1
2 /*
3 * Copyright (c) 1998, 2001, Oracle and/or its affiliates. All rights reserved.
4 * DO NOT ALTER OR REMOVE COPYRIGHT NOTICES OR THIS FILE HEADER.
5 *
6 * This code is free software; you can redistribute it and/or modify it
7 * under the terms of the GNU General Public License version 2 only, as
8 * published by the Free Software Foundation. Oracle designates this
9 * particular file as subject to the "Classpath" exception as provided
10 * by Oracle in the LICENSE file that accompanied this code.
11 *
12 * This code is distributed in the hope that it will be useful, but WITHOUT
13 * ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or
14 * FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License
15 * version 2 for more details (a copy is included in the LICENSE file that
16 * accompanied this code).
17 *
18 * You should have received a copy of the GNU General Public License version
19 * 2 along with this work; if not, write to the Free Software Foundation,
20 * Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA.
21 *
22 * Please contact Oracle, 500 Oracle Parkway, Redwood Shores, CA 94065 USA
23 * or visit www.oracle.com if you need additional information or have any
24 * questions.
25 */
26
27 /* __ieee754_j0(x), __ieee754_y0(x)
28 * Bessel function of the first and second kinds of order zero.
29 * Method -- j0(x):
30 * 1. For tiny x, we use j0(x) = 1 - x^2/4 + x^4/64 - ...
31 * 2. Reduce x to |x| since j0(x)=j0(-x), and
32 * for x in (0,2)
33 * j0(x) = 1-z/4+ z^2*R0/S0, where z = x*x;
34 * (precision: |j0-1+z/4-z^2R0/S0 |<2**-63.67 )
35 * for x in (2,inf)
36 * j0(x) = sqrt(2/(pi*x))*(p0(x)*cos(x0)-q0(x)*sin(x0))
37 * where x0 = x-pi/4. It is better to compute sin(x0),cos(x0)
38 * as follow:
39 * cos(x0) = cos(x)cos(pi/4)+sin(x)sin(pi/4)
40 * = 1/sqrt(2) * (cos(x) + sin(x))
41 * sin(x0) = sin(x)cos(pi/4)-cos(x)sin(pi/4)
42 * = 1/sqrt(2) * (sin(x) - cos(x))
43 * (To avoid cancellation, use
44 * sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
45 * to compute the worse one.)
46 *
47 * 3 Special cases
48 * j0(nan)= nan
49 * j0(0) = 1
50 * j0(inf) = 0
51 *
52 * Method -- y0(x):
53 * 1. For x<2.
54 * Since
55 * y0(x) = 2/pi*(j0(x)*(ln(x/2)+Euler) + x^2/4 - ...)
56 * therefore y0(x)-2/pi*j0(x)*ln(x) is an even function.
57 * We use the following function to approximate y0,
58 * y0(x) = U(z)/V(z) + (2/pi)*(j0(x)*ln(x)), z= x^2
59 * where
60 * U(z) = u00 + u01*z + ... + u06*z^6
61 * V(z) = 1 + v01*z + ... + v04*z^4
62 * with absolute approximation error bounded by 2**-72.
63 * Note: For tiny x, U/V = u0 and j0(x)~1, hence
64 * y0(tiny) = u0 + (2/pi)*ln(tiny), (choose tiny<2**-27)
65 * 2. For x>=2.
66 * y0(x) = sqrt(2/(pi*x))*(p0(x)*cos(x0)+q0(x)*sin(x0))
67 * where x0 = x-pi/4. It is better to compute sin(x0),cos(x0)
68 * by the method mentioned above.
69 * 3. Special cases: y0(0)=-inf, y0(x<0)=NaN, y0(inf)=0.
70 */
71
72 #include "fdlibm.h"
73
74 #ifdef __STDC__
75 static double pzero(double), qzero(double);
76 #else
77 static double pzero(), qzero();
78 #endif
79
80 #ifdef __STDC__
81 static const double
82 #else
83 static double
84 #endif
85 huge = 1e300,
86 one = 1.0,
87 invsqrtpi= 5.64189583547756279280e-01, /* 0x3FE20DD7, 0x50429B6D */
88 tpi = 6.36619772367581382433e-01, /* 0x3FE45F30, 0x6DC9C883 */
89 /* R0/S0 on [0, 2.00] */
90 R02 = 1.56249999999999947958e-02, /* 0x3F8FFFFF, 0xFFFFFFFD */
91 R03 = -1.89979294238854721751e-04, /* 0xBF28E6A5, 0xB61AC6E9 */
92 R04 = 1.82954049532700665670e-06, /* 0x3EBEB1D1, 0x0C503919 */
93 R05 = -4.61832688532103189199e-09, /* 0xBE33D5E7, 0x73D63FCE */
94 S01 = 1.56191029464890010492e-02, /* 0x3F8FFCE8, 0x82C8C2A4 */
95 S02 = 1.16926784663337450260e-04, /* 0x3F1EA6D2, 0xDD57DBF4 */
96 S03 = 5.13546550207318111446e-07, /* 0x3EA13B54, 0xCE84D5A9 */
97 S04 = 1.16614003333790000205e-09; /* 0x3E1408BC, 0xF4745D8F */
98
99 static double zero = 0.0;
100
101 #ifdef __STDC__
102 double __ieee754_j0(double x)
103 #else
104 double __ieee754_j0(x)
105 double x;
106 #endif
107 {
108 double z, s,c,ss,cc,r,u,v;
109 int hx,ix;
110
111 hx = __HI(x);
112 ix = hx&0x7fffffff;
113 if(ix>=0x7ff00000) return one/(x*x);
114 x = fabs(x);
115 if(ix >= 0x40000000) { /* |x| >= 2.0 */
116 s = sin(x);
117 c = cos(x);
118 ss = s-c;
119 cc = s+c;
120 if(ix<0x7fe00000) { /* make sure x+x not overflow */
121 z = -cos(x+x);
122 if ((s*c)<zero) cc = z/ss;
123 else ss = z/cc;
124 }
125 /*
126 * j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x)
127 * y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x)
128 */
129 if(ix>0x48000000) z = (invsqrtpi*cc)/sqrt(x);
130 else {
131 u = pzero(x); v = qzero(x);
132 z = invsqrtpi*(u*cc-v*ss)/sqrt(x);
133 }
134 return z;
135 }
136 if(ix<0x3f200000) { /* |x| < 2**-13 */
137 if(huge+x>one) { /* raise inexact if x != 0 */
138 if(ix<0x3e400000) return one; /* |x|<2**-27 */
139 else return one - 0.25*x*x;
140 }
141 }
142 z = x*x;
143 r = z*(R02+z*(R03+z*(R04+z*R05)));
144 s = one+z*(S01+z*(S02+z*(S03+z*S04)));
145 if(ix < 0x3FF00000) { /* |x| < 1.00 */
146 return one + z*(-0.25+(r/s));
147 } else {
148 u = 0.5*x;
149 return((one+u)*(one-u)+z*(r/s));
150 }
151 }
152
153 #ifdef __STDC__
154 static const double
155 #else
156 static double
157 #endif
158 u00 = -7.38042951086872317523e-02, /* 0xBFB2E4D6, 0x99CBD01F */
159 u01 = 1.76666452509181115538e-01, /* 0x3FC69D01, 0x9DE9E3FC */
160 u02 = -1.38185671945596898896e-02, /* 0xBF8C4CE8, 0xB16CFA97 */
161 u03 = 3.47453432093683650238e-04, /* 0x3F36C54D, 0x20B29B6B */
162 u04 = -3.81407053724364161125e-06, /* 0xBECFFEA7, 0x73D25CAD */
163 u05 = 1.95590137035022920206e-08, /* 0x3E550057, 0x3B4EABD4 */
164 u06 = -3.98205194132103398453e-11, /* 0xBDC5E43D, 0x693FB3C8 */
165 v01 = 1.27304834834123699328e-02, /* 0x3F8A1270, 0x91C9C71A */
166 v02 = 7.60068627350353253702e-05, /* 0x3F13ECBB, 0xF578C6C1 */
167 v03 = 2.59150851840457805467e-07, /* 0x3E91642D, 0x7FF202FD */
168 v04 = 4.41110311332675467403e-10; /* 0x3DFE5018, 0x3BD6D9EF */
169
170 #ifdef __STDC__
171 double __ieee754_y0(double x)
172 #else
173 double __ieee754_y0(x)
174 double x;
175 #endif
176 {
177 double z, s,c,ss,cc,u,v;
178 int hx,ix,lx;
179
180 hx = __HI(x);
181 ix = 0x7fffffff&hx;
182 lx = __LO(x);
183 /* Y0(NaN) is NaN, y0(-inf) is Nan, y0(inf) is 0 */
184 if(ix>=0x7ff00000) return one/(x+x*x);
185 if((ix|lx)==0) return -one/zero;
186 if(hx<0) return zero/zero;
187 if(ix >= 0x40000000) { /* |x| >= 2.0 */
188 /* y0(x) = sqrt(2/(pi*x))*(p0(x)*sin(x0)+q0(x)*cos(x0))
189 * where x0 = x-pi/4
190 * Better formula:
191 * cos(x0) = cos(x)cos(pi/4)+sin(x)sin(pi/4)
192 * = 1/sqrt(2) * (sin(x) + cos(x))
193 * sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4)
194 * = 1/sqrt(2) * (sin(x) - cos(x))
195 * To avoid cancellation, use
196 * sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
197 * to compute the worse one.
198 */
199 s = sin(x);
200 c = cos(x);
201 ss = s-c;
202 cc = s+c;
203 /*
204 * j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x)
205 * y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x)
206 */
207 if(ix<0x7fe00000) { /* make sure x+x not overflow */
208 z = -cos(x+x);
209 if ((s*c)<zero) cc = z/ss;
210 else ss = z/cc;
211 }
212 if(ix>0x48000000) z = (invsqrtpi*ss)/sqrt(x);
213 else {
214 u = pzero(x); v = qzero(x);
215 z = invsqrtpi*(u*ss+v*cc)/sqrt(x);
216 }
217 return z;
218 }
219 if(ix<=0x3e400000) { /* x < 2**-27 */
220 return(u00 + tpi*__ieee754_log(x));
221 }
222 z = x*x;
223 u = u00+z*(u01+z*(u02+z*(u03+z*(u04+z*(u05+z*u06)))));
224 v = one+z*(v01+z*(v02+z*(v03+z*v04)));
225 return(u/v + tpi*(__ieee754_j0(x)*__ieee754_log(x)));
226 }
227
228 /* The asymptotic expansions of pzero is
229 * 1 - 9/128 s^2 + 11025/98304 s^4 - ..., where s = 1/x.
230 * For x >= 2, We approximate pzero by
231 * pzero(x) = 1 + (R/S)
232 * where R = pR0 + pR1*s^2 + pR2*s^4 + ... + pR5*s^10
233 * S = 1 + pS0*s^2 + ... + pS4*s^10
234 * and
235 * | pzero(x)-1-R/S | <= 2 ** ( -60.26)
236 */
237 #ifdef __STDC__
238 static const double pR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
239 #else
240 static double pR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
241 #endif
242 0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */
243 -7.03124999999900357484e-02, /* 0xBFB1FFFF, 0xFFFFFD32 */
244 -8.08167041275349795626e+00, /* 0xC02029D0, 0xB44FA779 */
245 -2.57063105679704847262e+02, /* 0xC0701102, 0x7B19E863 */
246 -2.48521641009428822144e+03, /* 0xC0A36A6E, 0xCD4DCAFC */
247 -5.25304380490729545272e+03, /* 0xC0B4850B, 0x36CC643D */
248 };
249 #ifdef __STDC__
250 static const double pS8[5] = {
251 #else
252 static double pS8[5] = {
253 #endif
254 1.16534364619668181717e+02, /* 0x405D2233, 0x07A96751 */
255 3.83374475364121826715e+03, /* 0x40ADF37D, 0x50596938 */
256 4.05978572648472545552e+04, /* 0x40E3D2BB, 0x6EB6B05F */
257 1.16752972564375915681e+05, /* 0x40FC810F, 0x8F9FA9BD */
258 4.76277284146730962675e+04, /* 0x40E74177, 0x4F2C49DC */
259 };
260
261 #ifdef __STDC__
262 static const double pR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
263 #else
264 static double pR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
265 #endif
266 -1.14125464691894502584e-11, /* 0xBDA918B1, 0x47E495CC */
267 -7.03124940873599280078e-02, /* 0xBFB1FFFF, 0xE69AFBC6 */
268 -4.15961064470587782438e+00, /* 0xC010A370, 0xF90C6BBF */
269 -6.76747652265167261021e+01, /* 0xC050EB2F, 0x5A7D1783 */
270 -3.31231299649172967747e+02, /* 0xC074B3B3, 0x6742CC63 */
271 -3.46433388365604912451e+02, /* 0xC075A6EF, 0x28A38BD7 */
272 };
273 #ifdef __STDC__
274 static const double pS5[5] = {
275 #else
276 static double pS5[5] = {
277 #endif
278 6.07539382692300335975e+01, /* 0x404E6081, 0x0C98C5DE */
279 1.05125230595704579173e+03, /* 0x40906D02, 0x5C7E2864 */
280 5.97897094333855784498e+03, /* 0x40B75AF8, 0x8FBE1D60 */
281 9.62544514357774460223e+03, /* 0x40C2CCB8, 0xFA76FA38 */
282 2.40605815922939109441e+03, /* 0x40A2CC1D, 0xC70BE864 */
283 };
284
285 #ifdef __STDC__
286 static const double pR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
287 #else
288 static double pR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
289 #endif
290 -2.54704601771951915620e-09, /* 0xBE25E103, 0x6FE1AA86 */
291 -7.03119616381481654654e-02, /* 0xBFB1FFF6, 0xF7C0E24B */
292 -2.40903221549529611423e+00, /* 0xC00345B2, 0xAEA48074 */
293 -2.19659774734883086467e+01, /* 0xC035F74A, 0x4CB94E14 */
294 -5.80791704701737572236e+01, /* 0xC04D0A22, 0x420A1A45 */
295 -3.14479470594888503854e+01, /* 0xC03F72AC, 0xA892D80F */
296 };
297 #ifdef __STDC__
298 static const double pS3[5] = {
299 #else
300 static double pS3[5] = {
301 #endif
302 3.58560338055209726349e+01, /* 0x4041ED92, 0x84077DD3 */
303 3.61513983050303863820e+02, /* 0x40769839, 0x464A7C0E */
304 1.19360783792111533330e+03, /* 0x4092A66E, 0x6D1061D6 */
305 1.12799679856907414432e+03, /* 0x40919FFC, 0xB8C39B7E */
306 1.73580930813335754692e+02, /* 0x4065B296, 0xFC379081 */
307 };
308
309 #ifdef __STDC__
310 static const double pR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
311 #else
312 static double pR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
313 #endif
314 -8.87534333032526411254e-08, /* 0xBE77D316, 0xE927026D */
315 -7.03030995483624743247e-02, /* 0xBFB1FF62, 0x495E1E42 */
316 -1.45073846780952986357e+00, /* 0xBFF73639, 0x8A24A843 */
317 -7.63569613823527770791e+00, /* 0xC01E8AF3, 0xEDAFA7F3 */
318 -1.11931668860356747786e+01, /* 0xC02662E6, 0xC5246303 */
319 -3.23364579351335335033e+00, /* 0xC009DE81, 0xAF8FE70F */
320 };
321 #ifdef __STDC__
322 static const double pS2[5] = {
323 #else
324 static double pS2[5] = {
325 #endif
326 2.22202997532088808441e+01, /* 0x40363865, 0x908B5959 */
327 1.36206794218215208048e+02, /* 0x4061069E, 0x0EE8878F */
328 2.70470278658083486789e+02, /* 0x4070E786, 0x42EA079B */
329 1.53875394208320329881e+02, /* 0x40633C03, 0x3AB6FAFF */
330 1.46576176948256193810e+01, /* 0x402D50B3, 0x44391809 */
331 };
332
333 #ifdef __STDC__
334 static double pzero(double x)
335 #else
336 static double pzero(x)
337 double x;
338 #endif
339 {
340 #ifdef __STDC__
341 const double *p=(void*)0,*q=(void*)0;
342 #else
343 double *p,*q;
344 #endif
345 double z,r,s;
346 int ix;
347 ix = 0x7fffffff&__HI(x);
348 if(ix>=0x40200000) {p = pR8; q= pS8;}
349 else if(ix>=0x40122E8B){p = pR5; q= pS5;}
350 else if(ix>=0x4006DB6D){p = pR3; q= pS3;}
351 else if(ix>=0x40000000){p = pR2; q= pS2;}
352 z = one/(x*x);
353 r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
354 s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*q[4]))));
355 return one+ r/s;
356 }
357
358
359 /* For x >= 8, the asymptotic expansions of qzero is
360 * -1/8 s + 75/1024 s^3 - ..., where s = 1/x.
361 * We approximate pzero by
362 * qzero(x) = s*(-1.25 + (R/S))
363 * where R = qR0 + qR1*s^2 + qR2*s^4 + ... + qR5*s^10
364 * S = 1 + qS0*s^2 + ... + qS5*s^12
365 * and
366 * | qzero(x)/s +1.25-R/S | <= 2 ** ( -61.22)
367 */
368 #ifdef __STDC__
369 static const double qR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
370 #else
371 static double qR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
372 #endif
373 0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */
374 7.32421874999935051953e-02, /* 0x3FB2BFFF, 0xFFFFFE2C */
375 1.17682064682252693899e+01, /* 0x40278952, 0x5BB334D6 */
376 5.57673380256401856059e+02, /* 0x40816D63, 0x15301825 */
377 8.85919720756468632317e+03, /* 0x40C14D99, 0x3E18F46D */
378 3.70146267776887834771e+04, /* 0x40E212D4, 0x0E901566 */
379 };
380 #ifdef __STDC__
381 static const double qS8[6] = {
382 #else
383 static double qS8[6] = {
384 #endif
385 1.63776026895689824414e+02, /* 0x406478D5, 0x365B39BC */
386 8.09834494656449805916e+03, /* 0x40BFA258, 0x4E6B0563 */
387 1.42538291419120476348e+05, /* 0x41016652, 0x54D38C3F */
388 8.03309257119514397345e+05, /* 0x412883DA, 0x83A52B43 */
389 8.40501579819060512818e+05, /* 0x4129A66B, 0x28DE0B3D */
390 -3.43899293537866615225e+05, /* 0xC114FD6D, 0x2C9530C5 */
391 };
392
393 #ifdef __STDC__
394 static const double qR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
395 #else
396 static double qR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
397 #endif
398 1.84085963594515531381e-11, /* 0x3DB43D8F, 0x29CC8CD9 */
399 7.32421766612684765896e-02, /* 0x3FB2BFFF, 0xD172B04C */
400 5.83563508962056953777e+00, /* 0x401757B0, 0xB9953DD3 */
401 1.35111577286449829671e+02, /* 0x4060E392, 0x0A8788E9 */
402 1.02724376596164097464e+03, /* 0x40900CF9, 0x9DC8C481 */
403 1.98997785864605384631e+03, /* 0x409F17E9, 0x53C6E3A6 */
404 };
405 #ifdef __STDC__
406 static const double qS5[6] = {
407 #else
408 static double qS5[6] = {
409 #endif
410 8.27766102236537761883e+01, /* 0x4054B1B3, 0xFB5E1543 */
411 2.07781416421392987104e+03, /* 0x40A03BA0, 0xDA21C0CE */
412 1.88472887785718085070e+04, /* 0x40D267D2, 0x7B591E6D */
413 5.67511122894947329769e+04, /* 0x40EBB5E3, 0x97E02372 */
414 3.59767538425114471465e+04, /* 0x40E19118, 0x1F7A54A0 */
415 -5.35434275601944773371e+03, /* 0xC0B4EA57, 0xBEDBC609 */
416 };
417
418 #ifdef __STDC__
419 static const double qR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
420 #else
421 static double qR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
422 #endif
423 4.37741014089738620906e-09, /* 0x3E32CD03, 0x6ADECB82 */
424 7.32411180042911447163e-02, /* 0x3FB2BFEE, 0x0E8D0842 */
425 3.34423137516170720929e+00, /* 0x400AC0FC, 0x61149CF5 */
426 4.26218440745412650017e+01, /* 0x40454F98, 0x962DAEDD */
427 1.70808091340565596283e+02, /* 0x406559DB, 0xE25EFD1F */
428 1.66733948696651168575e+02, /* 0x4064D77C, 0x81FA21E0 */
429 };
430 #ifdef __STDC__
431 static const double qS3[6] = {
432 #else
433 static double qS3[6] = {
434 #endif
435 4.87588729724587182091e+01, /* 0x40486122, 0xBFE343A6 */
436 7.09689221056606015736e+02, /* 0x40862D83, 0x86544EB3 */
437 3.70414822620111362994e+03, /* 0x40ACF04B, 0xE44DFC63 */
438 6.46042516752568917582e+03, /* 0x40B93C6C, 0xD7C76A28 */
439 2.51633368920368957333e+03, /* 0x40A3A8AA, 0xD94FB1C0 */
440 -1.49247451836156386662e+02, /* 0xC062A7EB, 0x201CF40F */
441 };
442
443 #ifdef __STDC__
444 static const double qR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
445 #else
446 static double qR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
447 #endif
448 1.50444444886983272379e-07, /* 0x3E84313B, 0x54F76BDB */
449 7.32234265963079278272e-02, /* 0x3FB2BEC5, 0x3E883E34 */
450 1.99819174093815998816e+00, /* 0x3FFFF897, 0xE727779C */
451 1.44956029347885735348e+01, /* 0x402CFDBF, 0xAAF96FE5 */
452 3.16662317504781540833e+01, /* 0x403FAA8E, 0x29FBDC4A */
453 1.62527075710929267416e+01, /* 0x403040B1, 0x71814BB4 */
454 };
455 #ifdef __STDC__
456 static const double qS2[6] = {
457 #else
458 static double qS2[6] = {
459 #endif
460 3.03655848355219184498e+01, /* 0x403E5D96, 0xF7C07AED */
461 2.69348118608049844624e+02, /* 0x4070D591, 0xE4D14B40 */
462 8.44783757595320139444e+02, /* 0x408A6645, 0x22B3BF22 */
463 8.82935845112488550512e+02, /* 0x408B977C, 0x9C5CC214 */
464 2.12666388511798828631e+02, /* 0x406A9553, 0x0E001365 */
465 -5.31095493882666946917e+00, /* 0xC0153E6A, 0xF8B32931 */
466 };
467
468 #ifdef __STDC__
469 static double qzero(double x)
470 #else
471 static double qzero(x)
472 double x;
473 #endif
474 {
475 #ifdef __STDC__
476 const double *p=(void*)0,*q=(void*)0;
477 #else
478 double *p,*q;
479 #endif
480 double s,r,z;
481 int ix;
482 ix = 0x7fffffff&__HI(x);
483 if(ix>=0x40200000) {p = qR8; q= qS8;}
484 else if(ix>=0x40122E8B){p = qR5; q= qS5;}
485 else if(ix>=0x4006DB6D){p = qR3; q= qS3;}
486 else if(ix>=0x40000000){p = qR2; q= qS2;}
487 z = one/(x*x);
488 r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
489 s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*(q[4]+z*q[5])))));
490 return (-.125 + r/s)/x;
491 }