1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
|
/* See the import.pl script for potential modifications */
/* e_jnf.c -- Simple version of e_jn.c.
* Conversion to Simple by Ian Lance Taylor, Cygnus Support, ian@cygnus.com.
*/
/*
* ====================================================
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
*
* Developed at SunPro, a Sun Microsystems, Inc. business.
* Permission to use, copy, modify, and distribute this
* software is freely granted, provided that this notice
* is preserved.
* ====================================================
*/
#if defined(LIBM_SCCS) && !defined(lint)
static char rcsid[] = "$NetBSD: e_jnf.c,v 1.5f 1995/05/10 20:45:37 jtc Exp $";
#endif
#include "math.h"
#include "math_private.h"
namespace streflop_libm {
#ifdef __STDC__
static const Simple
#else
static Simple
#endif
two = 2.0000000000e+00f, /* 0x40000000 */
one = 1.0000000000e+00f; /* 0x3F800000 */
#ifdef __STDC__
static const Simple zero = 0.0000000000e+00f;
#else
static Simple zero = 0.0000000000e+00f;
#endif
#ifdef __STDC__
Simple __ieee754_jnf(int n, Simple x)
#else
Simple __ieee754_jnf(n,x)
int n; Simple x;
#endif
{
int32_t i,hx,ix, sgn;
Simple a, b, temp, di;
Simple z, w;
/* J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x)
* Thus, J(-n,x) = J(n,-x)
*/
GET_FLOAT_WORD(hx,x);
ix = 0x7fffffff&hx;
/* if J(n,NaN) is NaN */
if(ix>0x7f800000) return x+x;
if(n<0){
n = -n;
x = -x;
hx ^= 0x80000000;
}
if(n==0) return(__ieee754_j0f(x));
if(n==1) return(__ieee754_j1f(x));
sgn = (n&1)&(hx>>31); /* even n -- 0, odd n -- sign(x) */
x = fabsf(x);
if(ix==0||ix>=0x7f800000) /* if x is 0 or inf */
b = zero;
else if((Simple)n<=x) {
/* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */
a = __ieee754_j0f(x);
b = __ieee754_j1f(x);
for(i=1;i<n;i++){
temp = b;
b = b*((Simple)(i+i)/x) - a; /* avoid underflow */
a = temp;
}
} else {
if(ix<0x30800000) { /* x < 2**-29 */
/* x is tiny, return the first Taylor expansion of J(n,x)
* J(n,x) = 1/n!*(x/2)^n - ...
*/
if(n>33) /* underflow */
b = zero;
else {
temp = x*(Simple)0.5f; b = temp;
for (a=one,i=2;i<=n;i++) {
a *= (Simple)i; /* a = n! */
b *= temp; /* b = (x/2)^n */
}
b = b/a;
}
} else {
/* use backward recurrence */
/* x x^2 x^2
* J(n,x)/J(n-1,x) = ---- ------ ------ .....
* 2n - 2(n+1) - 2(n+2)
*
* 1 1 1
* (for large x) = ---- ------ ------ .....
* 2n 2(n+1) 2(n+2)
* -- - ------ - ------ -
* x x x
*
* Let w = 2n/x and h=2/x, then the above quotient
* is equal to the continued fraction:
* 1
* = -----------------------
* 1
* w - -----------------
* 1
* w+h - ---------
* w+2h - ...
*
* To determine how many terms needed, let
* Q(0) = w, Q(1) = w(w+h) - 1,
* Q(k) = (w+k*h)*Q(k-1) - Q(k-2),
* When Q(k) > 1e4 good for single
* When Q(k) > 1e9 good for Double
* When Q(k) > 1e17 good for quadruple
*/
/* determine k */
Simple t,v;
Simple q0,q1,h,tmp; int32_t k,m;
w = (n+n)/(Simple)x; h = (Simple)2.0f/(Simple)x;
q0 = w; z = w+h; q1 = w*z - (Simple)1.0f; k=1;
while(q1<(Simple)1.0e9f) {
k += 1; z += h;
tmp = z*q1 - q0;
q0 = q1;
q1 = tmp;
}
m = n+n;
for(t=zero, i = 2*(n+k); i>=m; i -= 2) t = one/(i/x-t);
a = t;
b = one;
/* estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n)
* Hence, if n*(log(2n/x)) > ...
* single 8.8722839355e+01f
* Double 7.09782712893383973096e+02f
* Extended 1.1356523406294143949491931077970765006170e+04f
* then recurrent value may overflow and the result is
* likely underflow to zero
*/
tmp = n;
v = two/x;
tmp = tmp*__ieee754_logf(fabsf(v*tmp));
if(tmp<(Simple)8.8721679688e+01f) {
for(i=n-1,di=(Simple)(i+i);i>0;i--){
temp = b;
b *= di;
b = b/x - a;
a = temp;
di -= two;
}
} else {
for(i=n-1,di=(Simple)(i+i);i>0;i--){
temp = b;
b *= di;
b = b/x - a;
a = temp;
di -= two;
/* scale b to avoid spurious overflow */
if(b>(Simple)1e10f) {
a /= b;
t /= b;
b = one;
}
}
}
b = (t*__ieee754_j0f(x)/b);
}
}
if(sgn==1) return -b; else return b;
}
#ifdef __STDC__
Simple __ieee754_ynf(int n, Simple x)
#else
Simple __ieee754_ynf(n,x)
int n; Simple x;
#endif
{
int32_t i,hx,ix;
u_int32_t ib;
int32_t sign;
Simple a, b, temp;
GET_FLOAT_WORD(hx,x);
ix = 0x7fffffff&hx;
/* if Y(n,NaN) is NaN */
if(ix>0x7f800000) return x+x;
if(ix==0) return -HUGE_VALF+x; /* -inf and overflow exception. */
if(hx<0) return zero/(zero*x);
sign = 1;
if(n<0){
n = -n;
sign = 1 - ((n&1)<<1);
}
if(n==0) return(__ieee754_y0f(x));
if(n==1) return(sign*__ieee754_y1f(x));
if(ix==0x7f800000) return zero;
a = __ieee754_y0f(x);
b = __ieee754_y1f(x);
/* quit if b is -inf */
GET_FLOAT_WORD(ib,b);
for(i=1;i<n&&ib!=0xff800000;i++){
temp = b;
b = ((Simple)(i+i)/x)*b - a;
GET_FLOAT_WORD(ib,b);
a = temp;
}
if(sign>0) return b; else return -b;
}
}
|
