ieee754.h File Reference

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Classes
union	ieee_double_shape_type

Macros
#define	EXTRACT_WORDS(ix0, ix1, d)
#define	GET_HIGH_WORD(i, d)
#define	GET_LOW_WORD(i, d)

Typedefs
typedef __int32	int32_t
typedef unsigned __int32	u_int32_t

Functions
static __inline double	__ieee754_sqrt (double x)
static __inline double	__ieee754_log (double x)
static __inline double	__cos (double x)
static __inline void	__sincos (double x, double *s, double *c)
double	__ieee754_j0 (double)
double	__ieee754_j1 (double)
double	__ieee754_jn (int, double)
double	__ieee754_y0 (double)
double	__ieee754_y1 (double)
double	__ieee754_yn (int, double)

Macro Definition Documentation

EXTRACT_WORDS

#define EXTRACT_WORDS	(		ix0,
			ix1,
			d
	)

Value:

do {                                \
  ieee_double_shape_type ew_u;      \
  ew_u.value = (d);                 \
  (ix0) = ew_u.parts.msw;           \
  (ix1) = ew_u.parts.lsw;           \
} while (0)

Definition at line 16 of file ieee754.h.

GET_HIGH_WORD

#define GET_HIGH_WORD	(		i,
			d
	)

Value:

do {                                \
  ieee_double_shape_type gh_u;      \
  gh_u.value = (d);                 \
  (i) = gh_u.parts.msw;             \
} while (0)

Definition at line 26 of file ieee754.h.

GET_LOW_WORD

#define GET_LOW_WORD	(		i,
			d
	)

Value:

do {                                \
  ieee_double_shape_type gl_u;      \
  gl_u.value = (d);                 \
  (i) = gl_u.parts.lsw;             \
} while (0)

Definition at line 33 of file ieee754.h.

Typedef Documentation

int32_t

typedef __int32 int32_t

Definition at line 3 of file ieee754.h.

u_int32_t

typedef unsigned __int32 u_int32_t

Definition at line 4 of file ieee754.h.

Function Documentation

__cos()

static __inline double __cos ( double  x )

static

Definition at line 42 of file ieee754.h.

{return cos(x);}

Referenced by __ieee754_j0(), __ieee754_j1(), __ieee754_y0(), and __ieee754_y1().

__ieee754_j0()

double __ieee754_j0

(

double

x

)

Definition at line 102 of file j0_y0.c.

{
    double z, s,c,ss,cc,r,u,v,r1,r2,s1,s2,z2,z4;
    int32_t hx,ix;
 
    GET_HIGH_WORD(hx,x);
    ix = hx&0x7fffffff;
    if(ix>=0x7ff00000) return one/(x*x);
    x = fabs(x);
    if(ix >= 0x40000000) {  /* |x| >= 2.0 */
        __sincos (x, &s, &c);
        ss = s-c;
        cc = s+c;
        if(ix<0x7fe00000) {  /* make sure x+x not overflow */
            z = -__cos(x+x);
            if ((s*c)<zero) cc = z/ss;
            else        ss = z/cc;
        }
    /*
     * j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x)
     * y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x)
     */
        if(ix>0x48000000) z = (invsqrtpi*cc)/__ieee754_sqrt(x);
        else {
            u = pzero(x); v = qzero(x);
            z = invsqrtpi*(u*cc-v*ss)/__ieee754_sqrt(x);
        }
        return z;
    }
    if(ix<0x3f200000) { /* |x| < 2**-13 */
        if(huge+x>one) {    /* raise inexact if x != 0 */
            if(ix<0x3e400000) return one;   /* |x|<2**-27 */
            else          return one - 0.25*x*x;
        }
    }
    z = x*x;
#ifdef DO_NOT_USE_THIS
    r =  z*(R02+z*(R03+z*(R04+z*R05)));
    s =  one+z*(S01+z*(S02+z*(S03+z*S04)));
#else
    r1 = z*R[2]; z2=z*z;
    r2 = R[3]+z*R[4]; z4=z2*z2;
    r  = r1 + z2*r2 + z4*R[5];
    s1 = one+z*S[1];
    s2 = S[2]+z*S[3];
    s = s1 + z2*s2 + z4*S[4];
#endif
    if(ix < 0x3FF00000) {   /* |x| < 1.00 */
        return one + z*(-0.25+(r/s));
    } else {
        u = 0.5*x;
        return((one+u)*(one-u)+z*(r/s));
    }
}

Referenced by __ieee754_jn(), __ieee754_y0(), and _j0().

__ieee754_j1()

double __ieee754_j1

(

double

x

)

Definition at line 103 of file j1_y1.c.

{
    double z, s,c,ss,cc,r,u,v,y,r1,r2,s1,s2,s3,z2,z4;
    int32_t hx,ix;
 
    GET_HIGH_WORD(hx,x);
    ix = hx&0x7fffffff;
    if(ix>=0x7ff00000) return one/x;
    y = fabs(x);
    if(ix >= 0x40000000) {  /* |x| >= 2.0 */
        __sincos (y, &s, &c);
        ss = -s-c;
        cc = s-c;
        if(ix<0x7fe00000) {  /* make sure y+y not overflow */
            z = __cos(y+y);
            if ((s*c)>zero) cc = z/ss;
            else        ss = z/cc;
        }
    /*
     * j1(x) = 1/sqrt(pi) * (P(1,x)*cc - Q(1,x)*ss) / sqrt(x)
     * y1(x) = 1/sqrt(pi) * (P(1,x)*ss + Q(1,x)*cc) / sqrt(x)
     */
        if(ix>0x48000000) z = (invsqrtpi*cc)/__ieee754_sqrt(y);
        else {
            u = pone(y); v = qone(y);
            z = invsqrtpi*(u*cc-v*ss)/__ieee754_sqrt(y);
        }
        if(hx<0) return -z;
        else     return  z;
    }
    if(ix<0x3e400000) { /* |x|<2**-27 */
        if(huge+x>one) return 0.5*x;/* inexact if x!=0 necessary */
    }
    z = x*x;
#ifdef DO_NOT_USE_THIS
    r =  z*(r00+z*(r01+z*(r02+z*r03)));
    s =  one+z*(s01+z*(s02+z*(s03+z*(s04+z*s05))));
    r *= x;
#else
    r1 = z*R[0]; z2=z*z;
    r2 = R[1]+z*R[2]; z4=z2*z2;
    r = r1 + z2*r2 + z4*R[3];
    r *= x;
    s1 = one+z*S[1];
    s2 = S[2]+z*S[3];
    s3 = S[4]+z*S[5];
    s = s1 + z2*s2 + z4*s3;
#endif
    return(x*0.5+r/s);
}

Referenced by __ieee754_jn(), __ieee754_y1(), and _j1().

__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_log()

static __inline double __ieee754_log ( double  x )

static

Definition at line 41 of file ieee754.h.

{return log(x);}

Referenced by __ieee754_jn(), __ieee754_y0(), and __ieee754_y1().

__ieee754_sqrt()

static __inline double __ieee754_sqrt ( double  x )

static

Definition at line 40 of file ieee754.h.

{return sqrt(x);}

Referenced by __ieee754_j0(), __ieee754_j1(), __ieee754_jn(), __ieee754_y0(), __ieee754_y1(), and __ieee754_yn().

__ieee754_y0()

double __ieee754_y0

(

double

x

)

Definition at line 179 of file j0_y0.c.

{
    double z, s,c,ss,cc,u,v,z2,z4,z6,u1,u2,u3,v1,v2;
    int32_t hx,ix,lx;
 
    EXTRACT_WORDS(hx,lx,x);
        ix = 0x7fffffff&hx;
    /* Y0(NaN) is NaN, y0(-inf) is Nan, y0(inf) is 0, y0(0) is -inf.  */
    if(ix>=0x7ff00000) return  one/(x+x*x);
        if((ix|lx)==0) return -HUGE_VAL+x; /* -inf and overflow exception.  */
        if(hx<0) return zero/(zero*x);
        if(ix >= 0x40000000) {  /* |x| >= 2.0 */
        /* y0(x) = sqrt(2/(pi*x))*(p0(x)*sin(x0)+q0(x)*cos(x0))
         * where x0 = x-pi/4
         *      Better formula:
         *              cos(x0) = cos(x)cos(pi/4)+sin(x)sin(pi/4)
         *                      =  1/sqrt(2) * (sin(x) + cos(x))
         *              sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4)
         *                      =  1/sqrt(2) * (sin(x) - cos(x))
         * To avoid cancellation, use
         *              sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
         * to compute the worse one.
         */
        __sincos (x, &s, &c);
                ss = s-c;
                cc = s+c;
    /*
     * j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x)
     * y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x)
     */
                if(ix<0x7fe00000) {  /* make sure x+x not overflow */
                    z = -__cos(x+x);
                    if ((s*c)<zero) cc = z/ss;
                    else            ss = z/cc;
                }
                if(ix>0x48000000) z = (invsqrtpi*ss)/__ieee754_sqrt(x);
                else {
                    u = pzero(x); v = qzero(x);
                    z = invsqrtpi*(u*ss+v*cc)/__ieee754_sqrt(x);
                }
                return z;
    }
    if(ix<=0x3e400000) {    /* x < 2**-27 */
        return(U[0] + tpi*__ieee754_log(x));
    }
    z = x*x;
#ifdef DO_NOT_USE_THIS
    u = u00+z*(u01+z*(u02+z*(u03+z*(u04+z*(u05+z*u06)))));
    v = one+z*(v01+z*(v02+z*(v03+z*v04)));
#else
    u1 = U[0]+z*U[1]; z2=z*z;
    u2 = U[2]+z*U[3]; z4=z2*z2;
    u3 = U[4]+z*U[5]; z6=z4*z2;
    u = u1 + z2*u2 + z4*u3 + z6*U[6];
    v1 = one+z*V[0];
    v2 = V[1]+z*V[2];
    v = v1 + z2*v2 + z4*V[3];
#endif
    return(u/v + tpi*(__ieee754_j0(x)*__ieee754_log(x)));
}

Referenced by __ieee754_yn(), and _y0().

__ieee754_y1()

double __ieee754_y1

(

double

x

)

Definition at line 182 of file j1_y1.c.

{
    double z, s,c,ss,cc,u,v,u1,u2,v1,v2,v3,z2,z4;
    int32_t hx,ix,lx;
 
    EXTRACT_WORDS(hx,lx,x);
        ix = 0x7fffffff&hx;
    /* if Y1(NaN) is NaN, Y1(-inf) is NaN, Y1(inf) is 0 */
    if(ix>=0x7ff00000) return  one/(x+x*x);
        if((ix|lx)==0) return -HUGE_VAL+x; /* -inf and overflow exception.  */;
        if(hx<0) return zero/(zero*x);
        if(ix >= 0x40000000) {  /* |x| >= 2.0 */
        __sincos (x, &s, &c);
                ss = -s-c;
                cc = s-c;
                if(ix<0x7fe00000) {  /* make sure x+x not overflow */
                    z = __cos(x+x);
                    if ((s*c)>zero) cc = z/ss;
                    else            ss = z/cc;
                }
        /* y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x0)+q1(x)*cos(x0))
         * where x0 = x-3pi/4
         *      Better formula:
         *              cos(x0) = cos(x)cos(3pi/4)+sin(x)sin(3pi/4)
         *                      =  1/sqrt(2) * (sin(x) - cos(x))
         *              sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4)
         *                      = -1/sqrt(2) * (cos(x) + sin(x))
         * To avoid cancellation, use
         *              sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
         * to compute the worse one.
         */
                if(ix>0x48000000) z = (invsqrtpi*ss)/__ieee754_sqrt(x);
                else {
                    u = pone(x); v = qone(x);
                    z = invsqrtpi*(u*ss+v*cc)/__ieee754_sqrt(x);
                }
                return z;
        }
        if(ix<=0x3c900000) {    /* x < 2**-54 */
            return(-tpi/x);
        }
        z = x*x;
#ifdef DO_NOT_USE_THIS
        u = U0[0]+z*(U0[1]+z*(U0[2]+z*(U0[3]+z*U0[4])));
        v = one+z*(V0[0]+z*(V0[1]+z*(V0[2]+z*(V0[3]+z*V0[4]))));
#else
    u1 = U0[0]+z*U0[1];z2=z*z;
    u2 = U0[2]+z*U0[3];z4=z2*z2;
    u  = u1 + z2*u2 + z4*U0[4];
    v1 = one+z*V0[0];
    v2 = V0[1]+z*V0[2];
    v3 = V0[3]+z*V0[4];
    v = v1 + z2*v2 + z4*v3;
#endif
        return(x*(u/v) + tpi*(__ieee754_j1(x)*__ieee754_log(x)-one/x));
}

Referenced by __ieee754_yn(), and _y1().

__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().

__sincos()

static __inline void __sincos ( double  x, double *  s, double *  c  )

static

Definition at line 43 of file ieee754.h.

{
    *s = sin(x);
    *c = cos(x);
}

Referenced by __ieee754_j0(), __ieee754_j1(), __ieee754_jn(), __ieee754_y0(), __ieee754_y1(), and __ieee754_yn().