nsz Git - musl/blob - src/math/jnf.c

   1 /* origin: FreeBSD /usr/src/lib/msun/src/e_jnf.c */
   2 /*
   3  * Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com.
   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 "libm.h"
  17
  18 static const float
  19 two = 2.0000000000e+00, /* 0x40000000 */
  20 one = 1.0000000000e+00; /* 0x3F800000 */
  21
  22 static const float zero = 0.0000000000e+00;
  23
  24 float jnf(int n, float x)
  25 {
  26         int32_t i,hx,ix, sgn;
  27         float a, b, temp, di;
  28         float z, w;
  29
  30         /* J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x)
  31          * Thus, J(-n,x) = J(n,-x)
  32          */
  33         GET_FLOAT_WORD(hx, x);
  34         ix = 0x7fffffff & hx;
  35         /* if J(n,NaN) is NaN */
  36         if (ix > 0x7f800000)
  37                 return x+x;
  38         if (n < 0) {
  39                 n = -n;
  40                 x = -x;
  41                 hx ^= 0x80000000;
  42         }
  43         if (n == 0) return j0f(x);
  44         if (n == 1) return j1f(x);
  45
  46         sgn = (n&1)&(hx>>31);  /* even n -- 0, odd n -- sign(x) */
  47         x = fabsf(x);
  48         if (ix == 0 || ix >= 0x7f800000)  /* if x is 0 or inf */
  49                 b = zero;
  50         else if((float)n <= x) {
  51                 /* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */
  52                 a = j0f(x);
  53                 b = j1f(x);
  54                 for (i=1; i<n; i++){
  55                         temp = b;
  56                         b = b*((float)(i+i)/x) - a; /* avoid underflow */
  57                         a = temp;
  58                 }
  59         } else {
  60                 if (ix < 0x30800000) { /* x < 2**-29 */
  61                         /* x is tiny, return the first Taylor expansion of J(n,x)
  62                          * J(n,x) = 1/n!*(x/2)^n  - ...
  63                          */
  64                         if (n > 33)  /* underflow */
  65                                 b = zero;
  66                         else {
  67                                 temp = 0.5f * x;
  68                                 b = temp;
  69                                 for (a=one,i=2; i<=n; i++) {
  70                                         a *= (float)i;    /* a = n! */
  71                                         b *= temp;        /* b = (x/2)^n */
  72                                 }
  73                                 b = b/a;
  74                         }
  75                 } else {
  76                         /* use backward recurrence */
  77                         /*                      x      x^2      x^2
  78                          *  J(n,x)/J(n-1,x) =  ----   ------   ------   .....
  79                          *                      2n  - 2(n+1) - 2(n+2)
  80                          *
  81                          *                      1      1        1
  82                          *  (for large x)   =  ----  ------   ------   .....
  83                          *                      2n   2(n+1)   2(n+2)
  84                          *                      -- - ------ - ------ -
  85                          *                       x     x         x
  86                          *
  87                          * Let w = 2n/x and h=2/x, then the above quotient
  88                          * is equal to the continued fraction:
  89                          *                  1
  90                          *      = -----------------------
  91                          *                     1
  92                          *         w - -----------------
  93                          *                        1
  94                          *              w+h - ---------
  95                          *                     w+2h - ...
  96                          *
  97                          * To determine how many terms needed, let
  98                          * Q(0) = w, Q(1) = w(w+h) - 1,
  99                          * Q(k) = (w+k*h)*Q(k-1) - Q(k-2),
 100                          * When Q(k) > 1e4      good for single
 101                          * When Q(k) > 1e9      good for double
 102                          * When Q(k) > 1e17     good for quadruple
 103                          */
 104                         /* determine k */
 105                         float t,v;
 106                         float q0,q1,h,tmp;
 107                         int32_t k,m;
 108
 109                         w = (n+n)/x;
 110                         h = 2.0f/x;
 111                         z = w+h;
 112                         q0 = w;
 113                         q1 = w*z - 1.0f;
 114                         k = 1;
 115                         while (q1 < 1.0e9f) {
 116                                 k += 1;
 117                                 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)
 124                                 t = one/(i/x-t);
 125                         a = t;
 126                         b = one;
 127                         /*  estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n)
 128                          *  Hence, if n*(log(2n/x)) > ...
 129                          *  single 8.8722839355e+01
 130                          *  double 7.09782712893383973096e+02
 131                          *  long double 1.1356523406294143949491931077970765006170e+04
 132                          *  then recurrent value may overflow and the result is
 133                          *  likely underflow to zero
 134                          */
 135                         tmp = n;
 136                         v = two/x;
 137                         tmp = tmp*logf(fabsf(v*tmp));
 138                         if (tmp < 88.721679688f) {
 139                                 for (i=n-1,di=(float)(i+i); i>0; i--) {
 140                                         temp = b;
 141                                         b *= di;
 142                                         b = b/x - a;
 143                                         a = temp;
 144                                         di -= two;
 145                                 }
 146                         } else {
 147                                 for (i=n-1,di=(float)(i+i); i>0; i--){
 148                                         temp = b;
 149                                         b *= di;
 150                                         b = b/x - a;
 151                                         a = temp;
 152                                         di -= two;
 153                                         /* scale b to avoid spurious overflow */
 154                                         if (b > 1e10f) {
 155                                                 a /= b;
 156                                                 t /= b;
 157                                                 b = one;
 158                                         }
 159                                 }
 160                         }
 161                         z = j0f(x);
 162                         w = j1f(x);
 163                         if (fabsf(z) >= fabsf(w))
 164                                 b = t*z/b;
 165                         else
 166                                 b = t*w/a;
 167                 }
 168         }
 169         if (sgn == 1) return -b;
 170         return b;
 171 }
 172
 173 float ynf(int n, float x)
 174 {
 175         int32_t i,hx,ix,ib;
 176         int32_t sign;
 177         float a, b, temp;
 178
 179         GET_FLOAT_WORD(hx, x);
 180         ix = 0x7fffffff & hx;
 181         /* if Y(n,NaN) is NaN */
 182         if (ix > 0x7f800000)
 183                 return x+x;
 184         if (ix == 0)
 185                 return -one/zero;
 186         if (hx < 0)
 187                 return zero/zero;
 188         sign = 1;
 189         if (n < 0) {
 190                 n = -n;
 191                 sign = 1 - ((n&1)<<1);
 192         }
 193         if (n == 0)
 194                 return y0f(x);
 195         if (n == 1)
 196                 return sign*y1f(x);
 197         if (ix == 0x7f800000)
 198                 return zero;
 199
 200         a = y0f(x);
 201         b = y1f(x);
 202         /* quit if b is -inf */
 203         GET_FLOAT_WORD(ib,b);
 204         for (i = 1; i < n && ib != 0xff800000; i++){
 205                 temp = b;
 206                 b = ((float)(i+i)/x)*b - a;
 207                 GET_FLOAT_WORD(ib, b);
 208                 a = temp;
 209         }
 210         if (sign > 0)
 211                 return b;
 212         return -b;
 213 }