1 /* e_jnf.c -- float version of e_jn.c.
2 * Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com.
6 * ====================================================
7 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
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
13 * ====================================================
17 static char rcsid[] = "$FreeBSD$";
21 #include "math_private.h"
28 invsqrtpi= 5.6418961287e-01, /* 0x3f106ebb */
29 two = 2.0000000000e+00, /* 0x40000000 */
30 one = 1.0000000000e+00; /* 0x3F800000 */
33 static const float zero = 0.0000000000e+00;
35 static float zero = 0.0000000000e+00;
39 float __ieee754_jnf(int n, float x)
41 float __ieee754_jnf(n,x)
49 /* J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x)
50 * Thus, J(-n,x) = J(n,-x)
54 /* if J(n,NaN) is NaN */
55 if(ix>0x7f800000) return x+x;
61 if(n==0) return(__ieee754_j0f(x));
62 if(n==1) return(__ieee754_j1f(x));
63 sgn = (n&1)&(hx>>31); /* even n -- 0, odd n -- sign(x) */
65 if(ix==0||ix>=0x7f800000) /* if x is 0 or inf */
67 else if((float)n<=x) {
68 /* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */
73 b = b*((float)(i+i)/x) - a; /* avoid underflow */
77 if(ix<0x30800000) { /* x < 2**-29 */
78 /* x is tiny, return the first Taylor expansion of J(n,x)
79 * J(n,x) = 1/n!*(x/2)^n - ...
81 if(n>33) /* underflow */
84 temp = x*(float)0.5; b = temp;
85 for (a=one,i=2;i<=n;i++) {
86 a *= (float)i; /* a = n! */
87 b *= temp; /* b = (x/2)^n */
92 /* use backward recurrence */
94 * J(n,x)/J(n-1,x) = ---- ------ ------ .....
95 * 2n - 2(n+1) - 2(n+2)
98 * (for large x) = ---- ------ ------ .....
100 * -- - ------ - ------ -
103 * Let w = 2n/x and h=2/x, then the above quotient
104 * is equal to the continued fraction:
106 * = -----------------------
108 * w - -----------------
113 * To determine how many terms needed, let
114 * Q(0) = w, Q(1) = w(w+h) - 1,
115 * Q(k) = (w+k*h)*Q(k-1) - Q(k-2),
116 * When Q(k) > 1e4 good for single
117 * When Q(k) > 1e9 good for double
118 * When Q(k) > 1e17 good for quadruple
122 float q0,q1,h,tmp; int32_t k,m;
123 w = (n+n)/(float)x; h = (float)2.0/(float)x;
124 q0 = w; z = w+h; q1 = w*z - (float)1.0; k=1;
125 while(q1<(float)1.0e9) {
132 for(t=zero, i = 2*(n+k); i>=m; i -= 2) t = one/(i/x-t);
135 /* estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n)
136 * Hence, if n*(log(2n/x)) > ...
137 * single 8.8722839355e+01
138 * double 7.09782712893383973096e+02
139 * long double 1.1356523406294143949491931077970765006170e+04
140 * then recurrent value may overflow and the result is
141 * likely underflow to zero
145 tmp = tmp*__ieee754_logf(fabsf(v*tmp));
146 if(tmp<(float)8.8721679688e+01) {
147 for(i=n-1,di=(float)(i+i);i>0;i--){
155 for(i=n-1,di=(float)(i+i);i>0;i--){
161 /* scale b to avoid spurious overflow */
169 b = (t*__ieee754_j0f(x)/b);
172 if(sgn==1) return -b; else return b;
176 float __ieee754_ynf(int n, float x)
178 float __ieee754_ynf(n,x)
186 GET_FLOAT_WORD(hx,x);
188 /* if Y(n,NaN) is NaN */
189 if(ix>0x7f800000) return x+x;
190 if(ix==0) return -one/zero;
191 if(hx<0) return zero/zero;
195 sign = 1 - ((n&1)<<1);
197 if(n==0) return(__ieee754_y0f(x));
198 if(n==1) return(sign*__ieee754_y1f(x));
199 if(ix==0x7f800000) return zero;
201 a = __ieee754_y0f(x);
202 b = __ieee754_y1f(x);
203 /* quit if b is -inf */
204 GET_FLOAT_WORD(ib,b);
205 for(i=1;i<n&&ib!=0xff800000;i++){
207 b = ((float)(i+i)/x)*b - a;
208 GET_FLOAT_WORD(ib,b);
211 if(sign>0) return b; else return -b;