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 * ====================================================
16 #include <sys/cdefs.h>
17 __FBSDID("$FreeBSD$");
20 #include "math_private.h"
23 two = 2.0000000000e+00, /* 0x40000000 */
24 one = 1.0000000000e+00; /* 0x3F800000 */
26 static const float zero = 0.0000000000e+00;
29 __ieee754_jnf(int n, float x)
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)
40 /* if J(n,NaN) is NaN */
41 if(ix>0x7f800000) return x+x;
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) */
51 if(ix==0||ix>=0x7f800000) /* if x is 0 or inf */
53 else if((float)n<=x) {
54 /* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */
59 b = b*((float)(i+i)/x) - a; /* avoid underflow */
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 - ...
67 if(n>33) /* underflow */
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 */
78 /* use backward recurrence */
80 * J(n,x)/J(n-1,x) = ---- ------ ------ .....
81 * 2n - 2(n+1) - 2(n+2)
84 * (for large x) = ---- ------ ------ .....
86 * -- - ------ - ------ -
89 * Let w = 2n/x and h=2/x, then the above quotient
90 * is equal to the continued fraction:
92 * = -----------------------
94 * w - -----------------
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
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) {
118 for(t=zero, i = 2*(n+k); i>=m; i -= 2) t = one/(i/x-t);
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
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--){
141 for(i=n-1,di=(float)(i+i);i>0;i--){
147 /* scale b to avoid spurious overflow */
155 z = __ieee754_j0f(x);
156 w = __ieee754_j1f(x);
157 if (fabsf(z) >= fabsf(w))
163 if(sgn==1) return -b; else return b;
167 __ieee754_ynf(int n, float x)
173 GET_FLOAT_WORD(hx,x);
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;
182 sign = 1 - ((n&1)<<1);
184 if(n==0) return(__ieee754_y0f(x));
185 if(n==1) return(sign*__ieee754_y1f(x));
186 if(ix==0x7f800000) return zero;
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++){
194 b = ((float)(i+i)/x)*b - a;
195 GET_FLOAT_WORD(ib,b);
198 if(sign>0) return b; else return -b;