jn_yn.c File Reference

#include "math.h"
#include "ieee754.h"

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Functions
double	__ieee754_jn (int n, double x)
double	__ieee754_yn (int n, double x)

Variables
static double	invsqrtpi = 5.64189583547756279280e-01
static double	two = 2.00000000000000000000e+00
static double	one = 1.00000000000000000000e+00
static double	zero = 0.00000000000000000000e+00

Function Documentation

__ieee754_jn()

double __ieee754_jn	(	int	n,
		double	x
	)

Definition at line 64 of file jn_yn.c.

{
    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|((u_int32_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
     */
        double s;
        double c;
        __sincos (x, &s, &c);
        switch(n&3) {
            case 0: temp =  c + s; break;
            case 1: temp = -c + s; break;
            case 2: temp = -c - s; break;
            case 3: temp =  c - s; break;
        }
        b = invsqrtpi*temp/__ieee754_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;
}

Referenced by _jn().

__ieee754_yn()

double __ieee754_yn	(	int	n,
		double	x
	)

Definition at line 227 of file jn_yn.c.

{
    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|((u_int32_t)(lx|-lx))>>31)>0x7ff00000) return x+x;
    if((ix|lx)==0) return -HUGE_VAL+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_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
     */
        double c;
        double s;
        __sincos (x, &s, &c);
        switch(n&3) {
            case 0: temp =  s - c; break;
            case 1: temp = -s - c; break;
            case 2: temp = -s + c; break;
            case 3: temp =  s + c; break;
        }
        b = invsqrtpi*temp/__ieee754_sqrt(x);
    } else {
        u_int32_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;
}

Referenced by _yn().

Variable Documentation

invsqrtpi

double invsqrtpi = 5.64189583547756279280e-01

static

Definition at line 51 of file jn_yn.c.

Referenced by __ieee754_jn(), and __ieee754_yn().

one

double one = 1.00000000000000000000e+00

static

Definition at line 53 of file jn_yn.c.

Referenced by __ieee754_jn().

two

double two = 2.00000000000000000000e+00

static

Definition at line 52 of file jn_yn.c.

Referenced by __ieee754_jn(), DrawHorzCardStrip(), DrawVertCardStrip(), gen_color_correction(), hash_matches_blob(), pfr_lookup_bitmap_data(), shader_arb_get_src_param(), shader_hw_mov(), SymFindFileInPath(), SymFindFileInPathW(), StringTest::template_expression(), StringTest::template_wexpression(), test_HlinkClone(), test_HlinkGetSetMonikerReference(), test_HlinkGetSetStringReference(), test_PathCombineA(), test_vsnwprintf(), and vshader_add_footer().

zero

double zero = 0.00000000000000000000e+00

static

Definition at line 58 of file jn_yn.c.

Referenced by __ieee754_jn(), and __ieee754_yn().