e_jnf.c revision 97413 
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#ifndef lint
17static char rcsid[] = "$FreeBSD: head/lib/msun/src/e_jnf.c 97413 2002-05-28 18:15:04Z alfred $";
18#endif
19
20#include "math.h"
21#include "math_private.h"
22
23static const float
24invsqrtpi=  5.6418961287e-01, /* 0x3f106ebb */
25two   =  2.0000000000e+00, /* 0x40000000 */
26one   =  1.0000000000e+00; /* 0x3F800000 */
27
28static const float zero  =  0.0000000000e+00;
29
30float
31__ieee754_jnf(int n, float x)
32{
33	int32_t i,hx,ix, sgn;
34	float a, b, temp, di;
35	float z, w;
36
37    /* J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x)
38     * Thus, J(-n,x) = J(n,-x)
39     */
40	GET_FLOAT_WORD(hx,x);
41	ix = 0x7fffffff&hx;
42    /* if J(n,NaN) is NaN */
43	if(ix>0x7f800000) return x+x;
44	if(n<0){
45		n = -n;
46		x = -x;
47		hx ^= 0x80000000;
48	}
49	if(n==0) return(__ieee754_j0f(x));
50	if(n==1) return(__ieee754_j1f(x));
51	sgn = (n&1)&(hx>>31);	/* even n -- 0, odd n -- sign(x) */
52	x = fabsf(x);
53	if(ix==0||ix>=0x7f800000) 	/* if x is 0 or inf */
54	    b = zero;
55	else if((float)n<=x) {
56		/* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */
57	    a = __ieee754_j0f(x);
58	    b = __ieee754_j1f(x);
59	    for(i=1;i<n;i++){
60		temp = b;
61		b = b*((float)(i+i)/x) - a; /* avoid underflow */
62		a = temp;
63	    }
64	} else {
65	    if(ix<0x30800000) {	/* x < 2**-29 */
66    /* x is tiny, return the first Taylor expansion of J(n,x)
67     * J(n,x) = 1/n!*(x/2)^n  - ...
68     */
69		if(n>33)	/* underflow */
70		    b = zero;
71		else {
72		    temp = x*(float)0.5; b = temp;
73		    for (a=one,i=2;i<=n;i++) {
74			a *= (float)i;		/* a = n! */
75			b *= temp;		/* b = (x/2)^n */
76		    }
77		    b = b/a;
78		}
79	    } else {
80		/* use backward recurrence */
81		/* 			x      x^2      x^2
82		 *  J(n,x)/J(n-1,x) =  ----   ------   ------   .....
83		 *			2n  - 2(n+1) - 2(n+2)
84		 *
85		 * 			1      1        1
86		 *  (for large x)   =  ----  ------   ------   .....
87		 *			2n   2(n+1)   2(n+2)
88		 *			-- - ------ - ------ -
89		 *			 x     x         x
90		 *
91		 * Let w = 2n/x and h=2/x, then the above quotient
92		 * is equal to the continued fraction:
93		 *		    1
94		 *	= -----------------------
95		 *		       1
96		 *	   w - -----------------
97		 *			  1
98		 * 	        w+h - ---------
99		 *		       w+2h - ...
100		 *
101		 * To determine how many terms needed, let
102		 * Q(0) = w, Q(1) = w(w+h) - 1,
103		 * Q(k) = (w+k*h)*Q(k-1) - Q(k-2),
104		 * When Q(k) > 1e4	good for single
105		 * When Q(k) > 1e9	good for double
106		 * When Q(k) > 1e17	good for quadruple
107		 */
108	    /* determine k */
109		float t,v;
110		float q0,q1,h,tmp; int32_t k,m;
111		w  = (n+n)/(float)x; h = (float)2.0/(float)x;
112		q0 = w;  z = w+h; q1 = w*z - (float)1.0; k=1;
113		while(q1<(float)1.0e9) {
114			k += 1; z += h;
115			tmp = z*q1 - q0;
116			q0 = q1;
117			q1 = tmp;
118		}
119		m = n+n;
120		for(t=zero, i = 2*(n+k); i>=m; i -= 2) t = one/(i/x-t);
121		a = t;
122		b = one;
123		/*  estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n)
124		 *  Hence, if n*(log(2n/x)) > ...
125		 *  single 8.8722839355e+01
126		 *  double 7.09782712893383973096e+02
127		 *  long double 1.1356523406294143949491931077970765006170e+04
128		 *  then recurrent value may overflow and the result is
129		 *  likely underflow to zero
130		 */
131		tmp = n;
132		v = two/x;
133		tmp = tmp*__ieee754_logf(fabsf(v*tmp));
134		if(tmp<(float)8.8721679688e+01) {
135	    	    for(i=n-1,di=(float)(i+i);i>0;i--){
136		        temp = b;
137			b *= di;
138			b  = b/x - a;
139		        a = temp;
140			di -= two;
141	     	    }
142		} else {
143	    	    for(i=n-1,di=(float)(i+i);i>0;i--){
144		        temp = b;
145			b *= di;
146			b  = b/x - a;
147		        a = temp;
148			di -= two;
149		    /* scale b to avoid spurious overflow */
150			if(b>(float)1e10) {
151			    a /= b;
152			    t /= b;
153			    b  = one;
154			}
155	     	    }
156		}
157	    	b = (t*__ieee754_j0f(x)/b);
158	    }
159	}
160	if(sgn==1) return -b; else return b;
161}
162
163float
164__ieee754_ynf(int n, float x)
165{
166	int32_t i,hx,ix,ib;
167	int32_t sign;
168	float a, b, temp;
169
170	GET_FLOAT_WORD(hx,x);
171	ix = 0x7fffffff&hx;
172    /* if Y(n,NaN) is NaN */
173	if(ix>0x7f800000) return x+x;
174	if(ix==0) return -one/zero;
175	if(hx<0) return zero/zero;
176	sign = 1;
177	if(n<0){
178		n = -n;
179		sign = 1 - ((n&1)<<1);
180	}
181	if(n==0) return(__ieee754_y0f(x));
182	if(n==1) return(sign*__ieee754_y1f(x));
183	if(ix==0x7f800000) return zero;
184
185	a = __ieee754_y0f(x);
186	b = __ieee754_y1f(x);
187	/* quit if b is -inf */
188	GET_FLOAT_WORD(ib,b);
189	for(i=1;i<n&&ib!=0xff800000;i++){
190	    temp = b;
191	    b = ((float)(i+i)/x)*b - a;
192	    GET_FLOAT_WORD(ib,b);
193	    a = temp;
194	}
195	if(sign>0) return b; else return -b;
196}
197