1 /* @(#)e_j0.c 5.1 93/09/24 */
3 * ====================================================
4 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
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
10 * ====================================================
13 #if defined(LIBM_SCCS) && !defined(lint)
14 static char rcsid
[] = "$NetBSD: e_j0.c,v 1.8 1995/05/10 20:45:23 jtc Exp $";
17 /* __ieee754_j0(x), __ieee754_y0(x)
18 * Bessel function of the first and second kinds of order zero.
20 * 1. For tiny x, we use j0(x) = 1 - x^2/4 + x^4/64 - ...
21 * 2. Reduce x to |x| since j0(x)=j0(-x), and
23 * j0(x) = 1-z/4+ z^2*R0/S0, where z = x*x;
24 * (precision: |j0-1+z/4-z^2R0/S0 |<2**-63.67 )
26 * j0(x) = sqrt(2/(pi*x))*(p0(x)*cos(x0)-q0(x)*sin(x0))
27 * where x0 = x-pi/4. It is better to compute sin(x0),cos(x0)
29 * cos(x0) = cos(x)cos(pi/4)+sin(x)sin(pi/4)
30 * = 1/sqrt(2) * (cos(x) + sin(x))
31 * sin(x0) = sin(x)cos(pi/4)-cos(x)sin(pi/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.)
45 * y0(x) = 2/pi*(j0(x)*(ln(x/2)+Euler) + x^2/4 - ...)
46 * therefore y0(x)-2/pi*j0(x)*ln(x) is an even function.
47 * We use the following function to approximate y0,
48 * y0(x) = U(z)/V(z) + (2/pi)*(j0(x)*ln(x)), z= x^2
50 * U(z) = u00 + u01*z + ... + u06*z^6
51 * V(z) = 1 + v01*z + ... + v04*z^4
52 * with absolute approximation error bounded by 2**-72.
53 * Note: For tiny x, U/V = u0 and j0(x)~1, hence
54 * y0(tiny) = u0 + (2/pi)*ln(tiny), (choose tiny<2**-27)
56 * y0(x) = sqrt(2/(pi*x))*(p0(x)*cos(x0)+q0(x)*sin(x0))
57 * where x0 = x-pi/4. It is better to compute sin(x0),cos(x0)
58 * by the method mentioned above.
59 * 3. Special cases: y0(0)=-inf, y0(x<0)=NaN, y0(inf)=0.
63 #include "math_private.h"
67 static double pzero(double), qzero(double);
69 static double pzero(), qzero();
79 invsqrtpi
= 5.64189583547756279280e-01, /* 0x3FE20DD7, 0x50429B6D */
80 tpi
= 6.36619772367581382433e-01, /* 0x3FE45F30, 0x6DC9C883 */
81 /* R0/S0 on [0, 2.00] */
82 R02
= 1.56249999999999947958e-02, /* 0x3F8FFFFF, 0xFFFFFFFD */
83 R03
= -1.89979294238854721751e-04, /* 0xBF28E6A5, 0xB61AC6E9 */
84 R04
= 1.82954049532700665670e-06, /* 0x3EBEB1D1, 0x0C503919 */
85 R05
= -4.61832688532103189199e-09, /* 0xBE33D5E7, 0x73D63FCE */
86 S01
= 1.56191029464890010492e-02, /* 0x3F8FFCE8, 0x82C8C2A4 */
87 S02
= 1.16926784663337450260e-04, /* 0x3F1EA6D2, 0xDD57DBF4 */
88 S03
= 5.13546550207318111446e-07, /* 0x3EA13B54, 0xCE84D5A9 */
89 S04
= 1.16614003333790000205e-09; /* 0x3E1408BC, 0xF4745D8F */
92 static const double zero
= 0.0;
94 static double zero
= 0.0;
98 double attribute_hidden
__ieee754_j0(double x
)
100 double attribute_hidden
__ieee754_j0(x
)
104 double z
, s
,c
,ss
,cc
,r
,u
,v
;
109 if(ix
>=0x7ff00000) return one
/(x
*x
);
111 if(ix
>= 0x40000000) { /* |x| >= 2.0 */
116 if(ix
<0x7fe00000) { /* make sure x+x not overflow */
118 if ((s
*c
)<zero
) cc
= z
/ss
;
122 * j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x)
123 * y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x)
125 if(ix
>0x48000000) z
= (invsqrtpi
*cc
)/sqrt(x
);
127 u
= pzero(x
); v
= qzero(x
);
128 z
= invsqrtpi
*(u
*cc
-v
*ss
)/sqrt(x
);
132 if(ix
<0x3f200000) { /* |x| < 2**-13 */
133 if(huge
+x
>one
) { /* raise inexact if x != 0 */
134 if(ix
<0x3e400000) return one
; /* |x|<2**-27 */
135 else return one
- 0.25*x
*x
;
139 r
= z
*(R02
+z
*(R03
+z
*(R04
+z
*R05
)));
140 s
= one
+z
*(S01
+z
*(S02
+z
*(S03
+z
*S04
)));
141 if(ix
< 0x3FF00000) { /* |x| < 1.00 */
142 return one
+ z
*(-0.25+(r
/s
));
145 return((one
+u
)*(one
-u
)+z
*(r
/s
));
154 u00
= -7.38042951086872317523e-02, /* 0xBFB2E4D6, 0x99CBD01F */
155 u01
= 1.76666452509181115538e-01, /* 0x3FC69D01, 0x9DE9E3FC */
156 u02
= -1.38185671945596898896e-02, /* 0xBF8C4CE8, 0xB16CFA97 */
157 u03
= 3.47453432093683650238e-04, /* 0x3F36C54D, 0x20B29B6B */
158 u04
= -3.81407053724364161125e-06, /* 0xBECFFEA7, 0x73D25CAD */
159 u05
= 1.95590137035022920206e-08, /* 0x3E550057, 0x3B4EABD4 */
160 u06
= -3.98205194132103398453e-11, /* 0xBDC5E43D, 0x693FB3C8 */
161 v01
= 1.27304834834123699328e-02, /* 0x3F8A1270, 0x91C9C71A */
162 v02
= 7.60068627350353253702e-05, /* 0x3F13ECBB, 0xF578C6C1 */
163 v03
= 2.59150851840457805467e-07, /* 0x3E91642D, 0x7FF202FD */
164 v04
= 4.41110311332675467403e-10; /* 0x3DFE5018, 0x3BD6D9EF */
167 double attribute_hidden
__ieee754_y0(double x
)
169 double attribute_hidden
__ieee754_y0(x
)
173 double z
, s
,c
,ss
,cc
,u
,v
;
176 EXTRACT_WORDS(hx
,lx
,x
);
178 /* Y0(NaN) is NaN, y0(-inf) is Nan, y0(inf) is 0 */
179 if(ix
>=0x7ff00000) return one
/(x
+x
*x
);
180 if((ix
|lx
)==0) return -one
/zero
;
181 if(hx
<0) return zero
/zero
;
182 if(ix
>= 0x40000000) { /* |x| >= 2.0 */
183 /* y0(x) = sqrt(2/(pi*x))*(p0(x)*sin(x0)+q0(x)*cos(x0))
186 * cos(x0) = cos(x)cos(pi/4)+sin(x)sin(pi/4)
187 * = 1/sqrt(2) * (sin(x) + cos(x))
188 * sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4)
189 * = 1/sqrt(2) * (sin(x) - cos(x))
190 * To avoid cancellation, use
191 * sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
192 * to compute the worse one.
199 * j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x)
200 * y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x)
202 if(ix
<0x7fe00000) { /* make sure x+x not overflow */
204 if ((s
*c
)<zero
) cc
= z
/ss
;
207 if(ix
>0x48000000) z
= (invsqrtpi
*ss
)/sqrt(x
);
209 u
= pzero(x
); v
= qzero(x
);
210 z
= invsqrtpi
*(u
*ss
+v
*cc
)/sqrt(x
);
214 if(ix
<=0x3e400000) { /* x < 2**-27 */
215 return(u00
+ tpi
*__ieee754_log(x
));
218 u
= u00
+z
*(u01
+z
*(u02
+z
*(u03
+z
*(u04
+z
*(u05
+z
*u06
)))));
219 v
= one
+z
*(v01
+z
*(v02
+z
*(v03
+z
*v04
)));
220 return(u
/v
+ tpi
*(__ieee754_j0(x
)*__ieee754_log(x
)));
223 /* The asymptotic expansions of pzero is
224 * 1 - 9/128 s^2 + 11025/98304 s^4 - ..., where s = 1/x.
225 * For x >= 2, We approximate pzero by
226 * pzero(x) = 1 + (R/S)
227 * where R = pR0 + pR1*s^2 + pR2*s^4 + ... + pR5*s^10
228 * S = 1 + pS0*s^2 + ... + pS4*s^10
230 * | pzero(x)-1-R/S | <= 2 ** ( -60.26)
233 static const double pR8
[6] = { /* for x in [inf, 8]=1/[0,0.125] */
235 static double pR8
[6] = { /* for x in [inf, 8]=1/[0,0.125] */
237 0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */
238 -7.03124999999900357484e-02, /* 0xBFB1FFFF, 0xFFFFFD32 */
239 -8.08167041275349795626e+00, /* 0xC02029D0, 0xB44FA779 */
240 -2.57063105679704847262e+02, /* 0xC0701102, 0x7B19E863 */
241 -2.48521641009428822144e+03, /* 0xC0A36A6E, 0xCD4DCAFC */
242 -5.25304380490729545272e+03, /* 0xC0B4850B, 0x36CC643D */
245 static const double pS8
[5] = {
247 static double pS8
[5] = {
249 1.16534364619668181717e+02, /* 0x405D2233, 0x07A96751 */
250 3.83374475364121826715e+03, /* 0x40ADF37D, 0x50596938 */
251 4.05978572648472545552e+04, /* 0x40E3D2BB, 0x6EB6B05F */
252 1.16752972564375915681e+05, /* 0x40FC810F, 0x8F9FA9BD */
253 4.76277284146730962675e+04, /* 0x40E74177, 0x4F2C49DC */
257 static const double pR5
[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
259 static double pR5
[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
261 -1.14125464691894502584e-11, /* 0xBDA918B1, 0x47E495CC */
262 -7.03124940873599280078e-02, /* 0xBFB1FFFF, 0xE69AFBC6 */
263 -4.15961064470587782438e+00, /* 0xC010A370, 0xF90C6BBF */
264 -6.76747652265167261021e+01, /* 0xC050EB2F, 0x5A7D1783 */
265 -3.31231299649172967747e+02, /* 0xC074B3B3, 0x6742CC63 */
266 -3.46433388365604912451e+02, /* 0xC075A6EF, 0x28A38BD7 */
269 static const double pS5
[5] = {
271 static double pS5
[5] = {
273 6.07539382692300335975e+01, /* 0x404E6081, 0x0C98C5DE */
274 1.05125230595704579173e+03, /* 0x40906D02, 0x5C7E2864 */
275 5.97897094333855784498e+03, /* 0x40B75AF8, 0x8FBE1D60 */
276 9.62544514357774460223e+03, /* 0x40C2CCB8, 0xFA76FA38 */
277 2.40605815922939109441e+03, /* 0x40A2CC1D, 0xC70BE864 */
281 static const double pR3
[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
283 static double pR3
[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
285 -2.54704601771951915620e-09, /* 0xBE25E103, 0x6FE1AA86 */
286 -7.03119616381481654654e-02, /* 0xBFB1FFF6, 0xF7C0E24B */
287 -2.40903221549529611423e+00, /* 0xC00345B2, 0xAEA48074 */
288 -2.19659774734883086467e+01, /* 0xC035F74A, 0x4CB94E14 */
289 -5.80791704701737572236e+01, /* 0xC04D0A22, 0x420A1A45 */
290 -3.14479470594888503854e+01, /* 0xC03F72AC, 0xA892D80F */
293 static const double pS3
[5] = {
295 static double pS3
[5] = {
297 3.58560338055209726349e+01, /* 0x4041ED92, 0x84077DD3 */
298 3.61513983050303863820e+02, /* 0x40769839, 0x464A7C0E */
299 1.19360783792111533330e+03, /* 0x4092A66E, 0x6D1061D6 */
300 1.12799679856907414432e+03, /* 0x40919FFC, 0xB8C39B7E */
301 1.73580930813335754692e+02, /* 0x4065B296, 0xFC379081 */
305 static const double pR2
[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
307 static double pR2
[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
309 -8.87534333032526411254e-08, /* 0xBE77D316, 0xE927026D */
310 -7.03030995483624743247e-02, /* 0xBFB1FF62, 0x495E1E42 */
311 -1.45073846780952986357e+00, /* 0xBFF73639, 0x8A24A843 */
312 -7.63569613823527770791e+00, /* 0xC01E8AF3, 0xEDAFA7F3 */
313 -1.11931668860356747786e+01, /* 0xC02662E6, 0xC5246303 */
314 -3.23364579351335335033e+00, /* 0xC009DE81, 0xAF8FE70F */
317 static const double pS2
[5] = {
319 static double pS2
[5] = {
321 2.22202997532088808441e+01, /* 0x40363865, 0x908B5959 */
322 1.36206794218215208048e+02, /* 0x4061069E, 0x0EE8878F */
323 2.70470278658083486789e+02, /* 0x4070E786, 0x42EA079B */
324 1.53875394208320329881e+02, /* 0x40633C03, 0x3AB6FAFF */
325 1.46576176948256193810e+01, /* 0x402D50B3, 0x44391809 */
329 static double pzero(double x
)
331 static double pzero(x
)
336 const double *p
= 0,*q
= 0;
344 if(ix
>=0x40200000) {p
= pR8
; q
= pS8
;}
345 else if(ix
>=0x40122E8B){p
= pR5
; q
= pS5
;}
346 else if(ix
>=0x4006DB6D){p
= pR3
; q
= pS3
;}
347 else if(ix
>=0x40000000){p
= pR2
; q
= pS2
;}
349 r
= p
[0]+z
*(p
[1]+z
*(p
[2]+z
*(p
[3]+z
*(p
[4]+z
*p
[5]))));
350 s
= one
+z
*(q
[0]+z
*(q
[1]+z
*(q
[2]+z
*(q
[3]+z
*q
[4]))));
355 /* For x >= 8, the asymptotic expansions of qzero is
356 * -1/8 s + 75/1024 s^3 - ..., where s = 1/x.
357 * We approximate pzero by
358 * qzero(x) = s*(-1.25 + (R/S))
359 * where R = qR0 + qR1*s^2 + qR2*s^4 + ... + qR5*s^10
360 * S = 1 + qS0*s^2 + ... + qS5*s^12
362 * | qzero(x)/s +1.25-R/S | <= 2 ** ( -61.22)
365 static const double qR8
[6] = { /* for x in [inf, 8]=1/[0,0.125] */
367 static double qR8
[6] = { /* for x in [inf, 8]=1/[0,0.125] */
369 0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */
370 7.32421874999935051953e-02, /* 0x3FB2BFFF, 0xFFFFFE2C */
371 1.17682064682252693899e+01, /* 0x40278952, 0x5BB334D6 */
372 5.57673380256401856059e+02, /* 0x40816D63, 0x15301825 */
373 8.85919720756468632317e+03, /* 0x40C14D99, 0x3E18F46D */
374 3.70146267776887834771e+04, /* 0x40E212D4, 0x0E901566 */
377 static const double qS8
[6] = {
379 static double qS8
[6] = {
381 1.63776026895689824414e+02, /* 0x406478D5, 0x365B39BC */
382 8.09834494656449805916e+03, /* 0x40BFA258, 0x4E6B0563 */
383 1.42538291419120476348e+05, /* 0x41016652, 0x54D38C3F */
384 8.03309257119514397345e+05, /* 0x412883DA, 0x83A52B43 */
385 8.40501579819060512818e+05, /* 0x4129A66B, 0x28DE0B3D */
386 -3.43899293537866615225e+05, /* 0xC114FD6D, 0x2C9530C5 */
390 static const double qR5
[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
392 static double qR5
[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
394 1.84085963594515531381e-11, /* 0x3DB43D8F, 0x29CC8CD9 */
395 7.32421766612684765896e-02, /* 0x3FB2BFFF, 0xD172B04C */
396 5.83563508962056953777e+00, /* 0x401757B0, 0xB9953DD3 */
397 1.35111577286449829671e+02, /* 0x4060E392, 0x0A8788E9 */
398 1.02724376596164097464e+03, /* 0x40900CF9, 0x9DC8C481 */
399 1.98997785864605384631e+03, /* 0x409F17E9, 0x53C6E3A6 */
402 static const double qS5
[6] = {
404 static double qS5
[6] = {
406 8.27766102236537761883e+01, /* 0x4054B1B3, 0xFB5E1543 */
407 2.07781416421392987104e+03, /* 0x40A03BA0, 0xDA21C0CE */
408 1.88472887785718085070e+04, /* 0x40D267D2, 0x7B591E6D */
409 5.67511122894947329769e+04, /* 0x40EBB5E3, 0x97E02372 */
410 3.59767538425114471465e+04, /* 0x40E19118, 0x1F7A54A0 */
411 -5.35434275601944773371e+03, /* 0xC0B4EA57, 0xBEDBC609 */
415 static const double qR3
[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
417 static double qR3
[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
419 4.37741014089738620906e-09, /* 0x3E32CD03, 0x6ADECB82 */
420 7.32411180042911447163e-02, /* 0x3FB2BFEE, 0x0E8D0842 */
421 3.34423137516170720929e+00, /* 0x400AC0FC, 0x61149CF5 */
422 4.26218440745412650017e+01, /* 0x40454F98, 0x962DAEDD */
423 1.70808091340565596283e+02, /* 0x406559DB, 0xE25EFD1F */
424 1.66733948696651168575e+02, /* 0x4064D77C, 0x81FA21E0 */
427 static const double qS3
[6] = {
429 static double qS3
[6] = {
431 4.87588729724587182091e+01, /* 0x40486122, 0xBFE343A6 */
432 7.09689221056606015736e+02, /* 0x40862D83, 0x86544EB3 */
433 3.70414822620111362994e+03, /* 0x40ACF04B, 0xE44DFC63 */
434 6.46042516752568917582e+03, /* 0x40B93C6C, 0xD7C76A28 */
435 2.51633368920368957333e+03, /* 0x40A3A8AA, 0xD94FB1C0 */
436 -1.49247451836156386662e+02, /* 0xC062A7EB, 0x201CF40F */
440 static const double qR2
[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
442 static double qR2
[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
444 1.50444444886983272379e-07, /* 0x3E84313B, 0x54F76BDB */
445 7.32234265963079278272e-02, /* 0x3FB2BEC5, 0x3E883E34 */
446 1.99819174093815998816e+00, /* 0x3FFFF897, 0xE727779C */
447 1.44956029347885735348e+01, /* 0x402CFDBF, 0xAAF96FE5 */
448 3.16662317504781540833e+01, /* 0x403FAA8E, 0x29FBDC4A */
449 1.62527075710929267416e+01, /* 0x403040B1, 0x71814BB4 */
452 static const double qS2
[6] = {
454 static double qS2
[6] = {
456 3.03655848355219184498e+01, /* 0x403E5D96, 0xF7C07AED */
457 2.69348118608049844624e+02, /* 0x4070D591, 0xE4D14B40 */
458 8.44783757595320139444e+02, /* 0x408A6645, 0x22B3BF22 */
459 8.82935845112488550512e+02, /* 0x408B977C, 0x9C5CC214 */
460 2.12666388511798828631e+02, /* 0x406A9553, 0x0E001365 */
461 -5.31095493882666946917e+00, /* 0xC0153E6A, 0xF8B32931 */
465 static double qzero(double x
)
467 static double qzero(x
)
472 const double *p
=0,*q
=0;
480 if(ix
>=0x40200000) {p
= qR8
; q
= qS8
;}
481 else if(ix
>=0x40122E8B){p
= qR5
; q
= qS5
;}
482 else if(ix
>=0x4006DB6D){p
= qR3
; q
= qS3
;}
483 else if(ix
>=0x40000000){p
= qR2
; q
= qS2
;}
485 r
= p
[0]+z
*(p
[1]+z
*(p
[2]+z
*(p
[3]+z
*(p
[4]+z
*p
[5]))));
486 s
= one
+z
*(q
[0]+z
*(q
[1]+z
*(q
[2]+z
*(q
[3]+z
*(q
[4]+z
*q
[5])))));
487 return (-.125 + r
/s
)/x
;