e_jnf.c revision 22993 
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[] = "$Id$";
18#endif
19
20#include "math.h"
21#include "math_private.h"
22
23#ifdef __STDC__
24static const float
25#else
26static float
27#endif
28invsqrtpi=  5.6418961287e-01, /* 0x3f106ebb */
29two   =  2.0000000000e+00, /* 0x40000000 */
30one   =  1.0000000000e+00; /* 0x3F800000 */
31
32#ifdef __STDC__
33static const float zero  =  0.0000000000e+00;
34#else
35static float zero  =  0.0000000000e+00;
36#endif
37
38#ifdef __STDC__
39	float __ieee754_jnf(int n, float x)
40#else
41	float __ieee754_jnf(n,x)
42	int n; float x;
43#endif
44{
45	int32_t i,hx,ix, sgn;
46	float a, b, temp, di;
47	float z, w;
48
49    /* J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x)
50     * Thus, J(-n,x) = J(n,-x)
51     */
52	GET_FLOAT_WORD(hx,x);
53	ix = 0x7fffffff&hx;
54    /* if J(n,NaN) is NaN */
55	if(ix>0x7f800000) return x+x;
56	if(n<0){
57		n = -n;
58		x = -x;
59		hx ^= 0x80000000;
60	}
61	if(n==0) return(__ieee754_j0f(x));
62	if(n==1) return(__ieee754_j1f(x));
63	sgn = (n&1)&(hx>>31);	/* even n -- 0, odd n -- sign(x) */
64	x = fabsf(x);
65	if(ix==0||ix>=0x7f800000) 	/* if x is 0 or inf */
66	    b = zero;
67	else if((float)n<=x) {
68		/* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */
69	    a = __ieee754_j0f(x);
70	    b = __ieee754_j1f(x);
71	    for(i=1;i<n;i++){
72		temp = b;
73		b = b*((float)(i+i)/x) - a; /* avoid underflow */
74		a = temp;
75	    }
76	} else {
77	    if(ix<0x30800000) {	/* x < 2**-29 */
78    /* x is tiny, return the first Taylor expansion of J(n,x)
79     * J(n,x) = 1/n!*(x/2)^n  - ...
80     */
81		if(n>33)	/* underflow */
82		    b = zero;
83		else {
84		    temp = x*(float)0.5; b = temp;
85		    for (a=one,i=2;i<=n;i++) {
86			a *= (float)i;		/* a = n! */
87			b *= temp;		/* b = (x/2)^n */
88		    }
89		    b = b/a;
90		}
91	    } else {
92		/* use backward recurrence */
93		/* 			x      x^2      x^2
94		 *  J(n,x)/J(n-1,x) =  ----   ------   ------   .....
95		 *			2n  - 2(n+1) - 2(n+2)
96		 *
97		 * 			1      1        1
98		 *  (for large x)   =  ----  ------   ------   .....
99		 *			2n   2(n+1)   2(n+2)
100		 *			-- - ------ - ------ -
101		 *			 x     x         x
102		 *
103		 * Let w = 2n/x and h=2/x, then the above quotient
104		 * is equal to the continued fraction:
105		 *		    1
106		 *	= -----------------------
107		 *		       1
108		 *	   w - -----------------
109		 *			  1
110		 * 	        w+h - ---------
111		 *		       w+2h - ...
112		 *
113		 * To determine how many terms needed, let
114		 * Q(0) = w, Q(1) = w(w+h) - 1,
115		 * Q(k) = (w+k*h)*Q(k-1) - Q(k-2),
116		 * When Q(k) > 1e4	good for single
117		 * When Q(k) > 1e9	good for double
118		 * When Q(k) > 1e17	good for quadruple
119		 */
120	    /* determine k */
121		float t,v;
122		float q0,q1,h,tmp; int32_t k,m;
123		w  = (n+n)/(float)x; h = (float)2.0/(float)x;
124		q0 = w;  z = w+h; q1 = w*z - (float)1.0; k=1;
125		while(q1<(float)1.0e9) {
126			k += 1; z += h;
127			tmp = z*q1 - q0;
128			q0 = q1;
129			q1 = tmp;
130		}
131		m = n+n;
132		for(t=zero, i = 2*(n+k); i>=m; i -= 2) t = one/(i/x-t);
133		a = t;
134		b = one;
135		/*  estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n)
136		 *  Hence, if n*(log(2n/x)) > ...
137		 *  single 8.8722839355e+01
138		 *  double 7.09782712893383973096e+02
139		 *  long double 1.1356523406294143949491931077970765006170e+04
140		 *  then recurrent value may overflow and the result is
141		 *  likely underflow to zero
142		 */
143		tmp = n;
144		v = two/x;
145		tmp = tmp*__ieee754_logf(fabsf(v*tmp));
146		if(tmp<(float)8.8721679688e+01) {
147	    	    for(i=n-1,di=(float)(i+i);i>0;i--){
148		        temp = b;
149			b *= di;
150			b  = b/x - a;
151		        a = temp;
152			di -= two;
153	     	    }
154		} else {
155	    	    for(i=n-1,di=(float)(i+i);i>0;i--){
156		        temp = b;
157			b *= di;
158			b  = b/x - a;
159		        a = temp;
160			di -= two;
161		    /* scale b to avoid spurious overflow */
162			if(b>(float)1e10) {
163			    a /= b;
164			    t /= b;
165			    b  = one;
166			}
167	     	    }
168		}
169	    	b = (t*__ieee754_j0f(x)/b);
170	    }
171	}
172	if(sgn==1) return -b; else return b;
173}
174
175#ifdef __STDC__
176	float __ieee754_ynf(int n, float x)
177#else
178	float __ieee754_ynf(n,x)
179	int n; float x;
180#endif
181{
182	int32_t i,hx,ix,ib;
183	int32_t sign;
184	float a, b, temp;
185
186	GET_FLOAT_WORD(hx,x);
187	ix = 0x7fffffff&hx;
188    /* if Y(n,NaN) is NaN */
189	if(ix>0x7f800000) return x+x;
190	if(ix==0) return -one/zero;
191	if(hx<0) return zero/zero;
192	sign = 1;
193	if(n<0){
194		n = -n;
195		sign = 1 - ((n&1)<<1);
196	}
197	if(n==0) return(__ieee754_y0f(x));
198	if(n==1) return(sign*__ieee754_y1f(x));
199	if(ix==0x7f800000) return zero;
200
201	a = __ieee754_y0f(x);
202	b = __ieee754_y1f(x);
203	/* quit if b is -inf */
204	GET_FLOAT_WORD(ib,b);
205	for(i=1;i<n&&ib!=0xff800000;i++){
206	    temp = b;
207	    b = ((float)(i+i)/x)*b - a;
208	    GET_FLOAT_WORD(ib,b);
209	    a = temp;
210	}
211	if(sign>0) return b; else return -b;
212}
213