lib/libm/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
   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  * $NetBSD: e_jnf.c,v 1.9 2002/05/26 22:01:50 wiz Exp $
  16  * $DragonFly: src/lib/libm/src/e_jnf.c,v 1.1 2005/07/26 21:15:20 joerg Exp $
  17  */
  18
  19 #include <math.h>
  20 #include "math_private.h"
  21
  22 static const float
  23 #if 0
  24 invsqrtpi=  5.6418961287e-01, /* 0x3f106ebb */
  25 #endif
  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                 b = (t*j0f(x)/b);
 159             }
 160         }
 161         if(sgn==1) return -b; else return b;
 162 }
 163
 164 float
 165 ynf(int n, float x)
 166 {
 167         int32_t i,hx,ix,ib;
 168         int32_t sign;
 169         float a, b, temp;
 170
 171         GET_FLOAT_WORD(hx,x);
 172         ix = 0x7fffffff&hx;
 173     /* if Y(n,NaN) is NaN */
 174         if(ix>0x7f800000) return x+x;
 175         if(ix==0) return -one/zero;
 176         if(hx<0) return zero/zero;
 177         sign = 1;
 178         if(n<0){
 179                 n = -n;
 180                 sign = 1 - ((n&1)<<1);
 181         }
 182         if(n==0) return(y0f(x));
 183         if(n==1) return(sign*y1f(x));
 184         if(ix==0x7f800000) return zero;
 185
 186         a = y0f(x);
 187         b = y1f(x);
 188         /* quit if b is -inf */
 189         GET_FLOAT_WORD(ib,b);
 190         for(i=1;i<n&&ib!=0xff800000;i++){
 191             temp = b;
 192             b = ((float)(i+i)/x)*b - a;
 193             GET_FLOAT_WORD(ib,b);
 194             a = temp;
 195         }
 196         if(sign>0) return b; else return -b;
 197 }