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[] = "$FreeBSD$"; 18 #endif 19 20 #include "math.h" 21 #include "math_private.h" 22 23 static const float 24 invsqrtpi= 5.6418961287e-01, /* 0x3f106ebb */ 25 two = 2.0000000000e+00, /* 0x40000000 */ 26 one = 1.0000000000e+00; /* 0x3F800000 */ 27 28 static const float zero = 0.0000000000e+00; 29 30 float 31 __ieee754_jnf(int n, float x) 32 { 33 int32_t i,hx,ix, sgn; 34 float a, b, temp, di; 35 float z, w; 36 37 /* J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x) 38 * Thus, J(-n,x) = J(n,-x) 39 */ 40 GET_FLOAT_WORD(hx,x); 41 ix = 0x7fffffff&hx; 42 /* if J(n,NaN) is NaN */ 43 if(ix>0x7f800000) return x+x; 44 if(n<0){ 45 n = -n; 46 x = -x; 47 hx ^= 0x80000000; 48 } 49 if(n==0) return(__ieee754_j0f(x)); 50 if(n==1) return(__ieee754_j1f(x)); 51 sgn = (n&1)&(hx>>31); /* even n -- 0, odd n -- sign(x) */ 52 x = fabsf(x); 53 if(ix==0||ix>=0x7f800000) /* if x is 0 or inf */ 54 b = zero; 55 else if((float)n<=x) { 56 /* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */ 57 a = __ieee754_j0f(x); 58 b = __ieee754_j1f(x); 59 for(i=1;i<n;i++){ 60 temp = b; 61 b = b*((float)(i+i)/x) - a; /* avoid underflow */ 62 a = temp; 63 } 64 } else { 65 if(ix<0x30800000) { /* x < 2**-29 */ 66 /* x is tiny, return the first Taylor expansion of J(n,x) 67 * J(n,x) = 1/n!*(x/2)^n - ... 68 */ 69 if(n>33) /* underflow */ 70 b = zero; 71 else { 72 temp = x*(float)0.5; b = temp; 73 for (a=one,i=2;i<=n;i++) { 74 a *= (float)i; /* a = n! */ 75 b *= temp; /* b = (x/2)^n */ 76 } 77 b = b/a; 78 } 79 } else { 80 /* use backward recurrence */ 81 /* x x^2 x^2 82 * J(n,x)/J(n-1,x) = ---- ------ ------ ..... 83 * 2n - 2(n+1) - 2(n+2) 84 * 85 * 1 1 1 86 * (for large x) = ---- ------ ------ ..... 87 * 2n 2(n+1) 2(n+2) 88 * -- - ------ - ------ - 89 * x x x 90 * 91 * Let w = 2n/x and h=2/x, then the above quotient 92 * is equal to the continued fraction: 93 * 1 94 * = ----------------------- 95 * 1 96 * w - ----------------- 97 * 1 98 * w+h - --------- 99 * w+2h - ... 100 * 101 * To determine how many terms needed, let 102 * Q(0) = w, Q(1) = w(w+h) - 1, 103 * Q(k) = (w+k*h)*Q(k-1) - Q(k-2), 104 * When Q(k) > 1e4 good for single 105 * When Q(k) > 1e9 good for double 106 * When Q(k) > 1e17 good for quadruple 107 */ 108 /* determine k */ 109 float t,v; 110 float q0,q1,h,tmp; int32_t k,m; 111 w = (n+n)/(float)x; h = (float)2.0/(float)x; 112 q0 = w; z = w+h; q1 = w*z - (float)1.0; k=1; 113 while(q1<(float)1.0e9) { 114 k += 1; z += h; 115 tmp = z*q1 - q0; 116 q0 = q1; 117 q1 = tmp; 118 } 119 m = n+n; 120 for(t=zero, i = 2*(n+k); i>=m; i -= 2) t = one/(i/x-t); 121 a = t; 122 b = one; 123 /* estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n) 124 * Hence, if n*(log(2n/x)) > ... 125 * single 8.8722839355e+01 126 * double 7.09782712893383973096e+02 127 * long double 1.1356523406294143949491931077970765006170e+04 128 * then recurrent value may overflow and the result is 129 * likely underflow to zero 130 */ 131 tmp = n; 132 v = two/x; 133 tmp = tmp*__ieee754_logf(fabsf(v*tmp)); 134 if(tmp<(float)8.8721679688e+01) { 135 for(i=n-1,di=(float)(i+i);i>0;i--){ 136 temp = b; 137 b *= di; 138 b = b/x - a; 139 a = temp; 140 di -= two; 141 } 142 } else { 143 for(i=n-1,di=(float)(i+i);i>0;i--){ 144 temp = b; 145 b *= di; 146 b = b/x - a; 147 a = temp; 148 di -= two; 149 /* scale b to avoid spurious overflow */ 150 if(b>(float)1e10) { 151 a /= b; 152 t /= b; 153 b = one; 154 } 155 } 156 } 157 b = (t*__ieee754_j0f(x)/b); 158 } 159 } 160 if(sgn==1) return -b; else return b; 161 } 162 163 float 164 __ieee754_ynf(int n, float x) 165 { 166 int32_t i,hx,ix,ib; 167 int32_t sign; 168 float a, b, temp; 169 170 GET_FLOAT_WORD(hx,x); 171 ix = 0x7fffffff&hx; 172 /* if Y(n,NaN) is NaN */ 173 if(ix>0x7f800000) return x+x; 174 if(ix==0) return -one/zero; 175 if(hx<0) return zero/zero; 176 sign = 1; 177 if(n<0){ 178 n = -n; 179 sign = 1 - ((n&1)<<1); 180 } 181 if(n==0) return(__ieee754_y0f(x)); 182 if(n==1) return(sign*__ieee754_y1f(x)); 183 if(ix==0x7f800000) return zero; 184 185 a = __ieee754_y0f(x); 186 b = __ieee754_y1f(x); 187 /* quit if b is -inf */ 188 GET_FLOAT_WORD(ib,b); 189 for(i=1;i<n&&ib!=0xff800000;i++){ 190 temp = b; 191 b = ((float)(i+i)/x)*b - a; 192 GET_FLOAT_WORD(ib,b); 193 a = temp; 194 } 195 if(sign>0) return b; else return -b; 196 } 197