compiler/stdc/math/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: src/lib/msun/src/e_jnf.c,v 1.11 2010/11/13 10:54:10 uqs Exp $";
  18 #endif
  19
  20 /*
  21  * See e_jn.c for complete comments.
  22  */
  23
  24 #include "math.h"
  25 #include "math_private.h"
  26
  27 static const volatile float vone __attribute__ ((__section__(".rodata,\"a\" " SECTIONCOMMENT))) = 1.0, vzero __attribute__ ((__section__(".rodata,\"a\" " SECTIONCOMMENT))) = 0.0;
  28
  29 static const float
  30 two   =  2.0000000000e+00, /* 0x40000000 */
  31 one   =  1.0000000000e+00; /* 0x3F800000 */
  32
  33 static const float zero  =  0.0000000000e+00;
  34
  35 float
  36 __ieee754_jnf(int n, float x)
  37 {
  38         int32_t i,hx,ix, sgn;
  39         float a, b, temp, di;
  40         float z, w;
  41
  42     /* J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x)
  43      * Thus, J(-n,x) = J(n,-x)
  44      */
  45         GET_FLOAT_WORD(hx,x);
  46         ix = 0x7fffffff&hx;
  47     /* if J(n,NaN) is NaN */
  48         if(ix>0x7f800000) return x+x;
  49         if(n<0){
  50                 n = -n;
  51                 x = -x;
  52                 hx ^= 0x80000000;
  53         }
  54         if(n==0) return(__ieee754_j0f(x));
  55         if(n==1) return(__ieee754_j1f(x));
  56         sgn = (n&1)&(hx>>31);   /* even n -- 0, odd n -- sign(x) */
  57         x = fabsf(x);
  58         if(ix==0||ix>=0x7f800000)       /* if x is 0 or inf */
  59             b = zero;
  60         else if((float)n<=x) {
  61                 /* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */
  62             a = __ieee754_j0f(x);
  63             b = __ieee754_j1f(x);
  64             for(i=1;i<n;i++){
  65                 temp = b;
  66                 b = b*((float)(i+i)/x) - a; /* avoid underflow */
  67                 a = temp;
  68             }
  69         } else {
  70             if(ix<0x30800000) { /* x < 2**-29 */
  71     /* x is tiny, return the first Taylor expansion of J(n,x)
  72      * J(n,x) = 1/n!*(x/2)^n  - ...
  73      */
  74                 if(n>33)        /* underflow */
  75                     b = zero;
  76                 else {
  77                     temp = x*(float)0.5; b = temp;
  78                     for (a=one,i=2;i<=n;i++) {
  79                         a *= (float)i;          /* a = n! */
  80                         b *= temp;              /* b = (x/2)^n */
  81                     }
  82                     b = b/a;
  83                 }
  84             } else {
  85                 /* use backward recurrence */
  86                 /*                      x      x^2      x^2
  87                  *  J(n,x)/J(n-1,x) =  ----   ------   ------   .....
  88                  *                      2n  - 2(n+1) - 2(n+2)
  89                  *
  90                  *                      1      1        1
  91                  *  (for large x)   =  ----  ------   ------   .....
  92                  *                      2n   2(n+1)   2(n+2)
  93                  *                      -- - ------ - ------ -
  94                  *                       x     x         x
  95                  *
  96                  * Let w = 2n/x and h=2/x, then the above quotient
  97                  * is equal to the continued fraction:
  98                  *                  1
  99                  *      = -----------------------
 100                  *                     1
 101                  *         w - -----------------
 102                  *                        1
 103                  *              w+h - ---------
 104                  *                     w+2h - ...
 105                  *
 106                  * To determine how many terms needed, let
 107                  * Q(0) = w, Q(1) = w(w+h) - 1,
 108                  * Q(k) = (w+k*h)*Q(k-1) - Q(k-2),
 109                  * When Q(k) > 1e4      good for single
 110                  * When Q(k) > 1e9      good for double
 111                  * When Q(k) > 1e17     good for quadruple
 112                  */
 113             /* determine k */
 114                 float t,v;
 115                 float q0,q1,h,tmp; int32_t k,m;
 116                 w  = (n+n)/(float)x; h = (float)2.0/(float)x;
 117                 q0 = w;  z = w+h; q1 = w*z - (float)1.0; k=1;
 118                 while(q1<(float)1.0e9) {
 119                         k += 1; z += h;
 120                         tmp = z*q1 - q0;
 121                         q0 = q1;
 122                         q1 = tmp;
 123                 }
 124                 m = n+n;
 125                 for(t=zero, i = 2*(n+k); i>=m; i -= 2) t = one/(i/x-t);
 126                 a = t;
 127                 b = one;
 128                 /*  estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n)
 129                  *  Hence, if n*(log(2n/x)) > ...
 130                  *  single 8.8722839355e+01
 131                  *  double 7.09782712893383973096e+02
 132                  *  long double 1.1356523406294143949491931077970765006170e+04
 133                  *  then recurrent value may overflow and the result is
 134                  *  likely underflow to zero
 135                  */
 136                 tmp = n;
 137                 v = two/x;
 138                 tmp = tmp*__ieee754_logf(fabsf(v*tmp));
 139                 if(tmp<(float)8.8721679688e+01) {
 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                     }
 147                 } else {
 148                     for(i=n-1,di=(float)(i+i);i>0;i--){
 149                         temp = b;
 150                         b *= di;
 151                         b  = b/x - a;
 152                         a = temp;
 153                         di -= two;
 154                     /* scale b to avoid spurious overflow */
 155                         if(b>(float)1e10) {
 156                             a /= b;
 157                             t /= b;
 158                             b  = one;
 159                         }
 160                     }
 161                 }
 162                 z = __ieee754_j0f(x);
 163                 w = __ieee754_j1f(x);
 164                 if (fabsf(z) >= fabsf(w))
 165                     b = (t*z/b);
 166                 else
 167                     b = (t*w/a);
 168             }
 169         }
 170         if(sgn==1) return -b; else return b;
 171 }
 172
 173 float
 174 __ieee754_ynf(int n, float x)
 175 {
 176         int32_t i,hx,ix,ib;
 177         int32_t sign;
 178         float a, b, temp;
 179
 180         GET_FLOAT_WORD(hx,x);
 181         ix = 0x7fffffff&hx;
 182         if(ix>0x7f800000) return x+x;
 183         if(ix==0) return -one/vzero;
 184         if(hx<0) return vzero/vzero;
 185         sign = 1;
 186         if(n<0){
 187                 n = -n;
 188                 sign = 1 - ((n&1)<<1);
 189         }
 190         if(n==0) return(__ieee754_y0f(x));
 191         if(n==1) return(sign*__ieee754_y1f(x));
 192         if(ix==0x7f800000) return zero;
 193
 194         a = __ieee754_y0f(x);
 195         b = __ieee754_y1f(x);
 196         /* quit if b is -inf */
 197         GET_FLOAT_WORD(ib,b);
 198         for(i=1;i<n&&ib!=0xff800000;i++){
 199             temp = b;
 200             b = ((float)(i+i)/x)*b - a;
 201             GET_FLOAT_WORD(ib,b);
 202             a = temp;
 203         }
 204         if(sign>0) return b; else return -b;
 205 }