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 /* 18 * See e_jn.c for complete comments. 19 */ 20 21 #include "math.h" 22 #include "math_private.h" 23 24 static const volatile float vone = 1, vzero = 0; 25 26 static const float 27 two = 2.0000000000e+00, /* 0x40000000 */ 28 one = 1.0000000000e+00; /* 0x3F800000 */ 29 30 static const float zero = 0.0000000000e+00; 31 32 float 33 jnf(int n, float x) 34 { 35 int32_t i,hx,ix, sgn; 36 float a, b, temp, di; 37 float z, w; 38 39 /* J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x) 40 * Thus, J(-n,x) = J(n,-x) 41 */ 42 GET_FLOAT_WORD(hx,x); 43 ix = 0x7fffffff&hx; 44 /* if J(n,NaN) is NaN */ 45 if(ix>0x7f800000) return x+x; 46 if(n<0){ 47 n = -n; 48 x = -x; 49 hx ^= 0x80000000; 50 } 51 if(n==0) return(j0f(x)); 52 if(n==1) return(j1f(x)); 53 sgn = (n&1)&(hx>>31); /* even n -- 0, odd n -- sign(x) */ 54 x = fabsf(x); 55 if(ix==0||ix>=0x7f800000) /* if x is 0 or inf */ 56 b = zero; 57 else if((float)n<=x) { 58 /* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */ 59 a = j0f(x); 60 b = j1f(x); 61 for(i=1;i<n;i++){ 62 temp = b; 63 b = b*((float)(i+i)/x) - a; /* avoid underflow */ 64 a = temp; 65 } 66 } else { 67 if(ix<0x30800000) { /* x < 2**-29 */ 68 /* x is tiny, return the first Taylor expansion of J(n,x) 69 * J(n,x) = 1/n!*(x/2)^n - ... 70 */ 71 if(n>33) /* underflow */ 72 b = zero; 73 else { 74 temp = x*(float)0.5; b = temp; 75 for (a=one,i=2;i<=n;i++) { 76 a *= (float)i; /* a = n! */ 77 b *= temp; /* b = (x/2)^n */ 78 } 79 b = b/a; 80 } 81 } else { 82 /* use backward recurrence */ 83 /* x x^2 x^2 84 * J(n,x)/J(n-1,x) = ---- ------ ------ ..... 85 * 2n - 2(n+1) - 2(n+2) 86 * 87 * 1 1 1 88 * (for large x) = ---- ------ ------ ..... 89 * 2n 2(n+1) 2(n+2) 90 * -- - ------ - ------ - 91 * x x x 92 * 93 * Let w = 2n/x and h=2/x, then the above quotient 94 * is equal to the continued fraction: 95 * 1 96 * = ----------------------- 97 * 1 98 * w - ----------------- 99 * 1 100 * w+h - --------- 101 * w+2h - ... 102 * 103 * To determine how many terms needed, let 104 * Q(0) = w, Q(1) = w(w+h) - 1, 105 * Q(k) = (w+k*h)*Q(k-1) - Q(k-2), 106 * When Q(k) > 1e4 good for single 107 * When Q(k) > 1e9 good for double 108 * When Q(k) > 1e17 good for quadruple 109 */ 110 /* determine k */ 111 float t,v; 112 float q0,q1,h,tmp; int32_t k,m; 113 w = (n+n)/(float)x; h = (float)2.0/(float)x; 114 q0 = w; z = w+h; q1 = w*z - (float)1.0; k=1; 115 while(q1<(float)1.0e9) { 116 k += 1; z += h; 117 tmp = z*q1 - q0; 118 q0 = q1; 119 q1 = tmp; 120 } 121 m = n+n; 122 for(t=zero, i = 2*(n+k); i>=m; i -= 2) t = one/(i/x-t); 123 a = t; 124 b = one; 125 /* estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n) 126 * Hence, if n*(log(2n/x)) > ... 127 * single 8.8722839355e+01 128 * double 7.09782712893383973096e+02 129 * long double 1.1356523406294143949491931077970765006170e+04 130 * then recurrent value may overflow and the result is 131 * likely underflow to zero 132 */ 133 tmp = n; 134 v = two/x; 135 tmp = tmp*logf(fabsf(v*tmp)); 136 if(tmp<(float)8.8721679688e+01) { 137 for(i=n-1,di=(float)(i+i);i>0;i--){ 138 temp = b; 139 b *= di; 140 b = b/x - a; 141 a = temp; 142 di -= two; 143 } 144 } else { 145 for(i=n-1,di=(float)(i+i);i>0;i--){ 146 temp = b; 147 b *= di; 148 b = b/x - a; 149 a = temp; 150 di -= two; 151 /* scale b to avoid spurious overflow */ 152 if(b>(float)1e10) { 153 a /= b; 154 t /= b; 155 b = one; 156 } 157 } 158 } 159 z = j0f(x); 160 w = j1f(x); 161 if (fabsf(z) >= fabsf(w)) 162 b = (t*z/b); 163 else 164 b = (t*w/a); 165 } 166 } 167 if(sgn==1) return -b; else return b; 168 } 169 170 float 171 ynf(int n, float x) 172 { 173 int32_t i,hx,ix,ib; 174 int32_t sign; 175 float a, b, temp; 176 177 GET_FLOAT_WORD(hx,x); 178 ix = 0x7fffffff&hx; 179 if(ix>0x7f800000) return x+x; 180 if(ix==0) return -one/vzero; 181 if(hx<0) return vzero/vzero; 182 sign = 1; 183 if(n<0){ 184 n = -n; 185 sign = 1 - ((n&1)<<1); 186 } 187 if(n==0) return(y0f(x)); 188 if(n==1) return(sign*y1f(x)); 189 if(ix==0x7f800000) return zero; 190 191 a = y0f(x); 192 b = y1f(x); 193 /* quit if b is -inf */ 194 GET_FLOAT_WORD(ib,b); 195 for(i=1;i<n&&ib!=0xff800000;i++){ 196 temp = b; 197 b = ((float)(i+i)/x)*b - a; 198 GET_FLOAT_WORD(ib,b); 199 a = temp; 200 } 201 if(sign>0) return b; else return -b; 202 } 203