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glibc  2.9
Functions | Variables
e_jnf.c File Reference
#include "math.h"
#include "math_private.h"

Go to the source code of this file.

Functions

float __ieee754_jnf (int n, float x)
float __ieee754_ynf (int n, float x)

Variables

static float two = 2.0000000000e+00
static float one = 1.0000000000e+00
static float zero = 0.0000000000e+00

Function Documentation

float __ieee754_jnf ( int  n,
float  x 
)

Definition at line 40 of file e_jnf.c.

{
       int32_t i,hx,ix, sgn;
       float a, b, temp, di;
       float z, w;

    /* J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x)
     * Thus, J(-n,x) = J(n,-x)
     */
       GET_FLOAT_WORD(hx,x);
       ix = 0x7fffffff&hx;
    /* if J(n,NaN) is NaN */
       if(ix>0x7f800000) return x+x;
       if(n<0){
              n = -n;
              x = -x;
              hx ^= 0x80000000;
       }
       if(n==0) return(__ieee754_j0f(x));
       if(n==1) return(__ieee754_j1f(x));
       sgn = (n&1)&(hx>>31);       /* even n -- 0, odd n -- sign(x) */
       x = fabsf(x);
       if(ix==0||ix>=0x7f800000)   /* if x is 0 or inf */
           b = zero;
       else if((float)n<=x) {
              /* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */
           a = __ieee754_j0f(x);
           b = __ieee754_j1f(x);
           for(i=1;i<n;i++){
              temp = b;
              b = b*((float)(i+i)/x) - a; /* avoid underflow */
              a = temp;
           }
       } else {
           if(ix<0x30800000) {     /* x < 2**-29 */
    /* x is tiny, return the first Taylor expansion of J(n,x)
     * J(n,x) = 1/n!*(x/2)^n  - ...
     */
              if(n>33)      /* underflow */
                  b = zero;
              else {
                  temp = x*(float)0.5; b = temp;
                  for (a=one,i=2;i<=n;i++) {
                     a *= (float)i;              /* a = n! */
                     b *= temp;           /* b = (x/2)^n */
                  }
                  b = b/a;
              }
           } else {
              /* use backward recurrence */
              /*                   x      x^2      x^2
               *  J(n,x)/J(n-1,x) =  ----   ------   ------   .....
               *                   2n  - 2(n+1) - 2(n+2)
               *
               *                   1      1        1
               *  (for large x)   =  ----  ------   ------   .....
               *                   2n   2(n+1)   2(n+2)
               *                   -- - ------ - ------ -
               *                    x     x         x
               *
               * Let w = 2n/x and h=2/x, then the above quotient
               * is equal to the continued fraction:
               *                1
               *     = -----------------------
               *                   1
               *        w - -----------------
               *                     1
               *             w+h - ---------
               *                   w+2h - ...
               *
               * To determine how many terms needed, let
               * Q(0) = w, Q(1) = w(w+h) - 1,
               * Q(k) = (w+k*h)*Q(k-1) - Q(k-2),
               * When Q(k) > 1e4   good for single
               * When Q(k) > 1e9   good for double
               * When Q(k) > 1e17  good for quadruple
               */
           /* determine k */
              float t,v;
              float q0,q1,h,tmp; int32_t k,m;
              w  = (n+n)/(float)x; h = (float)2.0/(float)x;
              q0 = w;  z = w+h; q1 = w*z - (float)1.0; k=1;
              while(q1<(float)1.0e9) {
                     k += 1; z += h;
                     tmp = z*q1 - q0;
                     q0 = q1;
                     q1 = tmp;
              }
              m = n+n;
              for(t=zero, i = 2*(n+k); i>=m; i -= 2) t = one/(i/x-t);
              a = t;
              b = one;
              /*  estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n)
               *  Hence, if n*(log(2n/x)) > ...
               *  single 8.8722839355e+01
               *  double 7.09782712893383973096e+02
               *  long double 1.1356523406294143949491931077970765006170e+04
               *  then recurrent value may overflow and the result is
               *  likely underflow to zero
               */
              tmp = n;
              v = two/x;
              tmp = tmp*__ieee754_logf(fabsf(v*tmp));
              if(tmp<(float)8.8721679688e+01) {
                  for(i=n-1,di=(float)(i+i);i>0;i--){
                      temp = b;
                     b *= di;
                     b  = b/x - a;
                      a = temp;
                     di -= two;
                  }
              } else {
                  for(i=n-1,di=(float)(i+i);i>0;i--){
                      temp = b;
                     b *= di;
                     b  = b/x - a;
                      a = temp;
                     di -= two;
                  /* scale b to avoid spurious overflow */
                     if(b>(float)1e10) {
                         a /= b;
                         t /= b;
                         b  = one;
                     }
                  }
              }
              b = (t*__ieee754_j0f(x)/b);
           }
       }
       if(sgn==1) return -b; else return b;
}

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float __ieee754_ynf ( int  n,
float  x 
)

Definition at line 177 of file e_jnf.c.

{
       int32_t i,hx,ix;
       u_int32_t ib;
       int32_t sign;
       float a, b, temp;

       GET_FLOAT_WORD(hx,x);
       ix = 0x7fffffff&hx;
    /* if Y(n,NaN) is NaN */
       if(ix>0x7f800000) return x+x;
       if(ix==0) return -HUGE_VALF+x;  /* -inf and overflow exception.  */
       if(hx<0) return zero/(zero*x);
       sign = 1;
       if(n<0){
              n = -n;
              sign = 1 - ((n&1)<<1);
       }
       if(n==0) return(__ieee754_y0f(x));
       if(n==1) return(sign*__ieee754_y1f(x));
       if(ix==0x7f800000) return zero;

       a = __ieee754_y0f(x);
       b = __ieee754_y1f(x);
       /* quit if b is -inf */
       GET_FLOAT_WORD(ib,b);
       for(i=1;i<n&&ib!=0xff800000;i++){
           temp = b;
           b = ((float)(i+i)/x)*b - a;
           GET_FLOAT_WORD(ib,b);
           a = temp;
       }
       if(sign>0) return b; else return -b;
}

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Variable Documentation

float one = 1.0000000000e+00 [static]

Definition at line 29 of file e_jnf.c.

float two = 2.0000000000e+00 [static]

Definition at line 28 of file e_jnf.c.

float zero = 0.0000000000e+00 [static]

Definition at line 34 of file e_jnf.c.