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