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e_j0.c
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1 
2 /* @(#)e_j0.c 1.3 95/01/18 */
3 /*
4  * ====================================================
5  * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
6  *
7  * Developed at SunSoft, a Sun Microsystems, Inc. business.
8  * Permission to use, copy, modify, and distribute this
9  * software is freely granted, provided that this notice
10  * is preserved.
11  * ====================================================
12  */
13 
14 /* __ieee754_j0(x), __ieee754_y0(x)
15  * Bessel function of the first and second kinds of order zero.
16  * Method -- j0(x):
17  * 1. For tiny x, we use j0(x) = 1 - x^2/4 + x^4/64 - ...
18  * 2. Reduce x to |x| since j0(x)=j0(-x), and
19  * for x in (0,2)
20  * j0(x) = 1-z/4+ z^2*R0/S0, where z = x*x;
21  * (precision: |j0-1+z/4-z^2R0/S0 |<2**-63.67 )
22  * for x in (2,inf)
23  * j0(x) = sqrt(2/(pi*x))*(p0(x)*cos(x0)-q0(x)*sin(x0))
24  * where x0 = x-pi/4. It is better to compute sin(x0),cos(x0)
25  * as follow:
26  * cos(x0) = cos(x)cos(pi/4)+sin(x)sin(pi/4)
27  * = 1/sqrt(2) * (cos(x) + sin(x))
28  * sin(x0) = sin(x)cos(pi/4)-cos(x)sin(pi/4)
29  * = 1/sqrt(2) * (sin(x) - cos(x))
30  * (To avoid cancellation, use
31  * sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
32  * to compute the worse one.)
33  *
34  * 3 Special cases
35  * j0(nan)= nan
36  * j0(0) = 1
37  * j0(inf) = 0
38  *
39  * Method -- y0(x):
40  * 1. For x<2.
41  * Since
42  * y0(x) = 2/pi*(j0(x)*(ln(x/2)+Euler) + x^2/4 - ...)
43  * therefore y0(x)-2/pi*j0(x)*ln(x) is an even function.
44  * We use the following function to approximate y0,
45  * y0(x) = U(z)/V(z) + (2/pi)*(j0(x)*ln(x)), z= x^2
46  * where
47  * U(z) = u00 + u01*z + ... + u06*z^6
48  * V(z) = 1 + v01*z + ... + v04*z^4
49  * with absolute approximation error bounded by 2**-72.
50  * Note: For tiny x, U/V = u0 and j0(x)~1, hence
51  * y0(tiny) = u0 + (2/pi)*ln(tiny), (choose tiny<2**-27)
52  * 2. For x>=2.
53  * y0(x) = sqrt(2/(pi*x))*(p0(x)*cos(x0)+q0(x)*sin(x0))
54  * where x0 = x-pi/4. It is better to compute sin(x0),cos(x0)
55  * by the method mentioned above.
56  * 3. Special cases: y0(0)=-inf, y0(x<0)=NaN, y0(inf)=0.
57  */
58 
59 #include "fdlibm.h"
60 
61 #ifdef __STDC__
62 static double pzero(double), qzero(double);
63 #else
64 static double pzero(), qzero();
65 #endif
66 
67 #ifdef __STDC__
68 static const double
69 #else
70 static double
71 #endif
72 huge = 1e300,
73 one = 1.0,
74 invsqrtpi= 5.64189583547756279280e-01, /* 0x3FE20DD7, 0x50429B6D */
75 tpi = 6.36619772367581382433e-01, /* 0x3FE45F30, 0x6DC9C883 */
76  /* R0/S0 on [0, 2.00] */
77 R02 = 1.56249999999999947958e-02, /* 0x3F8FFFFF, 0xFFFFFFFD */
78 R03 = -1.89979294238854721751e-04, /* 0xBF28E6A5, 0xB61AC6E9 */
79 R04 = 1.82954049532700665670e-06, /* 0x3EBEB1D1, 0x0C503919 */
80 R05 = -4.61832688532103189199e-09, /* 0xBE33D5E7, 0x73D63FCE */
81 S01 = 1.56191029464890010492e-02, /* 0x3F8FFCE8, 0x82C8C2A4 */
82 S02 = 1.16926784663337450260e-04, /* 0x3F1EA6D2, 0xDD57DBF4 */
83 S03 = 5.13546550207318111446e-07, /* 0x3EA13B54, 0xCE84D5A9 */
84 S04 = 1.16614003333790000205e-09; /* 0x3E1408BC, 0xF4745D8F */
85 
86 static double zero = 0.0;
87 
88 #ifdef __STDC__
89  double __ieee754_j0(double x)
90 #else
91  double __ieee754_j0(x)
92  double x;
93 #endif
94 {
95  double z, s,c,ss,cc,r,u,v;
96  int hx,ix;
97 
98  hx = __HI(x);
99  ix = hx&0x7fffffff;
100  if(ix>=0x7ff00000) return one/(x*x);
101  x = fabs(x);
102  if(ix >= 0x40000000) { /* |x| >= 2.0 */
103  s = sin(x);
104  c = cos(x);
105  ss = s-c;
106  cc = s+c;
107  if(ix<0x7fe00000) { /* make sure x+x not overflow */
108  z = -cos(x+x);
109  if ((s*c)<zero) cc = z/ss;
110  else ss = z/cc;
111  }
112  /*
113  * j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x)
114  * y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x)
115  */
116  if(ix>0x48000000) z = (invsqrtpi*cc)/sqrt(x);
117  else {
118  u = pzero(x); v = qzero(x);
119  z = invsqrtpi*(u*cc-v*ss)/sqrt(x);
120  }
121  return z;
122  }
123  if(ix<0x3f200000) { /* |x| < 2**-13 */
124  if(huge+x>one) { /* raise inexact if x != 0 */
125  if(ix<0x3e400000) return one; /* |x|<2**-27 */
126  else return one - 0.25*x*x;
127  }
128  }
129  z = x*x;
130  r = z*(R02+z*(R03+z*(R04+z*R05)));
131  s = one+z*(S01+z*(S02+z*(S03+z*S04)));
132  if(ix < 0x3FF00000) { /* |x| < 1.00 */
133  return one + z*(-0.25+(r/s));
134  } else {
135  u = 0.5*x;
136  return((one+u)*(one-u)+z*(r/s));
137  }
138 }
139 
140 #ifdef __STDC__
141 static const double
142 #else
143 static double
144 #endif
145 u00 = -7.38042951086872317523e-02, /* 0xBFB2E4D6, 0x99CBD01F */
146 u01 = 1.76666452509181115538e-01, /* 0x3FC69D01, 0x9DE9E3FC */
147 u02 = -1.38185671945596898896e-02, /* 0xBF8C4CE8, 0xB16CFA97 */
148 u03 = 3.47453432093683650238e-04, /* 0x3F36C54D, 0x20B29B6B */
149 u04 = -3.81407053724364161125e-06, /* 0xBECFFEA7, 0x73D25CAD */
150 u05 = 1.95590137035022920206e-08, /* 0x3E550057, 0x3B4EABD4 */
151 u06 = -3.98205194132103398453e-11, /* 0xBDC5E43D, 0x693FB3C8 */
152 v01 = 1.27304834834123699328e-02, /* 0x3F8A1270, 0x91C9C71A */
153 v02 = 7.60068627350353253702e-05, /* 0x3F13ECBB, 0xF578C6C1 */
154 v03 = 2.59150851840457805467e-07, /* 0x3E91642D, 0x7FF202FD */
155 v04 = 4.41110311332675467403e-10; /* 0x3DFE5018, 0x3BD6D9EF */
156 
157 #ifdef __STDC__
158  double __ieee754_y0(double x)
159 #else
160  double __ieee754_y0(x)
161  double x;
162 #endif
163 {
164  double z, s,c,ss,cc,u,v;
165  int hx,ix,lx;
166 
167  hx = __HI(x);
168  ix = 0x7fffffff&hx;
169  lx = __LO(x);
170  /* Y0(NaN) is NaN, y0(-inf) is Nan, y0(inf) is 0 */
171  if(ix>=0x7ff00000) return one/(x+x*x);
172  if((ix|lx)==0) return -one/zero;
173  if(hx<0) return zero/zero;
174  if(ix >= 0x40000000) { /* |x| >= 2.0 */
175  /* y0(x) = sqrt(2/(pi*x))*(p0(x)*sin(x0)+q0(x)*cos(x0))
176  * where x0 = x-pi/4
177  * Better formula:
178  * cos(x0) = cos(x)cos(pi/4)+sin(x)sin(pi/4)
179  * = 1/sqrt(2) * (sin(x) + cos(x))
180  * sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4)
181  * = 1/sqrt(2) * (sin(x) - cos(x))
182  * To avoid cancellation, use
183  * sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
184  * to compute the worse one.
185  */
186  s = sin(x);
187  c = cos(x);
188  ss = s-c;
189  cc = s+c;
190  /*
191  * j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x)
192  * y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x)
193  */
194  if(ix<0x7fe00000) { /* make sure x+x not overflow */
195  z = -cos(x+x);
196  if ((s*c)<zero) cc = z/ss;
197  else ss = z/cc;
198  }
199  if(ix>0x48000000) z = (invsqrtpi*ss)/sqrt(x);
200  else {
201  u = pzero(x); v = qzero(x);
202  z = invsqrtpi*(u*ss+v*cc)/sqrt(x);
203  }
204  return z;
205  }
206  if(ix<=0x3e400000) { /* x < 2**-27 */
207  return(u00 + tpi*__ieee754_log(x));
208  }
209  z = x*x;
210  u = u00+z*(u01+z*(u02+z*(u03+z*(u04+z*(u05+z*u06)))));
211  v = one+z*(v01+z*(v02+z*(v03+z*v04)));
212  return(u/v + tpi*(__ieee754_j0(x)*__ieee754_log(x)));
213 }
214 
215 /* The asymptotic expansions of pzero is
216  * 1 - 9/128 s^2 + 11025/98304 s^4 - ..., where s = 1/x.
217  * For x >= 2, We approximate pzero by
218  * pzero(x) = 1 + (R/S)
219  * where R = pR0 + pR1*s^2 + pR2*s^4 + ... + pR5*s^10
220  * S = 1 + pS0*s^2 + ... + pS4*s^10
221  * and
222  * | pzero(x)-1-R/S | <= 2 ** ( -60.26)
223  */
224 #ifdef __STDC__
225 static const double pR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
226 #else
227 static double pR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
228 #endif
229  0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */
230  -7.03124999999900357484e-02, /* 0xBFB1FFFF, 0xFFFFFD32 */
231  -8.08167041275349795626e+00, /* 0xC02029D0, 0xB44FA779 */
232  -2.57063105679704847262e+02, /* 0xC0701102, 0x7B19E863 */
233  -2.48521641009428822144e+03, /* 0xC0A36A6E, 0xCD4DCAFC */
234  -5.25304380490729545272e+03, /* 0xC0B4850B, 0x36CC643D */
235 };
236 #ifdef __STDC__
237 static const double pS8[5] = {
238 #else
239 static double pS8[5] = {
240 #endif
241  1.16534364619668181717e+02, /* 0x405D2233, 0x07A96751 */
242  3.83374475364121826715e+03, /* 0x40ADF37D, 0x50596938 */
243  4.05978572648472545552e+04, /* 0x40E3D2BB, 0x6EB6B05F */
244  1.16752972564375915681e+05, /* 0x40FC810F, 0x8F9FA9BD */
245  4.76277284146730962675e+04, /* 0x40E74177, 0x4F2C49DC */
246 };
247 
248 #ifdef __STDC__
249 static const double pR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
250 #else
251 static double pR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
252 #endif
253  -1.14125464691894502584e-11, /* 0xBDA918B1, 0x47E495CC */
254  -7.03124940873599280078e-02, /* 0xBFB1FFFF, 0xE69AFBC6 */
255  -4.15961064470587782438e+00, /* 0xC010A370, 0xF90C6BBF */
256  -6.76747652265167261021e+01, /* 0xC050EB2F, 0x5A7D1783 */
257  -3.31231299649172967747e+02, /* 0xC074B3B3, 0x6742CC63 */
258  -3.46433388365604912451e+02, /* 0xC075A6EF, 0x28A38BD7 */
259 };
260 #ifdef __STDC__
261 static const double pS5[5] = {
262 #else
263 static double pS5[5] = {
264 #endif
265  6.07539382692300335975e+01, /* 0x404E6081, 0x0C98C5DE */
266  1.05125230595704579173e+03, /* 0x40906D02, 0x5C7E2864 */
267  5.97897094333855784498e+03, /* 0x40B75AF8, 0x8FBE1D60 */
268  9.62544514357774460223e+03, /* 0x40C2CCB8, 0xFA76FA38 */
269  2.40605815922939109441e+03, /* 0x40A2CC1D, 0xC70BE864 */
270 };
271 
272 #ifdef __STDC__
273 static const double pR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
274 #else
275 static double pR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
276 #endif
277  -2.54704601771951915620e-09, /* 0xBE25E103, 0x6FE1AA86 */
278  -7.03119616381481654654e-02, /* 0xBFB1FFF6, 0xF7C0E24B */
279  -2.40903221549529611423e+00, /* 0xC00345B2, 0xAEA48074 */
280  -2.19659774734883086467e+01, /* 0xC035F74A, 0x4CB94E14 */
281  -5.80791704701737572236e+01, /* 0xC04D0A22, 0x420A1A45 */
282  -3.14479470594888503854e+01, /* 0xC03F72AC, 0xA892D80F */
283 };
284 #ifdef __STDC__
285 static const double pS3[5] = {
286 #else
287 static double pS3[5] = {
288 #endif
289  3.58560338055209726349e+01, /* 0x4041ED92, 0x84077DD3 */
290  3.61513983050303863820e+02, /* 0x40769839, 0x464A7C0E */
291  1.19360783792111533330e+03, /* 0x4092A66E, 0x6D1061D6 */
292  1.12799679856907414432e+03, /* 0x40919FFC, 0xB8C39B7E */
293  1.73580930813335754692e+02, /* 0x4065B296, 0xFC379081 */
294 };
295 
296 #ifdef __STDC__
297 static const double pR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
298 #else
299 static double pR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
300 #endif
301  -8.87534333032526411254e-08, /* 0xBE77D316, 0xE927026D */
302  -7.03030995483624743247e-02, /* 0xBFB1FF62, 0x495E1E42 */
303  -1.45073846780952986357e+00, /* 0xBFF73639, 0x8A24A843 */
304  -7.63569613823527770791e+00, /* 0xC01E8AF3, 0xEDAFA7F3 */
305  -1.11931668860356747786e+01, /* 0xC02662E6, 0xC5246303 */
306  -3.23364579351335335033e+00, /* 0xC009DE81, 0xAF8FE70F */
307 };
308 #ifdef __STDC__
309 static const double pS2[5] = {
310 #else
311 static double pS2[5] = {
312 #endif
313  2.22202997532088808441e+01, /* 0x40363865, 0x908B5959 */
314  1.36206794218215208048e+02, /* 0x4061069E, 0x0EE8878F */
315  2.70470278658083486789e+02, /* 0x4070E786, 0x42EA079B */
316  1.53875394208320329881e+02, /* 0x40633C03, 0x3AB6FAFF */
317  1.46576176948256193810e+01, /* 0x402D50B3, 0x44391809 */
318 };
319 
320 #ifdef __STDC__
321  static double pzero(double x)
322 #else
323  static double pzero(x)
324  double x;
325 #endif
326 {
327 #ifdef __STDC__
328  const double *p=NULL,*q=NULL;
329 #else
330  double *p,*q;
331 #endif
332  double z,r,s;
333  int ix;
334  ix = 0x7fffffff&__HI(x);
335  if(ix>=0x40200000) {p = pR8; q= pS8;}
336  else if(ix>=0x40122E8B){p = pR5; q= pS5;}
337  else if(ix>=0x4006DB6D){p = pR3; q= pS3;}
338  else if(ix>=0x40000000){p = pR2; q= pS2;}
339  z = one/(x*x);
340  r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
341  s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*q[4]))));
342  return one+ r/s;
343 }
344 
345 
346 /* For x >= 8, the asymptotic expansions of qzero is
347  * -1/8 s + 75/1024 s^3 - ..., where s = 1/x.
348  * We approximate pzero by
349  * qzero(x) = s*(-1.25 + (R/S))
350  * where R = qR0 + qR1*s^2 + qR2*s^4 + ... + qR5*s^10
351  * S = 1 + qS0*s^2 + ... + qS5*s^12
352  * and
353  * | qzero(x)/s +1.25-R/S | <= 2 ** ( -61.22)
354  */
355 #ifdef __STDC__
356 static const double qR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
357 #else
358 static double qR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
359 #endif
360  0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */
361  7.32421874999935051953e-02, /* 0x3FB2BFFF, 0xFFFFFE2C */
362  1.17682064682252693899e+01, /* 0x40278952, 0x5BB334D6 */
363  5.57673380256401856059e+02, /* 0x40816D63, 0x15301825 */
364  8.85919720756468632317e+03, /* 0x40C14D99, 0x3E18F46D */
365  3.70146267776887834771e+04, /* 0x40E212D4, 0x0E901566 */
366 };
367 #ifdef __STDC__
368 static const double qS8[6] = {
369 #else
370 static double qS8[6] = {
371 #endif
372  1.63776026895689824414e+02, /* 0x406478D5, 0x365B39BC */
373  8.09834494656449805916e+03, /* 0x40BFA258, 0x4E6B0563 */
374  1.42538291419120476348e+05, /* 0x41016652, 0x54D38C3F */
375  8.03309257119514397345e+05, /* 0x412883DA, 0x83A52B43 */
376  8.40501579819060512818e+05, /* 0x4129A66B, 0x28DE0B3D */
377  -3.43899293537866615225e+05, /* 0xC114FD6D, 0x2C9530C5 */
378 };
379 
380 #ifdef __STDC__
381 static const double qR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
382 #else
383 static double qR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
384 #endif
385  1.84085963594515531381e-11, /* 0x3DB43D8F, 0x29CC8CD9 */
386  7.32421766612684765896e-02, /* 0x3FB2BFFF, 0xD172B04C */
387  5.83563508962056953777e+00, /* 0x401757B0, 0xB9953DD3 */
388  1.35111577286449829671e+02, /* 0x4060E392, 0x0A8788E9 */
389  1.02724376596164097464e+03, /* 0x40900CF9, 0x9DC8C481 */
390  1.98997785864605384631e+03, /* 0x409F17E9, 0x53C6E3A6 */
391 };
392 #ifdef __STDC__
393 static const double qS5[6] = {
394 #else
395 static double qS5[6] = {
396 #endif
397  8.27766102236537761883e+01, /* 0x4054B1B3, 0xFB5E1543 */
398  2.07781416421392987104e+03, /* 0x40A03BA0, 0xDA21C0CE */
399  1.88472887785718085070e+04, /* 0x40D267D2, 0x7B591E6D */
400  5.67511122894947329769e+04, /* 0x40EBB5E3, 0x97E02372 */
401  3.59767538425114471465e+04, /* 0x40E19118, 0x1F7A54A0 */
402  -5.35434275601944773371e+03, /* 0xC0B4EA57, 0xBEDBC609 */
403 };
404 
405 #ifdef __STDC__
406 static const double qR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
407 #else
408 static double qR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
409 #endif
410  4.37741014089738620906e-09, /* 0x3E32CD03, 0x6ADECB82 */
411  7.32411180042911447163e-02, /* 0x3FB2BFEE, 0x0E8D0842 */
412  3.34423137516170720929e+00, /* 0x400AC0FC, 0x61149CF5 */
413  4.26218440745412650017e+01, /* 0x40454F98, 0x962DAEDD */
414  1.70808091340565596283e+02, /* 0x406559DB, 0xE25EFD1F */
415  1.66733948696651168575e+02, /* 0x4064D77C, 0x81FA21E0 */
416 };
417 #ifdef __STDC__
418 static const double qS3[6] = {
419 #else
420 static double qS3[6] = {
421 #endif
422  4.87588729724587182091e+01, /* 0x40486122, 0xBFE343A6 */
423  7.09689221056606015736e+02, /* 0x40862D83, 0x86544EB3 */
424  3.70414822620111362994e+03, /* 0x40ACF04B, 0xE44DFC63 */
425  6.46042516752568917582e+03, /* 0x40B93C6C, 0xD7C76A28 */
426  2.51633368920368957333e+03, /* 0x40A3A8AA, 0xD94FB1C0 */
427  -1.49247451836156386662e+02, /* 0xC062A7EB, 0x201CF40F */
428 };
429 
430 #ifdef __STDC__
431 static const double qR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
432 #else
433 static double qR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
434 #endif
435  1.50444444886983272379e-07, /* 0x3E84313B, 0x54F76BDB */
436  7.32234265963079278272e-02, /* 0x3FB2BEC5, 0x3E883E34 */
437  1.99819174093815998816e+00, /* 0x3FFFF897, 0xE727779C */
438  1.44956029347885735348e+01, /* 0x402CFDBF, 0xAAF96FE5 */
439  3.16662317504781540833e+01, /* 0x403FAA8E, 0x29FBDC4A */
440  1.62527075710929267416e+01, /* 0x403040B1, 0x71814BB4 */
441 };
442 #ifdef __STDC__
443 static const double qS2[6] = {
444 #else
445 static double qS2[6] = {
446 #endif
447  3.03655848355219184498e+01, /* 0x403E5D96, 0xF7C07AED */
448  2.69348118608049844624e+02, /* 0x4070D591, 0xE4D14B40 */
449  8.44783757595320139444e+02, /* 0x408A6645, 0x22B3BF22 */
450  8.82935845112488550512e+02, /* 0x408B977C, 0x9C5CC214 */
451  2.12666388511798828631e+02, /* 0x406A9553, 0x0E001365 */
452  -5.31095493882666946917e+00, /* 0xC0153E6A, 0xF8B32931 */
453 };
454 
455 #ifdef __STDC__
456  static double qzero(double x)
457 #else
458  static double qzero(x)
459  double x;
460 #endif
461 {
462 #ifdef __STDC__
463  const double *p=NULL,*q=NULL;
464 #else
465  double *p,*q;
466 #endif
467  double s,r,z;
468  int ix;
469  ix = 0x7fffffff&__HI(x);
470  if(ix>=0x40200000) {p = qR8; q= qS8;}
471  else if(ix>=0x40122E8B){p = qR5; q= qS5;}
472  else if(ix>=0x4006DB6D){p = qR3; q= qS3;}
473  else if(ix>=0x40000000){p = qR2; q= qS2;}
474  z = one/(x*x);
475  r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
476  s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*(q[4]+z*q[5])))));
477  return (-.125 + r/s)/x;
478 }