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 "cdefs-compat.h" 17 //__FBSDID("$FreeBSD: src/lib/msun/src/e_jnf.c,v 1.11 2010/11/13 10:54:10 uqs Exp $"); 18 19 #include <openlibm_math.h> 20 21 #include "math_private.h" 22 23 static const float 24 two = 2.0000000000e+00, /* 0x40000000 */ 25 one = 1.0000000000e+00; /* 0x3F800000 */ 26 27 static const float zero = 0.0000000000e+00; 28 29 DLLEXPORT float 30 __ieee754_jnf(int n, float x) 31 { 32 int32_t i,hx,ix, sgn; 33 float a, b, temp, di; 34 float z, w; 35 36 /* J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x) 37 * Thus, J(-n,x) = J(n,-x) 38 */ 39 GET_FLOAT_WORD(hx,x); 40 ix = 0x7fffffff&hx; 41 /* if J(n,NaN) is NaN */ 42 if(ix>0x7f800000) return x+x; 43 if(n<0){ 44 n = -n; 45 x = -x; 46 hx ^= 0x80000000; 47 } 48 if(n==0) return(__ieee754_j0f(x)); 49 if(n==1) return(__ieee754_j1f(x)); 50 sgn = (n&1)&(hx>>31); /* even n -- 0, odd n -- sign(x) */ 51 x = fabsf(x); 52 if(ix==0||ix>=0x7f800000) /* if x is 0 or inf */ 53 b = zero; 54 else if((float)n<=x) { 55 /* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */ 56 a = __ieee754_j0f(x); 57 b = __ieee754_j1f(x); 58 for(i=1;i<n;i++){ 59 temp = b; 60 b = b*((float)(i+i)/x) - a; /* avoid underflow */ 61 a = temp; 62 } 63 } else { 64 if(ix<0x30800000) { /* x < 2**-29 */ 65 /* x is tiny, return the first Taylor expansion of J(n,x) 66 * J(n,x) = 1/n!*(x/2)^n - ... 67 */ 68 if(n>33) /* underflow */ 69 b = zero; 70 else { 71 temp = x*(float)0.5; b = temp; 72 for (a=one,i=2;i<=n;i++) { 73 a *= (float)i; /* a = n! */ 74 b *= temp; /* b = (x/2)^n */ 75 } 76 b = b/a; 77 } 78 } else { 79 /* use backward recurrence */ 80 /* x x^2 x^2 81 * J(n,x)/J(n-1,x) = ---- ------ ------ ..... 82 * 2n - 2(n+1) - 2(n+2) 83 * 84 * 1 1 1 85 * (for large x) = ---- ------ ------ ..... 86 * 2n 2(n+1) 2(n+2) 87 * -- - ------ - ------ - 88 * x x x 89 * 90 * Let w = 2n/x and h=2/x, then the above quotient 91 * is equal to the continued fraction: 92 * 1 93 * = ----------------------- 94 * 1 95 * w - ----------------- 96 * 1 97 * w+h - --------- 98 * w+2h - ... 99 * 100 * To determine how many terms needed, let 101 * Q(0) = w, Q(1) = w(w+h) - 1, 102 * Q(k) = (w+k*h)*Q(k-1) - Q(k-2), 103 * When Q(k) > 1e4 good for single 104 * When Q(k) > 1e9 good for double 105 * When Q(k) > 1e17 good for quadruple 106 */ 107 /* determine k */ 108 float t,v; 109 float q0,q1,h,tmp; int32_t k,m; 110 w = (n+n)/(float)x; h = (float)2.0/(float)x; 111 q0 = w; z = w+h; q1 = w*z - (float)1.0; k=1; 112 while(q1<(float)1.0e9) { 113 k += 1; z += h; 114 tmp = z*q1 - q0; 115 q0 = q1; 116 q1 = tmp; 117 } 118 m = n+n; 119 for(t=zero, i = 2*(n+k); i>=m; i -= 2) t = one/(i/x-t); 120 a = t; 121 b = one; 122 /* estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n) 123 * Hence, if n*(log(2n/x)) > ... 124 * single 8.8722839355e+01 125 * double 7.09782712893383973096e+02 126 * long double 1.1356523406294143949491931077970765006170e+04 127 * then recurrent value may overflow and the result is 128 * likely underflow to zero 129 */ 130 tmp = n; 131 v = two/x; 132 tmp = tmp*__ieee754_logf(fabsf(v*tmp)); 133 if(tmp<(float)8.8721679688e+01) { 134 for(i=n-1,di=(float)(i+i);i>0;i--){ 135 temp = b; 136 b *= di; 137 b = b/x - a; 138 a = temp; 139 di -= two; 140 } 141 } else { 142 for(i=n-1,di=(float)(i+i);i>0;i--){ 143 temp = b; 144 b *= di; 145 b = b/x - a; 146 a = temp; 147 di -= two; 148 /* scale b to avoid spurious overflow */ 149 if(b>(float)1e10) { 150 a /= b; 151 t /= b; 152 b = one; 153 } 154 } 155 } 156 z = __ieee754_j0f(x); 157 w = __ieee754_j1f(x); 158 if (fabsf(z) >= fabsf(w)) 159 b = (t*z/b); 160 else 161 b = (t*w/a); 162 } 163 } 164 if(sgn==1) return -b; else return b; 165 } 166 167 DLLEXPORT float 168 __ieee754_ynf(int n, float x) 169 { 170 int32_t i,hx,ix,ib; 171 int32_t sign; 172 float a, b, temp; 173 174 GET_FLOAT_WORD(hx,x); 175 ix = 0x7fffffff&hx; 176 /* if Y(n,NaN) is NaN */ 177 if(ix>0x7f800000) return x+x; 178 if(ix==0) return -one/zero; 179 if(hx<0) return zero/zero; 180 sign = 1; 181 if(n<0){ 182 n = -n; 183 sign = 1 - ((n&1)<<1); 184 } 185 if(n==0) return(__ieee754_y0f(x)); 186 if(n==1) return(sign*__ieee754_y1f(x)); 187 if(ix==0x7f800000) return zero; 188 189 a = __ieee754_y0f(x); 190 b = __ieee754_y1f(x); 191 /* quit if b is -inf */ 192 GET_FLOAT_WORD(ib,b); 193 for(i=1;i<n&&ib!=0xff800000;i++){ 194 temp = b; 195 b = ((float)(i+i)/x)*b - a; 196 GET_FLOAT_WORD(ib,b); 197 a = temp; 198 } 199 if(sign>0) return b; else return -b; 200 } 201