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
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
|
/* e_jnf.c -- float version of e_jn.c.
* Conversion to float 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.
* ====================================================
*/
#include <errno.h>
#include <float.h>
#include <math.h>
#include <math_private.h>
static const float
two = 2.0000000000e+00, /* 0x40000000 */
one = 1.0000000000e+00; /* 0x3F800000 */
static const float zero = 0.0000000000e+00;
float
__ieee754_jnf(int n, float x)
{
float ret;
{
int32_t i,hx,ix, sgn;
float a, b, temp, di;
float 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(__builtin_expect(ix>0x7f800000, 0)) 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);
SET_RESTORE_ROUNDF (FE_TONEAREST);
if(__builtin_expect(ix==0||ix>=0x7f800000, 0)) /* if x is 0 or inf */
return sgn == 1 ? -zero : zero;
else if((float)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*((double)(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*(float)0.5; b = temp;
for (a=one,i=2;i<=n;i++) {
a *= (float)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 */
float t,v;
float q0,q1,h,tmp; int32_t k,m;
w = (n+n)/(float)x; h = (float)2.0/(float)x;
q0 = w; z = w+h; q1 = w*z - (float)1.0; k=1;
while(q1<(float)1.0e9) {
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+01
* double 7.09782712893383973096e+02
* long double 1.1356523406294143949491931077970765006170e+04
* 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<(float)8.8721679688e+01) {
for(i=n-1,di=(float)(i+i);i>0;i--){
temp = b;
b *= di;
b = b/x - a;
a = temp;
di -= two;
}
} else {
for(i=n-1,di=(float)(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>(float)1e10) {
a /= b;
t /= b;
b = one;
}
}
}
/* j0() and j1() suffer enormous loss of precision at and
* near zero; however, we know that their zero points never
* coincide, so just choose the one further away from zero.
*/
z = __ieee754_j0f (x);
w = __ieee754_j1f (x);
if (fabsf (z) >= fabsf (w))
b = (t * z / b);
else
b = (t * w / a);
}
}
if(sgn==1) ret = -b; else ret = b;
ret = math_narrow_eval (ret);
}
if (ret == 0)
{
ret = math_narrow_eval (__copysignf (FLT_MIN, ret) * FLT_MIN);
__set_errno (ERANGE);
}
else
math_check_force_underflow (ret);
return ret;
}
strong_alias (__ieee754_jnf, __jnf_finite)
float
__ieee754_ynf(int n, float x)
{
float ret;
{
int32_t i,hx,ix;
uint32_t ib;
int32_t sign;
float a, b, temp;
GET_FLOAT_WORD(hx,x);
ix = 0x7fffffff&hx;
/* if Y(n,NaN) is NaN */
if(__builtin_expect(ix>0x7f800000, 0)) return x+x;
sign = 1;
if(n<0){
n = -n;
sign = 1 - ((n&1)<<1);
}
if(n==0) return(__ieee754_y0f(x));
if(__builtin_expect(ix==0, 0))
return -sign/zero;
if(__builtin_expect(hx<0, 0)) return zero/(zero*x);
SET_RESTORE_ROUNDF (FE_TONEAREST);
if(n==1) {
ret = sign*__ieee754_y1f(x);
goto out;
}
if(__builtin_expect(ix==0x7f800000, 0)) 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 = ((double)(i+i)/x)*b - a;
GET_FLOAT_WORD(ib,b);
a = temp;
}
/* If B is +-Inf, set up errno accordingly. */
if (! isfinite (b))
__set_errno (ERANGE);
if(sign>0) ret = b; else ret = -b;
}
out:
if (isinf (ret))
ret = __copysignf (FLT_MAX, ret) * FLT_MAX;
return ret;
}
strong_alias (__ieee754_ynf, __ynf_finite)
|
