1 /* @(#)e_jn.c 5.1 93/09/24 */
3 * ====================================================
4 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
6 * Developed at SunPro, a Sun Microsystems, Inc. business.
7 * Permission to use, copy, modify, and distribute this
8 * software is freely granted, provided that this notice
10 * ====================================================
14 * __ieee754_jn(n, x), __ieee754_yn(n, x)
15 * floating point Bessel's function of the 1st and 2nd kind
19 * y0(0)=y1(0)=yn(n,0) = -inf with overflow signal;
20 * y0(-ve)=y1(-ve)=yn(n,-ve) are NaN with invalid signal.
21 * Note 2. About jn(n,x), yn(n,x)
22 * For n=0, j0(x) is called,
23 * for n=1, j1(x) is called,
24 * for n<x, forward recursion us used starting
25 * from values of j0(x) and j1(x).
26 * for n>x, a continued fraction approximation to
27 * j(n,x)/j(n-1,x) is evaluated and then backward
28 * recursion is used starting from a supposed value
29 * for j(n,x). The resulting value of j(0,x) is
30 * compared with the actual value to correct the
31 * supposed value of j(n,x).
33 * yn(n,x) is similar in all respects, except
34 * that forward recursion is used for all
42 #include <math-narrow-eval.h>
43 #include <math_private.h>
44 #include <fenv_private.h>
45 #include <math-underflow.h>
46 #include <libm-alias-finite.h>
49 invsqrtpi
= 5.64189583547756279280e-01, /* 0x3FE20DD7, 0x50429B6D */
50 two
= 2.00000000000000000000e+00, /* 0x40000000, 0x00000000 */
51 one
= 1.00000000000000000000e+00; /* 0x3FF00000, 0x00000000 */
53 static const double zero
= 0.00000000000000000000e+00;
56 __ieee754_jn (int n
, double x
)
58 int32_t i
, hx
, ix
, lx
, sgn
;
59 double a
, b
, temp
, di
, ret
;
62 /* J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x)
63 * Thus, J(-n,x) = J(n,-x)
65 EXTRACT_WORDS (hx
, lx
, x
);
67 /* if J(n,NaN) is NaN */
68 if (__glibc_unlikely ((ix
| ((uint32_t) (lx
| -lx
)) >> 31) > 0x7ff00000))
77 return (__ieee754_j0 (x
));
79 return (__ieee754_j1 (x
));
80 sgn
= (n
& 1) & (hx
>> 31); /* even n -- 0, odd n -- sign(x) */
83 SET_RESTORE_ROUND (FE_TONEAREST
);
84 if (__glibc_unlikely ((ix
| lx
) == 0 || ix
>= 0x7ff00000))
85 /* if x is 0 or inf */
86 return sgn
== 1 ? -zero
: zero
;
87 else if ((double) n
<= x
)
89 /* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */
90 if (ix
>= 0x52D00000) /* x > 2**302 */
92 * Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
93 * Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
94 * Let s=sin(x), c=cos(x),
95 * xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
97 * n sin(xn)*sqt2 cos(xn)*sqt2
98 * ----------------------------------
106 __sincos (x
, &s
, &c
);
109 case 0: temp
= c
+ s
; break;
110 case 1: temp
= -c
+ s
; break;
111 case 2: temp
= -c
- s
; break;
112 case 3: temp
= c
- s
; break;
113 default: __builtin_unreachable ();
115 b
= invsqrtpi
* temp
/ sqrt (x
);
119 a
= __ieee754_j0 (x
);
120 b
= __ieee754_j1 (x
);
121 for (i
= 1; i
< n
; i
++)
124 b
= b
* ((double) (i
+ i
) / x
) - a
; /* avoid underflow */
131 if (ix
< 0x3e100000) /* x < 2**-29 */
132 { /* x is tiny, return the first Taylor expansion of J(n,x)
133 * J(n,x) = 1/n!*(x/2)^n - ...
135 if (n
> 33) /* underflow */
139 temp
= x
* 0.5; b
= temp
;
140 for (a
= one
, i
= 2; i
<= n
; i
++)
142 a
*= (double) i
; /* a = n! */
143 b
*= temp
; /* b = (x/2)^n */
150 /* use backward recurrence */
152 * J(n,x)/J(n-1,x) = ---- ------ ------ .....
153 * 2n - 2(n+1) - 2(n+2)
156 * (for large x) = ---- ------ ------ .....
158 * -- - ------ - ------ -
161 * Let w = 2n/x and h=2/x, then the above quotient
162 * is equal to the continued fraction:
164 * = -----------------------
166 * w - -----------------
171 * To determine how many terms needed, let
172 * Q(0) = w, Q(1) = w(w+h) - 1,
173 * Q(k) = (w+k*h)*Q(k-1) - Q(k-2),
174 * When Q(k) > 1e4 good for single
175 * When Q(k) > 1e9 good for double
176 * When Q(k) > 1e17 good for quadruple
180 double q0
, q1
, h
, tmp
; int32_t k
, m
;
181 w
= (n
+ n
) / (double) x
; h
= 2.0 / (double) x
;
182 q0
= w
; z
= w
+ h
; q1
= w
* z
- 1.0; k
= 1;
191 for (t
= zero
, i
= 2 * (n
+ k
); i
>= m
; i
-= 2)
192 t
= one
/ (i
/ x
- t
);
195 /* estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n)
196 * Hence, if n*(log(2n/x)) > ...
197 * single 8.8722839355e+01
198 * double 7.09782712893383973096e+02
199 * long double 1.1356523406294143949491931077970765006170e+04
200 * then recurrent value may overflow and the result is
201 * likely underflow to zero
205 tmp
= tmp
* __ieee754_log (fabs (v
* tmp
));
206 if (tmp
< 7.09782712893383973096e+02)
208 for (i
= n
- 1, di
= (double) (i
+ i
); i
> 0; i
--)
219 for (i
= n
- 1, di
= (double) (i
+ i
); i
> 0; i
--)
226 /* scale b to avoid spurious overflow */
235 /* j0() and j1() suffer enormous loss of precision at and
236 * near zero; however, we know that their zero points never
237 * coincide, so just choose the one further away from zero.
239 z
= __ieee754_j0 (x
);
240 w
= __ieee754_j1 (x
);
241 if (fabs (z
) >= fabs (w
))
251 ret
= math_narrow_eval (ret
);
255 ret
= math_narrow_eval (copysign (DBL_MIN
, ret
) * DBL_MIN
);
256 __set_errno (ERANGE
);
259 math_check_force_underflow (ret
);
262 libm_alias_finite (__ieee754_jn
, __jn
)
265 __ieee754_yn (int n
, double x
)
267 int32_t i
, hx
, ix
, lx
;
269 double a
, b
, temp
, ret
;
271 EXTRACT_WORDS (hx
, lx
, x
);
272 ix
= 0x7fffffff & hx
;
273 /* if Y(n,NaN) is NaN */
274 if (__glibc_unlikely ((ix
| ((uint32_t) (lx
| -lx
)) >> 31) > 0x7ff00000))
280 sign
= 1 - ((n
& 1) << 1);
283 return (__ieee754_y0 (x
));
284 if (__glibc_unlikely ((ix
| lx
) == 0))
286 /* -inf and overflow exception. */;
287 if (__glibc_unlikely (hx
< 0))
288 return zero
/ (zero
* x
);
290 SET_RESTORE_ROUND (FE_TONEAREST
);
293 ret
= sign
* __ieee754_y1 (x
);
296 if (__glibc_unlikely (ix
== 0x7ff00000))
298 if (ix
>= 0x52D00000) /* x > 2**302 */
300 * Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
301 * Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
302 * Let s=sin(x), c=cos(x),
303 * xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
305 * n sin(xn)*sqt2 cos(xn)*sqt2
306 * ----------------------------------
314 __sincos (x
, &s
, &c
);
317 case 0: temp
= s
- c
; break;
318 case 1: temp
= -s
- c
; break;
319 case 2: temp
= -s
+ c
; break;
320 case 3: temp
= s
+ c
; break;
321 default: __builtin_unreachable ();
323 b
= invsqrtpi
* temp
/ sqrt (x
);
328 a
= __ieee754_y0 (x
);
329 b
= __ieee754_y1 (x
);
330 /* quit if b is -inf */
331 GET_HIGH_WORD (high
, b
);
332 for (i
= 1; i
< n
&& high
!= 0xfff00000; i
++)
335 b
= ((double) (i
+ i
) / x
) * b
- a
;
336 GET_HIGH_WORD (high
, b
);
339 /* If B is +-Inf, set up errno accordingly. */
341 __set_errno (ERANGE
);
350 ret
= copysign (DBL_MAX
, ret
) * DBL_MAX
;
353 libm_alias_finite (__ieee754_yn
, __yn
)