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
  16 #include <sys/cdefs.h>
  17 #if defined(LIBM_SCCS) && !defined(lint)
  18 __RCSID("$NetBSD: e_jnf.c,v 1.11 2010/11/29 15:10:06 drochner Exp $");
  19 #endif
  20
  21 #include "math.h"
  22 #include "math_private.h"
  23
  24 static const float
  25 #if 0
  26 invsqrtpi=  5.6418961287e-01, /* 0x3f106ebb */
  27 #endif
  28 two   =  2.0000000000e+00, /* 0x40000000 */
  29 one   =  1.0000000000e+00; /* 0x3F800000 */
  30
  31 static const float zero  =  0.0000000000e+00;
  32
  33 float
  34 __ieee754_jnf(int n, float x)
  35 {
  36         int32_t i,hx,ix, sgn;
  37         float a, b, temp, di;
  38         float z, w;
  39
  40     /* J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x)
  41      * Thus, J(-n,x) = J(n,-x)
  42      */
  43         GET_FLOAT_WORD(hx,x);
  44         ix = 0x7fffffff&hx;
  45     /* if J(n,NaN) is NaN */
  46         if(ix>0x7f800000) return x+x;
  47         if(n<0){
  48                 n = -n;
  49                 x = -x;
  50                 hx ^= 0x80000000;
  51         }
  52         if(n==0) return(__ieee754_j0f(x));
  53         if(n==1) return(__ieee754_j1f(x));
  54         sgn = (n&1)&(hx>>31);   /* even n -- 0, odd n -- sign(x) */
  55         x = fabsf(x);
  56         if(ix==0||ix>=0x7f800000)       /* if x is 0 or inf */
  57             b = zero;
  58         else if((float)n<=x) {
  59                 /* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */
  60             a = __ieee754_j0f(x);
  61             b = __ieee754_j1f(x);
  62             for(i=1;i<n;i++){
  63                 temp = b;
  64                 b = b*((float)(i+i)/x) - a; /* avoid underflow */
  65                 a = temp;
  66             }
  67         } else {
  68             if(ix<0x30800000) { /* x < 2**-29 */
  69     /* x is tiny, return the first Taylor expansion of J(n,x)
  70      * J(n,x) = 1/n!*(x/2)^n  - ...
  71      */
  72                 if(n>33)        /* underflow */
  73                     b = zero;
  74                 else {
  75                     temp = x*(float)0.5; b = temp;
  76                     for (a=one,i=2;i<=n;i++) {
  77                         a *= (float)i;          /* a = n! */
  78                         b *= temp;              /* b = (x/2)^n */
  79                     }
  80                     b = b/a;
  81                 }
  82             } else {
  83                 /* use backward recurrence */
  84                 /*                      x      x^2      x^2
  85                  *  J(n,x)/J(n-1,x) =  ----   ------   ------   .....
  86                  *                      2n  - 2(n+1) - 2(n+2)
  87                  *
  88                  *                      1      1        1
  89                  *  (for large x)   =  ----  ------   ------   .....
  90                  *                      2n   2(n+1)   2(n+2)
  91                  *                      -- - ------ - ------ -
  92                  *                       x     x         x
  93                  *
  94                  * Let w = 2n/x and h=2/x, then the above quotient
  95                  * is equal to the continued fraction:
  96                  *                  1
  97                  *      = -----------------------
  98                  *                     1
  99                  *         w - -----------------
 100                  *                        1
 101                  *              w+h - ---------
 102                  *                     w+2h - ...
 103                  *
 104                  * To determine how many terms needed, let
 105                  * Q(0) = w, Q(1) = w(w+h) - 1,
 106                  * Q(k) = (w+k*h)*Q(k-1) - Q(k-2),
 107                  * When Q(k) > 1e4      good for single
 108                  * When Q(k) > 1e9      good for double
 109                  * When Q(k) > 1e17     good for quadruple
 110                  */
 111             /* determine k */
 112                 float t,v;
 113                 float q0,q1,h,tmp; int32_t k,m;
 114                 w  = (n+n)/(float)x; h = (float)2.0/(float)x;
 115                 q0 = w;  z = w+h; q1 = w*z - (float)1.0; k=1;
 116                 while(q1<(float)1.0e9) {
 117                         k += 1; z += h;
 118                         tmp = z*q1 - q0;
 119                         q0 = q1;
 120                         q1 = tmp;
 121                 }
 122                 m = n+n;
 123                 for(t=zero, i = 2*(n+k); i>=m; i -= 2) t = one/(i/x-t);
 124                 a = t;
 125                 b = one;
 126                 /*  estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n)
 127                  *  Hence, if n*(log(2n/x)) > ...
 128                  *  single 8.8722839355e+01
 129                  *  double 7.09782712893383973096e+02
 130                  *  long double 1.1356523406294143949491931077970765006170e+04
 131                  *  then recurrent value may overflow and the result is
 132                  *  likely underflow to zero
 133                  */
 134                 tmp = n;
 135                 v = two/x;
 136                 tmp = tmp*__ieee754_logf(fabsf(v*tmp));
 137                 if(tmp<(float)8.8721679688e+01) {
 138                     for(i=n-1,di=(float)(i+i);i>0;i--){
 139                         temp = b;
 140                         b *= di;
 141                         b  = b/x - a;
 142                         a = temp;
 143                         di -= two;
 144                     }
 145                 } else {
 146                     for(i=n-1,di=(float)(i+i);i>0;i--){
 147                         temp = b;
 148                         b *= di;
 149                         b  = b/x - a;
 150                         a = temp;
 151                         di -= two;
 152                     /* scale b to avoid spurious overflow */
 153                         if(b>(float)1e10) {
 154                             a /= b;
 155                             t /= b;
 156                             b  = one;
 157                         }
 158                     }
 159                 }
 160                 z = __ieee754_j0f(x);
 161                 w = __ieee754_j1f(x);
 162                 if (fabsf(z) >= fabsf(w))
 163                         b = (t*z/b);
 164                 else
 165                         b = (t*w/a);
 166             }
 167         }
 168         if(sgn==1) return -b; else return b;
 169 }
 170
 171 float
 172 __ieee754_ynf(int n, float x)
 173 {
 174         int32_t i,hx,ix,ib;
 175         int32_t sign;
 176         float a, b, temp;
 177
 178         GET_FLOAT_WORD(hx,x);
 179         ix = 0x7fffffff&hx;
 180     /* if Y(n,NaN) is NaN */
 181         if(ix>0x7f800000) return x+x;
 182         if(ix==0) return -one/zero;
 183         if(hx<0) return zero/zero;
 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&&(uint32_t)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 }