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