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 <https://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>
63 #include <fenv_private.h>
64 #include <math-underflow.h>
65 #include <libm-alias-finite.h>
67 static const long double
68 invsqrtpi
= 5.64189583547756286948079e-1L, two
= 2.0e0L
, one
= 1.0e0L
;
70 static const long double zero
= 0.0L;
73 __ieee754_jnl (int n
, long double x
)
77 long double a
, b
, temp
, di
, ret
;
80 /* J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x)
81 * Thus, J(-n,x) = J(n,-x)
84 GET_LDOUBLE_WORDS (se
, i0
, i1
, x
);
87 /* if J(n,NaN) is NaN */
88 if (__glibc_unlikely ((ix
== 0x7fff) && ((i0
& 0x7fffffff) != 0)))
97 return (__ieee754_j0l (x
));
99 return (__ieee754_j1l (x
));
100 sgn
= (n
& 1) & (se
>> 15); /* even n -- 0, odd n -- sign(x) */
103 SET_RESTORE_ROUNDL (FE_TONEAREST
);
104 if (__glibc_unlikely ((ix
| i0
| i1
) == 0 || ix
>= 0x7fff))
105 /* if x is 0 or inf */
106 return sgn
== 1 ? -zero
: zero
;
107 else if ((long double) n
<= x
)
109 /* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */
113 /* ??? This might be a futile gesture.
114 If x exceeds X_TLOSS anyway, the wrapper function
115 will set the result to zero. */
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
123 * n sin(xn)*sqt2 cos(xn)*sqt2
124 * ----------------------------------
132 __sincosl (x
, &s
, &c
);
148 __builtin_unreachable ();
150 b
= invsqrtpi
* temp
/ sqrtl (x
);
154 a
= __ieee754_j0l (x
);
155 b
= __ieee754_j1l (x
);
156 for (i
= 1; i
< n
; i
++)
159 b
= b
* ((long double) (i
+ i
) / x
) - a
; /* avoid underflow */
168 /* x is tiny, return the first Taylor expansion of J(n,x)
169 * J(n,x) = 1/n!*(x/2)^n - ...
171 if (n
>= 400) /* underflow, result < 10^-4952 */
177 for (a
= one
, i
= 2; i
<= n
; i
++)
179 a
*= (long double) i
; /* a = n! */
180 b
*= temp
; /* b = (x/2)^n */
187 /* use backward recurrence */
189 * J(n,x)/J(n-1,x) = ---- ------ ------ .....
190 * 2n - 2(n+1) - 2(n+2)
193 * (for large x) = ---- ------ ------ .....
195 * -- - ------ - ------ -
198 * Let w = 2n/x and h=2/x, then the above quotient
199 * is equal to the continued fraction:
201 * = -----------------------
203 * w - -----------------
208 * To determine how many terms needed, let
209 * Q(0) = w, Q(1) = w(w+h) - 1,
210 * Q(k) = (w+k*h)*Q(k-1) - Q(k-2),
211 * When Q(k) > 1e4 good for single
212 * When Q(k) > 1e9 good for double
213 * When Q(k) > 1e17 good for quadruple
217 long double q0
, q1
, h
, tmp
;
219 w
= (n
+ n
) / (long double) x
;
220 h
= 2.0L / (long double) x
;
234 for (t
= zero
, i
= 2 * (n
+ k
); i
>= m
; i
-= 2)
235 t
= one
/ (i
/ x
- t
);
238 /* estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n)
239 * Hence, if n*(log(2n/x)) > ...
240 * single 8.8722839355e+01
241 * double 7.09782712893383973096e+02
242 * long double 1.1356523406294143949491931077970765006170e+04
243 * then recurrent value may overflow and the result is
244 * likely underflow to zero
248 tmp
= tmp
* __ieee754_logl (fabsl (v
* tmp
));
250 if (tmp
< 1.1356523406294143949491931077970765006170e+04L)
252 for (i
= n
- 1, di
= (long double) (i
+ i
); i
> 0; i
--)
263 for (i
= n
- 1, di
= (long double) (i
+ i
); i
> 0; i
--)
270 /* scale b to avoid spurious overflow */
279 /* j0() and j1() suffer enormous loss of precision at and
280 * near zero; however, we know that their zero points never
281 * coincide, so just choose the one further away from zero.
283 z
= __ieee754_j0l (x
);
284 w
= __ieee754_j1l (x
);
285 if (fabsl (z
) >= fabsl (w
))
298 ret
= copysignl (LDBL_MIN
, ret
) * LDBL_MIN
;
299 __set_errno (ERANGE
);
302 math_check_force_underflow (ret
);
305 libm_alias_finite (__ieee754_jnl
, __jnl
)
308 __ieee754_ynl (int n
, long double x
)
313 long double a
, b
, temp
, ret
;
316 GET_LDOUBLE_WORDS (se
, i0
, i1
, x
);
318 /* if Y(n,NaN) is NaN */
319 if (__builtin_expect ((ix
== 0x7fff) && ((i0
& 0x7fffffff) != 0), 0))
321 if (__builtin_expect ((ix
| i0
| i1
) == 0, 0))
322 /* -inf or inf and divide-by-zero exception. */
323 return ((n
< 0 && (n
& 1) != 0) ? 1.0L : -1.0L) / 0.0L;
324 if (__builtin_expect (se
& 0x8000, 0))
325 return zero
/ (zero
* x
);
330 sign
= 1 - ((n
& 1) << 1);
333 return (__ieee754_y0l (x
));
335 SET_RESTORE_ROUNDL (FE_TONEAREST
);
338 ret
= sign
* __ieee754_y1l (x
);
341 if (__glibc_unlikely (ix
== 0x7fff))
346 /* ??? See comment above on the possible futility of this. */
349 * Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
350 * Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
351 * Let s=sin(x), c=cos(x),
352 * xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
354 * n sin(xn)*sqt2 cos(xn)*sqt2
355 * ----------------------------------
363 __sincosl (x
, &s
, &c
);
379 __builtin_unreachable ();
381 b
= invsqrtpi
* temp
/ sqrtl (x
);
385 a
= __ieee754_y0l (x
);
386 b
= __ieee754_y1l (x
);
387 /* quit if b is -inf */
388 GET_LDOUBLE_WORDS (se
, i0
, i1
, b
);
389 /* Use 0xffffffff since GET_LDOUBLE_WORDS sign-extends SE. */
390 for (i
= 1; i
< n
&& se
!= 0xffffffff; i
++)
393 b
= ((long double) (i
+ i
) / x
) * b
- a
;
394 GET_LDOUBLE_WORDS (se
, i0
, i1
, b
);
398 /* If B is +-Inf, set up errno accordingly. */
400 __set_errno (ERANGE
);
408 ret
= copysignl (LDBL_MAX
, ret
) * LDBL_MAX
;
411 libm_alias_finite (__ieee754_ynl
, __ynl
)