lib/msun/src/e_jnf.c

   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  * $FreeBSD: src/lib/msun/src/e_jnf.c,v 1.6 1999/08/28 00:06:34 peter Exp $
   5  * $DragonFly: src/lib/msun/src/Attic/e_jnf.c,v 1.2 2003/06/17 04:26:52 dillon Exp $
   6  */
   7
   8 /*
   9  * ====================================================
  10  * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
  11  *
  12  * Developed at SunPro, a Sun Microsystems, Inc. business.
  13  * Permission to use, copy, modify, and distribute this
  14  * software is freely granted, provided that this notice
  15  * is preserved.
  16  * ====================================================
  17  */
  18
  19 #include "math.h"
  20 #include "math_private.h"
  21
  22 #ifdef __STDC__
  23 static const float
  24 #else
  25 static float
  26 #endif
  27 invsqrtpi=  5.6418961287e-01, /* 0x3f106ebb */
  28 two   =  2.0000000000e+00, /* 0x40000000 */
  29 one   =  1.0000000000e+00; /* 0x3F800000 */
  30
  31 #ifdef __STDC__
  32 static const float zero  =  0.0000000000e+00;
  33 #else
  34 static float zero  =  0.0000000000e+00;
  35 #endif
  36
  37 #ifdef __STDC__
  38         float __ieee754_jnf(int n, float x)
  39 #else
  40         float __ieee754_jnf(n,x)
  41         int n; float x;
  42 #endif
  43 {
  44         int32_t i,hx,ix, sgn;
  45         float a, b, temp, di;
  46         float z, w;
  47
  48     /* J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x)
  49      * Thus, J(-n,x) = J(n,-x)
  50      */
  51         GET_FLOAT_WORD(hx,x);
  52         ix = 0x7fffffff&hx;
  53     /* if J(n,NaN) is NaN */
  54         if(ix>0x7f800000) return x+x;
  55         if(n<0){
  56                 n = -n;
  57                 x = -x;
  58                 hx ^= 0x80000000;
  59         }
  60         if(n==0) return(__ieee754_j0f(x));
  61         if(n==1) return(__ieee754_j1f(x));
  62         sgn = (n&1)&(hx>>31);   /* even n -- 0, odd n -- sign(x) */
  63         x = fabsf(x);
  64         if(ix==0||ix>=0x7f800000)       /* if x is 0 or inf */
  65             b = zero;
  66         else if((float)n<=x) {
  67                 /* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */
  68             a = __ieee754_j0f(x);
  69             b = __ieee754_j1f(x);
  70             for(i=1;i<n;i++){
  71                 temp = b;
  72                 b = b*((float)(i+i)/x) - a; /* avoid underflow */
  73                 a = temp;
  74             }
  75         } else {
  76             if(ix<0x30800000) { /* x < 2**-29 */
  77     /* x is tiny, return the first Taylor expansion of J(n,x)
  78      * J(n,x) = 1/n!*(x/2)^n  - ...
  79      */
  80                 if(n>33)        /* underflow */
  81                     b = zero;
  82                 else {
  83                     temp = x*(float)0.5; b = temp;
  84                     for (a=one,i=2;i<=n;i++) {
  85                         a *= (float)i;          /* a = n! */
  86                         b *= temp;              /* b = (x/2)^n */
  87                     }
  88                     b = b/a;
  89                 }
  90             } else {
  91                 /* use backward recurrence */
  92                 /*                      x      x^2      x^2
  93                  *  J(n,x)/J(n-1,x) =  ----   ------   ------   .....
  94                  *                      2n  - 2(n+1) - 2(n+2)
  95                  *
  96                  *                      1      1        1
  97                  *  (for large x)   =  ----  ------   ------   .....
  98                  *                      2n   2(n+1)   2(n+2)
  99                  *                      -- - ------ - ------ -
 100                  *                       x     x         x
 101                  *
 102                  * Let w = 2n/x and h=2/x, then the above quotient
 103                  * is equal to the continued fraction:
 104                  *                  1
 105                  *      = -----------------------
 106                  *                     1
 107                  *         w - -----------------
 108                  *                        1
 109                  *              w+h - ---------
 110                  *                     w+2h - ...
 111                  *
 112                  * To determine how many terms needed, let
 113                  * Q(0) = w, Q(1) = w(w+h) - 1,
 114                  * Q(k) = (w+k*h)*Q(k-1) - Q(k-2),
 115                  * When Q(k) > 1e4      good for single
 116                  * When Q(k) > 1e9      good for double
 117                  * When Q(k) > 1e17     good for quadruple
 118                  */
 119             /* determine k */
 120                 float t,v;
 121                 float q0,q1,h,tmp; int32_t k,m;
 122                 w  = (n+n)/(float)x; h = (float)2.0/(float)x;
 123                 q0 = w;  z = w+h; q1 = w*z - (float)1.0; k=1;
 124                 while(q1<(float)1.0e9) {
 125                         k += 1; z += h;
 126                         tmp = z*q1 - q0;
 127                         q0 = q1;
 128                         q1 = tmp;
 129                 }
 130                 m = n+n;
 131                 for(t=zero, i = 2*(n+k); i>=m; i -= 2) t = one/(i/x-t);
 132                 a = t;
 133                 b = one;
 134                 /*  estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n)
 135                  *  Hence, if n*(log(2n/x)) > ...
 136                  *  single 8.8722839355e+01
 137                  *  double 7.09782712893383973096e+02
 138                  *  long double 1.1356523406294143949491931077970765006170e+04
 139                  *  then recurrent value may overflow and the result is
 140                  *  likely underflow to zero
 141                  */
 142                 tmp = n;
 143                 v = two/x;
 144                 tmp = tmp*__ieee754_logf(fabsf(v*tmp));
 145                 if(tmp<(float)8.8721679688e+01) {
 146                     for(i=n-1,di=(float)(i+i);i>0;i--){
 147                         temp = b;
 148                         b *= di;
 149                         b  = b/x - a;
 150                         a = temp;
 151                         di -= two;
 152                     }
 153                 } else {
 154                     for(i=n-1,di=(float)(i+i);i>0;i--){
 155                         temp = b;
 156                         b *= di;
 157                         b  = b/x - a;
 158                         a = temp;
 159                         di -= two;
 160                     /* scale b to avoid spurious overflow */
 161                         if(b>(float)1e10) {
 162                             a /= b;
 163                             t /= b;
 164                             b  = one;
 165                         }
 166                     }
 167                 }
 168                 b = (t*__ieee754_j0f(x)/b);
 169             }
 170         }
 171         if(sgn==1) return -b; else return b;
 172 }
 173
 174 #ifdef __STDC__
 175         float __ieee754_ynf(int n, float x)
 176 #else
 177         float __ieee754_ynf(n,x)
 178         int n; float x;
 179 #endif
 180 {
 181         int32_t i,hx,ix,ib;
 182         int32_t sign;
 183         float a, b, temp;
 184
 185         GET_FLOAT_WORD(hx,x);
 186         ix = 0x7fffffff&hx;
 187     /* if Y(n,NaN) is NaN */
 188         if(ix>0x7f800000) return x+x;
 189         if(ix==0) return -one/zero;
 190         if(hx<0) return zero/zero;
 191         sign = 1;
 192         if(n<0){
 193                 n = -n;
 194                 sign = 1 - ((n&1)<<1);
 195         }
 196         if(n==0) return(__ieee754_y0f(x));
 197         if(n==1) return(sign*__ieee754_y1f(x));
 198         if(ix==0x7f800000) return zero;
 199
 200         a = __ieee754_y0f(x);
 201         b = __ieee754_y1f(x);
 202         /* quit if b is -inf */
 203         GET_FLOAT_WORD(ib,b);
 204         for(i=1;i<n&&ib!=0xff800000;i++){
 205             temp = b;
 206             b = ((float)(i+i)/x)*b - a;
 207             GET_FLOAT_WORD(ib,b);
 208             a = temp;
 209         }
 210         if(sign>0) return b; else return -b;
 211 }