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 #define _GNU_SOURCE
  17 #include "libm.h"
  18
  19 static const float
  20 two = 2.0000000000e+00, /* 0x40000000 */
  21 one = 1.0000000000e+00; /* 0x3F800000 */
  22
  23 static const float zero = 0.0000000000e+00;
  24
  25 float jnf(int n, float x)
  26 {
  27         int32_t i,hx,ix, sgn;
  28         float a, b, temp, di;
  29         float z, w;
  30
  31         /* J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x)
  32          * Thus, J(-n,x) = J(n,-x)
  33          */
  34         GET_FLOAT_WORD(hx, x);
  35         ix = 0x7fffffff & hx;
  36         /* if J(n,NaN) is NaN */
  37         if (ix > 0x7f800000)
  38                 return x+x;
  39         if (n < 0) {
  40                 n = -n;
  41                 x = -x;
  42                 hx ^= 0x80000000;
  43         }
  44         if (n == 0) return j0f(x);
  45         if (n == 1) return j1f(x);
  46
  47         sgn = (n&1)&(hx>>31);  /* even n -- 0, odd n -- sign(x) */
  48         x = fabsf(x);
  49         if (ix == 0 || ix >= 0x7f800000)  /* if x is 0 or inf */
  50                 b = zero;
  51         else if((float)n <= x) {
  52                 /* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */
  53                 a = j0f(x);
  54                 b = j1f(x);
  55                 for (i=1; i<n; i++){
  56                         temp = b;
  57                         b = b*((float)(i+i)/x) - a; /* avoid underflow */
  58                         a = temp;
  59                 }
  60         } else {
  61                 if (ix < 0x30800000) { /* x < 2**-29 */
  62                         /* x is tiny, return the first Taylor expansion of J(n,x)
  63                          * J(n,x) = 1/n!*(x/2)^n  - ...
  64                          */
  65                         if (n > 33)  /* underflow */
  66                                 b = zero;
  67                         else {
  68                                 temp = 0.5f * x;
  69                                 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;
 108                         int32_t k,m;
 109
 110                         w = (n+n)/x;
 111                         h = 2.0f/x;
 112                         z = w+h;
 113                         q0 = w;
 114                         q1 = w*z - 1.0f;
 115                         k = 1;
 116                         while (q1 < 1.0e9f) {
 117                                 k += 1;
 118                                 z += h;
 119                                 tmp = z*q1 - q0;
 120                                 q0 = q1;
 121                                 q1 = tmp;
 122                         }
 123                         m = n+n;
 124                         for (t=zero, i = 2*(n+k); i>=m; i -= 2)
 125                                 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*logf(fabsf(v*tmp));
 139                         if (tmp < 88.721679688f) {
 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 > 1e10f) {
 156                                                 a /= b;
 157                                                 t /= b;
 158                                                 b = one;
 159                                         }
 160                                 }
 161                         }
 162                         z = j0f(x);
 163                         w = 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;
 171         return b;
 172 }
 173
 174 float 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 Y(n,NaN) is NaN */
 183         if (ix > 0x7f800000)
 184                 return x+x;
 185         if (ix == 0)
 186                 return -one/zero;
 187         if (hx < 0)
 188                 return zero/zero;
 189         sign = 1;
 190         if (n < 0) {
 191                 n = -n;
 192                 sign = 1 - ((n&1)<<1);
 193         }
 194         if (n == 0)
 195                 return y0f(x);
 196         if (n == 1)
 197                 return sign*y1f(x);
 198         if (ix == 0x7f800000)
 199                 return zero;
 200
 201         a = y0f(x);
 202         b = y1f(x);
 203         /* quit if b is -inf */
 204         GET_FLOAT_WORD(ib,b);
 205         for (i = 1; i < n && ib != 0xff800000; i++){
 206                 temp = b;
 207                 b = ((float)(i+i)/x)*b - a;
 208                 GET_FLOAT_WORD(ib, b);
 209                 a = temp;
 210         }
 211         if (sign > 0)
 212                 return b;
 213         return -b;
 214 }