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 <errno.h>
17#include <float.h>
18#include <math.h>
19#include <math-narrow-eval.h>
20#include <math_private.h>
21#include <math-underflow.h>
22
23static const float
24two = 2.0000000000e+00, /* 0x40000000 */
25one = 1.0000000000e+00; /* 0x3F800000 */
26
27static const float zero = 0.0000000000e+00;
28
29float
30__ieee754_jnf(int n, float x)
31{
32 float ret;
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(__builtin_expect(ix>0x7f800000, 0)) return x+x;
45 if(n<0){
46 n = -n;
47 x = -x;
48 hx ^= 0x80000000;
49 }
50 if(n==0) return(__ieee754_j0f(x));
51 if(n==1) return(__ieee754_j1f(x));
52 sgn = (n&1)&(hx>>31); /* even n -- 0, odd n -- sign(x) */
53 x = fabsf(x);
54 SET_RESTORE_ROUNDF (FE_TONEAREST);
55 if(__builtin_expect(ix==0||ix>=0x7f800000, 0)) /* if x is 0 or inf */
56 return sgn == 1 ? -zero : zero;
57 else if((float)n<=x) {
58 /* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */
59 a = __ieee754_j0f(x);
60 b = __ieee754_j1f(x);
61 for(i=1;i<n;i++){
62 temp = b;
63 b = b*((double)(i+i)/x) - a; /* avoid underflow */
64 a = temp;
65 }
66 } else {
67 if(ix<0x30800000) { /* x < 2**-29 */
68 /* x is tiny, return the first Taylor expansion of J(n,x)
69 * J(n,x) = 1/n!*(x/2)^n - ...
70 */
71 if(n>33) /* underflow */
72 b = zero;
73 else {
74 temp = x*(float)0.5; b = temp;
75 for (a=one,i=2;i<=n;i++) {
76 a *= (float)i; /* a = n! */
77 b *= temp; /* b = (x/2)^n */
78 }
79 b = b/a;
80 }
81 } else {
82 /* use backward recurrence */
83 /* x x^2 x^2
84 * J(n,x)/J(n-1,x) = ---- ------ ------ .....
85 * 2n - 2(n+1) - 2(n+2)
86 *
87 * 1 1 1
88 * (for large x) = ---- ------ ------ .....
89 * 2n 2(n+1) 2(n+2)
90 * -- - ------ - ------ -
91 * x x x
92 *
93 * Let w = 2n/x and h=2/x, then the above quotient
94 * is equal to the continued fraction:
95 * 1
96 * = -----------------------
97 * 1
98 * w - -----------------
99 * 1
100 * w+h - ---------
101 * w+2h - ...
102 *
103 * To determine how many terms needed, let
104 * Q(0) = w, Q(1) = w(w+h) - 1,
105 * Q(k) = (w+k*h)*Q(k-1) - Q(k-2),
106 * When Q(k) > 1e4 good for single
107 * When Q(k) > 1e9 good for double
108 * When Q(k) > 1e17 good for quadruple
109 */
110 /* determine k */
111 float t,v;
112 float q0,q1,h,tmp; int32_t k,m;
113 w = (n+n)/(float)x; h = (float)2.0/(float)x;
114 q0 = w; z = w+h; q1 = w*z - (float)1.0; k=1;
115 while(q1<(float)1.0e9) {
116 k += 1; z += h;
117 tmp = z*q1 - q0;
118 q0 = q1;
119 q1 = tmp;
120 }
121 m = n+n;
122 for(t=zero, i = 2*(n+k); i>=m; i -= 2) t = one/(i/x-t);
123 a = t;
124 b = one;
125 /* estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n)
126 * Hence, if n*(log(2n/x)) > ...
127 * single 8.8722839355e+01
128 * double 7.09782712893383973096e+02
129 * long double 1.1356523406294143949491931077970765006170e+04
130 * then recurrent value may overflow and the result is
131 * likely underflow to zero
132 */
133 tmp = n;
134 v = two/x;
135 tmp = tmp*__ieee754_logf(fabsf(v*tmp));
136 if(tmp<(float)8.8721679688e+01) {
137 for(i=n-1,di=(float)(i+i);i>0;i--){
138 temp = b;
139 b *= di;
140 b = b/x - a;
141 a = temp;
142 di -= two;
143 }
144 } else {
145 for(i=n-1,di=(float)(i+i);i>0;i--){
146 temp = b;
147 b *= di;
148 b = b/x - a;
149 a = temp;
150 di -= two;
151 /* scale b to avoid spurious overflow */
152 if(b>(float)1e10) {
153 a /= b;
154 t /= b;
155 b = one;
156 }
157 }
158 }
159 /* j0() and j1() suffer enormous loss of precision at and
160 * near zero; however, we know that their zero points never
161 * coincide, so just choose the one further away from zero.
162 */
163 z = __ieee754_j0f (x);
164 w = __ieee754_j1f (x);
165 if (fabsf (z) >= fabsf (w))
166 b = (t * z / b);
167 else
168 b = (t * w / a);
169 }
170 }
171 if(sgn==1) ret = -b; else ret = b;
172 ret = math_narrow_eval (ret);
173 }
174 if (ret == 0)
175 {
176 ret = math_narrow_eval (__copysignf (FLT_MIN, ret) * FLT_MIN);
177 __set_errno (ERANGE);
178 }
179 else
180 math_check_force_underflow (ret);
181 return ret;
182}
183strong_alias (__ieee754_jnf, __jnf_finite)
184
185float
186__ieee754_ynf(int n, float x)
187{
188 float ret;
189 {
190 int32_t i,hx,ix;
191 uint32_t ib;
192 int32_t sign;
193 float a, b, temp;
194
195 GET_FLOAT_WORD(hx,x);
196 ix = 0x7fffffff&hx;
197 /* if Y(n,NaN) is NaN */
198 if(__builtin_expect(ix>0x7f800000, 0)) return x+x;
199 sign = 1;
200 if(n<0){
201 n = -n;
202 sign = 1 - ((n&1)<<1);
203 }
204 if(n==0) return(__ieee754_y0f(x));
205 if(__builtin_expect(ix==0, 0))
206 return -sign/zero;
207 if(__builtin_expect(hx<0, 0)) return zero/(zero*x);
208 SET_RESTORE_ROUNDF (FE_TONEAREST);
209 if(n==1) {
210 ret = sign*__ieee754_y1f(x);
211 goto out;
212 }
213 if(__builtin_expect(ix==0x7f800000, 0)) return zero;
214
215 a = __ieee754_y0f(x);
216 b = __ieee754_y1f(x);
217 /* quit if b is -inf */
218 GET_FLOAT_WORD(ib,b);
219 for(i=1;i<n&&ib!=0xff800000;i++){
220 temp = b;
221 b = ((double)(i+i)/x)*b - a;
222 GET_FLOAT_WORD(ib,b);
223 a = temp;
224 }
225 /* If B is +-Inf, set up errno accordingly. */
226 if (! isfinite (b))
227 __set_errno (ERANGE);
228 if(sign>0) ret = b; else ret = -b;
229 }
230 out:
231 if (isinf (ret))
232 ret = __copysignf (FLT_MAX, ret) * FLT_MAX;
233 return ret;
234}
235strong_alias (__ieee754_ynf, __ynf_finite)
236