2 * ====================================================
3 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
5 * Developed at SunPro, a Sun Microsystems, Inc. business.
6 * Permission to use, copy, modify, and distribute this
7 * software is freely granted, provided that this notice
9 * ====================================================
12 /* __ieee754_j1(x), __ieee754_y1(x)
13 * Bessel function of the first and second kinds of order zero.
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
18 * j1(x) = x/2 + x*z*R0/S0, where z = x*x;
19 * (precision: |j1/x - 1/2 - R0/S0 |<2**-61.51 )
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)
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.)
39 * 1. screen out x<=0 cases: y1(0)=-inf, y1(x<0)=NaN
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)
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.
58 #include "math_private.h"
60 static double pone(double), qone(double);
65 invsqrtpi
= 5.64189583547756279280e-01, /* 0x3FE20DD7, 0x50429B6D */
66 tpi
= 6.36619772367581382433e-01, /* 0x3FE45F30, 0x6DC9C883 */
68 r00
= -6.25000000000000000000e-02, /* 0xBFB00000, 0x00000000 */
69 r01
= 1.40705666955189706048e-03, /* 0x3F570D9F, 0x98472C61 */
70 r02
= -1.59955631084035597520e-05, /* 0xBEF0C5C6, 0xBA169668 */
71 r03
= 4.96727999609584448412e-08, /* 0x3E6AAAFA, 0x46CA0BD9 */
72 s01
= 1.91537599538363460805e-02, /* 0x3F939D0B, 0x12637E53 */
73 s02
= 1.85946785588630915560e-04, /* 0x3F285F56, 0xB9CDF664 */
74 s03
= 1.17718464042623683263e-06, /* 0x3EB3BFF8, 0x333F8498 */
75 s04
= 5.04636257076217042715e-09, /* 0x3E35AC88, 0xC97DFF2C */
76 s05
= 1.23542274426137913908e-11; /* 0x3DAB2ACF, 0xCFB97ED8 */
78 static const double zero
= 0.0;
80 double __ieee754_j1(double x
)
82 double z
, s
,c
,ss
,cc
,r
,u
,v
,y
;
87 if(ix
>=0x7ff00000) return one
/x
;
89 if(ix
>= 0x40000000) { /* |x| >= 2.0 */
94 if(ix
<0x7fe00000) { /* make sure y+y not overflow */
96 if ((s
*c
)>zero
) cc
= z
/ss
;
100 * j1(x) = 1/sqrt(pi) * (P(1,x)*cc - Q(1,x)*ss) / sqrt(x)
101 * y1(x) = 1/sqrt(pi) * (P(1,x)*ss + Q(1,x)*cc) / sqrt(x)
103 if(ix
>0x48000000) z
= (invsqrtpi
*cc
)/sqrt(y
);
105 u
= pone(y
); v
= qone(y
);
106 z
= invsqrtpi
*(u
*cc
-v
*ss
)/sqrt(y
);
111 if(ix
<0x3e400000) { /* |x|<2**-27 */
112 if(huge
+x
>one
) return 0.5*x
;/* inexact if x!=0 necessary */
115 r
= z
*(r00
+z
*(r01
+z
*(r02
+z
*r03
)));
116 s
= one
+z
*(s01
+z
*(s02
+z
*(s03
+z
*(s04
+z
*s05
))));
127 double z
= __ieee754_j1(x
);
128 if (_LIB_VERSION
== _IEEE_
|| isnan(x
))
130 if (fabs(x
) > X_TLOSS
)
131 return __kernel_standard(x
, x
, 36); /* j1(|x|>X_TLOSS) */
135 strong_alias(__ieee754_j1
, j1
)
138 static const double U0
[5] = {
139 -1.96057090646238940668e-01, /* 0xBFC91866, 0x143CBC8A */
140 5.04438716639811282616e-02, /* 0x3FA9D3C7, 0x76292CD1 */
141 -1.91256895875763547298e-03, /* 0xBF5F55E5, 0x4844F50F */
142 2.35252600561610495928e-05, /* 0x3EF8AB03, 0x8FA6B88E */
143 -9.19099158039878874504e-08, /* 0xBE78AC00, 0x569105B8 */
145 static const double V0
[5] = {
146 1.99167318236649903973e-02, /* 0x3F94650D, 0x3F4DA9F0 */
147 2.02552581025135171496e-04, /* 0x3F2A8C89, 0x6C257764 */
148 1.35608801097516229404e-06, /* 0x3EB6C05A, 0x894E8CA6 */
149 6.22741452364621501295e-09, /* 0x3E3ABF1D, 0x5BA69A86 */
150 1.66559246207992079114e-11, /* 0x3DB25039, 0xDACA772A */
153 double __ieee754_y1(double x
)
155 double z
, s
,c
,ss
,cc
,u
,v
;
158 EXTRACT_WORDS(hx
,lx
,x
);
160 /* if Y1(NaN) is NaN, Y1(-inf) is NaN, Y1(inf) is 0 */
161 if(ix
>=0x7ff00000) return one
/(x
+x
*x
);
162 if((ix
|lx
)==0) return -one
/zero
;
163 if(hx
<0) return zero
/zero
;
164 if(ix
>= 0x40000000) { /* |x| >= 2.0 */
169 if(ix
<0x7fe00000) { /* make sure x+x not overflow */
171 if ((s
*c
)>zero
) cc
= z
/ss
;
174 /* y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x0)+q1(x)*cos(x0))
177 * cos(x0) = cos(x)cos(3pi/4)+sin(x)sin(3pi/4)
178 * = 1/sqrt(2) * (sin(x) - cos(x))
179 * sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4)
180 * = -1/sqrt(2) * (cos(x) + sin(x))
181 * To avoid cancellation, use
182 * sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
183 * to compute the worse one.
185 if(ix
>0x48000000) z
= (invsqrtpi
*ss
)/sqrt(x
);
187 u
= pone(x
); v
= qone(x
);
188 z
= invsqrtpi
*(u
*ss
+v
*cc
)/sqrt(x
);
192 if(ix
<=0x3c900000) { /* x < 2**-54 */
196 u
= U0
[0]+z
*(U0
[1]+z
*(U0
[2]+z
*(U0
[3]+z
*U0
[4])));
197 v
= one
+z
*(V0
[0]+z
*(V0
[1]+z
*(V0
[2]+z
*(V0
[3]+z
*V0
[4]))));
198 return(x
*(u
/v
) + tpi
*(__ieee754_j1(x
)*__ieee754_log(x
)-one
/x
));
207 double z
= __ieee754_y1(x
);
208 if (_LIB_VERSION
== _IEEE_
|| isnan(x
))
211 if (x
== 0.0) /* d = -one/(x-x); */
212 return __kernel_standard(x
, x
, 10);
213 /* d = zero/(x-x); */
214 return __kernel_standard(x
, x
, 11);
217 return __kernel_standard(x
, x
, 37); /* y1(x>X_TLOSS) */
221 strong_alias(__ieee754_y1
, y1
)
224 /* For x >= 8, the asymptotic expansions of pone is
225 * 1 + 15/128 s^2 - 4725/2^15 s^4 - ..., where s = 1/x.
226 * We approximate pone by
227 * pone(x) = 1 + (R/S)
228 * where R = pr0 + pr1*s^2 + pr2*s^4 + ... + pr5*s^10
229 * S = 1 + ps0*s^2 + ... + ps4*s^10
231 * | pone(x)-1-R/S | <= 2 ** ( -60.06)
234 static const double pr8
[6] = { /* for x in [inf, 8]=1/[0,0.125] */
235 0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */
236 1.17187499999988647970e-01, /* 0x3FBDFFFF, 0xFFFFFCCE */
237 1.32394806593073575129e+01, /* 0x402A7A9D, 0x357F7FCE */
238 4.12051854307378562225e+02, /* 0x4079C0D4, 0x652EA590 */
239 3.87474538913960532227e+03, /* 0x40AE457D, 0xA3A532CC */
240 7.91447954031891731574e+03, /* 0x40BEEA7A, 0xC32782DD */
242 static const double ps8
[5] = {
243 1.14207370375678408436e+02, /* 0x405C8D45, 0x8E656CAC */
244 3.65093083420853463394e+03, /* 0x40AC85DC, 0x964D274F */
245 3.69562060269033463555e+04, /* 0x40E20B86, 0x97C5BB7F */
246 9.76027935934950801311e+04, /* 0x40F7D42C, 0xB28F17BB */
247 3.08042720627888811578e+04, /* 0x40DE1511, 0x697A0B2D */
250 static const double pr5
[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
251 1.31990519556243522749e-11, /* 0x3DAD0667, 0xDAE1CA7D */
252 1.17187493190614097638e-01, /* 0x3FBDFFFF, 0xE2C10043 */
253 6.80275127868432871736e+00, /* 0x401B3604, 0x6E6315E3 */
254 1.08308182990189109773e+02, /* 0x405B13B9, 0x452602ED */
255 5.17636139533199752805e+02, /* 0x40802D16, 0xD052D649 */
256 5.28715201363337541807e+02, /* 0x408085B8, 0xBB7E0CB7 */
258 static const double ps5
[5] = {
259 5.92805987221131331921e+01, /* 0x404DA3EA, 0xA8AF633D */
260 9.91401418733614377743e+02, /* 0x408EFB36, 0x1B066701 */
261 5.35326695291487976647e+03, /* 0x40B4E944, 0x5706B6FB */
262 7.84469031749551231769e+03, /* 0x40BEA4B0, 0xB8A5BB15 */
263 1.50404688810361062679e+03, /* 0x40978030, 0x036F5E51 */
266 static const double pr3
[6] = {
267 3.02503916137373618024e-09, /* 0x3E29FC21, 0xA7AD9EDD */
268 1.17186865567253592491e-01, /* 0x3FBDFFF5, 0x5B21D17B */
269 3.93297750033315640650e+00, /* 0x400F76BC, 0xE85EAD8A */
270 3.51194035591636932736e+01, /* 0x40418F48, 0x9DA6D129 */
271 9.10550110750781271918e+01, /* 0x4056C385, 0x4D2C1837 */
272 4.85590685197364919645e+01, /* 0x4048478F, 0x8EA83EE5 */
274 static const double ps3
[5] = {
275 3.47913095001251519989e+01, /* 0x40416549, 0xA134069C */
276 3.36762458747825746741e+02, /* 0x40750C33, 0x07F1A75F */
277 1.04687139975775130551e+03, /* 0x40905B7C, 0x5037D523 */
278 8.90811346398256432622e+02, /* 0x408BD67D, 0xA32E31E9 */
279 1.03787932439639277504e+02, /* 0x4059F26D, 0x7C2EED53 */
282 static const double pr2
[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
283 1.07710830106873743082e-07, /* 0x3E7CE9D4, 0xF65544F4 */
284 1.17176219462683348094e-01, /* 0x3FBDFF42, 0xBE760D83 */
285 2.36851496667608785174e+00, /* 0x4002F2B7, 0xF98FAEC0 */
286 1.22426109148261232917e+01, /* 0x40287C37, 0x7F71A964 */
287 1.76939711271687727390e+01, /* 0x4031B1A8, 0x177F8EE2 */
288 5.07352312588818499250e+00, /* 0x40144B49, 0xA574C1FE */
290 static const double ps2
[5] = {
291 2.14364859363821409488e+01, /* 0x40356FBD, 0x8AD5ECDC */
292 1.25290227168402751090e+02, /* 0x405F5293, 0x14F92CD5 */
293 2.32276469057162813669e+02, /* 0x406D08D8, 0xD5A2DBD9 */
294 1.17679373287147100768e+02, /* 0x405D6B7A, 0xDA1884A9 */
295 8.36463893371618283368e+00, /* 0x4020BAB1, 0xF44E5192 */
298 static double pone(double x
)
300 const double *p
=0,*q
=0;
305 if(ix
>=0x40200000) {p
= pr8
; q
= ps8
;}
306 else if(ix
>=0x40122E8B){p
= pr5
; q
= ps5
;}
307 else if(ix
>=0x4006DB6D){p
= pr3
; q
= ps3
;}
308 else if(ix
>=0x40000000){p
= pr2
; q
= ps2
;}
310 r
= p
[0]+z
*(p
[1]+z
*(p
[2]+z
*(p
[3]+z
*(p
[4]+z
*p
[5]))));
311 s
= one
+z
*(q
[0]+z
*(q
[1]+z
*(q
[2]+z
*(q
[3]+z
*q
[4]))));
316 /* For x >= 8, the asymptotic expansions of qone is
317 * 3/8 s - 105/1024 s^3 - ..., where s = 1/x.
318 * We approximate pone by
319 * qone(x) = s*(0.375 + (R/S))
320 * where R = qr1*s^2 + qr2*s^4 + ... + qr5*s^10
321 * S = 1 + qs1*s^2 + ... + qs6*s^12
323 * | qone(x)/s -0.375-R/S | <= 2 ** ( -61.13)
326 static const double qr8
[6] = { /* for x in [inf, 8]=1/[0,0.125] */
327 0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */
328 -1.02539062499992714161e-01, /* 0xBFBA3FFF, 0xFFFFFDF3 */
329 -1.62717534544589987888e+01, /* 0xC0304591, 0xA26779F7 */
330 -7.59601722513950107896e+02, /* 0xC087BCD0, 0x53E4B576 */
331 -1.18498066702429587167e+04, /* 0xC0C724E7, 0x40F87415 */
332 -4.84385124285750353010e+04, /* 0xC0E7A6D0, 0x65D09C6A */
334 static const double qs8
[6] = {
335 1.61395369700722909556e+02, /* 0x40642CA6, 0xDE5BCDE5 */
336 7.82538599923348465381e+03, /* 0x40BE9162, 0xD0D88419 */
337 1.33875336287249578163e+05, /* 0x4100579A, 0xB0B75E98 */
338 7.19657723683240939863e+05, /* 0x4125F653, 0x72869C19 */
339 6.66601232617776375264e+05, /* 0x412457D2, 0x7719AD5C */
340 -2.94490264303834643215e+05, /* 0xC111F969, 0x0EA5AA18 */
343 static const double qr5
[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
344 -2.08979931141764104297e-11, /* 0xBDB6FA43, 0x1AA1A098 */
345 -1.02539050241375426231e-01, /* 0xBFBA3FFF, 0xCB597FEF */
346 -8.05644828123936029840e+00, /* 0xC0201CE6, 0xCA03AD4B */
347 -1.83669607474888380239e+02, /* 0xC066F56D, 0x6CA7B9B0 */
348 -1.37319376065508163265e+03, /* 0xC09574C6, 0x6931734F */
349 -2.61244440453215656817e+03, /* 0xC0A468E3, 0x88FDA79D */
351 static const double qs5
[6] = {
352 8.12765501384335777857e+01, /* 0x405451B2, 0xFF5A11B2 */
353 1.99179873460485964642e+03, /* 0x409F1F31, 0xE77BF839 */
354 1.74684851924908907677e+04, /* 0x40D10F1F, 0x0D64CE29 */
355 4.98514270910352279316e+04, /* 0x40E8576D, 0xAABAD197 */
356 2.79480751638918118260e+04, /* 0x40DB4B04, 0xCF7C364B */
357 -4.71918354795128470869e+03, /* 0xC0B26F2E, 0xFCFFA004 */
360 static const double qr3
[6] = {
361 -5.07831226461766561369e-09, /* 0xBE35CFA9, 0xD38FC84F */
362 -1.02537829820837089745e-01, /* 0xBFBA3FEB, 0x51AEED54 */
363 -4.61011581139473403113e+00, /* 0xC01270C2, 0x3302D9FF */
364 -5.78472216562783643212e+01, /* 0xC04CEC71, 0xC25D16DA */
365 -2.28244540737631695038e+02, /* 0xC06C87D3, 0x4718D55F */
366 -2.19210128478909325622e+02, /* 0xC06B66B9, 0x5F5C1BF6 */
368 static const double qs3
[6] = {
369 4.76651550323729509273e+01, /* 0x4047D523, 0xCCD367E4 */
370 6.73865112676699709482e+02, /* 0x40850EEB, 0xC031EE3E */
371 3.38015286679526343505e+03, /* 0x40AA684E, 0x448E7C9A */
372 5.54772909720722782367e+03, /* 0x40B5ABBA, 0xA61D54A6 */
373 1.90311919338810798763e+03, /* 0x409DBC7A, 0x0DD4DF4B */
374 -1.35201191444307340817e+02, /* 0xC060E670, 0x290A311F */
377 static const double qr2
[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
378 -1.78381727510958865572e-07, /* 0xBE87F126, 0x44C626D2 */
379 -1.02517042607985553460e-01, /* 0xBFBA3E8E, 0x9148B010 */
380 -2.75220568278187460720e+00, /* 0xC0060484, 0x69BB4EDA */
381 -1.96636162643703720221e+01, /* 0xC033A9E2, 0xC168907F */
382 -4.23253133372830490089e+01, /* 0xC04529A3, 0xDE104AAA */
383 -2.13719211703704061733e+01, /* 0xC0355F36, 0x39CF6E52 */
385 static const double qs2
[6] = {
386 2.95333629060523854548e+01, /* 0x403D888A, 0x78AE64FF */
387 2.52981549982190529136e+02, /* 0x406F9F68, 0xDB821CBA */
388 7.57502834868645436472e+02, /* 0x4087AC05, 0xCE49A0F7 */
389 7.39393205320467245656e+02, /* 0x40871B25, 0x48D4C029 */
390 1.55949003336666123687e+02, /* 0x40637E5E, 0x3C3ED8D4 */
391 -4.95949898822628210127e+00, /* 0xC013D686, 0xE71BE86B */
394 static double qone(double x
)
396 const double *p
=0,*q
=0;
401 if(ix
>=0x40200000) {p
= qr8
; q
= qs8
;}
402 else if(ix
>=0x40122E8B){p
= qr5
; q
= qs5
;}
403 else if(ix
>=0x4006DB6D){p
= qr3
; q
= qs3
;}
404 else if(ix
>=0x40000000){p
= qr2
; q
= qs2
;}
406 r
= p
[0]+z
*(p
[1]+z
*(p
[2]+z
*(p
[3]+z
*(p
[4]+z
*p
[5]))));
407 s
= one
+z
*(q
[0]+z
*(q
[1]+z
*(q
[2]+z
*(q
[3]+z
*(q
[4]+z
*q
[5])))));
408 return (.375 + r
/s
)/x
;