projects / thirdparty / glibc.git / blame
| Commit | Line | Data |
|---|---|---|
| 2e6f4694 AJ |
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 contributed by | |
| 13 | Stephen L. Moshier <moshier@na-net.ornl.gov> */ | |
| 14 | ||
| 15 | /* | |
| 16 | * __ieee754_jn(n, x), __ieee754_yn(n, x) | |
| 17 | * floating point Bessel's function of the 1st and 2nd kind | |
| 18 | * of order n | |
| 19 | * | |
| 20 | * Special cases: | |
| 21 | * y0(0)=y1(0)=yn(n,0) = -inf with division by zero signal; | |
| 22 | * y0(-ve)=y1(-ve)=yn(n,-ve) are NaN with invalid signal. | |
| 23 | * Note 2. About jn(n,x), yn(n,x) | |
| 24 | * For n=0, j0(x) is called, | |
| 25 | * for n=1, j1(x) is called, | |
| 26 | * for n<x, forward recursion us used starting | |
| 27 | * from values of j0(x) and j1(x). | |
| 28 | * for n>x, a continued fraction approximation to | |
| 29 | * j(n,x)/j(n-1,x) is evaluated and then backward | |
| 30 | * recursion is used starting from a supposed value | |
| 31 | * for j(n,x). The resulting value of j(0,x) is | |
| 32 | * compared with the actual value to correct the | |
| 33 | * supposed value of j(n,x). | |
| 34 | * | |
| 35 | * yn(n,x) is similar in all respects, except | |
| 36 | * that forward recursion is used for all | |
| 37 | * values of n>1. | |
| 38 | * | |
| 39 | */ | |
| 40 | ||
| 41 | #include "math.h" | |
| 42 | #include "math_private.h" | |
| 43 | ||
| 44 | #ifdef __STDC__ | |
| 45 | static const long double | |
| 46 | #else | |
| 47 | static long double | |
| 48 | #endif | |
| 49 | invsqrtpi = 5.6418958354775628694807945156077258584405E-1L, | |
| 50 | two = 2.0e0L, | |
| 51 | one = 1.0e0L, | |
| 52 | zero = 0.0L; | |
| 53 | ||
| 54 | ||
| 55 | #ifdef __STDC__ | |
| 56 | long double | |
| 57 | __ieee754_jnl (int n, long double x) | |
| 58 | #else | |
| 59 | long double | |
| 60 | __ieee754_jnl (n, x) | |
| 61 | int n; | |
| 62 | long double x; | |
| 63 | #endif | |
| 64 | { | |
| 65 | u_int32_t se; | |
| 66 | int32_t i, ix, sgn; | |
| 67 | long double a, b, temp, di; | |
| 68 | long double z, w; | |
| 69 | ieee854_long_double_shape_type u; | |
| 70 | ||
| 71 | ||
| 72 | /* J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x) | |
| 73 | * Thus, J(-n,x) = J(n,-x) | |
| 74 | */ | |
| 75 | ||
| 76 | u.value = x; | |
| 77 | se = u.parts32.w0; | |
| 78 | ix = se & 0x7fffffff; | |
| 79 | ||
| 80 | /* if J(n,NaN) is NaN */ | |
| 81 | if (ix >= 0x7fff0000) | |
| 82 | { | |
| 83 | if ((u.parts32.w0 & 0xffff) | u.parts32.w1 | u.parts32.w2 | u.parts32.w3) | |
| 84 | return x + x; | |
| 85 | } | |
| 86 | ||
| 87 | if (n < 0) | |
| 88 | { | |
| 89 | n = -n; | |
| 90 | x = -x; | |
| 91 | se ^= 0x80000000; | |
| 92 | } | |
| 93 | if (n == 0) | |
| 94 | return (__ieee754_j0l (x)); | |
| 95 | if (n == 1) | |
| 96 | return (__ieee754_j1l (x)); | |
| 97 | sgn = (n & 1) & (se >> 31); /* even n -- 0, odd n -- sign(x) */ | |
| 98 | x = fabsl (x); | |
| 99 | ||
| 100 | if (x == 0.0L || ix >= 0x7fff0000) /* if x is 0 or inf */ | |
| 101 | b = zero; | |
| 102 | else if ((long double) n <= x) | |
| 103 | { | |
| 104 | /* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */ | |
| 105 | if (ix >= 0x412D0000) | |
| 106 | { /* x > 2**302 */ | |
| 107 | ||
| 108 | /* ??? Could use an expansion for large x here. */ | |
| 109 | ||
| 110 | /* (x >> n**2) | |
| 111 | * Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi) | |
| 112 | * Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi) | |
| 113 | * Let s=sin(x), c=cos(x), | |
| 114 | * xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then | |
| 115 | * | |
| 116 | * n sin(xn)*sqt2 cos(xn)*sqt2 | |
| 117 | * ---------------------------------- | |
| 118 | * 0 s-c c+s | |
| 119 | * 1 -s-c -c+s | |
| 120 | * 2 -s+c -c-s | |
| 121 | * 3 s+c c-s | |
| 122 | */ | |
| 123 | long double s; | |
| 124 | long double c; | |
| 125 | __sincosl (x, &s, &c); | |
| 126 | switch (n & 3) | |
| 127 | { | |
| 128 | case 0: | |
| 129 | temp = c + s; | |
| 130 | break; | |
| 131 | case 1: | |
| 132 | temp = -c + s; | |
| 133 | break; | |
| 134 | case 2: | |
| 135 | temp = -c - s; | |
| 136 | break; | |
| 137 | case 3: | |
| 138 | temp = c - s; | |
| 139 | break; | |
| 140 | } | |
| 141 | b = invsqrtpi * temp / __ieee754_sqrtl (x); | |
| 142 | } | |
| 143 | else | |
| 144 | { | |
| 145 | a = __ieee754_j0l (x); | |
| 146 | b = __ieee754_j1l (x); | |
| 147 | for (i = 1; i < n; i++) | |
| 148 | { | |
| 149 | temp = b; | |
| 150 | b = b * ((long double) (i + i) / x) - a; /* avoid underflow */ | |
| 151 | a = temp; | |
| 152 | } | |
| 153 | } | |
| 154 | } | |
| 155 | else | |
| 156 | { | |
| 157 | if (ix < 0x3fc60000) | |
| 158 | { /* x < 2**-57 */ | |
| 159 | /* x is tiny, return the first Taylor expansion of J(n,x) | |
| 160 | * J(n,x) = 1/n!*(x/2)^n - ... | |
| 161 | */ | |
| 162 | if (n >= 400) /* underflow, result < 10^-4952 */ | |
| 163 | b = zero; | |
| 164 | else | |
| 165 | { | |
| 166 | temp = x * 0.5; | |
| 167 | b = temp; | |
| 168 | for (a = one, i = 2; i <= n; i++) | |
| 169 | { | |
| 170 | a *= (long double) i; /* a = n! */ | |
| 171 | b *= temp; /* b = (x/2)^n */ | |
| 172 | } | |
| 173 | b = b / a; | |
| 174 | } | |
| 175 | } | |
| 176 | else | |
| 177 | { | |
| 178 | /* use backward recurrence */ | |
| 179 | /* x x^2 x^2 | |
| 180 | * J(n,x)/J(n-1,x) = ---- ------ ------ ..... | |
| 181 | * 2n - 2(n+1) - 2(n+2) | |
| 182 | * | |
| 183 | * 1 1 1 | |
| 184 | * (for large x) = ---- ------ ------ ..... | |
| 185 | * 2n 2(n+1) 2(n+2) | |
| 186 | * -- - ------ - ------ - | |
| 187 | * x x x | |
| 188 | * | |
| 189 | * Let w = 2n/x and h=2/x, then the above quotient | |
| 190 | * is equal to the continued fraction: | |
| 191 | * 1 | |
| 192 | * = ----------------------- | |
| 193 | * 1 | |
| 194 | * w - ----------------- | |
| 195 | * 1 | |
| 196 | * w+h - --------- | |
| 197 | * w+2h - ... | |
| 198 | * | |
| 199 | * To determine how many terms needed, let | |
| 200 | * Q(0) = w, Q(1) = w(w+h) - 1, | |
| 201 | * Q(k) = (w+k*h)*Q(k-1) - Q(k-2), | |
| 202 | * When Q(k) > 1e4 good for single | |
| 203 | * When Q(k) > 1e9 good for double | |
| 204 | * When Q(k) > 1e17 good for quadruple | |
| 205 | */ | |
| 206 | /* determine k */ | |
| 207 | long double t, v; | |
| 208 | long double q0, q1, h, tmp; | |
| 209 | int32_t k, m; | |
| 210 | w = (n + n) / (long double) x; | |
| 211 | h = 2.0L / (long double) x; | |
| 212 | q0 = w; | |
| 213 | z = w + h; | |
| 214 | q1 = w * z - 1.0L; | |
| 215 | k = 1; | |
| 216 | while (q1 < 1.0e17L) | |
| 217 | { | |
| 218 | k += 1; | |
| 219 | z += h; | |
| 220 | tmp = z * q1 - q0; | |
| 221 | q0 = q1; | |
| 222 | q1 = tmp; | |
| 223 | } | |
| 224 | m = n + n; | |
| 225 | for (t = zero, i = 2 * (n + k); i >= m; i -= 2) | |
| 226 | t = one / (i / x - t); | |
| 227 | a = t; | |
| 228 | b = one; | |
| 229 | /* estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n) | |
| 230 | * Hence, if n*(log(2n/x)) > ... | |
| 231 | * single 8.8722839355e+01 | |
| 232 | * double 7.09782712893383973096e+02 | |
| 233 | * long double 1.1356523406294143949491931077970765006170e+04 | |
| 234 | * then recurrent value may overflow and the result is | |
| 235 | * likely underflow to zero | |
| 236 | */ | |
| 237 | tmp = n; | |
| 238 | v = two / x; | |
| 239 | tmp = tmp * __ieee754_logl (fabsl (v * tmp)); | |
| 240 | ||
| 241 | if (tmp < 1.1356523406294143949491931077970765006170e+04L) | |
| 242 | { | |
| 243 | for (i = n - 1, di = (long double) (i + i); i > 0; i--) | |
| 244 | { | |
| 245 | temp = b; | |
| 246 | b *= di; | |
| 247 | b = b / x - a; | |
| 248 | a = temp; | |
| 249 | di -= two; | |
| 250 | } | |
| 251 | } | |
| 252 | else | |
| 253 | { | |
| 254 | for (i = n - 1, di = (long double) (i + i); i > 0; i--) | |
| 255 | { | |
| 256 | temp = b; | |
| 257 | b *= di; | |
| 258 | b = b / x - a; | |
| 259 | a = temp; | |
| 260 | di -= two; | |
| 261 | /* scale b to avoid spurious overflow */ | |
| 262 | if (b > 1e100L) | |
| 263 | { | |
| 264 | a /= b; | |
| 265 | t /= b; | |
| 266 | b = one; | |
| 267 | } | |
| 268 | } | |
| 269 | } | |
| 270 | b = (t * __ieee754_j0l (x) / b); | |
| 271 | } | |
| 272 | } | |
| 273 | if (sgn == 1) | |
| 274 | return -b; | |
| 275 | else | |
| 276 | return b; | |
| 277 | } | |
| 278 | ||
| 279 | #ifdef __STDC__ | |
| 280 | long double | |
| 281 | __ieee754_ynl (int n, long double x) | |
| 282 | #else | |
| 283 | long double | |
| 284 | __ieee754_ynl (n, x) | |
| 285 | int n; | |
| 286 | long double x; | |
| 287 | #endif | |
| 288 | { | |
| 289 | u_int32_t se; | |
| 290 | int32_t i, ix; | |
| 291 | int32_t sign; | |
| 292 | long double a, b, temp; | |
| 293 | ieee854_long_double_shape_type u; | |
| 294 | ||
| 295 | u.value = x; | |
| 296 | se = u.parts32.w0; | |
| 297 | ix = se & 0x7fffffff; | |
| 298 | ||
| 299 | /* if Y(n,NaN) is NaN */ | |
| 300 | if (ix >= 0x7fff0000) | |
| 301 | { | |
| 302 | if ((u.parts32.w0 & 0xffff) | u.parts32.w1 | u.parts32.w2 | u.parts32.w3) | |
| 303 | return x + x; | |
| 304 | } | |
| 305 | if (x <= 0.0L) | |
| 306 | { | |
| 307 | if (x == 0.0L) | |
| 308 | return -one / zero; | |
| 309 | if (se & 0x80000000) | |
| 310 | return zero / zero; | |
| 311 | } | |
| 312 | sign = 1; | |
| 313 | if (n < 0) | |
| 314 | { | |
| 315 | n = -n; | |
| 316 | sign = 1 - ((n & 1) << 1); | |
| 317 | } | |
| 318 | if (n == 0) | |
| 319 | return (__ieee754_y0l (x)); | |
| 320 | if (n == 1) | |
| 321 | return (sign * __ieee754_y1l (x)); | |
| 322 | if (ix >= 0x7fff0000) | |
| 323 | return zero; | |
| 324 | if (ix >= 0x412D0000) | |
| 325 | { /* x > 2**302 */ | |
| 326 | ||
| 327 | /* ??? See comment above on the possible futility of this. */ | |
| 328 | ||
| 329 | /* (x >> n**2) | |
| 330 | * Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi) | |
| 331 | * Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi) | |
| 332 | * Let s=sin(x), c=cos(x), | |
| 333 | * xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then | |
| 334 | * | |
| 335 | * n sin(xn)*sqt2 cos(xn)*sqt2 | |
| 336 | * ---------------------------------- | |
| 337 | * 0 s-c c+s | |
| 338 | * 1 -s-c -c+s | |
| 339 | * 2 -s+c -c-s | |
| 340 | * 3 s+c c-s | |
| 341 | */ | |
| 342 | long double s; | |
| 343 | long double c; | |
| 344 | __sincosl (x, &s, &c); | |
| 345 | switch (n & 3) | |
| 346 | { | |
| 347 | case 0: | |
| 348 | temp = s - c; | |
| 349 | break; | |
| 350 | case 1: | |
| 351 | temp = -s - c; | |
| 352 | break; | |
| 353 | case 2: | |
| 354 | temp = -s + c; | |
| 355 | break; | |
| 356 | case 3: | |
| 357 | temp = s + c; | |
| 358 | break; | |
| 359 | } | |
| 360 | b = invsqrtpi * temp / __ieee754_sqrtl (x); | |
| 361 | } | |
| 362 | else | |
| 363 | { | |
| 364 | a = __ieee754_y0l (x); | |
| 365 | b = __ieee754_y1l (x); | |
| 366 | /* quit if b is -inf */ | |
| 367 | u.value = b; | |
| 368 | se = u.parts32.w0 & 0xffff0000; | |
| 369 | for (i = 1; i < n && se != 0xffff0000; i++) | |
| 370 | { | |
| 371 | temp = b; | |
| 372 | b = ((long double) (i + i) / x) * b - a; | |
| 373 | u.value = b; | |
| 374 | se = u.parts32.w0 & 0xffff0000; | |
| 375 | a = temp; | |
| 376 | } | |
| 377 | } | |
| 378 | if (sign > 0) | |
| 379 | return b; | |
| 380 | else | |
| 381 | return -b; | |
| 382 | } |