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  * $FreeBSD: head/lib/msun/src/e_jnf.c 215237 2010-11-13 10:54:10Z uqs $
  16  */
  17
  18 #include "math.h"
  19 #include "math_private.h"
  20
  21 static const float
  22 two   =  2.0000000000e+00, /* 0x40000000 */
  23 one   =  1.0000000000e+00; /* 0x3F800000 */
  24
  25 static const float zero  =  0.0000000000e+00;
  26
  27 float
  28 __ieee754_jnf(int n, float x)
  29 {
  30         int32_t i,hx,ix, sgn;
  31         float a, b, temp, di;
  32         float z, w;
  33
  34     /* J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x)
  35      * Thus, J(-n,x) = J(n,-x)
  36      */
  37         GET_FLOAT_WORD(hx,x);
  38         ix = 0x7fffffff&hx;
  39     /* if J(n,NaN) is NaN */
  40         if(ix>0x7f800000) return x+x;
  41         if(n<0){
  42                 n = -n;
  43                 x = -x;
  44                 hx ^= 0x80000000;
  45         }
  46         if(n==0) return(__ieee754_j0f(x));
  47         if(n==1) return(__ieee754_j1f(x));
  48         sgn = (n&1)&(hx>>31);   /* even n -- 0, odd n -- sign(x) */
  49         x = fabsf(x);
  50         if(ix==0||ix>=0x7f800000)       /* if x is 0 or inf */
  51             b = zero;
  52         else if((float)n<=x) {
  53                 /* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */
  54             a = __ieee754_j0f(x);
  55             b = __ieee754_j1f(x);
  56             for(i=1;i<n;i++){
  57                 temp = b;
  58                 b = b*((float)(i+i)/x) - a; /* avoid underflow */
  59                 a = temp;
  60             }
  61         } else {
  62             if(ix<0x30800000) { /* x < 2**-29 */
  63     /* x is tiny, return the first Taylor expansion of J(n,x)
  64      * J(n,x) = 1/n!*(x/2)^n  - ...
  65      */
  66                 if(n>33)        /* underflow */
  67                     b = zero;
  68                 else {
  69                     temp = x*(float)0.5; b = temp;
  70                     for (a=one,i=2;i<=n;i++) {
  71                         a *= (float)i;          /* a = n! */
  72                         b *= temp;              /* b = (x/2)^n */
  73                     }
  74                     b = b/a;
  75                 }
  76             } else {
  77                 /* use backward recurrence */
  78                 /*                      x      x^2      x^2
  79                  *  J(n,x)/J(n-1,x) =  ----   ------   ------   .....
  80                  *                      2n  - 2(n+1) - 2(n+2)
  81                  *
  82                  *                      1      1        1
  83                  *  (for large x)   =  ----  ------   ------   .....
  84                  *                      2n   2(n+1)   2(n+2)
  85                  *                      -- - ------ - ------ -
  86                  *                       x     x         x
  87                  *
  88                  * Let w = 2n/x and h=2/x, then the above quotient
  89                  * is equal to the continued fraction:
  90                  *                  1
  91                  *      = -----------------------
  92                  *                     1
  93                  *         w - -----------------
  94                  *                        1
  95                  *              w+h - ---------
  96                  *                     w+2h - ...
  97                  *
  98                  * To determine how many terms needed, let
  99                  * Q(0) = w, Q(1) = w(w+h) - 1,
 100                  * Q(k) = (w+k*h)*Q(k-1) - Q(k-2),
 101                  * When Q(k) > 1e4      good for single
 102                  * When Q(k) > 1e9      good for double
 103                  * When Q(k) > 1e17     good for quadruple
 104                  */
 105             /* determine k */
 106                 float t,v;
 107                 float q0,q1,h,tmp; int32_t k,m;
 108                 w  = (n+n)/(float)x; h = (float)2.0/(float)x;
 109                 q0 = w;  z = w+h; q1 = w*z - (float)1.0; k=1;
 110                 while(q1<(float)1.0e9) {
 111                         k += 1; z += h;
 112                         tmp = z*q1 - q0;
 113                         q0 = q1;
 114                         q1 = tmp;
 115                 }
 116                 m = n+n;
 117                 for(t=zero, i = 2*(n+k); i>=m; i -= 2) t = one/(i/x-t);
 118                 a = t;
 119                 b = one;
 120                 /*  estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n)
 121                  *  Hence, if n*(log(2n/x)) > ...
 122                  *  single 8.8722839355e+01
 123                  *  double 7.09782712893383973096e+02
 124                  *  long double 1.1356523406294143949491931077970765006170e+04
 125                  *  then recurrent value may overflow and the result is
 126                  *  likely underflow to zero
 127                  */
 128                 tmp = n;
 129                 v = two/x;
 130                 tmp = tmp*__ieee754_logf(fabsf(v*tmp));
 131                 if(tmp<(float)8.8721679688e+01) {
 132                     for(i=n-1,di=(float)(i+i);i>0;i--){
 133                         temp = b;
 134                         b *= di;
 135                         b  = b/x - a;
 136                         a = temp;
 137                         di -= two;
 138                     }
 139                 } else {
 140                     for(i=n-1,di=(float)(i+i);i>0;i--){
 141                         temp = b;
 142                         b *= di;
 143                         b  = b/x - a;
 144                         a = temp;
 145                         di -= two;
 146                     /* scale b to avoid spurious overflow */
 147                         if(b>(float)1e10) {
 148                             a /= b;
 149                             t /= b;
 150                             b  = one;
 151                         }
 152                     }
 153                 }
 154                 z = __ieee754_j0f(x);
 155                 w = __ieee754_j1f(x);
 156                 if (fabsf(z) >= fabsf(w))
 157                     b = (t*z/b);
 158                 else
 159                     b = (t*w/a);
 160             }
 161         }
 162         if(sgn==1) return -b; else return b;
 163 }
 164
 165 float
 166 __ieee754_ynf(int n, float x)
 167 {
 168         int32_t i,hx,ix,ib;
 169         int32_t sign;
 170         float a, b, temp;
 171
 172         GET_FLOAT_WORD(hx,x);
 173         ix = 0x7fffffff&hx;
 174     /* if Y(n,NaN) is NaN */
 175         if(ix>0x7f800000) return x+x;
 176         if(ix==0) return -one/zero;
 177         if(hx<0) return zero/zero;
 178         sign = 1;
 179         if(n<0){
 180                 n = -n;
 181                 sign = 1 - ((n&1)<<1);
 182         }
 183         if(n==0) return(__ieee754_y0f(x));
 184         if(n==1) return(sign*__ieee754_y1f(x));
 185         if(ix==0x7f800000) return zero;
 186
 187         a = __ieee754_y0f(x);
 188         b = __ieee754_y1f(x);
 189         /* quit if b is -inf */
 190         GET_FLOAT_WORD(ib,b);
 191         for(i=1;i<n&&ib!=0xff800000;i++){
 192             temp = b;
 193             b = ((float)(i+i)/x)*b - a;
 194             GET_FLOAT_WORD(ib,b);
 195             a = temp;
 196         }
 197         if(sign>0) return b; else return -b;
 198 }