src/libm/k_tan.c
changeset 8840 9b6ddcbdea65
child 11683 48bcba563d9c
     1.1 --- /dev/null	Thu Jan 01 00:00:00 1970 +0000
     1.2 +++ b/src/libm/k_tan.c	Sat Jun 07 18:20:01 2014 -0700
     1.3 @@ -0,0 +1,118 @@
     1.4 +/*
     1.5 + * ====================================================
     1.6 + * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
     1.7 + *
     1.8 + * Developed at SunPro, a Sun Microsystems, Inc. business.
     1.9 + * Permission to use, copy, modify, and distribute this
    1.10 + * software is freely granted, provided that this notice
    1.11 + * is preserved.
    1.12 + * ====================================================
    1.13 + */
    1.14 +
    1.15 +/* __kernel_tan( x, y, k )
    1.16 + * kernel tan function on [-pi/4, pi/4], pi/4 ~ 0.7854
    1.17 + * Input x is assumed to be bounded by ~pi/4 in magnitude.
    1.18 + * Input y is the tail of x.
    1.19 + * Input k indicates whether tan (if k=1) or
    1.20 + * -1/tan (if k= -1) is returned.
    1.21 + *
    1.22 + * Algorithm
    1.23 + *	1. Since tan(-x) = -tan(x), we need only to consider positive x.
    1.24 + *	2. if x < 2^-28 (hx<0x3e300000 0), return x with inexact if x!=0.
    1.25 + *	3. tan(x) is approximated by a odd polynomial of degree 27 on
    1.26 + *	   [0,0.67434]
    1.27 + *		  	         3             27
    1.28 + *	   	tan(x) ~ x + T1*x + ... + T13*x
    1.29 + *	   where
    1.30 + *
    1.31 + * 	        |tan(x)         2     4            26   |     -59.2
    1.32 + * 	        |----- - (1+T1*x +T2*x +.... +T13*x    )| <= 2
    1.33 + * 	        |  x 					|
    1.34 + *
    1.35 + *	   Note: tan(x+y) = tan(x) + tan'(x)*y
    1.36 + *		          ~ tan(x) + (1+x*x)*y
    1.37 + *	   Therefore, for better accuracy in computing tan(x+y), let
    1.38 + *		     3      2      2       2       2
    1.39 + *		r = x *(T2+x *(T3+x *(...+x *(T12+x *T13))))
    1.40 + *	   then
    1.41 + *		 		    3    2
    1.42 + *		tan(x+y) = x + (T1*x + (x *(r+y)+y))
    1.43 + *
    1.44 + *      4. For x in [0.67434,pi/4],  let y = pi/4 - x, then
    1.45 + *		tan(x) = tan(pi/4-y) = (1-tan(y))/(1+tan(y))
    1.46 + *		       = 1 - 2*(tan(y) - (tan(y)^2)/(1+tan(y)))
    1.47 + */
    1.48 +
    1.49 +#include "math_libm.h"
    1.50 +#include "math_private.h"
    1.51 +
    1.52 +static const double
    1.53 +one   =  1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
    1.54 +pio4  =  7.85398163397448278999e-01, /* 0x3FE921FB, 0x54442D18 */
    1.55 +pio4lo=  3.06161699786838301793e-17, /* 0x3C81A626, 0x33145C07 */
    1.56 +T[] =  {
    1.57 +  3.33333333333334091986e-01, /* 0x3FD55555, 0x55555563 */
    1.58 +  1.33333333333201242699e-01, /* 0x3FC11111, 0x1110FE7A */
    1.59 +  5.39682539762260521377e-02, /* 0x3FABA1BA, 0x1BB341FE */
    1.60 +  2.18694882948595424599e-02, /* 0x3F9664F4, 0x8406D637 */
    1.61 +  8.86323982359930005737e-03, /* 0x3F8226E3, 0xE96E8493 */
    1.62 +  3.59207910759131235356e-03, /* 0x3F6D6D22, 0xC9560328 */
    1.63 +  1.45620945432529025516e-03, /* 0x3F57DBC8, 0xFEE08315 */
    1.64 +  5.88041240820264096874e-04, /* 0x3F4344D8, 0xF2F26501 */
    1.65 +  2.46463134818469906812e-04, /* 0x3F3026F7, 0x1A8D1068 */
    1.66 +  7.81794442939557092300e-05, /* 0x3F147E88, 0xA03792A6 */
    1.67 +  7.14072491382608190305e-05, /* 0x3F12B80F, 0x32F0A7E9 */
    1.68 + -1.85586374855275456654e-05, /* 0xBEF375CB, 0xDB605373 */
    1.69 +  2.59073051863633712884e-05, /* 0x3EFB2A70, 0x74BF7AD4 */
    1.70 +};
    1.71 +
    1.72 +double __kernel_tan(double x, double y, int iy)
    1.73 +{
    1.74 +	double z,r,v,w,s;
    1.75 +	int32_t ix,hx;
    1.76 +	GET_HIGH_WORD(hx,x);
    1.77 +	ix = hx&0x7fffffff;	/* high word of |x| */
    1.78 +	if(ix<0x3e300000)			/* x < 2**-28 */
    1.79 +	    {if((int)x==0) {			/* generate inexact */
    1.80 +	        u_int32_t low;
    1.81 +		GET_LOW_WORD(low,x);
    1.82 +		if(((ix|low)|(iy+1))==0) return one/fabs(x);
    1.83 +		else return (iy==1)? x: -one/x;
    1.84 +	    }
    1.85 +	    }
    1.86 +	if(ix>=0x3FE59428) { 			/* |x|>=0.6744 */
    1.87 +	    if(hx<0) {x = -x; y = -y;}
    1.88 +	    z = pio4-x;
    1.89 +	    w = pio4lo-y;
    1.90 +	    x = z+w; y = 0.0;
    1.91 +	}
    1.92 +	z	=  x*x;
    1.93 +	w 	=  z*z;
    1.94 +    /* Break x^5*(T[1]+x^2*T[2]+...) into
    1.95 +     *	  x^5(T[1]+x^4*T[3]+...+x^20*T[11]) +
    1.96 +     *	  x^5(x^2*(T[2]+x^4*T[4]+...+x^22*[T12]))
    1.97 +     */
    1.98 +	r = T[1]+w*(T[3]+w*(T[5]+w*(T[7]+w*(T[9]+w*T[11]))));
    1.99 +	v = z*(T[2]+w*(T[4]+w*(T[6]+w*(T[8]+w*(T[10]+w*T[12])))));
   1.100 +	s = z*x;
   1.101 +	r = y + z*(s*(r+v)+y);
   1.102 +	r += T[0]*s;
   1.103 +	w = x+r;
   1.104 +	if(ix>=0x3FE59428) {
   1.105 +	    v = (double)iy;
   1.106 +	    return (double)(1-((hx>>30)&2))*(v-2.0*(x-(w*w/(w+v)-r)));
   1.107 +	}
   1.108 +	if(iy==1) return w;
   1.109 +	else {		/* if allow error up to 2 ulp,
   1.110 +			   simply return -1.0/(x+r) here */
   1.111 +     /*  compute -1.0/(x+r) accurately */
   1.112 +	    double a,t;
   1.113 +	    z  = w;
   1.114 +	    SET_LOW_WORD(z,0);
   1.115 +	    v  = r-(z - x); 	/* z+v = r+x */
   1.116 +	    t = a  = -1.0/w;	/* a = -1.0/w */
   1.117 +	    SET_LOW_WORD(t,0);
   1.118 +	    s  = 1.0+t*z;
   1.119 +	    return t+a*(s+t*v);
   1.120 +	}
   1.121 +}