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
   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("$FreeBSD$");
  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 }