1 | /* |
2 | * ==================================================== |
3 | * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. |
4 | * |
5 | * Developed at SunPro, a Sun Microsystems, Inc. business. |
6 | * Permission to use, copy, modify, and distribute this |
7 | * software is freely granted, provided that this notice |
8 | * is preserved. |
9 | * ==================================================== |
10 | */ |
11 | |
12 | /* Long double expansions are |
13 | Copyright (C) 2001 Stephen L. Moshier <moshier@na-net.ornl.gov> |
14 | and are incorporated herein by permission of the author. The author |
15 | reserves the right to distribute this material elsewhere under different |
16 | copying permissions. These modifications are distributed here under |
17 | the following terms: |
18 | |
19 | This library is free software; you can redistribute it and/or |
20 | modify it under the terms of the GNU Lesser General Public |
21 | License as published by the Free Software Foundation; either |
22 | version 2.1 of the License, or (at your option) any later version. |
23 | |
24 | This library is distributed in the hope that it will be useful, |
25 | but WITHOUT ANY WARRANTY; without even the implied warranty of |
26 | MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU |
27 | Lesser General Public License for more details. |
28 | |
29 | You should have received a copy of the GNU Lesser General Public |
30 | License along with this library; if not, see |
31 | <http://www.gnu.org/licenses/>. */ |
32 | |
33 | /* __ieee754_j0(x), __ieee754_y0(x) |
34 | * Bessel function of the first and second kinds of order zero. |
35 | * Method -- j0(x): |
36 | * 1. For tiny x, we use j0(x) = 1 - x^2/4 + x^4/64 - ... |
37 | * 2. Reduce x to |x| since j0(x)=j0(-x), and |
38 | * for x in (0,2) |
39 | * j0(x) = 1 - z/4 + z^2*R0/S0, where z = x*x; |
40 | * for x in (2,inf) |
41 | * j0(x) = sqrt(2/(pi*x))*(p0(x)*cos(x0)-q0(x)*sin(x0)) |
42 | * where x0 = x-pi/4. It is better to compute sin(x0),cos(x0) |
43 | * as follow: |
44 | * cos(x0) = cos(x)cos(pi/4)+sin(x)sin(pi/4) |
45 | * = 1/sqrt(2) * (cos(x) + sin(x)) |
46 | * sin(x0) = sin(x)cos(pi/4)-cos(x)sin(pi/4) |
47 | * = 1/sqrt(2) * (sin(x) - cos(x)) |
48 | * (To avoid cancellation, use |
49 | * sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x)) |
50 | * to compute the worse one.) |
51 | * |
52 | * 3 Special cases |
53 | * j0(nan)= nan |
54 | * j0(0) = 1 |
55 | * j0(inf) = 0 |
56 | * |
57 | * Method -- y0(x): |
58 | * 1. For x<2. |
59 | * Since |
60 | * y0(x) = 2/pi*(j0(x)*(ln(x/2)+Euler) + x^2/4 - ...) |
61 | * therefore y0(x)-2/pi*j0(x)*ln(x) is an even function. |
62 | * We use the following function to approximate y0, |
63 | * y0(x) = U(z)/V(z) + (2/pi)*(j0(x)*ln(x)), z= x^2 |
64 | * |
65 | * Note: For tiny x, U/V = u0 and j0(x)~1, hence |
66 | * y0(tiny) = u0 + (2/pi)*ln(tiny), (choose tiny<2**-27) |
67 | * 2. For x>=2. |
68 | * y0(x) = sqrt(2/(pi*x))*(p0(x)*cos(x0)+q0(x)*sin(x0)) |
69 | * where x0 = x-pi/4. It is better to compute sin(x0),cos(x0) |
70 | * by the method mentioned above. |
71 | * 3. Special cases: y0(0)=-inf, y0(x<0)=NaN, y0(inf)=0. |
72 | */ |
73 | |
74 | #include <math.h> |
75 | #include <math-barriers.h> |
76 | #include <math_private.h> |
77 | |
78 | static long double pzero (long double), qzero (long double); |
79 | |
80 | static const long double |
81 | huge = 1e4930L, |
82 | one = 1.0L, |
83 | invsqrtpi = 5.6418958354775628694807945156077258584405e-1L, |
84 | tpi = 6.3661977236758134307553505349005744813784e-1L, |
85 | |
86 | /* J0(x) = 1 - x^2 / 4 + x^4 R0(x^2) / S0(x^2) |
87 | 0 <= x <= 2 |
88 | peak relative error 1.41e-22 */ |
89 | R[5] = { |
90 | 4.287176872744686992880841716723478740566E7L, |
91 | -6.652058897474241627570911531740907185772E5L, |
92 | 7.011848381719789863458364584613651091175E3L, |
93 | -3.168040850193372408702135490809516253693E1L, |
94 | 6.030778552661102450545394348845599300939E-2L, |
95 | }, |
96 | |
97 | S[4] = { |
98 | 2.743793198556599677955266341699130654342E9L, |
99 | 3.364330079384816249840086842058954076201E7L, |
100 | 1.924119649412510777584684927494642526573E5L, |
101 | 6.239282256012734914211715620088714856494E2L, |
102 | /* 1.000000000000000000000000000000000000000E0L,*/ |
103 | }; |
104 | |
105 | static const long double zero = 0.0; |
106 | |
107 | long double |
108 | __ieee754_j0l (long double x) |
109 | { |
110 | long double z, s, c, ss, cc, r, u, v; |
111 | int32_t ix; |
112 | uint32_t se; |
113 | |
114 | GET_LDOUBLE_EXP (se, x); |
115 | ix = se & 0x7fff; |
116 | if (__glibc_unlikely (ix >= 0x7fff)) |
117 | return one / (x * x); |
118 | x = fabsl (x); |
119 | if (ix >= 0x4000) /* |x| >= 2.0 */ |
120 | { |
121 | __sincosl (x, &s, &c); |
122 | ss = s - c; |
123 | cc = s + c; |
124 | if (ix < 0x7ffe) |
125 | { /* make sure x+x not overflow */ |
126 | z = -__cosl (x + x); |
127 | if ((s * c) < zero) |
128 | cc = z / ss; |
129 | else |
130 | ss = z / cc; |
131 | } |
132 | /* |
133 | * j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x) |
134 | * y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x) |
135 | */ |
136 | if (__glibc_unlikely (ix > 0x4080)) /* 2^129 */ |
137 | z = (invsqrtpi * cc) / sqrtl (x); |
138 | else |
139 | { |
140 | u = pzero (x); |
141 | v = qzero (x); |
142 | z = invsqrtpi * (u * cc - v * ss) / sqrtl (x); |
143 | } |
144 | return z; |
145 | } |
146 | if (__glibc_unlikely (ix < 0x3fef)) /* |x| < 2**-16 */ |
147 | { |
148 | /* raise inexact if x != 0 */ |
149 | math_force_eval (huge + x); |
150 | if (ix < 0x3fde) /* |x| < 2^-33 */ |
151 | return one; |
152 | else |
153 | return one - 0.25 * x * x; |
154 | } |
155 | z = x * x; |
156 | r = z * (R[0] + z * (R[1] + z * (R[2] + z * (R[3] + z * R[4])))); |
157 | s = S[0] + z * (S[1] + z * (S[2] + z * (S[3] + z))); |
158 | if (ix < 0x3fff) |
159 | { /* |x| < 1.00 */ |
160 | return (one - 0.25 * z + z * (r / s)); |
161 | } |
162 | else |
163 | { |
164 | u = 0.5 * x; |
165 | return ((one + u) * (one - u) + z * (r / s)); |
166 | } |
167 | } |
168 | strong_alias (__ieee754_j0l, __j0l_finite) |
169 | |
170 | |
171 | /* y0(x) = 2/pi ln(x) J0(x) + U(x^2)/V(x^2) |
172 | 0 < x <= 2 |
173 | peak relative error 1.7e-21 */ |
174 | static const long double |
175 | U[6] = { |
176 | -1.054912306975785573710813351985351350861E10L, |
177 | 2.520192609749295139432773849576523636127E10L, |
178 | -1.856426071075602001239955451329519093395E9L, |
179 | 4.079209129698891442683267466276785956784E7L, |
180 | -3.440684087134286610316661166492641011539E5L, |
181 | 1.005524356159130626192144663414848383774E3L, |
182 | }; |
183 | static const long double |
184 | V[5] = { |
185 | 1.429337283720789610137291929228082613676E11L, |
186 | 2.492593075325119157558811370165695013002E9L, |
187 | 2.186077620785925464237324417623665138376E7L, |
188 | 1.238407896366385175196515057064384929222E5L, |
189 | 4.693924035211032457494368947123233101664E2L, |
190 | /* 1.000000000000000000000000000000000000000E0L */ |
191 | }; |
192 | |
193 | long double |
194 | __ieee754_y0l (long double x) |
195 | { |
196 | long double z, s, c, ss, cc, u, v; |
197 | int32_t ix; |
198 | uint32_t se, i0, i1; |
199 | |
200 | GET_LDOUBLE_WORDS (se, i0, i1, x); |
201 | ix = se & 0x7fff; |
202 | /* Y0(NaN) is NaN, y0(-inf) is Nan, y0(inf) is 0 */ |
203 | if (__glibc_unlikely (se & 0x8000)) |
204 | return zero / (zero * x); |
205 | if (__glibc_unlikely (ix >= 0x7fff)) |
206 | return one / (x + x * x); |
207 | if (__glibc_unlikely ((i0 | i1) == 0)) |
208 | return -HUGE_VALL + x; /* -inf and overflow exception. */ |
209 | if (ix >= 0x4000) |
210 | { /* |x| >= 2.0 */ |
211 | |
212 | /* y0(x) = sqrt(2/(pi*x))*(p0(x)*sin(x0)+q0(x)*cos(x0)) |
213 | * where x0 = x-pi/4 |
214 | * Better formula: |
215 | * cos(x0) = cos(x)cos(pi/4)+sin(x)sin(pi/4) |
216 | * = 1/sqrt(2) * (sin(x) + cos(x)) |
217 | * sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4) |
218 | * = 1/sqrt(2) * (sin(x) - cos(x)) |
219 | * To avoid cancellation, use |
220 | * sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x)) |
221 | * to compute the worse one. |
222 | */ |
223 | __sincosl (x, &s, &c); |
224 | ss = s - c; |
225 | cc = s + c; |
226 | /* |
227 | * j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x) |
228 | * y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x) |
229 | */ |
230 | if (ix < 0x7ffe) |
231 | { /* make sure x+x not overflow */ |
232 | z = -__cosl (x + x); |
233 | if ((s * c) < zero) |
234 | cc = z / ss; |
235 | else |
236 | ss = z / cc; |
237 | } |
238 | if (__glibc_unlikely (ix > 0x4080)) /* 1e39 */ |
239 | z = (invsqrtpi * ss) / sqrtl (x); |
240 | else |
241 | { |
242 | u = pzero (x); |
243 | v = qzero (x); |
244 | z = invsqrtpi * (u * ss + v * cc) / sqrtl (x); |
245 | } |
246 | return z; |
247 | } |
248 | if (__glibc_unlikely (ix <= 0x3fde)) /* x < 2^-33 */ |
249 | { |
250 | z = -7.380429510868722527629822444004602747322E-2L |
251 | + tpi * __ieee754_logl (x); |
252 | return z; |
253 | } |
254 | z = x * x; |
255 | u = U[0] + z * (U[1] + z * (U[2] + z * (U[3] + z * (U[4] + z * U[5])))); |
256 | v = V[0] + z * (V[1] + z * (V[2] + z * (V[3] + z * (V[4] + z)))); |
257 | return (u / v + tpi * (__ieee754_j0l (x) * __ieee754_logl (x))); |
258 | } |
259 | strong_alias (__ieee754_y0l, __y0l_finite) |
260 | |
261 | /* The asymptotic expansions of pzero is |
262 | * 1 - 9/128 s^2 + 11025/98304 s^4 - ..., where s = 1/x. |
263 | * For x >= 2, We approximate pzero by |
264 | * pzero(x) = 1 + s^2 R(s^2) / S(s^2) |
265 | */ |
266 | static const long double pR8[7] = { |
267 | /* 8 <= x <= inf |
268 | Peak relative error 4.62 */ |
269 | -4.094398895124198016684337960227780260127E-9L, |
270 | -8.929643669432412640061946338524096893089E-7L, |
271 | -6.281267456906136703868258380673108109256E-5L, |
272 | -1.736902783620362966354814353559382399665E-3L, |
273 | -1.831506216290984960532230842266070146847E-2L, |
274 | -5.827178869301452892963280214772398135283E-2L, |
275 | -2.087563267939546435460286895807046616992E-2L, |
276 | }; |
277 | static const long double pS8[6] = { |
278 | 5.823145095287749230197031108839653988393E-8L, |
279 | 1.279281986035060320477759999428992730280E-5L, |
280 | 9.132668954726626677174825517150228961304E-4L, |
281 | 2.606019379433060585351880541545146252534E-2L, |
282 | 2.956262215119520464228467583516287175244E-1L, |
283 | 1.149498145388256448535563278632697465675E0L, |
284 | /* 1.000000000000000000000000000000000000000E0L, */ |
285 | }; |
286 | |
287 | static const long double pR5[7] = { |
288 | /* 4.54541015625 <= x <= 8 |
289 | Peak relative error 6.51E-22 */ |
290 | -2.041226787870240954326915847282179737987E-7L, |
291 | -2.255373879859413325570636768224534428156E-5L, |
292 | -7.957485746440825353553537274569102059990E-4L, |
293 | -1.093205102486816696940149222095559439425E-2L, |
294 | -5.657957849316537477657603125260701114646E-2L, |
295 | -8.641175552716402616180994954177818461588E-2L, |
296 | -1.354654710097134007437166939230619726157E-2L, |
297 | }; |
298 | static const long double pS5[6] = { |
299 | 2.903078099681108697057258628212823545290E-6L, |
300 | 3.253948449946735405975737677123673867321E-4L, |
301 | 1.181269751723085006534147920481582279979E-2L, |
302 | 1.719212057790143888884745200257619469363E-1L, |
303 | 1.006306498779212467670654535430694221924E0L, |
304 | 2.069568808688074324555596301126375951502E0L, |
305 | /* 1.000000000000000000000000000000000000000E0L, */ |
306 | }; |
307 | |
308 | static const long double pR3[7] = { |
309 | /* 2.85711669921875 <= x <= 4.54541015625 |
310 | peak relative error 5.25e-21 */ |
311 | -5.755732156848468345557663552240816066802E-6L, |
312 | -3.703675625855715998827966962258113034767E-4L, |
313 | -7.390893350679637611641350096842846433236E-3L, |
314 | -5.571922144490038765024591058478043873253E-2L, |
315 | -1.531290690378157869291151002472627396088E-1L, |
316 | -1.193350853469302941921647487062620011042E-1L, |
317 | -8.567802507331578894302991505331963782905E-3L, |
318 | }; |
319 | static const long double pS3[6] = { |
320 | 8.185931139070086158103309281525036712419E-5L, |
321 | 5.398016943778891093520574483111255476787E-3L, |
322 | 1.130589193590489566669164765853409621081E-1L, |
323 | 9.358652328786413274673192987670237145071E-1L, |
324 | 3.091711512598349056276917907005098085273E0L, |
325 | 3.594602474737921977972586821673124231111E0L, |
326 | /* 1.000000000000000000000000000000000000000E0L, */ |
327 | }; |
328 | |
329 | static const long double pR2[7] = { |
330 | /* 2 <= x <= 2.85711669921875 |
331 | peak relative error 2.64e-21 */ |
332 | -1.219525235804532014243621104365384992623E-4L, |
333 | -4.838597135805578919601088680065298763049E-3L, |
334 | -5.732223181683569266223306197751407418301E-2L, |
335 | -2.472947430526425064982909699406646503758E-1L, |
336 | -3.753373645974077960207588073975976327695E-1L, |
337 | -1.556241316844728872406672349347137975495E-1L, |
338 | -5.355423239526452209595316733635519506958E-3L, |
339 | }; |
340 | static const long double pS2[6] = { |
341 | 1.734442793664291412489066256138894953823E-3L, |
342 | 7.158111826468626405416300895617986926008E-2L, |
343 | 9.153839713992138340197264669867993552641E-1L, |
344 | 4.539209519433011393525841956702487797582E0L, |
345 | 8.868932430625331650266067101752626253644E0L, |
346 | 6.067161890196324146320763844772857713502E0L, |
347 | /* 1.000000000000000000000000000000000000000E0L, */ |
348 | }; |
349 | |
350 | static long double |
351 | pzero (long double x) |
352 | { |
353 | const long double *p, *q; |
354 | long double z, r, s; |
355 | int32_t ix; |
356 | uint32_t se, i0, i1; |
357 | |
358 | GET_LDOUBLE_WORDS (se, i0, i1, x); |
359 | ix = se & 0x7fff; |
360 | /* ix >= 0x4000 for all calls to this function. */ |
361 | if (ix >= 0x4002) |
362 | { |
363 | p = pR8; |
364 | q = pS8; |
365 | } /* x >= 8 */ |
366 | else |
367 | { |
368 | i1 = (ix << 16) | (i0 >> 16); |
369 | if (i1 >= 0x40019174) /* x >= 4.54541015625 */ |
370 | { |
371 | p = pR5; |
372 | q = pS5; |
373 | } |
374 | else if (i1 >= 0x4000b6db) /* x >= 2.85711669921875 */ |
375 | { |
376 | p = pR3; |
377 | q = pS3; |
378 | } |
379 | else /* x >= 2 */ |
380 | { |
381 | p = pR2; |
382 | q = pS2; |
383 | } |
384 | } |
385 | z = one / (x * x); |
386 | r = |
387 | p[0] + z * (p[1] + |
388 | z * (p[2] + z * (p[3] + z * (p[4] + z * (p[5] + z * p[6]))))); |
389 | s = |
390 | q[0] + z * (q[1] + z * (q[2] + z * (q[3] + z * (q[4] + z * (q[5] + z))))); |
391 | return (one + z * r / s); |
392 | } |
393 | |
394 | |
395 | /* For x >= 8, the asymptotic expansions of qzero is |
396 | * -1/8 s + 75/1024 s^3 - ..., where s = 1/x. |
397 | * We approximate qzero by |
398 | * qzero(x) = s*(-.125 + R(s^2) / S(s^2)) |
399 | */ |
400 | static const long double qR8[7] = { |
401 | /* 8 <= x <= inf |
402 | peak relative error 2.23e-21 */ |
403 | 3.001267180483191397885272640777189348008E-10L, |
404 | 8.693186311430836495238494289942413810121E-8L, |
405 | 8.496875536711266039522937037850596580686E-6L, |
406 | 3.482702869915288984296602449543513958409E-4L, |
407 | 6.036378380706107692863811938221290851352E-3L, |
408 | 3.881970028476167836382607922840452192636E-2L, |
409 | 6.132191514516237371140841765561219149638E-2L, |
410 | }; |
411 | static const long double qS8[7] = { |
412 | 4.097730123753051126914971174076227600212E-9L, |
413 | 1.199615869122646109596153392152131139306E-6L, |
414 | 1.196337580514532207793107149088168946451E-4L, |
415 | 5.099074440112045094341500497767181211104E-3L, |
416 | 9.577420799632372483249761659674764460583E-2L, |
417 | 7.385243015344292267061953461563695918646E-1L, |
418 | 1.917266424391428937962682301561699055943E0L, |
419 | /* 1.000000000000000000000000000000000000000E0L, */ |
420 | }; |
421 | |
422 | static const long double qR5[7] = { |
423 | /* 4.54541015625 <= x <= 8 |
424 | peak relative error 1.03e-21 */ |
425 | 3.406256556438974327309660241748106352137E-8L, |
426 | 4.855492710552705436943630087976121021980E-6L, |
427 | 2.301011739663737780613356017352912281980E-4L, |
428 | 4.500470249273129953870234803596619899226E-3L, |
429 | 3.651376459725695502726921248173637054828E-2L, |
430 | 1.071578819056574524416060138514508609805E-1L, |
431 | 7.458950172851611673015774675225656063757E-2L, |
432 | }; |
433 | static const long double qS5[7] = { |
434 | 4.650675622764245276538207123618745150785E-7L, |
435 | 6.773573292521412265840260065635377164455E-5L, |
436 | 3.340711249876192721980146877577806687714E-3L, |
437 | 7.036218046856839214741678375536970613501E-2L, |
438 | 6.569599559163872573895171876511377891143E-1L, |
439 | 2.557525022583599204591036677199171155186E0L, |
440 | 3.457237396120935674982927714210361269133E0L, |
441 | /* 1.000000000000000000000000000000000000000E0L,*/ |
442 | }; |
443 | |
444 | static const long double qR3[7] = { |
445 | /* 2.85711669921875 <= x <= 4.54541015625 |
446 | peak relative error 5.24e-21 */ |
447 | 1.749459596550816915639829017724249805242E-6L, |
448 | 1.446252487543383683621692672078376929437E-4L, |
449 | 3.842084087362410664036704812125005761859E-3L, |
450 | 4.066369994699462547896426554180954233581E-2L, |
451 | 1.721093619117980251295234795188992722447E-1L, |
452 | 2.538595333972857367655146949093055405072E-1L, |
453 | 8.560591367256769038905328596020118877936E-2L, |
454 | }; |
455 | static const long double qS3[7] = { |
456 | 2.388596091707517488372313710647510488042E-5L, |
457 | 2.048679968058758616370095132104333998147E-3L, |
458 | 5.824663198201417760864458765259945181513E-2L, |
459 | 6.953906394693328750931617748038994763958E-1L, |
460 | 3.638186936390881159685868764832961092476E0L, |
461 | 7.900169524705757837298990558459547842607E0L, |
462 | 5.992718532451026507552820701127504582907E0L, |
463 | /* 1.000000000000000000000000000000000000000E0L, */ |
464 | }; |
465 | |
466 | static const long double qR2[7] = { |
467 | /* 2 <= x <= 2.85711669921875 |
468 | peak relative error 1.58e-21 */ |
469 | 6.306524405520048545426928892276696949540E-5L, |
470 | 3.209606155709930950935893996591576624054E-3L, |
471 | 5.027828775702022732912321378866797059604E-2L, |
472 | 3.012705561838718956481911477587757845163E-1L, |
473 | 6.960544893905752937420734884995688523815E-1L, |
474 | 5.431871999743531634887107835372232030655E-1L, |
475 | 9.447736151202905471899259026430157211949E-2L, |
476 | }; |
477 | static const long double qS2[7] = { |
478 | 8.610579901936193494609755345106129102676E-4L, |
479 | 4.649054352710496997203474853066665869047E-2L, |
480 | 8.104282924459837407218042945106320388339E-1L, |
481 | 5.807730930825886427048038146088828206852E0L, |
482 | 1.795310145936848873627710102199881642939E1L, |
483 | 2.281313316875375733663657188888110605044E1L, |
484 | 1.011242067883822301487154844458322200143E1L, |
485 | /* 1.000000000000000000000000000000000000000E0L, */ |
486 | }; |
487 | |
488 | static long double |
489 | qzero (long double x) |
490 | { |
491 | const long double *p, *q; |
492 | long double s, r, z; |
493 | int32_t ix; |
494 | uint32_t se, i0, i1; |
495 | |
496 | GET_LDOUBLE_WORDS (se, i0, i1, x); |
497 | ix = se & 0x7fff; |
498 | /* ix >= 0x4000 for all calls to this function. */ |
499 | if (ix >= 0x4002) /* x >= 8 */ |
500 | { |
501 | p = qR8; |
502 | q = qS8; |
503 | } |
504 | else |
505 | { |
506 | i1 = (ix << 16) | (i0 >> 16); |
507 | if (i1 >= 0x40019174) /* x >= 4.54541015625 */ |
508 | { |
509 | p = qR5; |
510 | q = qS5; |
511 | } |
512 | else if (i1 >= 0x4000b6db) /* x >= 2.85711669921875 */ |
513 | { |
514 | p = qR3; |
515 | q = qS3; |
516 | } |
517 | else /* x >= 2 */ |
518 | { |
519 | p = qR2; |
520 | q = qS2; |
521 | } |
522 | } |
523 | z = one / (x * x); |
524 | r = |
525 | p[0] + z * (p[1] + |
526 | z * (p[2] + z * (p[3] + z * (p[4] + z * (p[5] + z * p[6]))))); |
527 | s = |
528 | q[0] + z * (q[1] + |
529 | z * (q[2] + |
530 | z * (q[3] + z * (q[4] + z * (q[5] + z * (q[6] + z)))))); |
531 | return (-.125 + z * r / s) / x; |
532 | } |
533 | |