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
;
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) */
99 if (__glibc_unlikely ((ix
| i0
| i1
) == 0 || ix
>= 0x7fff))
100 /* if x is 0 or inf */
102 else if ((long double) n
<= x
)
104 /* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */
108 /* ??? This might be a futile gesture.
109 If x exceeds X_TLOSS anyway, the wrapper function
110 will set the result to zero. */
113 * Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
114 * Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
115 * Let s=sin(x), c=cos(x),
116 * xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
118 * n sin(xn)*sqt2 cos(xn)*sqt2
119 * ----------------------------------
127 __sincosl (x
, &s
, &c
);
143 b
= invsqrtpi
* temp
/ __ieee754_sqrtl (x
);
147 a
= __ieee754_j0l (x
);
148 b
= __ieee754_j1l (x
);
149 for (i
= 1; i
< n
; i
++)
152 b
= b
* ((long double) (i
+ i
) / x
) - a
; /* avoid underflow */
161 /* x is tiny, return the first Taylor expansion of J(n,x)
162 * J(n,x) = 1/n!*(x/2)^n - ...
164 if (n
>= 400) /* underflow, result < 10^-4952 */
170 for (a
= one
, i
= 2; i
<= n
; i
++)
172 a
*= (long double) i
; /* a = n! */
173 b
*= temp
; /* b = (x/2)^n */
180 /* use backward recurrence */
182 * J(n,x)/J(n-1,x) = ---- ------ ------ .....
183 * 2n - 2(n+1) - 2(n+2)
186 * (for large x) = ---- ------ ------ .....
188 * -- - ------ - ------ -
191 * Let w = 2n/x and h=2/x, then the above quotient
192 * is equal to the continued fraction:
194 * = -----------------------
196 * w - -----------------
201 * To determine how many terms needed, let
202 * Q(0) = w, Q(1) = w(w+h) - 1,
203 * Q(k) = (w+k*h)*Q(k-1) - Q(k-2),
204 * When Q(k) > 1e4 good for single
205 * When Q(k) > 1e9 good for double
206 * When Q(k) > 1e17 good for quadruple
210 long double q0
, q1
, h
, tmp
;
212 w
= (n
+ n
) / (long double) x
;
213 h
= 2.0L / (long double) x
;
227 for (t
= zero
, i
= 2 * (n
+ k
); i
>= m
; i
-= 2)
228 t
= one
/ (i
/ x
- t
);
231 /* estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n)
232 * Hence, if n*(log(2n/x)) > ...
233 * single 8.8722839355e+01
234 * double 7.09782712893383973096e+02
235 * long double 1.1356523406294143949491931077970765006170e+04
236 * then recurrent value may overflow and the result is
237 * likely underflow to zero
241 tmp
= tmp
* __ieee754_logl (fabsl (v
* tmp
));
243 if (tmp
< 1.1356523406294143949491931077970765006170e+04L)
245 for (i
= n
- 1, di
= (long double) (i
+ i
); i
> 0; i
--)
256 for (i
= n
- 1, di
= (long double) (i
+ i
); i
> 0; i
--)
263 /* scale b to avoid spurious overflow */
272 /* j0() and j1() suffer enormous loss of precision at and
273 * near zero; however, we know that their zero points never
274 * coincide, so just choose the one further away from zero.
276 z
= __ieee754_j0l (x
);
277 w
= __ieee754_j1l (x
);
278 if (fabsl (z
) >= fabsl (w
))
289 strong_alias (__ieee754_jnl
, __jnl_finite
)
292 __ieee754_ynl (int n
, long double x
)
294 u_int32_t se
, i0
, i1
;
297 long double a
, b
, temp
, ret
;
300 GET_LDOUBLE_WORDS (se
, i0
, i1
, x
);
302 /* if Y(n,NaN) is NaN */
303 if (__builtin_expect ((ix
== 0x7fff) && ((i0
& 0x7fffffff) != 0), 0))
305 if (__builtin_expect ((ix
| i0
| i1
) == 0, 0))
306 /* -inf or inf and divide-by-zero exception. */
307 return ((n
< 0 && (n
& 1) != 0) ? 1.0L : -1.0L) / 0.0L;
308 if (__builtin_expect (se
& 0x8000, 0))
309 return zero
/ (zero
* x
);
314 sign
= 1 - ((n
& 1) << 1);
317 return (__ieee754_y0l (x
));
319 SET_RESTORE_ROUNDL (FE_TONEAREST
);
322 ret
= sign
* __ieee754_y1l (x
);
325 if (__glibc_unlikely (ix
== 0x7fff))
330 /* ??? See comment above on the possible futility of this. */
333 * Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
334 * Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
335 * Let s=sin(x), c=cos(x),
336 * xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
338 * n sin(xn)*sqt2 cos(xn)*sqt2
339 * ----------------------------------
347 __sincosl (x
, &s
, &c
);
363 b
= invsqrtpi
* temp
/ __ieee754_sqrtl (x
);
367 a
= __ieee754_y0l (x
);
368 b
= __ieee754_y1l (x
);
369 /* quit if b is -inf */
370 GET_LDOUBLE_WORDS (se
, i0
, i1
, b
);
371 /* Use 0xffffffff since GET_LDOUBLE_WORDS sign-extends SE. */
372 for (i
= 1; i
< n
&& se
!= 0xffffffff; i
++)
375 b
= ((long double) (i
+ i
) / x
) * b
- a
;
376 GET_LDOUBLE_WORDS (se
, i0
, i1
, b
);
380 /* If B is +-Inf, set up errno accordingly. */
382 __set_errno (ERANGE
);
390 ret
= __copysignl (LDBL_MAX
, ret
) * LDBL_MAX
;
393 strong_alias (__ieee754_ynl
, __ynl_finite
)