text: 
  /* @(#)e_jn.c 1.4 95/01/18 */
  /*
   * ====================================================
   * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
   *
   * Developed at SunSoft, a Sun Microsystems, Inc. business.
   * Permission to use, copy, modify, and distribute this
   * software is freely granted, provided that this notice 
   * is preserved.
   * ====================================================
   */
  
  
  /*
   * __ieee754_jn(n, x), __ieee754_yn(n, x)
   * floating point Bessel's function of the 1st and 2nd kind
   * of order n
   *          
   * Special cases:
   *	y0(0)=y1(0)=yn(n,0) = -inf with division by zero signal;
   *	y0(-ve)=y1(-ve)=yn(n,-ve) are NaN with invalid signal.
   * Note 2. About jn(n,x), yn(n,x)
   *	For n=0, j0(x) is called,
   *	for n=1, j1(x) is called,
   *	for n<x, forward recursion us used starting
   *	from values of j0(x) and j1(x).
   *	for n>x, a continued fraction approximation to
   *	j(n,x)/j(n-1,x) is evaluated and then backward
   *	recursion is used starting from a supposed value
   *	for j(n,x). The resulting value of j(0,x) is
   *	compared with the actual value to correct the
   *	supposed value of j(n,x).
   *
   *	yn(n,x) is similar in all respects, except
   *	that forward recursion is used for all
   *	values of n>1.
   *	
   */
  
  #include "math.h"
  #include "math_private.h"
  
  static const double
  invsqrtpi=  5.64189583547756279280e-01, /* 0x3FE20DD7, 0x50429B6D */
  two   =  2.00000000000000000000e+00, /* 0x40000000, 0x00000000 */
  one   =  1.00000000000000000000e+00; /* 0x3FF00000, 0x00000000 */
  
  static const double zero  =  0.00000000000000000000e+00;
  
  double
  __ieee754_jn(int n, double x)
  {
  	int32_t i,hx,ix,lx, sgn;
  	double a, b, temp, di;
  	double 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)
       */
  	EXTRACT_WORDS(hx,lx,x);
  	ix = 0x7fffffff&hx;
      /* if J(n,NaN) is NaN */
  	if((ix|((uint32_t)(lx|-lx))>>31)>0x7ff00000) return x+x;
  	if(n<0){		
  		n = -n;
  		x = -x;
  		hx ^= 0x80000000;
  	}
  	if(n==0) return(__ieee754_j0(x));
  	if(n==1) return(__ieee754_j1(x));
  	sgn = (n&1)&(hx>>31);	/* even n -- 0, odd n -- sign(x) */
  	x = fabs(x);
  	if((ix|lx)==0||ix>=0x7ff00000) 	/* if x is 0 or inf */
  	    b = zero;
  	else if((double)n<=x) {   
  		/* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */
  	    if(ix>=0x52D00000) { /* x > 2**302 */
      /* (x >> n**2) 
       *	    Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
       *	    Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
       *	    Let s=sin(x), c=cos(x), 
       *		xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
       *
       *		   n	sin(xn)*sqt2	cos(xn)*sqt2
       *		----------------------------------
       *		   0	 s-c		 c+s
       *		   1	-s-c 		-c+s
       *		   2	-s+c		-c-s
       *		   3	 s+c		 c-s
       */
  		switch(n&3) {
  		    case 0: temp =  cos(x)+sin(x); break;
  		    case 1: temp = -cos(x)+sin(x); break;
  		    case 2: temp = -cos(x)-sin(x); break;
  		    case 3: temp =  cos(x)-sin(x); break;
  		}
  		b = invsqrtpi*temp/sqrt(x);
  	    } else {	
  	        a = __ieee754_j0(x);
  	        b = __ieee754_j1(x);
  	        for(i=1;i<n;i++){
  		    temp = b;
  		    b = b*((double)(i+i)/x) - a; /* avoid underflow */
  		    a = temp;
  	        }
  	    }
  	} else {
  	    if(ix<0x3e100000) {	/* 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*0.5; b = temp;
  		    for (a=one,i=2;i<=n;i++) {
  			a *= (double)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 */
  		double t,v;
  		double q0,q1,h,tmp; int32_t k,m;
  		w  = (n+n)/(double)x; h = 2.0/(double)x;
  		q0 = w;  z = w+h; q1 = w*z - 1.0; k=1;
  		while(q1<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_log(fabs(v*tmp));
  		if(tmp<7.09782712893383973096e+02) {
  	    	    for(i=n-1,di=(double)(i+i);i>0;i--){
  		        temp = b;
  			b *= di;
  			b  = b/x - a;
  		        a = temp;
  			di -= two;
  	     	    }
  		} else {
  	    	    for(i=n-1,di=(double)(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>1e100) {
  			    a /= b;
  			    t /= b;
  			    b  = one;
  			}
  	     	    }
  		}
  	    	b = (t*__ieee754_j0(x)/b);
  	    }
  	}
  	if(sgn==1) return -b; else return b;
  }
  
  double
  __ieee754_yn(int n, double x)
  {
  	int32_t i,hx,ix,lx;
  	int32_t sign;
  	double a, b, temp;
  
  	EXTRACT_WORDS(hx,lx,x);
  	ix = 0x7fffffff&hx;
      /* if Y(n,NaN) is NaN */
  	if((ix|((uint32_t)(lx|-lx))>>31)>0x7ff00000) return x+x;
  	if((ix|lx)==0) return -one/zero;
  	if(hx<0) return zero/zero;
  	sign = 1;
  	if(n<0){
  		n = -n;
  		sign = 1 - ((n&1)<<1);
  	}
  	if(n==0) return(__ieee754_y0(x));
  	if(n==1) return(sign*__ieee754_y1(x));
  	if(ix==0x7ff00000) return zero;
  	if(ix>=0x52D00000) { /* x > 2**302 */
      /* (x >> n**2) 
       *	    Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
       *	    Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
       *	    Let s=sin(x), c=cos(x), 
       *		xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
       *
       *		   n	sin(xn)*sqt2	cos(xn)*sqt2
       *		----------------------------------
       *		   0	 s-c		 c+s
       *		   1	-s-c 		-c+s
       *		   2	-s+c		-c-s
       *		   3	 s+c		 c-s
       */
  		switch(n&3) {
  		    case 0: temp =  sin(x)-cos(x); break;
  		    case 1: temp = -sin(x)-cos(x); break;
  		    case 2: temp = -sin(x)+cos(x); break;
  		    case 3: temp =  sin(x)+cos(x); break;
  		}
  		b = invsqrtpi*temp/sqrt(x);
  	} else {
  	    uint32_t high;
  	    a = __ieee754_y0(x);
  	    b = __ieee754_y1(x);
  	/* quit if b is -inf */
  	    GET_HIGH_WORD(high,b);
  	    for(i=1;i<n&&high!=0xfff00000;i++){
  		temp = b;
  		b = ((double)(i+i)/x)*b - a;
  		GET_HIGH_WORD(high,b);
  		a = temp;
  	    }
  	}
  	if(sign>0) return b; else return -b;
  }
  