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 #if defined(LIBM_SCCS) && !defined(lint) 18 __RCSID("$NetBSD: e_jnf.c,v 1.9 2002/05/26 22:01:50 wiz Exp $"); 19 #endif 20 21 #include "math.h" 22 #include "math_private.h" 23 24 static const float 25 #if 0 26 invsqrtpi= 5.6418961287e-01, /* 0x3f106ebb */ 27 #endif 28 two = 2.0000000000e+00, /* 0x40000000 */ 29 one = 1.0000000000e+00; /* 0x3F800000 */ 30 31 static const float zero = 0.0000000000e+00; 32 33 float 34 __ieee754_jnf(int n, float x) 35 { 36 int32_t i,hx,ix, sgn; 37 float a, b, temp, di; 38 float z, w; 39 40 /* J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x) 41 * Thus, J(-n,x) = J(n,-x) 42 */ 43 GET_FLOAT_WORD(hx,x); 44 ix = 0x7fffffff&hx; 45 /* if J(n,NaN) is NaN */ 46 if(ix>0x7f800000) return x+x; 47 if(n<0){ 48 n = -n; 49 x = -x; 50 hx ^= 0x80000000; 51 } 52 if(n==0) return(__ieee754_j0f(x)); 53 if(n==1) return(__ieee754_j1f(x)); 54 sgn = (n&1)&(hx>>31); /* even n -- 0, odd n -- sign(x) */ 55 x = fabsf(x); 56 if(ix==0||ix>=0x7f800000) /* if x is 0 or inf */ 57 b = zero; 58 else if((float)n<=x) { 59 /* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */ 60 a = __ieee754_j0f(x); 61 b = __ieee754_j1f(x); 62 for(i=1;i<n;i++){ 63 temp = b; 64 b = b*((float)(i+i)/x) - a; /* avoid underflow */ 65 a = temp; 66 } 67 } else { 68 if(ix<0x30800000) { /* x < 2**-29 */ 69 /* x is tiny, return the first Taylor expansion of J(n,x) 70 * J(n,x) = 1/n!*(x/2)^n - ... 71 */ 72 if(n>33) /* underflow */ 73 b = zero; 74 else { 75 temp = x*(float)0.5; b = temp; 76 for (a=one,i=2;i<=n;i++) { 77 a *= (float)i; /* a = n! */ 78 b *= temp; /* b = (x/2)^n */ 79 } 80 b = b/a; 81 } 82 } else { 83 /* use backward recurrence */ 84 /* x x^2 x^2 85 * J(n,x)/J(n-1,x) = ---- ------ ------ ..... 86 * 2n - 2(n+1) - 2(n+2) 87 * 88 * 1 1 1 89 * (for large x) = ---- ------ ------ ..... 90 * 2n 2(n+1) 2(n+2) 91 * -- - ------ - ------ - 92 * x x x 93 * 94 * Let w = 2n/x and h=2/x, then the above quotient 95 * is equal to the continued fraction: 96 * 1 97 * = ----------------------- 98 * 1 99 * w - ----------------- 100 * 1 101 * w+h - --------- 102 * w+2h - ... 103 * 104 * To determine how many terms needed, let 105 * Q(0) = w, Q(1) = w(w+h) - 1, 106 * Q(k) = (w+k*h)*Q(k-1) - Q(k-2), 107 * When Q(k) > 1e4 good for single 108 * When Q(k) > 1e9 good for double 109 * When Q(k) > 1e17 good for quadruple 110 */ 111 /* determine k */ 112 float t,v; 113 float q0,q1,h,tmp; int32_t k,m; 114 w = (n+n)/(float)x; h = (float)2.0/(float)x; 115 q0 = w; z = w+h; q1 = w*z - (float)1.0; k=1; 116 while(q1<(float)1.0e9) { 117 k += 1; z += h; 118 tmp = z*q1 - q0; 119 q0 = q1; 120 q1 = tmp; 121 } 122 m = n+n; 123 for(t=zero, i = 2*(n+k); i>=m; i -= 2) t = one/(i/x-t); 124 a = t; 125 b = one; 126 /* estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n) 127 * Hence, if n*(log(2n/x)) > ... 128 * single 8.8722839355e+01 129 * double 7.09782712893383973096e+02 130 * long double 1.1356523406294143949491931077970765006170e+04 131 * then recurrent value may overflow and the result is 132 * likely underflow to zero 133 */ 134 tmp = n; 135 v = two/x; 136 tmp = tmp*__ieee754_logf(fabsf(v*tmp)); 137 if(tmp<(float)8.8721679688e+01) { 138 for(i=n-1,di=(float)(i+i);i>0;i--){ 139 temp = b; 140 b *= di; 141 b = b/x - a; 142 a = temp; 143 di -= two; 144 } 145 } else { 146 for(i=n-1,di=(float)(i+i);i>0;i--){ 147 temp = b; 148 b *= di; 149 b = b/x - a; 150 a = temp; 151 di -= two; 152 /* scale b to avoid spurious overflow */ 153 if(b>(float)1e10) { 154 a /= b; 155 t /= b; 156 b = one; 157 } 158 } 159 } 160 b = (t*__ieee754_j0f(x)/b); 161 } 162 } 163 if(sgn==1) return -b; else return b; 164 } 165 166 float 167 __ieee754_ynf(int n, float x) 168 { 169 int32_t i,hx,ix,ib; 170 int32_t sign; 171 float a, b, temp; 172 173 GET_FLOAT_WORD(hx,x); 174 ix = 0x7fffffff&hx; 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; 179 sign = 1; 180 if(n<0){ 181 n = -n; 182 sign = 1 - ((n&1)<<1); 183 } 184 if(n==0) return(__ieee754_y0f(x)); 185 if(n==1) return(sign*__ieee754_y1f(x)); 186 if(ix==0x7f800000) return zero; 187 188 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++){ 193 temp = b; 194 b = ((float)(i+i)/x)*b - a; 195 GET_FLOAT_WORD(ib,b); 196 a = temp; 197 } 198 if(sign>0) return b; else return -b; 199 } 200