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