1 //===========================================================================
5 // Part of the standard mathematical function library
7 //===========================================================================
8 //####ECOSGPLCOPYRIGHTBEGIN####
9 // -------------------------------------------
10 // This file is part of eCos, the Embedded Configurable Operating System.
11 // Copyright (C) 1998, 1999, 2000, 2001, 2002 Red Hat, Inc.
13 // eCos is free software; you can redistribute it and/or modify it under
14 // the terms of the GNU General Public License as published by the Free
15 // Software Foundation; either version 2 or (at your option) any later version.
17 // eCos is distributed in the hope that it will be useful, but WITHOUT ANY
18 // WARRANTY; without even the implied warranty of MERCHANTABILITY or
19 // FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License
22 // You should have received a copy of the GNU General Public License along
23 // with eCos; if not, write to the Free Software Foundation, Inc.,
24 // 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA.
26 // As a special exception, if other files instantiate templates or use macros
27 // or inline functions from this file, or you compile this file and link it
28 // with other works to produce a work based on this file, this file does not
29 // by itself cause the resulting work to be covered by the GNU General Public
30 // License. However the source code for this file must still be made available
31 // in accordance with section (3) of the GNU General Public License.
33 // This exception does not invalidate any other reasons why a work based on
34 // this file might be covered by the GNU General Public License.
36 // Alternative licenses for eCos may be arranged by contacting Red Hat, Inc.
37 // at http://sources.redhat.com/ecos/ecos-license/
38 // -------------------------------------------
39 //####ECOSGPLCOPYRIGHTEND####
40 //===========================================================================
41 //#####DESCRIPTIONBEGIN####
43 // Author(s): jlarmour
44 // Contributors: jlarmour
50 //####DESCRIPTIONEND####
52 //===========================================================================
56 #include <pkgconf/libm.h> // Configuration header
58 // Include the Math library?
61 // Derived from code with the following copyright
64 /* @(#)e_j1.c 1.3 95/01/18 */
66 * ====================================================
67 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
69 * Developed at SunSoft, a Sun Microsystems, Inc. business.
70 * Permission to use, copy, modify, and distribute this
71 * software is freely granted, provided that this notice
73 * ====================================================
76 /* __ieee754_j1(x), __ieee754_y1(x)
77 * Bessel function of the first and second kinds of order zero.
79 * 1. For tiny x, we use j1(x) = x/2 - x^3/16 + x^5/384 - ...
80 * 2. Reduce x to |x| since j1(x)=-j1(-x), and
82 * j1(x) = x/2 + x*z*R0/S0, where z = x*x;
83 * (precision: |j1/x - 1/2 - R0/S0 |<2**-61.51 )
85 * j1(x) = sqrt(2/(pi*x))*(p1(x)*cos(x1)-q1(x)*sin(x1))
86 * y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x1)+q1(x)*cos(x1))
87 * where x1 = x-3*pi/4. It is better to compute sin(x1),cos(x1)
89 * cos(x1) = cos(x)cos(3pi/4)+sin(x)sin(3pi/4)
90 * = 1/sqrt(2) * (sin(x) - cos(x))
91 * sin(x1) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4)
92 * = -1/sqrt(2) * (sin(x) + cos(x))
93 * (To avoid cancellation, use
94 * sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
95 * to compute the worse one.)
103 * 1. screen out x<=0 cases: y1(0)=-inf, y1(x<0)=NaN
106 * y1(x) = 2/pi*(j1(x)*(ln(x/2)+Euler)-1/x-x/2+5/64*x^3-...)
107 * therefore y1(x)-2/pi*j1(x)*ln(x)-1/x is an odd function.
108 * We use the following function to approximate y1,
109 * y1(x) = x*U(z)/V(z) + (2/pi)*(j1(x)*ln(x)-1/x), z= x^2
110 * where for x in [0,2] (abs err less than 2**-65.89)
111 * U(z) = U0[0] + U0[1]*z + ... + U0[4]*z^4
112 * V(z) = 1 + v0[0]*z + ... + v0[4]*z^5
113 * Note: For tiny x, 1/x dominate y1 and hence
114 * y1(tiny) = -2/pi/tiny, (choose tiny<2**-54)
116 * y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x1)+q1(x)*cos(x1))
117 * where x1 = x-3*pi/4. It is better to compute sin(x1),cos(x1)
118 * by method mentioned above.
121 #include "mathincl/fdlibm.h"
123 static double pone(double), qone(double);
128 invsqrtpi= 5.64189583547756279280e-01, /* 0x3FE20DD7, 0x50429B6D */
129 tpi = 6.36619772367581382433e-01, /* 0x3FE45F30, 0x6DC9C883 */
131 r00 = -6.25000000000000000000e-02, /* 0xBFB00000, 0x00000000 */
132 r01 = 1.40705666955189706048e-03, /* 0x3F570D9F, 0x98472C61 */
133 r02 = -1.59955631084035597520e-05, /* 0xBEF0C5C6, 0xBA169668 */
134 r03 = 4.96727999609584448412e-08, /* 0x3E6AAAFA, 0x46CA0BD9 */
135 s01 = 1.91537599538363460805e-02, /* 0x3F939D0B, 0x12637E53 */
136 s02 = 1.85946785588630915560e-04, /* 0x3F285F56, 0xB9CDF664 */
137 s03 = 1.17718464042623683263e-06, /* 0x3EB3BFF8, 0x333F8498 */
138 s04 = 5.04636257076217042715e-09, /* 0x3E35AC88, 0xC97DFF2C */
139 s05 = 1.23542274426137913908e-11; /* 0x3DAB2ACF, 0xCFB97ED8 */
141 static double zero = 0.0;
143 double __ieee754_j1(double x)
145 double z, s,c,ss,cc,r,u,v,y;
150 if(ix>=0x7ff00000) return one/x;
152 if(ix >= 0x40000000) { /* |x| >= 2.0 */
157 if(ix<0x7fe00000) { /* make sure y+y not overflow */
159 if ((s*c)>zero) cc = z/ss;
163 * j1(x) = 1/sqrt(pi) * (P(1,x)*cc - Q(1,x)*ss) / sqrt(x)
164 * y1(x) = 1/sqrt(pi) * (P(1,x)*ss + Q(1,x)*cc) / sqrt(x)
166 if(ix>0x48000000) z = (invsqrtpi*cc)/sqrt(y);
168 u = pone(y); v = qone(y);
169 z = invsqrtpi*(u*cc-v*ss)/sqrt(y);
174 if(ix<0x3e400000) { /* |x|<2**-27 */
175 if(huge+x>one) return 0.5*x;/* inexact if x!=0 necessary */
178 r = z*(r00+z*(r01+z*(r02+z*r03)));
179 s = one+z*(s01+z*(s02+z*(s03+z*(s04+z*s05))));
184 static const double U0[5] = {
185 -1.96057090646238940668e-01, /* 0xBFC91866, 0x143CBC8A */
186 5.04438716639811282616e-02, /* 0x3FA9D3C7, 0x76292CD1 */
187 -1.91256895875763547298e-03, /* 0xBF5F55E5, 0x4844F50F */
188 2.35252600561610495928e-05, /* 0x3EF8AB03, 0x8FA6B88E */
189 -9.19099158039878874504e-08, /* 0xBE78AC00, 0x569105B8 */
191 static const double V0[5] = {
192 1.99167318236649903973e-02, /* 0x3F94650D, 0x3F4DA9F0 */
193 2.02552581025135171496e-04, /* 0x3F2A8C89, 0x6C257764 */
194 1.35608801097516229404e-06, /* 0x3EB6C05A, 0x894E8CA6 */
195 6.22741452364621501295e-09, /* 0x3E3ABF1D, 0x5BA69A86 */
196 1.66559246207992079114e-11, /* 0x3DB25039, 0xDACA772A */
199 double __ieee754_y1(double x)
201 double z, s,c,ss,cc,u,v;
207 /* if Y1(NaN) is NaN, Y1(-inf) is NaN, Y1(inf) is 0 */
208 if(ix>=0x7ff00000) return one/(x+x*x);
209 if((ix|lx)==0) return -one/zero;
210 if(hx<0) return zero/zero;
211 if(ix >= 0x40000000) { /* |x| >= 2.0 */
216 if(ix<0x7fe00000) { /* make sure x+x not overflow */
218 if ((s*c)>zero) cc = z/ss;
221 /* y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x0)+q1(x)*cos(x0))
224 * cos(x0) = cos(x)cos(3pi/4)+sin(x)sin(3pi/4)
225 * = 1/sqrt(2) * (sin(x) - cos(x))
226 * sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4)
227 * = -1/sqrt(2) * (cos(x) + sin(x))
228 * To avoid cancellation, use
229 * sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
230 * to compute the worse one.
232 if(ix>0x48000000) z = (invsqrtpi*ss)/sqrt(x);
234 u = pone(x); v = qone(x);
235 z = invsqrtpi*(u*ss+v*cc)/sqrt(x);
239 if(ix<=0x3c900000) { /* x < 2**-54 */
243 u = U0[0]+z*(U0[1]+z*(U0[2]+z*(U0[3]+z*U0[4])));
244 v = one+z*(V0[0]+z*(V0[1]+z*(V0[2]+z*(V0[3]+z*V0[4]))));
245 return(x*(u/v) + tpi*(__ieee754_j1(x)*__ieee754_log(x)-one/x));
248 /* For x >= 8, the asymptotic expansions of pone is
249 * 1 + 15/128 s^2 - 4725/2^15 s^4 - ..., where s = 1/x.
250 * We approximate pone by
251 * pone(x) = 1 + (R/S)
252 * where R = pr0 + pr1*s^2 + pr2*s^4 + ... + pr5*s^10
253 * S = 1 + ps0*s^2 + ... + ps4*s^10
255 * | pone(x)-1-R/S | <= 2 ** ( -60.06)
258 static const double pr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
259 0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */
260 1.17187499999988647970e-01, /* 0x3FBDFFFF, 0xFFFFFCCE */
261 1.32394806593073575129e+01, /* 0x402A7A9D, 0x357F7FCE */
262 4.12051854307378562225e+02, /* 0x4079C0D4, 0x652EA590 */
263 3.87474538913960532227e+03, /* 0x40AE457D, 0xA3A532CC */
264 7.91447954031891731574e+03, /* 0x40BEEA7A, 0xC32782DD */
266 static const double ps8[5] = {
267 1.14207370375678408436e+02, /* 0x405C8D45, 0x8E656CAC */
268 3.65093083420853463394e+03, /* 0x40AC85DC, 0x964D274F */
269 3.69562060269033463555e+04, /* 0x40E20B86, 0x97C5BB7F */
270 9.76027935934950801311e+04, /* 0x40F7D42C, 0xB28F17BB */
271 3.08042720627888811578e+04, /* 0x40DE1511, 0x697A0B2D */
274 static const double pr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
275 1.31990519556243522749e-11, /* 0x3DAD0667, 0xDAE1CA7D */
276 1.17187493190614097638e-01, /* 0x3FBDFFFF, 0xE2C10043 */
277 6.80275127868432871736e+00, /* 0x401B3604, 0x6E6315E3 */
278 1.08308182990189109773e+02, /* 0x405B13B9, 0x452602ED */
279 5.17636139533199752805e+02, /* 0x40802D16, 0xD052D649 */
280 5.28715201363337541807e+02, /* 0x408085B8, 0xBB7E0CB7 */
282 static const double ps5[5] = {
283 5.92805987221131331921e+01, /* 0x404DA3EA, 0xA8AF633D */
284 9.91401418733614377743e+02, /* 0x408EFB36, 0x1B066701 */
285 5.35326695291487976647e+03, /* 0x40B4E944, 0x5706B6FB */
286 7.84469031749551231769e+03, /* 0x40BEA4B0, 0xB8A5BB15 */
287 1.50404688810361062679e+03, /* 0x40978030, 0x036F5E51 */
290 static const double pr3[6] = {
291 3.02503916137373618024e-09, /* 0x3E29FC21, 0xA7AD9EDD */
292 1.17186865567253592491e-01, /* 0x3FBDFFF5, 0x5B21D17B */
293 3.93297750033315640650e+00, /* 0x400F76BC, 0xE85EAD8A */
294 3.51194035591636932736e+01, /* 0x40418F48, 0x9DA6D129 */
295 9.10550110750781271918e+01, /* 0x4056C385, 0x4D2C1837 */
296 4.85590685197364919645e+01, /* 0x4048478F, 0x8EA83EE5 */
298 static const double ps3[5] = {
299 3.47913095001251519989e+01, /* 0x40416549, 0xA134069C */
300 3.36762458747825746741e+02, /* 0x40750C33, 0x07F1A75F */
301 1.04687139975775130551e+03, /* 0x40905B7C, 0x5037D523 */
302 8.90811346398256432622e+02, /* 0x408BD67D, 0xA32E31E9 */
303 1.03787932439639277504e+02, /* 0x4059F26D, 0x7C2EED53 */
306 static const double pr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
307 1.07710830106873743082e-07, /* 0x3E7CE9D4, 0xF65544F4 */
308 1.17176219462683348094e-01, /* 0x3FBDFF42, 0xBE760D83 */
309 2.36851496667608785174e+00, /* 0x4002F2B7, 0xF98FAEC0 */
310 1.22426109148261232917e+01, /* 0x40287C37, 0x7F71A964 */
311 1.76939711271687727390e+01, /* 0x4031B1A8, 0x177F8EE2 */
312 5.07352312588818499250e+00, /* 0x40144B49, 0xA574C1FE */
314 static const double ps2[5] = {
315 2.14364859363821409488e+01, /* 0x40356FBD, 0x8AD5ECDC */
316 1.25290227168402751090e+02, /* 0x405F5293, 0x14F92CD5 */
317 2.32276469057162813669e+02, /* 0x406D08D8, 0xD5A2DBD9 */
318 1.17679373287147100768e+02, /* 0x405D6B7A, 0xDA1884A9 */
319 8.36463893371618283368e+00, /* 0x4020BAB1, 0xF44E5192 */
322 static double pone(double x)
327 ix = 0x7fffffff&CYG_LIBM_HI(x);
328 if(ix>=0x40200000) {p = pr8; q= ps8;}
329 else if(ix>=0x40122E8B){p = pr5; q= ps5;}
330 else if(ix>=0x4006DB6D){p = pr3; q= ps3;}
331 else {p = pr2; q= ps2;} /* if(ix>=0x40000000) */
333 r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
334 s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*q[4]))));
339 /* For x >= 8, the asymptotic expansions of qone is
340 * 3/8 s - 105/1024 s^3 - ..., where s = 1/x.
341 * We approximate pone by
342 * qone(x) = s*(0.375 + (R/S))
343 * where R = qr1*s^2 + qr2*s^4 + ... + qr5*s^10
344 * S = 1 + qs1*s^2 + ... + qs6*s^12
346 * | qone(x)/s -0.375-R/S | <= 2 ** ( -61.13)
349 static const double qr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
350 0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */
351 -1.02539062499992714161e-01, /* 0xBFBA3FFF, 0xFFFFFDF3 */
352 -1.62717534544589987888e+01, /* 0xC0304591, 0xA26779F7 */
353 -7.59601722513950107896e+02, /* 0xC087BCD0, 0x53E4B576 */
354 -1.18498066702429587167e+04, /* 0xC0C724E7, 0x40F87415 */
355 -4.84385124285750353010e+04, /* 0xC0E7A6D0, 0x65D09C6A */
357 static const double qs8[6] = {
358 1.61395369700722909556e+02, /* 0x40642CA6, 0xDE5BCDE5 */
359 7.82538599923348465381e+03, /* 0x40BE9162, 0xD0D88419 */
360 1.33875336287249578163e+05, /* 0x4100579A, 0xB0B75E98 */
361 7.19657723683240939863e+05, /* 0x4125F653, 0x72869C19 */
362 6.66601232617776375264e+05, /* 0x412457D2, 0x7719AD5C */
363 -2.94490264303834643215e+05, /* 0xC111F969, 0x0EA5AA18 */
366 static const double qr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
367 -2.08979931141764104297e-11, /* 0xBDB6FA43, 0x1AA1A098 */
368 -1.02539050241375426231e-01, /* 0xBFBA3FFF, 0xCB597FEF */
369 -8.05644828123936029840e+00, /* 0xC0201CE6, 0xCA03AD4B */
370 -1.83669607474888380239e+02, /* 0xC066F56D, 0x6CA7B9B0 */
371 -1.37319376065508163265e+03, /* 0xC09574C6, 0x6931734F */
372 -2.61244440453215656817e+03, /* 0xC0A468E3, 0x88FDA79D */
374 static const double qs5[6] = {
375 8.12765501384335777857e+01, /* 0x405451B2, 0xFF5A11B2 */
376 1.99179873460485964642e+03, /* 0x409F1F31, 0xE77BF839 */
377 1.74684851924908907677e+04, /* 0x40D10F1F, 0x0D64CE29 */
378 4.98514270910352279316e+04, /* 0x40E8576D, 0xAABAD197 */
379 2.79480751638918118260e+04, /* 0x40DB4B04, 0xCF7C364B */
380 -4.71918354795128470869e+03, /* 0xC0B26F2E, 0xFCFFA004 */
383 static const double qr3[6] = {
384 -5.07831226461766561369e-09, /* 0xBE35CFA9, 0xD38FC84F */
385 -1.02537829820837089745e-01, /* 0xBFBA3FEB, 0x51AEED54 */
386 -4.61011581139473403113e+00, /* 0xC01270C2, 0x3302D9FF */
387 -5.78472216562783643212e+01, /* 0xC04CEC71, 0xC25D16DA */
388 -2.28244540737631695038e+02, /* 0xC06C87D3, 0x4718D55F */
389 -2.19210128478909325622e+02, /* 0xC06B66B9, 0x5F5C1BF6 */
391 static const double qs3[6] = {
392 4.76651550323729509273e+01, /* 0x4047D523, 0xCCD367E4 */
393 6.73865112676699709482e+02, /* 0x40850EEB, 0xC031EE3E */
394 3.38015286679526343505e+03, /* 0x40AA684E, 0x448E7C9A */
395 5.54772909720722782367e+03, /* 0x40B5ABBA, 0xA61D54A6 */
396 1.90311919338810798763e+03, /* 0x409DBC7A, 0x0DD4DF4B */
397 -1.35201191444307340817e+02, /* 0xC060E670, 0x290A311F */
400 static const double qr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
401 -1.78381727510958865572e-07, /* 0xBE87F126, 0x44C626D2 */
402 -1.02517042607985553460e-01, /* 0xBFBA3E8E, 0x9148B010 */
403 -2.75220568278187460720e+00, /* 0xC0060484, 0x69BB4EDA */
404 -1.96636162643703720221e+01, /* 0xC033A9E2, 0xC168907F */
405 -4.23253133372830490089e+01, /* 0xC04529A3, 0xDE104AAA */
406 -2.13719211703704061733e+01, /* 0xC0355F36, 0x39CF6E52 */
408 static const double qs2[6] = {
409 2.95333629060523854548e+01, /* 0x403D888A, 0x78AE64FF */
410 2.52981549982190529136e+02, /* 0x406F9F68, 0xDB821CBA */
411 7.57502834868645436472e+02, /* 0x4087AC05, 0xCE49A0F7 */
412 7.39393205320467245656e+02, /* 0x40871B25, 0x48D4C029 */
413 1.55949003336666123687e+02, /* 0x40637E5E, 0x3C3ED8D4 */
414 -4.95949898822628210127e+00, /* 0xC013D686, 0xE71BE86B */
417 static double qone(double x)
422 ix = 0x7fffffff&CYG_LIBM_HI(x);
423 if(ix>=0x40200000) {p = qr8; q= qs8;}
424 else if(ix>=0x40122E8B){p = qr5; q= qs5;}
425 else if(ix>=0x4006DB6D){p = qr3; q= qs3;}
426 else {p = qr2; q= qs2;} /* if(ix>=0x40000000) */
428 r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
429 s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*(q[4]+z*q[5])))));
430 return (.375 + r/s)/x;
433 #endif // ifdef CYGPKG_LIBM