e_jnf.c revision 176451 
1/* e_jnf.c -- float version of e_jn.c.
2 * Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com.
3 */
4
5/*
6 * ====================================================
7 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
8 *
9 * Developed at SunPro, a Sun Microsystems, Inc. business.
10 * Permission to use, copy, modify, and distribute this
11 * software is freely granted, provided that this notice
12 * is preserved.
13 * ====================================================
14 */
15
16#include <sys/cdefs.h>
17__FBSDID("$FreeBSD: head/lib/msun/src/e_jnf.c 176451 2008-02-22 02:30:36Z das $");
18
19#include "math.h"
20#include "math_private.h"
21
22static const float
23invsqrtpi=  5.6418961287e-01, /* 0x3f106ebb */
24two   =  2.0000000000e+00, /* 0x40000000 */
25one   =  1.0000000000e+00; /* 0x3F800000 */
26
27static const float zero  =  0.0000000000e+00;
28
29float
30__ieee754_jnf(int n, float x)
31{
32	int32_t i,hx,ix, sgn;
33	float a, b, temp, di;
34	float z, w;
35
36    /* J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x)
37     * Thus, J(-n,x) = J(n,-x)
38     */
39	GET_FLOAT_WORD(hx,x);
40	ix = 0x7fffffff&hx;
41    /* if J(n,NaN) is NaN */
42	if(ix>0x7f800000) return x+x;
43	if(n<0){
44		n = -n;
45		x = -x;
46		hx ^= 0x80000000;
47	}
48	if(n==0) return(__ieee754_j0f(x));
49	if(n==1) return(__ieee754_j1f(x));
50	sgn = (n&1)&(hx>>31);	/* even n -- 0, odd n -- sign(x) */
51	x = fabsf(x);
52	if(ix==0||ix>=0x7f800000) 	/* if x is 0 or inf */
53	    b = zero;
54	else if((float)n<=x) {
55		/* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */
56	    a = __ieee754_j0f(x);
57	    b = __ieee754_j1f(x);
58	    for(i=1;i<n;i++){
59		temp = b;
60		b = b*((float)(i+i)/x) - a; /* avoid underflow */
61		a = temp;
62	    }
63	} else {
64	    if(ix<0x30800000) {	/* x < 2**-29 */
65    /* x is tiny, return the first Taylor expansion of J(n,x)
66     * J(n,x) = 1/n!*(x/2)^n  - ...
67     */
68		if(n>33)	/* underflow */
69		    b = zero;
70		else {
71		    temp = x*(float)0.5; b = temp;
72		    for (a=one,i=2;i<=n;i++) {
73			a *= (float)i;		/* a = n! */
74			b *= temp;		/* b = (x/2)^n */
75		    }
76		    b = b/a;
77		}
78	    } else {
79		/* use backward recurrence */
80		/* 			x      x^2      x^2
81		 *  J(n,x)/J(n-1,x) =  ----   ------   ------   .....
82		 *			2n  - 2(n+1) - 2(n+2)
83		 *
84		 * 			1      1        1
85		 *  (for large x)   =  ----  ------   ------   .....
86		 *			2n   2(n+1)   2(n+2)
87		 *			-- - ------ - ------ -
88		 *			 x     x         x
89		 *
90		 * Let w = 2n/x and h=2/x, then the above quotient
91		 * is equal to the continued fraction:
92		 *		    1
93		 *	= -----------------------
94		 *		       1
95		 *	   w - -----------------
96		 *			  1
97		 * 	        w+h - ---------
98		 *		       w+2h - ...
99		 *
100		 * To determine how many terms needed, let
101		 * Q(0) = w, Q(1) = w(w+h) - 1,
102		 * Q(k) = (w+k*h)*Q(k-1) - Q(k-2),
103		 * When Q(k) > 1e4	good for single
104		 * When Q(k) > 1e9	good for double
105		 * When Q(k) > 1e17	good for quadruple
106		 */
107	    /* determine k */
108		float t,v;
109		float q0,q1,h,tmp; int32_t k,m;
110		w  = (n+n)/(float)x; h = (float)2.0/(float)x;
111		q0 = w;  z = w+h; q1 = w*z - (float)1.0; k=1;
112		while(q1<(float)1.0e9) {
113			k += 1; z += h;
114			tmp = z*q1 - q0;
115			q0 = q1;
116			q1 = tmp;
117		}
118		m = n+n;
119		for(t=zero, i = 2*(n+k); i>=m; i -= 2) t = one/(i/x-t);
120		a = t;
121		b = one;
122		/*  estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n)
123		 *  Hence, if n*(log(2n/x)) > ...
124		 *  single 8.8722839355e+01
125		 *  double 7.09782712893383973096e+02
126		 *  long double 1.1356523406294143949491931077970765006170e+04
127		 *  then recurrent value may overflow and the result is
128		 *  likely underflow to zero
129		 */
130		tmp = n;
131		v = two/x;
132		tmp = tmp*__ieee754_logf(fabsf(v*tmp));
133		if(tmp<(float)8.8721679688e+01) {
134	    	    for(i=n-1,di=(float)(i+i);i>0;i--){
135		        temp = b;
136			b *= di;
137			b  = b/x - a;
138		        a = temp;
139			di -= two;
140	     	    }
141		} else {
142	    	    for(i=n-1,di=(float)(i+i);i>0;i--){
143		        temp = b;
144			b *= di;
145			b  = b/x - a;
146		        a = temp;
147			di -= two;
148		    /* scale b to avoid spurious overflow */
149			if(b>(float)1e10) {
150			    a /= b;
151			    t /= b;
152			    b  = one;
153			}
154	     	    }
155		}
156	    	b = (t*__ieee754_j0f(x)/b);
157	    }
158	}
159	if(sgn==1) return -b; else return b;
160}
161
162float
163__ieee754_ynf(int n, float x)
164{
165	int32_t i,hx,ix,ib;
166	int32_t sign;
167	float a, b, temp;
168
169	GET_FLOAT_WORD(hx,x);
170	ix = 0x7fffffff&hx;
171    /* if Y(n,NaN) is NaN */
172	if(ix>0x7f800000) return x+x;
173	if(ix==0) return -one/zero;
174	if(hx<0) return zero/zero;
175	sign = 1;
176	if(n<0){
177		n = -n;
178		sign = 1 - ((n&1)<<1);
179	}
180	if(n==0) return(__ieee754_y0f(x));
181	if(n==1) return(sign*__ieee754_y1f(x));
182	if(ix==0x7f800000) return zero;
183
184	a = __ieee754_y0f(x);
185	b = __ieee754_y1f(x);
186	/* quit if b is -inf */
187	GET_FLOAT_WORD(ib,b);
188	for(i=1;i<n&&ib!=0xff800000;i++){
189	    temp = b;
190	    b = ((float)(i+i)/x)*b - a;
191	    GET_FLOAT_WORD(ib,b);
192	    a = temp;
193	}
194	if(sign>0) return b; else return -b;
195}
196