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 #ifndef lint 17 static char rcsid[] = "$Id$"; 18 #endif 19 20 #include "math.h" 21 #include "math_private.h" 22 23 #ifdef __STDC__ 24 static const float 25 #else 26 static float 27 #endif 28 invsqrtpi= 5.6418961287e-01, /* 0x3f106ebb */ 29 two = 2.0000000000e+00, /* 0x40000000 */ 30 one = 1.0000000000e+00; /* 0x3F800000 */ 31 32 #ifdef __STDC__ 33 static const float zero = 0.0000000000e+00; 34 #else 35 static float zero = 0.0000000000e+00; 36 #endif 37 38 #ifdef __STDC__ 39 float __ieee754_jnf(int n, float x) 40 #else 41 float __ieee754_jnf(n,x) 42 int n; float x; 43 #endif 44 { 45 int32_t i,hx,ix, sgn; 46 float a, b, temp, di; 47 float z, w; 48 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) 51 */ 52 GET_FLOAT_WORD(hx,x); 53 ix = 0x7fffffff&hx; 54 /* if J(n,NaN) is NaN */ 55 if(ix>0x7f800000) return x+x; 56 if(n<0){ 57 n = -n; 58 x = -x; 59 hx ^= 0x80000000; 60 } 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) */ 64 x = fabsf(x); 65 if(ix==0||ix>=0x7f800000) /* if x is 0 or inf */ 66 b = zero; 67 else if((float)n<=x) { 68 /* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */ 69 a = __ieee754_j0f(x); 70 b = __ieee754_j1f(x); 71 for(i=1;i<n;i++){ 72 temp = b; 73 b = b*((float)(i+i)/x) - a; /* avoid underflow */ 74 a = temp; 75 } 76 } else { 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 - ... 80 */ 81 if(n>33) /* underflow */ 82 b = zero; 83 else { 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 */ 88 } 89 b = b/a; 90 } 91 } else { 92 /* use backward recurrence */ 93 /* x x^2 x^2 94 * J(n,x)/J(n-1,x) = ---- ------ ------ ..... 95 * 2n - 2(n+1) - 2(n+2) 96 * 97 * 1 1 1 98 * (for large x) = ---- ------ ------ ..... 99 * 2n 2(n+1) 2(n+2) 100 * -- - ------ - ------ - 101 * x x x 102 * 103 * Let w = 2n/x and h=2/x, then the above quotient 104 * is equal to the continued fraction: 105 * 1 106 * = ----------------------- 107 * 1 108 * w - ----------------- 109 * 1 110 * w+h - --------- 111 * w+2h - ... 112 * 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 119 */ 120 /* determine k */ 121 float t,v; 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) { 126 k += 1; z += h; 127 tmp = z*q1 - q0; 128 q0 = q1; 129 q1 = tmp; 130 } 131 m = n+n; 132 for(t=zero, i = 2*(n+k); i>=m; i -= 2) t = one/(i/x-t); 133 a = t; 134 b = one; 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 142 */ 143 tmp = n; 144 v = two/x; 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--){ 148 temp = b; 149 b *= di; 150 b = b/x - a; 151 a = temp; 152 di -= two; 153 } 154 } else { 155 for(i=n-1,di=(float)(i+i);i>0;i--){ 156 temp = b; 157 b *= di; 158 b = b/x - a; 159 a = temp; 160 di -= two; 161 /* scale b to avoid spurious overflow */ 162 if(b>(float)1e10) { 163 a /= b; 164 t /= b; 165 b = one; 166 } 167 } 168 } 169 b = (t*__ieee754_j0f(x)/b); 170 } 171 } 172 if(sgn==1) return -b; else return b; 173 } 174 175 #ifdef __STDC__ 176 float __ieee754_ynf(int n, float x) 177 #else 178 float __ieee754_ynf(n,x) 179 int n; float x; 180 #endif 181 { 182 int32_t i,hx,ix,ib; 183 int32_t sign; 184 float a, b, temp; 185 186 GET_FLOAT_WORD(hx,x); 187 ix = 0x7fffffff&hx; 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; 192 sign = 1; 193 if(n<0){ 194 n = -n; 195 sign = 1 - ((n&1)<<1); 196 } 197 if(n==0) return(__ieee754_y0f(x)); 198 if(n==1) return(sign*__ieee754_y1f(x)); 199 if(ix==0x7f800000) return zero; 200 201 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++){ 206 temp = b; 207 b = ((float)(i+i)/x)*b - a; 208 GET_FLOAT_WORD(ib,b); 209 a = temp; 210 } 211 if(sign>0) return b; else return -b; 212 } 213