sysdeps/libm-ieee754/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 #if defined(LIBM_SCCS) && !defined(lint)
  17 static char rcsid[] = "$NetBSD: e_jnf.c,v 1.5 1995/05/10 20:45:37 jtc Exp $";
  18 #endif
  19
  20 #include "math.h"
  21 #include "math_private.h"
  22
  23 #ifdef __STDC__
  24 static const float
  25 #else
  26 static float
  27 #endif
  28 invsqrtpi=  5.6418961287e-01, /* 0x3f106ebb */
  29 two   =  2.0000000000e+00, /* 0x40000000 */
  30 one   =  1.0000000000e+00; /* 0x3F800000 */
  31
  32 #ifdef __STDC__
  33 static const float zero  =  0.0000000000e+00;
  34 #else
  35 static float zero  =  0.0000000000e+00;
  36 #endif
  37
  38 #ifdef __STDC__
  39         float __ieee754_jnf(int n, float x)
  40 #else
  41         float __ieee754_jnf(n,x)
  42         int n; float x;
  43 #endif
  44 {
  45         int32_t i,hx,ix, sgn;
  46         float a, b, temp, di;
  47         float z, w;
  48
  49     /* J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x)
  50      * Thus, J(-n,x) = J(n,-x)
  51      */
  52         GET_FLOAT_WORD(hx,x);
  53         ix = 0x7fffffff&hx;
  54     /* if J(n,NaN) is NaN */
  55         if(ix>0x7f800000) return x+x;
  56         if(n<0){
  57                 n = -n;
  58                 x = -x;
  59                 hx ^= 0x80000000;
  60         }
  61         if(n==0) return(__ieee754_j0f(x));
  62         if(n==1) return(__ieee754_j1f(x));
  63         sgn = (n&1)&(hx>>31);   /* even n -- 0, odd n -- sign(x) */
  64         x = fabsf(x);
  65         if(ix==0||ix>=0x7f800000)       /* if x is 0 or inf */
  66             b = zero;
  67         else if((float)n<=x) {
  68                 /* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */
  69             a = __ieee754_j0f(x);
  70             b = __ieee754_j1f(x);
  71             for(i=1;i<n;i++){
  72                 temp = b;
  73                 b = b*((float)(i+i)/x) - a; /* avoid underflow */
  74                 a = temp;
  75             }
  76         } else {
  77             if(ix<0x30800000) { /* x < 2**-29 */
  78     /* x is tiny, return the first Taylor expansion of J(n,x)
  79      * J(n,x) = 1/n!*(x/2)^n  - ...
  80      */
  81                 if(n>33)        /* underflow */
  82                     b = zero;
  83                 else {
  84                     temp = x*(float)0.5; b = temp;
  85                     for (a=one,i=2;i<=n;i++) {
  86                         a *= (float)i;          /* a = n! */
  87                         b *= temp;              /* b = (x/2)^n */
  88                     }
  89                     b = b/a;
  90                 }
  91             } else {
  92                 /* use backward recurrence */
  93                 /*                      x      x^2      x^2
  94                  *  J(n,x)/J(n-1,x) =  ----   ------   ------   .....
  95                  *                      2n  - 2(n+1) - 2(n+2)
  96                  *
  97                  *                      1      1        1
  98                  *  (for large x)   =  ----  ------   ------   .....
  99                  *                      2n   2(n+1)   2(n+2)
 100                  *                      -- - ------ - ------ -
 101                  *                       x     x         x
 102                  *
 103                  * Let w = 2n/x and h=2/x, then the above quotient
 104                  * is equal to the continued fraction:
 105                  *                  1
 106                  *      = -----------------------
 107                  *                     1
 108                  *         w - -----------------
 109                  *                        1
 110                  *              w+h - ---------
 111                  *                     w+2h - ...
 112                  *
 113                  * To determine how many terms needed, let
 114                  * Q(0) = w, Q(1) = w(w+h) - 1,
 115                  * Q(k) = (w+k*h)*Q(k-1) - Q(k-2),
 116                  * When Q(k) > 1e4      good for single
 117                  * When Q(k) > 1e9      good for double
 118                  * When Q(k) > 1e17     good for quadruple
 119                  */
 120             /* determine k */
 121                 float t,v;
 122                 float q0,q1,h,tmp; int32_t k,m;
 123                 w  = (n+n)/(float)x; h = (float)2.0/(float)x;
 124                 q0 = w;  z = w+h; q1 = w*z - (float)1.0; k=1;
 125                 while(q1<(float)1.0e9) {
 126                         k += 1; z += h;
 127                         tmp = z*q1 - q0;
 128                         q0 = q1;
 129                         q1 = tmp;
 130                 }
 131                 m = n+n;
 132                 for(t=zero, i = 2*(n+k); i>=m; i -= 2) t = one/(i/x-t);
 133                 a = t;
 134                 b = one;
 135                 /*  estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n)
 136                  *  Hence, if n*(log(2n/x)) > ...
 137                  *  single 8.8722839355e+01
 138                  *  double 7.09782712893383973096e+02
 139                  *  long double 1.1356523406294143949491931077970765006170e+04
 140                  *  then recurrent value may overflow and the result is
 141                  *  likely underflow to zero
 142                  */
 143                 tmp = n;
 144                 v = two/x;
 145                 tmp = tmp*__ieee754_logf(fabsf(v*tmp));
 146                 if(tmp<(float)8.8721679688e+01) {
 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                     }
 154                 } else {
 155                     for(i=n-1,di=(float)(i+i);i>0;i--){
 156                         temp = b;
 157                         b *= di;
 158                         b  = b/x - a;
 159                         a = temp;
 160                         di -= two;
 161                     /* scale b to avoid spurious overflow */
 162                         if(b>(float)1e10) {
 163                             a /= b;
 164                             t /= b;
 165                             b  = one;
 166                         }
 167                     }
 168                 }
 169                 b = (t*__ieee754_j0f(x)/b);
 170             }
 171         }
 172         if(sgn==1) return -b; else return b;
 173 }
 174
 175 #ifdef __STDC__
 176         float __ieee754_ynf(int n, float x)
 177 #else
 178         float __ieee754_ynf(n,x)
 179         int n; float x;
 180 #endif
 181 {
 182         int32_t i,hx,ix;
 183         u_int32_t ib;
 184         int32_t sign;
 185         float a, b, temp;
 186
 187         GET_FLOAT_WORD(hx,x);
 188         ix = 0x7fffffff&hx;
 189     /* if Y(n,NaN) is NaN */
 190         if(ix>0x7f800000) return x+x;
 191         if(ix==0) return -one/zero;
 192         if(hx<0) return zero/zero;
 193         sign = 1;
 194         if(n<0){
 195                 n = -n;
 196                 sign = 1 - ((n&1)<<1);
 197         }
 198         if(n==0) return(__ieee754_y0f(x));
 199         if(n==1) return(sign*__ieee754_y1f(x));
 200         if(ix==0x7f800000) return zero;
 201
 202         a = __ieee754_y0f(x);
 203         b = __ieee754_y1f(x);
 204         /* quit if b is -inf */
 205         GET_FLOAT_WORD(ib,b);
 206         for(i=1;i<n&&ib!=0xff800000;i++){
 207             temp = b;
 208             b = ((float)(i+i)/x)*b - a;
 209             GET_FLOAT_WORD(ib,b);
 210             a = temp;
 211         }
 212         if(sign>0) return b; else return -b;
 213 }