1 /* e_jnf.c -- float version of e_jn.c. 2 * Conversion to float by Ian Lance Taylor, Cygnus Support, ian (at) 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: head/lib/msun/src/e_jnf.c 279856 2015-03-10 17:10:54Z kargl $"); 18 19 /* 20 * See e_jn.c for complete comments. 21 */ 22 23 #include "math.h" 24 #include "math_private.h" 25 26 static const volatile float vone = 1, vzero = 0; 27 28 static const float 29 two = 2.0000000000e+00, /* 0x40000000 */ 30 one = 1.0000000000e+00; /* 0x3F800000 */ 31 32 static const float zero = 0.0000000000e+00; 33 34 float 35 __ieee754_jnf(int n, float x) 36 { 37 int32_t i,hx,ix, sgn; 38 float a, b, temp, di; 39 float z, w; 40 41 /* J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x) 42 * Thus, J(-n,x) = J(n,-x) 43 */ 44 GET_FLOAT_WORD(hx,x); 45 ix = 0x7fffffff&hx; 46 /* if J(n,NaN) is NaN */ 47 if(ix>0x7f800000) return x+x; 48 if(n<0){ 49 n = -n; 50 x = -x; 51 hx ^= 0x80000000; 52 } 53 if(n==0) return(__ieee754_j0f(x)); 54 if(n==1) return(__ieee754_j1f(x)); 55 sgn = (n&1)&(hx>>31); /* even n -- 0, odd n -- sign(x) */ 56 x = fabsf(x); 57 if(ix==0||ix>=0x7f800000) /* if x is 0 or inf */ 58 b = zero; 59 else if((float)n<=x) { 60 /* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */ 61 a = __ieee754_j0f(x); 62 b = __ieee754_j1f(x); 63 for(i=1;i<n;i++){ 64 temp = b; 65 b = b*((float)(i+i)/x) - a; /* avoid underflow */ 66 a = temp; 67 } 68 } else { 69 if(ix<0x30800000) { /* x < 2**-29 */ 70 /* x is tiny, return the first Taylor expansion of J(n,x) 71 * J(n,x) = 1/n!*(x/2)^n - ... 72 */ 73 if(n>33) /* underflow */ 74 b = zero; 75 else { 76 temp = x*(float)0.5; b = temp; 77 for (a=one,i=2;i<=n;i++) { 78 a *= (float)i; /* a = n! */ 79 b *= temp; /* b = (x/2)^n */ 80 } 81 b = b/a; 82 } 83 } else { 84 /* use backward recurrence */ 85 /* x x^2 x^2 86 * J(n,x)/J(n-1,x) = ---- ------ ------ ..... 87 * 2n - 2(n+1) - 2(n+2) 88 * 89 * 1 1 1 90 * (for large x) = ---- ------ ------ ..... 91 * 2n 2(n+1) 2(n+2) 92 * -- - ------ - ------ - 93 * x x x 94 * 95 * Let w = 2n/x and h=2/x, then the above quotient 96 * is equal to the continued fraction: 97 * 1 98 * = ----------------------- 99 * 1 100 * w - ----------------- 101 * 1 102 * w+h - --------- 103 * w+2h - ... 104 * 105 * To determine how many terms needed, let 106 * Q(0) = w, Q(1) = w(w+h) - 1, 107 * Q(k) = (w+k*h)*Q(k-1) - Q(k-2), 108 * When Q(k) > 1e4 good for single 109 * When Q(k) > 1e9 good for double 110 * When Q(k) > 1e17 good for quadruple 111 */ 112 /* determine k */ 113 float t,v; 114 float q0,q1,h,tmp; int32_t k,m; 115 w = (n+n)/(float)x; h = (float)2.0/(float)x; 116 q0 = w; z = w+h; q1 = w*z - (float)1.0; k=1; 117 while(q1<(float)1.0e9) { 118 k += 1; z += h; 119 tmp = z*q1 - q0; 120 q0 = q1; 121 q1 = tmp; 122 } 123 m = n+n; 124 for(t=zero, i = 2*(n+k); i>=m; i -= 2) t = one/(i/x-t); 125 a = t; 126 b = one; 127 /* estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n) 128 * Hence, if n*(log(2n/x)) > ... 129 * single 8.8722839355e+01 130 * double 7.09782712893383973096e+02 131 * long double 1.1356523406294143949491931077970765006170e+04 132 * then recurrent value may overflow and the result is 133 * likely underflow to zero 134 */ 135 tmp = n; 136 v = two/x; 137 tmp = tmp*__ieee754_logf(fabsf(v*tmp)); 138 if(tmp<(float)8.8721679688e+01) { 139 for(i=n-1,di=(float)(i+i);i>0;i--){ 140 temp = b; 141 b *= di; 142 b = b/x - a; 143 a = temp; 144 di -= two; 145 } 146 } else { 147 for(i=n-1,di=(float)(i+i);i>0;i--){ 148 temp = b; 149 b *= di; 150 b = b/x - a; 151 a = temp; 152 di -= two; 153 /* scale b to avoid spurious overflow */ 154 if(b>(float)1e10) { 155 a /= b; 156 t /= b; 157 b = one; 158 } 159 } 160 } 161 z = __ieee754_j0f(x); 162 w = __ieee754_j1f(x); 163 if (fabsf(z) >= fabsf(w)) 164 b = (t*z/b); 165 else 166 b = (t*w/a); 167 } 168 } 169 if(sgn==1) return -b; else return b; 170 } 171 172 float 173 __ieee754_ynf(int n, float x) 174 { 175 int32_t i,hx,ix,ib; 176 int32_t sign; 177 float a, b, temp; 178 179 GET_FLOAT_WORD(hx,x); 180 ix = 0x7fffffff&hx; 181 if(ix>0x7f800000) return x+x; 182 if(ix==0) return -one/vzero; 183 if(hx<0) return vzero/vzero; 184 sign = 1; 185 if(n<0){ 186 n = -n; 187 sign = 1 - ((n&1)<<1); 188 } 189 if(n==0) return(__ieee754_y0f(x)); 190 if(n==1) return(sign*__ieee754_y1f(x)); 191 if(ix==0x7f800000) return zero; 192 193 a = __ieee754_y0f(x); 194 b = __ieee754_y1f(x); 195 /* quit if b is -inf */ 196 GET_FLOAT_WORD(ib,b); 197 for(i=1;i<n&&ib!=0xff800000;i++){ 198 temp = b; 199 b = ((float)(i+i)/x)*b - a; 200 GET_FLOAT_WORD(ib,b); 201 a = temp; 202 } 203 if(sign>0) return b; else return -b; 204 } 205
