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 /*
17 * See e_jn.c for complete comments.
18 */
19
20 #include "math.h"
21 #include "math_private.h"
22
23 static const volatile float vone = 1, vzero = 0;
24
25 static const float
26 two = 2.0000000000e+00, /* 0x40000000 */
27 one = 1.0000000000e+00; /* 0x3F800000 */
28
29 static const float zero = 0.0000000000e+00;
30
31 float
jnf(int n,float x)32 jnf(int n, float x)
33 {
34 int32_t i,hx,ix, sgn;
35 float a, b, temp, di;
36 float z, w;
37
38 /* J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x)
39 * Thus, J(-n,x) = J(n,-x)
40 */
41 GET_FLOAT_WORD(hx,x);
42 ix = 0x7fffffff&hx;
43 /* if J(n,NaN) is NaN */
44 if(ix>0x7f800000) return x+x;
45 if(n<0){
46 n = -n;
47 x = -x;
48 hx ^= 0x80000000;
49 }
50 if(n==0) return(j0f(x));
51 if(n==1) return(j1f(x));
52 sgn = (n&1)&(hx>>31); /* even n -- 0, odd n -- sign(x) */
53 x = fabsf(x);
54 if(ix==0||ix>=0x7f800000) /* if x is 0 or inf */
55 b = zero;
56 else if((float)n<=x) {
57 /* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */
58 a = j0f(x);
59 b = j1f(x);
60 for(i=1;i<n;i++){
61 temp = b;
62 b = b*((float)(i+i)/x) - a; /* avoid underflow */
63 a = temp;
64 }
65 } else {
66 if(ix<0x30800000) { /* x < 2**-29 */
67 /* x is tiny, return the first Taylor expansion of J(n,x)
68 * J(n,x) = 1/n!*(x/2)^n - ...
69 */
70 if(n>33) /* underflow */
71 b = zero;
72 else {
73 temp = x*(float)0.5; b = temp;
74 for (a=one,i=2;i<=n;i++) {
75 a *= (float)i; /* a = n! */
76 b *= temp; /* b = (x/2)^n */
77 }
78 b = b/a;
79 }
80 } else {
81 /* use backward recurrence */
82 /* x x^2 x^2
83 * J(n,x)/J(n-1,x) = ---- ------ ------ .....
84 * 2n - 2(n+1) - 2(n+2)
85 *
86 * 1 1 1
87 * (for large x) = ---- ------ ------ .....
88 * 2n 2(n+1) 2(n+2)
89 * -- - ------ - ------ -
90 * x x x
91 *
92 * Let w = 2n/x and h=2/x, then the above quotient
93 * is equal to the continued fraction:
94 * 1
95 * = -----------------------
96 * 1
97 * w - -----------------
98 * 1
99 * w+h - ---------
100 * w+2h - ...
101 *
102 * To determine how many terms needed, let
103 * Q(0) = w, Q(1) = w(w+h) - 1,
104 * Q(k) = (w+k*h)*Q(k-1) - Q(k-2),
105 * When Q(k) > 1e4 good for single
106 * When Q(k) > 1e9 good for double
107 * When Q(k) > 1e17 good for quadruple
108 */
109 /* determine k */
110 float t,v;
111 float q0,q1,h,tmp; int32_t k,m;
112 w = (n+n)/(float)x; h = (float)2.0/(float)x;
113 q0 = w; z = w+h; q1 = w*z - (float)1.0; k=1;
114 while(q1<(float)1.0e9) {
115 k += 1; z += h;
116 tmp = z*q1 - q0;
117 q0 = q1;
118 q1 = tmp;
119 }
120 m = n+n;
121 for(t=zero, i = 2*(n+k); i>=m; i -= 2) t = one/(i/x-t);
122 a = t;
123 b = one;
124 /* estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n)
125 * Hence, if n*(log(2n/x)) > ...
126 * single 8.8722839355e+01
127 * double 7.09782712893383973096e+02
128 * long double 1.1356523406294143949491931077970765006170e+04
129 * then recurrent value may overflow and the result is
130 * likely underflow to zero
131 */
132 tmp = n;
133 v = two/x;
134 tmp = tmp*logf(fabsf(v*tmp));
135 if(tmp<(float)8.8721679688e+01) {
136 for(i=n-1,di=(float)(i+i);i>0;i--){
137 temp = b;
138 b *= di;
139 b = b/x - a;
140 a = temp;
141 di -= two;
142 }
143 } else {
144 for(i=n-1,di=(float)(i+i);i>0;i--){
145 temp = b;
146 b *= di;
147 b = b/x - a;
148 a = temp;
149 di -= two;
150 /* scale b to avoid spurious overflow */
151 if(b>(float)1e10) {
152 a /= b;
153 t /= b;
154 b = one;
155 }
156 }
157 }
158 z = j0f(x);
159 w = j1f(x);
160 if (fabsf(z) >= fabsf(w))
161 b = (t*z/b);
162 else
163 b = (t*w/a);
164 }
165 }
166 if(sgn==1) return -b; else return b;
167 }
168
169 float
ynf(int n,float x)170 ynf(int n, float x)
171 {
172 int32_t i,hx,ix,ib;
173 int32_t sign;
174 float a, b, temp;
175
176 GET_FLOAT_WORD(hx,x);
177 ix = 0x7fffffff&hx;
178 if(ix>0x7f800000) return x+x;
179 if(ix==0) return -one/vzero;
180 if(hx<0) return vzero/vzero;
181 sign = 1;
182 if(n<0){
183 n = -n;
184 sign = 1 - ((n&1)<<1);
185 }
186 if(n==0) return(y0f(x));
187 if(n==1) return(sign*y1f(x));
188 if(ix==0x7f800000) return zero;
189
190 a = y0f(x);
191 b = y1f(x);
192 /* quit if b is -inf */
193 GET_FLOAT_WORD(ib,b);
194 for(i=1;i<n&&ib!=0xff800000;i++){
195 temp = b;
196 b = ((float)(i+i)/x)*b - a;
197 GET_FLOAT_WORD(ib,b);
198 a = temp;
199 }
200 if(sign>0) return b; else return -b;
201 }
202