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 #include "math.h"
  20 #include "math_private.h"
  21
  22 static const float
  23 two   =  2.0000000000e+00, /* 0x40000000 */
  24 one   =  1.0000000000e+00; /* 0x3F800000 */
  25
  26 static const float zero  =  0.0000000000e+00;
  27
  28 float
  29 __ieee754_jnf(int n, float x)
  30 {
  31         int32_t i,hx,ix, sgn;
  32         float a, b, temp, di;
  33         float z, w;
  34
  35     /* J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x)
  36      * Thus, J(-n,x) = J(n,-x)
  37      */
  38         GET_FLOAT_WORD(hx,x);
  39         ix = 0x7fffffff&hx;
  40     /* if J(n,NaN) is NaN */
  41         if(ix>0x7f800000) return x+x;
  42         if(n<0){
  43                 n = -n;
  44                 x = -x;
  45                 hx ^= 0x80000000;
  46         }
  47         if(n==0) return(__ieee754_j0f(x));
  48         if(n==1) return(__ieee754_j1f(x));
  49         sgn = (n&1)&(hx>>31);   /* even n -- 0, odd n -- sign(x) */
  50         x = fabsf(x);
  51         if(ix==0||ix>=0x7f800000)       /* if x is 0 or inf */
  52             b = zero;
  53         else if((float)n<=x) {
  54                 /* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */
  55             a = __ieee754_j0f(x);
  56             b = __ieee754_j1f(x);
  57             for(i=1;i<n;i++){
  58                 temp = b;
  59                 b = b*((float)(i+i)/x) - a; /* avoid underflow */
  60                 a = temp;
  61             }
  62         } else {
  63             if(ix<0x30800000) { /* x < 2**-29 */
  64     /* x is tiny, return the first Taylor expansion of J(n,x)
  65      * J(n,x) = 1/n!*(x/2)^n  - ...
  66      */
  67                 if(n>33)        /* underflow */
  68                     b = zero;
  69                 else {
  70                     temp = x*(float)0.5; b = temp;
  71                     for (a=one,i=2;i<=n;i++) {
  72                         a *= (float)i;          /* a = n! */
  73                         b *= temp;              /* b = (x/2)^n */
  74                     }
  75                     b = b/a;
  76                 }
  77             } else {
  78                 /* use backward recurrence */
  79                 /*                      x      x^2      x^2
  80                  *  J(n,x)/J(n-1,x) =  ----   ------   ------   .....
  81                  *                      2n  - 2(n+1) - 2(n+2)
  82                  *
  83                  *                      1      1        1
  84                  *  (for large x)   =  ----  ------   ------   .....
  85                  *                      2n   2(n+1)   2(n+2)
  86                  *                      -- - ------ - ------ -
  87                  *                       x     x         x
  88                  *
  89                  * Let w = 2n/x and h=2/x, then the above quotient
  90                  * is equal to the continued fraction:
  91                  *                  1
  92                  *      = -----------------------
  93                  *                     1
  94                  *         w - -----------------
  95                  *                        1
  96                  *              w+h - ---------
  97                  *                     w+2h - ...
  98                  *
  99                  * To determine how many terms needed, let
 100                  * Q(0) = w, Q(1) = w(w+h) - 1,
 101                  * Q(k) = (w+k*h)*Q(k-1) - Q(k-2),
 102                  * When Q(k) > 1e4      good for single
 103                  * When Q(k) > 1e9      good for double
 104                  * When Q(k) > 1e17     good for quadruple
 105                  */
 106             /* determine k */
 107                 float t,v;
 108                 float q0,q1,h,tmp; int32_t k,m;
 109                 w  = (n+n)/(float)x; h = (float)2.0/(float)x;
 110                 q0 = w;  z = w+h; q1 = w*z - (float)1.0; k=1;
 111                 while(q1<(float)1.0e9) {
 112                         k += 1; z += h;
 113                         tmp = z*q1 - q0;
 114                         q0 = q1;
 115                         q1 = tmp;
 116                 }
 117                 m = n+n;
 118                 for(t=zero, i = 2*(n+k); i>=m; i -= 2) t = one/(i/x-t);
 119                 a = t;
 120                 b = one;
 121                 /*  estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n)
 122                  *  Hence, if n*(log(2n/x)) > ...
 123                  *  single 8.8722839355e+01
 124                  *  double 7.09782712893383973096e+02
 125                  *  long double 1.1356523406294143949491931077970765006170e+04
 126                  *  then recurrent value may overflow and the result is
 127                  *  likely underflow to zero
 128                  */
 129                 tmp = n;
 130                 v = two/x;
 131                 tmp = tmp*__ieee754_logf(fabsf(v*tmp));
 132                 if(tmp<(float)8.8721679688e+01) {
 133                     for(i=n-1,di=(float)(i+i);i>0;i--){
 134                         temp = b;
 135                         b *= di;
 136                         b  = b/x - a;
 137                         a = temp;
 138                         di -= two;
 139                     }
 140                 } else {
 141                     for(i=n-1,di=(float)(i+i);i>0;i--){
 142                         temp = b;
 143                         b *= di;
 144                         b  = b/x - a;
 145                         a = temp;
 146                         di -= two;
 147                     /* scale b to avoid spurious overflow */
 148                         if(b>(float)1e10) {
 149                             a /= b;
 150                             t /= b;
 151                             b  = one;
 152                         }
 153                     }
 154                 }
 155                 z = __ieee754_j0f(x);
 156                 w = __ieee754_j1f(x);
 157                 if (fabsf(z) >= fabsf(w))
 158                     b = (t*z/b);
 159                 else
 160                     b = (t*w/a);
 161             }
 162         }
 163         if(sgn==1) return -b; else return b;
 164 }
 165
 166 float
 167 __ieee754_ynf(int n, float x)
 168 {
 169         int32_t i,hx,ix,ib;
 170         int32_t sign;
 171         float a, b, temp;
 172
 173         GET_FLOAT_WORD(hx,x);
 174         ix = 0x7fffffff&hx;
 175     /* if Y(n,NaN) is NaN */
 176         if(ix>0x7f800000) return x+x;
 177         if(ix==0) return -one/zero;
 178         if(hx<0) return zero/zero;
 179         sign = 1;
 180         if(n<0){
 181                 n = -n;
 182                 sign = 1 - ((n&1)<<1);
 183         }
 184         if(n==0) return(__ieee754_y0f(x));
 185         if(n==1) return(sign*__ieee754_y1f(x));
 186         if(ix==0x7f800000) return zero;
 187
 188         a = __ieee754_y0f(x);
 189         b = __ieee754_y1f(x);
 190         /* quit if b is -inf */
 191         GET_FLOAT_WORD(ib,b);
 192         for(i=1;i<n&&ib!=0xff800000;i++){
 193             temp = b;
 194             b = ((float)(i+i)/x)*b - a;
 195             GET_FLOAT_WORD(ib,b);
 196             a = temp;
 197         }
 198         if(sign>0) return b; else return -b;
 199 }