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