix>=0x40000000){p = pr2; q= ps2;}
275
13820
        z = one/(x*x);
276
13820
        r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
277
13820
        s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*q[4]))));
278
13820
        return one+ r/s;
279
}
280
281
282
/* For x >= 8, the asymptotic expansions of qone is
283
 *	3/8 s - 105/1024 s^3 - ..., where s = 1/x.
284
 * We approximate pone by
285
 * 	qone(x) = s*(0.375 + (R/S))
286
 * where  R = qr1*s^2 + qr2*s^4 + ... + qr5*s^10
287
 * 	  S = 1 + qs1*s^2 + ... + qs6*s^12
288
 * and
289
 *	| qone(x)/s -0.375-R/S | <= 2  ** ( -61.13)
290
 */
291
292
static const double qr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
293
  0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */
294
 -1.02539062499992714161e-01, /* 0xBFBA3FFF, 0xFFFFFDF3 */
295
 -1.62717534544589987888e+01, /* 0xC0304591, 0xA26779F7 */
296
 -7.59601722513950107896e+02, /* 0xC087BCD0, 0x53E4B576 */
297
 -1.18498066702429587167e+04, /* 0xC0C724E7, 0x40F87415 */
298
 -4.84385124285750353010e+04, /* 0xC0E7A6D0, 0x65D09C6A */
299
};
300
static const double qs8[6] = {
301
  1.61395369700722909556e+02, /* 0x40642CA6, 0xDE5BCDE5 */
302
  7.82538599923348465381e+03, /* 0x40BE9162, 0xD0D88419 */
303
  1.33875336287249578163e+05, /* 0x4100579A, 0xB0B75E98 */
304
  7.19657723683240939863e+05, /* 0x4125F653, 0x72869C19 */
305
  6.66601232617776375264e+05, /* 0x412457D2, 0x7719AD5C */
306
 -2.94490264303834643215e+05, /* 0xC111F969, 0x0EA5AA18 */
307
};
308
309
static const double qr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
310
 -2.08979931141764104297e-11, /* 0xBDB6FA43, 0x1AA1A098 */
311
 -1.02539050241375426231e-01, /* 0xBFBA3FFF, 0xCB597FEF */
312
 -8.05644828123936029840e+00, /* 0xC0201CE6, 0xCA03AD4B */
313
 -1.83669607474888380239e+02, /* 0xC066F56D, 0x6CA7B9B0 */
314
 -1.37319376065508163265e+03, /* 0xC09574C6, 0x6931734F */
315
 -2.61244440453215656817e+03, /* 0xC0A468E3, 0x88FDA79D */
316
};
317
static const double qs5[6] = {
318
  8.12765501384335777857e+01, /* 0x405451B2, 0xFF5A11B2 */
319
  1.99179873460485964642e+03, /* 0x409F1F31, 0xE77BF839 */
320
  1.74684851924908907677e+04, /* 0x40D10F1F, 0x0D64CE29 */
321
  4.98514270910352279316e+04, /* 0x40E8576D, 0xAABAD197 */
322
  2.79480751638918118260e+04, /* 0x40DB4B04, 0xCF7C364B */
323
 -4.71918354795128470869e+03, /* 0xC0B26F2E, 0xFCFFA004 */
324
};
325
326
static const double qr3[6] = {
327
 -5.07831226461766561369e-09, /* 0xBE35CFA9, 0xD38FC84F */
328
 -1.02537829820837089745e-01, /* 0xBFBA3FEB, 0x51AEED54 */
329
 -4.61011581139473403113e+00, /* 0xC01270C2, 0x3302D9FF */
330
 -5.78472216562783643212e+01, /* 0xC04CEC71, 0xC25D16DA */
331
 -2.28244540737631695038e+02, /* 0xC06C87D3, 0x4718D55F */
332
 -2.19210128478909325622e+02, /* 0xC06B66B9, 0x5F5C1BF6 */
333
};
334
static const double qs3[6] = {
335
  4.76651550323729509273e+01, /* 0x4047D523, 0xCCD367E4 */
336
  6.73865112676699709482e+02, /* 0x40850EEB, 0xC031EE3E */
337
  3.38015286679526343505e+03, /* 0x40AA684E, 0x448E7C9A */
338
  5.54772909720722782367e+03, /* 0x40B5ABBA, 0xA61D54A6 */
339
  1.90311919338810798763e+03, /* 0x409DBC7A, 0x0DD4DF4B */
340
 -1.35201191444307340817e+02, /* 0xC060E670, 0x290A311F */
341
};
342
343
static const double qr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
344
 -1.78381727510958865572e-07, /* 0xBE87F126, 0x44C626D2 */
345
 -1.02517042607985553460e-01, /* 0xBFBA3E8E, 0x9148B010 */
346
 -2.75220568278187460720e+00, /* 0xC0060484, 0x69BB4EDA */
347
 -1.96636162643703720221e+01, /* 0xC033A9E2, 0xC168907F */
348
 -4.23253133372830490089e+01, /* 0xC04529A3, 0xDE104AAA */
349
 -2.13719211703704061733e+01, /* 0xC0355F36, 0x39CF6E52 */
350
};
351
static const double qs2[6] = {
352
  2.95333629060523854548e+01, /* 0x403D888A, 0x78AE64FF */
353
  2.52981549982190529136e+02, /* 0x406F9F68, 0xDB821CBA */
354
  7.57502834868645436472e+02, /* 0x4087AC05, 0xCE49A0F7 */
355
  7.39393205320467245656e+02, /* 0x40871B25, 0x48D4C029 */
356
  1.55949003336666123687e+02, /* 0x40637E5E, 0x3C3ED8D4 */
357
 -4.95949898822628210127e+00, /* 0xC013D686, 0xE71BE86B */
358
};
359
360
static double
361
qone(double x)
362
{
363
	const double *p,*q;
364
	double  s,r,z;
365
	int32_t ix;
366
27640
	GET_HIGH_WORD(ix,x);
367
13820
	ix &= 0x7fffffff;
368
23420
	if(ix>=0x40200000)     {p = qr8; q= qs8;}
369
6040
	else if(ix>=0x40122E8B){p = qr5; q= qs5;}
370
3820
	else if(ix>=0x4006DB6D){p = qr3; q= qs3;}
371
1960
	else if(ix>=0x40000000){p = qr2; q= qs2;}
372
13820
	z = one/(x*x);
373
13820
	r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
374
13820
	s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*(q[4]+z*q[5])))));
375
13820
	return (.375 + r/s)/x;
376
}