ktan.c File Reference

Kernel tan function. More...

#include "prim.h"

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Macros
#define	one   xxx[13]
#define	pio4   xxx[14]
#define	pio4lo   xxx[15]
#define	T   xxx

Functions
double	__kernel_tan (double x, double y, int iy)
	Kernel tan function. More...

Variables
static const double	xxx []

Detailed Description

Kernel tan function.

Definition in file ktan.c.

Macro Definition Documentation

one

#define one   xxx[13]

Definition at line 67 of file ktan.c.

pio4

#define pio4   xxx[14]

Definition at line 68 of file ktan.c.

pio4lo

#define pio4lo   xxx[15]

Definition at line 69 of file ktan.c.

T

#define T   xxx

Definition at line 70 of file ktan.c.

Function Documentation

__kernel_tan()

double __kernel_tan	(	double	x,
		double	y,
		int	iy
	)

Kernel tan function.

Kernel tan function on [-pi/4, pi/4], pi/4 ~ 0.7854
Input x is assumed to be bounded by ~pi/4 in magnitude.
Input y is the tail of x.
Input k indicates whether tan (if k = 1) or -1/tan (if k = -1) is returned.

Algorithm
 1. Since tan(-x) = -tan(x), we need only to consider positive x.
 2. if x < 2^-28 (hx<0x3e300000 0), return x with inexact if x!=0.
 3. tan(x) is approximated by a odd polynomial of degree 27 on
    [0,0.67434]
                 3             27
        tan(x) ~ x + T1*x + ... + T13*x
    where
    |tan(x)         2     4            26   |     -59.2
    |----- - (1+T1*x +T2*x +.... +T13*x    )| <= 2
    |  x                    |

    Note: tan(x+y) = tan(x) + tan'(x)*y
              ~ tan(x) + (1+x*x)*y
    Therefore, for better accuracy in computing tan(x+y), let
         3      2      2       2       2
    r = x *(T2+x *(T3+x *(...+x *(T12+x *T13))))
    then
                3    2
    tan(x+y) = x + (T1*x + (x *(r+y)+y))

     4. For x in [0.67434,pi/4],  let y = pi/4 - x, then
    tan(x) = tan(pi/4-y) = (1-tan(y))/(1+tan(y))
           = 1 - 2*(tan(y) - (tan(y)^2)/(1+tan(y)))

Definition at line 108 of file ktan.c.

References fabs().

Referenced by tan().

{
    double z, r, v, w, s;
    double a, t;
    int32_t ix, hx;
    uint32_t low;

    // high word of x
    GET_HIGH_WORD(hx, x);

    // high word of |x|
    ix = hx & 0x7FFFFFFF;

    // x < 2**-28
    if (ix < 0x3E300000)
    {
        // generate inexact
        if ((int32_t)x == 0)
        {
            GET_LOW_WORD(low, x);

            if (((ix | low) | (iy + 1)) == 0)
            {
                return one / fabs(x);
            }

            if (iy == 1)
            {
                return x;
            }

            // compute -1 / (x+y) carefully
            z = w = x + y;
            SET_LOW_WORD(z, 0);
            v = y - (z - x);
            t = a = -one / w;
            SET_LOW_WORD(t, 0);
            s = one + t * z;
            return t + a * (s + t * v);
        }
    }

    // |x| >= 0.6744
    if (ix >= 0x3FE59428)
    {
        if (hx < 0)
        {
            x = -x;
            y = -y;
        }
        z = pio4 - x;
        w = pio4lo - y;
        x = z + w;
        y = 0.0;
    }

    z = x * x;
    w = z * z;

    /*
     * Break x^5*(T[1]+x^2*T[2]+...) into
     * x^5(T[1]+x^4*T[3]+...+x^20*T[11]) +
     * x^5(x^2*(T[2]+x^4*T[4]+...+x^22*[T12]))
     */
    r = T[1] + w * (T[3] + w * (T[5] + w * (T[7] + w * (T[9] + w * T[11]))));
    v = z * (T[2] + w * (T[4] + w * (T[6] + w * (T[8] + w * (T[10] + w * T[12])))));
    s = z * x;
    r = y + z * (s * (r + v) + y);
    r += T[0] * s;
    w = x + r;

    if (ix >= 0x3FE59428)
    {
        v = (double)iy;
        return (double)(1 - ((hx >> 30) & 2)) *
               (v - 2.0 * (x - (w * w / (w + v) - r)));
    }

    if (iy == 1)
    {
        return w;
    }

    // compute -1.0 / (x+r) accurately
    z = w;
    SET_LOW_WORD(z, 0);
    v = r - (z - x);  // z+v = r+x
    t = a = -1.0 / w; // a = -1.0/w
    SET_LOW_WORD(t, 0);
    s = 1.0 + t * z;
    return t + a * (s + t * v);
}

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Variable Documentation

xxx

const double xxx[]

static

Initial value:

= {
    3.33333333333334091986e-01,             
    1.33333333333201242699e-01,             
    5.39682539762260521377e-02,             
    2.18694882948595424599e-02,             
    8.86323982359930005737e-03,             
    3.59207910759131235356e-03,             
    1.45620945432529025516e-03,             
    5.88041240820264096874e-04,             
    2.46463134818469906812e-04,             
    7.81794442939557092300e-05,             
    7.14072491382608190305e-05,             
    -1.85586374855275456654e-05,            
    2.59073051863633712884e-05,             
     1.00000000000000000000e+00,   
     7.85398163397448278999e-01,  
     3.06161699786838301793e-17 
}

Definition at line 48 of file ktan.c.