Print this page
11175 libm should use signbit() correctly
11188 c99 math macros should return strictly backward compatible values


  51  *
  52  *      yn(n,x) is similar in all respects, except
  53  *      that forward recursion is used for all
  54  *      values of n>1.
  55  *
  56  */
  57 
  58 #include "libm.h"
  59 #include "longdouble.h"
  60 #include <float.h>        /* LDBL_MAX */
  61 
  62 #define GENERIC long double
  63 
  64 static const GENERIC
  65 invsqrtpi = 5.641895835477562869480794515607725858441e-0001L,
  66 two  = 2.0L,
  67 zero = 0.0L,
  68 one  = 1.0L;
  69 
  70 GENERIC
  71 jnl(n, x) int n; GENERIC x; {

  72         int i, sgn;
  73         GENERIC a, b, temp = 0, z, w;
  74 
  75         /*
  76          * J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x)
  77          * Thus, J(-n,x) = J(n,-x)
  78          */
  79         if (n < 0) {
  80                 n = -n;
  81                 x = -x;
  82         }
  83         if (n == 0) return (j0l(x));
  84         if (n == 1) return (j1l(x));
  85         if (x != x) return x+x;



  86         if ((n&1) == 0)
  87                 sgn = 0;                        /* even n */
  88         else
  89                 sgn = signbitl(x);      /* old n  */
  90         x = fabsl(x);
  91         if (x == zero || !finitel(x)) b = zero;
  92         else if ((GENERIC)n <= x) {
  93                         /*
  94                          * Safe to use
  95                          * J(n+1,x)=2n/x *J(n,x)-J(n-1,x)
  96                          */
  97             if (x > 1.0e91L) {
  98                                 /*
  99                                  * x >> n**2
 100                                  *  Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
 101                                  *  Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
 102                                  *  Let s=sin(x), c=cos(x),
 103                                  *  xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
 104                                  *
 105                                  *         n    sin(xn)*sqt2    cos(xn)*sqt2
 106                                  *      ----------------------------------
 107                                  *         0     s-c             c+s
 108                                  *         1    -s-c            -c+s
 109                                  *         2    -s+c            -c-s
 110                                  *         3     s+c             c-s
 111                                  */
 112                 switch (n&3) {
 113                     case 0: temp =  cosl(x)+sinl(x); break;
 114                     case 1: temp = -cosl(x)+sinl(x); break;
 115                     case 2: temp = -cosl(x)-sinl(x); break;
 116                     case 3: temp =  cosl(x)-sinl(x); break;








 117                 }
 118                 b = invsqrtpi*temp/sqrtl(x);
 119             } else {
 120                         a = j0l(x);
 121                         b = j1l(x);
 122                         for (i = 1; i < n; i++) {
 123                     temp = b;
 124                     b = b*((GENERIC)(i+i)/x) - a; /* avoid underflow */

 125                     a = temp;
 126                         }
 127             }
 128         } else {
 129             if (x < 1e-17L) {        /* use J(n,x) = 1/n!*(x/2)^n */
 130                 b = powl(0.5L*x, (GENERIC) n);
 131                 if (b != zero) {
 132                     for (a = one, i = 1; i <= n; i++) a *= (GENERIC)i;

 133                     b = b/a;
 134                 }
 135             } else {
 136                 /*
 137                  * use backward recurrence
 138                  *                      x      x^2      x^2
 139                  *  J(n,x)/J(n-1,x) =  ----   ------   ------   .....
 140                  *                      2n  - 2(n+1) - 2(n+2)
 141                  *
 142                  *                      1      1        1
 143                  *  (for large x)   =  ----  ------   ------   .....
 144                  *                      2n   2(n+1)   2(n+2)
 145                  *                      -- - ------ - ------ -
 146                  *                       x     x         x
 147                  *
 148                  * Let w = 2n/x and h=2/x, then the above quotient
 149                  * is equal to the continued fraction:
 150                  *                  1
 151                  *      = -----------------------
 152                  *                     1
 153                  *         w - -----------------
 154                  *                        1
 155                  *              w+h - ---------
 156                  *                     w+2h - ...
 157                  *
 158                  * To determine how many terms needed, let
 159                  * Q(0) = w, Q(1) = w(w+h) - 1,
 160                  * Q(k) = (w+k*h)*Q(k-1) - Q(k-2),
 161                  * When Q(k) > 1e4   good for single
 162                  * When Q(k) > 1e9   good for double
 163                  * When Q(k) > 1e17  good for quaduple
 164                  */
 165             /* determin k */
 166                 GENERIC t, v;
 167                 double q0, q1, h, tmp; int k, m;
 168                 w  = (n+n)/(double)x; h = 2.0/(double)x;
 169                 q0 = w;  z = w+h; q1 = w*z - 1.0; k = 1;





 170                 while (q1 < 1.0e17) {
 171                         k += 1; z += h;

 172                         tmp = z*q1 - q0;
 173                         q0 = q1;
 174                         q1 = tmp;
 175                 }
 176                 m = n+n;
 177                 for (t = zero, i = 2*(n+k); i >= m; i -= 2) t = one/(i/x-t);

 178                 a = t;
 179                 b = one;
 180                         /*
 181                          * Estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n)
 182                          * hence, if n*(log(2n/x)) > ...
 183                          * single 8.8722839355e+01
 184                          * double 7.09782712893383973096e+02
 185                          * long double 1.1356523406294143949491931077970765006170e+04
 186                          * then recurrent value may overflow and the result is
 187                          * likely underflow to zero.
 188                          */
 189                 tmp = n;
 190                 v = two/x;
 191                 tmp = tmp*logl(fabsl(v*tmp));
 192                 if (tmp < 1.1356523406294143949491931077970765e+04L) {
 193                                 for (i = n-1; i > 0; i--) {
 194                                 temp = b;
 195                                 b = ((i+i)/x)*b - a;
 196                                 a = temp;
 197                                 }
 198                 } else {
 199                                 for (i = n-1; i > 0; i--) {
 200                                 temp = b;
 201                                 b = ((i+i)/x)*b - a;
 202                                 a = temp;
 203                         if (b > 1e1000L) {
 204                                                 a /= b;
 205                                                 t /= b;
 206                                                 b  = 1.0;
 207                                         }
 208                                 }
 209                 }
 210                         b = (t*j0l(x)/b);
 211             }
 212         }
 213         if (sgn == 1)
 214                 return -b;
 215         else
 216                 return b;
 217 }
 218 
 219 GENERIC
 220 ynl(n, x) int n; GENERIC x; {

 221         int i;
 222         int sign;
 223         GENERIC a, b, temp = 0;
 224 
 225         if (x != x)
 226                 return x+x;
 227         if (x <= zero) {
 228                 if (x == zero)
 229                         return -one/zero;
 230                 else
 231                         return zero/zero;
 232         }
 233         sign = 1;
 234         if (n < 0) {
 235                 n = -n;
 236                 if ((n&1) == 1) sign = -1;

 237         }
 238         if (n == 0) return (y0l(x));
 239         if (n == 1) return (sign*y1l(x));
 240         if (!finitel(x)) return zero;



 241 
 242         if (x > 1.0e91L) {
 243                                 /*
 244                                  * x >> n**2
 245                                  * Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
 246                                  *   Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
 247                                  *   Let s=sin(x), c=cos(x),
 248                                  * xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
 249                                  *
 250                                  *         n    sin(xn)*sqt2    cos(xn)*sqt2
 251                                  *      ----------------------------------
 252                                  *         0     s-c             c+s
 253                                  *         1    -s-c            -c+s
 254                                  *         2    -s+c            -c-s
 255                                  *         3     s+c             c-s
 256                                  */
 257                 switch (n&3) {
 258                     case 0: temp =  sinl(x)-cosl(x); break;
 259                     case 1: temp = -sinl(x)-cosl(x); break;
 260                     case 2: temp = -sinl(x)+cosl(x); break;
 261                     case 3: temp =  sinl(x)+cosl(x); break;








 262                 }
 263                 b = invsqrtpi*temp/sqrtl(x);
 264         } else {
 265                 a = y0l(x);
 266                 b = y1l(x);
 267                 /*
 268                  * fix 1262058 and take care of non-default rounding
 269                  */
 270                 for (i = 1; i < n; i++) {
 271                         temp = b;
 272                         b *= (GENERIC) (i + i) / x;
 273                         if (b <= -LDBL_MAX)
 274                                 break;
 275                         b -= a;
 276                         a = temp;
 277                 }
 278         }
 279         if (sign > 0)
 280                 return b;
 281         else
 282                 return -b;
 283 }


  51  *
  52  *      yn(n,x) is similar in all respects, except
  53  *      that forward recursion is used for all
  54  *      values of n>1.
  55  *
  56  */
  57 
  58 #include "libm.h"
  59 #include "longdouble.h"
  60 #include <float.h>        /* LDBL_MAX */
  61 
  62 #define GENERIC long double
  63 
  64 static const GENERIC
  65 invsqrtpi = 5.641895835477562869480794515607725858441e-0001L,
  66 two  = 2.0L,
  67 zero = 0.0L,
  68 one  = 1.0L;
  69 
  70 GENERIC
  71 jnl(int n, GENERIC x)
  72 {
  73         int i, sgn;
  74         GENERIC a, b, temp = 0, z, w;
  75 
  76         /*
  77          * J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x)
  78          * Thus, J(-n,x) = J(n,-x)
  79          */
  80         if (n < 0) {
  81                 n = -n;
  82                 x = -x;
  83         }
  84         if (n == 0)
  85                 return (j0l(x));
  86         if (n == 1)
  87                 return (j1l(x));
  88         if (x != x)
  89                 return (x+x);
  90         if ((n&1) == 0)
  91                 sgn = 0;                        /* even n */
  92         else
  93                 sgn = signbitl(x);      /* old n  */
  94         x = fabsl(x);
  95         if (x == zero || !finitel(x)) b = zero;
  96         else if ((GENERIC)n <= x) {
  97                         /*
  98                          * Safe to use
  99                          * J(n+1,x)=2n/x *J(n,x)-J(n-1,x)
 100                          */
 101                 if (x > 1.0e91L) {
 102                                 /*
 103                                  * x >> n**2
 104                                  *  Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
 105                                  *  Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
 106                                  *  Let s=sin(x), c=cos(x),
 107                                  *  xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
 108                                  *
 109                                  *         n    sin(xn)*sqt2    cos(xn)*sqt2
 110                                  *      ----------------------------------
 111                                  *         0     s-c             c+s
 112                                  *         1    -s-c            -c+s
 113                                  *         2    -s+c            -c-s
 114                                  *         3     s+c             c-s
 115                                  */
 116                         switch (n&3) {
 117                         case 0:
 118                                 temp =  cosl(x)+sinl(x);
 119                                 break;
 120                         case 1:
 121                                 temp = -cosl(x)+sinl(x);
 122                                 break;
 123                         case 2:
 124                                 temp = -cosl(x)-sinl(x);
 125                                 break;
 126                         case 3:
 127                                 temp =  cosl(x)-sinl(x);
 128                                 break;
 129                         }
 130                         b = invsqrtpi*temp/sqrtl(x);
 131                 } else {
 132                         a = j0l(x);
 133                         b = j1l(x);
 134                         for (i = 1; i < n; i++) {
 135                                 temp = b;
 136                                 /* avoid underflow */
 137                                 b = b*((GENERIC)(i+i)/x) - a;
 138                                 a = temp;
 139                         }
 140                 }
 141         } else {
 142                 if (x < 1e-17L) {    /* use J(n,x) = 1/n!*(x/2)^n */
 143                         b = powl(0.5L*x, (GENERIC)n);
 144                         if (b != zero) {
 145                                 for (a = one, i = 1; i <= n; i++)
 146                                         a *= (GENERIC)i;
 147                                 b = b/a;
 148                         }
 149                 } else {
 150                         /*
 151                          * use backward recurrence
 152                          *                      x      x^2      x^2
 153                          *  J(n,x)/J(n-1,x) =  ----   ------   ------   .....
 154                          *                      2n  - 2(n+1) - 2(n+2)
 155                          *
 156                          *                      1      1        1
 157                          *  (for large x)   =  ----  ------   ------   .....
 158                          *                      2n   2(n+1)   2(n+2)
 159                          *                      -- - ------ - ------ -
 160                          *                       x     x         x
 161                          *
 162                          * Let w = 2n/x and h=2/x, then the above quotient
 163                          * is equal to the continued fraction:
 164                          *                  1
 165                          *      = -----------------------
 166                          *                     1
 167                          *         w - -----------------
 168                          *                        1
 169                          *              w+h - ---------
 170                          *                     w+2h - ...
 171                          *
 172                          * To determine how many terms needed, let
 173                          * Q(0) = w, Q(1) = w(w+h) - 1,
 174                          * Q(k) = (w+k*h)*Q(k-1) - Q(k-2),
 175                          * When Q(k) > 1e4   good for single
 176                          * When Q(k) > 1e9   good for double
 177                          * When Q(k) > 1e17  good for quaduple
 178                          */
 179                         /* determine k */
 180                         GENERIC t, v;
 181                         double q0, q1, h, tmp;
 182                         int k, m;
 183                         w  = (n+n)/(double)x;
 184                         h = 2.0/(double)x;
 185                         q0 = w;
 186                         z = w+h;
 187                         q1 = w*z - 1.0;
 188                         k = 1;
 189                         while (q1 < 1.0e17) {
 190                                 k += 1;
 191                                 z += h;
 192                                 tmp = z*q1 - q0;
 193                                 q0 = q1;
 194                                 q1 = tmp;
 195                         }
 196                         m = n+n;
 197                         for (t = zero, i = 2*(n+k); i >= m; i -= 2)
 198                                 t = one/(i/x-t);
 199                         a = t;
 200                         b = one;
 201                         /*
 202                          * Estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n)
 203                          * hence, if n*(log(2n/x)) > ...
 204                          * single 8.8722839355e+01
 205                          * double 7.09782712893383973096e+02
 206                          * long double 1.1356523406294143949491931077970765006170e+04
 207                          * then recurrent value may overflow and the result is
 208                          * likely underflow to zero.
 209                          */
 210                         tmp = n;
 211                         v = two/x;
 212                         tmp = tmp*logl(fabsl(v*tmp));
 213                         if (tmp < 1.1356523406294143949491931077970765e+04L) {
 214                                 for (i = n-1; i > 0; i--) {
 215                                         temp = b;
 216                                         b = ((i+i)/x)*b - a;
 217                                         a = temp;
 218                                 }
 219                         } else {
 220                                 for (i = n-1; i > 0; i--) {
 221                                         temp = b;
 222                                         b = ((i+i)/x)*b - a;
 223                                         a = temp;
 224                                         if (b > 1e1000L) {
 225                                                 a /= b;
 226                                                 t /= b;
 227                                                 b  = 1.0;
 228                                         }
 229                                 }
 230                         }
 231                         b = (t*j0l(x)/b);
 232                 }
 233         }
 234         if (sgn != 0)
 235                 return (-b);
 236         else
 237                 return (b);
 238 }
 239 
 240 GENERIC
 241 ynl(int n, GENERIC x)
 242 {
 243         int i;
 244         int sign;
 245         GENERIC a, b, temp = 0;
 246 
 247         if (x != x)
 248                 return (x+x);
 249         if (x <= zero) {
 250                 if (x == zero)
 251                         return (-one/zero);
 252                 else
 253                         return (zero/zero);
 254         }
 255         sign = 1;
 256         if (n < 0) {
 257                 n = -n;
 258                 if ((n&1) == 1)
 259                         sign = -1;
 260         }
 261         if (n == 0)
 262                 return (y0l(x));
 263         if (n == 1)
 264                 return (sign*y1l(x));
 265         if (!finitel(x))
 266                 return (zero);
 267 
 268         if (x > 1.0e91L) {
 269                                 /*
 270                                  * x >> n**2
 271                                  * Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
 272                                  *   Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
 273                                  *   Let s=sin(x), c=cos(x),
 274                                  * xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
 275                                  *
 276                                  *         n    sin(xn)*sqt2    cos(xn)*sqt2
 277                                  *      ----------------------------------
 278                                  *         0     s-c             c+s
 279                                  *         1    -s-c            -c+s
 280                                  *         2    -s+c            -c-s
 281                                  *         3     s+c             c-s
 282                                  */
 283                 switch (n&3) {
 284                 case 0:
 285                         temp =  sinl(x)-cosl(x);
 286                         break;
 287                 case 1:
 288                         temp = -sinl(x)-cosl(x);
 289                         break;
 290                 case 2:
 291                         temp = -sinl(x)+cosl(x);
 292                         break;
 293                 case 3:
 294                         temp =  sinl(x)+cosl(x);
 295                         break;
 296                 }
 297                 b = invsqrtpi*temp/sqrtl(x);
 298         } else {
 299                 a = y0l(x);
 300                 b = y1l(x);
 301                 /*
 302                  * fix 1262058 and take care of non-default rounding
 303                  */
 304                 for (i = 1; i < n; i++) {
 305                         temp = b;
 306                         b *= (GENERIC) (i + i) / x;
 307                         if (b <= -LDBL_MAX)
 308                                 break;
 309                         b -= a;
 310                         a = temp;
 311                 }
 312         }
 313         if (sign > 0)
 314                 return (b);
 315         else
 316                 return (-b);
 317 }