2 * ====================================================
3 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
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
9 * ====================================================
12 /* Modifications for 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
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.
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.
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/>. */
34 * __ieee754_jn(n, x), __ieee754_yn(n, x)
35 * floating point Bessel's function of the 1st and 2nd kind
39 * y0(0)=y1(0)=yn(n,0) = -inf with overflow 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).
53 * yn(n,x) is similar in all respects, except
54 * that forward recursion is used for all
62 #include <math_private.h>
64 static const long double
65 invsqrtpi
= 5.64189583547756286948079e-1L, two
= 2.0e0L
, one
= 1.0e0L
;
67 static const long double zero
= 0.0L;
70 __ieee754_jnl (int n
, long double x
)
74 long double a
, b
, temp
, di
, ret
;
77 /* J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x)
78 * Thus, J(-n,x) = J(n,-x)
81 GET_LDOUBLE_WORDS (se
, i0
, i1
, x
);
84 /* if J(n,NaN) is NaN */
85 if (__glibc_unlikely ((ix
== 0x7fff) && ((i0
& 0x7fffffff) != 0)))
94 return (__ieee754_j0l (x
));
96 return (__ieee754_j1l (x
));
97 sgn
= (n
& 1) & (se
>> 15); /* even n -- 0, odd n -- sign(x) */
100 SET_RESTORE_ROUNDL (FE_TONEAREST
);
101 if (__glibc_unlikely ((ix
| i0
| i1
) == 0 || ix
>= 0x7fff))
102 /* if x is 0 or inf */
103 return sgn
== 1 ? -zero
: zero
;
104 else if ((long double) n
<= x
)
106 /* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */
110 /* ??? This might be a futile gesture.
111 If x exceeds X_TLOSS anyway, the wrapper function
112 will set the result to zero. */
115 * Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
116 * Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
117 * Let s=sin(x), c=cos(x),
118 * xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
120 * n sin(xn)*sqt2 cos(xn)*sqt2
121 * ----------------------------------
129 __sincosl (x
, &s
, &c
);
145 b
= invsqrtpi
* temp
/ __ieee754_sqrtl (x
);
149 a
= __ieee754_j0l (x
);
150 b
= __ieee754_j1l (x
);
151 for (i
= 1; i
< n
; i
++)
154 b
= b
* ((long double) (i
+ i
) / x
) - a
; /* avoid underflow */
163 /* x is tiny, return the first Taylor expansion of J(n,x)
164 * J(n,x) = 1/n!*(x/2)^n - ...
166 if (n
>= 400) /* underflow, result < 10^-4952 */
172 for (a
= one
, i
= 2; i
<= n
; i
++)
174 a
*= (long double) i
; /* a = n! */
175 b
*= temp
; /* b = (x/2)^n */
182 /* use backward recurrence */
184 * J(n,x)/J(n-1,x) = ---- ------ ------ .....
185 * 2n - 2(n+1) - 2(n+2)
188 * (for large x) = ---- ------ ------ .....
190 * -- - ------ - ------ -
193 * Let w = 2n/x and h=2/x, then the above quotient
194 * is equal to the continued fraction:
196 * = -----------------------
198 * w - -----------------
203 * To determine how many terms needed, let
204 * Q(0) = w, Q(1) = w(w+h) - 1,
205 * Q(k) = (w+k*h)*Q(k-1) - Q(k-2),
206 * When Q(k) > 1e4 good for single
207 * When Q(k) > 1e9 good for double
208 * When Q(k) > 1e17 good for quadruple
212 long double q0
, q1
, h
, tmp
;
214 w
= (n
+ n
) / (long double) x
;
215 h
= 2.0L / (long double) x
;
229 for (t
= zero
, i
= 2 * (n
+ k
); i
>= m
; i
-= 2)
230 t
= one
/ (i
/ x
- t
);
233 /* estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n)
234 * Hence, if n*(log(2n/x)) > ...
235 * single 8.8722839355e+01
236 * double 7.09782712893383973096e+02
237 * long double 1.1356523406294143949491931077970765006170e+04
238 * then recurrent value may overflow and the result is
239 * likely underflow to zero
243 tmp
= tmp
* __ieee754_logl (fabsl (v
* tmp
));
245 if (tmp
< 1.1356523406294143949491931077970765006170e+04L)
247 for (i
= n
- 1, di
= (long double) (i
+ i
); i
> 0; i
--)
258 for (i
= n
- 1, di
= (long double) (i
+ i
); i
> 0; i
--)
265 /* scale b to avoid spurious overflow */
274 /* j0() and j1() suffer enormous loss of precision at and
275 * near zero; however, we know that their zero points never
276 * coincide, so just choose the one further away from zero.
278 z
= __ieee754_j0l (x
);
279 w
= __ieee754_j1l (x
);
280 if (fabsl (z
) >= fabsl (w
))
293 ret
= __copysignl (LDBL_MIN
, ret
) * LDBL_MIN
;
294 __set_errno (ERANGE
);
297 math_check_force_underflow (ret
);
300 strong_alias (__ieee754_jnl
, __jnl_finite
)
303 __ieee754_ynl (int n
, long double x
)
305 u_int32_t se
, i0
, i1
;
308 long double a
, b
, temp
, ret
;
311 GET_LDOUBLE_WORDS (se
, i0
, i1
, x
);
313 /* if Y(n,NaN) is NaN */
314 if (__builtin_expect ((ix
== 0x7fff) && ((i0
& 0x7fffffff) != 0), 0))
316 if (__builtin_expect ((ix
| i0
| i1
) == 0, 0))
317 /* -inf or inf and divide-by-zero exception. */
318 return ((n
< 0 && (n
& 1) != 0) ? 1.0L : -1.0L) / 0.0L;
319 if (__builtin_expect (se
& 0x8000, 0))
320 return zero
/ (zero
* x
);
325 sign
= 1 - ((n
& 1) << 1);
328 return (__ieee754_y0l (x
));
330 SET_RESTORE_ROUNDL (FE_TONEAREST
);
333 ret
= sign
* __ieee754_y1l (x
);
336 if (__glibc_unlikely (ix
== 0x7fff))
341 /* ??? See comment above on the possible futility of this. */
344 * Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
345 * Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
346 * Let s=sin(x), c=cos(x),
347 * xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
349 * n sin(xn)*sqt2 cos(xn)*sqt2
350 * ----------------------------------
358 __sincosl (x
, &s
, &c
);
374 b
= invsqrtpi
* temp
/ __ieee754_sqrtl (x
);
378 a
= __ieee754_y0l (x
);
379 b
= __ieee754_y1l (x
);
380 /* quit if b is -inf */
381 GET_LDOUBLE_WORDS (se
, i0
, i1
, b
);
382 /* Use 0xffffffff since GET_LDOUBLE_WORDS sign-extends SE. */
383 for (i
= 1; i
< n
&& se
!= 0xffffffff; i
++)
386 b
= ((long double) (i
+ i
) / x
) * b
- a
;
387 GET_LDOUBLE_WORDS (se
, i0
, i1
, b
);
391 /* If B is +-Inf, set up errno accordingly. */
393 __set_errno (ERANGE
);
401 ret
= __copysignl (LDBL_MAX
, ret
) * LDBL_MAX
;
404 strong_alias (__ieee754_ynl
, __ynl_finite
)