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 /* 17 * See e_jn.c for complete comments. 18 */ 19 20 #include "math.h" 21 #include "math_private.h" 22 23 static const volatile float vone = 1, vzero = 0; 24 25 static const float 26 two = 2.0000000000e+00, /* 0x40000000 */ 27 one = 1.0000000000e+00; /* 0x3F800000 */ 28 29 static const float zero = 0.0000000000e+00; 30 31 float 32 jnf(int n, float x) 33 { 34 int32_t i,hx,ix, sgn; 35 float a, b, temp, di; 36 float z, w; 37 38 /* J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x) 39 * Thus, J(-n,x) = J(n,-x) 40 */ 41 GET_FLOAT_WORD(hx,x); 42 ix = 0x7fffffff&hx; 43 /* if J(n,NaN) is NaN */ 44 if(ix>0x7f800000) return x+x; 45 if(n<0){ 46 n = -n; 47 x = -x; 48 hx ^= 0x80000000; 49 } 50 if(n==0) return(j0f(x)); 51 if(n==1) return(j1f(x)); 52 sgn = (n&1)&(hx>>31); /* even n -- 0, odd n -- sign(x) */ 53 x = fabsf(x); 54 if(ix==0||ix>=0x7f800000) /* if x is 0 or inf */ 55 b = zero; 56 else if((float)n<=x) { 57 /* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */ 58 a = j0f(x); 59 b = j1f(x); 60 for(i=1;i<n;i++){ 61 temp = b; 62 b = b*((float)(i+i)/x) - a; /* avoid underflow */ 63 a = temp; 64 } 65 } else { 66 if(ix<0x30800000) { /* x < 2**-29 */ 67 /* x is tiny, return the first Taylor expansion of J(n,x) 68 * J(n,x) = 1/n!*(x/2)^n - ... 69 */ 70 if(n>33) /* underflow */ 71 b = zero; 72 else { 73 temp = x*(float)0.5; b = temp; 74 for (a=one,i=2;i<=n;i++) { 75 a *= (float)i; /* a = n! */ 76 b *= temp; /* b = (x/2)^n */ 77 } 78 b = b/a; 79 } 80 } else { 81 /* use backward recurrence */ 82 /* x x^2 x^2 83 * J(n,x)/J(n-1,x) = ---- ------ ------ ..... 84 * 2n - 2(n+1) - 2(n+2) 85 * 86 * 1 1 1 87 * (for large x) = ---- ------ ------ ..... 88 * 2n 2(n+1) 2(n+2) 89 * -- - ------ - ------ - 90 * x x x 91 * 92 * Let w = 2n/x and h=2/x, then the above quotient 93 * is equal to the continued fraction: 94 * 1 95 * = ----------------------- 96 * 1 97 * w - ----------------- 98 * 1 99 * w+h - --------- 100 * w+2h - ... 101 * 102 * To determine how many terms needed, let 103 * Q(0) = w, Q(1) = w(w+h) - 1, 104 * Q(k) = (w+k*h)*Q(k-1) - Q(k-2), 105 * When Q(k) > 1e4 good for single 106 * When Q(k) > 1e9 good for double 107 * When Q(k) > 1e17 good for quadruple 108 */ 109 /* determine k */ 110 float t,v; 111 float q0,q1,h,tmp; int32_t k,m; 112 w = (n+n)/(float)x; h = (float)2.0/(float)x; 113 q0 = w; z = w+h; q1 = w*z - (float)1.0; k=1; 114 while(q1<(float)1.0e9) { 115 k += 1; z += h; 116 tmp = z*q1 - q0; 117 q0 = q1; 118 q1 = tmp; 119 } 120 m = n+n; 121 for(t=zero, i = 2*(n+k); i>=m; i -= 2) t = one/(i/x-t); 122 a = t; 123 b = one; 124 /* estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n) 125 * Hence, if n*(log(2n/x)) > ... 126 * single 8.8722839355e+01 127 * double 7.09782712893383973096e+02 128 * long double 1.1356523406294143949491931077970765006170e+04 129 * then recurrent value may overflow and the result is 130 * likely underflow to zero 131 */ 132 tmp = n; 133 v = two/x; 134 tmp = tmp*logf(fabsf(v*tmp)); 135 if(tmp<(float)8.8721679688e+01) { 136 for(i=n-1,di=(float)(i+i);i>0;i--){ 137 temp = b; 138 b *= di; 139 b = b/x - a; 140 a = temp; 141 di -= two; 142 } 143 } else { 144 for(i=n-1,di=(float)(i+i);i>0;i--){ 145 temp = b; 146 b *= di; 147 b = b/x - a; 148 a = temp; 149 di -= two; 150 /* scale b to avoid spurious overflow */ 151 if(b>(float)1e10) { 152 a /= b; 153 t /= b; 154 b = one; 155 } 156 } 157 } 158 z = j0f(x); 159 w = j1f(x); 160 if (fabsf(z) >= fabsf(w)) 161 b = (t*z/b); 162 else 163 b = (t*w/a); 164 } 165 } 166 if(sgn==1) return -b; else return b; 167 } 168 169 float 170 ynf(int n, float x) 171 { 172 int32_t i,hx,ix,ib; 173 int32_t sign; 174 float a, b, temp; 175 176 GET_FLOAT_WORD(hx,x); 177 ix = 0x7fffffff&hx; 178 if(ix>0x7f800000) return x+x; 179 if(ix==0) return -one/vzero; 180 if(hx<0) return vzero/vzero; 181 sign = 1; 182 if(n<0){ 183 n = -n; 184 sign = 1 - ((n&1)<<1); 185 } 186 if(n==0) return(y0f(x)); 187 if(n==1) return(sign*y1f(x)); 188 if(ix==0x7f800000) return zero; 189 190 a = y0f(x); 191 b = y1f(x); 192 /* quit if b is -inf */ 193 GET_FLOAT_WORD(ib,b); 194 for(i=1;i<n&&ib!=0xff800000;i++){ 195 temp = b; 196 b = ((float)(i+i)/x)*b - a; 197 GET_FLOAT_WORD(ib,b); 198 a = temp; 199 } 200 if(sign>0) return b; else return -b; 201 } 202