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#if defined(LIBM_SCCS) && !defined(lint)
18__RCSID("$NetBSD: e_jnf.c,v 1.10 2009/01/19 05:58:27 lukem Exp $");
19#endif
20
21#include "math.h"
22#include "math_private.h"
23
24static const float
25#if 0
26invsqrtpi=  5.6418961287e-01, /* 0x3f106ebb */
27#endif
28two   =  2.0000000000e+00, /* 0x40000000 */
29one   =  1.0000000000e+00; /* 0x3F800000 */
30
31static const float zero  =  0.0000000000e+00;
32
33float
34__ieee754_jnf(int n, float x)
35{
36	int32_t i,hx,ix, sgn;
37	float a, b, temp, di;
38	float z, w;
39
40    /* J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x)
41     * Thus, J(-n,x) = J(n,-x)
42     */
43	GET_FLOAT_WORD(hx,x);
44	ix = 0x7fffffff&hx;
45    /* if J(n,NaN) is NaN */
46	if(ix>0x7f800000) return x+x;
47	if(n<0){
48		n = -n;
49		x = -x;
50		hx ^= 0x80000000;
51	}
52	if(n==0) return(__ieee754_j0f(x));
53	if(n==1) return(__ieee754_j1f(x));
54	sgn = (n&1)&(hx>>31);	/* even n -- 0, odd n -- sign(x) */
55	x = fabsf(x);
56	if(ix==0||ix>=0x7f800000) 	/* if x is 0 or inf */
57	    b = zero;
58	else if((float)n<=x) {
59		/* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */
60	    a = __ieee754_j0f(x);
61	    b = __ieee754_j1f(x);
62	    for(i=1;i<n;i++){
63		temp = b;
64		b = b*((float)(i+i)/x) - a; /* avoid underflow */
65		a = temp;
66	    }
67	} else {
68	    if(ix<0x30800000) {	/* x < 2**-29 */
69    /* x is tiny, return the first Taylor expansion of J(n,x)
70     * J(n,x) = 1/n!*(x/2)^n  - ...
71     */
72		if(n>33)	/* underflow */
73		    b = zero;
74		else {
75		    temp = x*(float)0.5; b = temp;
76		    for (a=one,i=2;i<=n;i++) {
77			a *= (float)i;		/* a = n! */
78			b *= temp;		/* b = (x/2)^n */
79		    }
80		    b = b/a;
81		}
82	    } else {
83		/* use backward recurrence */
84		/* 			x      x^2      x^2
85		 *  J(n,x)/J(n-1,x) =  ----   ------   ------   .....
86		 *			2n  - 2(n+1) - 2(n+2)
87		 *
88		 * 			1      1        1
89		 *  (for large x)   =  ----  ------   ------   .....
90		 *			2n   2(n+1)   2(n+2)
91		 *			-- - ------ - ------ -
92		 *			 x     x         x
93		 *
94		 * Let w = 2n/x and h=2/x, then the above quotient
95		 * is equal to the continued fraction:
96		 *		    1
97		 *	= -----------------------
98		 *		       1
99		 *	   w - -----------------
100		 *			  1
101		 * 	        w+h - ---------
102		 *		       w+2h - ...
103		 *
104		 * To determine how many terms needed, let
105		 * Q(0) = w, Q(1) = w(w+h) - 1,
106		 * Q(k) = (w+k*h)*Q(k-1) - Q(k-2),
107		 * When Q(k) > 1e4	good for single
108		 * When Q(k) > 1e9	good for double
109		 * When Q(k) > 1e17	good for quadruple
110		 */
111	    /* determine k */
112		float t,v;
113		float q0,q1,h,tmp; int32_t k,m;
114		w  = (n+n)/(float)x; h = (float)2.0/(float)x;
115		q0 = w;  z = w+h; q1 = w*z - (float)1.0; k=1;
116		while(q1<(float)1.0e9) {
117			k += 1; z += h;
118			tmp = z*q1 - q0;
119			q0 = q1;
120			q1 = tmp;
121		}
122		m = n+n;
123		for(t=zero, i = 2*(n+k); i>=m; i -= 2) t = one/(i/x-t);
124		a = t;
125		b = one;
126		/*  estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n)
127		 *  Hence, if n*(log(2n/x)) > ...
128		 *  single 8.8722839355e+01
129		 *  double 7.09782712893383973096e+02
130		 *  long double 1.1356523406294143949491931077970765006170e+04
131		 *  then recurrent value may overflow and the result is
132		 *  likely underflow to zero
133		 */
134		tmp = n;
135		v = two/x;
136		tmp = tmp*__ieee754_logf(fabsf(v*tmp));
137		if(tmp<(float)8.8721679688e+01) {
138	    	    for(i=n-1,di=(float)(i+i);i>0;i--){
139		        temp = b;
140			b *= di;
141			b  = b/x - a;
142		        a = temp;
143			di -= two;
144	     	    }
145		} else {
146	    	    for(i=n-1,di=(float)(i+i);i>0;i--){
147		        temp = b;
148			b *= di;
149			b  = b/x - a;
150		        a = temp;
151			di -= two;
152		    /* scale b to avoid spurious overflow */
153			if(b>(float)1e10) {
154			    a /= b;
155			    t /= b;
156			    b  = one;
157			}
158	     	    }
159		}
160		z = __ieee754_j0f(x);
161		w = __ieee754_j1f(x);
162		if (fabsf(z) >= fabsf(w))
163			b = (t*z/b);
164		else
165			b = (t*w/a);
166	    }
167	}
168	if(sgn==1) return -b; else return b;
169}
170
171float
172__ieee754_ynf(int n, float x)
173{
174	int32_t i,hx,ix,ib;
175	int32_t sign;
176	float a, b, temp;
177
178	GET_FLOAT_WORD(hx,x);
179	ix = 0x7fffffff&hx;
180    /* if Y(n,NaN) is NaN */
181	if(ix>0x7f800000) return x+x;
182	if(ix==0) return -one/zero;
183	if(hx<0) return zero/zero;
184	sign = 1;
185	if(n<0){
186		n = -n;
187		sign = 1 - ((n&1)<<1);
188	}
189	if(n==0) return(__ieee754_y0f(x));
190	if(n==1) return(sign*__ieee754_y1f(x));
191	if(ix==0x7f800000) return zero;
192
193	a = __ieee754_y0f(x);
194	b = __ieee754_y1f(x);
195	/* quit if b is -inf */
196	GET_FLOAT_WORD(ib,b);
197	for(i=1;i<n&&(uint32_t)ib!=0xff800000;i++){
198	    temp = b;
199	    b = ((float)(i+i)/x)*b - a;
200	    GET_FLOAT_WORD(ib,b);
201	    a = temp;
202	}
203	if(sign>0) return b; else return -b;
204}
205