1/* origin: FreeBSD /usr/src/lib/msun/src/e_jnf.c */
2/*
3 * Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com.
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#define _GNU_SOURCE
17#include "libm.h"
18
19float jnf(int n, float x)
20{
21	uint32_t ix;
22	int nm1, sign, i;
23	float a, b, temp;
24
25	GET_FLOAT_WORD(ix, x);
26	sign = ix>>31;
27	ix &= 0x7fffffff;
28	if (ix > 0x7f800000) /* nan */
29		return x;
30
31	/* J(-n,x) = J(n,-x), use |n|-1 to avoid overflow in -n */
32	if (n == 0)
33		return j0f(x);
34	if (n < 0) {
35		nm1 = -(n+1);
36		x = -x;
37		sign ^= 1;
38	} else
39		nm1 = n-1;
40	if (nm1 == 0)
41		return j1f(x);
42
43	sign &= n;  /* even n: 0, odd n: signbit(x) */
44	x = fabsf(x);
45	if (ix == 0 || ix == 0x7f800000)  /* if x is 0 or inf */
46		b = 0.0f;
47	else if (nm1 < x) {
48		/* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */
49		a = j0f(x);
50		b = j1f(x);
51		for (i=0; i<nm1; ){
52			i++;
53			temp = b;
54			b = b*(2.0f*i/x) - a;
55			a = temp;
56		}
57	} else {
58		if (ix < 0x35800000) { /* x < 2**-20 */
59			/* x is tiny, return the first Taylor expansion of J(n,x)
60			 * J(n,x) = 1/n!*(x/2)^n  - ...
61			 */
62			if (nm1 > 8)  /* underflow */
63				nm1 = 8;
64			temp = 0.5f * x;
65			b = temp;
66			a = 1.0f;
67			for (i=2; i<=nm1+1; i++) {
68				a *= (float)i;    /* a = n! */
69				b *= temp;        /* b = (x/2)^n */
70			}
71			b = b/a;
72		} else {
73			/* use backward recurrence */
74			/*                      x      x^2      x^2
75			 *  J(n,x)/J(n-1,x) =  ----   ------   ------   .....
76			 *                      2n  - 2(n+1) - 2(n+2)
77			 *
78			 *                      1      1        1
79			 *  (for large x)   =  ----  ------   ------   .....
80			 *                      2n   2(n+1)   2(n+2)
81			 *                      -- - ------ - ------ -
82			 *                       x     x         x
83			 *
84			 * Let w = 2n/x and h=2/x, then the above quotient
85			 * is equal to the continued fraction:
86			 *                  1
87			 *      = -----------------------
88			 *                     1
89			 *         w - -----------------
90			 *                        1
91			 *              w+h - ---------
92			 *                     w+2h - ...
93			 *
94			 * To determine how many terms needed, let
95			 * Q(0) = w, Q(1) = w(w+h) - 1,
96			 * Q(k) = (w+k*h)*Q(k-1) - Q(k-2),
97			 * When Q(k) > 1e4      good for single
98			 * When Q(k) > 1e9      good for double
99			 * When Q(k) > 1e17     good for quadruple
100			 */
101			/* determine k */
102			float t,q0,q1,w,h,z,tmp,nf;
103			int k;
104
105			nf = nm1+1.0f;
106			w = 2*nf/x;
107			h = 2/x;
108			z = w+h;
109			q0 = w;
110			q1 = w*z - 1.0f;
111			k = 1;
112			while (q1 < 1.0e4f) {
113				k += 1;
114				z += h;
115				tmp = z*q1 - q0;
116				q0 = q1;
117				q1 = tmp;
118			}
119			for (t=0.0f, i=k; i>=0; i--)
120				t = 1.0f/(2*(i+nf)/x-t);
121			a = t;
122			b = 1.0f;
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 = nf*logf(fabsf(w));
132			if (tmp < 88.721679688f) {
133				for (i=nm1; i>0; i--) {
134					temp = b;
135					b = 2.0f*i*b/x - a;
136					a = temp;
137				}
138			} else {
139				for (i=nm1; i>0; i--){
140					temp = b;
141					b = 2.0f*i*b/x - a;
142					a = temp;
143					/* scale b to avoid spurious overflow */
144					if (b > 0x1p60f) {
145						a /= b;
146						t /= b;
147						b = 1.0f;
148					}
149				}
150			}
151			z = j0f(x);
152			w = j1f(x);
153			if (fabsf(z) >= fabsf(w))
154				b = t*z/b;
155			else
156				b = t*w/a;
157		}
158	}
159	return sign ? -b : b;
160}
161
162float ynf(int n, float x)
163{
164	uint32_t ix, ib;
165	int nm1, sign, i;
166	float a, b, temp;
167
168	GET_FLOAT_WORD(ix, x);
169	sign = ix>>31;
170	ix &= 0x7fffffff;
171	if (ix > 0x7f800000) /* nan */
172		return x;
173	if (sign && ix != 0) /* x < 0 */
174		return 0/0.0f;
175	if (ix == 0x7f800000)
176		return 0.0f;
177
178	if (n == 0)
179		return y0f(x);
180	if (n < 0) {
181		nm1 = -(n+1);
182		sign = n&1;
183	} else {
184		nm1 = n-1;
185		sign = 0;
186	}
187	if (nm1 == 0)
188		return sign ? -y1f(x) : y1f(x);
189
190	a = y0f(x);
191	b = y1f(x);
192	/* quit if b is -inf */
193	GET_FLOAT_WORD(ib,b);
194	for (i = 0; i < nm1 && ib != 0xff800000; ) {
195		i++;
196		temp = b;
197		b = (2.0f*i/x)*b - a;
198		GET_FLOAT_WORD(ib, b);
199		a = temp;
200	}
201	return sign ? -b : b;
202}
203