1 /* ef_jn.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 "fdlibm.h" 17 18 #ifdef __STDC__ 19 static const float 20 #else 21 static float 22 #endif 23 invsqrtpi= 5.6418961287e-01, /* 0x3f106ebb */ 24 two = 2.0000000000e+00, /* 0x40000000 */ 25 one = 1.0000000000e+00; /* 0x3F800000 */ 26 27 #ifdef __STDC__ 28 static const float zero = 0.0000000000e+00; 29 #else 30 static float zero = 0.0000000000e+00; 31 #endif 32 33 #ifdef __STDC__ __ieee754_jnf(int n,float x)34 float __ieee754_jnf(int n, float x) 35 #else 36 float __ieee754_jnf(n,x) 37 int n; float x; 38 #endif 39 { 40 __int32_t i,hx,ix, sgn; 41 float a, b, temp, di; 42 float z, w; 43 44 /* J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x) 45 * Thus, J(-n,x) = J(n,-x) 46 */ 47 GET_FLOAT_WORD(hx,x); 48 ix = 0x7fffffff&hx; 49 /* if J(n,NaN) is NaN */ 50 if(FLT_UWORD_IS_NAN(ix)) return x+x; 51 if(n<0){ 52 n = -n; 53 x = -x; 54 hx ^= 0x80000000; 55 } 56 if(n==0) return(__ieee754_j0f(x)); 57 if(n==1) return(__ieee754_j1f(x)); 58 sgn = (n&1)&(hx>>31); /* even n -- 0, odd n -- sign(x) */ 59 x = fabsf(x); 60 if(FLT_UWORD_IS_ZERO(ix)||FLT_UWORD_IS_INFINITE(ix)) 61 b = zero; 62 else if((float)n<=x) { 63 /* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */ 64 a = __ieee754_j0f(x); 65 b = __ieee754_j1f(x); 66 for(i=1;i<n;i++){ 67 temp = b; 68 b = b*((float)(i+i)/x) - a; /* avoid underflow */ 69 a = temp; 70 } 71 } else { 72 if(ix<0x30800000) { /* x < 2**-29 */ 73 /* x is tiny, return the first Taylor expansion of J(n,x) 74 * J(n,x) = 1/n!*(x/2)^n - ... 75 */ 76 if(n>33) /* underflow */ 77 b = zero; 78 else { 79 temp = x*(float)0.5; b = temp; 80 for (a=one,i=2;i<=n;i++) { 81 a *= (float)i; /* a = n! */ 82 b *= temp; /* b = (x/2)^n */ 83 } 84 b = b/a; 85 } 86 } else { 87 /* use backward recurrence */ 88 /* x x^2 x^2 89 * J(n,x)/J(n-1,x) = ---- ------ ------ ..... 90 * 2n - 2(n+1) - 2(n+2) 91 * 92 * 1 1 1 93 * (for large x) = ---- ------ ------ ..... 94 * 2n 2(n+1) 2(n+2) 95 * -- - ------ - ------ - 96 * x x x 97 * 98 * Let w = 2n/x and h=2/x, then the above quotient 99 * is equal to the continued fraction: 100 * 1 101 * = ----------------------- 102 * 1 103 * w - ----------------- 104 * 1 105 * w+h - --------- 106 * w+2h - ... 107 * 108 * To determine how many terms needed, let 109 * Q(0) = w, Q(1) = w(w+h) - 1, 110 * Q(k) = (w+k*h)*Q(k-1) - Q(k-2), 111 * When Q(k) > 1e4 good for single 112 * When Q(k) > 1e9 good for double 113 * When Q(k) > 1e17 good for quadruple 114 */ 115 /* determine k */ 116 float t,v; 117 float q0,q1,h,tmp; __int32_t k,m; 118 w = (n+n)/(float)x; h = (float)2.0/(float)x; 119 q0 = w; z = w+h; q1 = w*z - (float)1.0; k=1; 120 while(q1<(float)1.0e9) { 121 k += 1; z += h; 122 tmp = z*q1 - q0; 123 q0 = q1; 124 q1 = tmp; 125 } 126 m = n+n; 127 for(t=zero, i = 2*(n+k); i>=m; i -= 2) t = one/(i/x-t); 128 a = t; 129 b = one; 130 /* estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n) 131 * Hence, if n*(log(2n/x)) > ... 132 * single 8.8722839355e+01 133 * double 7.09782712893383973096e+02 134 * long double 1.1356523406294143949491931077970765006170e+04 135 * then recurrent value may overflow and the result is 136 * likely underflow to zero 137 */ 138 tmp = n; 139 v = two/x; 140 tmp = tmp*__ieee754_logf(fabsf(v*tmp)); 141 if(tmp<(float)8.8721679688e+01) { 142 for(i=n-1,di=(float)(i+i);i>0;i--){ 143 temp = b; 144 b *= di; 145 b = b/x - a; 146 a = temp; 147 di -= two; 148 } 149 } else { 150 for(i=n-1,di=(float)(i+i);i>0;i--){ 151 temp = b; 152 b *= di; 153 b = b/x - a; 154 a = temp; 155 di -= two; 156 /* scale b to avoid spurious overflow */ 157 if(b>(float)1e10) { 158 a /= b; 159 t /= b; 160 b = one; 161 } 162 } 163 } 164 b = (t*__ieee754_j0f(x)/b); 165 } 166 } 167 if(sgn==1) return -b; else return b; 168 } 169 170 #ifdef __STDC__ __ieee754_ynf(int n,float x)171 float __ieee754_ynf(int n, float x) 172 #else 173 float __ieee754_ynf(n,x) 174 int n; float x; 175 #endif 176 { 177 __int32_t i,hx,ix,ib; 178 __int32_t sign; 179 float a, b, temp; 180 181 GET_FLOAT_WORD(hx,x); 182 ix = 0x7fffffff&hx; 183 /* if Y(n,NaN) is NaN */ 184 if(FLT_UWORD_IS_NAN(ix)) return x+x; 185 if(FLT_UWORD_IS_ZERO(ix)) return -one/zero; 186 if(hx<0) return zero/zero; 187 sign = 1; 188 if(n<0){ 189 n = -n; 190 sign = 1 - ((n&1)<<1); 191 } 192 if(n==0) return(__ieee754_y0f(x)); 193 if(n==1) return(sign*__ieee754_y1f(x)); 194 if(FLT_UWORD_IS_INFINITE(ix)) return zero; 195 196 a = __ieee754_y0f(x); 197 b = __ieee754_y1f(x); 198 /* quit if b is -inf */ 199 GET_FLOAT_WORD(ib,b); 200 for(i=1;i<n&&ib!=0xff800000;i++){ 201 temp = b; 202 b = ((float)(i+i)/x)*b - a; 203 GET_FLOAT_WORD(ib,b); 204 a = temp; 205 } 206 if(sign>0) return b; else return -b; 207 } 208