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: head/lib/msun/src/e_jnf.c 279856 2015-03-10 17:10:54Z kargl $");
18
19 /*
20 * See e_jn.c for complete comments.
21 */
22
23 #include "math.h"
24 #include "math_private.h"
25
26 static const volatile float vone = 1, vzero = 0;
27
28 static const float
29 two = 2.0000000000e+00, /* 0x40000000 */
30 one = 1.0000000000e+00; /* 0x3F800000 */
31
32 static const float zero = 0.0000000000e+00;
33
34 float
__ieee754_jnf(int n,float x)35 __ieee754_jnf(int n, float x)
36 {
37 int32_t i,hx,ix, sgn;
38 float a, b, temp, di;
39 float z, w;
40
41 /* J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x)
42 * Thus, J(-n,x) = J(n,-x)
43 */
44 GET_FLOAT_WORD(hx,x);
45 ix = 0x7fffffff&hx;
46 /* if J(n,NaN) is NaN */
47 if(ix>0x7f800000) return x+x;
48 if(n<0){
49 n = -n;
50 x = -x;
51 hx ^= 0x80000000;
52 }
53 if(n==0) return(__ieee754_j0f(x));
54 if(n==1) return(__ieee754_j1f(x));
55 sgn = (n&1)&(hx>>31); /* even n -- 0, odd n -- sign(x) */
56 x = fabsf(x);
57 if(ix==0||ix>=0x7f800000) /* if x is 0 or inf */
58 b = zero;
59 else if((float)n<=x) {
60 /* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */
61 a = __ieee754_j0f(x);
62 b = __ieee754_j1f(x);
63 for(i=1;i<n;i++){
64 temp = b;
65 b = b*((float)(i+i)/x) - a; /* avoid underflow */
66 a = temp;
67 }
68 } else {
69 if(ix<0x30800000) { /* x < 2**-29 */
70 /* x is tiny, return the first Taylor expansion of J(n,x)
71 * J(n,x) = 1/n!*(x/2)^n - ...
72 */
73 if(n>33) /* underflow */
74 b = zero;
75 else {
76 temp = x*(float)0.5; b = temp;
77 for (a=one,i=2;i<=n;i++) {
78 a *= (float)i; /* a = n! */
79 b *= temp; /* b = (x/2)^n */
80 }
81 b = b/a;
82 }
83 } else {
84 /* use backward recurrence */
85 /* x x^2 x^2
86 * J(n,x)/J(n-1,x) = ---- ------ ------ .....
87 * 2n - 2(n+1) - 2(n+2)
88 *
89 * 1 1 1
90 * (for large x) = ---- ------ ------ .....
91 * 2n 2(n+1) 2(n+2)
92 * -- - ------ - ------ -
93 * x x x
94 *
95 * Let w = 2n/x and h=2/x, then the above quotient
96 * is equal to the continued fraction:
97 * 1
98 * = -----------------------
99 * 1
100 * w - -----------------
101 * 1
102 * w+h - ---------
103 * w+2h - ...
104 *
105 * To determine how many terms needed, let
106 * Q(0) = w, Q(1) = w(w+h) - 1,
107 * Q(k) = (w+k*h)*Q(k-1) - Q(k-2),
108 * When Q(k) > 1e4 good for single
109 * When Q(k) > 1e9 good for double
110 * When Q(k) > 1e17 good for quadruple
111 */
112 /* determine k */
113 float t,v;
114 float q0,q1,h,tmp; int32_t k,m;
115 w = (n+n)/(float)x; h = (float)2.0/(float)x;
116 q0 = w; z = w+h; q1 = w*z - (float)1.0; k=1;
117 while(q1<(float)1.0e9) {
118 k += 1; z += h;
119 tmp = z*q1 - q0;
120 q0 = q1;
121 q1 = tmp;
122 }
123 m = n+n;
124 for(t=zero, i = 2*(n+k); i>=m; i -= 2) t = one/(i/x-t);
125 a = t;
126 b = one;
127 /* estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n)
128 * Hence, if n*(log(2n/x)) > ...
129 * single 8.8722839355e+01
130 * double 7.09782712893383973096e+02
131 * long double 1.1356523406294143949491931077970765006170e+04
132 * then recurrent value may overflow and the result is
133 * likely underflow to zero
134 */
135 tmp = n;
136 v = two/x;
137 tmp = tmp*__ieee754_logf(fabsf(v*tmp));
138 if(tmp<(float)8.8721679688e+01) {
139 for(i=n-1,di=(float)(i+i);i>0;i--){
140 temp = b;
141 b *= di;
142 b = b/x - a;
143 a = temp;
144 di -= two;
145 }
146 } else {
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 /* scale b to avoid spurious overflow */
154 if(b>(float)1e10) {
155 a /= b;
156 t /= b;
157 b = one;
158 }
159 }
160 }
161 z = __ieee754_j0f(x);
162 w = __ieee754_j1f(x);
163 if (fabsf(z) >= fabsf(w))
164 b = (t*z/b);
165 else
166 b = (t*w/a);
167 }
168 }
169 if(sgn==1) return -b; else return b;
170 }
171
172 float
__ieee754_ynf(int n,float x)173 __ieee754_ynf(int n, float x)
174 {
175 int32_t i,hx,ix,ib;
176 int32_t sign;
177 float a, b, temp;
178
179 GET_FLOAT_WORD(hx,x);
180 ix = 0x7fffffff&hx;
181 if(ix>0x7f800000) return x+x;
182 if(ix==0) return -one/vzero;
183 if(hx<0) return vzero/vzero;
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&&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