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 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
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 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).
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 _Float128
68 invsqrtpi
= L(5.6418958354775628694807945156077258584405E-1),
75 __ieee754_jnl (int n
, _Float128 x
)
79 _Float128 a
, b
, temp
, di
, ret
;
81 ieee854_long_double_shape_type u
;
84 /* J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x)
85 * Thus, J(-n,x) = J(n,-x)
92 /* if J(n,NaN) is NaN */
95 if ((u
.parts32
.w0
& 0xffff) | u
.parts32
.w1
| u
.parts32
.w2
| u
.parts32
.w3
)
106 return (__ieee754_j0l (x
));
108 return (__ieee754_j1l (x
));
109 sgn
= (n
& 1) & (se
>> 31); /* even n -- 0, odd n -- sign(x) */
113 SET_RESTORE_ROUNDL (FE_TONEAREST
);
114 if (x
== 0 || ix
>= 0x7fff0000) /* if x is 0 or inf */
115 return sgn
== 1 ? -zero
: zero
;
116 else if ((_Float128
) n
<= x
)
118 /* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */
119 if (ix
>= 0x412D0000)
122 /* ??? Could use an expansion for large x here. */
125 * Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
126 * Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
127 * Let s=sin(x), c=cos(x),
128 * xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
130 * n sin(xn)*sqt2 cos(xn)*sqt2
131 * ----------------------------------
139 __sincosl (x
, &s
, &c
);
155 __builtin_unreachable ();
157 b
= invsqrtpi
* temp
/ sqrtl (x
);
161 a
= __ieee754_j0l (x
);
162 b
= __ieee754_j1l (x
);
163 for (i
= 1; i
< n
; i
++)
166 b
= b
* ((_Float128
) (i
+ i
) / x
) - a
; /* avoid underflow */
175 /* x is tiny, return the first Taylor expansion of J(n,x)
176 * J(n,x) = 1/n!*(x/2)^n - ...
178 if (n
>= 400) /* underflow, result < 10^-4952 */
184 for (a
= one
, i
= 2; i
<= n
; i
++)
186 a
*= (_Float128
) i
; /* a = n! */
187 b
*= temp
; /* b = (x/2)^n */
194 /* use backward recurrence */
196 * J(n,x)/J(n-1,x) = ---- ------ ------ .....
197 * 2n - 2(n+1) - 2(n+2)
200 * (for large x) = ---- ------ ------ .....
202 * -- - ------ - ------ -
205 * Let w = 2n/x and h=2/x, then the above quotient
206 * is equal to the continued fraction:
208 * = -----------------------
210 * w - -----------------
215 * To determine how many terms needed, let
216 * Q(0) = w, Q(1) = w(w+h) - 1,
217 * Q(k) = (w+k*h)*Q(k-1) - Q(k-2),
218 * When Q(k) > 1e4 good for single
219 * When Q(k) > 1e9 good for double
220 * When Q(k) > 1e17 good for quadruple
224 _Float128 q0
, q1
, h
, tmp
;
226 w
= (n
+ n
) / (_Float128
) x
;
227 h
= 2 / (_Float128
) x
;
232 while (q1
< L(1.0e17
))
241 for (t
= zero
, i
= 2 * (n
+ k
); i
>= m
; i
-= 2)
242 t
= one
/ (i
/ x
- t
);
245 /* estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n)
246 * Hence, if n*(log(2n/x)) > ...
247 * single 8.8722839355e+01
248 * double 7.09782712893383973096e+02
249 * long double 1.1356523406294143949491931077970765006170e+04
250 * then recurrent value may overflow and the result is
251 * likely underflow to zero
255 tmp
= tmp
* __ieee754_logl (fabsl (v
* tmp
));
257 if (tmp
< L(1.1356523406294143949491931077970765006170e+04))
259 for (i
= n
- 1, di
= (_Float128
) (i
+ i
); i
> 0; i
--)
270 for (i
= n
- 1, di
= (_Float128
) (i
+ i
); i
> 0; i
--)
277 /* scale b to avoid spurious overflow */
286 /* j0() and j1() suffer enormous loss of precision at and
287 * near zero; however, we know that their zero points never
288 * coincide, so just choose the one further away from zero.
290 z
= __ieee754_j0l (x
);
291 w
= __ieee754_j1l (x
);
292 if (fabsl (z
) >= fabsl (w
))
305 ret
= copysignl (LDBL_MIN
, ret
) * LDBL_MIN
;
306 __set_errno (ERANGE
);
309 math_check_force_underflow (ret
);
312 libm_alias_finite (__ieee754_jnl
, __jnl
)
315 __ieee754_ynl (int n
, _Float128 x
)
320 _Float128 a
, b
, temp
, ret
;
321 ieee854_long_double_shape_type u
;
325 ix
= se
& 0x7fffffff;
327 /* if Y(n,NaN) is NaN */
328 if (ix
>= 0x7fff0000)
330 if ((u
.parts32
.w0
& 0xffff) | u
.parts32
.w1
| u
.parts32
.w2
| u
.parts32
.w3
)
336 return ((n
< 0 && (n
& 1) != 0) ? 1 : -1) / L(0.0);
338 return zero
/ (zero
* x
);
344 sign
= 1 - ((n
& 1) << 1);
347 return (__ieee754_y0l (x
));
349 SET_RESTORE_ROUNDL (FE_TONEAREST
);
352 ret
= sign
* __ieee754_y1l (x
);
355 if (ix
>= 0x7fff0000)
357 if (ix
>= 0x412D0000)
360 /* ??? See comment above on the possible futility of this. */
363 * Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
364 * Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
365 * Let s=sin(x), c=cos(x),
366 * xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
368 * n sin(xn)*sqt2 cos(xn)*sqt2
369 * ----------------------------------
377 __sincosl (x
, &s
, &c
);
393 __builtin_unreachable ();
395 b
= invsqrtpi
* temp
/ sqrtl (x
);
399 a
= __ieee754_y0l (x
);
400 b
= __ieee754_y1l (x
);
401 /* quit if b is -inf */
403 se
= u
.parts32
.w0
& 0xffff0000;
404 for (i
= 1; i
< n
&& se
!= 0xffff0000; i
++)
407 b
= ((_Float128
) (i
+ i
) / x
) * b
- a
;
409 se
= u
.parts32
.w0
& 0xffff0000;
413 /* If B is +-Inf, set up errno accordingly. */
415 __set_errno (ERANGE
);
423 ret
= copysignl (LDBL_MAX
, ret
) * LDBL_MAX
;
426 libm_alias_finite (__ieee754_ynl
, __ynl
)