2 * Copyright (c) 1992, 1993
3 * The Regents of the University of California. All rights reserved.
5 * Redistribution and use in source and binary forms, with or without
6 * modification, are permitted provided that the following conditions
8 * 1. Redistributions of source code must retain the above copyright
9 * notice, this list of conditions and the following disclaimer.
10 * 2. Redistributions in binary form must reproduce the above copyright
11 * notice, this list of conditions and the following disclaimer in the
12 * documentation and/or other materials provided with the distribution.
13 * 3. All advertising materials mentioning features or use of this software
14 * must display the following acknowledgement:
15 * This product includes software developed by the University of
16 * California, Berkeley and its contributors.
17 * 4. Neither the name of the University nor the names of its contributors
18 * may be used to endorse or promote products derived from this software
19 * without specific prior written permission.
21 * THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND
22 * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
23 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
24 * ARE DISCLAIMED. IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE
25 * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
26 * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
27 * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
28 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
29 * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
30 * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
33 * @(#)j1.c 8.2 (Berkeley) 11/30/93
38 * Minor modifications by Peter McIlroy to adapt non-IEEE architecture.
42 * ====================================================
43 * Copyright (C) 1992 by Sun Microsystems, Inc.
45 * Developed at SunPro, a Sun Microsystems, Inc. business.
46 * Permission to use, copy, modify, and distribute this
47 * software is freely granted, provided that this notice
49 * ====================================================
51 * ******************* WARNING ********************
52 * This is an alpha version of SunPro's FDLIBM (Freely
53 * Distributable Math Library) for IEEE double precision
54 * arithmetic. FDLIBM is a basic math library written
55 * in C that runs on machines that conform to IEEE
56 * Standard 754/854. This alpha version is distributed
57 * for testing purpose. Those who use this software
58 * should report any bugs to
60 * fdlibm-comments@sunpro.eng.sun.com
62 * -- K.C. Ng, Oct 12, 1992
63 * ************************************************
66 /* double j1(double x), y1(double x)
67 * Bessel function of the first and second kinds of order zero.
69 * 1. For tiny x, we use j1(x) = x/2 - x^3/16 + x^5/384 - ...
70 * 2. Reduce x to |x| since j1(x)=-j1(-x), and
72 * j1(x) = x/2 + x*z*R0/S0, where z = x*x;
73 * (precision: |j1/x - 1/2 - R0/S0 |<2**-61.51 )
75 * j1(x) = sqrt(2/(pi*x))*(p1(x)*cos(x1)-q1(x)*sin(x1))
76 * y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x1)+q1(x)*cos(x1))
77 * where x1 = x-3*pi/4. It is better to compute sin(x1),cos(x1)
79 * cos(x1) = cos(x)cos(3pi/4)+sin(x)sin(3pi/4)
80 * = 1/sqrt(2) * (sin(x) - cos(x))
81 * sin(x1) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4)
82 * = -1/sqrt(2) * (sin(x) + cos(x))
83 * (To avoid cancellation, use
84 * sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
85 * to compute the worse one.)
93 * 1. screen out x<=0 cases: y1(0)=-inf, y1(x<0)=NaN
96 * y1(x) = 2/pi*(j1(x)*(ln(x/2)+Euler)-1/x-x/2+5/64*x^3-...)
97 * therefore y1(x)-2/pi*j1(x)*ln(x)-1/x is an odd function.
98 * We use the following function to approximate y1,
99 * y1(x) = x*U(z)/V(z) + (2/pi)*(j1(x)*ln(x)-1/x), z= x^2
100 * where for x in [0,2] (abs err less than 2**-65.89)
101 * U(z) = u0 + u1*z + ... + u4*z^4
102 * V(z) = 1 + v1*z + ... + v5*z^5
103 * Note: For tiny x, 1/x dominate y1 and hence
104 * y1(tiny) = -2/pi/tiny, (choose tiny<2**-54)
106 * y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x1)+q1(x)*cos(x1))
107 * where x1 = x-3*pi/4. It is better to compute sin(x1),cos(x1)
108 * by method mentioned above.
114 #if defined(vax) || defined(tahoe)
118 #define infnan(x) (0.0)
121 static double pone(), qone();
127 invsqrtpi= 5.641895835477562869480794515607725858441e-0001,
128 tpi = 0.636619772367581343075535053490057448,
131 r00 = -6.250000000000000020842322918309200910191e-0002,
132 r01 = 1.407056669551897148204830386691427791200e-0003,
133 r02 = -1.599556310840356073980727783817809847071e-0005,
134 r03 = 4.967279996095844750387702652791615403527e-0008,
135 s01 = 1.915375995383634614394860200531091839635e-0002,
136 s02 = 1.859467855886309024045655476348872850396e-0004,
137 s03 = 1.177184640426236767593432585906758230822e-0006,
138 s04 = 5.046362570762170559046714468225101016915e-0009,
139 s05 = 1.235422744261379203512624973117299248281e-0011;
141 #define two_129 6.80564733841876926e+038 /* 2^129 */
142 #define two_m54 5.55111512312578270e-017 /* 2^-54 */
146 double z, s,c,ss,cc,r,u,v,y;
148 if (!finite(x)) /* Inf or NaN */
152 return (copysign(x, zero));
154 if (y >= 2) /* |x| >= 2.0 */
160 if (y < .5*DBL_MAX) { /* make sure y+y not overflow */
162 if ((s*c)<zero) cc = z/ss;
166 * j1(x) = 1/sqrt(pi) * (P(1,x)*cc - Q(1,x)*ss) / sqrt(x)
167 * y1(x) = 1/sqrt(pi) * (P(1,x)*ss + Q(1,x)*cc) / sqrt(x)
169 #if !defined(vax) && !defined(tahoe)
170 if (y > two_129) /* x > 2^129 */
171 z = (invsqrtpi*cc)/sqrt(y);
173 #endif /* defined(vax) || defined(tahoe) */
175 u = pone(y); v = qone(y);
176 z = invsqrtpi*(u*cc-v*ss)/sqrt(y);
178 if (x < 0) return -z;
181 if (y < 7.450580596923828125e-009) { /* |x|<2**-27 */
182 if(huge+x>one) return 0.5*x;/* inexact if x!=0 necessary */
185 r = z*(r00+z*(r01+z*(r02+z*r03)));
186 s = one+z*(s01+z*(s02+z*(s03+z*(s04+z*s05))));
191 static double u0[5] = {
192 -1.960570906462389484206891092512047539632e-0001,
193 5.044387166398112572026169863174882070274e-0002,
194 -1.912568958757635383926261729464141209569e-0003,
195 2.352526005616105109577368905595045204577e-0005,
196 -9.190991580398788465315411784276789663849e-0008,
198 static double v0[5] = {
199 1.991673182366499064031901734535479833387e-0002,
200 2.025525810251351806268483867032781294682e-0004,
201 1.356088010975162198085369545564475416398e-0006,
202 6.227414523646214811803898435084697863445e-0009,
203 1.665592462079920695971450872592458916421e-0011,
209 double z, s, c, ss, cc, u, v;
210 /* if Y1(NaN) is NaN, Y1(-inf) is NaN, Y1(inf) is 0 */
212 if (!_IEEE) return (infnan(EDOM));
220 if (_IEEE && x == 0) return -one/zero;
221 else if(x == 0) return(infnan(-ERANGE));
222 else if(_IEEE) return (zero/zero);
223 else return(infnan(EDOM));
225 if (x >= 2) /* |x| >= 2.0 */
231 if (x < .5 * DBL_MAX) /* make sure x+x not overflow */
234 if ((s*c)>zero) cc = z/ss;
237 /* y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x0)+q1(x)*cos(x0))
240 * cos(x0) = cos(x)cos(3pi/4)+sin(x)sin(3pi/4)
241 * = 1/sqrt(2) * (sin(x) - cos(x))
242 * sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4)
243 * = -1/sqrt(2) * (cos(x) + sin(x))
244 * To avoid cancellation, use
245 * sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
246 * to compute the worse one.
248 if (_IEEE && x>two_129)
249 z = (invsqrtpi*ss)/sqrt(x);
251 u = pone(x); v = qone(x);
252 z = invsqrtpi*(u*ss+v*cc)/sqrt(x);
256 if (x <= two_m54) { /* x < 2**-54 */
260 u = u0[0]+z*(u0[1]+z*(u0[2]+z*(u0[3]+z*u0[4])));
261 v = one+z*(v0[0]+z*(v0[1]+z*(v0[2]+z*(v0[3]+z*v0[4]))));
262 return (x*(u/v) + tpi*(j1(x)*log(x)-one/x));
265 /* For x >= 8, the asymptotic expansions of pone is
266 * 1 + 15/128 s^2 - 4725/2^15 s^4 - ..., where s = 1/x.
267 * We approximate pone by
268 * pone(x) = 1 + (R/S)
269 * where R = pr0 + pr1*s^2 + pr2*s^4 + ... + pr5*s^10
270 * S = 1 + ps0*s^2 + ... + ps4*s^10
272 * | pone(x)-1-R/S | <= 2 ** ( -60.06)
275 static double pr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
277 1.171874999999886486643746274751925399540e-0001,
278 1.323948065930735690925827997575471527252e+0001,
279 4.120518543073785433325860184116512799375e+0002,
280 3.874745389139605254931106878336700275601e+0003,
281 7.914479540318917214253998253147871806507e+0003,
283 static double ps8[5] = {
284 1.142073703756784104235066368252692471887e+0002,
285 3.650930834208534511135396060708677099382e+0003,
286 3.695620602690334708579444954937638371808e+0004,
287 9.760279359349508334916300080109196824151e+0004,
288 3.080427206278887984185421142572315054499e+0004,
291 static double pr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
292 1.319905195562435287967533851581013807103e-0011,
293 1.171874931906140985709584817065144884218e-0001,
294 6.802751278684328781830052995333841452280e+0000,
295 1.083081829901891089952869437126160568246e+0002,
296 5.176361395331997166796512844100442096318e+0002,
297 5.287152013633375676874794230748055786553e+0002,
299 static double ps5[5] = {
300 5.928059872211313557747989128353699746120e+0001,
301 9.914014187336144114070148769222018425781e+0002,
302 5.353266952914879348427003712029704477451e+0003,
303 7.844690317495512717451367787640014588422e+0003,
304 1.504046888103610723953792002716816255382e+0003,
307 static double pr3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
308 3.025039161373736032825049903408701962756e-0009,
309 1.171868655672535980750284752227495879921e-0001,
310 3.932977500333156527232725812363183251138e+0000,
311 3.511940355916369600741054592597098912682e+0001,
312 9.105501107507812029367749771053045219094e+0001,
313 4.855906851973649494139275085628195457113e+0001,
315 static double ps3[5] = {
316 3.479130950012515114598605916318694946754e+0001,
317 3.367624587478257581844639171605788622549e+0002,
318 1.046871399757751279180649307467612538415e+0003,
319 8.908113463982564638443204408234739237639e+0002,
320 1.037879324396392739952487012284401031859e+0002,
323 static double pr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
324 1.077108301068737449490056513753865482831e-0007,
325 1.171762194626833490512746348050035171545e-0001,
326 2.368514966676087902251125130227221462134e+0000,
327 1.224261091482612280835153832574115951447e+0001,
328 1.769397112716877301904532320376586509782e+0001,
329 5.073523125888185399030700509321145995160e+0000,
331 static double ps2[5] = {
332 2.143648593638214170243114358933327983793e+0001,
333 1.252902271684027493309211410842525120355e+0002,
334 2.322764690571628159027850677565128301361e+0002,
335 1.176793732871470939654351793502076106651e+0002,
336 8.364638933716182492500902115164881195742e+0000,
339 static double pone(x)
343 if (x >= 8.0) {p = pr8; q= ps8;}
344 else if (x >= 4.54545211791992188) {p = pr5; q= ps5;}
345 else if (x >= 2.85714149475097656) {p = pr3; q= ps3;}
346 else /* if (x >= 2.0) */ {p = pr2; q= ps2;}
348 r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
349 s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*q[4]))));
354 /* For x >= 8, the asymptotic expansions of qone is
355 * 3/8 s - 105/1024 s^3 - ..., where s = 1/x.
356 * We approximate pone by
357 * qone(x) = s*(0.375 + (R/S))
358 * where R = qr1*s^2 + qr2*s^4 + ... + qr5*s^10
359 * S = 1 + qs1*s^2 + ... + qs6*s^12
361 * | qone(x)/s -0.375-R/S | <= 2 ** ( -61.13)
364 static double qr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
366 -1.025390624999927207385863635575804210817e-0001,
367 -1.627175345445899724355852152103771510209e+0001,
368 -7.596017225139501519843072766973047217159e+0002,
369 -1.184980667024295901645301570813228628541e+0004,
370 -4.843851242857503225866761992518949647041e+0004,
372 static double qs8[6] = {
373 1.613953697007229231029079421446916397904e+0002,
374 7.825385999233484705298782500926834217525e+0003,
375 1.338753362872495800748094112937868089032e+0005,
376 7.196577236832409151461363171617204036929e+0005,
377 6.666012326177764020898162762642290294625e+0005,
378 -2.944902643038346618211973470809456636830e+0005,
381 static double qr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
382 -2.089799311417640889742251585097264715678e-0011,
383 -1.025390502413754195402736294609692303708e-0001,
384 -8.056448281239359746193011295417408828404e+0000,
385 -1.836696074748883785606784430098756513222e+0002,
386 -1.373193760655081612991329358017247355921e+0003,
387 -2.612444404532156676659706427295870995743e+0003,
389 static double qs5[6] = {
390 8.127655013843357670881559763225310973118e+0001,
391 1.991798734604859732508048816860471197220e+0003,
392 1.746848519249089131627491835267411777366e+0004,
393 4.985142709103522808438758919150738000353e+0004,
394 2.794807516389181249227113445299675335543e+0004,
395 -4.719183547951285076111596613593553911065e+0003,
398 static double qr3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
399 -5.078312264617665927595954813341838734288e-0009,
400 -1.025378298208370901410560259001035577681e-0001,
401 -4.610115811394734131557983832055607679242e+0000,
402 -5.784722165627836421815348508816936196402e+0001,
403 -2.282445407376317023842545937526967035712e+0002,
404 -2.192101284789093123936441805496580237676e+0002,
406 static double qs3[6] = {
407 4.766515503237295155392317984171640809318e+0001,
408 6.738651126766996691330687210949984203167e+0002,
409 3.380152866795263466426219644231687474174e+0003,
410 5.547729097207227642358288160210745890345e+0003,
411 1.903119193388108072238947732674639066045e+0003,
412 -1.352011914443073322978097159157678748982e+0002,
415 static double qr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
416 -1.783817275109588656126772316921194887979e-0007,
417 -1.025170426079855506812435356168903694433e-0001,
418 -2.752205682781874520495702498875020485552e+0000,
419 -1.966361626437037351076756351268110418862e+0001,
420 -4.232531333728305108194363846333841480336e+0001,
421 -2.137192117037040574661406572497288723430e+0001,
423 static double qs2[6] = {
424 2.953336290605238495019307530224241335502e+0001,
425 2.529815499821905343698811319455305266409e+0002,
426 7.575028348686454070022561120722815892346e+0002,
427 7.393932053204672479746835719678434981599e+0002,
428 1.559490033366661142496448853793707126179e+0002,
429 -4.959498988226281813825263003231704397158e+0000,
432 static double qone(x)
436 if (x >= 8.0) {p = qr8; q= qs8;}
437 else if (x >= 4.54545211791992188) {p = qr5; q= qs5;}
438 else if (x >= 2.85714149475097656) {p = qr3; q= qs3;}
439 else /* if (x >= 2.0) */ {p = qr2; q= qs2;}
441 r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
442 s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*(q[4]+z*q[5])))));
443 return (.375 + r/s)/x;