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 | /* Modifications for 128-bit long double 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, write to the Free Software |

31 | Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA */ |

32 | |

33 | /* |

34 | * __ieee754_jn(n, x), __ieee754_yn(n, x) |

35 | * floating point Bessel's function of the 1st and 2nd kind |

36 | * of order n |

37 | * |

38 | * Special cases: |

39 | * y0(0)=y1(0)=yn(n,0) = -inf with division by zero signal; |

40 | * y0(-ve)=y1(-ve)=yn(n,-ve) are NaN with invalid signal. |

41 | * Note 2. About jn(n,x), yn(n,x) |

42 | * For n=0, j0(x) is called, |

43 | * for n=1, j1(x) is called, |

44 | * for n<x, forward recursion us used starting |

45 | * from values of j0(x) and j1(x). |

46 | * for n>x, a continued fraction approximation to |

47 | * j(n,x)/j(n-1,x) is evaluated and then backward |

48 | * recursion is used starting from a supposed value |

49 | * for j(n,x). The resulting value of j(0,x) is |

50 | * compared with the actual value to correct the |

51 | * supposed value of j(n,x). |

52 | * |

53 | * yn(n,x) is similar in all respects, except |

54 | * that forward recursion is used for all |

55 | * values of n>1. |

56 | * |

57 | */ |

58 | |

59 | #include <errno.h> |

60 | #include "quadmath-imp.h" |

61 | |

62 | static const __float128 |

63 | invsqrtpi = 5.6418958354775628694807945156077258584405E-1Q, |

64 | two = 2.0e0Q, |

65 | one = 1.0e0Q, |

66 | zero = 0.0Q; |

67 | |

68 | |

69 | __float128 |

70 | jnq (int n, __float128 x) |

71 | { |

72 | uint32_t se; |

73 | int32_t i, ix, sgn; |

74 | __float128 a, b, temp, di; |

75 | __float128 z, w; |

76 | ieee854_float128 u; |

77 | |

78 | |

79 | /* J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x) |

80 | * Thus, J(-n,x) = J(n,-x) |

81 | */ |

82 | |

83 | u.value = x; |

84 | se = u.words32.w0; |

85 | ix = se & 0x7fffffff; |

86 | |

87 | /* if J(n,NaN) is NaN */ |

88 | if (ix >= 0x7fff0000) |

89 | { |

90 | if ((u.words32.w0 & 0xffff) | u.words32.w1 | u.words32.w2 | u.words32.w3) |

91 | return x + x; |

92 | } |

93 | |

94 | if (n < 0) |

95 | { |

96 | n = -n; |

97 | x = -x; |

98 | se ^= 0x80000000; |

99 | } |

100 | if (n == 0) |

101 | return (j0q (x)); |

102 | if (n == 1) |

103 | return (j1q (x)); |

104 | sgn = (n & 1) & (se >> 31); /* even n -- 0, odd n -- sign(x) */ |

105 | x = fabsq (x); |

106 | |

107 | if (x == 0.0Q || ix >= 0x7fff0000) /* if x is 0 or inf */ |

108 | b = zero; |

109 | else if ((__float128) n <= x) |

110 | { |

111 | /* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */ |

112 | if (ix >= 0x412D0000) |

113 | { /* x > 2**302 */ |

114 | |

115 | /* ??? Could use an expansion for large x here. */ |

116 | |

117 | /* (x >> n**2) |

118 | * Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi) |

119 | * Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi) |

120 | * Let s=sin(x), c=cos(x), |

121 | * xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then |

122 | * |

123 | * n sin(xn)*sqt2 cos(xn)*sqt2 |

124 | * ---------------------------------- |

125 | * 0 s-c c+s |

126 | * 1 -s-c -c+s |

127 | * 2 -s+c -c-s |

128 | * 3 s+c c-s |

129 | */ |

130 | __float128 s; |

131 | __float128 c; |

132 | sincosq (x, &s, &c); |

133 | switch (n & 3) |

134 | { |

135 | case 0: |

136 | temp = c + s; |

137 | break; |

138 | case 1: |

139 | temp = -c + s; |

140 | break; |

141 | case 2: |

142 | temp = -c - s; |

143 | break; |

144 | case 3: |

145 | temp = c - s; |

146 | break; |

147 | } |

148 | b = invsqrtpi * temp / sqrtq (x); |

149 | } |

150 | else |

151 | { |

152 | a = j0q (x); |

153 | b = j1q (x); |

154 | for (i = 1; i < n; i++) |

155 | { |

156 | temp = b; |

157 | b = b * ((__float128) (i + i) / x) - a; /* avoid underflow */ |

158 | a = temp; |

159 | } |

160 | } |

161 | } |

162 | else |

163 | { |

164 | if (ix < 0x3fc60000) |

165 | { /* x < 2**-57 */ |

166 | /* x is tiny, return the first Taylor expansion of J(n,x) |

167 | * J(n,x) = 1/n!*(x/2)^n - ... |

168 | */ |

169 | if (n >= 400) /* underflow, result < 10^-4952 */ |

170 | b = zero; |

171 | else |

172 | { |

173 | temp = x * 0.5; |

174 | b = temp; |

175 | for (a = one, i = 2; i <= n; i++) |

176 | { |

177 | a *= (__float128) i; /* a = n! */ |

178 | b *= temp; /* b = (x/2)^n */ |

179 | } |

180 | b = b / a; |

181 | } |

182 | } |

183 | else |

184 | { |

185 | /* use backward recurrence */ |

186 | /* x x^2 x^2 |

187 | * J(n,x)/J(n-1,x) = ---- ------ ------ ..... |

188 | * 2n - 2(n+1) - 2(n+2) |

189 | * |

190 | * 1 1 1 |

191 | * (for large x) = ---- ------ ------ ..... |

192 | * 2n 2(n+1) 2(n+2) |

193 | * -- - ------ - ------ - |

194 | * x x x |

195 | * |

196 | * Let w = 2n/x and h=2/x, then the above quotient |

197 | * is equal to the continued fraction: |

198 | * 1 |

199 | * = ----------------------- |

200 | * 1 |

201 | * w - ----------------- |

202 | * 1 |

203 | * w+h - --------- |

204 | * w+2h - ... |

205 | * |

206 | * To determine how many terms needed, let |

207 | * Q(0) = w, Q(1) = w(w+h) - 1, |

208 | * Q(k) = (w+k*h)*Q(k-1) - Q(k-2), |

209 | * When Q(k) > 1e4 good for single |

210 | * When Q(k) > 1e9 good for double |

211 | * When Q(k) > 1e17 good for quadruple |

212 | */ |

213 | /* determine k */ |

214 | __float128 t, v; |

215 | __float128 q0, q1, h, tmp; |

216 | int32_t k, m; |

217 | w = (n + n) / (__float128) x; |

218 | h = 2.0Q / (__float128) x; |

219 | q0 = w; |

220 | z = w + h; |

221 | q1 = w * z - 1.0Q; |

222 | k = 1; |

223 | while (q1 < 1.0e17Q) |

224 | { |

225 | k += 1; |

226 | z += h; |

227 | tmp = z * q1 - q0; |

228 | q0 = q1; |

229 | q1 = tmp; |

230 | } |

231 | m = n + n; |

232 | for (t = zero, i = 2 * (n + k); i >= m; i -= 2) |

233 | t = one / (i / x - t); |

234 | a = t; |

235 | b = one; |

236 | /* estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n) |

237 | * Hence, if n*(log(2n/x)) > ... |

238 | * single 8.8722839355e+01 |

239 | * double 7.09782712893383973096e+02 |

240 | * __float128 1.1356523406294143949491931077970765006170e+04 |

241 | * then recurrent value may overflow and the result is |

242 | * likely underflow to zero |

243 | */ |

244 | tmp = n; |

245 | v = two / x; |

246 | tmp = tmp * logq (fabsq (v * tmp)); |

247 | |

248 | if (tmp < 1.1356523406294143949491931077970765006170e+04Q) |

249 | { |

250 | for (i = n - 1, di = (__float128) (i + i); i > 0; i--) |

251 | { |

252 | temp = b; |

253 | b *= di; |

254 | b = b / x - a; |

255 | a = temp; |

256 | di -= two; |

257 | } |

258 | } |

259 | else |

260 | { |

261 | for (i = n - 1, di = (__float128) (i + i); i > 0; i--) |

262 | { |

263 | temp = b; |

264 | b *= di; |

265 | b = b / x - a; |

266 | a = temp; |

267 | di -= two; |

268 | /* scale b to avoid spurious overflow */ |

269 | if (b > 1e100Q) |

270 | { |

271 | a /= b; |

272 | t /= b; |

273 | b = one; |

274 | } |

275 | } |

276 | } |

277 | /* j0() and j1() suffer enormous loss of precision at and |

278 | * near zero; however, we know that their zero points never |

279 | * coincide, so just choose the one further away from zero. |

280 | */ |

281 | z = j0q (x); |

282 | w = j1q (x); |

283 | if (fabsq (z) >= fabsq (w)) |

284 | b = (t * z / b); |

285 | else |

286 | b = (t * w / a); |

287 | } |

288 | } |

289 | if (sgn == 1) |

290 | return -b; |

291 | else |

292 | return b; |

293 | } |

294 | |

295 | __float128 |

296 | ynq (int n, __float128 x) |

297 | { |

298 | uint32_t se; |

299 | int32_t i, ix; |

300 | int32_t sign; |

301 | __float128 a, b, temp; |

302 | ieee854_float128 u; |

303 | |

304 | u.value = x; |

305 | se = u.words32.w0; |

306 | ix = se & 0x7fffffff; |

307 | |

308 | /* if Y(n,NaN) is NaN */ |

309 | if (ix >= 0x7fff0000) |

310 | { |

311 | if ((u.words32.w0 & 0xffff) | u.words32.w1 | u.words32.w2 | u.words32.w3) |

312 | return x + x; |

313 | } |

314 | if (x <= 0.0Q) |

315 | { |

316 | if (x == 0.0Q) |

317 | return -HUGE_VALQ + x; |

318 | if (se & 0x80000000) |

319 | return zero / (zero * x); |

320 | } |

321 | sign = 1; |

322 | if (n < 0) |

323 | { |

324 | n = -n; |

325 | sign = 1 - ((n & 1) << 1); |

326 | } |

327 | if (n == 0) |

328 | return (y0q (x)); |

329 | if (n == 1) |

330 | return (sign * y1q (x)); |

331 | if (ix >= 0x7fff0000) |

332 | return zero; |

333 | if (ix >= 0x412D0000) |

334 | { /* x > 2**302 */ |

335 | |

336 | /* ??? See comment above on the possible futility of this. */ |

337 | |

338 | /* (x >> n**2) |

339 | * Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi) |

340 | * Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi) |

341 | * Let s=sin(x), c=cos(x), |

342 | * xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then |

343 | * |

344 | * n sin(xn)*sqt2 cos(xn)*sqt2 |

345 | * ---------------------------------- |

346 | * 0 s-c c+s |

347 | * 1 -s-c -c+s |

348 | * 2 -s+c -c-s |

349 | * 3 s+c c-s |

350 | */ |

351 | __float128 s; |

352 | __float128 c; |

353 | sincosq (x, &s, &c); |

354 | switch (n & 3) |

355 | { |

356 | case 0: |

357 | temp = s - c; |

358 | break; |

359 | case 1: |

360 | temp = -s - c; |

361 | break; |

362 | case 2: |

363 | temp = -s + c; |

364 | break; |

365 | case 3: |

366 | temp = s + c; |

367 | break; |

368 | } |

369 | b = invsqrtpi * temp / sqrtq (x); |

370 | } |

371 | else |

372 | { |

373 | a = y0q (x); |

374 | b = y1q (x); |

375 | /* quit if b is -inf */ |

376 | u.value = b; |

377 | se = u.words32.w0 & 0xffff0000; |

378 | for (i = 1; i < n && se != 0xffff0000; i++) |

379 | { |

380 | temp = b; |

381 | b = ((__float128) (i + i) / x) * b - a; |

382 | u.value = b; |

383 | se = u.words32.w0 & 0xffff0000; |

384 | a = temp; |

385 | } |

386 | } |

387 | /* If B is +-Inf, set up errno accordingly. */ |

388 | if (! finiteq (b)) |

389 | errno = ERANGE; |

390 | if (sign > 0) |

391 | return b; |

392 | else |

393 | return -b; |

394 | } |

395 |