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 1.7 1999/07/02 15:37:40 simonb Exp $"); 19 #endif 20 21 #include "math.h" 22 #include "math_private.h" 23 24 #ifdef __STDC__ 25 static const float 26 #else 27 static float 28 #endif 29 invsqrtpi= 5.6418961287e-01, /* 0x3f106ebb */ 30 two = 2.0000000000e+00, /* 0x40000000 */ 31 one = 1.0000000000e+00; /* 0x3F800000 */ 32 33 #ifdef __STDC__ 34 static const float zero = 0.0000000000e+00; 35 #else 36 static float zero = 0.0000000000e+00; 37 #endif 38 39 #ifdef __STDC__ 40 float __ieee754_jnf(int n, float x) 41 #else 42 float __ieee754_jnf(n,x) 43 int n; float x; 44 #endif 45 { 46 int32_t i,hx,ix, sgn; 47 float a, b, temp, di; 48 float z, w; 49 50 /* J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x) 51 * Thus, J(-n,x) = J(n,-x) 52 */ 53 GET_FLOAT_WORD(hx,x); 54 ix = 0x7fffffff&hx; 55 /* if J(n,NaN) is NaN */ 56 if(ix>0x7f800000) return x+x; 57 if(n<0){ 58 n = -n; 59 x = -x; 60 hx ^= 0x80000000; 61 } 62 if(n==0) return(__ieee754_j0f(x)); 63 if(n==1) return(__ieee754_j1f(x)); 64 sgn = (n&1)&(hx>>31); /* even n -- 0, odd n -- sign(x) */ 65 x = fabsf(x); 66 if(ix==0||ix>=0x7f800000) /* if x is 0 or inf */ 67 b = zero; 68 else if((float)n<=x) { 69 /* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */ 70 a = __ieee754_j0f(x); 71 b = __ieee754_j1f(x); 72 for(i=1;i<n;i++){ 73 temp = b; 74 b = b*((float)(i+i)/x) - a; /* avoid underflow */ 75 a = temp; 76 } 77 } else { 78 if(ix<0x30800000) { /* x < 2**-29 */ 79 /* x is tiny, return the first Taylor expansion of J(n,x) 80 * J(n,x) = 1/n!*(x/2)^n - ... 81 */ 82 if(n>33) /* underflow */ 83 b = zero; 84 else { 85 temp = x*(float)0.5; b = temp; 86 for (a=one,i=2;i<=n;i++) { 87 a *= (float)i; /* a = n! */ 88 b *= temp; /* b = (x/2)^n */ 89 } 90 b = b/a; 91 } 92 } else { 93 /* use backward recurrence */ 94 /* x x^2 x^2 95 * J(n,x)/J(n-1,x) = ---- ------ ------ ..... 96 * 2n - 2(n+1) - 2(n+2) 97 * 98 * 1 1 1 99 * (for large x) = ---- ------ ------ ..... 100 * 2n 2(n+1) 2(n+2) 101 * -- - ------ - ------ - 102 * x x x 103 * 104 * Let w = 2n/x and h=2/x, then the above quotient 105 * is equal to the continued fraction: 106 * 1 107 * = ----------------------- 108 * 1 109 * w - ----------------- 110 * 1 111 * w+h - --------- 112 * w+2h - ... 113 * 114 * To determine how many terms needed, let 115 * Q(0) = w, Q(1) = w(w+h) - 1, 116 * Q(k) = (w+k*h)*Q(k-1) - Q(k-2), 117 * When Q(k) > 1e4 good for single 118 * When Q(k) > 1e9 good for double 119 * When Q(k) > 1e17 good for quadruple 120 */ 121 /* determine k */ 122 float t,v; 123 float q0,q1,h,tmp; int32_t k,m; 124 w = (n+n)/(float)x; h = (float)2.0/(float)x; 125 q0 = w; z = w+h; q1 = w*z - (float)1.0; k=1; 126 while(q1<(float)1.0e9) { 127 k += 1; z += h; 128 tmp = z*q1 - q0; 129 q0 = q1; 130 q1 = tmp; 131 } 132 m = n+n; 133 for(t=zero, i = 2*(n+k); i>=m; i -= 2) t = one/(i/x-t); 134 a = t; 135 b = one; 136 /* estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n) 137 * Hence, if n*(log(2n/x)) > ... 138 * single 8.8722839355e+01 139 * double 7.09782712893383973096e+02 140 * long double 1.1356523406294143949491931077970765006170e+04 141 * then recurrent value may overflow and the result is 142 * likely underflow to zero 143 */ 144 tmp = n; 145 v = two/x; 146 tmp = tmp*__ieee754_logf(fabsf(v*tmp)); 147 if(tmp<(float)8.8721679688e+01) { 148 for(i=n-1,di=(float)(i+i);i>0;i--){ 149 temp = b; 150 b *= di; 151 b = b/x - a; 152 a = temp; 153 di -= two; 154 } 155 } else { 156 for(i=n-1,di=(float)(i+i);i>0;i--){ 157 temp = b; 158 b *= di; 159 b = b/x - a; 160 a = temp; 161 di -= two; 162 /* scale b to avoid spurious overflow */ 163 if(b>(float)1e10) { 164 a /= b; 165 t /= b; 166 b = one; 167 } 168 } 169 } 170 b = (t*__ieee754_j0f(x)/b); 171 } 172 } 173 if(sgn==1) return -b; else return b; 174 } 175 176 #ifdef __STDC__ 177 float __ieee754_ynf(int n, float x) 178 #else 179 float __ieee754_ynf(n,x) 180 int n; float x; 181 #endif 182 { 183 int32_t i,hx,ix,ib; 184 int32_t sign; 185 float a, b, temp; 186 187 GET_FLOAT_WORD(hx,x); 188 ix = 0x7fffffff&hx; 189 /* if Y(n,NaN) is NaN */ 190 if(ix>0x7f800000) return x+x; 191 if(ix==0) return -one/zero; 192 if(hx<0) return zero/zero; 193 sign = 1; 194 if(n<0){ 195 n = -n; 196 sign = 1 - ((n&1)<<1); 197 } 198 if(n==0) return(__ieee754_y0f(x)); 199 if(n==1) return(sign*__ieee754_y1f(x)); 200 if(ix==0x7f800000) return zero; 201 202 a = __ieee754_y0f(x); 203 b = __ieee754_y1f(x); 204 /* quit if b is -inf */ 205 GET_FLOAT_WORD(ib,b); 206 for(i=1;i<n&&ib!=0xff800000;i++){ 207 temp = b; 208 b = ((float)(i+i)/x)*b - a; 209 GET_FLOAT_WORD(ib,b); 210 a = temp; 211 } 212 if(sign>0) return b; else return -b; 213 } 214