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 float jnf(int n, float x)
  20 {
  21         uint32_t ix;
  22         int nm1, sign, i;
  23         float a, b, temp;
  24
  25         GET_FLOAT_WORD(ix, x);
  26         sign = ix>>31;
  27         ix &= 0x7fffffff;
  28         if (ix > 0x7f800000) /* nan */
  29                 return x;
  30
  31         /* J(-n,x) = J(n,-x), use |n|-1 to avoid overflow in -n */
  32         if (n == 0)
  33                 return j0f(x);
  34         if (n < 0) {
  35                 nm1 = -(n+1);
  36                 x = -x;
  37                 sign ^= 1;
  38         } else
  39                 nm1 = n-1;
  40         if (nm1 == 0)
  41                 return j1f(x);
  42
  43         sign &= n;  /* even n: 0, odd n: signbit(x) */
  44         x = fabsf(x);
  45         if (ix == 0 || ix == 0x7f800000)  /* if x is 0 or inf */
  46                 b = 0.0f;
  47         else if (nm1 < x) {
  48                 /* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */
  49                 a = j0f(x);
  50                 b = j1f(x);
  51                 for (i=0; i<nm1; ){
  52                         i++;
  53                         temp = b;
  54                         b = b*(2.0f*i/x) - a;
  55                         a = temp;
  56                 }
  57         } else {
  58                 if (ix < 0x35800000) { /* x < 2**-20 */
  59                         /* x is tiny, return the first Taylor expansion of J(n,x)
  60                          * J(n,x) = 1/n!*(x/2)^n  - ...
  61                          */
  62                         if (nm1 > 8)  /* underflow */
  63                                 nm1 = 8;
  64                         temp = 0.5f * x;
  65                         b = temp;
  66                         a = 1.0f;
  67                         for (i=2; i<=nm1+1; i++) {
  68                                 a *= (float)i;    /* a = n! */
  69                                 b *= temp;        /* b = (x/2)^n */
  70                         }
  71                         b = b/a;
  72                 } else {
  73                         /* use backward recurrence */
  74                         /*                      x      x^2      x^2
  75                          *  J(n,x)/J(n-1,x) =  ----   ------   ------   .....
  76                          *                      2n  - 2(n+1) - 2(n+2)
  77                          *
  78                          *                      1      1        1
  79                          *  (for large x)   =  ----  ------   ------   .....
  80                          *                      2n   2(n+1)   2(n+2)
  81                          *                      -- - ------ - ------ -
  82                          *                       x     x         x
  83                          *
  84                          * Let w = 2n/x and h=2/x, then the above quotient
  85                          * is equal to the continued fraction:
  86                          *                  1
  87                          *      = -----------------------
  88                          *                     1
  89                          *         w - -----------------
  90                          *                        1
  91                          *              w+h - ---------
  92                          *                     w+2h - ...
  93                          *
  94                          * To determine how many terms needed, let
  95                          * Q(0) = w, Q(1) = w(w+h) - 1,
  96                          * Q(k) = (w+k*h)*Q(k-1) - Q(k-2),
  97                          * When Q(k) > 1e4      good for single
  98                          * When Q(k) > 1e9      good for double
  99                          * When Q(k) > 1e17     good for quadruple
 100                          */
 101                         /* determine k */
 102                         float t,q0,q1,w,h,z,tmp,nf;
 103                         int k;
 104
 105                         nf = nm1+1.0f;
 106                         w = 2*nf/x;
 107                         h = 2/x;
 108                         z = w+h;
 109                         q0 = w;
 110                         q1 = w*z - 1.0f;
 111                         k = 1;
 112                         while (q1 < 1.0e4f) {
 113                                 k += 1;
 114                                 z += h;
 115                                 tmp = z*q1 - q0;
 116                                 q0 = q1;
 117                                 q1 = tmp;
 118                         }
 119                         for (t=0.0f, i=k; i>=0; i--)
 120                                 t = 1.0f/(2*(i+nf)/x-t);
 121                         a = t;
 122                         b = 1.0f;
 123                         /*  estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n)
 124                          *  Hence, if n*(log(2n/x)) > ...
 125                          *  single 8.8722839355e+01
 126                          *  double 7.09782712893383973096e+02
 127                          *  long double 1.1356523406294143949491931077970765006170e+04
 128                          *  then recurrent value may overflow and the result is
 129                          *  likely underflow to zero
 130                          */
 131                         tmp = nf*logf(fabsf(w));
 132                         if (tmp < 88.721679688f) {
 133                                 for (i=nm1; i>0; i--) {
 134                                         temp = b;
 135                                         b = 2.0f*i*b/x - a;
 136                                         a = temp;
 137                                 }
 138                         } else {
 139                                 for (i=nm1; i>0; i--){
 140                                         temp = b;
 141                                         b = 2.0f*i*b/x - a;
 142                                         a = temp;
 143                                         /* scale b to avoid spurious overflow */
 144                                         if (b > 0x1p60f) {
 145                                                 a /= b;
 146                                                 t /= b;
 147                                                 b = 1.0f;
 148                                         }
 149                                 }
 150                         }
 151                         z = j0f(x);
 152                         w = j1f(x);
 153                         if (fabsf(z) >= fabsf(w))
 154                                 b = t*z/b;
 155                         else
 156                                 b = t*w/a;
 157                 }
 158         }
 159         return sign ? -b : b;
 160 }
 161
 162 float ynf(int n, float x)
 163 {
 164         uint32_t ix, ib;
 165         int nm1, sign, i;
 166         float a, b, temp;
 167
 168         GET_FLOAT_WORD(ix, x);
 169         sign = ix>>31;
 170         ix &= 0x7fffffff;
 171         if (ix > 0x7f800000) /* nan */
 172                 return x;
 173         if (sign && ix != 0) /* x < 0 */
 174                 return 0/0.0f;
 175         if (ix == 0x7f800000)
 176                 return 0.0f;
 177
 178         if (n == 0)
 179                 return y0f(x);
 180         if (n < 0) {
 181                 nm1 = -(n+1);
 182                 sign = n&1;
 183         } else {
 184                 nm1 = n-1;
 185                 sign = 0;
 186         }
 187         if (nm1 == 0)
 188                 return sign ? -y1f(x) : y1f(x);
 189
 190         a = y0f(x);
 191         b = y1f(x);
 192         /* quit if b is -inf */
 193         GET_FLOAT_WORD(ib,b);
 194         for (i = 0; i < nm1 && ib != 0xff800000; ) {
 195                 i++;
 196                 temp = b;
 197                 b = (2.0f*i/x)*b - a;
 198                 GET_FLOAT_WORD(ib, b);
 199                 a = temp;
 200         }
 201         return sign ? -b : b;
 202 }